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The Micro-Doppler Effect in Radar

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The Micro-Doppler Effect in Radar

Victor C. Chen

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Cover design by Vicki Kane

ISBN 13: 978-1-60807-057-2

© 2011 ARTECH HOUSE685 Canton StreetNorwood, MA 02062

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10 9 8 7 6 5 4 3 2 1

v

Contents

Preface xi

1 Introduction 1

1.1 Doppler Effect 2

1.2 Relativistic Doppler Effect and Time Dilation 4

1.3 Doppler Effect Observed in Radar 7

1.4 Estimation and Analysis of Doppler Frequency Shifts 10

1.5 Cramer-Rao Bound of the Doppler Frequency Estimation 17

1.6 The Micro-Doppler Effect 18

1.7 Micro-Doppler Effect Observed in Radar 20

1.8 Estimation and Analysis of Micro-Doppler Frequency Shifts 20

1.8.1 Instantaneous Frequency Analysis 211.8.2 Joint Time-Frequency Analysis 23

1.9 The Micro-Doppler Signature of Objects 26

References 28

Appendix 1A 32

MATLAB Source Codes 32

vi The Micro-Doppler Effect in Radar

2 Basics of the Micro-Doppler Effect in Radar 35

2.1 Rigid Body Motion 35

2.1.1 Euler Angles 362.1.2 Quaternion 422.1.3 Equations of Motion 44

2.2 Nonrigid Body Motion 47

2.3 Electromagnetic Scattering from a Body with Motion 50

2.3.1 Radar Cross Section of a Target 502.3.2 RCS Prediction Methods 532.3.3 EM Scattering from a Body with Motion 54

2.4 Basic Mathematics for Calculating the Micro-Doppler Effect 56

2.4.1 Micro-Doppler Induced by a Target with Micro Motion 562.4.2 Vibration-Induced Micro-Doppler Shift 602.4.3 Rotation-Induced Micro-Doppler Shift 632.4.4 Coning Motion-Induced Micro-Doppler Shift 66

2.5 Bistatic Micro-Doppler Effect 71

2.6 Multistatic Micro-Doppler Effect 77

2.7 Cramer-Rao Bound of the Micro-Doppler Estimation 79

References 79

Appendix 2A 81

Appendix 2B 83


3 The Micro-Doppler Effect of the Rigid Body Motion 93

3.1 Pendulum Oscillation 94

3.1.1 Modeling Nonlinear Motion Dynamic of a Pendulum 953.1.2 Modeling RCS of a Pendulum 1013.1.3 Radar Backscattering from an Oscillating Pendulum 1023.1.4 Micro-Doppler Signatures Generated by an Oscillating Pendulum 105

3.2 Helicopter Rotor Blades 105

3.2.1 Mathematic Model of Rotating Rotor Blades 107

Contents vii

3.2.2 RCS Model of Rotating Rotor Blades 1123.2.3 PO Facet Prediction Model 1143.2.4 Radar Backscattering from Rotor Blades 1163.2.5 Micro-Doppler Signatures of Rotor Blades 1203.2.6 Required Minimum PRF 1233.2.7 Analysis and Interpretation of the Micro-Doppler Signature of Rotor Blades 123

3.3 Spinning Symmetric Top 127

3.3.1 Force-Free Rotation of a Symmetric Top 1303.3.2 Torque-Induced Rotation of a Symmetric Top 1323.3.3 RCS Model of a Symmetric Top 1333.3.4 Radar Backscattering from a Symmetric Top 1353.3.5 Micro-Doppler Signatures Generated by a Precession Top 1363.3.6 Analysis and Interpretation of the Micro-Doppler Signature of a Precession Top 136

3.4 Wind Turbines 139

3.4.1 Micro-Doppler Signatures of Wind Turbines 1403.4.2 Analysis and Interpretation of the Micro-Doppler Signature of Wind Turbines 140

References 141

Appendix 3A 143


4 The Micro-Doppler Effect of the Nonrigid Body Motion 157

4.1 Human Body Articulated Motion 159

4.1.1 Human Walking 1594.1.2 Description of the Periodic Motion of Human Walking 1614.1.3 Simulation of Human Movements 1624.1.4 Human Body Segment Parameters 1624.1.5 Human Walking Model Derived from Empirical Mathematical Parameterizations 1644.1.6 Capturing Human Motion Kinematic Parameters 1774.1.7 Three-Dimensional Kinematic Data Collection 1824.1.8 Characteristics of Angular Kinematics Using the Angle-Cyclogram Pattern 1844.1.9 Radar Backscattering from a Walking Human 184

viii The Micro-Doppler Effect in Radar

4.1.10 Human Movement Data Processing 1874.1.11 Human Movement–Induced Radar Micro-Doppler Signatures 189

4.2 Bird Wing Flapping 194

4.2.1 Bird Wing Flapping Kinematics 1954.2.2 Doppler Observations of the Bird Wing Flapping 1984.2.3 Simulation of the Bird Wing Flapping 199

4.3 Quadrupedal Animal Motion 202

4.3.1 Modeling of Quadrupedal Locomotion 2044.3.2 Micro-Doppler Signatures of Quadrupedal Locomotion 2054.3.3 Summary 205

References 207

Appendix 4A 209


Appendix 4B 238


5 Analysis and Interpretation of Micro-Doppler Signatures 247

5.1 Biological Motion Perception 248

5.2 Decomposition of Biological Motion 250

5.2.1 Statistics-Based Decomposition 2515.2.2 Decomposition of Micro-Doppler Signatures in the Joint Time-Frequency Domain 2515.2.3 Physical Component–Based Decomposition 252

5.3 Extraction of Features from Micro-Doppler Signatures 256

5.4 Estimation of Kinematic Parameters from Micro-Doppler Signatures 257

5.5 Identifying Human Movements 262

5.5.1 Features Used for Identifying Human Movements 2635.5.2 Anomalous Human Behavior 2645.5.3 Summary 266

References 267

Contents ix

6 Summary, Challenges, and Perspectives 271

6.1 Summary 271

6.2 Challenges 272

6.2.1 Decomposing Micro-Doppler Signatures 2736.2.2 Feature Extraction and Target Identification Based on Micro-Doppler Signatures 273

6.3 Perspectives 275

6.3.1 Multistatic Micro-Doppler Analysis 2756.3.2 Micro-Doppler Signature-Based Classification 2766.3.3 Aural Methods for Micro-Doppler–Based Discrimination 2766.3.4 Through-the-Wall Micro-Doppler Signatures 277

References 278

About the Author 281

Index 283

xi

PrefaceThe micro-Doppler effect was originally introduced in a coherent laser system to measure the kinematic properties of an object, such as the vibration rate and the displacement of the vibration. Micro-Doppler frequency shifts can be characterized by the distinctive signature that represents the intricate features generated from structural components of the object. In laser radar systems, even with a very low vibration rate, a micro displacement of the object’s vibration can easily cause a large Doppler shift. In contrast with the laser radar, in the microwave radar systems in which we are interested, it is difficult to observe micro-Doppler modulations induced by micro vibrations due to much longer wavelengths. However, if the oscillation rate times the displacement of the os-cillation and the product is high enough, the micro-Doppler modulation of an oscillation may be observed. For example, micro-Doppler shifts generated by rotations of rotor blades may be detectable because of their longer rotation arms and, thus, higher tip speeds. Although the Doppler frequency shifts induced by rotating rotor blades have been observed for a long time, not enough attention was paid to the time-varying characteristic of the Doppler frequency shifts until the joint time-frequency analysis was introduced in radar signal analysis.

In 1998, when I was working on applications of the joint time-frequency analysis to radar imaging and signal analysis, I was given an experimental radar data of a walking human where the radar is stationary and the human is mov-ing. The radar data collected by the stationary radar is actually inverse synthetic aperture radar (ISAR) data, and a sequence of range-Doppler ISAR images of the walking human can be generated.

There are a variety of algorithms available for generating focused ISAR images. However, any uncompensated motion will cause smearing along the range domain in the image, and, if the duration of the coherent processing

xii The Micro-Doppler Effect in Radar

interval is long, uncompensated motion may generate time-varying Doppler frequency shifts that make ISAR images smeared in the Doppler domain. From the sequence of range-Doppler images of the walking human, we can clearly see a focused hot spot of the human moving along the range at a stable Doppler frequency shift related to the walking speed. However, in the range-Doppler images, there is smearing along the Doppler domain caused by the uncompen-sated swinging arms and legs of the walking human. To exploit the details of the swinging arms and legs, I applied the joint time-frequency analysis to the radar range profiles around the smeared portion. As an immediate result, the time-frequency representation clearly showed Doppler oscillations of the arms and legs around the Doppler frequency shift of the human body’s translational motion. That was the first micro-Doppler signature of a walking human that I analyzed and represented in the joint time-frequency domain. The signature clearly shows the time-varying frequency distribution caused by the periodic motions of the structural components of the human body, such as feet, hands, arms, and legs.

The exploitation of micro-Doppler signatures does not necessarily require range resolutions. Any continuous wave (CW) radar is good enough to produce the micro-Doppler signature of a target, but ISAR range-Doppler images hav-ing both range and Doppler resolutions are useful in revealing locations of the target or its structural components in the range-Doppler domain. That is why in this book we prefer to generate micro-Doppler signatures of targets using two-dimensional range profiles instead of using a one-dimensional time series. From two-dimensional range profiles, after applying motion compensation and autofocusing algorithms, a range-Doppler image of the target can be recon-structed. On the other hand, from uncompensated rotations of the structural components in the target, we can also extract micro-Doppler features of the target.

In the last decade, numerous articles related to the micro-Doppler ef-fect in radar have been published. The purpose of this book is to introduce principles and theories of the micro-Doppler effect in radar, exploit potential applications of micro-Doppler signatures, and provide a simple and easy tool for generating micro-Doppler signatures of targets of interest. Apart from the analysis of real-world radar data, simulation is an important method for study-ing the micro-Doppler effect in radar. Based on simulation examples provided in this book, readers are encouraged to make modifications and extend the simulations to other applications of interest.

In this book, Chapter 1 is an introduction to the micro-Doppler effect in radar. The basic concept and mathematics of the micro-Doppler effect in radar are given in Chapter 2. In Chapters 3 and 4, some available RCS prediction models are introduced for calculating micro-Doppler signatures of rigid and nonrigid motions of objects. Several typical examples along with MATLAB

Preface xiii

source codes are provided in the chapters. Chapter 5 is devoted to biological motion perception and the relationship between biological motion information and micro-Doppler signatures. This helps identify a target with kinematic mo-tion based on its micro-Doppler signature. Chapter 6 summarizes the micro-Doppler effect in radar and lists some challenges and perspectives in micro-Doppler research.

For educational purposes, this book provides MATLAB source codes on the accompanying DVD. The source codes are provided by the contributors on an as-is basis and no warranties are claimed. The contributors of the source codes will not be held liable for any damage caused.

Part of the MATLAB source codes provided in this book are attributed to the efforts of my students Yang Hai and Yinan Yang. I would like to express my sincere thanks to them. I also give my thanks to Raghu Raj for his work on the physical component–based decomposition method and related figures in Chapter 5. I especially thank David Tahmoush for his interesting work on micro-Doppler signatures of quadrupedal animal motions and the related fig-ure prepared for the book in Chapter 4.

I also wish to express my thanks to William Miceli, my longtime friend and the sponsor of my micro-Doppler research, for his constant support and helpful discussion during the past decade for the research work on the micro-Doppler effect in radar, especially the technical discussions on the cyclogram in Chapter 4 and the aural methods in Chapter 6.

I am grateful to the reviewer of the book for the constructive suggestions and also to the staff of Artech House for the interest and support in the publica-tion of this book.

1

1 IntroductionRadar transmits an electromagnetic (EM) signal to an object and receives a returned signal from the object. Based on the time delay of the received sig-nal, radar can measure the range of the object. If the object is moving, the frequency of the received signal will be shifted from the frequency of the trans-mitted signal, known as the Doppler effect [1, 2]. The Doppler frequency shift is determined by the radial velocity of the moving object, that is, the velocity component in the direction of the line of sight (LOS). Based on the Doppler frequency shift of the received signal, radar can measure the radial velocity of the moving object. If the object or any structural component of the object has an oscillatory motion in addition to the bulk motion of the object, the oscilla-tion will induce additional frequency modulation on the returned signal and generates side bands about the Doppler shifted frequency of the transmitted signal due to the bulk motion. The additional Doppler modulation is called the micro-Doppler effect [3–5].

The Doppler frequency shift is usually measured in the frequency domain by taking the Fourier transform of the received signal. In the Fourier spec-trum, the peak component indicates the Doppler frequency shift induced by the radial velocity of the object’s motion. The width of Doppler frequency shifts gives an estimate of the velocity dispersion due to the micro-Doppler effect. To accurately track the phase information in the radar received signals, the radar transmitter must be driven by a highly stable frequency source to maintain a full phase coherency.

2 The Micro-Doppler Effect in Radar

The micro-Doppler effect can be used to determine kinematic properties of an object. For example, the vibration generated by a vehicle engine can be detected from the surface vibration of the vehicle body. By measuring the mi-cro-Doppler characteristics of the surface vibration, the speed of the engine can be measured and used to identify a specific type of vehicle, such as a tank with a gas turbine engine or a bus with a diesel engine. The micro-Doppler effect observed in an object can be characterized by its signature (i.e., the distinctive characteristics of the object that represents the intricate frequency modulation generated from the structural components of the object and represented in the joint time and Doppler frequency domain).

1.1 Doppler Effect

In 1842, Austrian mathematician and physicist Christian Doppler stated a phe-nomenon on the colored light effect of stars [1]. The apparent color of the light source is changed by its motion. For a light source moving toward an observer, the color of the light would appear bluer, while moving away from an observer, the light would appear redder. For the first time, the phenomenon, known as the Doppler effect, was discovered. The effect claims that the observed frequen-cy (or wavelength) of a light source depends on the velocity of the source rela-tive to the observer. The motion of the source causes the waves in front of the source to be compressed and behind the source to be stretched (see Figure 1.1).

In 1843, the Doppler effect was experimentally proved by sound waves of a trumpeter of a train moving at different speeds. The wavelength of the sound source is defined by λ = csound

/f, where csound

is the propagation speed of the sound wave in a given medium and f is the frequency of the sound source. If only the source is moving at a velocity v

s relative to the medium, the frequency perceived

by the observer is

(1.1)

If vs /c

sound << 1, the Doppler shifted frequency perceived by the observer

is approximately

(1.2)

1

1 /sound

sound s s sound

cf f f

c v v c= =′

1(1 )

1 /s

s sound sound

vf f f

v c c= ≅ ±′

Introduction 3

If the source is stationary and the observer is moving with a velocity vo

relative to the medium, the frequency perceived by the observer becomes

(1 )sound o o

sound sound

c v vf f f

c c

±= = ±′ (1.3)

If both the source and the observer are moving, the frequency perceived by the observer becomes

(1.4)

When the source and the observer move toward each other, the upper set of signs in (1.4) is applied; when the source and the observer move away from each other, the lower set of signs is applied.

1 /

1 /sound o o sound

sound s s sound

c v v cf f f

c v v c

± ±= =′

Figure 1.1 In 1842, Christian Doppler fi rst discovered the phenomenon on the apparent color of a light source changed by its motion, known as the Doppler effect.


1.2 Relativistic Doppler Effect and Time Dilation

Compared to sound waves, there is no medium involved in light or EM wave propagation. The propagation speed of light or EM waves, c, viewed from both the source and the observer, is the same constant.

For light or EM waves, changes in the frequency or wavelength caused by the relative motion between the source and the observer should count the effect of the theory of special relativity [6]. Thus, the Doppler frequency shift must be modified to be consistent with the Lorentz transformation. The relativistic Doppler effect is different from the classical Doppler effect because it includes the time dilation effect of the special relativity and does not involve the medium of the wave propagation as a reference point.

When a light or EM source at a frequency, f, is moving with a velocity vs

at an angle θs relative to the direction from the source S to the observer O as

shown in Figure 1.2, the time interval between two successive crests of the wave emitted at t

1 and t

2 is determined by

(1.5)

where γ = − 2 2 1/21/(1 / )sv c is a factor that represents the relativistic time dilation and c is the propagation speed of the light or EM waves. Then the time interval between the arrivals of the two successive wave crests at the observer is

(1.6)

2 1st t tf

γΔ = − =

2 12 1

cos1o

s svr rt t t

c c f c

θγ ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞Δ = + − + = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Figure 1.2 The Doppler effect in the case that only the source S is moving with a velocity vs at an angle θs relative to the direction from the source S to the observer O.

Introduction 5

Thus, the corresponding observed frequency by the observer becomes

(1.7)

If the angle, sθ′ , between the moving direction of the source and the di-rection from the source to the observer is measured at the time when the wave arrives at the observer, the observed frequency by the observer becomes

(1.8)

Thus, the two angles θs and sθ′ are related by

(1.9)

or

(1.10)

If both the source and the observer move as illustrated in Figure 1.3 in the two-dimensional case, the observed frequency at the time when the wave is emitted is similar to (1.4) as

(1.11)

where θs and θ

o, as illustrated in Figure 1.3, are the angle of the source moving

and the angle of the observer moving at the time when the wave is emitted, respectively.

In general, given the relative motion between the source and the observer v, when the source and the observer move toward each other, the observed Doppler shifted frequency can be rewritten by

0

1 1cos

1 s s

ff

vtc

θγ= =′

Δ −

cos1 s sv

f fc

θγ

′⎛ ⎞= +′ ⎜ ⎟⎝ ⎠

cos /cos

cos1

s ss

s s

v cv

c

θθ

θ+′

= ′+

cos /cos

cos1

s ss

s s

v cv

c

θθ

θ−

=′−

cos11

cos1

o o

s s

vcf f

vc

θ

θγ

±=′


21 11

1 1 1

f ff f

ββ

γ β β β

+= = − =′

− − − (1.12)

where β = v/c. If the source is moving away from the observer, the observed frequency becomes

(1.13)

If the velocity v is much lower than the velocity of the EM wave propaga-tion c, that is, v << c or β = v/c ≈ 0, the relativistic Doppler frequency is the same as the classical Doppler frequency.

According to the MacLaurin series,

(1.14)

when the source and the observer are moving away from each other, the Dop-pler shifted frequency can be approximated by

(1.15)

1

1f f

β

β

−=′

+

211 ...

1 2

β ββ

β

−= − + −

+

(1 ) (1 )v

f f fc

β≅ − = −′

Figure 1.3 Doppler effect in the case that both the source and the observer move.

Introduction 7

This is the same as the classical Doppler frequency shift. Thus, the Dop-pler frequency shift between the emitted frequency of the source and the per-ceived frequency by the observer is

(1.16)

The Doppler frequency shift is proportional to the emitted frequency f of the wave source and the relative velocity v between the source and the observer.

1.3 Doppler Effect Observed in Radar

In radar, the velocity of a target, v, is usually much slower than the propagation speed of the EM wave propagation c, that is, v << c or β = v/c ≈ 0. In monos-tatic radar systems, where the wave source (radar transmitter) and the receiver are at the same location, the round-trip distance traveled by the EM wave is twice the distance between the transmitter and the target. In this case, the wave movement consists of two segments: traveling from the transmitter to the target that produces a Doppler shift (−f v/c), and traveling from the target back to the receiver that produces another Doppler shift (−f v/c), where f is the transmitted frequency. Thus, the total Doppler shift becomes

(1.17)

If the radar is stationary, v will be the radial velocity of the target along the LOS of the radar. Velocity is defined to be positive when the object is moving away from the radar. As a consequence, the Doppler shift becomes negative.

In a bistatic radar system as shown in the two-dimensional case in Figure 1.4, the transmitter and receiver are separated by a baseline distance L, which is comparable with the maximum range of targets with respect to the transmit-ter and the receiver. The range from the transmitter to the target is given by a vector, rT

, and the range from the receiver to the target is given by a vector, r

R, where a boldface letter is used to denote the vector. The bistatic angle ϕ is

defined by the angle between the transmitter-to-target line and the receiver-to-target line. The transmitter look angle is α

T and the receiver look angle is α

R

as illustrated in Figure 1.4. The look angle is defined by the angle from a cor-responding reference vector perpendicular to the transmitter-receiver baseline to the target’s LOS vector. The positive angle is defined in a counterclockwise direction. Thus, the bistatic angle ϕ = α

R − α

T .

D

vf f f f

c≅ − = −′

2D

vf f

c= −

Administrator

铅笔


If the distance from the transmitter to the target is known by rT = |rT|, the

distance from the receiver to the target is

(1.18)

and the receiver look angle becomes

(1.19)

When the target is moving with a velocity vector V, its component along the LOS direction from the transmitter to the target is

(1.20)

and the component along the LOS direction from the receiver to the target is

( )1/22 2 2 sinR R T T Tr L r r L α= = + −r

αα

α− ⎛ ⎞−

= ⎜ ⎟⎝ ⎠1 sin

tancosT T

RT T

L r

r

TT

T

v =r

Vr

Figure 1.4 The two-dimensional bistatic radar system confi guration.

Introduction 9

RR

R

v =r

Vr

(1.21)

Then, due to the target’s motion, the range from the transmitter to the target is a function of time

(1.22)

and the range from the receiver to the target is also a function of time

(1.23)

The phase change between the transmitted signal and the received signal is a function of the radar wavelength λ = c /f, the distance from the transmitter to the target rT

(t), and the distance from the target to the receiver rR(t):

(1.24)

Then the Doppler frequency shift is measured by the phase change rate. By taking the time derivative of the phase change, the bistatic Doppler fre-quency shift becomes

(1.25)

In order to track the phase changing with time, the phase of the transmit-ted signal must be exactly known. Thus, a fully coherent system is required to preserve and track the phase change in the received signal.

In a bistatic radar system, the Doppler shift depends on three factors [7]. The first factor is the maximum Doppler shift. If a target is moving with a ve-locity V, the maximum Doppler shift is

(1.26)

The second factor is related to the bistatic triangulation factor:

( ) ( 0)T T Tr t r t v t= = +

( ) ( 0)R R Rr t r t v t= = +

( ) ( )( ) T Rr t r tt

λ

+ΔΦ =

1 1 1 1 1( ) ( ) ( ) ( )

2 2 2BiD T R T R

d d df t r t r t v v

dt dt dtπ π λ π λ⎡ ⎤= ΔΦ = + = +⎢ ⎥⎣ ⎦

maxD

2 ff

c= V

Administrator

铅笔


α α ϕ−⎛ ⎞ ⎛ ⎞= = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠cos cos

2 2R TD (1.27)

The third factor is related to the angle δ between the direction in which the target is moving and the direction of the bisector: C = cosδ.

Thus, the Doppler shift of a bistatic radar system can be represented by

(1.28)

If two targets are separated in range and velocity, these targets can be re-solved by the range resolution and the Doppler resolution of the radar system. If in a monostatic radar system the range resolution is known as Δr

Mono and the

Doppler resolution is known as ΔMonoDf , then the range resolution and the Dop-

pler resolution of a bistatic radar system can be determined by the correspond-ing monostatic range resolution and Doppler resolution scaled by a function of the bistatic angle ϕ. Thus, the bistatic range resolution is

(1.29)

and the bistatic Doppler resolution is

(1.30)

However, in an extreme case where the bistatic angle is near 180°, the radar becomes a forward scattering radar [8]. The EM field scattered in the forward direction is 180° out of phase with the incident field. Thus, it removes power from the incident field and forms a shadow area behind the target. Based on (1.28), in the forward scattering case, the bistatic Doppler frequency shift becomes zero regardless the actual target velocities.

1.4 Estimation and Analysis of Doppler Frequency Shifts

Doppler radars utilize the Doppler effect to measure the radial velocity of a tar-get. The Doppler frequency shift can be extracted by a quadrature detector that produces an in-phase (I) component and a quadrature phase (Q) component from the input signal as shown in Figure 1.5.

max

2cos cos

2BiD D

ff f D C

c

ϕδ⎛ ⎞= = ⎜ ⎟⎝ ⎠V

1

cos( /2)Bi Monor rϕ

Δ = Δ

cos( /2)Bi MonoD Df fϕΔ = Δ

Administrator

铅笔

Administrator

铅笔

Introduction 11

In the quadrature detector, the received signal is split into two mixers called synchronous detectors. In the synchronous detector I, the received signal is mixed with a reference signal, the transmitted signal; in the other channel it is mixed with a 90° shift of the transmitted signal.

If the received signal is expressed as

(1.31)

where a is the amplitude of the received signal, f0 is the carrier frequency of the

transmitter, and ϕ(t) = 2πfDt is the phase shift on the received signal due to the

target’s motion. By mixing with the transmitted signal

(1.32)

the output of the synchronous detector I is

(1.33)

After lowpass filtering, the I-channel output is

[ ] [ ]0 0( ) cos 2 ( ) cos 2 ( )r Ds t a f f t a f t tπ π ϕ= + = +

0( ) cos(2 )ts t f tπ=

[ ]0( ) ( ) cos 4 ( ) cos ( )2 2r t

a as t s t f t t tπ ϕ ϕ= + +

Figure 1.5 Doppler shifts extracted by a quadrature detector.


(1.34)

By mixing with the 90° phase shifted transmitted signal,

(1.35)

the output of the synchronous detector ΙΙ is

(1.36)

After lowpass filtering, the Q-channel output is

(1.37)

Combining the I and Q outputs, a complex Doppler signal can be formed by

(1.38)

Thus, the Doppler frequency shift fD can be estimated from the complex

Doppler signal sD(t) by using a frequency measurement tool. To estimate the

Doppler frequency shift of a single sinusoidal signal, the periodogram can be used to calculate the spectral density of the signal. Then the maximum likeli-hood estimation can be applied to locate the maximum of the periodogram [9, 10].

(1.39)

When the number of samples in the analyzed signal is limited, to estimate the signal spectral, the simplest method is to use the fast Fourier transform (FFT), which is computationally efficient and easy to implement. However, its frequency resolution is limited to the reciprocal of the time interval of the signal and suffers from spectrum leakage associated with the time windowing.

( ) cos ( )2

aI t tϕ=

900( ) sin(2 )ts t f tπ=

[ ]900( ) ( ) sin 4 ( ) sin ( )

2 2r t

a as t s t f t t tπ ϕ ϕ= + −

( ) sin ( )2

aQ t tϕ= −

( ) ( ) ( ) exp[ ( )] exp( 2 )2 2D D

a as t I t jQ t j t j f tϕ π= + = − = −

2

( )1

ˆ max ( )exp( 2 ( )N

D Df kD k

f a k j f kπ=

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭∑

Introduction 13

Manually increasing the time window with zero padding corresponds to a high-er interpolation density in the frequency domain, but not a higher frequency resolution. The usual way to increase frequency resolution is to take FFT with a longer time duration of the analyzed signal without zero padding. However, the computation time of the FFT is in the order of O(N ×logN), where N is the number of samples in the analyzed signal. For a large number of samples N, the FFT is not computationally efficient.

To alleviate the limitations of the FFT, alternative spectral estimation methods were proposed [10, 11]. Autoregressive (AR) modeling and eigen-vector-based methods, such as the multiple signal classification (MUSIC) and other super-resolution methods for spectral analysis, can be used in the fre-quency estimation. However, they either require intensive matrix computations or iterative optimization techniques.

Because the frequency is determined by the time derivative of the phase function, the phase difference ϕ(t) between the received and the transmitted signal can be used to calculate the Doppler frequency shift fD

of the received signal

(1.40)

However, this frequency is an instantaneous frequency, which is only suit-able for monocomponent or single-tone signals, but not for those containing multiple components. To deal with a signal having multiple components, an approach that decomposes a multicomponent signal into multiple monocom-ponent signals may be used. Then the complete time-frequency distribution of the multicomponent signal can be derived by computing the instantaneous fre-quency for each of the monocomponent signals and adding these instantaneous frequencies of monocomponent signals together.

For a single-tone signal in additive Gaussian noise, if the phase differ-ence Δϕ(k), (k = 1, …, N − 1), from one sample to the next can be tracked, a weighted linear combination of these phase differences may be used to estimate the sinusoidal frequency tone [12]

(1.41)

where the weighting function w(k) is

1 ( )

2D

d tf

dt

ϕ

π=

1

1

1ˆ ( ) ( )2

N

Dk

f w k kϕπ

−

=

= Δ∑


2

6 ( )( )

( 1)

k N kw k

N N

−=

− (1.42)

This estimation shows that at high SNR, the frequency estimation attains the Cramer-Rao lower bound as described in Section 1.5.

To preserve and track the phase of the received signals, the frequency source in the transmitter must keep a very high phase stability. Therefore, the radar must be fully coherent to keep an accurate phase coherency.

From the estimated Doppler frequency, the radial velocity of the target is determined by

(1.43)

The I and Q outputs of the quadrature detector can also be used to deter-mine whether the target is approaching to or away from the radar. As illustrated in Figure 1.6, by comparing the relative phase between the I-channel and the 90°-shifted Q-channel, two flow channels—one is “approaching” to the radar and the other is “away” from the radar, can be produced.

Doppler radars include pure continuous wave (CW) radar without mod-ulations, frequency modulated continuous wave (FM-CW) radar, and coherent pulsed Doppler radar. Pure CW radars can only measure the velocity. FM-CW and coherent pulsed Doppler radars can have wide frequency bandwidth to

ˆ ˆ2 2D D

cv f f

f

λ= =

Figure 1.6 The relative phase between the I-channel and the 90°-shifted Q-channel to deter-mine whether the target is approaching or going away from the radar.

Introduction 15

gain a high-range resolution and measure both the range and Doppler infor-mation. Coherent Doppler radars retain the phase of the transmitted signals and track the phase changes in the received signals. Doppler frequency shift is proportional to the phase change rate. If the phase change is more than ±π, the estimated Doppler frequency becomes ambiguous and is called Doppler alias-ing. This is caused by discrete time sampling of a continuous time signal. Sam-pling process can be represented by the multiplication of the continuous time signal s(t) with a sequence of delta functions δ(t). Taking the Fourier transform of the time sampled signal, the discrete time sampled signal is transformed to the frequency domain represented by a discrete form of the Fourier transform:

(1.44)

where Δt is the time-sampling interval, the convolution operator ⊗ in the fre-quency domain makes the signal spectrum S(f ) to be replicated at a period of 1/Δt. If the frequency bandwidth of the signal spectrum is greater than the Nyquist frequency 1/(2Δt), or the sampling rate is lower than the half band-width of the signal, this replication will cause the signal spectrum to overlap and produce ambiguity, called aliasing.

Figure 1.7 illustrates the aliasing phenomenon. The frequency modulat-

ed signal is sampled with sampling rate lower than the Nyquist rate ±Δ1

2 t.

Spectrum aliasing can be seen clearly in Figure 1.7(a). Time-varying frequency

spectrum of the signal is shown in Figure 1.7(c), where modulated frequency

values in excess of the Nyquist rate ±Δ1

2 t are aliasing. The aliasing causes the

true frequency values to be offset by multiples of (1/Δt) until they fall into

the Nyquist cointerval. For example, if the Nyquist frequency is ±1,000 Hz, a frequency value of +1,500 Hz is aliased to +1,500 Hz − 2 × 1,000 Hz = −500 Hz and −1,500 Hz is aliased to −1,500 Hz + 2 × 1,000 Hz = +500 Hz. In this case, the true frequency values are offset by one multiple of (1/Δt) to fall into the Nyquist interval as illustrated in Figure 1.7(c).

To resolve the aliasing ambiguity, increasing the sampling rate or tech-niques that interpolate missing data points may be applied. Figure 1.7(b) is the spectrum of the same signal sampled with the twice sampling rate. The time-varying spectrum with twice the sampling rates is shown in Figure 1.7(d), where the full time-varying spectrum is restored.

Generally, the required unambiguous radial velocity must be at least the radial velocity with which the target moves. Thus, the unambiguous velocity that a radar can be measured depends on the transmitted frequency f or wave-length λ = c/f and the time interval Δt between two sampling points

( ) ( ) ( ) ( / )n m

s t t n t S f f m tδ δ× − Δ ⇒ ⊗ − Δ∑ ∑


max

max 2 / 4Df

vf c t

λ= = ±

Δ (1.45)

For the coherent pulsed radar, which measures both the velocity and range, the time interval Δt equals to 1/PRF, where the PRF is the pulse repeti-tion frequency (PRF). Thus, the maximum velocity that the pulsed radar can be measured is

(1.46)

Velocities greater than λ PRF/4 are folded into ± λ PRF/4, called the Ny-quist velocity.

max 4

PRFv

λ= ±

Figure 1.7 Illustration of the aliasing phenomenon. (a) Spectrum with aliasing; (b) The spec-trum of the same signal but sampled with twice sampling rate; (c) The time-varying spectrum of the signal used in (a); (d) The time-varying spectrum of the signal used in (b).

Introduction 17

The range that can be measured by the pulsed radar is limited by the maximum unambiguous range

max 2

cr

PRF= (1.47)

Ranges greater than the rmax

are folded into the first range region. Without additional information, the correct range information cannot be determined.

The PRF is proportional to the maximum unambiguous velocity, but in-versely proportional to the maximum range being measured. A compromise between the maximum range and maximum velocity is always desirable. The maximum velocity can be extended by using two alternating PRFs. Since the maximum velocity is related to the wavelength, a longer wavelength or a lower frequency can increase the limit of the maximum velocity.

However, the product of the maximum unambiguous velocity and the unambiguous range

(1.48)

is not directly related by the PRF. It is only determined by the frequency f or the wavelength λ. Given a frequency band, the product of the maximum un-ambiguous velocity and the unambiguous range is a constant. Increasing the maximum unambiguous velocity will decreasing the maximum unambiguous range and vice versa. The trade-off between the unambiguous velocity and un-ambiguous range is often referred to as the Doppler dilemma.

In pulsed radars, to avoid aliasing, very high PRF should be selected; to avoid ambiguous range, very low PRF is required. However, the low PRF also limits the extracted Doppler information. To have a suitable range ambiguity and aliasing, multiple PRFs are often used.

1.5 Cramer-Rao Bound of the Doppler Frequency Estimation

In practice, the Doppler frequency estimation is considered in the presence of noise. From the estimation theory, to estimate the value of an unknown param-eter θ from N noisy measurements, if the expected value of the estimate equals the true value of the parameter θ θ=ˆ{ }E , the estimator is said to be unbiased. Otherwise, the estimator is biased. When the estimator asymptotically con-

2

max max 8 8

c cv r

f

λ= ± = ±


verges in probability Pr{•} to the true value (i.e., θ θ ε→∞

− > =ˆlim Pr{| | } 0N

, where ε is an arbitrary small positive number), this is a consistent estimator.

The benchmark that evaluates the variance of a particular unbiased esti-mator can be described by the Cramer-Rao lower bound (CRLB) that provides a lower bound on the variance of a linear or nonlinear unbiased estimator and gives an insight into the performance of the estimator [13, 14]. It states that the variance of an unbiased estimator is at least as high as the inverse of the Fisher information.

If an unknown deterministic parameter θ is estimated from the N statisti-cal measurements x

k (k = 1, .., N) with the probability density function of p(x

k;

θ), the variance of the unbiased estimation θvar{ } is bounded by the inverse of the Fisher information I(θ), that is,

(1.49)

The Fisher information is defined by

(1.50)

where E{•} means the expectation value that is taken with respect to p(xk; θ) and

results in a function of θ. To estimate a single sinusoidal Doppler frequency in white Gaussian

noise, the Fisher information can be inverted easily. The CRLB of the Doppler frequency estimation can be derived as

(1.51)

where SNR is the signal-to-noise ratio and N is the number of samples of the signal [14].

1.6 The Micro-Doppler Effect

The micro-Doppler effect was originally introduced in coherent laser (light am-plification by stimulated emission of radiation) radar systems [5]. Laser detec-tion and ranging (LADAR) system transmits EM wave at optical frequencies to an object and receives the reflected or backscattered light wave to measure the

ˆvar{ } 1/ ( )Iθ θ≥

2 2

2( ) log ( ; ) ( ; )k kI p x p xθ θ θθ θ

⎧ ⎫ ⎧ ⎫∂ ∂⎪ ⎪⎡ ⎤= Ε = −Ε⎨ ⎬ ⎨ ⎬⎢ ⎥∂ ∂⎣ ⎦⎪ ⎪ ⎩ ⎭⎩ ⎭

2

6ˆvar( )( 1)Df

N N SNR≥

− ⋅

Introduction 19

object’s range, velocity, and other properties through modulations of its laser beam by amplitude, frequency, phase, and even polarization.

Coherent LADAR, which preserves the phase information of the scat-tered light wave with respect to a reference laser wave generated in the local oscillator, has greater sensitivity to phase changing and is capable of measuring object velocity from the phase change rate.

In a coherent system, because the phase of a returned signal from an ob-ject is sensitive to the variation in range, a half-wavelength change in range can cause 360° phase change. For LADAR with a wavelength of 2 μm, a 1-μm range variation can cause a 360° phase change. In the case of vibration, if the vibration frequency is f

v and the amplitude of the vibration is D

v, the maximum

Doppler frequency variation is determined by

(1.52)

As a consequence, in a high-frequency system, even with a very low vibra-tion rate f

v, a very small vibration amplitude D

v can cause a large phase change,

and thus, Doppler frequency shifts can be easily detected. In many cases, an object or any structural component of the object may

have oscillatory motion, which can be called the micro motion. The term “micro motion” defined here includes a broader usage of the “micro,” such that in ad-dition to the bulk motion of the object, any oscillatory motion of the object or any structural component of the object can be called the micro motion. The source of micro motion may be a rotating propeller of a fixed-wing aircraft, the rotating rotor blades of a helicopter, a rotating antenna, the flapping wings of birds, a walking person with swinging arms and legs, or other causes.

Human motion is an important topic in micro-Doppler study. Human articulated motion is accomplished by a series of motion of human body parts. It is a complex micro motion due to high articulation and flexibility. Walking is a typical example of human articulated motion.

Micro motion induces frequency modulations on the carrier frequency of radar transmitted signals. For a pure periodic vibration or rotation, micro motion generates side-band Doppler frequency shifts about the center of the Doppler shifted carrier frequency. The modulation contains harmonic frequen-cies determined by the carrier frequency, the vibration or rotation rate, and the angle between the direction of vibration and the direction of the incident wave. The frequency modulation enables us to determine the kinematic properties of the object of interest. Although the use of a propeller or rotor blade modulation and jet-engine modulation for target identification has been proposed for a long time, the methods of how to represent the time-varying frequency modulation as a signature of target, how to extract the kinematic information about the

{ }max (2/ )D v vf D fλ=


target from the signature, and how to use the signature for target identification are still new research tasks.

1.7 Micro-Doppler Effect Observed in Radar

The micro-Doppler effect is sensitive to the frequency band of the signal. For a radar system operating at the microwave frequency bands, the micro-Doppler effect may be observable if the product of the target’s oscillation rate and the displacement of the oscillation is high enough. For a radar operating at the X-band with a 3-cm wavelength, a vibration rate of 15 Hz with a displacement of 0.3 cm can induce a detectable maximum micro-Doppler frequency shift of 18.8 Hz. If the radar is operated at the L-band with a 10-cm wavelength, to achieve the same maximum micro-Doppler shift of 18.8 Hz, for the same vibration rate of 15 Hz, the required displacement must be 1 cm, which may be too large to be achieved in practice. Therefore, in radar systems operating at lower-frequency bands, micro-Doppler shifts generated by vibration may not be detectable. However, micro-Doppler shifts generated by rotations, such as rotating rotor blades, may be detectable because of their longer rotation arms and higher tip speeds.

The UHF-band radar operating at a frequency band of 300–1,000 MHz is widely used for foliage penetration (FOPEN) to detect targets under trees. In FOPEN radars, the micro-Doppler shift induced by target’s vibrations is usual-ly too small to be detected. However, it is still possible to detect micro-Doppler shifts generated by rotating rotor blades or propellers. For a radar operating at the UHF-band with a 0.6-m wavelength, if a helicopter’s rotor blade rotates with a tip speed of 200 m/s, its maximum micro-Doppler shift can reach 666 Hz and it is certainly detectable.

1.8 Estimation and Analysis of Micro-Doppler Frequency Shifts

The micro-Doppler shift is a time-varying frequency shift that can be extracted from the complex output signal of a quadrature detector used in the conven-tional Doppler processing. For analyzing time-varying frequency features, the Fourier transform is not suitable because it cannot provide time-dependent fre-quency information. The commonly used analysis methods to describe a signal simultaneously in the time and frequency domains are the instantaneous fre-quency analysis and the joint time-frequency analysis.

The terminology of the instantaneous frequency defined by the time de-rivative of the phase function in a time-varying signal has been argued decades ago because the amplitude and phase functions are not unique. A well-accepted

Introduction 21

instantaneous frequency definition uses a pair of Hilbert transform to form the real part and the imaginary part of an analytic signal [15]. Thus, the instanta-neous term means in the sense of the present time instant, and its measurement requires only the knowledge of the analyzed signal over the past and not from the future.

The instantaneous frequency derived by the time-derivative operation yields only one value of frequency at a given time instant. This means it is only suitable for monocomponent signals but not for multicomponent signals. A monocomponent signal is narrowband at any time and has energy in a contigu-ous portion in the joint time-frequency domain. Conversely, a multicomponent signal has energy in multiple isolated frequency bands at the same time instant. To deal with multicomponent signals, an obvious approach is to decompose the multicomponent signal into multiple addable monocomponent signal compo-nents [16]. The complete time-frequency distribution of the signal is obtained by computing the instantaneous frequencies for each component signal and adding these individual instantaneous frequencies together.

The joint time-frequency analysis has been used for decades for analyzing the time-varying frequency spectrum. It is designed to localize the energy distri-bution of a given signal in the two-dimensional time and frequency domains. It is quite suitable for not only monocomponent signals but also multicomponent signals.

1.8.1 Instantaneous Frequency Analysis

For a real valued signal s(t), its associated complex valued signal z(t) is defined by

(1.53)

where H{•} is the Hilbert transform of the signal given by

(1.54)

z(t) is called the analytic signal associated with s(t), a(t) is the amplitude func-tion, and ϕ(t) is the phase function of the analytic signal. In the frequency do-main, the Fourier transform of the analytic signal, Z(f ), is single-sided with zero values in the negative frequencies and double values in the positive frequencies. Thus, the instantaneous frequency of the signal z(t) is the time derivative of the uniquely defined phase function ϕ(t) of the analytic signal

{ }( ) ( ) ( ) ( )exp[ ( )]z t s t jH s t a t tϕ= + =

{ } 1 ( )( )

ss t d

t

ττ

π τ

∞

−∞

Η =−∫


(1.55)

In practice, a discrete real-valued signal s(n) and samples at time instants t = nΔt, n = 1, 2, …, N may be used. Then the discrete analytic signal z(n) becomes

(1.56)

For a discrete signal, the instantaneous frequency is similar to (1.55), but with discrete derivatives of the phase, which can be estimated by using the cen-tral finite difference equation of the phase function [17]:

(1.57)

where Δt is the sampling interval, [ • ]2π

represents the reduction modulo 2π, and n is the discrete number of time samples.

Instantaneous frequency only gives one value at a time and is only good for describing signals comprised of a single oscillating frequency component at one time. It is not suitable for signals having several different oscillating frequency components at one time. To distinguish frequency contributions of a multi-component signal, it is necessary to preprocess the multicomponent signal into its monocomponent elements. Huang et al. [16] introduced the concept of empirical mode decomposition (EMD) to separate a multicomponent signal into monocomponent constituents by a progressive sifting process to yield the bases called the intrinsic mode functions (IMFs). Later, Olhede and Walden [18] introduced a wavelet packet-based decomposition as a replacement of the EMD in preprocessing the multicomponent signal.

The EMD adaptively decomposes a signal into a limited number of zero-mean, narrowband IMFs. Then the instantaneous frequency of each IMF is calculated by using the normalized Hilbert transform, called the Hilbert-Huang transform (HHT) [16]. The combination of the Hilbert spectrum is the com-plete time-varying frequency spectrum.

The original formulation of the EMD can only be applied to real-valued signals. However, in radar applications, signals are always complex with I and Q parts. The extension of the EMD to handle complex values signals has been proposed in [19, 20]. The detailed procedure of calculating complex EMD and HHT with MATLAB codes can be obtained from [21].

1( ) ( )

2

df t t

dtϕ

π=

{ }( ) ( ) ( )z n s n j s n= + Η

[ ]2

1 1( ) ( 1) ( 1)

2 2f n n n

t πϕ ϕ

π= + − −

Δ

Introduction 23

1.8.2 Joint Time-Frequency Analysis

When the spectral composition of a signal varies as a function of time, the conventional Fourier transform cannot provide a time-dependent spectral de-scription. Thus, a joint time-frequency analysis provides more insight into the time-varying behavior of the signal.

Motivated by defining the information content in a signal, in 1948, Den-nis Gabor, a Hungarian Nobel laureate, proposed the first algorithm on time-frequency analysis of an arbitrary signal [22]. Gabor suggested that the time and frequency characteristics of a signal s(t) can be simultaneously observed by using the expansion

(1.58)

where G(g, n, m), called the Gabor function, is expressed in terms of a Gaussian window g(t) by

(1.59)

ΔT and ΔF are the time and frequency lattice intervals, respectively, and the Gaussian window is defined by

(1.60)

Gabor claimed that the basis functions G(•) used in this time-frequency decomposition have the minimum area in the joint time-frequency plane.

Spectrogram is a widely used method to display time-varying spectral density of a time-varying signal. It is a spectro-temporal representation and provides the actual change of frequency contents of a signal over time. The spectrogram is calculated by using the short-time Fourier transform (STFT) and represented by the squared magnitude of the STFT without keeping phase information of the signal

(1.61)

STFT performs the Fourier transform on a short-time window basis rath-er than taking the Fourier transform to the entire signal using one long-time window.

( ) ( , , )mnm n

s t a G g n m∞ ∞

= −∞ = −∞

= ∑ ∑

( , , ) ( ) jn FtG g n m g t m T e Δ= − Δ

2

21/4

1( ) exp

2

tg t

σπ σ

⎧ ⎫= −⎨ ⎬

⎩ ⎭

( ) ( ) 2Spectrogram , ,t f STFT t f=


With the time-limited window function, the resolution of the STFT is determined by the window size. There is a trade-off between the time resolution and the frequency resolution. A larger window has a higher-frequency resolu-tion but a poorer time resolution. The Gabor transform is a typical short-time Fourier transform using Gaussian windowing and has the minimal product of the time resolution and the frequency resolution.

To better analyze the time-varying micro-Doppler frequency characteris-tics and visualize the localized joint time and frequency information, the sig-nal must be analyzed by using a high-resolution time-frequency transform to characterize the spectral and temporal behavior of the signal. Bilinear trans-forms, such as the Wigner-Ville distribution (WVD), are high-resolution time-frequency transforms. The WVD of a signal s(t) is defined by the Fourier trans-form of the time-dependent autocorrelation function

(1.62)

where ′ ′⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠*

2 2

t ts t s t can be interpreted as a time-dependent autocorre-

lation function. The bilinear WVD has a better joint time-frequency resolu-

tion than any linear transform, such as the STFT. However, it suffers from the problem of cross-term interference (i.e., the WVD of the sum of two signals is not the sum of their individual WVDs). If a signal contains more than one component in the joint time-frequency domain, its WVD will contain cross-terms that occur halfway between each pair of autoterms. The magnitude of these oscillatory cross-terms can be twice as large as the autoterms. To reduce the cross-term interference, filtered WVDs have been used to preserve the use-ful properties of the time-frequency transform with a slightly reduced time-frequency resolution and a largely reduced cross-term interference. The WVD with a linear lowpass filter belongs to the Cohen class [23].

The general form of the Cohen class is defined by

(1.63)

The Fourier transform of the lowpass filter ψ(t, τ), denoted as Ψ(θ, τ), is called the kernel function. If Ψ(θ, τ) = 1, then ψ(t, τ) = δ(t) and the Cohen class reduces to the WVD. The Cohen class with different kernel functions, such as the pseudo-Wigner, the smoothed pseudo-Wigner-Ville (SPWV), the

{ }*( , ) exp 22 2

t tWVD t f s t s t j ft dtπ

′ ′⎛ ⎞ ⎛ ⎞= + − − ′ ′⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫

{ }*( , ) ( , )exp 22 2

C t f s u s u t u j f du dτ τ

ψ τ π τ τ⎛ ⎞ ⎛ ⎞= + − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫∫

Introduction 25

Choi-Williams distribution, and the cone kernel distribution, can be used to largely reduce the cross-term interference in the WVD.

Other useful high-resolution time-frequency transforms are the adaptive Gabor representation and the time-frequency distribution series [24]. They de-compose a signal into a family of basis functions, such as the Gabor function, which is well localized in both the time and the frequency domains and adap-tive to match the local behavior of the analyzed signal.

In contrast with the EMD method, the adaptive Gabor representation is a signal-adaptive decomposition. It decomposes a signal s(t) into the Gabor basis functions h

p(t) with an adjustable standard deviation σ

p and a time-frequency

center (tp, f

p):

(1.64)

where

(1.65)

The coefficients Bp are found by an iterative procedure beginning with

the stage p = 1 and choosing the parameters σp, t

p, and f

p such that h

p(t) is most

similar to s(t):

(1.66)

where s0(t) = s(t), that is, the analyzed signal is taken as the initial signal for p =

1. For p >1, sp(t) is the residual after the orthogonal projection of s

p−1(t) onto h

p

(t) has been removed from the signal:

(1.67)

This procedure is iterated to generate as many coefficients as needed to accurately represent the original signal. Finally, the time-dependent spectrum is obtained by

(1.68)

1

( ) ( )p pp

s t B h t∞

=

= ∑

( )2

1/422

( )( ) exp exp( 2 )

2p

p p pp

t th t j f tπσ π

σ

− ⎡ ⎤−= −⎢ ⎥

⎢ ⎥⎣ ⎦

, ,

22 *

1max ( ) ( )

t fp p pp pp

B s t h t dtσ −

= ∫

1( ) ( ) ( ) ( )p p p ps t s t B t h t−= −

( ) ( )2AdaptiveGabor , ,

pp hp

f t B WVD t f= ∑


The well-known MATLAB time-frequency toolbox developed at the CNRS (Centre National de la Recherche Scientique) of France is a collection of time-frequency analysis tools [25]. It includes many commonly used linear and bilinear time-frequency distributions and can be used to compute micro-Doppler signatures represented in the joint time-frequency domain. However, this time-frequency toolbox was designed for analytic signals. Thus, rescaling the frequency scale in the output time-frequency representations is needed if the input signal is a complex I and Q signal.

1.9 The Micro-Doppler Signature of Objects

The term “signature” is commonly used to refer to the characteristic expres-sion of an object or a process. For example, the characteristic mode in different ocean basins is called a signature of climate phenomenon, such as ENSO and El Nino. In Doppler weather radars, a special pattern of strong outbound and inbound winds is called the signature of a tornado.

When examining the Doppler phenomenon in an object, distinctive mi-cro-Doppler characteristics provide evidence of the identity of the movement of the object. The micro-Doppler signature is the distinctive characteristics of the object’s movement. It is an intricate frequency modulation represented in the joint time and Doppler frequency domain, and it is the distinctive characteris-tics that give an object its identity.

Figure 1.8 shows the micro-Doppler signature of a rotating air-launched cruise missile (ALCM) provided in the simulation software in [26, 27]. The length of the cruise missile is 6.4m and its wingspan is about 3.4m. A burst of a 1-μs chirp pulse radar operating in the X-band is assumed to simulate the EM backscattering field. The radar transmits 8,192 pulses with the pulse repetition interval of 67 μs during a period of 0.55 second to cover the total target’s rota-tion angle of 360°. The simulated radar I and Q data from the rotating ALCM is provided in the companion MATLAB micro-Doppler signature analysis tools on the DVD with this book.

The micro-Doppler features of the rotating ALCM can be observed in the frequency domain and with a much more clear view in the joint time-frequency domain [28]. Figure 1.8(b) shows the joint time-frequency micro-Doppler signature of the rotating ALCM. For comparison, the conventional Fourier spectrum is shown in Figure 1.8(a). Recall that the missile rotating rate is about 1.8 cycle/second because it takes 0.55 second to complete a rotation of 360°. For the ALCM model, (1) the missile head tip, (2) head joint, (3) wing joint, (4) turbine engine intake, (5) tail fin and tail plane, and (6) tail tip and engine exhaust are located at about −2.5m, −1.8m, 0.2m, 2.5m, 3.5m, and 4.2m, respectively, off the pivot point at 0. If these parts are considered to be the

Introduction 27

dominant scatterers, the maximum Doppler shifts induced by rotations of these parts would appear at those positions when their angular velocities are nearly in parallel with the radar LOS, when the missile is at a 90° or 270° aspect to the radar, or at the elapsed time of 0.14 second or 0.41 second, respectively, as shown in Figure 1.8(b). When the missile is at a 90° aspect, the induced Dop-pler shifts are −1,917 Hz, −1,380 Hz, 153 Hz, 1,917 Hz, 2,684 Hz, and 3,217 Hz, respectively. For these dominant scatterers, the traces of induced Doppler shifts by rotations are clearly shown in Figure 1.8(b). At the aspects from 180° to 90° and from 180° to 270°, the induced Doppler shifts of the ALCM model have the same magnitudes but with opposite signs. The same motion kinemat-ics is also seen at the missile aspects from 0° to 90° and from 360° to 270°. These Doppler shifts gradually reduce to zero when their moving directions are perpendicular to the radar LOS (i.e., when the missiles are approximately at 180° and 360° aspects). Notice that the induced Doppler shifts are further dispersed for two scatterers near the missile tail where two turbines are located. Thus, the missile kinematic motion can be well characterized by its micro-Doppler signature, which can be used to identify distinctive target features. The

Figure 1.8 The micro-Doppler signature of a simulated rotating ALCM. (After: [28].)


MATLAB source code for calculating the micro-Doppler signature of a rotating ALCM is listed in Appendix 1A and can be used as a micro-Doppler signature analysis tool.

In recent years, micro-Doppler signatures have been applied to various targets to extract features and detect and identify targets of interest. Among the numerous references on the micro-Doppler effect in radar, a few of them con-tributed to theoretical properties of the micro-Doppler effect in radar [5, 29–31] and some of them contributed to the extension from monostatic to bistatic and multistatic micro-Doppler features [32, 33]. Many other references con-tributed to processing and analyzing micro-Doppler signatures of rigid bodies with oscillatory motion and nonrigid bodies with articulated motion [34–54]. Target classification, recognition, and identification based on micro-Doppler signatures are also important topics and have been investigated [55–63].

The basic concept of the micro-Doppler effect in radar, the basic math-ematics of the micro-Doppler effect, and the mathematic representations of micro-Doppler shifts induced by the target’s vibration, rotation, and coning motion are introduced in Chapter 2.

In simulation study of the micro-Doppler effect in radar, the first step is to generate micro-Doppler signatures of a target of interest. To do this, a suitable motion model of the target must first be defined. The next step is to model the radar scattering from the target. Thus, a suitable radar cross section (RCS) model of the target should be determined. In Chapters 3 and 4, the simplest point-scatterer model and a more accurate RCS prediction model are introduced to calculate the micro-Doppler signatures of targets, and examples of MATLAB source codes for calculating radar scattering are provided.

Chapter 5 introduces the biological motion perception and the relation-ship between biological motion information and micro-Doppler signatures, which leads to the possible identification of targets’ movements with their mi-cro-Doppler signatures.

Chapter 6 summarizes the micro-Doppler effect in radar introduced in this book and lists some challenges and perspectives in micro-Doppler research.

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Introduction 29

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[23] Cohen, L., Time-Frequency Analysis, Englewood Cliffs, NJ: Prentice-Hall, 1995.


[24] Qian, S., and D. Chen, Introduction to Joint Time-Frequency Analysis: Methods and Applications, Englewood Cliffs, NJ: Prentice-Hall, 1996.

[25] Time-Frequency Toolbox, http://tftb.nongnu.org/.

[26] Shirman, Y. D., Computer Simulation of Aerial Target Radar Scattering, Recognition, Detection, and Tracking, Norwood, MA: Artech House, 2002.

[27] Gorshkov, S. A., et al., Radar Target Backscattering Simulation: Software and User’s Manual, Norwood, MA: Artech House, 2002.

[28] Chen, V. C., C. -T. Lin, and W. P. Pala, “Time-Varying Doppler Analysis of Electromagnetic Backscattering from Rotating Object,” The IEEE Radar Conference Record, Verona, NY, April 24–27, 2006, pp. 807–812.

[29] Setlur, P., M. Amin, and F. Ahmad, “Cramer-Rao Bounds for Range and Motion Parameter Estimations Using Dual Frequency Radars,” Proc. of the IEEE International Conference on Acoustics, Speech and Signal Processing, Honolulu, HI, April 15–20, 2007, pp. 813–816.

[30] Setlur, P., M. Amin, and F. Ahmad, “Optimal and Suboptimal Micro-Doppler Estimation Schemes Using Carrier Diverse Doppler Radars,” Proc. of the IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, April 19–24, 2009.

[31] Setlur, P., M. Amin, and F. Ahmad, “Dual Frequency Doppler Radars for Indoor Range Estimation: Cramér-Rao Bound Analysis,” IET Signal Processing, Vol. 4, No. 3, 2010, pp. 256–271.

[32] Smith, G. E., K. Woodbridge, and C. J. Baker, “Multistatic Micro-Doppler Signature of Personnel,” IEEE Radar Conference, Rome, Italy, May 2008.

[33] Chen, V. C., A. des Rosiers, and R. Lipps, “Bi-Static ISAR Range-Doppler Imaging and Resolution Analysis,” IEEE Radar Conference, Pasadena, CA, May 2009.

[34] Sparr, T., and B. Krane, “Micro-Doppler Analysis of Vibrating Targets in SAR,” IEE Radar, Sonar and Navigation, Vol. 150, No. 4, 2003, pp. 277–283.

[35] Chen, V. C., and A. P. des Rosiers, “Micro-Doppler Phenomenon and Radar Signature,” Proc. of the IEEE National Radar Conference, 2003, pp. 198–202.

[36] Chen, V. C., “Detection and Analysis of Human Motion by Radar,” IEEE Radar Conference, Rome, Italy, May 2008.

[37] Chen, V. C., “Doppler Signatures of Radar Backscattering from Objects with Micro-Motions,” IET Signal Processing, Vol. 2, No. 3, 2008, pp. 291–300.

[38] Ghaleb, A., L. Vignaud, and J. M. Nicolas, “Micro-Doppler Analysis of Pedestrians in ISAR Imaging,” IEEE Radar Conference, Rome, Italy, May 2008.

[39] Ghaleb, A., L. Vignaud, and J. M. Nicolas, “Micro-Doppler Analysis of Wheel and Pedestrians in ISAR Imaging,” IET Signal Processing, Vol. 2, No. 3, 2008, pp. 301–311.

[40] Luo, Y., et al., “Micro-Doppler Extraction of Frequency-Stepped Chirp Signal Based on the Hough Transform,” Proc. of the 8th International Symposium on Antenna, Propagation and EM Theory, 2008, pp. 408–411.

[41] Li, K., et al., “A New Separation Method for Micro-Doppler Information of a Target with Rotating Parts,” Proc. of International Conference on Communications, Circuits and Systems (ICCCAS), 2008, pp. 1365–1369.

Introduction 31

[42] Bai, X., et al., “Imaging of Micro Motion Targets with Rotating Parts Based on Empirical Mode Decomposition,” IEEE Transactions on Geoscience and Remote Sensing, Vol. 46, No. 11, 2008, pp. 3514–3523.

[43] Sun, H. X., and Z. Liu, “Micro-Doppler Feature Extraction for Ballistic Missile Warhead,” Proc. of the 2008 IEEE International Conference on Information and Automation (ICIA), 2008, pp. 1333–1336.

[44] Liu, Z., and H. X. Sun, “Micro-Doppler Analysis and Application of Radar Targets,” Proc. of the 2008 IEEE International Conference on Information and Automation (ICIA), 2008, pp. 1343–1347.

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[48] Ram, S. S., et al., “Doppler-Based Detection and Tracking of Humans in Indoor Environments,” Journal of the Franklin Institute, Vol. 345, No. 6, 2008, pp. 679–699.

[49] Zhang, Z., et al., “Acoustic Micro-Doppler Radar for Human Gait Imaging,” Journal of the Acoustical Society of America, Vol. 121, No. 3, 2007, pp. EL110–EL113.

[50] Li, B., et al., “ISAR Based on Micro-Doppler Analysis and Chirplet Parameter Separation,” Proc. of the First Asian and Pacifi c Conference on Synthetic Aperture Radar (APSAR), 2007, pp. 379–384.

[51] Ning, C., et al., “Modeling and Simulation of Micro-Motion in the Complex Warhead Target,” Proc. of SPIE: Second International Conference on Space Information Technology, Vol. 6795, 2007, pp. 679551.

[52] Anderson, M. G., and R. L. Rogers, “Micro-Doppler Analysis of Multiple Frequency Continuous Wave Radar Signatures,” Proc. of SPIE: Through-the-Wall and Human Detection Radar, Vol. 6547, 2007, pp. 65470A.

[53] Liu, Y., et al., “Radar Micro-Doppler Target Resolution,” Proc. of the CIE International Conference of Radar, 2007.

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[55] Kim, Y., and H. Ling, “Human Activity Classifi cation Based on Micro-Doppler Signatures Using an Artifi cial Neural Network,” IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, 2008.

[56] Kim, Y., and H. Ling, “Human Activity Classifi cation Based on Micro-Doppler Signatures Using a Support Vector Machine,” IEEE Transactions on Geoscience and Remote Sensing, Vol. 47, No. 5, 2009, pp.1328–1337.


[57] Smith, G. E., K. Woodbridge, and C. J. Baker, “Micro-Doppler Signature Classifi cation,” Proc. of the CIE International Conference of Radar, 2007.

[58] Smith, G. E., K. Woodbridge, and C. J. Baker, “Template Based Micro-Doppler Signature Classifi cation,” Proc. of the 3rd European Radar Conference (EuRAD), 2007, pp. 158–161.

[59] Smith, G. E., K. Woodbridge, and C. J. Baker, “Naïve Bayesian Radar Micro-Doppler Recognition,” Proc. of the 2008 International Conference on Radar, 2008, pp. 111–116.

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[62] Yang, Y., and C. Lu, “Human Identifi cations Using Micro-Doppler Signatures,” Proc. of the 5th IASTED International Conference on Antennas, Radar, and Wave Propagation, 2008, pp. 69–73.

[63] Yang, Y., et al., “Target Classifi cation and Pattern Recognition Using Micro-Doppler Radar Signatures,” Proc. of the 7th International Conference on Software Eng., Artifi c. Intelligence, Netw., and Parallel/Distributed Comput., 2006, pp. 213–217.

Appendix 1A

MATLAB Source Codes

ALCMSignature.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% micro-Doppler signature of rotating ALCM% % ALCM: length of 6.4 m and wingspan of 3.4 m% rotating (yaw): 0-359 degree;% Radar Signal: 3 cm wavelength % 1 us short pulse waveform % pulse duration 0.0533 us% Pulse repetition frequency (PRF): 15,000 Hz% ALCM translational velocity: 0% Number of range samples: 100% Range dimension: 20 m% Number of realization (pulses): 8192% Total rotating time period: 0.55 s% Data type: complex I&Q%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all% read the ALCM file: [filename] = ‘RotatingALCM.dat’;% read data header and data

Introduction 33

[np,t1,t2,t3,nr,t4]=textread(filename,’%d %s %s %s %d %s’,1);[c1,c2]=...textread(filename,’%n%n%*[^\n]’,’headerlines’,1,’delimiter’,’,’);for k=1:nr Cdata(k,1:np)=c1(k+[0:np-1]*nr,1)+j*c2(k+[0:np-1]*nr,1);end rngpro = fftshift(fft(Cdata),1); Cdata = Cdata/max(max(abs(Cdata))); figure(1)colormap(jet)imagesc([1 np],[1 nr],20*log10(abs(Cdata(:,1:10:end)+eps)));grid on; box on; zoom on;axis xy;xlabel(‘Pulses’);ylabel(‘Range cells’);title(‘Range Profiles (dBm^2)’)clim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-60 -10]);colorbar;drawnow % Micro-Doppler signaturePRF = 15000;PRI = 1/PRF;T = PRI*np;F = 1/PRI; % analyzed signalx = sum(Cdata); % divide long data into a number of shorter (512) data segmentswd = 512;wdd2 = wd/2;wdd8 = wd/8;ns = np/wd; % total number of segments % calculate short-time Fourier transform for each short data segmentdisp(‘Calculating STFT for segments ...’)for k = 1:ns disp(strcat(‘Progress: segment no.’,num2str(k),’/’,... num2str(round(ns)))) sig(1:wd,1) = x(1,(k-1)*wd+1:(k-1)*wd+wd); TMP = stft(sig,24); TF2(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);endTF = TF2;disp(‘Calculating STFT for time-shifted segments ...’)TF1 = zeros(size(TF));for k = 1:ns-1 disp(strcat(‘Progress: shifted no.’,num2str(k),’/’,... num2str(round(ns-1)))) sig(1:wd,1) = x(1,(k-1)*wd+1+wdd2:(k-1)*wd+wd+wdd2);


TMP = stft(sig,24); TF1(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);enddisp(‘Removing the edge effect ...’)for k = 1:ns-1 TF(:,k*wdd8-8:k*wdd8+8) = ... TF1(:,(k-1)*wdd8+wdd8/2-8:(k-1)*wdd8+wdd8/2+8);end % display final time-frequency micro-Doppler signaturefigure(2)colormap(jet)imagesc([0,T],[-F/2,F/2],20*log10(fftshift(abs(TF),1)+eps))xlabel(‘Time (s)’)ylabel(‘Doppler (Hz)’)title(‘Micro-Doppler Signature’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-60 -10]);colorbardrawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

35

2 Basics of the Micro-Doppler Effect in RadarThe micro-Doppler effect induced by the micro motion of an object or struc-tures on the object can be formulated from physics and mathematics points of view. The physics of the micro-Doppler effect is derived directly from the theory of the EM scattering field. The mathematics of the micro-Doppler effect is derived by introducing micro motion to the conventional Doppler analysis.

Before analyzing radar scattering from an object with micro motion, the basic principle of rigid body and nonrigid body motions and radar scattering from an object with motion will be introduced.

2.1 Rigid Body Motion

An object can be a rigid body or a nonrigid body. A rigid body is a solid body with a finite size, but without deformation (i.e., the distance between any two particles of the body does not vary during any motion). This is an idealization, but it efficiently simplifies simulation and analysis.

The mass of a rigid body, M, is the sum of its particle masses, = ∑ ,kk

M m

where mk is the mass of the k-particle. The general motion of a rigid body is

a combination of translations (i.e., the parallel motion of all particles in the body) and rotations (i.e., the circular motion of all particles in the body about an axis) [1–3].

To describe the motion of a rigid body, two coordinate systems are com-monly used, as shown in Figure 2.1: the global or space-fixed system (X, Y, Z ) and the local or body-fixed system (x, y, z), which is rigidly fixed in the body. The range vector R is from the origin of the spaced-fixed system to the origin


of the body-fixed system. Let the origin of the body-fixed system be the center of mass (CM) of the body. Then, the orientation of the axes of the body-fixed system relative to the axes of the space-fixed system is given by three indepen-dent angles. Therefore, the rigid body becomes a mechanical system with six degrees of freedom. Let r denote the position of an arbitrary particle P in the body-fixed system. Then its position in the space-fixed system is given by r + R, and its velocity is

(2.1)

where V is the translation velocity of the CM of the rigid body and � is the angular velocity of the body rotation. The direction of � is along the axis of rotation. Thus, a rigid body motion consists of a bulk translational motion and the rotation and/or vibration of the body. As defined in Chapter 1, the rotation and vibration of a body can also be called the micro motion of the body.

To represent the orientation of an object, Euler angles, rotation matrices, and quaternions are the commonly used representations.

2.1.1 Euler Angles

In a rigid body, the rotation about an axis can be described by the rotation axis and the rotation angle using a vector of angular velocity � . The direction of the vector is along the rotation axis. The rotation about an axis can also be de-scribed by three rotations about coordinate axes. Euler’s rotation theorem states

( )d

dt= + = + ×�v R r V r

Figure 2.1 Two coordinate systems: the space-fi xed system (X, Y, Z) and the body-fi xed sys-tem (x, y, z) used to describe the motion of an object.

Basics of the Micro-Doppler Effect in Radar 37

that any two independent orthonormal coordinates are related by a sequence of rotations about coordinate axes [1–4]. There are 12 different sequences avail-able to represent an orientation with Euler angles. They are x-y-z, x-z-y, x-y-x, x-z-x, y-x-z, y-z-x, y-x-y, y-z-y, z-x-y, z-y-x, z-x-z, and z-y-z. The sequence may use the same axis twice but not successively. The order in a rotation sequence is important because the matrix multiplication is not commutative.

The rotation angles (ϕ, θ, ψ) are called the Euler angles, where ϕ is de-fined as the counterclockwise rotation about the z-axis, θ is defined as the coun-terclockwise rotation about the y-axis, and ψ is defined as the counterclockwise rotation about the x-axis. Euler angles are commonly used to represent three successive rotations in a given rotation sequence. The first step of the rotation sequence rotates the coordinates (x, y, z) to the new coordinates (x

1, y

1, z

1).

The second step changes the new coordinates to (x2, y

2, z

2), and the third step

transforms this coordinates to the final coordinates (x3, y

3, z

3), as illustrated in

Figure 2.2.Some conventions regarding successive rotation sequence are most com-

monly used. In aerospace engineering, to describe a flying aircraft heading in the x-axis, with its left side toward the y-axis and the upper side to the z-axis, three angles rotating about the x-axis, y-axis, and z-axis, called the roll-pitch-yaw convention, are commonly used as illustrated in Figure 2.3. Pitch or attitude is defined by the rotation of θ between −π/2 and π/2 about the y-axis from the pilot’s right side toward the left side, and, thus, the nose of the aircraft pitches up or down. Roll or bank is defined by the rotation of ψ between –π and π about the longitudinal x-axis of the aircraft from the aircraft tail to its nose. Yaw or heading is defined by the rotation of ϕ between –π and π about the vertical z-axis of the aircraft from the bottom toward the top of the aircraft and perpen-dicular to the other two axes.

Another commonly used rotation sequence is called the x-convention in classical mechanics. It follows the z-x-z sequence that takes the first rotation by an angle about the z-axis, the second rotation by an angle about the x-axis, and the third rotation by an angle about the z-axis again.

Figure 2.2 Euler angles commonly used to represent three successive rotations.


Given a specific rotation sequence, the rotation matrix is a useful tool for calculating rigid body rotations. Any rigid body specified by its orientation and rotation can be described by its rotation matrix, which can be represented by a product of three elemental rotations defined by the three rotation angles.

For the roll-pitch-yaw or x-y-z sequence, the rotation is in the roll-pitch-yaw (ψ-θ-ϕ) sequence. The first step is rotating about the x-axis x = [1 0 0]T by an angle ψ defined by the elemental rotation matrix:

(2.2)

The second step is rotating about the new y-axis: y1 = [0 cosψ sinψ]T by an

angle θ defined by the elemental rotation matrix:

(2.3)

The third step is rotating about the new z-axis: z2 = [−sinθ cosθsinψ

cosθcosψ]T by an angle ϕ defined by the elemental rotation matrix:

1 0 0

0 cos sin

0 sin cosX ψ ψ

ψ ψ

⎡ ⎤⎢ ⎥ℜ = ⎢ ⎥⎢ ⎥−⎣ ⎦

cos 0 sin

0 1 0

sin 0 cosY

θ θ

θ θ

−⎡ ⎤⎢ ⎥ℜ = ⎢ ⎥⎢ ⎥⎣ ⎦

Figure 2.3 The roll-pitch-yaw convention used to describe a fl ying aircraft.


cos sin 0

sin cos 0

0 0 1Z

ϕ ϕ

ϕ ϕ

⎡ ⎤⎢ ⎥ℜ = −⎢ ⎥⎢ ⎥⎣ ⎦

(2.4)

Thus, the rotation matrix of the roll-pitch-yaw sequence is

(2.5)

where the components of the rotation matrix are

(2.6)

(2.7)

(2.8)

From the components of the rotation matrix, if cosθ ≠ 0 or |r11

|+|r12

| ≠ 0, three rotation angles (ψ, θ, ϕ) can be determined as the following:

(2.9)

If θ = /2 or −π/2, cosθ = 0, the gimbal lock phenomenon, which will be discussed later, occurs [4–7].

11 12 13

21 22 23

31 32 33

( )X Y Z Z Y X

r r r

r r r

r r r− −

⎡ ⎤⎢ ⎥ℜ = ℜ ⋅ ℜ ⋅ℜ = ⎢ ⎥⎢ ⎥⎣ ⎦

11

12

13

cos cos

sin sin cos cos sin

cos sin cos sin sin

r

r

r

θ ϕ

ψ θ ϕ ψ ϕ

ψ θ ϕ ψ ϕ

=⎧⎪ = +⎨⎪ = − +⎩

21

22

23

cos sin

sin sin sin cos cos

cos sin sin sin cos

r

r

r

θ ϕ

ψ θ ϕ ψ ϕ

ψ θ ϕ ψ ϕ

= −⎧⎪ = − +⎨⎪ = +⎩

31

32

33

sin

sin cos

cos cos

r

r

r

θ

ψ θ

ψ θ

=⎧⎪ = −⎨⎪ =⎩

132 33

131

121 11

tan ( / )

sin ( )

tan ( / )

r r

r

r r

ψ

θ

ϕ

−

−

−

⎧ = −⎪ =⎨⎪ = −⎩


For the x-convention with the z-x-z sequence, the first step is rotating about the z-axis z = [0 0 1]T by an angle ϕ defined by the elemental rotation matrix �

Z. The second step is rotating about the new x-axis: x

1 = [cosϕ −sinϕ

0]T by an angle defined by the elemental rotation matrix �X. The third step is

rotating about the new z-axis: z2 = [cosϕcosθ sinϕcosθ −sinϕ]T by an angle ψ

defined by the elemental rotation matrix �Z again. Then the rotation matrix of

the z-x-z sequence is

(2.10)

where the components of the rotation matrix are

(2.11)

(2.12)

(2.13)

If sinθ ≠ 0 or |r13

| + |r23

| ≠ 0, the three rotation angles can be determined from the components of the matrix as the following:

(2.14)

The rotation matrix ℜ is a 3-by-3 matrix and must satisfy the conditions that the product of the matrix and its transposed matrix is a 3-by-3 unit matrix, I, and the determinant of the rotation matrix is +1:

11 12 13

21 22 23

31 32 33

( )Z X Z Z X Z

r r r

r r r

r r r− −

⎡ ⎤⎢ ⎥ℜ = ℜ ⋅ ℜ ⋅ℜ = ⎢ ⎥⎢ ⎥⎣ ⎦

11

21

31

sin cos sin cos cos

cos cos sin sin cos

sin sin

r

r

r

ϕ θ ψ ϕ ψ

ϕ θ ψ ϕ ψ

θ ψ

= − +⎧⎪ = − −⎨⎪ =⎩

12

22

32

sin cos cos cos sin

cos cos cos sin sin

sin cos

r

r

r

ϕ θ ψ ϕ ψ

ϕ θ ψ ϕ ψ

θ ψ

= +⎧⎪ = −⎨⎪ = −⎩

13

23

33

sin sin

cos sin

cos

r

r

r

ϕ θ

ϕ θ

θ

=⎧⎪ =⎨⎪ =⎩

113 23

133

131 32

tan ( / )

cos ( )

tan ( / )

r r

r

r r

ϕ

θ

ψ

−

−

−

⎧ =⎪ =⎨⎪ = −⎩


This means that the three-column vector of the rotation matrix must be orthonormal.

Although the Euler angle rotation matrices are mathematically simpler to handle and easier to understand, they have a problem called the gimbal lock [4, 7]. The gimbal lock happens when two of the coordinate axes align to each other. It results in a temporary loss of one degree of freedom. This phenomenon happens on Earth when an object is at the North Pole or at the South Pole. For example, in the x-y-z sequence, when the rotation angle about the y-axis (pitch angle) θ = π/2, the y-axis, the x-axis and the z-axis will collapse onto one another, and therefore, one degree of freedom will be lost. Recall (2.5) through (2.8), when the rotation angle about the y-axis (pitch angle) θ = π/2, the rota-tion matrix becomes

(2.15)

If ψ = 0, the rotation matrix is

(2.16)

When ϕ = 0 and ψ = −ψ, the rotation matrix is

(2.17)

Thus, the value of the 3-by-3 rotation matrix π

ϕ− −⎛ ⎞⎜ ⎟⎝ ⎠� 0, ,

2X Y Z equals the

value of the rotation matrix π

ψ− −⎛ ⎞−⎜ ⎟⎝ ⎠� , ,0

2X Y Z , and one degree of freedom is

det 1

T I⎧ ℜ ℜ =⎨ ℜ = +⎩

( , , )2

0 0 1

sin cos cos sin sin sin cos cos 0

cos cos cos cos cos sin sin cos 0

X Y Z

πψ θ ϕ

ψ ϕ ψ ϕ ψ ϕ ψ ϕ

ψ ϕ ψ ϕ ψ ϕ ψ ϕ

− −ℜ = =

−⎡ ⎤⎢ ⎥− +⎢ ⎥⎢ ⎥+ −⎣ ⎦

0 0 1

( 0, , ) sin cos 02

cos sin 0X Y Z

πψ θ ϕ ϕ ϕ

ϕ ϕ− −

−⎡ ⎤⎢ ⎥ℜ = = = −⎢ ⎥⎢ ⎥⎣ ⎦

0 0 1

( , , 0) sin cos 02

cos sin 0X Y Z

πψ ψ θ ϕ ψ ψ

ψ ψ− −

−⎡ ⎤⎢ ⎥ℜ = − = = = −⎢ ⎥⎢ ⎥⎣ ⎦


lost. In this case, there are only two degrees of freedom. Thus, the gimbal lock makes an unexpected and limited movement.

However, an alternative that uses quaternion algebra instead of the Euler angle rotation matrices can eliminate the gimbal lock. Quaternions are com-monly used in computer graphics for representing orientation in 3-D space. Instead of using three rotation angles, the quaternion represents an arbitrary orientation by a rotation about a unit axis through a certain angle.

2.1.2 Quaternion

The problem associated with the Euler rotation matrices comes from the trigo-nometric function operation. When the Euler angle reaches ±π/2, the numeri-cal singularity may occur. Quaternion algebra alleviates the singularity by using rotation around an axis instead of using the yaw, pitch, and roll angular rotation [4, 7]. By using the quaternion, there is only a single axis of rotation and it does not require predefined Euler angle sequences. Thus, no degree of freedom can be lost.

A quaternion uses a four-component vector to represent 3-D orientation. The real part is a scalar rotation angle α, and the imaginary part is a vector with x, y, z coordinates. If a point at (x

1, y

1, z

1) in a 3-D space P

1 = x

1· i + y

1 · j + z

1·

k rotates through an angle α about a unit axis e = q1 · i + q

2 · j + q

3 · k = (q

1,

q2, q

3), the resulting point can be represented by the following transformation

formula [6]:

(2.18)

where q(α, e) = [cos(α/2), e ⋅ sin(α/2)] is the quaternion representing the rota-tion about the unit vector e by an angle α. A translation can be represented by P

2 = q + P

1.

The quaternion can also be represented by four components as

(2.19)

where q0 = cos(α/2), q

1 = xsin(α/2), q

2 = ysin(α/2), and q

3 = zsin(α/2) with the

constraint:

2 1 ( )conj= ⋅ ⋅P q P q

[ ]0 1 2 3 0 1 2 3[ , , , ]

cos( / 2), sin( / 2), sin( / 2), sin( / 2)

q q q q q q q q

x y zα α α α

= + + + =

=

q u i j k

2 2 2 20 1 2 3 1q q q q+ + + =


The conjugate of a quaternion, conj(q), is a quaternion that has the same magnitudes, but the signs of the imaginary parts are changed:

(2.20)

Multiplying a quaternion by its conjugate makes a real number: q · conj(q) = real number. The norm of a quaternion is defined by

(2.21)

and

(2.22)

where = + +2 2 2 1/21 2 3( )S q q q . The normalized quaternion is defined by |q| = 1.

The multiplication of two quaternions, p = [p0, p

1, p

2, p

3] and q = [q

0, q

1,

q2, q

3], is given by [4]

(2.23)

It should be noticed that quaternions are not commutative under the multiplication operation, that is, p · q ≠ q ⋅ p.

Given a rotation angle α and a unit axis e = (q1 ⋅ i, q

2 ⋅ j, q

3 ⋅ k), the unit

vector in (x, y, z) is defined by ⎡ ⎤= = ⋅ ⋅ ⋅ = ⎢ ⎥⎣ ⎦1 2 3[ , , ] , ,

q q qx y z

S S S S

eu i j k i j k and

the corresponding rotation matrix can be derived by the quaternion q(α, e) =

[cos(α/2), e ⋅ sin(α/2)] [6]:

(2.24)

[ ]( 0 1 2 3) , , ,

cos( / 2), sin( / 2), sin( / 2), sin( / 2)

q q q q

x y zα α α α

= − − −⎡ ⎤⎣ ⎦= − − −

conj q

[ ]1/2( )conj= ⋅q q q

2 2 2 2 2 2 21 2 3cos ( / 2) cos ( / 2)q q q Sα α= + + + = +q

0 0 1 1 2 2 3 3

0 1 2 3 0 1 2 3

2 3 3 2 3 1 1 3 1 2 2 1

( )

( ) ( )

( ) ( ) ( )

p q p q p q p q

p q q q q p p p

p q p q p q p q p q p q

⋅ = − + ++ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅+ − ⋅ + − ⋅ + − ⋅

p qi j k i j k

i j k

1 2 3 1 3 2

1 2 3 2 3 1

2

1 3 2 2 3 1 3

2 2

1 1

2 2

2 2

2

3

cos (1 ) (1 cos ) sin (1 cos ) sin

(1 cos ) sin cos (1 ) (1 cos ) sin

(1 cos ) sin (1 cos ) sin cos (1 )

( , )

q q q q q q q q

q q q q q q q q

q q q q q q q q

α α α α α

α α α α α

α α α α α

α

+ − − + − −

− − + − − +

− + − − + −

ℜ =

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

e


or represented by q = [q0, q

1, q

2, q

3] as

(2.25)

In some applications, if the Euler angles are already given, they can be used along with quaternions to take the advantage of the quaternion. Euler angles can be easily converted to quaternions. Any of the Euler angle conven-tions can be performed with quaternions. Using quaternion components, each roll, pitch, and yaw rotation is described by q

roll = [cos(ψ/2), sin(ψ/2), 0, 0], q

pitch

= [cos(θ/2), 0, sin(θ/2), 0] and qroll

= [cos(ϕ/2), 0, 0, sin(ϕ/2)].With the x-y-z sequence, given the roll, pitch, and yaw angles (ψ, θ, ϕ),

the quaternion is [6]:

(2.26)

where

(2.27)

The quaternion can also be converted to Euler angles by first converting the quaternion to a matrix and then converting the matrix to Euler angles.

However, if the quaternion combined with Euler angles is not used prop-erly, the gimbal lock can still occur because of the use of three rotations.

2.1.3 Equations of Motion

A rigid body in motion is described by its kinematics and dynamics [1, 3]. The kinematics of the rigid body motion describes the relation between the position, speed, and acceleration of the body motion without considering what forces cause the motion. The dynamics or kinetics of the rigid body motion de-

0 1 2 3

2 22 3 1 2 0 3 1 3 0 2

2 21 2 0 3 1 3 2 3 0 1

2 21 3 0 2 2 3 0 1 1 2

( , , , )

1 2 2 2 2 2 2

2 2 1 2 2 2 2

2 2 2 2 1 2 2

q q q q

q q q q q q q q q q

q q q q q q q q q q

q q q q q q q q q q

ℜ =

⎡ ⎤− − − +⎢ ⎥+ − − −⎢ ⎥⎢ ⎥− + − −⎣ ⎦

0 1 2 3( , , ) [ , , , ]x y z q q q qψ θ ϕ− − =q

0

1

2

3

cos cos cos sin sin sin ;2 2 2 2 2 2

sin cos cos cos sin sin ;2 2 2 2 2 2

cos sin cos sin cos sin ;2 2 2 2 2 2

cos cos sin sin sin cos .2 2 2 2 2 2

q

q

q

q

ψ θ ϕ ψ θ ϕ

ψ θ ϕ ψ θ ϕ

ψ θ ϕ ψ θ ϕ

ψ θ φ ψ θ ϕ

= +

= −

= +

= −


scribes motion of the body and the forces that affect the motion. The dynamic energy of a body is given by [1–3]

(2.28)

where mk, v

k, and r

k denote the mass, velocity, and the position of a point k in

the body-fixed system where =∑ 0k kk

m r is assumed, the V = ||V || is the norm

of the translation velocity of the rigid body, and Ω = ||Ω|| is the norm of the

angular velocity of the body rotation. Because of the constant = ∑, kk

V M m .

Thus, the dynamic energy of a rigid body is the sum of the dynamic energy of the translational motion of the CM, the E

CM, and the dynamic energy of the

rigid body rotation about the CM, ERot

. In the rigid body motion, the rotation plays a critical role. Given the an-

gular momentum about the CM of a rigid body defined by

(2.29)

the dynamic energy of the rotating body can be expressed as

(2.30)

where Ωj is the component of � along the j-axis of the body-fixed system, r

k is

expressed by [(rk)

1, (r

k)

2, (r

k)

3], and I

i,j is the inertia tensor defined by

(2.31)

Thus, the component angular momentum becomes

2 2

2 2 2 2

1 1( )

2 2

1 1( )

2 2

k k k kk k

k k CM Rotk

E m v m

MV r r E E

= = + Ω ×

⎡ ⎤= + Ω − Ω⋅ = +⎣ ⎦

∑ ∑

∑

V r

( ) ( )2k k k k k k k

k k

m m r⎡ ⎤= × Ω × = Ω − ⋅Ω⎣ ⎦∑ ∑L r r r r

2, ,

, ,

1 1( ) ( ) ( )

2 2Rot i j k k l i j k i k j i j i ji j k l i j

E m r r r Iδ⎡ ⎤

= Ω Ω − = Ω Ω⎢ ⎥⎣ ⎦

∑ ∑ ∑ ∑

2, ,

1( ) ( ) ( )

2i j k k l i j k l k jk l

I m r r rδ⎡ ⎤

= −⎢ ⎥⎣ ⎦

∑ ∑


2, ,( ) ( ) ( ) ]i i j j k k l i j k i k j j

j j k l

L I m r r rδ⎡ ⎤

= Ω = − Ω⎢ ⎥⎣ ⎦

∑ ∑∑ ∑ (2.32)

By an appropriate choice of the orientation of the body-fixed coordinate system, the moment inertia tensor can be reduced to a diagonal form. In this case, the directions of the axes are called the principal axes of the moment in-ertia tensor, and the diagonal components of the tensor are called the principal moments of inertia. Then the kinetic energy of the rotating body becomes

(2.33)

and the angular momentum vector L becomes

(2.34)

where [e1, e

2, e

3] is the unit vector in the directions of the principal axes, [Ω

1,

Ω2, Ω

3] and [L

1, L

2, L

3] are the angular velocity vector and the angular momen-

tum vector along the principal axes, respectively.The Euler equation of motion is derived by taking the time derivative of

the angular momentum. In the rotating body-fixed coordinates, the time de-rivative of the angular momentum is replaced by

(2.35)

In the space-fixed coordinates, the time derivative of the angular momen-tum equals the applied external force F:

(2.36)

Therefore, the Euler equations of motion become the following set of dif-ferential equations [3]:

( )2 2 21 1 2 2 3 3

1

2RotE I I I= Ω + Ω + Ω

1 1 2 2 3 3 1 1 1 2 2 2 3 3 3L L L I I I= + + = Ω + Ω + ΩL e e e e e e

Rot

d

dt⎛ ⎞ + Ω × =⎜ ⎟⎝ ⎠L L F

( )d d

dt dt= ⋅Ω =L I F


( )

( )

( )

11 3 2 2 3 1

22 1 3 3 1 2

33 2 1 1 2 3

dI I I F

dtd

I I I Fdt

dI I I F

dt

Ω+ − Ω Ω =

Ω+ − Ω Ω =

Ω+ − Ω Ω =

(2.37)

These differential equations that govern the motion will be used in calcu-lating rigid body nonlinear motion dynamics.

2.2 Nonrigid Body Motion

The nonrigid body is a deformable body, that is, a force acts on the body that will lead to the body changing its shape, called deformation. For a free-form de-formation, each particle in the body moves from an initial position P to a new position P

t = f (t, P) at time t, which is a function of t and P. Stress and strain

may occur everywhere in the body. To compute the deformation of a complex body, more complicated methods such as the finite difference method (FDM), the finite element method (FEM), and the boundary element method may be used [8].

In this book, any nonrigid body motion is modeled by jointly connected rigid body segments or parts [2, 8–10]. In robotics, the robot arm is considered to be the flexible part of a mechanical system. The joint connection of two rigid segments of the arm is defined by the kinematical constraint on the joint that restricts the relative motions of the two individual rigid segments. When study-ing radar scattering from a nonrigid body motion, the nonrigid body (such as a walking human or a flying bird) is modeled as several jointly connected rigid body segments. The motion of each segment is treated as a rigid body mo-tion. In mechanics, this type of nonrigid body system is defined as a multibody system.

The multibody system is widely used to model, simulate, analyze, and optimize motion kinematics and dynamic behavior of interconnected bodies in robotics and vehicle dynamics [8–10]. An example of the simplest multibody system is called the slider-crank mechanism. The mechanism is used to trans-form the circular rotation of the crank into a linear translation of the piston; alternately, when the piston is forced to move, the linear translation is converted into circular rotation. The slider is not allowed to rotate and three revolute joints are used to connect these interconnected bodies.

The crank, connecting-rod, and piston mechanism is shown in Figure 2.4. The kinematic analysis of the slider-crank mechanism gives the displace-ment (or position), velocity, and acceleration of the piston or the connecting


rod. From the geometry, the piston displacement x is determined by the length of the connecting rod L, the length of the crank R, the crank angle θ, and the connecting-rod angle ϕ

(2.38)

Since the connecting-rod angle ϕ and the crank angle θ are related by

(2.39)

or

(2.40)

and

cos cosx R Lθ φ= +

sin sinR Lθ ϕ= −

sin sinR

Lϕ θ= −

Figure 2.4 The slider-crank mechanism.


1/22

22cos 1 sin

R

Lϕ θ

⎛ ⎞= −⎜ ⎟⎝ ⎠

(2.41)

Thus, (2.38) can be rewritten as

(2.42)

Given the angular velocity of the rotating crank dθ/dt = Ω, by taking the time derivative of both sides of (2.38),

(2.43)

or

(2.44)

Thus, the angular velocity of the connecting rod is

(2.45)

The translational velocity of the slider becomes

(2.46)

The angular acceleration of the connecting rod and the slider translational acceleration are

(2.47)

( )( )

1/22

1/22 2 2

cos cos cos 1 sin

cos sin

x R L R L

R L R

θ ϕ θ ϕ

θ θ

= + = + −

= + −

cos cosd d

R Ldt dt

θ ϕθ ϕ= −

cos

cos

d R

dt L

ϕ θ

ϕ= − ⋅Ω

( )1/22 2 2

cos

sin

d R

dt L R

ϕ θ

θ= − ⋅Ω

−

( )2

1/22 2 2

sin cossin

sin

dx RR

dt L R

θ θθ

θ

⋅ ⋅= − ⋅ ⋅Ω − ⋅Ω

− ⋅

22

2

2

sin sin

cos

dR L

d dtdt L

ϕθ ϕ

ϕ

ϕ

⎛ ⎞⋅ ⋅Ω + ⋅ ⋅ ⎜ ⎟⎝ ⎠=

⋅


and

(2.48)

respectively.Given R = 1.0m, L = 3.0m, and Ω = 2π (rad/sec), Figure 2.5 shows the

variations in displacement, velocity, and acceleration of the piston as a function of time, and the variations in the connecting-rod angle, angular velocity, and angular acceleration as a function of time.

Based on the kinematic analysis, a robot may be modeled by a multibody system to simulate motion kinematics and dynamic behavior of the intercon-nected body segments. Similarly, a human body can also be modeled by a multi-body system, and each interconnected body segment is considered a rigid body.

2.3 Electromagnetic Scattering from a Body with Motion

Before introducing the physics of the micro-Doppler effect in radar and calcu-lating radar EM scattering from targets, the basics of EM scattering should be briefly introduced.

The simplest model of EM scattering mechanism of a target is the point scatterer model. The target is defined in terms of a three-dimensional reflectiv-ity density function characterized by point scatterers. The occlusion effect can also be implemented in the point scatterer model. Compared with other EM scattering models, the point scatterer model can easily incorporate the target’s motion in EM scattering and isolate the EM scattering from each individual motion component.

2.3.1 Radar Cross Section of a Target

EM scattering occurs when a target is illustrated by radar-transmitted EM waves. The incident EM waves induce electric and magnetic currents on the surface and/or within the volume of the target that will generate a scattered EM field. The scattered EM waves are transmitted to all possible directions. If the target is at a distance far enough from the radar, the incident wavefront can be treated as a plane wave. The power of scattered EM waves is measured by a bistatic scattering cross section of the target. If the direction is back to the radar, the bistatic scattering becomes backscattering and the cross section is a backscattering cross section, called the radar cross section (RCS).

According to [11], “the IEEE Dictionary of Electrical and Electronics Terms defines RCS as a measure of reflective strength of a target defined as 4π times

22 22

2 cos cos sind x d d

R L Ldt dt dt

ϕ ϕθ φ φ⎛ ⎞= − ⋅ ⋅Ω − ⋅ ⋅ − ⋅ ⋅⎜ ⎟⎝ ⎠


Figu

re 2

.5

(a, b

) The

var

iatio

ns in

dis

plac

emen

t, ve

loci

ty, a

nd a

ccel

erat

ion

of th

e pi

ston

as

a fu

nctio

n of

tim

e, a

nd th

e va

riatio

n in

the

conn

ectin

g-ro

d an

gle,

ang

ular

vel

ocity

, and

ang

ular

acc

eler

atio

n as

a fu

nctio

n of

tim

e.


the ratio of the power per unit solid angle scattered in a specified direction to

the power per unit area in a plane wave incident on the scatterer from a speci-

fied direction.” The RCS is formulated by 2

22lim 4 ,s

ri

Er

Eσ π

→∞= where E

s is the

intensity of the far-field scattered electric field, Ei is the intensity of the far-field

incident electric field, and r is the distance from the radar to the target.The RCS is defined to characterize the target characteristics. It is normal-

ized to the power density of the incident wave at the target in that it does not depend on the distance of the target from the radar. The RCS is dependent on the size, geometry, and material of the target, the frequency of the transmitter, the polarization of the transmitter and the receiver, and the aspect angles of the target relative to the radar transmit ter and the receiver, respectively [11–13].

The maximum detectable range of a target is proportional to the fourth root of its RCS. The RCS is usually described by the unit of square meters (m2). Typical RCS ranges are from 0.0005 m2 in insects, 0.01 m2 in small birds, 0.5 m2 in humans, and up to 100,000 m2 in a large ship.

Any complicated target can be decomposed into the basic geometric building blocks of simple shapes such as spheres, cylinders, and plates. Figure 2.6 illustrates an example of a human modeled by simple triangle-shaped sur-face geometric building blocks.

When a target is illuminated by EM waves, each building block produces a voltage. The vector sum of the building block voltages determines the total RCS of the target. It is defined by the square root of the magnitude of the vector

Figure 2.6 A human modeled by simple triangle-shaped surface building blocks.


sum. The accuracy of the RCS calculation depends on the accuracy of modeling these building blocks and the interactions between them. If the modeling is not accurate, the calculated RCS may not agree with the measured RCS.

The mechanism of the EM scattering from a target is a complicated pro-cess that includes reflections, diffractions, surface waves, ducting, and interac-tions between them. Reflection is from surfaces and has the highest RCS peak among other scattering mechanisms. Diffraction is from discontinuities (such as edges, corners, or vertices) and is less intense than the reflection. The surface wave is the current traveling along the surface of the target body. A flat surface produces leaky waves and curved surfaces produce creeping waves. Ducting oc-curs when a wave enters into a waveguide-like structure (such as the inlet cavity of an aircraft). Spiky features and lobes in the RCS may also associate with multiple reflections, diffraction, and other scattering mechanisms.

2.3.2 RCS Prediction Methods

The RCS prediction method is an analytical method of calculating the RCS. The incident wave induces a current on the target and, thus, radiates an EM field. If the distribution of current is known, it can be used in the radiation inte-grals to calculate the scattered field and the RCS. The commonly used methods for RCS prediction are the physical optics (PO), ray tracing, method of mo-ments, and finite-difference methods [13]. The PO method is a high-frequency approximation to estimate the surface current induced on a body. It is accurate in the specular direction, but inaccurate when computing at angles far from the specular directions or in the shadow regions. However, surface waves are not in-cluded in the PO method. To improve the accuracy for the current distribution near edges, the physical theory of diffraction (PTD) may be used.

The ray-tracing method is used to analyze large objects with arbitrary shape. Geometric optics (GO) is the classical theory of ray tracing and provides a formula for computing the reflected and refracted fields. It can also be supple-mented by the geometric theory of diffraction (GTD). Computer codes based on GTD, PTD, and their hybridizations have been developed for the predic-tion of high-frequency scattering from complex perfectly conducting objects.

The XPATCH code, based on the shooting-and-bouncing ray technique and PTD, has been widely used to generate a target’s RCS signatures for non-cooperative target recognition [14]. It allows the calculation of backscattering from complex geometries. Other computer codes for RCS prediction include the RAPPORT code developed by the Netherlands: numerical electromagnetic code (NEC), (electric field integral equation (EFIE), and finite-difference time domain (FDTD) [13, 15].

In this book, for some applications of the PO method will be used to es-timate scattered fields. A simple RCS prediction code based on the PO method


called the POFacet is available [16, 17]; this can calculate the monostatic RCS and the bistatic RCS of a static object. The bistatic RCS determines the EM power flux density scattered by an object in an arbitrary direction. In using the PO facet of the RCS prediction method, an object is usually approximated by a large number of subdivision surfaces (triangular meshes) called facets that produce a continuous surface of the object. The total RCS of the object is the superposition of the square root of the magnitude of each individual facet’s RCS. The scattered field of each triangle is computed by assuming that the triangle is isolated and other triangles are not present. Besides multiple reflec-tions, edge diffraction, and surface waves are not considered. Shadowing is only approximately included by considering a facet to be completely illuminated or completely shadowed by the incident wave. A standard spherical coordinate system is used in the PO facet RCS prediction to specify incident and scattering directions as shown in Figure 2.7, and the RCS is calculated at specific angles θ and ϕ.

2.3.3 EM Scattering from a Body with Motion

The characteristics of the EM scattering field from an object have been studied for decades. The characteristic of the scattering field and the RCS of objects are usually calculated under an assumption that the object is stationary. However, in most practical situations, an object or structures on the object are rarely sta-tionary and may have movements such as translation, rotation, or oscillation.

The EM field scattered from a moving object or an oscillating object has been studied in both theory and experiment [18–20]. Theoretical analysis indi-cates that the motion of an object modulates the phase function of the scattered EM waves. If the object oscillates linearly and periodically, the modulation

Figure 2.7 Coordinate system used in the RCS calculation.


generates sideband frequencies about the Doppler frequency induced by its translation.

For a translational object, the far electric field of the object can be derived as [19]

(2.49)

where k = 2π/λ is the wave number, uk is the unit vector of the incidence wave,

ur is the unit vector of the direction of observation, E(r) is the far electric field of

the object before moving, r = (X0, Y

0, Z

0) are the initial coordinates of the object

in the space-fixed coordinates (X, Y, Z), r ′ = (X1, Y

1, Z

1) are the coordinates of

the object in the space-fixed coordinates after translation, and r ′= r + r0, where

r0 is the translation vector, as illustrated in Figure 2.8.

The only difference in the electric field before and after the translation is the phase term, exp{jkr

0 ⋅ (u

k − u

r)}. If the translation is a function of time, r

0

= r0(t) = r

0(t)u

T , where uT is the unit vector of the translation, the phase factor

then becomes

(2.50)

For backscattering, the direction of observation is opposite to the direc-tion of the incidence wave. Thus, u

k = −u

r and

(2.51)

{ }0( ) exp ( ) ( )T k rj k= ⋅ −′E r r u u E r

{ }0exp{ ( )} exp ( ) ( )T k rj t jkr tΦ = ⋅ −u u u

{ }0exp{ ( )} exp 2 ( ) T kj t j kr tΦ = ⋅u u

Figure 2.8 Geometry of a translational object in a far EM fi eld.


If the translation direction is perpendicular to the direction of the inci-dence wave, that is, u

T ⋅ u

k, then exp{Φ(t)} = 1.

For a vibrating object, assuming r0(t) = AcosΩt, the phase factor becomes

a periodic function of the time with an angular frequency Ω

(2.52)

In general, when radar transmits an EM wave at a carrier frequency of f0,

the radar received signal can be expressed as

(2.53)

where the phase factor, exp{jkr0(t)(u

k − u

r)}, defines the modulation of the mi-

cro-Doppler effect caused by the time-varying motion r0(t).

2.4 Basic Mathematics for Calculating the Micro-Doppler Effect

The mathematics of the micro-Doppler can be derived from introducing the micro motion to the conventional Doppler analysis [21]. For simplicity, a radar target is represented by a set of point scatterers that are primary reflecting points on the target. The point scattering model simplifies the analysis while preserv-ing micro-Doppler features. In the simplified model, scatterers are assumed to be perfect reflectors, reflecting all the energy intercepted.

As shown in Figure 2.9, the radar is stationary and located at the origin Q of the radar-fixed coordinate system (X, Y, Z). The target is described in a local coordinate system (x, y, z) that is attached to the target and has translation and rotation with respect to the radar coordinates. To observe the target’s rotations, a reference coordinate system (X ′, Y ′, Z ′) is introduced, which shares the same origin with the target local coordinates and, thus, has the same translation as the target but no rotation with respect to the radar coordinates. The origin O of the reference coordinates is assumed to be at a distance R

0 from the radar.

2.4.1 Micro-Doppler Induced by a Target with Micro Motion

Suppose the target has a translation velocity v with respect to the radar and an angular rotation velocity �, which can be either represented in the target local coordinate system as � = (ω

x, ω

y, ω

z)T, or represented in the reference coordi-

nate system as � = (ωX, ω

Y, ω

Z)T. Thus, a point scatterer P at time t = 0 will

move to P ′ at time t. The movement consists of two steps: (1) translation from P to P ″, as shown in Figure 2.9, with a velocity v, that is, OO

t = vt, and (2)

rotation from P ″ to P ′ with an angular velocity �. If we observe the movement

{ }exp{ ( )} exp 2 cos T kj t j k A tΦ = Ω ⋅u u

{ }0 0( ) exp ( )( ) 2 ( )k rs t jk t j f tπ−= −r u u E r


in the reference coordinate system, the point scatterer P is located at r0 = (X

0,

Y0, Z

0)T, and the rotation from P ″ to P ′ is described by a rotation matrix ℜ

t.

Then, at time t the location of P ′ will be at

(2.54)

The range vector from the radar at Q to the scatterer at P ′ can be derived as

(2.55)

and the scalar range becomes

(2.56)

where ||•|| represents the Euclidean norm. If the radar transmits a sinusoidal waveform with a carrier frequency f, the

baseband of the signal returned from the point scatterer P is a function of r(t):

(2.57)

0t t t tO P O P= = ℜ = ℜ′ ′′r r

0 0t t tQP QO OO O P t= + + = + + ℜ′ ′ R v r

0 0( ) tr t t= + + ℜR v r

{ } [ ]{ }2 ( )( ) ( , , )exp 2 ( , , )exp ( )

r ts t x y z j f x y z j r t

cρ π ρ= = Φ

Figure 2.9 Geometry of the radar and a target with translation and rotation.


where ρ(x, y, z) is the reflectivity function of the point scatterer P described in the target local coordinates (x, y, z), c is the speed of the EM wave propagation, and the phase of the baseband signal is

(2.58)

By taking the time derivative of the phase, the Doppler frequency shift induced by the target’s motion can be derived [21]

(2.59)

where 0 0

0 0

t

t

t

t

+ +=

+ +�

�

R v rn

R v r is the unit vector of QP ′.

Before further deriving the Doppler shift induced by the rotation, a useful relationship × = uu r r is introduced. Given a vector u = (u

x, u

y, u

z)T and a skew

symmetric matrix defined by

(2.60)

the cross product of the vector u and any vector r can be computed through the matrix computation:

(2.61)

This equation is useful in the analysis of the special orthogonal matrix group or SO(3) group, also called the 3-D rotation matrix [2, 21].

We can now return to the rotation matrix in (2.11). In the reference coor-dinate system, the angular rotation velocity vector can be described by � = (ωX

,

2 ( )( ) 2

r tr f

cπΦ =

0 0 0 0

0

21 ( )( )

22 1

( ) ( )2 ( )

2( )

D

Tt t

T

t

fd t df r t

dt c dtf d

t tc r t dt

f d

c dt

π

Φ= =

⎡ ⎤= + + ℜ + + ℜ⎣ ⎦

⎡ ⎤= + ℜ⎢ ⎥⎣ ⎦

R v r R v r

v r n

0

ˆ 0

0

z y

z x

y x

u u

u u u

u u

⎡ ⎤−⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

0

ˆ0

0

y z z y z y x

z x x z z x y

x y y x y x z

u r u r u u r

u r u r u u r u

u r u r u u r

⎡ ⎤ ⎡ ⎤− − ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥× = − = − =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − ⎣ ⎦⎣ ⎦ ⎣ ⎦

u r r


ωY, ω

Z)T, and the target will rotate along the unit vector =′

��

� with a scalar

angular velocity Ω = ||�||. Assuming a high PRF and a relatively low angular velocity, the rotational motion during each time interval can be considered to be infinitesimal, and thus (see Appendix 2A)

(2.62)

where � is the skew symmetric matrix associated with �. Thus, the Doppler frequency shift in (2.59) becomes

(2.63)

If 0 ,tt>> + �R v r n can be approximated as = 0

0

Rn

R, which is the di-

rection of the radar LOS. The Doppler frequency shift is then approximately

(2.64)

where the first term is the Doppler shift due to the translation

(2.65)

and the second term is the micro-Doppler due to the rotation:

(2.66)

For a time-varying rotation, the angular rotation velocity is a function of time and can be expressed by a polynomial function:

(2.67)

ˆexp{ }t tωℜ =

ˆ ˆ0 0

2 2 ˆ( ) ( )

2 2ˆ( ) ( )

Tt t T

D

T T

f fdf e e

c dt cf f

c c

ω ωω

ω

⎡ ⎤= + = +⎢ ⎥⎣ ⎦

= + = + ×�

v r n v r n

v r n r nv

2[ ]D

ff

c= + × ⋅�v r n

2Trans

ff

c= ⋅v n

[ ]2mD

ff

c= × ⋅� r n

20 1 2( ) ...t t t= + + +� � � �


If no more than second-order terms are used, the micro-Doppler shift can be expressed by

(2.68)

where the vector operation (a × b) ⋅ c = a ⋅ (b × c) is applied.

2.4.2 Vibration-Induced Micro-Doppler Shift

As shown in Figure 2.10, the radar is located at the origin of the radar coordi-nate system (X, Y, Z) and a point scatterer P is vibrating about a center point O. This center point is also the origin of the reference coordinate system (X ′, Y ′, Z ′), which is translated from (X, Y, Z) to be situated at a distance R

0 from

the radar. We also assume that the center point O is stationary with respect to the radar. If the azimuth and elevation angle of the point O with respect to the radar are α and β, respectively, the point O is located at

(2.69)

in the radar coordinates (X, Y, Z). Then the unit vector of the radar LOS becomes

(2.70)

[ ] 20 1 2

2 2( ) ( ) ( ) ( )mD

f ff t t t

c c⎡ ⎤= × ⋅ = ⋅ × + ⋅ × + ⋅ ×⎣ ⎦� � � �r n r n r n r n

( )0 0 0cos cos , cos sin , sinR R Rβ α β α β

[ ]cos cos , sin cos , sinT

α β α β β=n

Figure 2.10 Geometry for the radar and a vibrating point target.


If one assumes that the scatterer P is vibrating at a frequency fv with an

amplitude Dv and that the azimuth and elevation angle of the vibration direc-

tion in the reference coordinates (X ′, Y ′, Z ′) are αP and β

P, respectively. Thus,

as shown in Figure 2.10, the vector from the radar to the scatterer P becomes R

t = R

0 + D

t and the range from the radar to the scatterer P can be expressed as

(2.71)

If R0 >> D

t, the range is approximately

(2.72)

If the azimuth angle α of the center point O and the elevation angle βP of

the scatterer P are all zero, and if R0 >> D

t, then we have

(2.73)

Because the angular frequency of the vibration rate is ωv and the ampli-

tude of the vibration is Dv, then D

t = D

vsinω

vt and the range of the scatterer

becomes

(2.74)

The radar received signal then becomes

(2.75)

where ρ is the reflectivity of the point scatterer, f0 is the carrier frequency of

the transmitted signal, λ is the wavelength, and Φ(t) = 4πR(t)/λ is the phase function.

Substituting (2.74) into (2.75) and denoting B = (4π/λ)Dv cosβ cosα

P ,

the received signal can be rewritten as

[ 2

0

20

1/220

( cos cos cos cos )

( cos sin cos sin )

( sin sin )

β α β α

β α β α

β β

= = +

+ +

⎤+ + ⎦

t t t P P

t P P

t P

R R D

R D

R D

R

{ }1/22 20 0

0

2 cos cos cos( ) sin sin

cos cos cos( ) sin sin

t t t p p p

t p p p

R R D R D

R D

β β α α β β

β β α α β β

⎡ ⎤= + + − +⎣ ⎦⎡ ⎤≈ + − +⎣ ⎦

( )1/22 20 0 02 cos cos cos cost t t P t PR R D R D R Dβ α β α= + + ≅ +

0( ) sin cos cost v v PR t R R D tω β α= = +

[ ]{ }0 0

( )( ) exp 2 4 exp 2 ( )R

R ts t j f t j f t tρ π π ρ π

λ⎧ ⎫⎡ ⎤= + = + Φ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭


{ } { }0 0

4( ) exp exp 2 sinR vs t j R j f t B t

πρ π ω

λ= + (2.76)

Equation (2.76) can be further expressed by the Bessel function of the first kind of order k:

(2.77)

and, thus,

(2.78)

Therefore, the micro-Doppler frequency spectrum consists of pairs of spectral lines around the center frequency f

0 and with a spacing ω

v /(2π) be-

tween adjacent lines.Because of the vibration, the point scatterer P, which is initially, at time t

= 0, located at [X0, Y

0, Z

0]T in (X ′, Y ′, Z ′), will, at time t, move to

(2.79)

The velocity of the scatterer P due to the vibration becomes

(2.80)

Based on the analysis in Section 2.4.1, the micro-Doppler shift induced by the vibration is

( ){ }1( ) exp sin

2kJ B j B u ku duπ

ππ −

= −∫

( )

( ){( ) ( )( ) ( )( )

0 0

0 0 0

1 0 1 0

2 0 2 0

3 0 3

4( ) exp ( )exp 2

4exp ( )exp 2

( )exp 2 ( )exp 2

( )exp 2 2 ( )exp 2 2

( )exp 2 3 ( )

R k vk

v v

v v

v

s t j R J B j f k t

j R J B j f t

J B j f t J B j f t

J B j f t J B j f t

J B j f t J B

πρ π ω

λ

πρ π

λ

π ω π ω

π ω π ω

π ω

∞

= −∞

⎛ ⎞ ⎡ ⎤= +⎜ ⎟ ⎣ ⎦⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤+ + − −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤+ + + −⎣ ⎦ ⎣ ⎦⎡ ⎤+ + −⎣ ⎦

∑

( )}

0exp 2 3 vj f tπ ω⎡ ⎤−⎣ ⎦+

1 0

1 0

1 0

cos cos

sin(2 ) sin cos

sin

P P

v v P P

P

X X

Y D f t Y

Z Z

α β

π α β

β

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

( )[ ]cos 2 cos cos , sin cos , sinT

v v v P P P P PD f f tπ α β α β β=v


[ ]

2( )

2cos( )cos cos sin sin cos(2 )

TmD

v vP P P v

ff

cf f D

f tc

α α β β β β π

= ⋅ =

− +

v n (2.81)

If the azimuth angle α and the elevation angle βP are both zero, we have

(2.82)

When the orientation of the vibrating scatterer is along the projection of the radar LOS direction, or α

P = 0, and the elevation angle β is also 0, the Dop-

pler frequency change reaches the maximum value of 2ffvD

v/c.

2.4.3 Rotation-Induced Micro-Doppler Shift

The geometry of the radar and a target with three-dimensional rotations is illus-trated in Figure 2.11. The radar coordinate system is (X, Y, Z), the target local coordinate system is (x, y, z), and the reference coordinate system (X ′, Y ′, Z ′) is parallel to the radar coordinates (X, Y, Z) and located at the origin of the target local coordinates. Assume that the azimuth and elevation angle of the target in the radar coordinates (X, Y, Z) are α and β, respectively, and the unit vector of the radar LOS is the same as (2.70).

Due to the target’s rotation, any point on the target described in the local coordinate system (x, y, z) will move to a new position in the reference coor-dinate system (X ′, Y ′, Z ′). The new position can be calculated from its initial position vector multiplied by an initial rotation matrix of the x-convention (z-x-z sequence) determined by the angles (ϕ

0, θ

0, ψ

0), where the angle ϕ

0 rotates

about the z-axis, the angle θ0 rotates about the x-axis, and the angle ψ

0 rotates

about the z-axis again. The corresponding initial rotation matrix is defined by

(2.83)

where

2cos cos cos(2 )v v

mD P v

f f Df f t

cβ α π=

11 12 13

0 0 0 21 22 23

31 32 33

( ) ( ) ( )Init Z X Z

r r r

r r r

r r r

φ θ ψ

⎡ ⎤⎢ ⎥ℜ = ℜ ⋅ℜ ⋅ℜ = ⎢ ⎥⎢ ⎥⎣ ⎦


(2.84)

(2.85)

(2.86)

11 0 0 0 0 0

21 0 0 0 0 0

31 0 0

sin cos sin cos cos

cos cos sin sin cos

sin sin

r

r

r

ϕ θ ψ ϕ ψ

ϕ θ ψ ϕ ψ

θ ψ

= − +⎧⎪ = − −⎨⎪ =⎩

12 0 0 0 0 0

22 0 0 0 0 0

32 0 0

sin cos cos cos sin

cos cos cos sin sin

sin cos

r

r

r

ϕ θ ψ ϕ ψ

ϕ θ ψ ϕ ψ

θ ψ

= +⎧⎪ = −⎨⎪ = −⎩

13 0 0

23 0 0

33 0

sin sin

cos sin

cos

r

r

r

ϕ θ

ϕ θ

θ

=⎧⎪ =⎨⎪ =⎩

Figure 2.11 (a) Geometry of the radar and a rotating target, and (b) the micro-Doppler signa-ture of the rotating target.


Viewed in the target local coordinate system, when a target rotates about its axes x, y, and z with an angular velocity � = (ω

x, ω

y, ω

z)T, a point scatterer P

at rP = (x

P, y

P, z

P)T represented in the target local coordinates (x, y, z) will move

to a new location in the reference coordinate system described by ℜInit

⋅ rP and

the unit vector of the rotation becomes

(2.87)

with the scalar angular velocity Ω = ||�||. Thus, according to the Rodrigues formula [2], at time t the rotation matrix becomes

(2.88)

where ω′ˆ is the skew symmetric matrix

(2.89)

Therefore, viewed in the reference coordinate system (X ′, Y ′, Z ′), at time t, the scatterer P will move from its initial location to a new location r = ℜ

t ⋅

ℜInit

⋅ rP. According to the discussion in Section 2.4.1, the micro-Doppler fre-

quency shift induced by the rotation is approximately

(2.90)

If the skew symmetric matrix ω′ˆ is defined by a unit vector, then ω ω= −′ ′3ˆ ˆ and the rotation induced micro-Doppler frequency becomes

(2.91)

Assume that the radar operates at 10 GHz and a target, located at (U = 1,000m, V = 5,000m, W = 5,000m), is rotating along the x, y, and z axes with

( , , )T Initx y zω ω ω

ℜ ⋅= =′ ′ ′

��

�

2ˆ ˆsin (1 cos )t I t tω ωℜ = + Ω + − Ω′ ′

0ˆ 0

0

z y

z x

y x

ω ω

ω ω ω

ω ω

⎡ ⎤− ′ ′⎢ ⎥= −′ ′ ′⎢ ⎥⎢ ⎥− ′ ′⎣ ⎦

[ ] ( ) [ ]

{ }2 3 2

2 2 2ˆ ˆ

2 ˆ ˆ ˆ ˆsin cos ( )

T T

mD t Initradial

T

Init

f f ff

c c cf

t tc

ω ω ω

ω ω ω ω

= Ω × = Ω ⋅ = Ω ℜ ⋅ℜ ⋅ ⋅′ ′ ′

Ω⎡ ⎤= Ω − Ω + + ℜ ⋅ ⋅′ ′ ′ ′⎣ ⎦

P

P

r r n r n

I r n

[ ]2 ˆ ˆ( sin cos )mD Init radial

ff t I t

cω ω

Ω= Ω + Ω ℜ ⋅′ ′ Pr


the initial Euler angles (ϕ = 30°, θ = 20°, ψ = 20°) and angular velocity � = [π, 2π, π]T rad/sec. Suppose that the target has three strong scatterer centers: scat-terer P

0 (the center of the rotation) is located at (x = 0m, y = 0m, z = 0m); scat-

terer P1 is located at (x = 1.0m, y = 0.6m, z = 0.8m); and scatterer P

2 is located

at (x = −1.0m, y = −0.6m, z = −0.8m). The theoretical micro-Doppler modu-lation calculated by (2.91) is shown in Figure 2.11(b). The micro-Doppler of the center point P

0 is the line at the zero frequency, and the micro-Doppler

modulations from the points P1 and P

2 are the two sinusoidal curves about the

zero frequency. The rotation period can be obtained from the rotation angular velocity as T

0 = 2π/||�|| = 0.8165 seconds.

2.4.4 Coning Motion-Induced Micro-Doppler Shift

The coning motion is a rotation about an axis that intersects with an axis of the local coordinates. A whipping top usually undergoes a coning motion while its body is spinning around its axis of symmetry with a fixed tip point and the axis of symmetry is rotating about another axis that intersects with the axis. If the axis of symmetry does not remain at a constant angle with the axis of coning, it will oscillate up and down between two limits, called nutation. In this case, the Euler angle ϕ is known as the spin angle, ψ is known as the precession angle, and θ is known as the nutation angle.

Without considering spinning and nutation, assume a target with pure coning motion along the axis SN, which intersects with the z-axis at the point S (x = 0, y = 0, z = z

0) of the local coordinates, as shown in Figure 2.12. The refer-

ence coordinate system (X ′, Y ′, Z ′) is parallel to the radar coordinates (X, Y, Z) and its origin is located at the point S. Assume that the azimuth and elevation angle of the target center O with respect to the radar are α and β, respectively, and the azimuth and elevation angle of the coning axis SN with respect to the reference coordinates (X ′, Y ′, Z ′) are α

N and β

N, respectively. Then the unit

vector of the radar LOS is

and the unit vector of the rotation axis in the reference coordinates (X ′, Y ′, Z ′) is

Assume that the initial position of a scatterer P is r = [x, y, z]T represented in the target local coordinates. Then the location of the point scatterer P in the reference coordinates (X ′, Y ′, Z ′) can be calculated through its local coordi-nates by subtracting the coordinates of the point S and multiplying the rotation


α β α β β=n


N N N N Nα β α β β=e


matrix ℜInit

defined by the initial Euler angles (ϕ, θ, ψ). Viewed from the refer-ence coordinates (X ′, Y ′, Z ′), the location of the scatterer P is at ℜ

Init⋅ [x, y, z

− z0]T. Suppose that the target has a coning motion with an angular velocity of

ω rad/sec. According to the Rodrigues formula, at time t the rotation matrix in the reference coordinates (X ′, Y ′, Z ′) becomes

Figure 2.12 (a) A target with pure coning motion along an axis that intersects with the z-axis at the origin of the local coordinates. (b) The micro-Doppler signature of the coning target.


(2.92)

where the skew symmetric matrix is defined by

(2.93)

Therefore, at time t the scatterer P will move to

(2.94)

If the point S is not too far from the target center of mass, the radar LOS can be approximated as the radial direction of the point S with respect to the radar. According to the mathematical formula described in Section 2.4.1, the micro-Doppler modulation induced by the coning motion is approximately

(2.95)

Assume that the radar operates at 10 GHz and a target is initially located at (X = 1,000m, Y = 5,000m, Z = 5,000m). Suppose that the initial Euler angles are ϕ = 10°, θ = 10°, and ψ = −20°, the target is coning with 2 Hz during T = 2.048 seconds, or with an angular velocity ω = (2 × 2π/T ) rad/sec, and the azimuth and elevation angle of the rotation axis are αN

= 160° and βP = 50°,

respectively. Thus, given the initial location of the point scatterer P0 at (x = 0m,

y = 0m, z = 0m), the scatterer P1 at (x = 0.3m, y = 0m, z = 0.6m), and the scat-

terer P2 at (x = −0.3m, y = 0m, z = 0.6m), the theoretical micro-Doppler modu-

lation calculated by (2.95) is shown in Figure 2.12(b). The micro-Doppler of the point P

0 is zero frequency, and the micro-Doppler modulations from the

points P1 and P

2 are the two sinusoidal curves about the zero frequency with

their amplitudes determined by their radial directions.When a coning body is also spinning around its axis of symmetry, the

motion becomes precession. The precession matrix ℜprec

is the product of the coning matrix ℜ

coning and the spinning matrix ℜ

spinning:

2ˆ ˆsin (1 cos )t I e t e tω ωℜ = + + −

0 sin sin cos

ˆ sin 0 cos cos

sin cos cos cos 0

N N N

N N N

N N N N

e

β α β

β α β

α β α β

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

[ ]0( ) , ,T

t Initt x y z z= ℜ ⋅ℜ ⋅ −r

{ }

0

20

2 2( ) [ , , ]

2ˆ ˆ( cos sin ) [ , , ]

T

TmD t Init

radial

TTInit

f fd df t x y z z

c dt c dt

fe t e t x y z z

c

ωω ω

⎧ ⎫⎡ ⎤ ⎡ ⎤= = ℜ ℜ ⋅ − ⋅⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

= + ℜ ⋅ − ⋅

r n

n


(2.96)

where the coning matrix is

(2.97)

and the spinning matrix is

(2.98)

The Ωconing

and Ωspinning

are the coning and spinning angular velocities, respectively. The skew symmetric matrix of the coning motion is defined by

(2.99)

and the skew symmetric matrix of the spinning motion is defined by

(2.100)

For a given point scatterer P at rP = [x

p, y

p, z

p]T represented in the targets,

the local coordinate system (x, y, z) will move to a new location in the reference coordinate system described by ℜ

Init ⋅ rP. Viewed in the reference coordinate

system (X ′, Y ′, Z ′), at time t, the scatterer P will move from its initial loca-tion to a new location r = ℜ

t ⋅ ℜ

Init ⋅ rP. The micro-Doppler modulation of the

point scatterer P in the target body due to the precession is formulated as the following:

(2.101)

prec coning spinningℜ = ℜ ⋅ℜ

2ˆ ˆsin( ) 1 cos( )coning coning coning coning coningI t tω ω ⎡ ⎤ℜ = + Ω + − Ω⎣ ⎦

( ) ( )2ˆ ˆsin 1 cosspinning spinning spinning spinning spinningI t tω ω ⎡ ⎤ℜ = + Ω + − Ω⎣ ⎦

0

ˆ 0

0

z y

z x

y x

coning coning

coning coning coning

coning coning

ω ω

ω ω ω

ω ω

−⎡ ⎤⎢ ⎥

= −⎢ ⎥⎢ ⎥−⎣ ⎦

0

ˆ 0

0

z y

z x

y x

spinning spinning

spinning spinning spinning

spinning spinning

ω ω

ω ω ω

ω ω

−⎡ ⎤⎢ ⎥

= −⎢ ⎥⎢ ⎥−⎣ ⎦

2 2| ( )

2( )

2

mD prec Initradial radial

coning spinning Initradial

coning spinning coning spinning Initradial

f fd df

c dt c dt

f d

c dt

f d d

c dt dt

⎡ ⎤ ⎡ ⎤= = ℜ ⋅ℜ ⋅⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤= ℜ ⋅ℜ ⋅ℜ ⋅⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞= ℜ ⋅ℜ + ℜ ⋅ ℜ ⋅ℜ ⋅⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

P P

P

P

r r

r

r


Assume that the radar operates at 10 GHz and a target is initially located at (X = 1,000m, Y = 5,000m, Z = 5,000m). Suppose that the initial Euler angles are ϕ = 30°, θ = 30°, and ψ = 20°, and the target is spinning with 8 Hz during T = 2.048 seconds, or with an angular velocity Ω

spinning = (8 × 2π/T) rad/sec.

Then, given the initial location of the point scatterer P0 at (x = 0m, y = 0m, z

= 0 m), the scatterer P1 at (x = 0.3m, y = 0m, z = 0.6m), and the scatterer P

2 at

(x = −0.3m, y = 0m, z = 0.6m), the theoretical micro-Doppler signature of the spinning target is shown in Figure 2.13.

Figure 2.13 (a) A target with spinning motion, and (b) the micro-Doppler signature of the spinning target.


Based on (2.101), the theoretical micro-Doppler signature of a precession target can be seen in Figure 2.14, where the target has the same initial condition as the coning and the spinning target as discussed above. However, the target is spinning with an angular velocity Ω

spinning = 3.906 rad/sec or spinning 8 cycles

during T = 2.048 seconds. The target is also coning with an angular velocity Ω

coning = 0.977 rad/sec or coning 2 cycles during T = 2.048 seconds. The preces-

sion micro-Doppler signature shows 8 cycles spinning modulated by 2 cycles coning.

2.5 Bistatic Micro-Doppler Effect

The micro-Doppler effect is dependent on the target aspect angle. For a target moving at angles 0° (head-on) and 180° (tail-on) to the radar LOS, micro-Dop-pler frequencies reach the maximum shift; and at angles ±90° to the radar LOS, micro-Doppler frequencies become zero. Besides, due to occlusions at certain aspect angles, some corresponding parts of the target may not be seen by radar.

A separated transmitter and receiver bistatic configuration offers more freedom of acquiring a complementary target’s information, avoiding blind

Figure 2.14 The micro-Doppler signature of a precession target.


velocities, and preventing monostatic RCS from the null or low monostatic positions.

In bistatic radar systems, to determine the target location, the azimuth and elevation of the target viewed at the transmitter and the timing of the trans-mitted signal must be known exactly. Synchronization between the receiver and transmitter must be maintained. The range resolution and Doppler resolution are both dependent on the bistatic angle. Rotational motions of targets can also induce a bistatic micro-Doppler effect depending on the triangulation geom-etry of the transmitter, the target, and the receiver.

In a bistatic radar, the transmitter and receiver are separated by a baseline distance that is comparable with the maximum range of targets with respect to the transmitter and the receiver. The separation brings an issue of synchroniza-tion between the two sites and requires phase synchronization between the local oscillator in the transmitter and the one in the receiver in order to accurately measure the target location and to perform range and Doppler processing.

The coordinate systems include global space-fixed coordinates, reference coordinates, and target local body-fixed coordinates, as shown in Figure 2.15 in a three-dimensional case. The bistatic plane is the one on which the transmit-ter, the receiver, and the target lie. The baseline (L) is the distance between the transmitter and receiver. The range from the transmitter to the target is a vector rT

and the range from the receiver to the target is a vector rR. The bistatic angle

ϕ is the angle between the transmitter-to-target line and the receiver-to-target line. Assume that the transmitter look angles (azimuth and elevation) are (α

T,

βT), the receiver look angles (α

R, β

R) can be obtained from the baseline distance,

the target range, and look angles relative to the transmitter. The positive angle

Figure 2.15 Three-dimensional bistatic radar system confi guration.


is defined in a counterclockwise direction. Thus, the distance from the receiver to the target becomes

(2.102)

and the receiver look angles are

(2.103)

(2.104)

The received signal from a point target P can be modeled as

(2.105)

where c is the propagation velocity. Then the received signal from a volume target is a volume integration of the point return over the whole target:

(2.106)

The phase term in the received signal is

(2.107)

where

(2.108)

( )1/22 2 2cos 2 cos sinR R T T T T Tr L r r Lβ β α= = + −r

1 cos sintan

cos cosT T T

RT T T

L r

r

β αα

β α− ⎛ ⎞−

= ⎜ ⎟⎝ ⎠

12 2 2 1/2

sintan

( cos 2 cos sin )T T

RT T T T T

r

L r r L

ββ

β β α− ⎛ ⎞

= ⎜ ⎟+ −⎝ ⎠

( ) ( ) ( ) ( )( ) ( )exp j2 T P R PP

r P

t t t ts t f

cρ π

⎧ ⎫+ + += ⎨ ⎬

⎩ ⎭

r r r rr

( ) ( )| | ( ) ( )( ) ( )exp j2 T P R P

r P P

t t t ts t f d

cρ π

⎧ ⎫+ + += ⎨ ⎬

⎩ ⎭∫∫∫

r r r rr

Target

r

[ ]

[ ]

,

( ) ( ) ( ) ( )( ) 2

( ) ( ) ( ) ( ) ( )2

( ) ( ) ( )( ) ( )2 2

( ) ( )

π

π

π π

Ω

+ + +Φ =

+ + + ⋅=

+ ⋅+= ⋅

= Φ ⋅Φ

T P R PP

T R T R P

T R PT R

V P

t t t tt f

cr t r t t t t

fc

t t tr t r tf f

c ct t

r r r r

r r r

r r r

( ) ( )( ) 2 T R

V

r t r tt f

cπ

+Φ =


is the phase term induced by translational motions and

(2.109)

is the phase term induced by rotational motions of the scatterer P.While the target has translational motion with a velocity V and rotation

with an initial Euler angle (ϕ0, θ

0, ψ

0) and an angular velocity vector � = (ω

x,

ωy, ω

z)T rotating about the target local-fixed axes x, y, and z, then the Doppler

frequency shift induced by the translation and rotation can be obtained by the time derivative of the phase function.

The Doppler shift consists of two parts: one is induced by the translation and the other is induced by the rotation,

(2.110)

where

(2.111)

is the translational Doppler shift and

(2.112)

is the rotation induced micro-Doppler, which is usually a periodic frequency function of time and distributed around the translational Doppler frequency.

If the target moves with a velocity V and an acceleration a, their compo-nents along the direction from the transmitter to the target are

(2.113)

and the components along the direction from the receiver to the target are

(2.114)

[ ],

( ) ( ) ( )( ) 2 T R P

P

t t tt f

cπΩ

+ ⋅Φ =

r r r

Bi Trans RotD D Df f f= +

[ ]( ) ( )TransD T R

f df r t r t

c dt= +

[ ]2( )

RotD P

ff t

c= ×� r

;T TT T

T T

V a= ⋅ = ⋅r r

V ar r

;R RR R

R R

V a= ⋅ = ⋅r r

V ar r


Then the range from the transmitter to the moving target becomes

(2.115)

and the range from the receiver to the target is

(2.116)

While the target has rotational motion, its rotation angles are determined by its initial angles �

0 and rotation rate �:

(2.117)

where

(2.118)

Therefore, the Doppler shift induced by the translational motion is

(2.119)

and that induced by the rotation becomes

(2.120)

Thus, the translational Doppler shift of the bistatic radar depends on three factors [22, 23]. The first factor is the maximum Doppler shift when a target moving with a velocity V:

(2.121)

The second factor is related to the bistatic triangulation factor

(2.122)

210 2 ...T T T Tr r V t a t= + + +

210 2 ...R R R Rr r V t a t= + + +

0 ...t= + +� � �

( , , )x y zω ω ω=�

[ ] [ ]( ) ( ) ( ) ,TransD T R T R T R

f fdf r t r t V V a a t

c dt c= + = + + +

[ ]{ }2( ) ( ) ( )D T R PRot

f df t t t

c dt

π= + ⋅r r r

max

2D

ff

c= V

cos cos2 2

R TDα α ϕ−⎛ ⎞ ⎛ ⎞= = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠


where ϕ = αR − α

Tis the bistatic angle.

The third factor is related to the angle δ between the moving direction of the target and the direction of the bisector:

(2.123)

Thus, the bistatic Doppler shift becomes

(2.124)

Similar to the Doppler shift induced by the radial velocity in the mo-nostatic radar case, for the bistatic radar case the Doppler shift is induced by the target’s bisector velocity. The bistatic Doppler shift is always smaller than the maximum monostatic Doppler because the term, cos(ϕ/2), is always less than 1.

If the radial velocity of the target with respect to the transmitter is very small or zero, the monostatic radar cannot produce an image of the target. However, according to (2.124), the target velocity with respect to the receiver must not be zero and the bistatic radar is able to produce a good image of the moving target.

As defined in the monostatic radar, in the bistatic case two targets may be separated in range, Doppler shift, and angle. The range resolution in the mo-nostatic radar is ΔrMono

and the Doppler resolution is ΔMonoDf . For bistatic radar,

the range resolution and Doppler resolution are both dependent on the bistatic angle ϕ. The range resolution becomes

(2.125)

and Doppler resolution is

(2.126)

If the bistatic angle is near 180°, this is the case of forward scattering radar. Cherniakov discussed the advantages and disadvantages of the forward-scattering radar [24]. For bistatic angles over 150°–160°, the bistatic RCS in-creases sharply. However, for bistatic angles near 180°, cos(φ/2) → 0. Thus, the bistatic range resolution is losing and there is no Doppler frequency resolution.

cosC δ=

max

2cos cos

2BiD D

ff f D C

c

ϕδ⎛ ⎞= ⋅ = ⎜ ⎟⎝ ⎠V

1

cos( /2)Bi Monor rϕ

Δ = Δ

cos( /2)D Bi DMonof fϕΔ = Δ


Similar to the bistatic Doppler shift generated by the bisector velocity, for a point scatterer P in the target, its bistatic micro-Doppler induced by the rota-tional motion of the target is also determined by its bisector component [22]:

(2.127)

The initial Euler angle (ϕ0, θ

0, ψ

0), the angular velocity vector Ω = (ω

x,

ωy, ω

z)T, and the point scatterer’s location determine the total micro-Doppler

shifts. According to (2.127), the bistatic micro-Doppler shift of a rotating target

can be expressed by

(2.128)

where

(2.129)

is the maximum micro-Doppler captured by monostatic radar.

2.6 Multistatic Micro-Doppler Effect

The multistatic radar configuration has multiple transmitters and receivers dis-tributed over several locations called nodes. It combines multiple measurements of a target viewed at different aspects to extract information about the target. Any individual node of the multistatic system may have both a transmitter and a receiver [24].

In a multistatic radar system, the range of a target is determined by a pair of a transmitter and a receiver and is measured from the time delay Δt between the transmitted and the received signals r

bistatic = r

T + r

R = c ⋅ Δt. However, the lo-

cation of the target measured by the bistatic radar is ambiguous and constrained to an ellipse with foci at the transmitter and the receiver.

The multistatic system can be treated as combinations of several bistatic systems. Figure 2.16 illustrates a multistatic system with four transmitters and four receivers. There is a total of N

Channel = N

Trans × N

Receiv = 16 possible chan-

nels to form the multistatic system: four monostatic systems and twelve bistatic

[ ]2( , ) ( , ) ( )mD D P BisectorBi Rot

ff t P f t P t

c= = ×� r

m cos2mD DBi Max

f fϕ⎛ ⎞= ⎜ ⎟⎝ ⎠

P

2( )mDMax

ff t

c= ×� r


systems. Because each bistatic system provides a different aspect view of the target, the multistatic system will provide a simultaneous multiple view of the micro-Doppler effect [25].

The processing in the multistatic system is to coherently combine the sig-nals received from each of these bistatic systems, model the change of the target position with time, and calculate the phase change Φ

n(t) in each bistatic system.

Then the received baseband signal can be expressed by

(2.130)

where An is the amplitude and Φ

n(t) is the phase function of the signal in the nth

channel, rT,n

(t) = ||rT,n

(t)|| is the distance from the nth transmitter to the target, and r

R,n(t) = ||r

R,n(t)|| is the distance from the target to the nth receiver.

If a point P on the target has rotation, the micro-Doppler induced by the rotation is given by

(2.131)

{ } , ,

1 1

2 [ ( ) ( )]( ) exp ( ) exp

N NT n R n

r n n nn n

f r t r ts t A j t A j

c

π

= =

+⎧ ⎫= − Φ = −⎨ ⎬⎩ ⎭

∑ ∑

[ ]2( , ) ( , ) ( )mD D P BisectorBi Rot

ff t P f t P t

c= = ×� r

Figure 2.16 A multistatic radar system with four transmitters and four receivers.


as derived in (2.128). Thus, the multistatic micro-Doppler is derived by taking the time deriva-

tive of the combined baseband signals

(2.132)

In the multistatic configuration, each node has its own aspect view of the target. The multistatic topology determines the multistatic micro-Doppler features. The combination of the number of nodes and the angular separation between these nodes determine the micro-Doppler signature of the target.

2.7 Cramer-Rao Bound of the Micro-Doppler Estimation

In a coherent radar system, the micro-Doppler modulation is embedded in the radar returns. The range profile can be expressed by

(2.133)

where R0 is the distance of the target, d is the displacement or the amplitude

of the micro-Doppler modulation, and ωmD

is the micro-Doppler modulation frequency. To estimate micro-Doppler frequency shifts, the signal-to-noise ratio (SNR), the total number of time samples (N ), the frequency of micro-Doppler modulation (ω

mD), the displacement of the micro-Doppler modulation (d ),

and the wavelength of the carrier frequency (λ) determine the lower bound on the micro-Doppler estimation precession. The Cramer-Rao lower bound of the micro-Doppler estimation can be found in [26, 27].

References

[1] Goldstein, H., Classical Mechanics, 2nd ed., Reading, MA: Addison-Wesley, 1980.

[2] Murray, R. M., Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipula-tion, Boca Raton, FL: CRC Press, 1994.

[3] Wittenburg, J., Dynamics of Systems of Rigid Bodies, Stuttgart: Teubner, 1977.

[4] Kuipers, J. B., Quaternions and Rotation Sequences, Princeton, NJ: Princeton University Press, 1999.

[5] Mukundand, R., “Quaternions: From Classical Mechanics to Computer Graphics, and Beyond,” Proc. of 7th Asian Technology Conference in Mathematics (ATCM), 2002, pp. 97–106.

1

( , ) ( , )Multi Bi

N

mD n mDn

f t P A f t P=

= ∑

0( ) cos mDr t R d tω= + ⋅


[6] Shoemake, K., “Animating Rotation with Quaternion Curves,” ACM Computer Graphics, Vol. 19, No. 3, 1985, pp. 245–254.

[7] Klumpp, A. R., “Singularity-Free Extraction of a Quaternion from a Direction-Cosine Matrix,” Journal of Spacecraft and Rockets, Vol. 13, 1976, pp. 754–755.

[8] Géradin, M., and A. Cardona, Flexible Multibody Dynamics: A Finite Element Approach, New York: John Wiley & Sons, 2001.

[9] Shabana, A. A., Dynamics of Multibody Systems, 3rd ed., Cambridge, U.K.: Cambridge University Press, 2005.

[10] Magnus, K., Dynamics of Multibody Systems, New York: Springer-Verlag, 1978.

[11] Knott, E. F., J. F. Schaffer, and M. T. Tuley, Radar Cross Section, 2nd ed., Norwood, MA: Artech House, 1993.

[12] Ruck, G. T., et al., Radar Cross Section Handbook, New York: Plenum Press, 1970.

[13] Jenn, D., “Radar Cross-Section,” in Encyclopedia of RF and Microwave Engineering, K. Chang, (ed.), New York: John Wiley & Sons, 2005.

[14] Ling, H., K. Chou, and S. Lee, “Shooting and Bouncing Rays: Calculating the RCS of an Arbitrarily Shaped Cavity,” IEEE Transactions on Antennas and Propagation, Vol. 37, No. 2, 1989, pp. 194–205.

[15] Kunz, K., and R. Luebbers, The Finite-Difference Time Domain Method for Electromagnetics, Boca Raton, FL: CRC Press, 1993.

[16] Chatzigeorgiadis, F., and D. Jenn, “A MATLAB Physical-Optics RCS Prediction Code,” IEEE Antennas and Propagation Magazine, Vol. 46, No. 4, 2004, pp. 137–139.

[17] Chatzigeorgiadis, F., “Development of Code for Physical Optics Radar Cross Section Prediction and Analysis Application,” Master’s Thesis, Naval Postgraduate School, Monterey, CA, September 2004.

[18] Cooper, J., “Scattering by Moving Bodies: The Quasi-Stationary Approximation,” Mathematical Methods in the Applied Sciences, Vol. 2, No. 2, 1980, pp. 131–148.

[19] Kleinman, R. E., and R. B. Mack, “Scattering by Linearly Vibrating Objects,” IEEE Transactions on Antennas and Propagation, Vol. 27, No. 3, 1979, pp. 344–352.

[20] Van Bladel, J., “Electromagnetic Fields in the Presence of Rotating Bodies,” Proc. of the IEEE, Vol. 64, No. 3, 1976, pp. 301–318.

[21] Chen, V. C., et al., “Micro-Doppler Effect in Radar: Phenomenon, Model, and Simulation Study,” IEEE Transactions on Aerospace and Electronics Systems, Vol. 42, No. 1, 2006, pp. 2–21.

[22] Chen, V. C., A. des Rosiers, and R. Lipps, “Bi-Static ISAR Range-Doppler Imaging and Resolution Analysis,” IEEE Radar Conference, Pasadena, CA, May 2009.

[23] Willis, N. J., Bistatic Radar, 2nd ed., Raleigh, NC: SciTech Publishing, 2005.

[24] Chernyak, V. S., Fundamentals of Multisite Radar Systems: Multistatic Radars and Multiradar Systems, Amsterdam, the Netherlands: Gordon and Breach Scientifi c Publishers, 1998.


[25] Chen, V. C., et al., “Radar Micro-Doppler Signatures for Characterization of Human Motion,” Chapter 15, in Through-the-Wall Radar Imaging, M. Amin, (ed.), Boca Raton, FL: CRC Press, 2010.

[26] Rao, C. R., “Information and Accuracy Attainable in the Estimation of Statistical Parameters,” Bull. Calcutta Math. Soc., No. 37, 1945, pp. 81–91.

[27] Setlur, P., M. Amin, and F. Ahmad, “Optimal and Suboptimal Micro-Doppler Estimation Schemes Using Carrier Diverse Doppler Radars,” Proc. of the IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, 2009.

Appendix 2A

For any vector u = [ux, u

y, u

z]T, the skew symmetric matrix is defined by

(2A.1)

The cross product of two vectors u and r can be derived through matrix computation as

(2A.2)

The cross product’s definition is useful in analyzing special orthogonal matrix groups called the SO(3) groups or the 3-D rotation matrix, defined by:

(2A.3)

Computing the derivative of the constraint R(t)RT(t) = I with respect to time t, a differential equation can be obtained as

(2A.4)

or

(2A.5)

0

ˆ 0

0

z y

z x

y x

u u

u u u

u u

⎡ ⎤−⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

0

ˆ0

0

y z z y z y x

z x x z z x y

x y y x y x z

u r u r u u r

u r u r u u r u

u r u r u u r

⎡ ⎤ ⎡ ⎤− − ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= × = − = − ⋅ = ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − ⎣ ⎦⎣ ⎦ ⎣ ⎦

p u r r

{ }3 3(3) | ,det( ) 1TSO R R R I R×= ∈ℜ = = +

( ) ( ) ( ) ( ) 0T TR t R t R t R t+ =

( ) ( ) ( ) ( )TT TR t R t R t R t⎡ ⎤= − ⎣ ⎦


The result reflects the fact that the matrix ×∈� 3 3( ) ( )TR t R t is a skew sym-metric matrix. Thus, there must be a vector ω ∈ ℜ3 such that:

(2A.6)

Multiplying both sides by R(t) to the right yields

(2A.7)

Assume that the vector ω ∈ ℜ3 is constant. According to the linear ordi-nary differential equation (ODE), the solution becomes

(2A.8)

where ωexp{ }t is the matrix exponential:

Assuming for the initial condition: R(0) = 1, (2A.8) becomes

(2A.9)

Thus, it can be conformed that the matrix ωexp{ }t is indeed a rotation matrix. Since ωexp{ }t

(2A.10)

and, thus, ω ω⋅ =ˆ ˆ[exp( )] exp( ) ,Tt t I from which one can obtain ω = ±ˆdet{exp( )} 1.tFurthermore,

(2A.11)

which shows that ω = ±ˆdet{exp( )} 1.t Therefore, matrix ω= ˆ( ) exp{ }R t t is the 3-D rotation matrix. Let Ω = ||�||. A physical interpretation of the equation

ω= ˆ( ) exp{ }R t t is simply a rotation around the axis � ∈ ℜ3 by Ω · t radians. If the rotation axis and the scalar angular velocity are given by a vector � ∈ ℜ3, the rotation matrix can be computed as ω= ˆ( ) exp{ }R t t at time t.

ˆ ( ) ( )TR t R tω =

ˆ( ) ( )R t R tω=

{ }ˆ( ) exp (0)R t t Rω=

2ˆ ˆ( ) ( )ˆ ˆexp2! !

nt tt I t

n

ω ωω ω

⎧ ⎫= + + + + +⎨ ⎬

⎩ ⎭

ω= ˆ( ) exp{ }R t t

[ ] [ ]1 ˆ ˆ ˆexp( ) exp( ) exp( ) exp( )TTt t t tω ω ω ω

− = − = = −

{ }2

ˆ ˆ ˆˆdet exp( ) det exp exp det exp 0

2 2 2

t t tt

ω ω ωω

⎡ ⎤⎧ ⎫ ⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⋅ = ≥⎨ ⎬ ⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭ ⎩ ⎭⎣ ⎦


The Rodrigues formula is one efficient way to compute the rotation ma-trix ω= ˆ( ) exp{ }R t t . Given 3R∈′� with 1=′� and = Ω⋅ ′� � , it is simple to verify that the power of ω′ˆ can be reduced by the following formula:

3ˆ ˆω ω= −′ ′ (2A.12)

Then the exponential series

(2A.13)

can be simplified as:

(2A.14)

Therefore,

(2A.15)

Appendix 2B

MATLAB Source Codes

spinning_theory.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mathematics of precession theory% % Object: three point scatterers% P1 = [0, 0, 0]; % first point scatterer% P2 = [0.3, 0, 0.6]; % second point scatterer% P3 = [-0.3, 0, 0.6]; % third point scatterer% spinning about its symmetric axis with 3 Hz spinning rate% coning about the other axis with 2 Hz coning rate%% Radar Signal: 10 GHz carrier frequency (X-band)%% The object is at (X0=1000;Y0=5000;Z0=5000) from the radar

ω ωω ω= + + + + +

2ˆ ˆ( ) ( )ˆ ˆexp( )2! !

nt tt I t

n

3 5

2 4 62

2

( ) ( )ˆ ˆexp( )3! 5!

( ) ( ) ( ) ˆ2! 4! 6!

ˆ ˆsin (1 cos )

t tt I t

t t t

I t t

ω ω

ω

ω ω

⎛ ⎞Ω Ω= + Ω − + − ′⎜ ⎟⎝ ⎠

⎛ ⎞Ω Ω Ω+ − + − ′⎜ ⎟⎝ ⎠

= + Ω + − Ω′ ′

2ˆ ˆ ˆ( ) exp( ) sin (1 cos )R t t I t tω ω ω= = + Ω + − Ω′ ′


% with initial roll, pitch and yaw angles (psi, theta, phi)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all f = 10e9;c = 3e8; % object model in the body fixed system (x-y-z)P1 = [0, 0, 0]; % first point scattererP2 = [0.3, 0, 0.6]; % second point scattererP3 = [-0.3, 0, 0.6]; % third point scattererP0 = [0, 0, 0]; % the origin of the body-fixed systemPax = [0, 0, 1]; % symmetric axismodel = [P1; P2; P3; P0; Pax];N_sctr = size(model,1); T = 2.048;dt = 0.001;t = 0:dt:T-dt;nt = length(t); % the origin of the object body-fixed system % in the radar coordinate system (X,Y,Z)X0 = 1000;Y0 = 5000;Z0 = 5000;n_radial = [X0;Y0;Z0]/norm([X0,Y0,Z0]); % radial direction %initial Euler anglesphi = 30*pi/180; % yaw angletheta = 30*pi/180; % pitch anglepsi = 20*pi/180; % roll angle% initial rotation matrix using R-P-Y convention (x-y-z sequence)Rz = [cos(phi) sin(phi) 0; -sin(phi) cos(phi) 0; 0 0 1];Ry = [cos(theta) 0 -sin(theta); 0 1 0; sin(theta) 0 cos(theta)];Rx = [1 0 0; 0 cos(psi) sin(psi); 0 -sin(psi) cos(psi)];Rinit = Rx*Ry*Rz; % the initial object model orientation in the reference coords.model_ini = model*Rinit; % orientation figure(1)plot3(model(1,1),model(1,2),model(1,3),’bo’)hold onplot3(model_ini(1,1),model_ini(1,2),model_ini(1,3),’bo’)plot3(model_ini(2,1),model_ini(2,2),model_ini(2,3),’go’)plot3(model_ini(3,1),model_ini(3,2),model_ini(3,3),’ro’) line([model_ini(2,1) model_ini(3,1)],[model_ini(2,2) model_ini(3,2)],...

[model_ini(2,3) model_ini(3,3)],... ‘color’,’k’,’LineWidth’,2.0,’LineStyle’,’-’)line([model_ini(2,1)/2+model_ini(3,1)/2 model_ini(1,1)],... [model_ini(2,2)/2+model_ini(3,2)/2 model_ini(1,2)],...


[model_ini(2,3)/2+model_ini(3,3)/2 model_ini(1,3)],... ‘color’,’k’,’LineWidth’,1.5,’LineStyle’,’--’)axis imageview(-10, 20)axis([-0.8 0.8, -0.8 0.8, -0.4 0.8])xlabel(‘X’)ylabel(‘Y’)zlabel(‘Z’)title(‘Trajectory of Three Spinning Scatterers’)griddrawnow % spinning axis: [ux,uy,uz]ux = model_ini(5,1);uy = model_ini(5,2);uz = model_ini(5,3); % spinning at fs frequencyfs = 8/T; % (Hz) spinning frequencyws_x = 2*pi*fs*ux;ws_y = 2*pi*fs*uy;ws_z = 2*pi*fs*uz; ws = [ws_x; ws_y; ws_z];Omiga_s = sqrt(ws_x*ws_x + ws_y*ws_y + ws_z*ws_z);%scalar angle velocity

ws_unit = ws/Omiga_s;% skew symmetric matrixWs = [0 -ws_unit(3) ws_unit(2); ... ws_unit(3) 0 -ws_unit(1); ... -ws_unit(2) ws_unit(1) 0 ]; model_new = model_ini;for i=1:100 Rs = eye(3)+Ws*sin(Omiga_s*t(i))+Ws*Ws*(1-cos(Omiga_s*t (i))); model_new = model_new*Rs; figure(1) plot3(model_new(1,1),model_new(1,2),model_new(1,3),’bo’) plot3(model_new(2,1),model_new(2,2),model_new(2,3),’go’) plot3(model_new(3,1),model_new(3,2),model_new(3,3),’ro’) drawnowendgrid on for i=1:nt Rs = eye(3)+Ws*sin(Omiga_s*t(i))+Ws*Ws*(1-cos(Omiga_s*t(i))); dRs = Omiga_s*(Ws*(Ws*sin(Omiga_s*t(i))+cos(Omiga_s*t(i)))); for j=1:size(model(1:3,:),1) MicroDoppler(j,i) = 2*(f/c)*(model_ini(j,:)*dRs)*n_radial; endend figure(2)plot(t, MicroDoppler(1,:),’b.’);hold onplot(t, MicroDoppler(2,:),’g.’);


plot(t, MicroDoppler(3,:),’r.’);title(‘Micro-Doppler Signature of a Spinning Target’);xlabel(‘Time (sec)’);ylabel(‘Doppler Frequency (Hz)’);axis([0 T -800 800])drawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

coning_theory.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mathematics of coning theory% % Object: three point scatterers% P1 = [0, 0, 0]; % first point scatterer% P2 = [0.3, 0, 0.6]; % second point scatterer% P3 = [-0.3, 0, 0.6]; % third point scatterer% coning about another axis with 2 Hz coning rate%% Radar Signal: 10 GHz carrier frequency (X-band)%% The object is at (X0=1000;Y0=5000;Z0=5000) from the radar% with initial roll, pitch and yaw angles (psi, theta, phi)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all f = 10e9;c = 3e8; % object model in the body fixed system (x-y-z)P1 = [0, 0, 0]; % first point scattererP2 = [0.3, 0, 0.6]; % second point scattererP3 = [-0.3, 0, 0.6]; % third point scattererP0 = [0, 0, 0]; % the origin of the body-fixed systemPax = [0, 0, 1]; % symmetric axismodel = [P1; P2; P3; P0; Pax];N_sctr = size(model,1); T = 2.048;dt = 0.001;t = 0:dt:T-dt;nt = length(t); % the origin of the object body-fixed system % in the radar coordinate system (X,Y,Z)X0 = 1000;Y0 = 5000;Z0 = 5000;n_radial = [X0;Y0;Z0]/norm([X0,Y0,Z0]); % radial direction % for coning motion, the object can be located anywhere % in the reference coordinate system (X’,Y’,Z’)


xc = 0;yc = 0;zc = 0;Cc = [xc, yc, zc];% the initial object model position in the reference coords.modelcc = model + repmat(Cc,size(model,1),size(Cc,1)); %position %initial Euler anglesphi = 10*pi/180; % yaw angletheta = 10*pi/180; % pitch anglepsi = -20*pi/180; % roll angle% initial rotation matrix using R-P-Y convention (x-y-z sequence)Rz = [cos(phi) sin(phi) 0; -sin(phi) cos(phi) 0; 0 0 1];Ry = [cos(theta) 0 -sin(theta); 0 1 0; sin(theta) 0 cos(theta)];Rx = [1 0 0; 0 cos(psi) sin(psi); 0 -sin(psi) cos(psi)];Rinit = Rx*Ry*Rz; % the initial object model orientation in the reference coords.model_ini = modelcc*Rinit; % orientationlocal_origin = [0; 0; 0]; % define the coning axis with given azimuth and elevation angles:alpha_c = 160*pi/180; beta_c = 50*pi/180;% the coning axis vector:e_vector = [cos(alpha_c)*cos(beta_c),sin(alpha_c)*cos(beta_c),sin(beta_c)];

% e_vector = [A,B,C]D = cos(beta_c);C = sin(beta_c);B = D*sin(alpha_c);A = D*cos(alpha_c); figure(1)plot3(model_ini(1,1),model_ini(1,2),model_ini(1,3),’bo’);hold onplot3(model_ini(2,1),model_ini(2,2),model_ini(2,3),’go’);plot3(model_ini(3,1),model_ini(3,2),model_ini(3,3),’ro’);line([model_ini(2,1) model_ini(3,1)],[model_ini(2,2) model_ini(3,2)],...

[model_ini(2,3) model_ini(3,3)],... ‘color’,’k’,’LineWidth’,2.0,’LineStyle’,’-’);line([model_ini(2,1)/2+model_ini(3,1)/2 model_ini(1,1)],... [model_ini(2,2)/2+model_ini(3,2)/2 model_ini(1,2)],... [model_ini(2,3)/2+model_ini(3,3)/2 model_ini(1,3)],... ‘color’,’k’,’LineWidth’,1.5,’LineStyle’,’--’);plot3(local_origin(1), local_origin(2), local_origin(3),’ko’)line([local_origin(1) e_vector(1)],... [local_origin(2) e_vector(2)],... [local_origin(3) e_vector(3)],... ‘color’,’k’,’LineWidth’,2.0,’LineStyle’,’-.’)grid onaxis imageview(-160, 15)axis([-1 1, -1 1, -0.5 1])xlabel(‘X’)ylabel(‘Y’)


zlabel(‘Z’)title(‘Trajectory of Three Coning Scatterers’)drawnow % rotating about the coning axis vector e_vector fc = 2/T; % (Hz) coning frequency: 2 cycles during Twc_x = 2*pi*fc*A;wc_y = 2*pi*fc*B;wc_z = 2*pi*fc*C;wc = [wc_x; wc_y; wc_z];Omiga_c = sqrt(wc_x*wc_x + wc_y*wc_y + wc_z*wc_z);wc_unit = wc/Omiga_c;% skew symmetric matrixWc = [0 -wc_unit(3) wc_unit(2); ... wc_unit(3) 0 -wc_unit(1); ... -wc_unit(2) wc_unit(1) 0 ]; model_new = model_ini;for i=1:100 Rc = eye(3)+Wc*sin(Omiga_c*t(i))+Wc*Wc*(1-cos(Omiga_c*t(i))); model_new = model_new*Rc; figure(1) plot3(model_new(1,1),model_new(1,2),model_new(1,3),’bo’) plot3(model_new(2,1),model_new(2,2),model_new(2,3),’go’) plot3(model_new(3,1),model_new(3,2),model_new(3,3),’ro’) drawnowend for i=1:nt Rc = eye(3)+Wc*sin(Omiga_c*t(i)+Wc*Wc*(1-cos(Omiga_c*t(i)))); dRc = Omiga_c*(Wc*(Wc*sin(Omiga_c*t(i))+cos(Omiga_c*t(i)))); for j=1:size(model(1:3,:),1) MicroDoppler(j,i) = 2*(f/c)*(model_ini(j,:)*dRc)*n_radial; endend figure(2)plot(t, MicroDoppler(1,:),’b.’);hold onplot(t, MicroDoppler(2,:),’g.’);plot(t, MicroDoppler(3,:),’r.’);title(‘Micro-Doppler Signature of a Coning Target’);xlabel(‘Time (sec)’);ylabel(‘Doppler Frequency (Hz)’);axis([0 T -800 800])drawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

precession_theory.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mathematics of precession theory%


% Object: three point scatterers% P1 = [0, 0, 0]; % first point scatterer% P2 = [0.3, 0, 0.6]; % second point scatterer% P3 = [-0.3, 0, 0.6]; % third point scatterer% precession about an axis with 2 Hz coning rate % and 3 Hz spinning rate%% Radar Signal: 10 GHz carrier frequency (X-band)%% The object at (X0=1000;Y0=5000;Z0=5000) from the radar% with initial roll, pitch and yaw angles (psi, theta, phi)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all f = 10e9;c = 3e8; % object model in the body fixed system (x-y-z)P1 = [0, 0, 0]; % first point scattererP2 = [0.3, 0, 0.6]; % second point scattererP3 = [-0.3, 0, 0.6]; % third point scattererP0 = [0, 0, 0]; % the origin of the body-fixed systemPax = [0, 0, 1]; % symmetric axismodel = [P1; P2; P3; P0; Pax];N_sctr = size(model,1); T = 2.048;dt = 0.001;t = 0:dt:T-dt;nt = length(t); % the origin of the object body-fixed system % in the radar coordinate system (X,Y,Z)X0 = 1000;Y0 = 5000;Z0 = 5000;n_radial = [X0;Y0;Z0]/norm([X0,Y0,Z0]); % radial direction %initial Euler anglesphi = 30*pi/180; % yawtheta = 30*pi/180; % pitchpsi = 45*pi/180; % roll% inital rotation matrix using R-P-Y convention (x-y-z sequence)Rz = [cos(phi) sin(phi) 0; -sin(phi) cos(phi) 0; 0 0 1];Ry = [cos(theta) 0 -sin(theta); 0 1 0; sin(theta) 0 cos(theta)];Rx = [1 0 0; 0 cos(psi) sin(psi); 0 -sin(psi) cos(psi)];Rinit = (Rx*Ry)*Rz; % the initial object model orientation in the reference coords.model_ini = model*Rinit; % orientationlocal_origin = [0; 0; 0]; % the coning axis with its azimuth and elevation angles:


alpha_c = 160*pi/180; beta_c = 50*pi/180;% the coning axis vector:e_vector = ... [cos(alpha_c)*cos(beta_c),sin(alpha_c)*cos(beta_c), sin(beta_c)]; % e_vector = [A,B,C]D = cos(beta_c);C = sin(beta_c);B = D*sin(alpha_c);A = D*cos(alpha_c); figure(1)hold onplot3(model_ini(1,1),model_ini(1,2),model_ini(1,3),’bo’);plot3(model_ini(2,1),model_ini(2,2),model_ini(2,3),’go’);plot3(model_ini(3,1),model_ini(3,2),model_ini(3,3),’ro’);line([model_ini(2,1) model_ini(3,1)],[model_ini(2,2) model_ini(3,2)],...

[model_ini(2,3) model_ini(3,3)],... ‘color’,’k’,’LineWidth’,2.0,’LineStyle’,’-’);line([model_ini(2,1)/2+model_ini(3,1)/2 model_ini(1,1)],... [model_ini(2,2)/2+model_ini(3,2)/2 model_ini(1,2)],... [model_ini(2,3)/2+model_ini(3,3)/2 model_ini(1,3)],... ‘color’,’k’,’LineWidth’,3.0,’LineStyle’,’-’);plot3(local_origin(1), local_origin(2), local_origin(3),’ko’)line([local_origin(1) e_vector(1)],... [local_origin(2) e_vector(2)],... [local_origin(3) e_vector(3)],... ‘color’,’b’,’LineWidth’,2.0,’LineStyle’,’-.’)grid onaxis imageview(-160, 15)axis([-1.5 1.5, -1.5 1.5, -1.5 1.5])xlabel(‘x’)ylabel(‘y’)zlabel(‘z’)title(‘Three Precession Scatterers’)drawnow % spinning axis: [ux,uy,uz]ux = model_ini(5,1);uy = model_ini(5,2);uz = model_ini(5,3);% spining at fs frequencyfs = 8/T; % (Hz) spinning frequency: 8 cycles during Tws_x = 2*pi*fs*ux;ws_y = 2*pi*fs*uy;ws_z = 2*pi*fs*uz; ws = [ws_x; ws_y; ws_z];Omiga_s = sqrt(ws_x*ws_x + ws_y*ws_y + ws_z*ws_z);%scalar angle-velocityws_unit = ws/Omiga_s;% skew symmetric matrixWs = [0 -ws_unit(3) ws_unit(2); ... ws_unit(3) 0 -ws_unit(1); ... -ws_unit(2) ws_unit(1) 0 ];


% rotating about the coning axis vector e_vector fc = 2/T; % (Hz) coning frequency: 2 cycles during Twc_x = 2*pi*fc*A;wc_y = 2*pi*fc*B;wc_z = 2*pi*fc*C; wc = [wc_x; wc_y; wc_z];Omiga_c = sqrt(wc_x*wc_x + wc_y*wc_y + wc_z*wc_z);wc_unit = wc/Omiga_c;Wc = [0 -wc_unit(3) wc_unit(2); ... wc_unit(3) 0 -wc_unit(1); ... -wc_unit(2) wc_unit(1) 0 ]; for i=1:nt Rc = eye(3)+Wc*sin(Omiga_c*t(i)+Wc*Wc*(1-cos(Omiga_c*t(i)))); dRc = Omiga_c*(Wc*(Wc*sin(Omiga_c*t(i))+cos(Omiga_c*t(i)))); Rs = eye(3)+Ws*sin(Omiga_s*t(i)+Ws*Ws*(1-cos(Omiga_s*t(i)))); dRs = Omiga_s*(Ws*(Ws*sin(Omiga_s*t(i))+cos(Omiga_s*t(i)))); for j=1:size(model(1:3,:),1) MicroDoppler(j,i) = ... 2*(f/c)*(model_ini(j,:)*(dRc*Rs+Rc*dRs))*n_radial; endend; figure(2)plot(t, MicroDoppler(1,:),’b.’);hold onplot(t, MicroDoppler(2,:),’g.’);plot(t, MicroDoppler(3,:),’r.’);title(‘Micro-Doppler Signature of a Precession Target’);xlabel(‘Time (sec)’);ylabel(‘Doppler Frequency (Hz)’);axis([0 2.048 -4000 4000])drawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

93

3 The Micro-Doppler Effect of the Rigid Body MotionThe rigid body is an idealization of a solid body without deformation (i.e., the distance between any two particles of the body remains constant while in mo-tion). Usually, the geometry of a body is described by its location and orienta-tion. The location is defined by the position of a reference point in the body, such as the center of mass or the centroid. The orientation is determined by its angular position. Thus, the motion of a rigid body is described by its kinematic and dynamic quantities, such as linear and angular velocity, linear and angular acceleration, linear and angular momentum, and the kinetic energy of the body. As described in Chapter 2, the orientation of a rigid body can be represented by a set of Euler angles in a three-dimensional Euclidean space, by a rotation matrix (called the direction cosine matrix), or by a quaternion.

When a rigid body moves, both its position and its orientation vary with time. The translation and rotation of the body are measured with respect to a reference coordinates system. In a rigid body, all particles of the body move with the same translational velocity. However, when rotating, all particles of the body (except those lying on the axis of rotation) change their position. Thus, the linear velocity of any two particles of the body may not be the same. How-ever, angular velocities of all particles are the same.

When an object has translational and/or rotational motion, the radar scat-tering from the target is subject to modulation in its amplitude and phase. Theoretical analysis indicates that the motion of an object can modulate the phase function of the scattered EM waves. If the object oscillates periodically, the modulation generates sideband frequencies about the frequency of the in-cident wave.


To incorporate any object’s motion in EM simulation, first, the trajectory and orientation of the object must be determined by using motion differen-tial equations and a rotation matrix of the object. Then, using the quasi-static method, the motion of the object is considered a sequence of snapshots taken at each time instant. Finally, using a suitable RCS prediction method, the scat-tering EM field is estimated.

The simplest model of the EM scattering mechanism is the point scat-terer model. With this model, an object is defined in terms of a three-dimen-sional reflectivity density function characterized by point scatterers. It is rather straightforward to incorporate an object’s motion in the point scatterer model simulation.

Another simple method of RCS modeling is to decompose an object into canonical geometric components, such as a sphere, an ellipsoid, or a cylinder. The RCS of each canonical component can be expressed by a mathematical formula.

A more accurate RCS modeling method is the physical optics (PO) model. It is a simple and convenient RCS prediction method for any three-dimensional object.

In this chapter, typical examples of an oscillating pendulum, rotating ro-tor blades, a spinning symmetric top, and wind turbines of windmills are in-troduced. Pendulum oscillation is a commonly used example to understand the basic principle of the nonlinear motion dynamic. Rotating helicopter rotor blades are one of the most popular subjects in radar target signature analy-sis. The spinning symmetric top has more complex nonlinear motion and has shown more interesting signatures than other rigid body motion. Wind tur-bines now become challenges to current radar systems. Large numbers of wind farms and large RCS of turbine blades have a significant impact on radar perfor-mance. In this chapter, details on the modeling of a nonlinear motion dynamic, the modeling of the RCS of a rigid body, the mathematical model of radar scat-tering from a rotating rigid body, and the micro-Doppler signature of a typical rigid body will be described, and the simulation of nonlinear motion and radar backscattering using MATLAB will be provided.

3.1 Pendulum Oscillation

Pendulum oscillation is a commonly used example to understand the basic principle of nonlinear motion dynamic. A simple pendulum is modeled by a weighted small bob, attached to one end of a weightless string, and the other end of the string is fixed to a pivot point as shown in Figure 3.1. Under the influence of the gravity g = 9.80665 m/s2, the small bob swings back and forth periodically about a fixed horizontal axis along the y-axis and at (x = 0, z = 0).

The Micro-Doppler Effect of the Rigid Body Motion 95

In a stable equilibrium condition, the center of mass of the pendulum is right below the axis of rotation at (x = 0, y = 0, z = L), where L is the length of the string.

3.1.1 Modeling Nonlinear Motion Dynamic of a Pendulum

Newton’s law states that the total force acting on the pendulum is equal to the product of the mass of the pendulum and its acceleration. If the pendulum is initially deviated from its stable position by a swinging angle θ, there are two forces acting on the mass of the pendulum: the downward gravitational force, mg, , where m is the mass of the pendulum, and the tension, T, in the string. However, the tension has no contribution to the torque because its line of action passes through the pivot point. From simple trigonometry, the line of action of the gravitational force passes a distance Lsinθ from the pivot point. Hence, the magnitude of the gravitational torque is mgLsinθ. The gravitational torque is a restoring torque (i.e., if the mass of the pendulum is displaced slightly from its equilibrium state at θ = 0, then the gravitational force acts to pull the pendulum back toward that state).

According to Newton’s second law of motion, the net torque is I(d 2θ/d 2t), where I is the moment of inertia, and (d 2θ/d 2t) is the angular acceleration. If the torque acting on the system is τ, then τ = I(d 2θ/d 2t) is the angular equation

Figure 3.1 A simple pendulum.


of motion. Given the length of the string L and the mass of the pendulum m, the moment of inertia about the pivot is

(3.1)

The vector torque � is the cross-product of the position vector L and the gravitational force vector mg (i.e., � = L × mg). The magnitude of the torque is τ = Lmg sinθ, and the net torque on the pendulum is

(3.2)

and, finally, the equation of pendulum becomes

(3.3)

This equation defines the relationship between the swinging angle θ and its second time derivatives d 2 θ/dt 2.

By denoting the angular velocity Ω = dθ/dt, the equation of pendulum can be rewritten as a set of two first-order ordinary differential equations (ODEs):

(3.4)

Figure 3.2 shows the oscillating angle and angular velocity of a simple pendulum with the mass m = 20g and the length L = 1.5m.

For a small angle θ, sinθ can be substituted by θ and the pendulum be-comes a linear oscillator. Thus, the differential equation of the pendulum mo-tion becomes

(3.5)

where ω0 = (g/L)1/2 is the angular frequency of the oscillating pendulum. Equa-

tion (3.5) is a harmonic equation and the solution of the swinging angle is

2I m L=

2 22

2 2sind d

Lm g I m Ldt dt

θ θθ− = =

2

2 sind

mL mgdt

θθ= −

,

sin

d

dtgd

dt L

θ

θ

⎧ = Ω⎪⎪⎨ Ω⎪ = −⎪⎩

2202 0

d

dt

θω θ+ ≅


0 0( ) sint tθ θ ω= (3.6)

and its angular velocity is

(3.7)

where θ 0 is the initial swinging angle of the pendulum, called the initial

amplitude.For a given initial amplitude θ

0, the period of the oscillating pendulum is

determined by

(3.8)

For a small initial amplitude θ0, the oscillating period is

0 0 0

( )( ) cos

d tt t

dt

θθ ω ωΩ = =

2 40 00

0

1 1 92 2 1 sin sin ...

4 2 64 2

LT

g

θ θπ π

ω⎛ ⎞= = + + +⎜ ⎟⎝ ⎠

Figure 3.2 Oscillating angle and angular velocity of a simple pendulum.


0 2

LT

gπ= (3.9)

or the frequency of the oscillation is f0 = 1/T

0.

The simple pendulum assumes that the string is weightless and the bob is small such that its angular momentum is negligible. However, a physical pen-dulum may have a large size and mass. Thus, it may have a significant moment of inertia I.

From (3.2), the equation of a physical pendulum can be written as

(3.11)

where Leffect

is the effective length of a physical pendulum, and the right-hand side of the equation is the net torque of the gravity. The physical pendulum equation can be simply expressed as

(3.12)

where 20 /effectmgL Iω = is the angular frequency of the physical pendulum. Thus,

the period of its swinging becomes

(3.13)

The physical pendulum equation with an effective length Leffect

is the same as the simple pendulum with its length L = L

effect. The physical pendulum (3.12)

is described by the same mathematic formula and is equivalent to the simple pendulum.

If linear friction exists in the oscillating pendulum, an additional term

2 ,d

dt

θγ− proportional to the angular velocity, must be added to the right side

of (3.3). Then the equation of pendulum becomes

(3.14)

2

2 sineffect

dI m g L

dt

θθ= −

2202 sin 0

d

dt

θω θ+ =

0 02 / 2effect

IT

mgLπ ω π= =

2202 2 sin 0

d d

dt dt

θ θγ ω θ+ + =


where ω

0 = (g/L)1/2 is the angular frequency of free oscillations, and γ is the

damping constant. Thus, the equation of pendulum with linear friction can be rewritten as a set of two first-order ODEs

(3.15)

For a small angle θ, sinθ ≈ θ and the pendulum equation is approximately

(3.16)

If the friction is weak such that γ < ω0, the solution of (3.16) is

(3.17)

where θ0 is the initial amplitude, ϕ

0 is the initial phase depending on the initial

excitation, and the exponential term θ0exp(−γ t) is a decreasing factor. The an-

gular frequency of the oscillation ω is given by 2 2 20 0 01 ( / ) .ω ω γ ω γ ω= − = −

When γ < ω0, the angular oscillation frequency and period become

(3.18)

which are close to the free oscillation frequency and period ω0 and T

0.

Figure 3.3(a) shows the oscillating angle and angular velocity of the damp-ing pendulum with damping constant γ = 0.07 and ω

0 = (g/L)1/2 = 2.56.

If there is a damping as well as a driving force in the pendulum oscillation, the equation of the pendulum must be modified to

(3.19)

where γ is the damping constant, ADr

is the amplitude of the driving, and fDr

is the driving frequency.

2 sin

d

dtgd

dt L

θ

γ θ

⎧ = Ω⎪⎪⎨ Ω⎪ + Ω = −⎪⎩

2202 2 0

d d

dt dt

θ θγ ω θ+ + =

0 0( ) cos( )t

t e tγ

θ θ ω ϕ−= +

20 0

2 20 0

/(2 );

1 / (2 )T T

ω ω γ ω

γ ω

≈ −

⎡ ⎤≈ +⎣ ⎦

( )2

2 2 sin cos 2DrDr

g Ad df t

dt dt L mL

θ θγ θ π+ + =


Figure 3.3 Oscillating angle and angular velocity of: (a) a damping pendulum and (b) a damp-ing driving pendulum.


If we let the angular velocity Ω = dθ/dt, the equation of pendulum with friction and driving force can be rewritten as a set of two first-order ODEs

(3.20)

Figure 3.3(b) shows the oscillating angle and angular velocity of the damp-ing driving pendulum with damping constant γ = 0.07, a driving amplitude A

Dr

= 15, and a normalized driving frequency fDr

= 0.2.

3.1.2 Modeling RCS of a Pendulum

The RCS measures the strength of an object’s reflectivity and is a function of the object’s orientation and the radar transmitted frequency. The small bob of a pendulum has a simple geometric shape, such as a sphere, an ellipsoid, or a cylinder, and the RCS of a simple geometric shape can be expressed by a math-ematical formula.

High-frequency RCS prediction methods and exact RCS prediction for-mulas can be found in [1] by Knott, Shaeffer, and Tuley. The computer simula-tion of radar backscattering used in this book is not an exact RCS prediction. Instead, the simulation is based on approximate and simplified complex scatter-ing solutions [2]. With the simplest component method, an object consists of a limited number of the simplest components, such as spheres, ellipsoids, and cylinders. The formulas of the simplest components are available, but are not exact solutions.

The RCS of a perfectly conducting sphere has three regions. In the optical region, which corresponds to a large sphere compared with the wavelength, the RCS is a constant and can be simply expressed by RCSsphere

= πr2, where r is the radius of the sphere and is much greater than the wavelength λ. In the Rayleigh region for a small sphere, the RCS is RCS

sphere = 9πr2(kr)4, where k = 2π/λ. The

region between the Rayleigh and optical regions is a resonance region called the Mie region [3].

An approximation for the RCS of an ellipsoid backscattering is given by [3]

(3.21)

( )2 sin cos 2DrDr

d

dtg Ad

f tdt L mL

θ

γ θ π

⎧ = Ω⎪⎪⎨ Ω⎪ + Ω = − +⎪⎩

( )2 2 2

22 2 2 2 2 2 2 2sin cos sin sin cosellip

a b cRCS

a b c

π

θ ϕ θ ϕ θ=

+ +


where a, b, and c represent the length of the three semi-axes of the ellipsoid in the x, y, and z directions, respectively. The incident aspect angle θ and the azimuth angle ϕ represent the orientation of the ellipsoid relative to the radar and are defined by

(3.22)

and

(3.23)

where the incident angle counts from the z-axis and the azimuth angle counts from the x-axis. If the ellipsoid is symmetric (i.e., a = b), the RCS will be inde-pendent of the azimuth angle ϕ.

The nonnormal incidence backscattered RCS for a symmetric cylinder due to a linear polarized incident wave is approximated by [3]

(3.24)

where r is the radius, θ is the incident aspect angle, and the RCS is independent of the azimuth angle ϕ.

These RCS formulas can be used to simulate radar backscattering from an oscillating pendulum.

3.1.3 Radar Backscattering from an Oscillating Pendulum

To calculate radar backscattering from an oscillating pendulum, ordinary dif-ferential equations are used for solving the swinging angle and the angular ve-locity. Therefore, at each time instant during a radar observation time interval, the location of the pendulum can be determined. Based on the location and orientation of the pendulum, the RCS of the pendulum and the radar received signal can be calculated.

If the radar transmits a sequence of narrow rectangular pulses with a trans-mitted frequency f

c, a pulse width Δ, and a pulse repetition interval ΔT, the

radar received baseband signal is

2 2

arctanx y

zθ

⎛ ⎞+= ⎜ ⎟⎜ ⎟⎝ ⎠

arctany

xϕ

⎛ ⎞= ⎜ ⎟⎝ ⎠

2

sin

8 coscylinder

rRCS

λ θ

π θ=


1

2 ( ) 2 ( )( ) ( ) exp 2

pn

P PB P c

k

R t R ts t t rect t k T j f

c cσ π

=

⎧ ⎫ ⎧ ⎫= − Δ − −⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭∑ (3.25)

where σP(t) is the RCS of the small bob at time t, n

p is the total number of pulses

received, RP(t) is the distance from the radar to the small bob at time t, and the

rectangular function rect is defined by

(3.26)

Given the location of the radar at (x = 10m, y = 0m, z = 0m), the pivot point of the pendulum is assumed at (x = 0m, y = 0m, z = 2m). The string length L = 1.5m and the mass of the small bob is 20g. In cases of damping and driving, let the damping constant be γ = 0.07 and the driving amplitude be A

Dr

= 15, and the normalized driving frequency fDr

= 0.2. The geometric configura-tion of the radar and the pendulum is illustrated in Figure 3.4.

Equations (3.4) and (3.15) are used to calculate the oscillating angle and angular velocity of the simple, the damping, and the damping and driving pen-dulums, respectively. In the rotation matrix of the pendulum, only the pitch angle varies and the roll and yaw angles are always zero. The RCS of the small bob is simulated by the point-scatterer model because the small bob can be seen as a point scatterer. After arranging the n

p range profiles, the 2-D pulse-range

profiles can be obtained. Figure 3.5(a) shows the 2-D range profiles of the sim-ple oscillating pendulum, where the radar wavelength is 0.03m at the X-band. The oscillating small bob can be seen around a distance of 10m from the radar.

1 0( )

0 otherwise

trect t

≤ ≤ Δ⎧= ⎨

⎩

Figure 3.4 Geometric confi guration of the radar and the pendulum.


Figure 3.5 (a) Range profi les and (b) micro-Doppler signature of the simple free oscillating pendulum.


3.1.4 Micro-Doppler Signatures Generated by an Oscillating Pendulum

The micro-Doppler signature of the oscillating simple pendulum, shown in Figure 3.5(b), is obtained from the summation of those range profiles that are within a range gate around 10m, where the small bob is located within the radar observation interval. The joint time-frequency transform used to generate the signature is a simple STFT. Other higher-resolution time-frequency transforms, such as the smoothed pseudo-Wigner-Ville distribution, may also be used.

Compared with the micro-Doppler signature a simple pendulum, Figure 3.6 shows the micro-Doppler signatures of a damping pendulum and a damp-ing and driving pendulum with L = 1.5m, m = 20g, γ = 0.07, A

Dr = 15, and f

Dr

= 0.2.From the micro-Doppler signature in Figure 3.6(a), an oscillating fre-

quency of 0.4 Hz can be measured and the damping constant γ is measured from the change of the amplitude of the Doppler modulation during the ob-servation time duration of 10 seconds. Due to the measured change of the am-plitude of the Doppler modulation being 101 Hz/202 Hz during a 10-second time interval, the damping constant is estimated as

which is consistent with the damping constant of 0.07 used in the simulation. The MATLAB code for calculating radar backscattering from an oscillating pendulum is listed in Appendix 3A.

3.2 Helicopter Rotor Blades

An airfoil of a helicopter rotor blade and its cross-section profile is shown in Figure 3.7 [4]. Different types of airfoils have various shapes and dimensions. A rotating aerofoil always has bending, flexing, and twisting. However, in the simulation study of a helicopter’s rotor blades, no bending, flexing, and twisting are considered.

Blades of helicopters are usually metallic or a composite material that produces strong radar reflectivity. EM scattering from an airfoil mainly includes specular reflections from its surfaces and leading edge, diffraction from its trail-ing edge, creeping waves around the leading edge, and traveling waves from the trailing edge. Radar returns from a helicopter have its unique spectral sig-nature [5, 6]. Figure 3.8 illustrates a general spectral signature of helicopters with rotating rotor blades. The spectral signature has spectral components from the fuselage, from the rotor hub, from the main rotor’s receding blades, and from approaching blades. The strongest spectral amplitude comes from the fu-selage. The spectral amplitude of the receding blade is different from that of

log (101/ 202)/10 0.069eγ = − =


Figure 3.6 The micro-Doppler signatures of (a) the damping oscillating pendulum and (b) the damping and driving pendulum.


the approaching blade because of the difference between the leading edge and the trailing edge. Among these spectral features, the receding blades and the approaching blades are especially interesting.

3.2.1 Mathematic Model of Rotating Rotor Blades

The geometry of the radar and rotating rotor blades is shown in Figure 3.9. The radar is located at the origin of the space-fixed coordinates (X, Y, Z ) and the ro-tor blades are centered at the origin of the body-fixed coordinates on the plane (x, y, z = 0) rotating about the z-axis with an angular rotation rate Ω. The refer-ence coordinates (X ′, Y ′, Z ′) is parallel to and translated from the space-fixed coordinates located at the same origin as the body-fixed coordinates. The dis-

Figure 3.7 An example of the airfoil of a helicopter’s rotor blade.

Figure 3.8 A general spectral signature of radar backscattering from a helicopter.


tance from the radar to the origin of the reference coordinates is R0. The radar

observed azimuth and elevation angles of the origin of the reference coordinates are α and β, respectively.

From the EM scattering point of view, each blade of the rotor consists of scatterer centers. Each scatterer center is considered a point with a certain reflectivity. For simplicity, the same reflectivity is assigned to all of the scatterer centers. Let α = β = 0; if a point scatterer P at (x

0, y

0, z

0 = 0) rotates about the

z-axis in the body-fixed coordinates with a constant angular rotation rate Ω, the distance from the origin of the body-fixed coordinates to the point scatterer P is 2 2 1/2

0 0( )Pl x y= + . If the initial rotation angle of the point P at t = 0 is ϕ0, then

at time t the rotation angle becomes ϕt = ϕ

0 + Ω

t and the point P rotates to (x

t,

yt, z

t = 0) as shown in Figure 3.9. Thus, the range from the radar to the point

scatterer P becomes

Figure 3.9 Geometry of the radar and the rotating rotor blades.


1/2

2 20 0 0

0 0 0

( ) 2 cos( )

cos cos sin sin

PP P

P P

R t R l l R t

R l t l t

ϕ

ϕ ϕ

⎡ ⎤= + + + Ω⎣ ⎦≅ + Ω + Ω

(3.27)

where assuming (lP/R

0)2 → 0 in the far field. Then the radar received signal from

the scatterer P is

(3.28)

where ( ) 4 ( ) /P Pt R tπ λΦ = is the phase function of the scatterer.If the elevation angle β and the height of rotor blades z

0 are not zero, the

phase function is modified as

(3.29)

and, thus, the returned signal from the point scatterer P becomes

(3.30)

Let ϕ0 = 0 and denote B = (4π/λ)l

Pcosβ; (3.30) can be expressed by the

Bessel function of the first kind. The spectrum of a scatterer on the blade con-sists of pairs of spectral lines around the center frequency f

0 and with spacing

Ω/(2π) between adjacent lines [7]. The baseband signal returned from the scatterer P is

(3.31)

By integrating (3.31) over the length of the blade L, the total baseband signal becomes [8, 9]

[ ]{ }0 0

4( ) exp 2 ( ) exp 2 ( )R P Ps t j f t R t j f t t

ππ π

λ⎧ ⎫⎡ ⎤= − + = − + Φ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

( )0 0 0 0

4( ) cos cos cos sin sin sinP P Pt R l t l t z

πβ ϕ ϕ β

λ⎡ ⎤Φ = + Ω + Ω +⎣ ⎦

[ ]{ } { }0 0 0 0

( )

4 4exp sin exp 2 cos cos( )

R

P

s t

j R z j f t l tπ π

β π β ϕλ λ

=

− + − − Ω +

[ ]{ } { }0 0 0

4 4( ) exp sin exp cos cos( )B Ps t j R z j l t

π πβ β ϕ

λ λ= − + − Ω +


[ ]{ } ( ){ }[ ]{ } ( ){ }

( ){ }

0 0 0

0

0 0 0

0

4 4( ) exp sin exp cos cos

4 4exp sin exp cos cos

2

4sinc cos cos

2

L

L P Ps t j R z j l t dl

LL j R z j t

Lt

π πβ β ϕ

λ λ

π πβ β ϕ

λ λ

πβ ϕ

λ

= − + − Ω +

= − + − Ω +

Ω +

∫ (3.32)

where sinc(•) is the sinc function: sinc(x) = 1 when x = 0; sinc(x) = sin(x)/x when x ≠ 0.

For a rotor with N blades, the N blades have N different initial rotation angles:

and the total received signal becomes

(3.33)

where the phase function is

(3.34)

The time-domain signature of rotor blades is defined by the magnitude of (3.33):

(3.35)

Assume a radar is operating at the C-band with a wavelength λ = 0.06m and a range resolution of 0.5m. The main rotor of a helicopter has two blades rotating with a constant rotation rate Ω = 4 revolution/second (r/s) (or 4 × 2π rad/sec). If the length of the blade from the rotor center to the blade tip is L = 6.5m, the range from the radar to the center of the rotor is 700m at an elevation

0 2 / , ( 0,1,2,... 1)k k N k Nθ θ π= + = −

[ ]{ }( ){ } { }

1

0 00

1

00

4( ) ( ) exp sin

4sinc cos cos 2 / exp ( )

2

k

N

Lk

N

kk

s t s t L j R z

Lt k N j t

πβ

λ

πβ ϕ π

λ

−

Σ=

−

=

= = − +

Ω + + − Φ

∑

∑

0

4( ) cos cos( 2 / ) ( 0,1,2,... 1)

2k

Lt t k N k N

πβ ϕ π

λΦ = Ω + + = −

[ ]{ }( ){ } { }

0 0

1

00

4( ) exp sin

4sinc cos cos 2 / exp ( )

2

N

kk

s t L j R z

Lt k N j t

πβ

λ

πβ ϕ π

λ

Σ

−

=

= − +

Ω + + − Φ∑


angle β = 45°. According to the length of the blade tip and its rotation rate, the velocity of the blade tip is V

tip = 2πLΩ = 163.4 m/s and, thus, the maximum

Doppler shift is {fD}

max = (2V

tip/λ)cosβ = 3.85 kHz. Thus, the Nyquist rate is

2 × 3.85 kHz = 7.7 kHz. With a 10-kHz sampling rate, which is higher than the Nyquist rate, the time-domain signature of the rotor blades is shown in Figure 3.10(a) and the frequency spectrum of the same signal is shown in Figure 3.10(b). There is no frequency aliasing.

The rotor blades’ return has short flashes when a blade has a specular reflection at the approaching or advancing points and the receding points [9]. The interval between flashes is related to the rotation rate of the rotor. The duration of the flash is determined by the blade length L, the wavelength λ, the elevation angle β, and the rotation rate Ω as described by the sinc function in (3.33). For a longer blade length and at a shorter wavelength, the duration of the flash is shorter. Because the number of blades is 2 and the rotation rate is 4 r/s, there are 8 flashes in 1.0 second for each of the blades, and the interval between flashes is TC

= 1/8 = 0.125 (second) as shown in Figure 3.10(a).

Figure 3.10 (a) Time-domain signature of the two-blade rotor from (3.35), and (b) the fre-quency spectrum of the two-blade rotating rotor derived from (3.33).


Because rotating rotor blades impart periodic modulations on the radar returned signal, the rotation-induced Doppler shifts occupy unique locations in the frequency domain relative to the Doppler shift of the fuselage. Figure 3.10(b) shows the spectral components of the rotating rotor blades without fuselage and rotor hub. The Doppler shift of the fuselage and the hub should be around zero Doppler if the helicopter has no translational motion.

Because the time derivative of the phase function of a signal is the in-stantaneous frequency of the signal, by taking the time derivative of the phase function Φ

k(t) in (3.34), the instantaneous Doppler frequency shift induced by

the kth rotor blade becomes

(3.36)

The Doppler frequency is modulated by the rotation rate Ω through two sinusoidal functions.

The rotation feature of rotor blades is considered an important feature for identifying helicopters of interest [9, 10]. The Doppler modulation induced by rotating rotor blades is regarded as such a unique signature of helicopters. Representing the Doppler modulation in the joint time-frequency domain, the micro-Doppler signature of the rotor blades can be seen. Figure 3.11 is the micro-Doppler signature of a rotating two-blade rotor, where 8 flashes from the blade no. 1 and 8 flashes from the blade no. 2 can be seen clearly.

For comparison, Figure 3.12 shows the micro-Doppler signature of a ro-tating three-blade rotor, where each blade has 8 flashes and the total number of flashes is 24.

3.2.2 RCS Model of Rotating Rotor Blades

To calculate EM scattering from rotating rotor blades, for simplicity, the blade in Figure 3.7 is simplified as a rigid, homogeneous, linear rectangular flat plate rotating about a fixed axis with a constant rotation rate and without consider-ing the leading edge and the trailing edge. No flapping, lagging, and feathering are considered in the rectangular flat. The geometry of the rotor blade and the radar is illustrated in Figure 3.13. For a perfectly-conducting rectangular flat plate, the mathematical formula of the RCS can be found in [3, 11]. An ap-proximation for the RCS of a rectangular flat plate is given by [3]. In the RCS formula, there are two terms: the peak RCS σ

Peak and the aspect factor σ

Aspect:

(3.37)

( ), 0 0( ) cos sin( 2 / )sin cos 2 / cosD k

Lf t k N t k N tβ ϕ π ϕ π

λ⎡ ⎤= Ω − + Ω + + Ω⎣ ⎦

22 2

2

4 sinsincos kk

Peak Aspectk k

a b yx

x y

πσ σ σ θ

λ

⎛ ⎞= = ⎜ ⎟⎝ ⎠


Figure 3.11 The micro-Doppler signature of the rotating two-blade rotor.

Figure 3.12 The micro-Doppler signature of the rotating three-blade rotor.


where 22 2

2

sinsin4, cos , sin sin ,kk

Peak Aspect k kk k

yxa bx k a y kb

x y

πσ σ θ θ ϕ

λ

⎛ ⎞= = = =⎜ ⎟⎝ ⎠

sin cos , and, 2 / .kθ ϕ π λ= Equation (3.37) is independent of the polarization and is accurate only for a small aspect angle θ ≤ 20°.

3.2.3 PO Facet Prediction Model

Physical optics (PO) is a convenient method for predicting the RCS of any three-dimensional object. It is a high-frequency region (or optical region) pre-diction and provides the best results for objects with a dimension much larger than the wavelength. The PO method applies to the illuminated surfaces, but does not apply edge diffractions, multiple reflections, or surface waves.

Any complex larger surface can be divided into many small surfaces, called facets. The facet used in the PO facet model is a triangular flat plate. The scat-tered field from each facet can be calculated as if it were isolated without con-sidering the effect of other facets. Thus, for a facet illuminated by the incident field, its surface current and scattered field can be calculated. For a shadowed surface, its surface current is set to zero.

Based on the incident and scattered fields, the RCS of a surface is deter-

mined by 2

22lim 4 s

Ri

Rσ π→∞

=E

E, where R is the range from the radar to the surface,

Figure 3.13 A rectangular fl at plate.


and |Es| and |E

i| are the amplitudes of the scattered and the incident electric

fields, respectively. As shown in Figure 3.14, an incident wave is described by the spheri-

cal coordinate angles θ and ϕ. The polarization of an incident wave can be decomposed into two orthogonal components in terms of the angles θ and ϕ to represent the incident field in the spherical system. Thus, the incident field is represented by E

i = E

θn

θ + E

ϕn

ϕ, where n

θ and n

ϕ are the unit vectors in the

spherical system.For calculating EM scattering from a facet defined by three vertices given

by points P1, P

2, and P

3 with an arbitrary orientation, the body-fixed coordinate

system is selected such that the triangular facet lies on the (x, y)-plane and the direction of its unit normal vector n is identical to the z-axis, as shown in Figure 3.15.

Because the radar is in the far zone and the size of the object is much smaller than the distance R

0 from the radar to the object, the range vector R

from the radar to the origin of the body-fixed coordinate (x, y, z) can be con-sidered to be parallel to the vector R

0. In the space-fixed coordinates, the unit

vector of the radar LOS is nXYZ

= [u, v, w], where u = sinθcosϕ, v = sin θsinϕ, and w = cosθ. In the body-fixed coordinates, any point P located at (x

P, y

P, z

P)

on the facet is represented by its position vector rP = [x

P, y

P, z

P]. Thus, the scat-

tered field from the facet is given by [12]

Figure 3.14 The spherical coordinate system used in PO facet computation.


(3.38)

where Js is the surface current, A is the area of the facet, ℜ

imp is the impedance

of free space, R is the range from the radar to the origin of the body-fixed coor-dinates, and k = 2π/λ. Thus, the scattered field from the facet can be calculated by integrating the surface current over the area of the facet. Therefore, the RCS of the facet as a function of R, θ, and ϕ is obtained. References [12–14] provide MATLAB codes for calculating the RCS of a triangular facet.

The same procedure can be applied to a collection of facets of an object. Thus, the total RCS of the object is a superposition of the RCSs of all of the facets.

3.2.4 Radar Backscattering from Rotor Blades

The rotation of a rotor blade can be easily obtained without using ordinary differential equations. The time-varying location and orientation is calculated using a rotation matrix with zero roll and pitch angles. The variation of its yaw angle is determined by the rotation rate and a given initial angle. Based on the location and orientation of a blade, the RCS and the reflected radar signal

[ ]( , , ) exp( ) exp ( )4

impS s P XYZ P

A

jkE R jkR jk ds

Rθ ϕ

π

− ℜ= − ⋅∫∫ J r n

Figure 3.15 Arbitrary oriented triangular facet defi ned in the space-fi xed system (X, Y, Z ) and in the body-fi xed local system (x, y, z).


from the blade can be calculated. Radar reflected signals from all blades can be obtained by the coherent superposition of the reflected signals from each individual blade.

If a coherent radar system transmits a sequence of narrow rectangular pulses with the pulse width Δ and the pulse repetition interval ΔT, the base-band signal in the receiver is

(3.39)

where NB is the total number of blades, n

p is the total number of pulses received

during the observation interval, fc is the radar transmitted frequency, R

n(t) is the

distance between the radar and the nth blade at time t, σn(t) is the RCS of the

nth rotor blade at time t, and rect is the rectangular function defined by rect(t) = 1 (0 < t ≤ Δ).

In (3.39), two variables, σn(t) and R

n(t), must be calculated. The RCS of

a perfectly-conducting rectangular flat plate, given by (3.37), can be used for the calculation of σ

n(t). Equation (3.37) is simple, but only accurate for aspect

angles θ ≤ 20°. The distance variable Rn(t) is from the radar to a scatterer center,

such as the centroid, of the rectangular plate. For a relatively large rectangular plate, the centroid can be far from the tip of the blade. Therefore, if the as-signed scattering center is the centroid, by using the approximate rectangular RCS formulas, the calculated radar received signal does not include Doppler components generated by the tips of blades. Therefore, the scatterer center may be assigned to the tip of blade.

The geometry of the radar and rotor blades is shown in Figure 3.16, where the radar is located at (X

1 = 500m, Y

1 = 0m, Z

1 = 500m) with a wavelength

of 0.0m at the C-band, the rotor center is located at (X0 = 0m, Y

0 = 0m, Z

0 =

0m), the length of the blade is L = 6m with its root L1 = 0.5m and its tip L

2 =

6.5m, the width of the blade is W = 1m, and the rotation rate is Ω = 4 r/s. The azimuth angle ϕ and aspect angle θ can be calculated from the radar location, the rotor location, and the blade geometry. By assigning the scatterer center of each blade to the tip of the blade, the baseband signal in the radar receiver is ob-tained from (3.39) and the RCS is calculated by (3.37) just for simplicity, even if it is not accurate. The accuracy of the RCS calculation only determines the magnitude’s distribution in the micro-Doppler signature; it does not affect the shape of the signature. After rearranging the n

p range profiles, 2-D pulse-range

profiles are shown in Figure 3.17(a) for a two-blade rotor, and its micro-Dop-pler signature is shown in Figure 3.17(b). The MATLAB code for calculating radar backscattering from rotor blades is listed in Appendix 3A.

1 1

2 ( ) 2 ( )( ) ( ) exp 2

p Bn N

n nB n c

k n


c cσ π

= =

⎧ ⎫ ⎧ ⎫= ⋅ − ⋅ Δ − ⋅ −⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭∑∑


From the range profiles, the rotating blades can be seen around range cell no. 1412 or about a 700-m distance from the radar. However, flashes cannot be seen because of the RCS is only assigned to one scatterer center. To see the flashes in the simulation, a more accurate RCS model is needed.

A simple but more accurate model for calculating radar backscattering from rotating rotor blades is the PO facet model. A rectangular blade is repre-sented by the arrays of triangular facets as shown in Figure 3.18. The scatterer center of each triangle is assumed to be the geometric centroid of its triangle vertices. With the PO facet model, the baseband signal in the radar receiver is modified as

(3.40)

where NB is the number of blades, N

F is the total number of facets in each blade,

np is the total number of pulses during the radar observation time interval, and

,,

1 1 1

,

2 ( )( )( )

2 ( )exp 2

p B Fn N N

n mn mB

k n m

n mc

R tt rect t k Ts t

c

R tj f

c

σ

π

= = =

⎧ ⎫− Δ −= ⎨ ⎬⎩ ⎭

⎧ ⎫−⎨ ⎬⎩ ⎭

∑∑∑

Figure 3.16 A geometry of the radar and rotating rotor blades.


Figure 3.17 RCS equation for a perfectly conducting rectangular fl at plane: (a) range profi les of rotating two-blade rotor, and (b) the micro-Doppler signature.


the RCS of each facet σn,m

(t) is calculated by using source codes provided in the POFacet [12–14].

Based on the geometry of a rotating three-blade rotor illustrated in Figure 3.18 and the same parameters used in the rotating two-blade rotor, the PO facet model-based radar range profiles and the micro-Doppler signature of the three-blade rotor are shown in Figure 3.19, where the flashes of the rotating blades are seen clearly.

3.2.5 Micro-Doppler Signatures of Rotor Blades

The micro-Doppler signature of the rotating three-blade’s rotor, shown in Fig-ure 3.19(b), is obtained from a summation of the range profiles within a range gate where the blades are located and shown in Figure 3.19(a). The joint time-frequency transform used to generate the signature is the STFT. Similarly, the range profiles and the micro-Doppler signature of a rotating two-blade rotor are shown in Figure 3.20, where, compared to the signature of the odd number of blades, different features for an even number of blades can be seen. With the PO facet model of rotor blades, the flashes appear in the micro-Doppler signa-tures. For the two-blade rotor shown in Figure 3.20 with a 4 r/s rotation rate, each blade has 8 flashes in 1.0 second, and the interval between flashes is 0.125

Figure 3.18 A rectangular blade represented by arrays of triangular facets.


Figu

re 3

.19

(a) T

he ra

nge

profi

les

and

(b) t

he m

icro

-Dop

pler

sig

natu

re o

f the

rota

ting

thre

e-bl

ade

roto

r.


Figu

re 3

.20

(a) T

he ra

nge

profi

les,

and

(b) m

icro

-Dop

pler

sig

natu

re o

f the

rota

ting

two-

blad

e ro

tor.


second. For the three-blade rotor in Figure 3.19, there are total of 24 flashes in 1.0 second, and the interval between two successive flashes is 0.0417 second.

Compared to the PO facet model prediction, Figure 3.21 shows the mi-cro-Doppler signature of a two-blade rotor on a scale model helicopter mea-sured by an X-band FMCW radar with the wavelength λ = 0.03m. Rotation rate of the rotor is about Ω = 2.33 r/s and the blade length is L = 0.2m. Thus, the tip velocity is V

tip = 2πLΩ = 2.93 m/s and the maximum Doppler shift is

{fD}

max = 195 Hz as shown in Figure 3.21. The signature is similar to that of the

PO facet model prediction with flashes. From the micro-Doppler signature, the number of blades, the length of the blade, and the rotation rate of the rotor can be estimated.

3.2.6 Required Minimum PRF

For a pulsed Doppler radar, its PRF determines the sampling rate. The required minimum sampling rate must satisfy the Nyquist rate to avoid frequency alias-ing. For actual helicopters, the range of their blade tip speed is around 200–230 m/s. With an X-band radar, a blade tip speed of V

tip = 230 m/s can generate

{fD}

max = 15 kHz Doppler shift. Therefore, the required minimum sampling

rate is 2 × {fD}

max = 30 kHz for a hovering helicopter. If the helicopter has a

translational motion with a radial velocity of 100 m/s, the maximum Doppler shift of the helicopter is 22 kHz, and the required minimum sampling rate can be 44 kHz [8].

Figure 3.22 demonstrates the impact of the sampling rate on the micro-Doppler signatures of rotor blades. In the demonstration, the parameters of a two-blade rotor are the same as described before but with a lower rotation rate, Ω = 1 r/s. In this case, the tip velocity of the blade will be V

tip = 40.84 m/s or the

Doppler shift of the tip is {fD}

max = 1.36 kHz. Figure 3.22(a) is the micro-Dop-

pler signature of the rotor blades under a sampling rate of 512 samples/second, where the periodic motion of blades cannot be seen. Figure 3.22(b) shows the micro-Doppler signature of the same rotor blades with two times higher sam-pling rate than 512 samples/second, and the periodic motion of blades begins to show up, but is incomplete. Figure 3.22(c) is the micro-Doppler signature of the rotor blades with 4 times the 512 sampling rate, and the periodic mo-tion of blades can be seen clearly and is almost completed. In this example, the required minimum sampling rate should be 2 × {f

D}

max = 2.72 kHz. Therefore,

to show the complete micro-Doppler signature, the sampling rate should be higher than 2.72 kHz.

3.2.7 Analysis and Interpretation of the Micro-Doppler Signature of Rotor Blades

Compared to the Doppler spectral signature of helicopter rotors shown in Fig-ure 3.8, the micro-Doppler signature of rotating rotor blades is represented in


Figu

re 3

.21

The

mic

ro-D

oppl

er s

igna

ture

of a

roto

r with

two

blad

es o

n a

mod

el h

elic

opte

r.


Figu

re 3

.22

The

mic

ro-D

oppl

er s

igna

ture

s of

a ro

tatin

g tw

o-bl

ade

roto

r und

er d

iffer

ent s

ampl

ing

rate

s: (a

) 512

sam

ples

/sec

ond,

(b) 1

,024

sam

ples

/sec

ond,

an

d (c

) 2,0

48 s

ampl

es/s

econ

d.


the joint time-frequency domain to better explore the time-varying Doppler features. The micro-Doppler features of rotating rotor blades with two blades and three blades are depicted in Figures 3.23(a, b), respectively. It is obviously that the Doppler patterns of the even number of blades and the odd number of blades are different. Even-number blades generate a symmetric Doppler pat-tern around the mean Doppler frequency, but odd-number blades generate an asymmetric pattern around it. From the micro-Doppler signature of the rotor blades represented in the joint time-frequency domain, the number of blades, the length of the blades, the rotation rate of the blades, and the speed of the tip can be estimated. These features are important for the classification of an unknown helicopter.

Figure 3.23 The micro-Doppler features of rotating rotor blades with (a) two blades and (b) three blades.


Figure 3.24 is the micro-Doppler signature from the scale model-helicop-ter with an X-band radar. From its symmetric Doppler pattern, the helicopter has two blades. Based on the estimated blade rotation period of T

C = 0.43

second and the peak Doppler of {fD}

max = 195 Hz, the rotation rate Ω, blade

diameter 2 × L, and tip velocity Vtip

can be estimated. Table 3.1 lists a few features of different helicopters. These estimated fea-

ture parameters are important for classifying the type of an unknown helicopter.

3.3 Spinning Symmetric Top

A top stands steadily on a fixed tip point on its symmetric axis and quickly spins about the axis. If the spin axis is inclined, it will rotate sweeping out a vertical cone in a 3-D space as illustrated in Figure 3.25. This type of motion of is called the torque-induced precession. The angle between the symmetric axis and the vertical axis, called the precession axis angle, usually varies with time, and the symmetric axis is bobbing up and down, known as nutation. In mechanics, nutation refers to irregularities in the precession caused by the torque applied to the top.

Figure 3.24 The micro-Doppler signature from the scale model helicopter using the X-band radar.


The motion dynamics of a spinning top can be solved by Euler’s motion differential equations. When a rigid top rotates about an arbitrary axis with Eulerian angles ψ, θ, and ϕ in the body-fixed coordinates, the changing rate of

Figure 3.25 Precession of a spinning top.

Table 3.1 Main Rotor Features of Typical Helicopters

Typical HelicopterNumberof Blades Diameter (m)

RotationRate (r/s)

Tip Velocity (m/s)

AH-1 HUEY COBRA 2 14.63 4.9 227

AH-64 APACHE 4 14.63 4.8 221

UH-60 BLACK HAWK 4 16.36 4.3 221

CH-53 STALLION 7 24.08 2.9 223

MD 500E DEFENDER 5 8.05 8.2 207

A 109 AGUSTA 4 11.0 6.4 222

AS 332 SUPER PUMA 4 15.6 4.4 217

SA 365 DAUPHIN 4 11.94 5.8 218


these angles described by the Eulerian angles’ derivatives vector [ , ]Tψ θ ϕ= ,� is related to the angular velocity vector � = [Ω

1, Ω

2, Ω

3]T through a 3-by-3 Euler

angle transform matrix [15]

(3.41)

where Ω 1, Ω

2, and Ω

3 are the instantaneous components of the angular velocity

with respect to the body-fixed coordinates and T denotes the transposed vector, such that

(3.42)

or

(3.43)

The inverse Euler angle transform matrix T−1 is [15]

(3.44)

and

(3.45)

If an external torque exists, the angular momentum will change, and its changing rate equals the torque. For a symmetric top spinning about its sym-metric axis and the torque applied about the axis, the angular momentum is L = I • �. The torque � equals to the change rate of the angular momentum:

dL ddt dt

= = ⋅ �� I (3.46)

sin sin cos 0

cos sin sin 0

cos 0 1

ϕ θ ϕ

ϕ θ ϕ

θ

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

T

=� ��

1

2

3

sin sin cos

cos sin sin

cos

ϕ θ ψ ϕ θ

ϕ θ ψ ϕ θ

θ ψ φ

⎡ ⎤+Ω⎡ ⎤ ⎢ ⎥⎢ ⎥Ω = −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥Ω +⎣ ⎦ ⎢ ⎥⎣ ⎦

1

sin / sin cos / sin 0

cos sin 0

sin cos / sin cos cos / sin 1

ϕ θ ϕ θ

ϕ ϕ

φ θ θ φ θ θ

−

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥− −⎣ ⎦

�

1−=� � �


where τ = [τ1, τ

2, τ

3]T and the inertia tensor I can be a diagonal matrix

(3.47)

if the principal axes are used as the coordinate axes.If only the external torque component τ

3 is applied such that the Euler

angle ϕ increases, according to the Lagrangian mechanics [16], the Lagrangian equation is

(3.48)

where 2 2 21 1 2 2 3 3

1( )

2RotE I I I= Ω + Ω + Ω is the kinetic energy of the rotating top

given by (2.33). Based on (3.43), (3.46) becomes

3 3 1 2 1 2 3( ) ,I I I τΩ − − Ω Ω = which is the Euler equation for one of the principal axes. The whole Euler equations for principal axes are derived as the differential equations in (2.37):

3.3.1 Force-Free Rotation of a Symmetric Top

For a symmetric top, the principal moment I1 is equal to I

2. If there is no exter-

nal torque on the top, the symmetric top will be rotating about an arbitrary axis with an angular velocity vector � = [Ω

1, Ω

2, Ω

3]T, where Ω

1, Ω

2, and Ω

3 are the

instantaneous components of its angular velocity with respect to the principal axes. Then the Euler equations become

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

0 0

0 0

0 0

I

I

I

I

3Rot Rotd E E

dtτ

ϕ ϕ

⎛ ⎞− =⎜ ⎟

⎝ ⎠

∂ ∂∂ ∂

11 3 2 2 3 1

22 1 3 3 1 2

33 2 1 1 2 3

( )

( )

( )

dI I I

dtd

I I Idt

dI I I

dt

τ

τ

τ

Ω+ − Ω Ω =

Ω+ − Ω Ω =

Ω+ − Ω Ω =


( )

( )

11 3 2 2 3

22 1 3 3 1

33

0

0

0

dI I I

dtd

I I Idt

dI

dt

Ω+ − Ω Ω =

Ω+ − Ω Ω =

Ω=

(3.49)

They can be rewritten as

(3.50)

From the third equation, Ω3 must be a constant: Ω

3 = C. Differentiating

the first equation and substituting from the second equation, and differentiat-ing the second equation and substituting from the first one, two simple har-monic motion equations can be derived:

(3.51)

The solutions of these simple harmonic motion equations are

(3.52)

where 2 2 1/21 2( )iniΩ = Ω + Ω is the initial amplitude, Ψ

ini is the initial phase angle

at t = 0, and Ψ is the precession angular velocity defined by ψ = Ω3(I

3 − I

1)/I

1.

Equation (3.52) indicates that a force-free symmetric top will rotate about the principal axes with an angular velocity vector � = [Ω

1, Ω

2, Ω

3]T, where Ω

3 is a

constant and 2 2 1/21 2( )Ω + Ω is also a constant.

3 113 2

1

3 123 1

1

3

( )

( )

0

I Id

dt I

I Id

dt I

d

dt

⎡ ⎤−Ω= − Ω Ω⎢ ⎥

⎣ ⎦⎡ ⎤−Ω

= Ω Ω⎢ ⎥⎣ ⎦

Ω=

223 11

3 121

223 12

3 221

( )

( )

I Id

dt I

I Id

dt I

⎡ ⎤−Ω= − Ω Ω⎢ ⎥

⎣ ⎦

⎡ ⎤−Ω= − Ω Ω⎢ ⎥

⎣ ⎦

( )( )

1

2

3

cos

sin

ini ini

ini ini

t

t

C

Ω = Ω Ψ + Ψ

Ω = Ω Ψ + ΨΩ =


3.3.2 Torque-Induced Rotation of a Symmetric Top

Figure 3.26 illustrates a system model of a spinning symmetric top standing steadily on a fixed tip point of the top. The mass of the top is m and the princi-pal moments of inertia with respect to the fixed-body coordinates are I

1, I

2, and

I3. If the distance from the center of mass to the fixed tip point is L, then, under

gravitational force, the Euler differential equations become [15]:

(3.53)

where ϕ is the spinning angle and θ is the nutation angle. The angle ψ in Figure 3.27 is the precession angle.

In order to incorporate the top’s motion into the EM simulation, the set of Euler’s equations (3.53) must be solved. Thus, the nonlinear dynamics of the top motion can be obtained. Under the gravity force, the spinning top with a fixed standing point should have a precession motion about an axis. Given the

2 211 2 3 2 3

2 222 3 1 1 3

33 1 2 1 2

( ) ( ) cos sin

( ) ( ) sin sin

( )

dI mL I I mL mgL

dtd

I mL I I mL mgLdt

dI I I

dt

ϕ θ

ϕ θ

Ω+ = − + Ω Ω +

Ω+ = − − Ω Ω −

Ω= − Ω Ω

Figure 3.26 System model of a force-induced motion of a symmetric top.


mass of the top m = 25 kg, the distance between the center of mass (CM) and the fixed standing point L = 0.563m, the moments of inertia I

1 = I

2 = 0.117 kg

m2 and I3 = 8.5 kg m2, the initial angle θ

0 = 20°, the initial spinning velocity

dϕ0/dt = 3 × 2π rad/sec, the initial precession velocity dψ

0/dt = 0.5 × 2π rad/

sec, and the initial nutation velocity dθ0/dt = 0, Figure 3.27 shows the angular

velocities and the dynamic Euler angles. Figure 3.28 shows the position of the CM and the trajectory of the CM of the top motion.

3.3.3 RCS Model of a Symmetric Top

A symmetric top can be any symmetric geometric shape, such as a cone, a truncated cone (frustum), a cylinder, a sphere, or an ellipsoid. The mathemati-cal formulas for calculating the RCS of these simple geometric shapes can be found in [3].

For a truncated cone as illustrated in Figure 3.29, the half cone angle α is determined by 2 1tan

r r

hα

−= , where h is the height of the truncated cone or

frustum. The monostatic RCS of the frustum is [3]

Figure 3.27 (a) Angular velocities and (b) dynamic Euler angles of a spinning top.


Figu

re 3

.28

(a) P

ositi

on o

f the

CM

and

(b) t

raje

ctor

y of

the

CM o

f the

top

mot

ion.


3/2 3/2 22 1

4

2

8 ( ) sin(at normal incidence)

9 (cos )tan

[tan( )] (for non-normal incidence)8 sin

frustum

z z

RCSz

π α

λ αλ α

θ απ θ

⎧ −⎪⎪= ⎨⎪ −⎪⎩

(3.54)

where λ is the wavelength, and z1, z

2 are indicated in Figure 3.29.

3.3.4 Radar Backscattering from a Symmetric Top

The location and orientation of any point in a rotating symmetric top can be calculated by a rotation matrix and are time-varying. Based on the calculated location and orientation at each time instant, the RCS of the top can be cal-culated in terms of a RCS model. Then, giving radar parameters and a signal waveform, the returned signal from the spinning top can be calculated. If a coherent radar system transmits a sequence of narrow rectangular pulses with a transmitted frequency f

c, a pulse width Δ, and a pulse repetition interval ΔT,

the baseband signal in the radar receiver is

Figure 3.29 The geometry of a frustum.


{ } { }1

2 ( ) 2 ( )( ) ( ) exp 2

pn

B ck


c cσ π

=

= − Δ − −∑ (3.55)

where σ(t) is the RCS of the top, np is the total number of pulses received dur-

ing the observation time, and R(t) is the distance between the radar and the top at time t.

The RCS formula (3.54) of a truncated cone is used for calculating σ(t), and the distance R(t) is calculated from the radar to the center of mass of the top.

3.3.5 Micro-Doppler Signatures Generated by a Precession Top

Given a radar location at (X = 20m, Y = 0m, Z = 0m) and the tip of the top at (X = 0m, Y = 0m, Z = 0m), the geometric configuration of the radar and the top is illustrated in Figure 3.30.

After rearranging the np range profiles, the range profiles of the spinning

precession top are shown in Figure 3.31(a), and the micro-Doppler signature of the top is shown in Figure 3.31(b). In the range profiles, the rotating top can be seen around the range cell no. 667 or about 20m of the distance from the radar. The micro-Doppler signature of the spinning top is obtained from the summation of those range profiles that are within the range gate around 20m of range, where the top is located. The joint time-frequency transform used to generate the signature is the simple STFT. The MATLAB code for calculating radar backscattering from a spinning top is listed in Appendix 3A.

3.3.6 Analysis and Interpretation of the Micro-Doppler Signature of a Precession Top

During the radar observation time interval, the simulated spinning and pre-cession top completed one cycle of precession and 27 cycles of nutation. The

Figure 3.30 Geometric confi guration of a radar and the top.


Doppler modulation by precession and nutations is shown clearly in the micro-Doppler signature in Figure 3.32. The RCS produced by the upper circular plate is not significant. However, the reflection from the edge of the upper circular plate, which is an interesting feature, is not considered in the simple RCS model.

A more accurate simulation that shows the edge reflection of the upper circular plate can be found in [17, 18], where the mass of the top is m = 25 kg, the distance between the center of mass and the fixed tip point is L = 0.563m, the moments of inertia I

1 = I

2 = 0.117 kg m2 and I

3 = 8.5 kg m2, and the initial

Figure 3.31 (a) Range profi les of the spinning precession top, and (b) the micro-Doppler sig-nature of the top.


nutation angle θ0 = 20°. The radar is an X-band radar at a 10-GHz transmitted

frequency and a 500-MHz bandwidth, located at a 12-m distance from the tip of the top. The more accurate RCS prediction model utilizes a multitude of dif-ferent backscattering algorithms, including geometrical optics, physical optics, the physical theory of diffraction, and the method of moments solutions.

The micro-Doppler signature of the above spinning top is shown in Fig-ure 3.33. From the signature, about one Doppler modulation cycle of the pre-cession during the observation time of 5.3 seconds can be seen clearly. There are also 12.5 Doppler modulation cycles of nutations as shown in Figure 3.33, and the Doppler modulation produced by the top upper circular plate disk is marked in Figure 3.33.

It should be emphasized that, in a precession top, the inertia ratio I1/I

3 is

an important characteristic. It can be estimated from the measured precession angular velocity, the spinning angular velocity, and the precession angle.

Figure 3.32 The micro-Doppler signature of a simulated precession and nutation top.


3.4 Wind Turbines

Since the use of wind energy has been dramatically increasing, the large num-bers of wind turbines and the large RCS of the wind turbine blade become challenges to current radar systems as illustrated in Figure 3.34. A typical tur-bine could have a RCS in the order of 60 dBms (or 106 m2) at the X-band [19]. Impacts of wind turbines on radar performance, including air traffic control systems, navigation systems, weather radar systems, and other primary or sec-ondary radar systems, have been investigated and reported [19–28].

A wind turbine consists of a tower, a power-generating nacelle, and tur-bine blades. The power-generating nacelle slowly rotates its direction to enable turbine blades to face to the wind. Even if it is slow rotating, the nacelle can still be considered a virtually stationary object. The actually moving parts of the wind turbine are the turbine blades. The blade is a large, aerodynamically shaped structure that operates like a rotor blade of a helicopter. Its motion ki-nematic and dynamic properties are similar to those of helicopter rotor blades. Thus, the mathematical model, motion dynamics, and EM scattering model of helicopter rotor blades are also suitable to the wind turbines.

Figure 3.33 The micro-Doppler signature produced by a spinning, precession, and nutation top with an upper circular plate disk. (After: [17].)


The Doppler frequency shift produced by the rotating blades of wind turbines has an impact on the ability of radars to discriminate the wind turbine from a flying aircraft. Even if the rotor rotation rate is low, however, a large blade diameter makes its tip velocity falling in the range of 50 to 150 m/s, which is within a speed range of an aircraft. The large physical size of the blade produces a substantial RCS and a broader spectrum. Consequently, the wind turbine blades viewing from radars appear as a moving aircraft.

3.4.1 Micro-Doppler Signatures of Wind Turbines

The wind turbine has a unique EM signature even that varies with environmen-tal conditions [21, 27, 28]. Using the PO facet method, the micro-Doppler sig-nature of a rotating three-blade turbine rotor is shown in Figure 3.35, where the length of the blade is 30m and the width of the blade is 1m, and the rotation rate is 0.25 r/s or rotating one cycle in 4 seconds. In the micro-Doppler signa-ture of the wind turbine blades, flashes at the receding and approaching points can be seen. There are a total of six flashes from three blades in 4 seconds. The interval between two flashes produced by two successive blades is 1.33 seconds.

Compared with a single turbine, the micro-Doppler signatures from multiple wind turbines are much more complicated. All turbines in a wind farm may not be aligned to the same direction and turbine directions may vary widely. The effects of multiple turbines on the RCS are beyond the purpose of this book.

3.4.2 Analysis and Interpretation of the Micro-Doppler Signature of Wind Turbines

Similar to the micro-Doppler signatures of helicopter rotors in Section 3.2.5, the micro-Doppler signature of wind turbines has strong components near zero

Figure 3.34 Wind turbines observed by radar systems.


Doppler due to strong the stationary reflections from the tower, nacelle, and other ground clutter. Its RCS is much higher than the RCS of the helicopter’s rotor blades. However, the oscillation rate is much lower than the helicopter rotors. It is easy to distinguish the micro-Doppler signatures of wind turbines from that of helicopters, especially when a helicopter is moving and the center line of its micro-Doppler signature is shifted from zero Doppler.

In addition, the micro-Doppler signatures of wind turbines can also have Doppler components of multiple bounces. The multibounce effect may occur if radar waves are reflected off two different surfaces before returning to the ra-dar receiver. In wind turbines, the multibounce occurs while radar-transmitted waves are reflected from large turbine blades to the turbine tower and then again to the blades before returning to the radar receiver.

References

[1] Knott, E. F., J. F. Schaffer, and M. T. Tuley, Radar Cross Section, 2nd ed., Norwood, MA: Artech House, 1993.

Figure 3.35 The micro-Doppler signature of a rotating three-blade turbine rotor.


[2] Shirman, Y. D., (ed.), Computer Simulation of Aerial Target Radar Scattering, Recognition, Detection, and Tracking, Norwood, MA: Artech House, 2002.

[3] Mahafza, B., Radar Systems Analysis and Design Using MATLAB, London, U.K.: Chapman & Hall/CRC, 2000.

[4] Youssef, N., “Radar Cross Section of Complex Targets,” Proc. of IEEE, Vol. 77, No. 5, 1989, pp. 722–734.

[5] MacKenzie, J. D., et al., “The Measurement of Radar Cross Section,” Proceedings of the Military Microwaves ’86 Conference, June 24–26, 1986, pp. 493–500.

[6] Shi, N. K., and F. Williams, “Radar Detection and Classification of Helicopters,” U.S. Patent No. 5,689,268, November 18, 1997.

[7] Chen, V. C., “Radar Signatures of Rotor Blades,” Proceedings of SPIE on Wavelet Applica-tions VIII, Vol. 4391, 2001, pp. 63–70.

[8] Martin, J., and B. Mulgrew, “Analysis of the Theoretical Radar Return Signal from Air-craft Propeller Blades,” IEEE 1990 International Radar Conference, 1990, pp. 569–572.

[9] Misiurewicz, J., K. Kulpa, and Z. Czekala, “Analysis of Recorded Helicopter Echo,” IEE Radar 97, Proceedings, 1997, pp. 449–453.

[10] Bullard, B. D., and P. C. Dowdy, “Pulse Doppler Signature of a Rotary-Wing Aircraft,” IEEE AES Systems Magazine, May 1991, pp. 28–30.

[11] Anderson, W. C., The Radar Cross Section of Perfectly Conducting Rectangular Flat Plates and Rectangular Cylinders: A Comparison of Physical Optics, GTD and UTD Solutions, Technical Report ERL-0344-TR DSTO, Australia, 1985.

[12] Chatzigeorgiadis, F., “Development of Code for Physical Optics Radar Cross Section Prediction and Analysis Application,” Master’s Thesis, Naval Postgraduate School, Monterey, CA, September 2004.

[13] Chatzigeorgiadis, F., and D. Jenn, “A MATLAB Physical-Optics RCS Prediction Code,” IEEE Antennas & Propagation Magazine, Vol. 46, No. 4, 2004, pp. 137–139.

[14] Garrido, E. E., “Graphical User Interface for Physical Optics Radar Cross Section Prediction Code,” Master’s Thesis, Naval Postgraduate School, Monterey, CA, September 2000.

[15] Goldstein, H., Classical Mechanics, 2nd ed., Reading, MA: Addison-Wesley, 1980.

[16] Trindade, M., and R. Sampaio, “On the Numerical Integration of Rigid Body Nonlinear Dynamics in Presence of Parameters Singularities,” Journal of the Brazilian Society of Mechanical Sciences, Vol. 23, No. 1, 2001.

[17] Chen, V. C., C. -T. Lin, and W. P. Pala, “Time-Varying Doppler Analysis of Electromagnetic Backscattering from Rotating Object,” The IEEE Radar Conference Record, Verona, NY, April 24–27, 2006, pp. 807–812.


[19] Rashid, L. S., and A. K. Brown, “Impact Modeling of Wind Farms on Marine Navigational Radar,” IET 2007 International Conference on Radar Systems, Edinburgh, U.K., October 5–18, 2007.


[20] Casanova, A. C., et al., “Wind Farming Interference Effects,” 2008 5th International Multi-Conference on Systems, Signals, and Devices, Amman-Jordan, July 20–23, 2008.

[21] Darcy, F., and D. de la Vega, “A Methodology for Calculating the Interference of a Wind Farm on Weather Radar,” 2009 Loughborough Antennas & Propagation Conference, 2009, pp. 665–667.

[22] Spera, D. A., (ed.), Wind Turbine Technology, Ch. 9, New York: The American Society of Mechanical Engineers, 1998.

[23] The Effect of Windmill Farms on Military Readiness, Offi ce of the Director of Defense Research and Engineering, Report to the Congressional Defense Committees, U.S. Department of Defense, 2006.

[24] Theil, A., and L. J. van Ewijk, “Radar Performance Degradation Due to the Presence of Wind Turbines,” IEEE 2007 Radar Conference, April 17–20, 2007, pp. 75–80.

[25] Johnson, K., et al., “Data Collection Plans for Investigating the Effect of Wind Farms on Federal Aviation Administration Air Traffi c Control Radar Installations,” Technical Memorandum OU/AEC 05-19TM 00012/4-1, Avionics Engineering Center, Ohio University, Athens, OH, January 2006.

[26] Feasibility of Mitigating the Effects of Wind Farms on Primary Radar, Alenia Marconi Systems Ltd., Report W/14/00623/REP, June 2003.

[27] Kent, B. M., et al., “Dynamic Radar Cross Section and Radar Doppler Measurements of Commercial General Electric Windmill Power Turbines Part 1: Predicted and Measured Radar Signatures,” IEEE Antennas & Propagation Magazine, Vol. 50, No. 2, 2008, pp. 211–219.

[28] Dabis, H. S., “Wind Turbine Electromagnetic Scatter Modeling Using Physical Optics Techniques,” Renewable Energy, Vol. 16, 1999, pp. 882–887.

Appendix 3A

MATLAB Source Codes

RadarPendulumReturns.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Radar returns from an oscillating pendulum%% Radar: X-band wavelength = 0.03 m% Range resolution 0.05 m% Observation time: T = 10 sec% Number of pulses 8192% Location: X = 10 m; Y = 0 m; Z = 2 m% Pendulum: Pivot point location X = 0; Y = 0; Z = 2 m% String length L = 1.5 m% Small bob size Lc = 0.3 m% Pendulum Mode: [1] free oscillating% [2] damping% [3] damping and driving%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all j = sqrt(-1); % radar parameters% time samplingT = 10; % time durationnt = 8192; % number of samplesdt = T/nt; % time intervalts = [0:dt:T-dt]; % time spanlambda = 0.03; % wavelength of transmitted radar signalc = 2.99792458e8;f0 = c/lambda;rangeres = 0.05; % designed range resolutionradarloc = [10, 0, 2]; % radar location% total number of range binsnr = floor(2*sqrt(radarloc(1)^2+radarloc(2)^2+radarloc(3)^2)/rangeres);

% Pendulum Motion Modelmode = input(‘[1]Free oscillating; [2]Damping; [3]Damping & Driving: ‘);

% initial valuetheta0 = -pi/8; % initial angleomega0 = 0; % initial angular velocitiesinitial_value = [theta0 omega0]; % calculate ODEif mode == 1 [t,x] = ode45(‘PendulumDynamic’,ts,initial_value);elseif mode == 2 [t,x] = ode45(‘PendulumDampingDynamic’,ts,initial_value);elseif mode == 3 [t,x] = ode45(‘PendulumDampingDrivingDynamic’,ts,initial_value);end % rotating angle - pitching theta = x(:,1); % angular velocityomega = x(:,2); % Pendulum geometric parametersL = 1.5; % string lengthLc = 0.3; % small bob size% ellipsoid parameterA = 0.15; B = 0.15; C = 0.15;% cylinder parameterRc = 0.15;H = Lc;RCS = 1; % point scatterer model % handle pendulum rotationx0 = zeros(1,nt);


y0 = zeros(1,nt);z0 = 2.0*ones(1,nt);centerpoint = [x0; y0; z0]; % string originpsi = zeros(1,nt); % rolltheta = theta’; % pitchphi = zeros(1,nt); % yaw % radar returns% prepare data collectiondata = zeros(nr,nt);for k = 1:nt Rxyz = RPYConvention(phi(k),theta(k),psi(k)); % Mass center: RotationMatrix + TranslationMatrix Pendul(:,k) = Rxyz’*[L*sin(theta(k)); 0; -L*cos(theta(k))]+... [centerpoint(1,k); centerpoint(2,k);centerpoint(3,k)]; % Upper center: RotationMatrix + TranslationMatrix PT1(:,k) = Rxyz’*[(L+Lc/2)*sin(theta(k)); ... 0; -(L+Lc/2)*cos(theta(k))]+... [centerpoint(1,k); centerpoint(2,k);centerpoint(3,k)]; % Lower center: RotationMatrix + TranslationMatrix PT2(:,k) = Rxyz’*[(L-Lc/2)*sin(theta(k)); ... 0; -(L-Lc/2)*cos(theta(k))]+... [centerpoint(1,k); centerpoint(2,k);centerpoint(3,k)]; % distance from radar to target center r_dist(1,k) = abs(Pendul(1,k)-radarloc(1)); r_dist(2,k) = abs(Pendul(2,k)-radarloc(2)); r_dist(3,k) = abs(Pendul(3,k)-radarloc(3)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k)); % aspect vector of the object aspct(1,k) = PT2(1,k)-Pendul(1,k); aspct(2,k) = PT2(2,k)-Pendul(2,k); aspct(3,k) = PT2(3,k)-Pendul(3,k); % calculate theta angle % vector 1: radar to target center V1 = [radarloc(1)-Pendul(1,k);... radarloc(2)-Pendul(2,k);radarloc(3)-Pendul(3,k)]; % vector 2: target orientation V2 = [aspct(1,k); aspct(2,k); aspct(3,k)]; V1dotV2 = dot(V1,V2,1); ThetaAngle(k) = acos(V1dotV2/(norm(V1)*norm(V2))); if(sign(V1dotV2)==-1) ThetaAngle(k) = -ThetaAngle(k); end PhiAngle(k) = phi(k); rcs(k) = RCS; amp(k) = sqrt(rcs(k));


PHs = amp(k)*(exp(j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end figure(1)for k = 1:12:nt clf hold on colormap([0.7 0.7 0.7]) % draw a ball [X,Y,Z] = ellipsoid(Pendul(1,k),Pendul(2,k),Pendul(3,k),A,B,C,30); surf(X,Y,Z) light lighting gouraud shading interp axis equal axis([-1,10,-1,1,0,2]) grid on set(gcf,’Color’,[1 1 1]) view([45,20]) xlabel(‘X’) ylabel(‘Y’) zlabel(‘Z’) title(‘Oscillating Pendulum’) % draw the string patch([centerpoint(1,k) Pendul(1,k)],[centerpoint(2,k) Pendul(2,k)],... [centerpoint(3,k) Pendul(3,k)],’k’,’linewidth’,1) % draw radar location plot3(radarloc(1),radarloc(2),radarloc(3),’-ro’,... ‘LineWidth’,2,... ‘MarkerEdgeColor’,’r’,... ‘MarkerFaceColor’,’y’,... ‘MarkerSize’,10) % draw a line from radar to the target center line([radarloc(1) Pendul(1,k)],[radarloc(2) Pendul(2,k)],... [radarloc(3) Pendul(3,k)],... ‘color’,[0.4 0.7 0.7],’LineWidth’,1.5,’LineStyle’,’-’) plot3(x0(1),y0(1),z0(1),’-ro’,... ‘LineWidth’,1,... ‘MarkerEdgeColor’,’k’,... ‘MarkerFaceColor’,’k’,... ‘MarkerSize’,5) drawnow end figure(2)colormap(jet(256))imagesc([1,nt],[0,nr*rangeres],20*log10(abs(data)+eps))xlabel(‘Pulses’)ylabel(‘Range (m)’)


title(‘Range Profiles of an Oscillating Pendulum’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-20 10]);colorbardrawnow % micro-Doppler signature x = sum(data);np = nt; dT = T/length(ts);F = 1/dT;dF = 1/T; wd = 512;wdd2 = wd/2;wdd8 = wd/8;ns = np/wd; % calculate time-frequency micro-Doppler signaturedisp(‘Calculating segments of TF distribution ...’)for k = 1:ns disp(strcat(‘ segment progress: ‘,num2str(k),’/’,num2str(round(ns)))) sig(1:wd,1) = x(1,(k-1)*wd+1:(k-1)*wd+wd); TMP = stft(sig,16); TF2(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);endTF = TF2;disp(‘Calculating shifted segments of TF distribution ...’)TF1 = zeros(size(TF));for k = 1:ns-1 disp(strcat(‘ shift progress: ‘,num2str(k),’/’,num2str(round(ns-1)))) sig(1:wd,1) = x(1,(k-1)*wd+1+wdd2:(k-1)*wd+wd+wdd2); TMP = stft(sig,16); TF1(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);enddisp(‘Removing edge effects ...’)for k = 1:ns-1 TF(:,k*wdd8-8:k*wdd8+8) = ... TF1(:,(k-1)*wdd8+wdd8/2-8:(k-1)*wdd8+wdd8/2+8);end % display final time-frequency signature figure(3)colormap(jet(256))imagesc([0,T],[-F/2,F/2],20*log10(fftshift(abs(TF),1)+eps))xlabel(‘Time (s)’)ylabel(‘Doppler (Hz)’)title(‘Micro-Doppler Signature of an Oscillating Pendulum’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-60 0]);


colorbardrawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

RadarRectBladeReturns.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Radar returns from rectangular rotor blades%% Radar: C-band wavelength = 0.06 m% Range resolution 0.5 m% Observation time: T = 1 sec% Number of pulses 10240% Location: X = 500 m; Y = 0 m; Z = 500 m% Rotor: Center location X = 0; Y = 0; Z = 0 m% Blade length L = 6 m (L1:0.5 - L2:6.5), wide W = 1 m% Rotation rate: 4 r/s%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all c = 2.99792458e8;j = sqrt(-1); % radar parametersradarloc = [500,0,500]; % radar locationxr = radarloc(1);yr = radarloc(2);zr = radarloc(3);rangeres = 0.5; % (m): designed range resolution% total number of range binsnr = floor(2*norm(radarloc)/rangeres); % time samplingT = 1; % time durationnt = 10240; % number of time samplests = linspace(0,T,nt); % time span f0 = 5e9;lambda = c/f0; % rotor bladesNb = 2; % number of blades (2 or 3)L = 6; % long (from 0.5 to 6.5)W = 1; % wideL1 = 0.5;L2 = 6.5;a = (L2-L1)/2;b = W/2; rotorloc = [0,0,0]; % rotor center location% direction of radar with respect to rotor center


radar_direction = atan2(radarloc(2)-rotorloc(2),radarloc(1)-rotorloc(1));

Omega = 4*2*pi; %Rotation rate % rotor blades’ rotation centerx0 = 0;y0 = 0;z0 = 0;rotor_center = [x0;y0;z0]; for k=1:nt psi(k) = 0; theta(k) = 0; phi(k) = Omega*ts(k);end data = zeros(nr,nt);for k = 1:nt % blade 1 Rxyz = XConvention(psi(k),theta(k),phi(k)); CM1 = Rxyz*[6.5;0;0]; xcm1 = CM1(1); ycm1 = CM1(2); zcm1 = CM1(3); % distance distance1 = sqrt((xr-xcm1)^2+(yr-ycm1)^2+(zr-zcm1)^2); % theta and phi angles Theta1 = atan2(sqrt((xr-xcm1)^2+(yr-ycm1)^2),zr-zcm1); Phi1 = -atan2(yr-ycm1,xr-xcm1); % triangle facet rcs1 = rcs_rect(a,b,Theta1, Phi1,f0); amp1 = sqrt(rcs1); PHs1 = amp1*(exp(j*4*pi*distance1/lambda)); data(floor(distance1/rangeres),k) = ... data(floor(distance1/rangeres),k)+ PHs1; % blade 2 Rxyz = XConvention(psi(k),theta(k),phi(k)+2*pi/Nb); CM2 = Rxyz*[6.5;0;0]; xcm2 = CM2(1); ycm2 = CM2(2); zcm2 = CM2(3); % distance distance2 = sqrt((xr-xcm2)^2+(yr-ycm2)^2+(zr-zcm2)^2); % theta and phi angles Theta2 = atan2(sqrt((xr-xcm2)^2+(yr-ycm2)^2),zr-zcm2); Phi2 = -atan2(yr-ycm2,xr-xcm2); % triangle facet rcs2 = rcs_rect(a,b,Theta2, Phi2,f0); amp2 = sqrt(rcs2); PHs2 = amp2*(exp(j*4*pi*distance2/lambda)); data(floor(distance2/rangeres),k) = ... data(floor(distance2/rangeres),k)+ PHs2; if Nb == 3


% blade 3 Rxyz = XConvention(psi(k),theta(k),phi(k)+2*2*pi/Nb); CM3 = Rxyz*[6.5;0;0]; xcm3 = CM3(1); ycm3 = CM3(2); zcm3 = CM3(3); % distance distance3 = sqrt((xr-xcm3)^2+(yr-ycm3)^2+(zr-zcm3)^2); % theta and phi angles Theta3 = atan2(sqrt((xr-xcm3)^2+(yr-ycm3)^2),zr-zcm3); Phi3 = -atan2(yr-ycm3,xr-xcm3); % triangle facet rcs3 = rcs_rect(a,b,Theta3, Phi3,f0); amp3 = sqrt(rcs3); PHs3 = amp3*(exp(j*4*pi*distance3/lambda)); data(floor(distance3/rangeres),k) = ... data(floor(distance3/rangeres),k)+ PHs3; endend figure(1)rngpro = 20*log10(abs(data)+eps);imagesc([1 nt],[nr/2-50 nr/2+50],rngpro(nr/2-50:nr/2+50,:))xlabel(‘Pulses’)ylabel(‘Range cells’)title(‘Range Profiles’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-50 0]);colorbardrawnow % micro-Doppler signaturef = sum(data);np = nt; dT = T/length(ts);F = 1/dT;dF = 1/T; wd = 512;wdd2 = wd/2;wdd8 = wd/8;ns = np/wd; % calculate time-frequency micro-Doppler signaturedisp(‘Calculating segments of TF distribution ...’)for k = 1:ns disp(strcat(‘ segment progress: ‘,num2str(k),’/’,num2str(round(ns))))

sig(1:wd,1) = f(1,(k-1)*wd+1:(k-1)*wd+wd); TMP = stft(sig,16); TF2(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);endTF = TF2;disp(‘Calculating shifted segments of TF distribution ...’)


TF1 = zeros(size(TF));for k = 1:ns-1 disp(strcat(‘ shift progress: ‘,num2str(k),’/’,num2str(round(ns-1)))) sig(1:wd,1) = f(1,(k-1)*wd+1+wdd2:(k-1)*wd+wd+wdd2); TMP = stft(sig,16); TF1(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);enddisp(‘Removing edge effects ...’)for k = 1:ns-1 TF(:,k*wdd8-8:k*wdd8+8) = ... TF1(:,(k-1)*wdd8+wdd8/2-8:(k-1)*wdd8+wdd8/2+8);end % display final time-frequency signature figure(2)colormap(jet)imagesc([0,T],[-F/2,F/2],20*log10(fftshift(abs(TF),1)+eps))xlabel(‘Time (s)’)ylabel(‘Doppler (Hz)’)title(‘Micro-Doppler Signature of Rotating Rotor Blades’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-50 0]);colorbardrawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

RadarTopReturns.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Radar returns from a spinning and precession top%% Radar: X-band wavelength = 0.03 m% Range resolution 0.03 m% Observation time: T = 8 sec% sampling interval: at = 0.005 sec % Number of samples: T/dt = 1600 % Location: X = 20 m; Y = 0 m; Z = 0 m% Top: Center location X = 0; Y = 0; Z = 0 m% I1 = 0.117 kg m^2: inertia% I2 = 0.117 kg m^2% I3 = 8.5 kg m^2% m = 25 kg% g = 9.8 m/sec^2% L = 0.563 m (position of center of mass)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all c = 2.99792458e8;j = sqrt(-1);


% time samplingT = 8; % time durationdt = 0.005; % sampling intervalts = [0:dt:T-dt]; % time spannt = length(ts); % number of time samples % radar parameterslambda = 0.03; % wavelength of transmitted radar signalf0 = c/lambda;rangeres = 0.03; % (m): designed range resolutionradarloc = [20,0,0]; % (x,y,z):radar location % total number of range binsnr = floor(2*norm(radarloc)/rangeres); % initial Euler anglespsi0 = 0; %precession theta0 = pi/9; %nutation phi0 = 0; %spin % Eulerangle transform matrixTm = EulerTransfMatrix(psi0,theta0,phi0); % initial angular velocitiesdpsi = pi; % rad/secdtheta = 0; % rad/secdphi = 6*pi; % rad/secOmega = Tm*[dpsi; dtheta; dphi]; initial_value = [Omega(1); Omega(2); Omega(3); psi0; theta0; phi0]; % set and calculate ODEoptions = odeset(‘RelTol’,1e-4,’AbsTol’, ... [1e-4 1e-4 1e-4 1e-4 1e-4 1e-4]);[t,x] = ode45(‘TopDynamic’,ts,initial_value,options); % x(1): Omega(1); x(2): Omega(2); x(3): Omega(3); % x(4): psi; x(5): theta; x(6):phipsi = x(:,4)’; % precession angletheta = x(:,5)’; % nutation anglephi = x(:,6)’; % spin angle % Position of center of massL = 0.563; % m % top rotationx0 = zeros(1,nt);y0 = zeros(1,nt);z0 = zeros(1,nt);centerpoint = [x0;y0;z0]; for k = 1:nt Rxyz = XConvention(psi(k),theta(k),phi(k)); %Rxyz = RPYConvention(psi(k),theta(k),phi(k));


% Mass center at L CM(:,k) = Rxyz*[0; 0; L]; % Bottom tip pointer at [0;0;0] PT1(:,k) = Rxyz*[0;0;0]; % Top pointer at [0;0;3*L] PT2(:,k) = Rxyz*[0;0;3*L]; % Top pointer at [0;0;3*L+0.01] PT3(:,k) = Rxyz*[0;0;3*L+0.01]; end figure(1)plot(t,CM(1,:),t,CM(2,:),t,CM(3,:))axis([0 max(ts) -Inf Inf])axis ‘auto y’xlabel(‘Time’)ylabel(‘Position of CM’)legend(‘Px’,’Py’,’Pz’,’Location’,’EastOutside’)drawnow figure(2)hold ontitle(‘Trajectory of CM’)view(50,25);plot3(CM(1,:),CM(2,:),CM(3,:))grid onaxis([-0.5 0.5 -0.5 0.5 0 1])xlabel(‘x’)ylabel(‘y’)zlabel(‘z’)plot3(0,0,0,’-ro’,’LineWidth’,2,’MarkerEdgeColor’,’r’,... ‘MarkerFaceColor’,’y’,’MarkerSize’,10)drawnow % calculate radar returns% mass center at L xt0 = CM(1,:);yt0 = CM(2,:);zt0 = CM(3,:); xt1 = PT1(1,:);yt1 = PT1(2,:);zt1 = PT1(3,:); xt2 = PT2(1,:);yt2 = PT2(2,:);zt2 = PT2(3,:); xt3 = PT3(1,:);yt3 = PT3(2,:);zt3 = PT3(3,:); data = zeros(nr,nt);for k = 1:nt


% distance from radar to object’s mass center r_dist(1,k) = abs(xt0(k)-radarloc(1)); r_dist(2,k) = abs(yt0(k)-radarloc(2)); r_dist(3,k) = abs(zt0(k)-radarloc(3)); distances(k) = norm(r_dist(:,k)); % aspect vector of the object aspct(1,k) = xt2(k)-xt0(k); aspct(2,k) = yt2(k)-yt0(k); aspct(3,k) = zt2(k)-zt0(k); % calculate theta angle % vector 1: radar to mass center V1 = [radarloc(1)-xt0(k);... radarloc(2)-yt0(k); radarloc(3)-zt0(k)]; % vector 3: radar to upper point V3 = [radarloc(1)-xt2(k);... radarloc(2)-yt2(k); radarloc(3)-zt2(k)]; % vector 2: object orientation V2 = [aspct(1,k); aspct(2,k); aspct(3,k)]; V1dotV2(k) = dot(V1,V2,1); ThetaAngle(k) = acos(V1dotV2(k)/(norm(V1)*norm(V2))); V1dotV3(k) = dot(V1,V3,1); ThetaAngle3(k) = acos(V1dotV3(k)/(norm(V1)*norm(V3))); PhiAngle(k) = rem(phi(k),2*pi); % circular plate rcs2(k) = rcscircplate(1.0,ThetaAngle3(k),f0); % frustum if V1dotV2(k)>=0 indicator =1; else indicator = 0; end rcs1(k) = rcsfrustum(0.01, 1.0, 3*L,ThetaAngle(k),f0,indicator); rcs(k) = rcs2(k) + rcs1(k); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(i*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end figure(3)colormap([0.7 0.7 0.7])for k = 1:6:nt clf hold on % draw a cylinder object


[X,Y,Z] = cylinder3([xt2(k),yt2(k),zt2(k)],... [0,0,0],[1.0 0.01],30); surf(X,Y,Z) % draw a circular plate [X2,Y2,Z2] = circularplate([xt2(k),yt2(k),zt2(k)],... [xt3(k),yt3(k),zt3(k)],[1.0 0.01],30); surf(X2,Y2,Z2) light lighting gouraud shading interp axis equal axis([-2,20,-2,2,-0.5,2.0]) grid on set(gcf,’Color’,[1 1 1]) view([40,15]) xlabel(‘X’) ylabel(‘Y’) zlabel(‘Z’) title(‘Top Precession’) % draw radar location plot3(radarloc(1),radarloc(2),radarloc(3),’-ro’,’LineWidth’,2,... ‘MarkerEdgeColor’,’r’,’MarkerFaceColor’,’y’,... ‘MarkerSize’,10) % draw a line from radar to the target center line([radarloc(1) xt0(k)],[radarloc(2) yt0(k)],... [radarloc(3) zt0(k)],... ‘color’,’red’,’LineWidth’,1.5,’LineStyle’,’-.’) text(20,0,1,’Radar’) drawnowend % range profilesfigure(4)colormap(jet(256))imagesc(20*log10(abs(data)+eps))xlabel(‘pulses’)ylabel(‘range cells’)title(‘Range Profiles of a Precession Top’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-20 0]);colorbardrawnow % micro-Doppler signaturex = sum(data);% up-samplingx1 = x;f = interp(x1,2); % interp itnp = nt*2; dT = T/length(ts);F = 1/dT;dF = 1/T;


wd = 512;wdd2 = wd/2;wdd8 = wd/8;ns = np/wd; % calculate time-frequency micro-Doppler signaturedisp(‘Calculating segments of TF distribution ...’)for k = 1:ns disp(strcat(‘ segment progress:’,num2str(k),’/’, ... num2str(round(ns)))) sig(1:wd,1) = f(1,(k-1)*wd+1:(k-1)*wd+wd); TMP = stft(sig,16); TF2(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);endTF = TF2;disp(‘Calculating shifted segments of TF distribution ...’)TF1 = zeros(size(TF));for k = 1:ns-1 disp(strcat(‘ shift progress: ‘,num2str(k),’/’, ... num2str(round(ns-1)))) sig(1:wd,1) = f(1,(k-1)*wd+1+wdd2:(k-1)*wd+wd+wdd2); TMP = stft(sig,16); TF1(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);enddisp(‘Removing edge effects ...’)for k = 1:ns-1 TF(:,k*wdd8-8:k*wdd8+8) = ... TF1(:,(k-1)*wdd8+wdd8/2-8:(k-1)*wdd8+wdd8/2+8);end % display final time-frequency signaturefigure(5)colormap(jet(256))imagesc([0,T],[-F/2,F/2],20*log10(fftshift(abs(TF),1)+eps))xlabel(‘Time (s)’)ylabel(‘Doppler (Hz)’)title(‘Micro-Doppler Signature of A Precession Top’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-20 0]);colorbardrawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

157

4The Micro-Doppler Effect of the Nonrigid Body MotionThe nonrigid body is a deformable body, that is, the distance between two points in the body could vary during body motion and thus the shape of the body could be changed. However, as mentioned in Chapter 2, when studying radar scattering from a nonrigid body motion, the body can be modeled as jointly connected rigid links or segments, and a nonrigid body motion can be treated as multiple rigid bodies’ motion.

The human gait has been studied in biomedical engineering, sports medi-cine, physiotherapy, medical diagnosis, and rehabilitation [1]. Motivated by athletic performance analysis, visual surveillance, and biometrics, the methods of how to extract and analyze various human body movements have attracted much attention. The most commonly used method for human movement anal-ysis uses visual image sequences [2]. However, visual perception of the human body motion can be affected by distance, variations in lighting, deformations of clothing, and occlusions on the appearance of human body segments. Radar, as an EM sensor, has been widely used for detecting, tracking, and imaging targets of interest because of its long-range capability, excellent day and night performance, and ability to penetrate wall and ground. Thus, radar has become a tool for detecting and tracking humans and animals.

Beside human body motion, animal motion is also an important nonrigid body motion. Compared to human bipedal motion, the four-legged animal’s motion has more choices for its feet striking the ground. In 1887, E. Muybridge documented animal locomotion using photography and published a book on animal locomotion that showed how lions, donkeys, dogs, deer, and elephants strode and ran [3]. Later, based on the understanding of animal locomotion,


legged machines appeared [4]. Due to the fixed motion pattern, the perfor-mance of such legged machines was very limited. Realizing the insufficiency of fixed motion patterns, a better walking machine using controlled legs was constructed [5].

With the understanding of the locomotion, the dynamic/kinematic char-acteristics and movement patterns of human and animal motions became hot topics in computer vision and computer graphics [2]. Point-light displays were used to demonstrate animated patterns of human and animal movements. Ob-servers can certainly identify human and animal motions through a limited number of animated point-light displays [6].

Radar has proven its ability to detect targets with small RCSs, such as humans and animals. However, the methods of how to analyze human or ani-mal dynamic/kinematic characteristics and extract motion patterns from radar returns are a challenge.

As a matter of fact, in most radar range-Doppler imagery, the Doppler modulation induced by the target’s rotation, vibration, or human body loco-motion has been observed; these show up as characteristic Doppler frequency distributions at those range cells that correspond to locations of these micro motion sources. Examples of such sources include the rotating antennas on a ship, the rotor blades of a helicopter, the swinging arms and legs of a human, or another oscillatory motion characteristic in a target. To generate a clear ra-dar imagery of a moving target, motion compensation and image autofocusing algorithms must be used to remove target translational motion and oscillatory motion components for reducing the induced Doppler distributions in the ra-dar imagery.

However, in order to extract vibration, rotation, or locomotion charac-teristics in radar returns, the induced Doppler distributions should not be re-moved. Instead, they should be further exploited. During the past decade, the radar micro-Doppler signatures of targets have been studied from experimental observations to theoretical analysis [7–14]. As discussed in Chapter 1, micro-Doppler signatures are represented in a joint time-frequency domain that pro-vides an additional time dimension to exploit time-varying micro-Doppler characteristics of the rotating or vibrating components in targets. The micro-Doppler characteristics reflect the motion kinematics of a target and provide a unique identification of the target’s movement. By carefully analyzing various attributes in the characteristic signature, information about the kinematics of the target can be extracted; this is the basis for discriminating the movement and characterizing the activity of the target. In Chapter 5, the methods of how to analyze micro-Doppler signatures and how to extract component signatures associated with the target’s structural component parts will be introduced and discussed.

The Micro-Doppler Effect of the Nonrigid Body Motion 159

The radar micro-Doppler signatures of human gaiting have been investi-gated since the late 1990s [7, 8]. However, very few works on radar micro-Dop-pler signatures of quadrupedal animals have been published so far. Most of the animal signatures simply illustrate the complicated micro-Doppler signatures of animal motion from collected real radar data. No animal locomotion models are available. Further studies on the theoretical basis and the simulations of animals’ motion are needed.

In this chapter, the biomechanical analysis methods and kinematics of typical nonrigid bodies’ motion are introduced. The kinematic model used for describing human motion is given. Based on the kinematic model, a simulation study and the micro-Doppler signature analysis for humans can be easily per-formed, simulated, and analyzed. Radar micro-Doppler signatures of human movements are extracted and analyzed. The micro-Doppler signatures of flap-ping birds are modeled, simulated, and analyzed in Section 4.2, and quadrupe-dal animals are introduced in Section 4.3.

4.1 Human Body Articulated Motion

The motion of human bodies is an articulated locomotion. The motion of limbs in a human body can be characterized by a repeated periodic movement. The human gait is a highly coordinated periodic movement with brain, muscles, nerves, joints, and bones.

Walking is a typical human articulated motion and can be decomposed into a periodic motion in the gait cycle. The human walking cycle consists of two phases: the stance phase and the swing phase. During the stance phase, the foot is on the ground with a heel strike and a toe off. In the swing phase, the foot is lifted from the ground with acceleration or deceleration. Methods used for human gait analysis can be a visual analysis, sensor measurements, and a kinematic system that measures displacements, velocities, accelerations, ori-entations of body segments, and angles of joints. Various human movements, such as walking, running, or jumping, have different body movement patterns. Radar micro-Doppler signatures are not sensitive to distance, light conditions, and background complexity, which are suffered by visual image sequences, and can be used to estimate periodicities of gaiting, the period of stance phase, and the period of the swing phase.

4.1.1 Human Walking

The human walk is a periodic motion with each foot from one position of support to the next position of support, periodically swinging arms and legs and periodically moving the body’s center of gravity up and down. Even if the


human walk has the same general manner, the individual human gait is still car-rying personalized characteristics. This is why people can recognize a friend at a distance from his or her walking style [14]. Therefore, human gait analysis may be also useful for personal identification through the way a person walks. In ad-dition, the emotional aspects of human gait are often observed in practice. For example, the gait of a cheerful person is quite different from that of a depressed person. Therefore, catching any emotion-like gaiting can help in detecting the anomalous behavior of the person. Another important aspect of human gait analysis is in medical applications, such as the medical diagnosis, sports medi-cine, physiotherapy, and rehabilitation.

Both the dynamic method and the kinematic method can be used to gen-erate human motion. If the motion is created without considering the forces involved, the kinematic method can be easily used to calculate positions of articulated body segments from joint angles as forward kinematics. Then in-verse kinematics can be used for determining joint angles from the segments’ positions.

Kinematic parameters are the essential parameters of human motion. These parameters are linear position (or displacement), linear velocity, linear acceleration, angular position, angular velocity, and angular acceleration. To completely describe any human motion in a 3-D Cartesian coordinate system, the linear kinematic parameters of position, velocity, and acceleration define the manner in which the position of any point in the human body changes over time. Velocity is the rate of the position change with respect to time. Accelera-tion defines the rate of velocity change with respect to time. These three kine-matic parameters can be used to understand the motion characteristics of any movement. The acceleration can be measured directly with an accelerometer. The corresponding velocity can be estimated by integrating the acceleration, and the corresponding position can be estimated by integrating the velocity.

Angular kinematic parameters include angular positions (or orientations) of body segments, called segment angles. Because the human body is consid-ered by a number of segments linked by joints, the joint angles are very useful parameters. Angular velocity is the rate of angle change with respect to time, and angular acceleration is the rate of angular velocity change with respect to time. These three angular kinematic parameters are used to describe the angular motion of human body segments.

When a rigid body segment undergoes an angular rotation, the linear velocity and acceleration of the rigid body can be calculated from the angular velocity and acceleration. In the global coordinate system, the angular motion of a rigid body is described by its angular velocity and acceleration. Thus, the linear velocity of a point in the rigid body can be determined by its tangential velocity and normal velocity. The tangential velocity of the point of interest can be derived from the angular velocity and its distance from the rotation center.


Then the tangential acceleration of the point is determined by the angular ac-celeration and the distance from the rotation center. Remember that the tan-gential and normal velocities and accelerations are given in the body-fixed local coordinate system. By using the body segment’s angular orientation with simple trigonometric identities, tangential and normal velocities and accelerations can be easily converted to the global coordinate system.

4.1.2 Description of the Periodic Motion of Human Walking

The nature of human walking is its periodicity. Figure 4.1 illustrates the move-ment of human walking in one cycle [1]. The stance phase occupies about 60% of the cycle, and the swing phase occupies the rest of the cycle. In the stance phase, one foot is in contact with the ground. In the swing phase, the foot is lifted from the ground and the leg is swinging and preparing next stride. This cyclic movement is repeated over and over again.

The stance phase consists of three periods: (1) first double support, where both feet are contacting with the ground; (2) single limb stance, where only one foot is in contact with the ground and the other foot is swinging; and (3) second double support, where both feet are in contact with the ground again.

During the stance phase, there are five events: heel strike, foot-flat, mid-stance, heel-off, and toe-off. The heel strike initiates the gait cycle, and the toe-off terminates the stance phase because the foot lifts from the ground.

During the swing phase, there are no double support periods and only single limb swinging. Three events are associated with the swing phase: forward acceleration of the leg, mid-swing when the foot passes directly beneath the body, and deceleration of the leg to stabilize the foot for preparing for the next heel strike.

Figure 4.1 Movement of human walking in one cycle.


4.1.3 Simulation of Human Movements

To simulate human movements, a movement model that describes the move-ment of interest and a body model that describes the human body are needed. The modeling of human movement can be a mathematical model or an empiri-cal model. Mathematical modeling is constructing a set of equations and simu-lating the human movement using a computer. An empirical model is based on large amounts of human motion data to formulate empirical human motion equations and construct a computer model of the human movement. Then human body model used in biomechanical engineering simplifies the human body and has only as many rigid body segments as needed that are controlled in their movements by joint moments. In the simulation of human movement, the simplified human body model will be used.

Even if the best way to collect human movement data is from human subjects, however, generating human movement data through computer simu-lation is still desirable. Simulation allows researchers to study a single parameter isolated from other parameters in the model or to study in conditions where human subjects cannot be tested. Therefore, simulations are important in hu-man movement studies.

4.1.4 Human Body Segment Parameters

The Denavit-Hartenberg convention (D-H convention) [15, 16] is a widely used kinematic representation that describes the positions of human body links and joints. The D-H convention states that each link has its own coordinate system with its z-axis in the direction of the joint axis, the x-axis aligned with the outgoing link, and the y-axis orthogonal to the x- and z-axes using a right-handed coordinate system as shown in Figure 4.2.

Once the coordinate systems are determined, the interlink transforma-tions are uniquely described by four parameters: θ is the joint angle about the previous z or z

1 from the old x or x

1 to the new x or x

2; d is the link offset along

the previous z to the common normal; a is the length of the common normal; and α is the angle about the common normal, from the old z-axis or z

1 to the

new z-axis or z2 as marked in Figure 4.2. Thus, every link-joint pair is described

as a coordinate transform from the previous coordinate system to the next co-ordinate system.

Any segment in a human body is assumed to be a rigid link, such that the segment size, shape, mass, center of gravity location, and moments of inertia do not change during movements. Under this assumption, a human body is modeled as joints and interconnected rigid links. The movement of such a rigid segment has six degrees of freedom (DOF): three positions in 3-D Cartesian coordinates and three Euler angles of rotation.


The kinematic model of the human body is a hierarchical model of the body links’ connectivity, where a family of parent-child spatial relationships are defined and joints become the nodes of the tree structured human body model. In this kinematic tree, all body coordinates are local coordinate systems relative to their parents. Any transformation of a node only affects its children nodes, but a transformation of the base (root) node will affect all children nodes in the body tree.

To estimate the kinematic parameters of body segments, Boulic, Thal-man, and Thalman [17] proposed a global human walking model based on an empirical mathematical parameterization using biomechanical experimental data. This global walking model averages out the personification of walking. This global human walking model will be introduced and used for human gait analysis and study of the micro-Doppler signatures of the human gait.

Motion capture methods use sensors to capture human movements. Sen-sors can be active sensors, such as accelerometers, gyroscopes, magnetometers, or passive sensors (e.g., video cameras). The Graphics Laboratory at the Carn-egie Mellon University uses 12 infrared cameras to capture motions with 41 markers placed on human body segments [18]. The markers’ positions and ori-entations in 3-D space are tracked with a 120-Hz frame rate and stored in a database. In this book, the simulation of radar returns from human walking is based on the empirical mathematical parameterization model [17], and more complex human movements, such as running, jumping, and other movements, are based on captured motion data provided in the Carnegie Mellon database [18].

Figure 4.2 The Denavit-Hartenberg (D-H) convention used to describe the positions of links and joints.


4.1.5 Human Walking Model Derived from Empirical Mathematical Parameterizations

Boulic, Thalmann, and Thalmman proposed a global human walk model based on empirical mathematical parameterizations derived from biomechanical ex-perimental data [17]. Because this model is based on averaging parameters from experimental measurements, it is an averaging human walking model without information about personalized motion features. Although the method is for modeling human walking, its principle is also applicable to other human move-ments if experimental data is available. In this section, the computer algorithm and the source codes for implementation of this human walking model will be described in detail and will be used to study the micro-Doppler signatures of human walking. To more easily follow Boulic, Thalmann, and Thalmman’s paper [17], the symbols used in this section are the same as those used in the paper. The MATLAB source code for the global human walking model in this chapter is based on [17]. A more detailed description of the empirical equations used in [17] may help to understand the MATLAB simulation of the global human walking model. The source code on the global human walking model is very long because of too many human body segments used. However, it may help readers to understand the whole procedure of the simulation.

This global walking model is derived based on a large number of experi-mental data but not from solving motion equations. It is intended to provide 3-D spatial positions and orientations of any segment of a walking human body as function of time. Specifically, the motion is described by 12 trajectories, 3 translations, and 14 rotations, five of which are duplicated for both sides of the body, as listed in Table 4.1. These translations and rotations describe one cycle of walking motion (i.e., from right heel strike to right heel strike). They are all dependent on the walking velocity.

According to [17], trajectories are described in three methods. Six trajec-tories are given by sinusoidal expressions (one of them by a piecewise function), and six trajectories are represented by cubic spline functions passing through control points located at the extremities of these trajectories.

Given a relative walking velocity VR in m/s (which is normalized by the

height of the leg, i.e., rescaled by a dimensionless value of Ht), the relative

length of one walking cycle is empirically expressed by 1.346C RR V= × in meters. Then the time duration of a cycle is defined by T

C = R

C/V

R in seconds,

and the relative time is normalized by a dimensionless value of TC is t

R = t/T

C in

seconds. The time duration of support is TS = 0.752T

C − 0.143, and the time

duration of double support is TDS

= 0.252TC − 0.143. The body-fixed local

coordinate system is centered at the origin of the spine. The height of the origin of the spine is about 58% of the human height H in meters.

Thus, the translational trajectories are:


1. Vertical translation: This is a vertical offset of the center of the spine from the height of the spine. The translation is

(4.1)

where av = 0.015V

R. The vertical translation function in meter is plot-

ted in Figure 4.3.

2. Lateral translation: This is a lateral oscillation of the center of the spine. The translation is

(4.2)

where

(4.3)

The lateral translation function is plotted in Figure 4.4.

( )verticalTr sin 2 2 0.35v v Ra a tπ⎡ ⎤= − + −⎣ ⎦

( )lateral lTr sin 2 0.1Ra tπ⎡ ⎤= −⎣ ⎦

2

l

0.128 0.128 ( 0.5)

0.032 ( 0.5)R R R

R

V V Va

V

⎧ − + <= ⎨− >⎩

Table 4.1Body Trajectories

Trajectory TranslationBody Rotation

Left Rotation

RightRotation

Vertical translation TV(t)

Lateral translation TL(t)

Translation forward/backward TFB(t)

Rotation forward/backward θFB(t)

Rotation left/right θLR(t)

Torsion rotation θTO (t)

Flexing at the hip θH(t) θH(t + 0.5)

Flexing at the knee θK(t) θK(t + 0.5)

Flexing at the ankle θA(t) θA(t + 0.5)

Motion of the thorax θTH(t)

Flexing at the shoulder θS(t) θS(t + 0.5)

Flexing at the elbow θE(t) θE(t + 0.5)


Figure 4.3 The vertical offset of the center of the spine.

Figure 4.4 The lateral oscillation of the center of the spine.


3. Translation forward/backward: This is the body acceleration and decel-eration when advancing a new step of a leg and stabilizing the leg. The translation is

(4.4)

where

(4.5)

and ϕF/B

= 0.625 − TS. The translation function is plotted in Figure

4.5.

The three trajectories of rotations are:

1. Rotation forward/backward: This is a fl exing movement of the back of the body relative to the pelvis before each step to make a forward mo-tion of the leg. The rotation expressed in degree is

(4.6)

where

(4.7)

The rotation function is plotted in Figure 4.6.

2. Rotation left/right: This is a fl exing movement that makes the pelvis fall on the side of the swinging leg. The piecewise function of the rotation is expressed by

(4.8)

( )/ / /sin 2 2 2F B F D R F BTr a tπ ϕ⎡ ⎤= +⎣ ⎦

2

/

0.084 0.084 ( 0.5)

0.021 ( 0.5)R R R

F BR

V V Va

V

⎧ − + <= ⎨− >⎩

( )/ / / sin 2 2 0.1F B F D F D RRo ar ar tπ⎡ ⎤= − + −⎣ ⎦

2

/

8 8 ( 0.5)

2 ( 0.5)R R R

F BR

V V Var

V

⎧ − + <= ⎨ >⎩

[ ]( ){ }

[ ]{ }[ ]{ }

/ /

/ /

/

/ /

/ /

cos 2 (10 /3) (0 0.15)

cos 2 10 0.15 /7 (0.15 0.5)

cos 2 10( 0.5)/3 (0.5 0.65)

cos 2 10( 0.65)/7 (0.65 1)

L R L R R R

L R L R R R

L R

L R L R R R

L R L R R R

ar ar t t

ar ar t tRo

ar ar t t

ar ar t t

π

π

π

π

⎧ − + ≤ <⎪

⎡ ⎤− − − ≤ <⎪ ⎣ ⎦⎪= ⎨− − − ≤ <⎪

⎪ − + − ≤ <⎪⎩


Figure 4.5 Translation forward and backward function.

Figure 4.6 Rotation forward and backward function.


where arL/R

= 1.66VR. The rotation function is plotted in Figure 4.7.

3. Torsion rotation: The pelvis rotates relative to the spine to make a step. The rotation expressed in degrees is

(4.9)

where arTor

= 4VR. The rotation function is plotted in Figure 4.8.

The six trajectories of flexing or torsion in the lower body and the upper body are:

1. Flexing at the hip: There are three control points to be fi tted. The fl ex-ing function of the hip in degrees is plotted in Figure 4.9.

2. Flexing at the knee: There are four control points to be fi tted. The fl ex-ing function of the knee is plotted in Figure 4.10.

3. Flexing at the ankle: There are fi ve control points to be fi tted. The fl ex-ing function of the ankle is plotted in Figure 4.11.

4. Motion of the thorax: There are four control points to be fi tted. The motion function of the thorax is plotted in Figure 4.12.

cos(2 )Tor Tor RRo ar tπ= −

Figure 4.7 Rotation left and right function.


Figure 4.8 Torsion rotation function.

Figure 4.9 The fl exing function of the hip.


Figure 4.10 The fl exing function of the knee.

Figure 4.11 The fl exing function of the ankle.


5. Flexing at the shoulder: The shoulder has left and right rotations and the fl exing function is

(4.10)

where arShould

= 9.88VR. The fl exing function of the shoulder is plotted

in Figure 4.13.

6. Flexing at the elbow: The fl exing function of the elbow is shape similar to that of the shoulder, but the fl exing angle of the elbow has no nega-tive value. The fl exing function is plotted in Figure 4.14.

Because these single equations and piecewise functions are differentiable through a multiple cycle of a time span, the method used to calculate the final result is to form a spline function by placing two additional sets of control points in that there is one cycle before and one cycle after the time span that represents the current cycle. Only the data from the middle cycle is used to guarantee continuity and differentiability when repeated periodically.

Having correctly calculated the necessary movement trajectories, a work-able walking model can be developed using these trajectories to calculate the

( )3 cos 2Should Should RRo ar tπ= −

Figure 4.12 The motion function of the torso.


Figure 4.13 The fl exing function of the shoulder.

Figure 4.14 The fl exing function of the elbow.


location of a series of 17 reference points on the human body in 3-D space: head, neck, base of spine, left and right shoulders, elbows, hands, hips, knees, ankles, and toes. The (x,y,z)-coordinates are defined as the positive x direction is forward, the positive y direction is right, the positive z direction is up, and the base of the spine is situated at the origin (see Figure 4.15).

The 3-D orientation of a body segment is determined by locating the 17 joint points based on the body reference coordinates centered at the origin of the spine as illustrated in Figure 4.16. From the flexing angle functions and the translations of the joint points described by the model of biomechanical experi-mental data, the Euler angle rotation matrix is used to calculate the positions of the 17 joint points at each frame time. By carefully handling the flexing and translation, the 3-D trajectories of these joint points are obtained. These linear and angular kinematic parameters of a walking human are used to simulate radar returns from the human.

In [19] a list of the length of each human body segment normalized by the height of the human is given. The human model used in this book has 17 reference joint points and segment lengths, as shown in Figure 4.17.

To calculate the location of each reference point based on the trajectories, the Euler rotation matrix is applied based on the XYZ convention, where the roll angle is ψ, the pitch angle is θ, and the yaw angle is ϕ. After the rotation transformations, the location of the reference points can be obtained. To allow for the accurate calculation of body reference points based on multiple angles,

Figure 4.15 Body reference points.


the outermost angles must be account for first (see Figure 4.18). Translations are then handled after contributions from angles have been accounted for.

Given a series of reference points over a period of time, animating a model using this data has confirmed the validity of the model. Animating this model shows that the model is able to produce a proper walking human model (see Figure 4.19). Appendix 4A lists the MATLAB source code to implement the human walking model proposed in [17] and to visualize an animated walking human.

The trajectories of individual body parts of the walking person in part are shown in Figure 4.20, and the corresponding radial velocities are calculated and shown in Figure 4.21. The radial velocity pattern of the person in Figure 4.22 is identical to the micro-Doppler signature of radar backscattering from a walking person shown in Section 4.1.9.

Figure 4.16 The 3-D orientations of human body segments are determined by 17 joint points based on the body reference coordinates centered at the origin of the spine (base).


Figure 4.17 The segment lengths used in the human model.

Figure 4.18 The order of the angle trajectory calculation.


4.1.6 Capturing Human Motion Kinematic Parameters

To capture human motion, the sensors used can be active or passive. Active sen-sors transmit signals to a human object and receive the reflected signals from the object. Passive sensors do not transmit any signal and only receive reflected sig-nals from objects illuminated by other sources. Markers used in motion capture systems can be passive or active [20, 21]. Light-point displays are useful active markers that appeared in the 1970s [22] and clearly demonstrated various mo-tion signatures of different animals.

For sensing 3-D motion, commonly used active sensors include acceler-ometers, gyroscopes, magnetometers, acoustic sensors, and even radar sensors. An accelerometer is a small sensor attached to an object for measuring its ac-celeration. It measures deflection caused by the movements of the sensor and

Figure 4.19 Animating the human walk.


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converts the deflection into an electrical signal. An electromagnetic sensor is attached to joints of any two connected segments and measures the orientation and position of the joint points with respect to the Earth’s magnetic field. These active sensors are now widely used to track the positions of joints or segments in a 3-D space.

However, accurate estimates of the corresponding displacement and ve-locity from the measured acceleration are critical. The velocity is determined by the integral of the acceleration with respect to time, and the displacement is the integral of the velocity with respect to time. For correctly performing the integration process, the measured acceleration histories must be integrated by iteratively adding successive changes in the velocity. Thus, the velocity history and displacement history are computed by

(4.11)

where Δt is the time interval between two successive measured acceleration samples, ai

is the measured acceleration at the sampling time i, vi is the esti-

11

11

2

2

i ii i

i ii i

a av t v

v vx t x

−−

−−

+= Δ +

+= Δ +

Figure 4.22 Radial velocities of body parts, which is identical to the micro-Doppler signature of a human walking towards the radar.


mated velocity at the sampling time i, xi is the estimated displacement at the

sampling time i, and i = 1, 2, …, N, where N is the total number of samples of measured acceleration. The first initial velocity and displacement must be known. For a movement activity starting statically, the first initial velocity and displacement can be set to zeros. Otherwise, they have to be measured by using other methods.

For sensing rotational motion, gyroscopes may be used. By combining a gyroscope with an accelerometer, full 6 DOF data are captured. Human body motion in 3-D space can be completely described by the 6 DOF: linear ac-celeration along each axis and angular rotation about each axis. Therefore, the combination of the accelerometer and the gyroscope may be used as a complete motion-sensing device for capturing human movement information.

A commonly used passive sensing for capturing human motion kinemat-ics is optical motion-caption system equipped with multiple cameras to record motions of optical markers attached to the moving parts of the human body to be measured. The orientation and arrangement of multiple cameras are illus-trated in Figure 4.23. Each camera captures data in 2-D coordinates and each optical sensor (marker) can be seen by at least two cameras. From these sets of 2-D coordinates, the motion data in 3-D coordinates can be calculated. Then the direct linear transform [23, 24] may be used to represent markers from 2-D camera coordinates to 3-D space.

Figure 4.23 Optical motion-caption system equipped with multiple cameras.


4.1.7 Three-Dimensional Kinematic Data Collection

The motion capture database collected by the Graphics Laboratory at the Carnegie Mellon University is available in the public domain and is very use-ful for studying human movements [18]. Twelve infrared cameras a with 120-Hz frame rate are placed around a rectangular area to capture human motion data. Humans wear a jumpsuit with 41 markers on it. The cameras can see the markers in infrared. The images from the 12 cameras are processed to gener-ate data of a 3-D human skeleton movement. The skeleton movement data is then stored in a pair of data files. The ASF (skeleton) file in the pair describes information about the skeleton and joints, and the AMC (motion capture) file of the pair contains the movement data.

In the ASF file, the lengths and directions of 30 bone segments are given. There are a total of 30 body segments read out from the AFS file. They are left and right hip joints, left and right femurs, left and right tibias, left and right feet, left and right toes, lower back, upper back, thorax, lower neck, upper neck, head, left and right clavicles, left and right humerus, left and right radius, left and right wrists, left and right hands, left and right fingers, and left and right thumbs. For studying human walking, running, leaping, or jumping, finger and thumb data may not be necessary.

In the motion capture database, the motion of a human object is in the x-z plane with the forwarding direction along the positive z-axis, which is dif-ferent from the coordinates defined in the human walking model derived in [17] where the human object moves in the x-y plane with the forwarding direc-tion along the positive x direction.

The root of the skeleton hierarchy is a special segment that does not have the direction and length. In the AMC file, the root only contains the starting position and rotation order information. The rotations and directions of other bone segments in the hierarchy are calculated in the AMC file. The rotation of a segment is defined by its axis. The direction of the segment defines the direction from the parent segment to the child segments.

To calculate the global transform for each segment, start by calculating the local transform matrix using the translation offset matrix from its parent segment and the rotation axis matrix. Using the motion data contained in the AMC file and the skeleton defined in the ASF file, the linear and angular ki-nematic parameters are available for the animation of human movement and calculating radar returns from human with movements.

From the captured kinematic database, 3-D motion trajectories of human body segments can be constructed. Figure 4.24 shows a human 2-D position trajectory extracted from the kinematic data of the human root point (i.e., the base of the spine), beginning with walking forward, then sideway stepping, then walking backward, and finally walking diagonally. Figure 4.25 shows the


reconstructed animated human model walking sideway stepping and walking backward.

However, the human movement trajectory in Figure 4.24 does not indi-cate if the human is walking or running, walking forward or walking backward,

Figure 4.24 The 2-D position trajectory of a walking person’s base of the spine when walking forward, then sideway stepping, then walking backward, and fi nally walking diagonally.

Figure 4.25 Reconstructed animated human model: (a) walking sideway stepping and (b) walking backward.


and walking upstairs or walking downstairs. Besides the reconstructed animated human movement as shown in Figure 4.25, the use of angle-cyclogram patterns is an additional method to identify detailed human movement. The angle cy-clogram is a phase-space representation of two joint angles.

4.1.8 Characteristics of Angular Kinematics Using the Angle-Cyclogram Pattern

A dynamic system, such as human locomotion, can be described by a set of state variables. The joint angles or joint velocities are these state variables and can be used to represent the human locomotion. Measurable locomotion descrip-tors, such as the step size, step frequency, and time durations of swing, stance, or double support, are important for the locomotion. To describe any repeti-tive movement activity, the cyclogram is a useful method [25]. Instead of de-scribing one individual joint kinematics, the cyclogram describes a coordinated movement of two joints linked by two or more segments. The perimeter of the cyclogram is a zero-order moment. The position of the center of mass of the cyclogram is a combination of the zero-order and the first-order moments [25].

The angle-angle cyclogram of human locomotion describes the posture of the leg and the coordination of the hip joint and the knee joint. It is a function of the slope and the speed of human locomotion. However, the angle-angle cyclogram does not describe the velocities involved in the leg (including the femur and the tibia). The angle-velocity phase diagram is a trajectory in the phase space and represents the dynamics of a joint. However, the angle-velocity phase gram has no information about the coordination between two joints. Therefore, the combination of the angle-angle cyclogram with the angle-veloci-ty phase diagram provides informative signatures of human locomotion.

Figure 4.26 depicts the joint angles defined in the human lower body. The hip joint angle can have positive and negative values. However, the knee joint angle can have only a single signed value, for which either a negative sign or a positive sign depends on how the angle is defined.

Figure 4.27 shows an example of joint trajectories (i.e., variations of joint angle with time) of a walking human. The hip and knee joint angles of the walking human are from the motion capture database.

In the cyclogram shown in Figure 4.28, the knee joint angle is plotted against the hip joint angle for a complete walking cycle with arrows that in-dicate time increases. The backward walking shows the revised time arrows as shown in Figure 4.29(b). Figures 4.29(c, d) demonstrate different cyclograms of a forward running and a jumping person, respectively.

4.1.9 Radar Backscattering from a Walking Human

Having animated a human movement model, it is easy to calculate radar back-scattering from the human. PO facet models can be used to calculate the RCSs


of human body segments. In this case, human body models should be more ac-curate computer-aided design (CAD) or 3-D graphics models that allow users to build more accurate human bodies.

For simplicity, in this book, human body segments are modeled by el-lipsoids. The RCS of an ellipsoid, RCS

ellip is given in (3.23), where a, b, and

c represent the length of the three semiaxes of the ellipsoid in the x, y, and z directions, respectively. The incident aspect angle θ and the azimuth angle ϕ represent the orientation of the ellipsoid relative to the radar and are illustrated in Figure 4.30. These RCS formulas are used to simulate radar backscattering from a human with movements. It should be mentioned that if human body segments are modeled by 3-D ellipsoids, the use of PO facet to calculate the RCSs of ellipsoids is not necessary.

Figure 4.31(a) illustrates the geometry of a radar and a walking human, where the radar is located at (X

1 = 10m, Y

1 = 0m, Z

1 = 2m) with a wavelength

of 0.02m and the starting point of the human base is located at (X0 = 0m, Y

0 =

0m, Z0 = 0m). The wavelength of the radar is 0.02m. Using the human walking

model derived in [11], assume that the relative velocity of the walking person is V

R = 1.0 sec−1, the height of the person is H = 1.8m, and the mean value of the

torso velocity of the person is 1.33 m/s. The corresponding Doppler frequency shift at a given wavelength of 0.02m is 2 × 1.33/0.02 = 133 Hz.

With the human modeled by ellipsoidal body segments, the radar back-scattering from the walking human can be calculated and the 2-D pulse-range profiles are shown in Figure 4.31(b). The micro-Doppler signature derived

Figure 4.26 The defi ned joint angles of lower body segments.


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and

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erso

n.


from the range profiles is shown in Figure 4.31(c), where the micro-Doppler components of the torso, foot, tibia, and clavicle are indicated through the simulation of separated human body segments.

4.1.10 Human Movement Data Processing

In measured radar data, due to background objects and unwanted moving ob-jects, range profiles show strong clutter. To extract the useful data in range profiles, the clutter must be suppressed. Fortunately, most background objects are stationary and the background clutter can be easily suppressed by a notch filter. Radar backscattering from unwanted moving objects may also be filtered as long as they can be distinguished by their ranges and speeds.

4.1.10.1 Clutter Suppression

Clutter suppression techniques utilize statistical properties of radar returns from stationary objects, which are usually near zero mean Doppler frequency

Figure 4.28 The hip-knee cyclogram of a walking person.


and have a smaller bandwidth of the spectrum. The returns from a moving human are offset from the zero Doppler shifts because of its radial velocity. As illustrated in Figure 4.32, a band-reject filter with a notch around zero velocity can reject most of the clutter without affecting the human motion signal as long as the mean velocity of the human motion is larger than the notch width. Figure 4.32(a) illustrates the Doppler spectrum of clutter and the Doppler spectrum of the human motion. Figure 4.32(b) shows the frequency response of the notch filter and Figure 4.32(d) shows the spectrum of the radar range profiles before notch filtering and after notch filtering. Figure 4.32(c) illustrates the Doppler spectrum after clutter suppression and the clutter cleaned range profiles are shown. The efficiency of the clutter suppression method depends on the notch depth and the relative width of the notch as well as the clutter properties. How-ever, the residue from a strong clutter may still produce a significant bias in

Figure 4.29 The hip-knee cyclogram patter of: (a) forward walking, (b) backward walking, (c) forward running, and (d) jumping.


estimating Doppler frequency of a human motion. The desired average clutter suppression should be more than 40 dB.

4.1.10.2 Time-Frequency Analysis of the Clutter Suppressed Data

The clutter suppressed data, as shown in Figure 4.32(c), should be used to calculate the time-frequency micro-Doppler signature of the walking human as shown in Figure 4.33. The micro-Doppler signatures represented in the joint time-frequency domain help to build a more comprehensive signature knowledge database for various micro-motion dynamics. The classification and identification of human motion behavior will be based on the micro-Doppler signature knowledge database.

4.1.11 Human Movement–Induced Radar Micro-Doppler Signatures

As shown in Figure 4.33, the radar micro-Doppler signature of a walking hu-man is derived by taking the time-frequency transform to the radar range pro-files. In the micro-Doppler signature, each forward leg swing appears as large peaks, and the left- and right-leg swings complete one gait cycle. The body torso motion that is the stronger component underneath the leg swings tends to have a slightly saw-tooth shape because the body speeds up and slows down during the swing as shown.

Figure 4.30 An illustration of the incident aspect angle θ and the azimuth angle ϕ represent-ing the orientation of the ellipsoidal human body segment relative to the radar.


Figu

re 4

.31

(a) T

he g

eom

etry

of r

adar

and

a w

alki

ng h

uman

, (b)

the

rada

r 2-D

pul

se-r

ange

pro

fi les

, and

(c) t

he m

icro

-Dop

pler

sig

natu

re o

f the

wal

king

hu

man

usi

ng th

e gl

obal

wal

king

mod

el p

ropo

sed

in [1

7].


The micro-Doppler signature is actually an integrated Doppler history of individual body segments during a given observation time. Unlike the captured kinematic data by motion sensing, the radar Doppler history data only carry radial velocity information. Therefore, from the micro-Doppler signatures, the animated body movement model cannot be reconstructed. However, the radar micro-Doppler signatures carry distinctive features of the body movements. It is

Figure 4.32 The illustration of clutter suppression by a band-reject fi lter with a notch around zero velocity. (a) The illustration of the Doppler spectrum of clutter and the Doppler spectrum of a human motion. (b) The frequency response of the notch fi lter. (c) The illustration of the Doppler spectrum after clutter suppression. (d) The spectrum of the radar range profi les be-fore notch fi ltering and after notch fi ltering. (After: [12].)


possible to classify and identify bodies and their movements based on the radar micro-Doppler signatures.

Figure 4.34 shows examples of the radar micro-Doppler signatures of hu-man walking, running, and crawling generated from the collected X-band radar data. Compared to the micro-Doppler signature of a walking person in Figure 4.34(a), the micro-Doppler signature of a running person in Figure 4.34(b) has a generally higher Doppler frequency shift and a short gait cycle. The crawling person’s Doppler frequency shift is much lower and the amplitude of the maxi-mum Doppler shift is also lower, as shown in Figure 4.34(c).

Figure 4.33 The time-frequency analysis of clutter suppressed data.


Figu

re 4

.34

(a) T

he m

icro

-Dop

pler

sig

natu

re o

f a w

alki

ng p

erso

n, (b

) the

mic

ro-D

oppl

er s

igna

ture

of a

runn

ing

pers

on, a

nd (c

) the

mic

ro-D

oppl

er s

igna

ture

of

a c

raw

ling

pers

on.


Figure 4.35 shows a person with arm and leg movements captured by the 3-D human motion database [18]. For simplicity, only 15 body parts were used in this model. Fingers and thumb are simply modeled as hands, toes are mod-eled as feet, the lower back, upper back, thorax, lower neck, and clavicles are simply modeled as the torso. The person is standing and performing nine ac-tions using arms and legs during a 9-second observation time period. The first five actions are simultaneously leg and arm motions. The last four actions are the arms’ swimming-like motion. The micro-Doppler signature of the person’s movement is shown in Figure 4.35.

The micro-Doppler signature clearly shows the nine actions. The first five actions show large peaks of Doppler shifts caused by the relatively large motion of the toes and hands. The last four actions show moderate peaks with similar shape caused by a swimming-like arm motion only. The forward and backward motions generate positive and negative Doppler shifts as shown in Figure 4.35. The relatively strong Doppler frequency around zero Doppler means that the person is standing and has no translational motion.

4.2 Bird Wing Flapping

Bird locomotion is a typical animal locomotion and has extracted much atten-tion for studying avian wings [26, 27]. A bird wing is shown in Figure 4.36, where the wing is defined by the wing span, chord, upper arm (humerus), fore-arm (ulna and radius), and hand (wrist, hand, and fingers) if using the terms of

Figure 4.35 The micro-Doppler signature of a person with arm and leg movements. (After: [12].)


human arms. Wing flapping is the elevation and depression of the upper arm or forearm around joints with certain flapping angles. Wing twisting is a rotation of the wing about its principal axis, which results in the elevation of the trailing edge and the depression of the leading edge. Sweeping is the forward protrac-tion or backward retraction of the shoulder. Flapping, twisting, and sweeping consist of the basic bird locomotion.

Wings can have vertical translation, flapping, sweeping, and twisting mo-tions. To study bird wing locomotion, a suitable kinematic model is needed [28–32]. Having a kinematic model, the bird flight movement can be analyzed. Ramakrishnananda and Wong proposed a model of the forward flapping flight of birds using sophisticated bird geometry with multijointed wings and used defined flapping wing parameters to achieve degrees of freedom [30].

4.2.1 Bird Wing Flapping Kinematics

To calculate bird locomotion, the D-H notation [15] is used to represent joint coordinates for a kinematic chain of revolute and translational joints. When analyzing articulated locomotion, harmonic oscillations are often assumed to describe the style of sinusoidal motion [32]. However, harmonic oscillations are the basis of analyzing complicated motion because any motion can be de-composed into a summation of a series of harmonic components with different amplitudes and frequencies by using the Fourier series.

By defining the flapping angle ψ, the twisting angle θ, and sweeping angle ϕ, the rotation matrices required for coordinate transformations are

Figure 4.36 Structure of a bird wing.


1. Wing fl ap matrix:

1 0 0

0 cos sin

0 sin cosflap ψ ψ

ψ ψ

⎡ ⎤⎢ ⎥ℜ = ⎢ ⎥⎢ ⎥−⎣ ⎦

(4.12)

2. Wing twist matrix:

(4.13)

3. Wing sweep matrix:

(4.14)

If a wing has a flapping motion, the flapping angle at a given flap fre-quency f

flap is defined as

(4.15)

the angular velocity is

(4.16)

and, thus, the linear velocity of the wing tip is

(4.17)

where r is the half wing span.During the flapping motion, the position of the wing tip in the body-

fixed local coordinate system is

(4.18)

The linear velocity vector in the body-fixed local coordinate system has only y and z components:

cos 0 sin

0 1 0

sin costwist

θ θ

θ θ

−⎡ ⎤⎢ ⎥ℜ = ⎢ ⎥⎢ ⎥−⎣ ⎦

cos sin 0

sin cos 0

0 0 1sweep

ϕ ϕ

ϕ ϕ

⎡ ⎤⎢ ⎥ℜ = −⎢ ⎥⎢ ⎥⎣ ⎦

( )( ) sin 2 flapt A f tψψ π=

( )( ) ( ) 2 cos 2flap flap

dt t f A f t

dtψ ψψ π πΩ = =

( )( ) ( ) 2 cos 2flap flapV t r t f r A f tψ ψ ψπ π= ⋅Ω = ⋅ ⋅

0

( ) cos ( )

sin ( )flapP t r t

r t

ψ

ψ

⎡ ⎤⎢ ⎥= ⋅⎢ ⎥⎢ ⎥⋅⎣ ⎦


( )( )

00

( ) ( ) 2 cos 2 sin ( )

( ) 2 cos 2 cos ( )

flap y flap flap

zflap flap

V t V t f rA f t t

V t f rA f t t

ψ

ψ

π π ψ

π π ψ

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

⎣ ⎦

(4.19)

If there is also wing twisting, the rotation transform involves angle flap-ping first and then angle twisting. With this order of rotations, the position vector of the wing tip is

(4.20)

and the velocity vector becomes

(4.21)

If the wing has flapping and sweeping, its velocity vector becomes

(4.22)

where the flapping angle and the sweeping angle are

(4.23)

and

(4.24)

( ) ( ( ))

cos 0 sin 1 0 0

0 1 0 0 cos sin ( )

sin 0 cos 0 sin cos

flap twist twist flap flap

flap

P t P t

P t

θ θ

ψ ψ

θ θ ψ ψ

− = ℜ ⋅ ℜ ⋅ =

⎛ ⎞−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎜ ⎟⋅ ⋅⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥−⎝ ⎠⎣ ⎦ ⎣ ⎦

( ) [ ( )]

cos 0 sin 1 0 0

0 1 0 0 cos sin ( )

sin 0 cos 0 sin cos

flap twist twist flap flap

flap

V t V t

V t

θ θ

ψ ψ

θ θ ψ ψ

− = ℜ ⋅ ℜ ⋅ =

⎛ ⎞−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎜ ⎟⋅ ⋅⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥−⎝ ⎠⎣ ⎦ ⎣ ⎦

cos sin 0

( ) ( ) sin cos 0 sin ( )

0 0 1 cos ( )

cos ( ) sin ( )sin ( )

sin ( ) sin ( )cos ( )

cos ( )

x

flap sweep sweep flap flap z

z

x z

x z

z

V

V t V t V t

V t

V t V t t

V t V t t

V t

ϕ ϕ

ϕ ϕ ψ

ψ

ϕ ψ ϕ

ϕ ψ ϕ

ψ

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤= ℜ ⋅ ℜ ⋅ = − ⋅⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦+⎡ ⎤

⎢ ⎥= − +⎢ ⎥⎢ ⎥⎣ ⎦

( )( ) sin 2 flapt A f tψψ π=

( )( ) cos 2 sweept A f tϕϕ π=


where Aψ and f

flap are the amplitude of the flap angle and the flapping frequency,

respectively, and Aϕ and f

sweep are the amplitude of the sweep angle and the

sweeping frequency, respectively. The angular velocities are defined by

(4.25)

and

(4.26)

Thus, the linear velocities at the wing tip are

(4.27)

and

(4.28)

where r is the half wing span.A more complicated wing structure may have two segments in the wing.

Segment 1 is the upper arm (i.e., the link from the shoulder joint to the elbow joint), and Segment 2 is the forearm (i.e., the link from the elbow to the wrist). For flapping, the elbow joint has only one degree of freedom.

4.2.2 Doppler Observations of the Bird Wing Flapping

Radar returns from flying birds carry Doppler modulation caused by flapping wings. The Doppler spread of bird wing flapping has been observed and re-ported [33–36]. Vaughn [33] cited a report [34] on the investigation of bird classification by using an X-band Doppler radar and showed an 11-second ra-dial velocity history profile of a snowy egret with a 0.97-m wingspan. It showed that the wing flapping rate estimated from the Doppler spectrogram is 4 Hz. From the Doppler spectrogram, the maximum radial velocity expected from an element of the wing can be calculated by

(4.29)

( )( ) ( ) 2 cos 2flap flap

dt t f A f t

dtψ ψψ π πΩ = =

( )( ) ( ) 2 sin 2sweep sweep

dt t f A f t

dtϕ ϕϕ π πΩ = =

( )( ) ( ) 2 cos 2flap flapV t r t f r A f tψ ψ ψπ π= ⋅Ω = ⋅ ⋅

( )( ) ( ) 2 sin 2sweep sweepV t r t f r A f tϕ ϕ ϕπ π= ⋅Ω = ⋅ ⋅

{ }max 2radial wingv Af d=


where A is the amplitude of the wing flapping during the down stroke, fwing

is the wing flapping rate, and d is the distance from the bird body center to the tip of the arm. While the bird is taking off, the amplitude A is relatively high (90°–135°). For a snowy egret, the distance d is 0.48m, and the maximum ra-dial velocity should be between 2.1 m/s and 3.7 m/s.

In 1974, Green and Balsley [35] first proposed a method for the identifi-cation of flying birds using the time-varying Doppler spectrum (i.e., the micro-Doppler signature). The time-varying Doppler spectrum of a Canadian goose shows the power of the radar returned signal from the flying goose at various Doppler frequencies when flying over time. The power spectrum shows that the Doppler shifts from the wings are more than 180 Hz higher than that from the body itself. The bandwidth of the time-varying Doppler spectrum varies with the size of the bird. Thus, the bandwidth may be used to discriminate birds.

Various types of birds, such as passerine-like flapping style or swift-like flapping style, have different patterns of the time-varying Doppler spectrum. The passerine-like flapping style shows repeated clusters of larger fluctuations [36].

4.2.3 Simulation of the Bird Wing Flapping

To study the radar returns from a wing-flapping bird, a simple kinematic model with two jointed wing segments is assumed as illustrated in Figure 4.37. In this

Figure 4.37 A simple kinematic model of a bird wing with two jointed wing segments.


simulation, the user-defined parameters are the flapping frequency fflap

= 1.0 Hz, the length of upper arm L

1 = 0.5m, the amplitude of the flapping angle

of the upper arm A1 = 40°, the lag of the flapping angle in the upper arm ψ

10

= 15°, the length of forearm L2 = 0.5m, the amplitude of flapping angle in the

forearm A2 = 30°, the lag of the flapping angle in the forearm ψ

20 = 40°, and the

amplitude of sweeping angle in the forearm C2 = 20°.

With these user-defined parameters, the flapping angle of the upper arm is a harmonic time-varying function given by

(4.30)

the flapping angle of the forearm is a harmonic time-varying function

(4.31)

and the twisting angle of the forearm is also a harmonic time-varying function

(4.32)

Therefore, the elbow joint position is P1 = [x

1, y

1, z

1], where

(4.33)

and the wrist joint position is P2 = [x

2, y

2, z

2], where

(4.34)

where 2 1 2( ) / cos[ ( ) ( )].d t t tϕ ψ ψ= −With this kinematic model of bird wing locomotion, the simulated bird

flying with wing flapping and sweeping is reconstructed. Figure 4.38 is the simulation results, where Figure 4.38(a) shows the wing flapping and sweeping, Figure 4.38(b) is the flight trajectory of two wing tips, and Figure 4.38(c) is the animated flying bird model. Having the simulation, the radar backscattering from the simulated flapping bird can be calculated. Assume that the X-band

1 1 10( ) cos(2 )flapt A f tψ π ψ= +

2 1 20( ) cos(2 )flapt A f tψ π ψ= +

2 2 20( ) cos(2 )flapt C f tϕ π ϕ= +

[ ][ ]

1

1 1 1

1 1 1

( ) 0

( ) cos ( ) /180 ;

( ) ( ) tan ( ) /180

x t

y t L t

z t y t t

ψ π

ψ π

== ⋅

= ⋅ ⋅

[ ][ ] [ ][ ] [ ]{ }

2 2 1

2 1 1 2 2 1 2

2 1 2 1 1 2

( ) ( ) ( ) tan( )

( ) cos ( ) /180 cos ( ) cos ( ) ( )

( ) ( ) ( ) ( ) tan ( ) ( ) /180

x t y t y t d

y t L t L t t t

z t z t y t y t t t

ψ π ϕ ψ ψ

ψ ψ π

= − − ⋅

= ⋅ + ⋅ −

= + − ⋅ − ⋅


Figu

re 4

.38

A si

mul

atio

n fl y

ing

bird

with

a s

impl

e ki

nem

atic

mod

el. (

a) T

he w

ing

fl app

ing

and

swee

ping

, (b)

the

fl igh

t tra

ject

ory

of tw

o w

ing

tips,

and

(c)

the

anim

ated

fl yi

ng b

ird m

odel

.


radar is located at X = 20m, Y = 0m, and Z = −10m and the bird is flying with a velocity 1.0 m/s. Figure 4.39 is the geometry of the radar and the flying bird. The radar range profiles are shown in Figure 4.40(a), and the micro-Doppler signature of the flying bird with flapping wings is shown in Figure 4.40(b). From the micro-Doppler signature, both the flapping upper arms and forearms with the flapping frequency 1.0 Hz can be seen. Appendix 4B lists the MAT-LAB source codes for the flying bird simulation.

4.3 Quadrupedal Animal Motion

Quadrupedal animals use four-legged motion and, thus, have more choices on their feet striking the ground than humans. They could strike ground with each foot separately (four-beat gaits), with two feet separately and other two feet together (three-beat gaits), with three feet together and the other one separately (two-beat gaits), they could strike in pairs (two-beat gaits), or they could strike with all four feet together (one-beat gait). The normal four-legged animals’ walking sequence is four evenly spaced beats with left hind, left fore, right hind, and right fore and without a suspension phase.

Muybridge found that all mammals followed the footfall sequence of the horse when walking on four legs. Figure 4.41 shows walking horses. For a half sequence of horse walking, beginning with an end of a three-legged support phase and the right front leg taking off the ground with the left rear leg pushing back, walking becomes alternately a two-legged support phase, a three-legged support phase, and a two-legged support phase. The other half of the walking sequence is exactly the same as the first half, but with the right and left legs reversed [3].

Figure 4.39 The geometry of a radar and the fl ying bird.


Figure 4.40 (a) Radar range profi les and (b) the micro-Doppler signature of the fl ying bird with fl apping wings.


To study fast animal movements, a high-speed video recorder should be used for recording 3-D motion information, just like the recording of human movements. At least two synchronized video recorders set at an angle to each other are needed for capturing 3-D motion information. More video recorders arranged separately around the animal movement area can avoid occlusions between animal body parts.

The point-light displays for human movement study were also used in studying animal movements [3]. It has been demonstrated that naive observers can identify animals through animated point-light displays [6].

4.3.1 Modeling of Quadrupedal Locomotion

As described the hierarchical human kinematic model in Section 4.1, a hierar-chical quadrupedal model of animal body links’ connectivity can also be used, where a family of parent-child spatial relationships between joints are defined. Figure 4.42 is a quadrupedal model of a dog, where 25 joints are selected. How-ever, a mathematical parameterization modeling of dog or other four-legged animal body segments is not available, and the motion-captured database for quadrupedal animals has not yet been compiled. Nevertheless, if the kinematic parameters of a quadrupedal animal are given, with the hierarchical quadrupe-dal model of animal body links, it is easy to reconstruct an animated quadrupe-dal animal movement.

From the flexing angle functions and the translations of the joint points described by the model, the positions of the 25 joint points can be calculated at

Figure 4.41 Walking horses.


each frame time. These linear and angular kinematic parameters of a quadru-pedal animal are used to simulate radar returns from the animal locomotion.

4.3.2 Micro-Doppler Signatures of Quadrupedal Locomotion

Quadrupedal animal motion can be easily distinguished from bipedal human motion by visualization. The radar micro-Doppler signatures of quadrupedal animals are quite different from the micro-Doppler signatures of bipedal hu-mans because four-legged animal locomotion can have four-beat gaits, three-beat gaits, two-beat gaits, or a one-beat gait. Therefore, the micro-Doppler components from four-legged motion are much more complicated than the micro-Doppler components of human bipedal motion. Figure 4.43 shows the micro-Doppler signature of a horse carrying a rider walking away from the ra-dar [37]. From the micro-Doppler signature, the estimated horse walking radial speed is about 1.5 m/s, and one walking gait cycle is about 1.3 seconds.

Figure 4.44 shows the micro-Doppler signature of a trotting horse carry-ing a rider [37]. From the micro-Doppler signature, the estimated horse radial speed is about 3 m/s, and one trotting cycle is about 0.8 second. The micro-Doppler signature of a dog approaching the X-band multiple frequency con-tinuous wave radar was also reported in [38].

4.3.3 Summary

Compared to human bipedal motion, the four-legged animals’ quadrupedal motion has more choices on feet striking. Thus, motion patterns of animal are more complicated. Unlike the global human walking model derived from a large number of experimental data, there is no similar model available for modeling a four-legged animal walking. Most of the radar micro-Doppler sig-

Figure 4.42 A hierarchical quadrupedal model of a dog.


Figure 4.44 The micro-Doppler signature of a trotting horse carrying a rider. (After: [37].)

Figure 4.43 The micro-Doppler signature of a horse carrying a rider walking away from ra-dar. (After: [37].)


natures of horses and dogs were generated from collected radar data. To further investigate radar scattering from animals with quadrupedal motion, a model of the quadrupedal locomotion is needed. This model can be either derived from experimental data or formulated using collected data by motion capture sensors. High-speed cinematographic technique is one of the best methods of recording an animal’s gait patterns. To completely describe the quadrupedal locomotion in a 3-D Cartesian coordinate system, kinematic parameters are needed to provide the position of any point in the animal body changes over time. These kinematic parameters can be used to understand locomotion char-acteristics of the quadrupedal movement.

After defining a suitable motion model for modeling quadrupedal lo-comotion, extracting the motion features of body component parts from the micro-Doppler signatures and identifying quadrupedal gaiting become feasible.

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Appendix 4A

MATLAB Source Codes

GlobalHumanWalk.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Global Human Walk:%% Based on “A Global Human Walking Model with Real-time Kinematic% Personification,” by R. Boulic, N. M. Thalmann, and D. Thalmann% The Visual Computer, Vol.6, 1990, pp. 344–358.% This model is based on biomechanical experimental data.% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all % human walking modelshowplots = ‘n’; % show translation and rotation of body segmentsformove = ‘y’; % forward walking


animove = ‘n’; % display animationgenmovie = ‘n’; % generate movie file % relative velocity defined by average walking velocity normalized by the% height from the toe to the hip: Htrv = 1.0; % relative velocity (from 0 to 3)nt = 2048; % number of frames per cycleif mod(nt,2) == 1 nt = nt+1;endnumcyc = 3; % number of cycleHeight = 1.8; [segment,seglength,T] = HumanWalkingModel(showplots, formove,... animove, genmovie, Height, rv, nt, numcyc); [data, TF] = RadarReturnsFromWalkingHuman(segment,seglength,T); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

HumanWalkingModel.mfunction [segment,seglength,T] = HumanWalkingModel(showplots,...formove,animove, genmovie, Height, rv, nt, numcyc)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Global Human Walking Model%% Based on “A global human walking model with real-time kinematic% personification,” by R. Boulic, N. M. Thalmann, and D. Thalmann% The Visual Computer, vol. 6, 1990, pp. 344-358.% This model is based on biomechanical experimental data.% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % total number of pulsesnp = nt*numcyc;fprintf(‘Total Number of Samples = %g\n’,np);

% spatial characteristicsrlc = 1.346*sqrt(rv); % relative length of a cycle% temporal characteristicsdc = rlc/rv; % duration of a cycleds = 0.752*dc-0.143; % duration of supportdsmod = ds/dc; % relative duration of supportT = dc*numcyc; % total time duration % body segments’ length (meter)headlen = 0.130*Height;shoulderlen = (0.259/2)*Height;torsolen = 0.288*Height;hiplen = (0.191/2)*Height;upperleglen = 0.245*Height;


lowerleglen = 0.246*Height;footlen = 0.143*Height;upperarmlen = 0.188*Height;lowerarmlen = 0.152*Height;Ht = upperleglen + lowerleglen;sprintf(‘The Walking Velocity is %g m/s’,rv*Ht); % time scalingdt = 1/nt;t = [0:dt:1-dt];t3 = [-1:dt:2-dt]; % designate gait characteristicif rv <= 0 error(‘Velocity must be positive’)elseif rv < 0.5 gait = ‘a’;elseif rv < 1.3 gait = ‘b’;elseif rv <= 3 gait = ‘c’;else error(‘Relative velocity must be less than 3’)end % Locations of body segments: % 3 translation trajectory coords. give the body segments location% relative to rv % calculate vertical translation: offset from the current height (Hs) % of the origin of the spine Osav = 0.015*rv;verttrans = -av+av*sin(2*pi*(2*t-0.35));maxvl = max(verttrans);minvl = min(verttrans);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,verttrans) title(‘Vertical translation offset from the origin of the spine’) xlabel(‘gait cycle’) ylabel([sprintf(‘Position (m) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % calculate lateral translation: Os oscillates laterally to ensure the% weight transfer from one leg to the other.if gait == ‘a’ al = -0.128*rv^2+0.128*rv;else al = -0.032;endlattrans = al*sin(2*pi*(t-0.1));maxvl = max(lattrans);


minvl = min(lattrans);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,lattrans) title(‘Lateral translation offset from the origin of the spine’) xlabel(‘gait cycle’) ylabel([sprintf(‘Offset Position (m) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % calculate translation forward/backward: acceleration and deceleration% phases. When rv grows this effect decreases.if gait == ‘a’ aa = -0.084*rv^2+0.084*rv;else aa = -0.021;endphia = 0.625-dsmod;transforback = aa*sin(2*pi*(2*t+2*phia));maxvl = max(transforback);minvl = min(transforback);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,transforback) title(‘Translation forward/backward from the origin of the spine’) xlabel(‘gait cycle’) ylabel([sprintf(‘Offset Position (m) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % two rotations of the pelvis % calculate rotation forward/backward: to make forward motion of % the leg, the center of gravity of the body must move. % To do this, flexing movement of the back relatively % to the pelvis must be done.if gait == ‘a’ a1 = -8*rv^2+8*rv;else a1 = 2;endrotforback = -a1+a1*sin(2*pi*(2*t-0.1));maxvl = max(rotforback);minvl = min(rotforback);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,rotforback) title(‘Rotation Forward/Backward’) xlabel(‘gait cycle’) ylabel([sprintf(‘Rotation forward/backward (degree)[rv = %g]’,rv)])


axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % calculate rotation left/right: the pelvis falls on the side of the% swinging leg.a2 = 1.66*rv;temp1 = -a2+a2*cos(2*pi*(10*t(1:round(nt*0.15))/3));temp2 = -a2-a2*cos(2*pi*(10*(t(round(nt*0.15)+1:round(nt*0.50))-0.15)/7));

temp3 = a2-a2*cos(2*pi*(10*(t(round(nt*0.50)+1:round(nt*0.65))-0.5)/3));

temp4 = a2+a2*cos(2*pi*(10*(t(round(nt*0.65)+1:nt)-0.65)/7));rotleftright = [temp1,temp2,temp3,temp4];maxvl = max(rotleftright);minvl = min(rotleftright);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,rotleftright) title(‘Rotation Left/Right’) xlabel(‘gait cycle’) ylabel([sprintf(‘Rotation left/right (degree) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % calculate torsion rotation: pelvis rotates relatively to the spine to% perform the stepa3 = 4*rv;torrot = -a3*cos(2*pi*t);maxvl = max(torrot);minvl = min(torrot);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,torrot) title(‘Torsion Rotation’) xlabel(‘gait cycle’) ylabel([sprintf(‘Torsion angle (degree) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % leg flexing/extension: at the hip, at the knee, and at the ankle. % calculate flexing at the hipif gait == ‘a’ x1 = -0.1; x2 = 0.5; x3 = 0.9; y1 = 50*rv; y2 = -30*rv; y3 = 50*rv;endif gait == ‘b’ x1 = -0.1;


x2 = 0.5; x3 = 0.9; y1 = 25; y2 = -15; y3 = 25;endif gait == ‘c’ x1 = 0.2*(rv-1.3)/1.7-0.1; x2 = 0.5; x3 = 0.9; y1 = 5*(rv-1.3)/1.7+25; y2 = -15; y3 = 6*(rv-1.3)/1.7+25;endif x1+1 == x3 x = [x1-1,x2-1,x1,x2,x3,x2+1,x3+1]; y = [y1,y2,y1,y2,y3,y2,y3];else x = [x1-1,x2-1,x3-1,x1,x2,x3,x1+1,x2+1,x3+1]; y = [y1,y2,y3,y1,y2,y3,y1,y2,y3];endtemp = pchip(x,y,t3);flexhip = temp(nt+1:2*nt);maxvl = max(flexhip);minvl = min(flexhip);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,flexhip) hold on plot(x1,y1,’ro’,x2,y2,’ro’,x3,y3,’ro’) title(‘Flexing at the Hip’) xlabel(‘gait cycle’) ylabel([sprintf(‘Flexing angle (degree) [rv = %g]’,rv)]) if x1 < 0 axis([x1 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) else axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) end drawnowend % calculate flexing at the knee: there are 4 control points.if gait == ‘a’ x1 = 0.17; x2 = 0.4; x3 = 0.75; x4 = 1; y1 = 3; y2 = 3; y3 = 140*rv; y4 = 3;endif gait == ‘b’ x1 = 0.17;


x2 = 0.4; x3 = 0.75; x4 = 1; y1 = 3; y2 = 3; y3 = 70; y4 = 3;endif gait == ‘c’ x1 = -0.05*(rv-1.3)/1.7+0.17; x2 = 0.4; x3 = -0.05*(rv-1.3)/1.7+0.75; x4 = -0.03*(rv-1.3)/1.7+1; y1 = 22*(rv-1.3)/1.7+3; y2 = 3; y3 = -5*(rv-1.3)/1.7+70; y4 = 3*(rv-1.3)/1.7+3;endx = [x1-1,x2-1,x3-1,x4-1,x1,x2,x3,x4,x1+1,x2+1,x3+1,x4+1];y = [y1,y2,y3,y4,y1,y2,y3,y4,y1,y2,y3,y4];temp = pchip(x,y,t3);flexknee = temp(nt+1:2*nt);maxvl = max(flexknee);minvl = min(flexknee);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,flexknee) hold on plot(x1,y1,’ro’,x2,y2,’ro’,x3,y3,’ro’,x4,y4,’ro’) title(‘Flexing at the Knee’) xlabel(‘gait cycle’) ylabel([sprintf(‘Flexing angle (degree) [rv = %g]’,rv)]) if x1 < 0 axis([x1 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) else axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) end drawnowend % calculate flexing at the ankle: there are 5 control pointsif gait == ‘a’ x1 = 0; x2 = 0.08; x3 = 0.5; x4 = dsmod; x5 = 0.85; y1 = -3; y2 = -30*rv-3; y3 = 22*rv-3; y4 = -34*rv-3; y5 = -3; end


if gait == ‘b’ x1 = 0; x2 = 0.08; x3 = 0.5; x4 = dsmod; x5 = 0.85; y1 = -3; y2 = -18; y3 = 8; y4 = -20; y5 = -3;endif gait == ‘c’ x1 = 0; x2 = 0.08; x3 = -0.1*(rv-1.3)/1.7+0.5; x4 = dsmod; x5 = 0.85; y1 = 5*(rv-1.3)/1.7-3; y2 = 4*(rv-1.3)/1.7-18; y3 = -3*(rv-1.3)/1.7+8; y4 = -8*(rv-1.3)/1.7-20; y5 = 5*(rv-1.3)/1.7-3;endx = [x1-1,x2-1,x3-1,x4-1,x5-1,x1,x2,x3,x4,x5,x1+1,x2+1,x3+1,x4+1,x5+1];

y = [y1,y2,y3,y4,y5,y1,y2,y3,y4,y5,y1,y2,y3,y4,y5];temp = pchip(x,y,t3);flexankle = temp(nt+1:2*nt);maxvl = max(flexankle);minvl = min(flexankle);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,flexankle) hold on plot(x1,y1,’ro’,x2,y2,’ro’,x3,y3,’ro’,x4,y4,’ro’,x5,y5,’ro’) title(‘Flexing at the Ankle’) xlabel(‘gait cycle’) ylabel([sprintf(‘Flexing angle (degree) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % trajectory of upper body % calculate motion (torsion) of the thorax: there are 4 control pointsx1 = 0.1;x2 = 0.4;x3 = 0.6;x4 = 0.9;y1 = (4/3)*rv;y2 = (-4.5/3)*rv;y3 = (-4/3)*rv;y4 = (4.5/3)*rv;x = [x1-1,x2-1,x3-1,x4-1,x1,x2,x3,x4,x1+1,x2+1,x3+1,x4+1];


y = [y1,y2,y3,y4,y1,y2,y3,y4,y1,y2,y3,y4];temp = pchip(x,y,t3);motthor = temp(nt+1:2*nt);maxvl = max(motthor);minvl = min(motthor);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,motthor) hold on plot(x1,y1,’ro’,x2,y2,’ro’,x3,y3,’ro’,x4,y4,’ro’) title(‘Torsion of the Thorax’) xlabel(‘gait cycle’) ylabel([sprintf(‘Torsion angle (degree) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % calculate flexing at the shoulderas = 9.88*rv;flexshoulder = 3-as/2-as*cos(2*pi*t);maxvl = max(flexshoulder);minvl = min(flexshoulder);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,flexshoulder) title(‘Flexing at the Shoulder’) xlabel(‘gait cycle’) ylabel([sprintf(‘Flexing angle (degree) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % calculate flexing at the elbowif gait == ‘a’ x1 = 0.05; x2 = 0.5; x3 = 0.9; y1 = 6*rv+3; y2 = 34*rv+3; y3 = 10*rv+3;endif gait == ‘b’ x1 = 0.05; x2 = 0.01*(rv-0.5)/0.8+0.5; x3 = 0.9; y1 = 8*(rv-0.5)/0.8+6; y2 = 24*(rv-0.5)/0.8+20; y3 = 9*(rv-0.5)/0.8+8;endif gait == ‘c’ x1 = 0.05; x2 = 0.04*(rv-1.3)/1.7+0.51; x3 = -0.1*(rv-1.3)/1.7+0.9;


y1 = -6*(rv-1.3)/1.7+14; y2 = 26*(rv-1.3)/1.7+44; y3 = -6*(rv-.13)/1.7+17;endx = [x1-1,x2-1,x3-1,x1,x2,x3,x1+1,x2+1,x3+1];y = [y1,y2,y3,y1,y2,y3,y1,y2,y3];temp = pchip(x,y,t3);flexelbow = temp(nt+1:2*nt);maxvl = max(flexelbow);minvl = min(flexelbow);diffvl = maxvl-minvl;if showplots == ‘y’ figure plot(t,flexelbow) hold on plot(x1,y1,’ro’,x2,y2,’ro’,x3,y3,’ro’) title(‘Flexing at the Elbow’) xlabel(‘gait cycle’) ylabel([sprintf(‘Flexing angle (degree) [rv = %g]’,rv)]) axis([0 1 minvl-0.2*diffvl maxvl+0.2*diffvl]) drawnowend % Handling lower body flexing at ankles, knees, and hips% handle flexing at the left anklepsi = linspace(0,0,nt);the(1:round(nt/2)) = flexankle(round(nt/2)+1:nt)*pi/180;the(round(nt/2)+1:nt) = flexankle(1:round(nt/2))*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[footlen;0;0]; ltoe(1,i) = temp(1); ltoe(2,i) = temp(2)+hiplen; ltoe(3,i) = temp(3)-(upperleglen+lowerleglen);end % handle flexing at the right anklepsi = linspace(0,0,nt);the = flexankle*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[footlen;0;0]; rtoe(1,i) = temp(1); rtoe(2,i) = temp(2)-hiplen; rtoe(3,i) = temp(3)-(upperleglen+lowerleglen);end % handle flexing at the left kneepsi = linspace(0,0,nt);the(1:round(nt/2)) = flexknee(round(nt/2)+1:nt)*pi/(-180);the(round(nt/2)+1:nt) = flexknee(1:round(nt/2))*pi/(-180);phi = linspace(0,0,nt);for i = 1:nt


Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-lowerleglen]; lankle(1,i) = temp(1); lankle(2,i) = temp(2)+hiplen; lankle(3,i) = temp(3)-upperleglen; temp = Rxyz*[ltoe(1,i);0;ltoe(3,i)+upperleglen]; ltoe(1,i) = temp(1); ltoe(2,i) = temp(2)+hiplen; ltoe(3,i) = temp(3)-upperleglen;end % handle flexing at the right kneepsi = linspace(0,0,nt);the = flexknee*pi/(-180);phi = linspace(0,0,nt);for i = 1:nt Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-lowerleglen]; rankle(1,i) = temp(1); rankle(2,i) = temp(2)-hiplen; rankle(3,i) = temp(3)-upperleglen; temp = Rxyz*[rtoe(1,i);0;rtoe(3,i)+upperleglen]; rtoe(1,i) = temp(1); rtoe(2,i) = temp(2)-hiplen; rtoe(3,i) = temp(3)-upperleglen;end % handle flexing at the left hippsi = linspace(0,0,nt);the(1:round(nt/2)) = flexhip(round(nt/2)+1:nt)*pi/180;the(round(nt/2)+1:nt) = flexhip(1:round(nt/2))*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-upperleglen]; lknee(1,i) = temp(1); lknee(2,i) = temp(2)+hiplen; lknee(3,i) = temp(3); temp = Rxyz*[lankle(1,i);0;lankle(3,i)]; lankle(1,i) = temp(1); lankle(2,i) = temp(2)+hiplen; lankle(3,i) = temp(3); temp = Rxyz*[ltoe(1,i);0;ltoe(3,i)]; ltoe(1,i) = temp(1); ltoe(2,i) = temp(2)+hiplen; ltoe(3,i) = temp(3);end % handle flexing at the right hippsi = linspace(0,0,nt);the = flexhip*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-upperleglen];


rknee(1,i) = temp(1); rknee(2,i) = temp(2)-hiplen; rknee(3,i) = temp(3); temp = Rxyz*[rankle(1,i);0;rankle(3,i)]; rankle(1,i) = temp(1); rankle(2,i) = temp(2)-hiplen; rankle(3,i) = temp(3); temp = Rxyz*[rtoe(1,i);0;rtoe(3,i)]; rtoe(1,i) = temp(1); rtoe(2,i) = temp(2)-hiplen; rtoe(3,i) = temp(3);end % Handling lower body rotationpsi = rotleftright*pi/(-180);the = linspace(0,0,nt);phi = torrot*pi/180;for i = 1:nt Rxyz = XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;hiplen;0]; lhip(1,i) = temp(1); lhip(2,i) = temp(2); lhip(3,i) = temp(3); temp = Rxyz*[0;-hiplen;0]; rhip(1,i) = temp(1); rhip(2,i) = temp(2); rhip(3,i) = temp(3); temp = Rxyz*[lknee(1,i);lknee(2,i);lknee(3,i)]; lknee(1,i) = temp(1); lknee(2,i) = temp(2); lknee(3,i) = temp(3); temp = Rxyz*[rknee(1,i);rknee(2,i);rknee(3,i)]; rknee(1,i) = temp(1); rknee(2,i) = temp(2); rknee(3,i) = temp(3); temp = Rxyz*[lankle(1,i);lankle(2,i);lankle(3,i)]; lankle(1,i) = temp(1); lankle(2,i) = temp(2); lankle(3,i) = temp(3); temp = Rxyz*[rankle(1,i);rankle(2,i);rankle(3,i)]; rankle(1,i) = temp(1); rankle(2,i) = temp(2); rankle(3,i) = temp(3); temp = Rxyz*[ltoe(1,i);ltoe(2,i);ltoe(3,i)]; ltoe(1,i) = temp(1); ltoe(2,i) = temp(2); ltoe(3,i) = temp(3); temp = Rxyz*[rtoe(1,i);rtoe(2,i);rtoe(3,i)]; rtoe(1,i) = temp(1); rtoe(2,i) = temp(2); rtoe(3,i) = temp(3);end % Handling upper body flexing at elbows and shoulders% handling flexing at the left elbow


psi = linspace(0,0,nt);the(1:round(nt/2)) = flexelbow(round(nt/2)+1:nt)*pi/180;the(round(nt/2)+1:nt) = flexelbow(1:round(nt/2))*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz =XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-lowerarmlen]; lhand(1,i) = temp(1); lhand(2,i) = temp(2)+shoulderlen; lhand(3,i) = temp(3)+(torsolen-upperarmlen);end % handle flexing at the right elbowpsi = linspace(0,0,nt);the = flexelbow*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz =XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-lowerarmlen]; rhand(1,i) = temp(1); rhand(2,i) = temp(2)-shoulderlen; rhand(3,i) = temp(3)+(torsolen-upperarmlen);end % handle flexing at the left shoulderpsi = linspace(0,0,nt);the(1:round(nt/2)) = flexshoulder(round(nt/2)+1:nt)*pi/180;the(round(nt/2)+1:nt) = flexshoulder(1:round(nt/2))*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz =XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-upperarmlen]; lelbow(1,i) = temp(1); lelbow(2,i) = temp(2)+shoulderlen; lelbow(3,i) = temp(3)+torsolen; temp = Rxyz*[lhand(1,i);0;lhand(3,i)-torsolen]; lhand(1,i) = temp(1); lhand(2,i) = temp(2)+shoulderlen; lhand(3,i) = temp(3)+torsolen;end % handle flexing at the right shoulderpsi = linspace(0,0,nt);the = flexshoulder*pi/180;phi = linspace(0,0,nt);for i = 1:nt Rxyz =XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;-upperarmlen]; relbow(1,i) = temp(1); relbow(2,i) = temp(2)-shoulderlen; relbow(3,i) = temp(3)+torsolen; temp = Rxyz*[rhand(1,i);0;rhand(3,i)-torsolen]; rhand(1,i) = temp(1); rhand(2,i) = temp(2)-shoulderlen; rhand(3,i) = temp(3)+torsolen;


end % Handling upper body rotationpsi = linspace(0,0,nt);the = rotforback*pi/180;phi = motthor*pi/180;for i = 1:nt Rxyz =XYZConvention(psi(i),the(i),phi(i)); temp = Rxyz*[0;0;torsolen+headlen]; head(1,i) = temp(1); head(2,i) = temp(2); head(3,i) = temp(3); temp = Rxyz*[0;0;torsolen]; neck(1,i) = temp(1); neck(2,i) = temp(2); neck(3,i) = temp(3); temp = Rxyz*[0;shoulderlen;torsolen]; lshoulder(1,i) = temp(1); lshoulder(2,i) = temp(2); lshoulder(3,i) = temp(3); temp = Rxyz*[0;-shoulderlen;torsolen]; rshoulder(1,i) = temp(1); rshoulder(2,i) = temp(2); rshoulder(3,i) = temp(3); temp = Rxyz*[lelbow(1,i);lelbow(2,i);lelbow(3,1)]; lelbow(1,i) = temp(1); lelbow(2,i) = temp(2); lelbow(3,i) = temp(3); temp = Rxyz*[relbow(1,i);relbow(2,i);relbow(3,1)]; relbow(1,i) = temp(1); relbow(2,i) = temp(2); relbow(3,i) = temp(3); temp = Rxyz*[lhand(1,i);lhand(2,i);lhand(3,i)]; lhand(1,i) = temp(1); lhand(2,i) = temp(2); lhand(3,i) = temp(3); temp = Rxyz*[rhand(1,i);rhand(2,i);rhand(3,i)]; rhand(1,i) = temp(1); rhand(2,i) = temp(2); rhand(3,i) = temp(3);end % The origin of the body coordinate systembase = [linspace(0,0,nt);linspace(0,0,nt);linspace(0,0,nt)]; % Handling translationbase = base+[transforback;lattrans;verttrans];neck = neck+[transforback;lattrans;verttrans];head = head+[transforback;lattrans;verttrans];lshoulder = lshoulder+[transforback;lattrans;verttrans];rshoulder = rshoulder+[transforback;lattrans;verttrans];lelbow = lelbow+[transforback;lattrans;verttrans];relbow = relbow+[transforback;lattrans;verttrans];lhand = lhand+[transforback;lattrans;verttrans];rhand = rhand+[transforback;lattrans;verttrans];


lhip = lhip+[transforback;lattrans;verttrans];rhip = rhip+[transforback;lattrans;verttrans];lknee = lknee+[transforback;lattrans;verttrans];rknee = rknee+[transforback;lattrans;verttrans];lankle = lankle+[transforback;lattrans;verttrans];rankle = rankle+[transforback;lattrans;verttrans];ltoe = ltoe+[transforback;lattrans;verttrans];rtoe = rtoe+[transforback;lattrans;verttrans]; % Visulizing human walking:if formove == ‘y’ base(1,:) = base(1,:)+linspace(0,rlc-rlc/(nt+1),nt); neck(1,:) = neck(1,:)+linspace(0,rlc-rlc/(nt+1),nt); head(1,:) = head(1,:)+linspace(0,rlc-rlc/(nt+1),nt); lshoulder(1,:) = lshoulder(1,:)+linspace(0,rlc-rlc/(nt+1),nt); rshoulder(1,:) = rshoulder(1,:)+linspace(0,rlc-rlc/(nt+1),nt); lelbow(1,:) = lelbow(1,:)+linspace(0,rlc-rlc/(nt+1),nt); relbow(1,:) = relbow(1,:)+linspace(0,rlc-rlc/(nt+1),nt); lhand(1,:) = lhand(1,:)+linspace(0,rlc-rlc/(nt+1),nt); rhand(1,:) = rhand(1,:)+linspace(0,rlc-rlc/(nt+1),nt); lhip(1,:) = lhip(1,:)+linspace(0,rlc-rlc/(nt+1),nt); rhip(1,:) = rhip(1,:)+linspace(0,rlc-rlc/(nt+1),nt); lknee(1,:) = lknee(1,:)+linspace(0,rlc-rlc/(nt+1),nt); rknee(1,:) = rknee(1,:)+linspace(0,rlc-rlc/(nt+1),nt); lankle(1,:) = lankle(1,:)+linspace(0,rlc-rlc/(nt+1),nt); rankle(1,:) = rankle(1,:)+linspace(0,rlc-rlc/(nt+1),nt); ltoe(1,:) = ltoe(1,:)+linspace(0,rlc-rlc/(nt+1),nt); rtoe(1,:) = rtoe(1,:)+linspace(0,rlc-rlc/(nt+1),nt);end if animove == ‘y’ framenum = 1; figure(1) colormap([0.7 0.7 0.7])end for i = 1:nt*numcyc if animove == ‘y’ && mod(i,50) == 0 clf hold on [x,y,z] = cylinder2P(neck(:,i)’,head(:,i)’,0.05,10); surf(x,y,z) % head [x,y,z] = ... ellipsoid2P(base(:,i)’,neck(:,i)’,0.15,0.15,torsolen/2,20); surf(x,y,z) % torso [x,y,z] = ...ellipsoid2P(lankle(:,i)’,lknee(:,i)’,0.06,0.06,lowerleglen/2,10); surf(x,y,z) % left lower leg [x,y,z] = ...ellipsoid2P(rankle(:,i)’,rknee(:,i)’,0.06,0.06,lowerleglen/2,10); surf(x,y,z) % right lower leg [x,y,z] = ...ellipsoid2P(ltoe(:,i)’,lankle(:,i)’,0.05,0.05,footlen/2,10); surf(x,y,z) % left foot [x,y,z] = ...


ellipsoid2P(rtoe(:,i)’,rankle(:,i)’,0.05,0.05,footlen/2,10); surf(x,y,z) % right foot [x,y,z] = ...ellipsoid2P(lknee(:,i)’,lhip(:,i)’,0.07,0.07,upperleglen/2,10); surf(x,y,z) % left upper leg [x,y,z] = ...ellipsoid2P(rknee(:,i)’,rhip(:,i)’,0.07,0.07,upperleglen/2,10); surf(x,y,z) % right upper leg [x,y,z] = ...ellipsoid2P(base(:,i)’,lhip(:,i)’,0.07,0.07,hiplen/2,10); surf(x,y,z) % left hip [x,y,z] = ...ellipsoid2P(base(:,i)’,rhip(:,i)’,0.07,0.07,hiplen/2,10); surf(x,y,z) % right hip [x,y,z] = ...ellipsoid2P(lelbow(:,i)’,lshoulder(:,i)’,0.06,0.06,upperarmlen/2,10);

surf(x,y,z) % left upper arm [x,y,z] = ...ellipsoid2P(relbow(:,i)’,rshoulder(:,i)’,0.06,0.06,upperarmlen/2,10);

surf(x,y,z) % right upper arm [x,y,z] = ...ellipsoid2P(neck(:,i)’,lshoulder(:,i)’,0.06,0.06, shoulderlen/2,10);

surf(x,y,z) % left shoulder [x,y,z] = ...ellipsoid2P(neck(:,i)’,rshoulder(:,i)’,0.06,0.06, shoulderlen/2,10);

surf(x,y,z) % right shoulder [x,y,z] = ...ellipsoid2P(lhand(:,i)’,lelbow(:,i)’,0.05,0.05,lowerarmlen/2,10);

surf(x,y,z) % left lower arm [x,y,z] = ...ellipsoid2P(rhand(:,i)’,relbow(:,i)’,0.05,0.05,lowerarmlen/2,10); surf(x,y,z) % right lower arm [x,y,z] = sphere(10); surf(x*0.02+ltoe(1,i),y*0.02+ltoe(2,i),z*0.01+ltoe(3,i)) % left toe [x,y,z] = sphere(10); surf(x*0.02+rtoe(1,i),y*0.02+rtoe(2,i),... z*0.01+rtoe(3,i)) % right toe [x,y,z] = sphere(10); surf(x*0.05+lankle(1,i),y*0.05+lankle(2,i),... z*0.05+lankle(3,i)) % lankle [x,y,z] = sphere(10); surf(x*0.05+rankle(1,i),y*0.05+rankle(2,i),... z*0.05+rankle(3,i)) % rankle [x,y,z] = sphere(10); surf(x*0.05+lknee(1,i),y*0.05+lknee(2,i),z*0.05+lknee(3,i)) % lknee [x,y,z] = sphere(10); surf(x*0.05+rknee(1,i),y*0.05+rknee(2,i),z*0.05+rknee(3,i)) % rknee [x,y,z] = sphere(10); surf(x*0.1+head(1,i),y*0.1+head(2,i),z*headlen/2+head(3,i)) % head [x,y,z] = sphere(10); surf(x*0.05+lhip(1,i),y*0.05+lhip(2,i),z*0.05+lhip(3,i)) % lhip [x,y,z] = sphere(10); surf(x*0.05+rhip(1,i),y*0.05+rhip(2,i),z*0.05+rhip(3,i)) % rhip [x,y,z] = sphere(10); % left shoulder


surf(x*0.05+lshoulder(1,i),y*0.05+lshoulder(2,i),... z*0.05+lshoulder(3,i)) [x,y,z] = sphere(10); % right shoulder surf(x*0.05+rshoulder(1,i),y*0.05+rshoulder(2,i),... z*0.05+rshoulder(3,i)) [x,y,z] = sphere(10); % lelbow surf(x*0.05+lelbow(1,i),y*0.05+lelbow(2,i),z*0.05+lelbow(3,i) [x,y,z] = sphere(10); % relbow surf(x*0.05+relbow(1,i),y*0.05+relbow(2,i),z*0.05+relbow(3,i)) [x,y,z] = sphere(10); % lhand surf(x*0.05+lhand(1,i),y*0.05+lhand(2,i),z*0.05+lhand(3,i)) [x,y,z] = sphere(10); % rhand surf(x*0.05+rhand(1,i),y*0.05+rhand(2,i),z*0.05+rhand(3,i)) light lighting gouraud shading interp axis equal if formove == ‘y’ axis([-2,2*numcyc,-2,2,-2,2]) else axis([-2,2,-2,2,-2,2]) end axis off grid off set(gcf,’Color’,[1 1 1]) view([45,20]) zoom(2) drawnow if genmovie == ‘y’ M(framenum) = getframe; framenum = framenum+1; end end if mod(i,nt) == 0 if formove == ‘y’ temp = base(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); base = [base,temp]; temp = neck(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); neck = [neck,temp]; temp = head(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); head = [head,temp]; temp = lshoulder(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); lshoulder = [lshoulder,temp]; temp = rshoulder(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); rshoulder = [rshoulder,temp]; temp = lelbow(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); lelbow = [lelbow,temp]; temp = relbow(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt);


relbow = [relbow,temp]; temp = lhand(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); lhand = [lhand,temp]; temp = rhand(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); rhand = [rhand,temp]; temp = lhip(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); lhip = [lhip,temp]; temp = rhip(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); rhip = [rhip,temp]; temp = lknee(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); lknee = [lknee,temp]; temp = rknee(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); rknee = [rknee,temp]; temp = lankle(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); lankle = [lankle,temp]; temp = rankle(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); rankle = [rankle,temp]; temp = ltoe(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); ltoe = [ltoe,temp]; temp = rtoe(:,i-nt+1:i); temp(1,:) = temp(1,:)+linspace(rlc,rlc,nt); rtoe = [rtoe,temp]; else temp = base(:,i-nt+1:i); base = [base,temp]; temp = neck(:,i-nt+1:i); neck = [neck,temp]; temp = head(:,i-nt+1:i); head = [head,temp]; temp = lshoulder(:,i-nt+1:i); lshoulder = [lshoulder,temp]; temp = rshoulder(:,i-nt+1:i); rshoulder = [rshoulder,temp]; temp = lelbow(:,i-nt+1:i); lelbow = [lelbow,temp]; temp = relbow(:,i-nt+1:i); relbow = [relbow,temp]; temp = lhand(:,i-nt+1:i); lhand = [lhand,temp]; temp = rhand(:,i-nt+1:i); rhand = [rhand,temp]; temp = lhip(:,i-nt+1:i); lhip = [lhip,temp]; temp = rhip(:,i-nt+1:i); rhip = [rhip,temp]; temp = lknee(:,i-nt+1:i);


lknee = [lknee,temp]; temp = rknee(:,i-nt+1:i); rknee = [rknee,temp]; temp = lankle(:,i-nt+1:i); lankle = [lankle,temp]; temp = rankle(:,i-nt+1:i); rankle = [rankle,temp]; temp = ltoe(:,i-nt+1:i); ltoe = [ltoe,temp]; temp = rtoe(:,i-nt+1:i); rtoe = [rtoe,temp]; end endend % generate movie fileif genmovie == ‘y’ movie2avi(M,’HumanWalkingModel.avi’,’FPS’,20,’compression’,’None’)end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

RadarReturnsFromWalkingHuman.m

function [data, TF] = RadarReturnsFromWalkingHuman(segment,seglength,T)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Radar Returns From Walking Human% % Read walking human kinematics data.%% Based on “A Global Human Walking Model with Real-Time Kinematic% Personification,” by R. Boulic, N. M. Thalmann, and D. Thalmann% The Visual Computer, vol.6, 1990, pp. 344-358.% This model is based on biomechanical experimental data.% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% input datafprintf(‘%s\n’,seglength(1).name);headlen = seglength(1).length;fprintf(‘%s\n’,seglength(2).name);shoulderlen = seglength(2).length;fprintf(‘%s\n’,seglength(3).name);torsolen = seglength(3).length;fprintf(‘%s\n’,seglength(4).name);hiplen = seglength(4).length;fprintf(‘%s\n’,seglength(5).name);upperleglen = seglength(5).length;fprintf(‘%s\n’,seglength(6).name);lowerleglen = seglength(6).length;fprintf(‘%s\n’,seglength(7).name);footlen = seglength(7).length;


fprintf(‘%s\n’,seglength(8).name);upperarmlen = seglength(8).length;fprintf(‘%s\n’,seglength(9).name);lowerarmlen = seglength(9).length;fprintf(‘%s\n’,segment(1).name);base = segment(1).PositionData;fprintf(‘%s\n’,segment(2).name);neck = segment(2).PositionData;fprintf(‘%s\n’,segment(3).name);head = segment(3).PositionData;fprintf(‘%s\n’,segment(4).name);lshoulder = segment(4).PositionData;fprintf(‘%s\n’,segment(5).name);rshoulder = segment(5).PositionData;fprintf(‘%s\n’,segment(6).name);lelbow = segment(6).PositionData;fprintf(‘%s\n’,segment(7).name);relbow = segment(7).PositionData;fprintf(‘%s\n’,segment(8).name);lhand = segment(8).PositionData;fprintf(‘%s\n’,segment(9).name);rhand = segment(9).PositionData;fprintf(‘%s\n’,segment(10).name);lhip = segment(10).PositionData;fprintf(‘%s\n’,segment(11).name);rhip = segment(11).PositionData;fprintf(‘%s\n’,segment(12).name);lknee = segment(12).PositionData;fprintf(‘%s\n’,segment(13).name);rknee = segment(13).PositionData;fprintf(‘%s\n’,segment(14).name);lankle = segment(14).PositionData;fprintf(‘%s\n’,segment(15).name);rankle = segment(15).PositionData;fprintf(‘%s\n’,segment(16).name);ltoe = segment(16).PositionData;fprintf(‘%s\n’,segment(17).name);rtoe = segment(17).PositionData; j = sqrt(-1); % radar parameterslambda = 0.02; % wave lengthrangeres = 0.01; % range resolutionradarloc = [10,0,2]; % radar locationnr = round(2*sqrt(radarloc(1)^2+radarloc(2)^2+radarloc(3)^2)/rangeres);

np = size(base,2); data = zeros(nr,np); % radar returns from the headfor k = 1:np % distance from radar to head r_dist(:,k) = abs(head(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2);


% aspect vector of the head aspct(:,k) = head(:,k)-neck(:,k); % calculate theta angle A = [radarloc(1)-head(1,k); radarloc(2)-head(2,k);... radarloc(3)-head(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-head(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.1; % ellipsoid parameter b = 0.1; c = headlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from torsofor k = 1:np % distance from radar to torso torso(:,k) = (neck(:,k)+base(:,k))/2; r_dist(:,k) = abs(torso(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the torso aspct(:,k) = neck(:,k)-base(:,k); % calculate theta angle A = [radarloc(1)-torso(1,k); radarloc(2)-torso(2,k);... radarloc(3)-torso(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-torso(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.15; b = 0.15; c = torsolen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from left shoulderfor k = 1:np % distance from radar to left shoulder r_dist(:,k) = abs(lshoulder(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2);


% aspect vector of the left shoulder aspct(:,k) = lshoulder(:,k)-neck(:,k); % calculate theta angle A = [radarloc(1)-lshoulder(1,k); radarloc(2)-lshoulder(2,k);... radarloc(3)-lshoulder(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-lshoulder(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.06; b = 0.06; c = shoulderlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right shoulderfor k = 1:np % distance from radar to right shoulder r_dist(:,k) = abs(rshoulder(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right shoulder aspct(:,k) = rshoulder(:,k)-neck(:,k); % calculate theta angle A = [radarloc(1)-rshoulder(1,k); radarloc(2)-rshoulder(2,k);... radarloc(3)-rshoulder(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rshoulder(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.06; b = 0.06; c = shoulderlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from left upper-armfor k = 1:np % distance from radar to left upper-arm lupperarm(:,k) = (lshoulder(:,k)+lelbow(:,k))/2; r_dist(:,k) = abs(lupperarm(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2);


% aspect vector of the left upper-arm aspct(:,k) = lshoulder(:,k)-lelbow(:,k); % calculate theta angle A = [radarloc(1)-lupperarm(1,k); radarloc(2)-lupperarm(2,k);... radarloc(3)-lupperarm(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-lupperarm(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.06; b = 0.06; c = upperarmlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right upper-armfor k = 1:np % distance from radar to right upper-arm rupperarm(:,k) = (rshoulder(:,k)+relbow(:,k))/2; r_dist(:,k) = abs(rupperarm(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right upper-arm aspct(:,k) = rshoulder(:,k)-relbow(:,k); % calculate theta angle A = [radarloc(1)-rupperarm(1,k); radarloc(2)-rupperarm(2,k);... radarloc(3)-rupperarm(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rupperarm(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.06; b = 0.06; c = upperarmlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from left lower-armfor k = 1:np % distance from radar to left lower-arm r_dist(:,k) = abs(lhand(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2);


% aspect vector of the left lower-arm aspct(:,k) = lelbow(:,k)-lhand(:,k); % calculate theta angle A = [radarloc(1)-lhand(1,k); radarloc(2)-lhand(2,k);... radarloc(3)-lhand(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-lhand(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.05; b = 0.05; c = lowerarmlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right lower-armfor k = 1:np % distance from radar to right lower-arm r_dist(:,k) = abs(rhand(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right lower-arm aspct(:,k) = relbow(:,k)-rhand(:,k); % calculate theta angle A = [radarloc(1)-rhand(1,k); radarloc(2)-rhand(2,k);... radarloc(3)-rhand(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rhand(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.05; b = 0.05; c = lowerarmlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from left hipfor k = 1:np % distance from radar to left hip r_dist(:,k) = abs(lhip(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2);


% aspect vector of the left hip aspct(:,k) = lhip(:,k)-base(:,k); % calculate theta angle A = [radarloc(1)-lhip(1,k); radarloc(2)-lhip(2,k);... radarloc(3)-lhip(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-lhip(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.07; b = 0.07; c = hiplen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right hipfor k = 1:np % distance from radar to right hip r_dist(:,k) = abs(rhip(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right hip aspct(:,k) = rhip(:,k)-base(:,k); % calculate theta angle A = [radarloc(1)-rhip(1,k); radarloc(2)-rhip(2,k);... radarloc(3)-rhip(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rhip(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.07; b = 0.07; c = hiplen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from left upper-legfor k = 1:np % distance from radar to left upper-leg lupperleg(:,k) = (lhip(:,k)+lknee(:,k))/2; r_dist(:,k) = abs(lupperleg(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2);


% aspect vector of the left upper-leg aspct(:,k) = lknee(:,k)-lhip(:,k); % calculate theta angle A = [radarloc(1)-lupperleg(1,k); radarloc(2)-lupperleg(2,k);... radarloc(3)-lupperleg(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-lupperleg(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.07; b = 0.07; c = upperleglen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right upper-legfor k = 1:np % distance from radar to right upper-leg rupperleg(:,k) = (rhip(:,k)+rknee(:,k))/2; r_dist(:,k) = abs(rupperleg(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right upper-leg aspct(:,k) = rknee(:,k)-rhip(:,k); % calculate theta angle A = [radarloc(1)-rupperleg(1,k); radarloc(2)-rupperleg(2,k);... radarloc(3)-rupperleg(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rupperleg(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.07; b = 0.07; c = upperleglen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end radar returns from left lower-legfor k = 1:np % distance from radar to left lower-leg llowerleg(:,k) = (lankle(:,k)+lknee(:,k))/2; r_dist(:,k) = abs(llowerleg(:,k)-radarloc(:));


distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the left lower-leg aspct(:,k) = lankle(:,k)-lknee(:,k); % calculate theta angle A = [radarloc(1)-llowerleg(1,k); radarloc(2)-llowerleg(2,k);... radarloc(3)-llowerleg(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-llowerleg(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.06; b = 0.06; c = upperleglen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right lower-legfor k = 1:np % distance from radar to right lower-leg rlowerleg(:,k) = (rankle(:,k)+rknee(:,k))/2; r_dist(:,k) = abs(rlowerleg(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right lower-leg aspct(:,k) = rankle(:,k)-rknee(:,k); % calculate theta angle A = [radarloc(1)-rlowerleg(1,k); radarloc(2)-rlowerleg(2,k);... radarloc(3)-rlowerleg(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rlowerleg(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.06; b = 0.06; c = upperleglen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from left footfor k = 1:np % distance from radar to left foot r_dist(:,k) = abs(ltoe(:,k)-radarloc(:));


distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the left foot aspct(:,k) = lankle(:,k)-ltoe(:,k); % calculate theta angle A = [radarloc(1)-ltoe(1,k); radarloc(2)-ltoe(2,k);... radarloc(3)-ltoe(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-ltoe(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.05; b = 0.05; c = footlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from right footfor k = 1:np % distance from radar to right foot r_dist(:,k) = abs(rtoe(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the right foot aspct(:,k) = rankle(:,k)-rtoe(:,k); % calculate theta angle A = [radarloc(1)-rtoe(1,k); radarloc(2)-rtoe(2,k);... radarloc(3)-rtoe(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-rtoe(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); a = 0.05; b = 0.05; c = footlen/2; rcs(k) = rcsellipsoid(a,b,c,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % display range profilesfigurecolormap(jet(256))imagesc([1,np],[0,nr*rangeres],20*log10(abs(data)+eps))xlabel(‘Pulses’)


ylabel(‘Range (m)’)title(‘Range Profiles’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-40 0]);colorbardrawnow % micro-Doppler signaturex = sum(data); % average over range cellsdT = T/np;F = 1/dT; % np/T; wd = 512;wdd2 = wd/2;wdd8 = wd/8;ns = np/wd; % calculate time-frequency micro-Doppler signaturedisp(‘Calculating segments of TF distribution ...’)for k = 1:ns disp(strcat(‘ segment progress: ‘,num2str(k),’/’,num2str(round(ns)))) sig(1:wd,1) = x(1,(k-1)*wd+1:(k-1)*wd+wd); TMP = stft(sig,16); TF2(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);endTF = TF2;disp(‘Calculating shifted segments of TF distribution ...’)TF1 = zeros(size(TF));for k = 1:ns-1 disp(strcat(‘ shift progress: ‘,num2str(k),’/’,num2str(round(ns-1)))) sig(1:wd,1) = x(1,(k-1)*wd+1+wdd2:(k-1)*wd+wd+wdd2); TMP = stft(sig,16); TF1(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);enddisp(‘Removing edge effects ...’)for k = 1:ns-1 TF(:,k*wdd8-8:k*wdd8+8) = ... TF1(:,(k-1)*wdd8+wdd8/2-8:(k-1)*wdd8+wdd8/2+8);end % display final time-frequency signature figurecolormap(jet(256))imagesc([0,T],[-F/2,F/2],20*log10(fftshift(abs(TF),1)+eps))xlabel(‘Time (s)’)ylabel(‘Doppler (Hz)’)title(‘Micro-Doppler Signature of Human Walk’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-45 0]);colorbardrawnow


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Appendix 4B

MATLAB Source Codes

BirdFlapping.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Simulation on Bird Flapping Model with Two Jointed Arms% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all t = 0:0.01:10-0.01;F0 = 1; % flapping frequencyV = 0.2; % forward translation velocity% segment 1 (upper arm) A1 = 40; % amplitude of flapping angel in degreeksi10 = 15; % lag flapping angle in degreeL1 = 0.5; % length of segment 1ksi1 = A1*cos(2*pi*F0*t)+ksi10; % flapping angle% the elbow joint position in local coordinatesx1 = V*t; %zeros(1,length(t));y1 = L1*cosd(ksi1);z1 = y1.*tand(ksi1); % segment 2 (forearm)A2 = 30; % amplitude of segment2 flapping angleksi20 = 40; % lag flapping angle in degreeL2 = 0.5; % length of segment 2C2 = 20; % amplitude of segment2 twisting angleksi2 = A2*cos(2*pi*F0*t)+ksi20;theta2 = C2*sin(2*pi*F0*t);% the wrist joint position in local coordinatesd = theta2./cosd(ksi1-ksi2);y2 = L1*cosd(ksi1)+L2*cosd(theta2).*cosd(ksi1-ksi2);x2 = x1-(y2-y1).*tand(d);z2 = z1+(y2-y1).*tand(ksi1-ksi2); figure(1)for k=1:5:60 hold on % body patch([-0.4 0.4],[0 0],[0 0],’k’,’linewidth’,8) % left wing patch([0 x1(k)],[0 y1(k)],[0 z1(k)],’k’,’linewidth’,2) patch([x1(k) x2(k)],[y1(k) y2(k)],[z1(k) z2(k)],’k’,’linewidth’,2) % right wing patch([0 x1(k)],[0 -y1(k)],[0 z1(k)],’k’,’linewidth’,2) patch([x1(k) x2(k)],[-y1(k) -y2(k)],[z1(k) z2(k)],’k’,’linewidth’,2) axis equal axis([-1,1,-1,1,-1,1])


grid on set(gcf,’Color’,[1 1 1]) view([80,2]) xlabel(‘X’) ylabel(‘Y’) zlabel(‘Z’) title(‘Bird Wing Flapping’) drawnow end figure(2)plot3(x2(1:2:end),y2(1:2:end),z2(1:2:end),’.’)hold onplot3(x2(1:2:end),-y2(1:2:end),z2(1:2:end),’.’)patch([-0.2+x1(end) 0.2+x1(end)],[0 0],[0 0],’k’,’linewidth’,8)patch([x1(end) x1(end)],[0 y1(end)],[0 z1(end)],’k’,’linewidth’,5)patch([x1(end) x2(end)],[y1(end) y2(end)],[z1(end) z2(end)],... ‘k’,’linewidth’,5)patch([x1(end) x1(end)],[0 -y1(end)],[0 z1(end)],’k’,’linewidth’,5)patch([x1(end) x2(end)],[-y1(end) -y2(end)],[z1(end) z2(end)],... ‘k’,’linewidth’,5)gridaxis imageaxis([-0.5 2.5 -1 1 -0.8 0.8])view(40,30)xlabel(‘x’)ylabel(‘y’)zlabel(‘z’)title(‘Flight Flapping Trajectory of Wing Tips’)drawnow figure(3)colormap([0.7 0.7 0.7])for k=1:10:length(t) clf hold on % body [x,y,z] = ellipsoid2P([-0.4+x1(k),0,0],[0.4+x1(k),0,0],0.1,0.1,0.4,30); surf(x,y,z) % left wing [x,y,z] = ellipsoid2P([x1(k),0,0],[x1(k),y1(k),z1(k)],... 0.05,0.05,0.25,30); surf(x,y,z) [x,y,z] = ellipsoid2P([x1(k),y1(k),z1(k)],[x2(k),y2(k),z2(k)],... 0.05,0.05,0.25,30); surf(x,y,z) % right wing [x,y,z] = ellipsoid2P([x1(k),0,0],[x1(k),-y1(k),z1(k)],... 0.05,0.05,0.25,30); surf(x,y,z) [x,y,z] = ellipsoid2P([x1(k),-y1(k),z1(k)],[x2(k),-y2(k),z2(k)],... 0.05,0.05,0.25,30); surf(x,y,z) light lighting gouraud


light(‘Position’,[10 5 10],’Style’,’infinite’); shading interp axis equal axis([-2,2,-2,2,-2,2]) axis off grid off set(gcf,’Color’,[1 1 1]) view([80,-10]) zoom(2) drawnow pause(0.1)end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

RadarBirdReturns.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Radar returns from bird flapping wings%% Radar: X-band wavelength = 0.03 m% Range resolution 0.05 m% Observation time: T = 10 sec% Number of pulses 8192% Location: X = 10 m; Y = 0 m; Z = 2 m% Bird: Body center location X = 0; Y = 0; Z = 0 m% Wing length L = 1 m% Length of the upper arm L1 = 0.5% Length of the forearm L2 = 0.5 m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all j = sqrt(-1); % radar parameters% time samplingT = 10; % time durationnt = 8192; % number of samplesdt = T/nt; % time intervalts = [0:dt:T-dt]; % time spanlambda = 0.03; % wavelength of transmitted radar signalc = 2.99792458e8;f0 = c/lambda;rangeres = 0.05; % designed range resolutionradarloc = [20, 0, -10]; % radar location% total number of range binsnr = floor(2*sqrt(radarloc(1)^2+radarloc(2)^2+radarloc(3)^2)/rangeres);

t = ts;F0 = 1.0; % flapping frequencyV = 1.0; % forward translation velocity% segment 1 (upper arm)


A1 = 40; % amplitude of flapping angle in degreeksi10 = 15; % lag flapping angle in degreeL1 = 0.5; % length of segment 1ksi1 = A1*cos(2*pi*F0*t)+ksi10; % flapping angle% the elbow joint position in local coordinatesx1 = V*t; %zeros(1,length(t));y1 = L1*cosd(ksi1);z1 = y1.*tand(ksi1); % segment 2 (forearm)A2 = 30; % amplitude of segment2 flapping angleksi20 = 40; % lag flapping angle in degreeL2 = 0.5; % length of segment 2C2 = 20; % amplitude of segment2 twisting angleksi2 = A2*cos(2*pi*F0*t)+ksi20;theta2 = C2*sin(2*pi*F0*t);% the wrist joint position in local coordinatesd = theta2./cosd(ksi1-ksi2);y2 = L1*cosd(ksi1)+L2*cosd(theta2).*cosd(ksi1-ksi2);x2 = x1-(y2-y1).*tand(d);z2 = z1+(y2-y1).*tand(ksi1-ksi2); % bird body positionPb1(1,:) = -0.4+x1(:);Pb1(2,:) = zeros(size(x1));Pb1(3,:) = zeros(size(x1));Pb2(1,:) = 0.4+x1(:);Pb2(2,:) = zeros(size(x1));Pb2(3,:) = zeros(size(x1)); % body center pointCb = (Pb1+Pb2)/2; % left upper arm positionPua11 = Cb;Pua21 = [x1; y1; z1];Puac1 = (Pua11+Pua21)/2; % upper arm center % left forearm positionPfa11 = Pua21;Pfa21 = [x2; y2; z2];Pfac1 = (Pfa11+Pfa21)/2; % forearm center % right upper arm positionPua12 = Cb;Pua22 = [x1; -y1; z1];Puac2 = (Pua12+Pua22)/2; % upper arm center % right forearm positionPfa12 = Pua22;Pfa22 = [x2; y2; z2];Pfac2 = (Pfa12+Pfa22)/2; % forearm center % ellipsoid parameter% body


Ba = 0.1;Bb = 0.1;Bc = 1.0;% wingWa = 0.05; Wb = 0.05; Wc = 0.25; % radar returns% prepare data collectiondata = zeros(nr,nt); % radar returns from the bodyfor k = 1:nt % distance from radar to bird body r_dist(:,k) = abs(Cb(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the body aspct(:,k) = Pb2(:,k)-Pb1(:,k); % calculate theta angle A = [radarloc(1)-Cb(1,k); radarloc(2)-Cb(2,k);... radarloc(3)-Cb(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-Cb(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); rcs(k) = rcsellipsoid(Ba,Bb,Bc,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from the left upper arm wingfor k = 1:nt % distance from radar to bird left upper arm r_dist(:,k) = abs(Puac1(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the body aspct(:,k) = Pua21(:,k)-Pua11(:,k); % calculate theta angle A = [radarloc(1)-Puac1(1,k); radarloc(2)-Puac1(2,k);... radarloc(3)-Puac1(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-Puac1(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); rcs(k) = rcsellipsoid(Wa,Wb,Wc,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k));


PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from the left forearm wingfor k = 1:nt % distance from radar to bird body r_dist(:,k) = abs(Pfac1(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the body aspct(:,k) = Pfa21(:,k)-Pfa11(:,k); % calculate theta angle A = [radarloc(1)-Pfac1(1,k); radarloc(2)-Pfac1(2,k);... radarloc(3)-Pfac1(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-Pfac1(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); rcs(k) = rcsellipsoid(Wa,Wb,Wc,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from the right upper arm wingfor k = 1:nt % distance from radar to bird body r_dist(:,k) = abs(Puac2(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the body aspct(:,k) = Pua22(:,k)-Pua12(:,k); % calculate theta angle A = [radarloc(1)-Puac2(1,k); radarloc(2)-Pua21(2,k);... radarloc(3)-Puac2(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-Puac2(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); rcs(k) = rcsellipsoid(Wa,Wb,Wc,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end % radar returns from the right forearm wing


for k = 1:nt % distance from radar to bird body r_dist(:,k) = abs(Pfac2(:,k)-radarloc(:)); distances(k) = sqrt(r_dist(1,k).^2+r_dist(2,k).^2+r_dist(3,k).^2); % aspect vector of the body aspct(:,k) = Pfa22(:,k)-Pfa12(:,k); % calculate theta angle A = [radarloc(1)-Pfac2(1,k); radarloc(2)-Pfac2(2,k);... radarloc(3)-Pfac2(3,k)]; B = [aspct(1,k); aspct(2,k); aspct(3,k)]; A_dot_B = dot(A,B,1); A_sum_sqrt = sqrt(sum(A.*A,1)); B_sum_sqrt = sqrt(sum(B.*B,1)); ThetaAngle(k) = acos(A_dot_B ./ (A_sum_sqrt .* B_sum_sqrt)); PhiAngle(k) = asin((radarloc(2)-Pfac2(2,k))./... sqrt(r_dist(1,k).^2+r_dist(2,k).^2)); rcs(k) = rcsellipsoid(Wa,Wb,Wc,PhiAngle(k),ThetaAngle(k)); amp(k) = sqrt(rcs(k)); PHs = amp(k)*(exp(-j*4*pi*distances(k)/lambda)); data(floor(distances(k)/rangeres),k) = ... data(floor(distances(k)/rangeres),k) + PHs; end figure(1)for k = 1:20:nt clf hold on colormap([0.7 0.7 0.7]) [x,y,z] = ellipsoid2P([-0.4+x1(k),0,0],[0.4+x1(k),0,0],0.1,0.1,0.4,30); surf(x,y,z) % left wing [x,y,z] = ellipsoid2P([x1(k),0,0],[x1(k),y1(k),z1(k)],... 0.05,0.05,0.25,30); surf(x,y,z) [x,y,z] = ellipsoid2P([x1(k),y1(k),z1(k)],[x2(k),y2(k),z2(k)],... 0.05,0.05,0.25,30); surf(x,y,z) % right wing [x,y,z] = ellipsoid2P([x1(k),0,0],[x1(k),-y1(k),z1(k)],... 0.05,0.05,0.25,30); surf(x,y,z) [x,y,z] = ellipsoid2P([x1(k),-y1(k),z1(k)],[x2(k),-y2(k),z2(k)],... 0.05,0.05,0.25,30); surf(x,y,z) light lighting gouraud light(‘Position’,[20 10 20],’Style’,’infinite’); shading interp axis equal axis([0,20,-5,5,-10,0]) axis on grid on set(gcf,’Color’,[1 1 1]) view([30,15]) % draw radar location


plot3(radarloc(1),radarloc(2),radarloc(3),’-ro’,... ‘LineWidth’,2,... ‘MarkerEdgeColor’,’r’,... ‘MarkerFaceColor’,’y’,... ‘MarkerSize’,10) % draw a line from radar to the target center line([radarloc(1) Cb(1,k)],[radarloc(2) Cb(2,k)],... [radarloc(3) Cb(3,k)],... ‘color’,[0.4 0.7 0.7],’LineWidth’,1.5,’LineStyle’,’-’) xlabel(‘X’) ylabel(‘Y’) zlabel(‘Z’) title(‘Radar Tracking a Flying Bird’) drawnow end figure(2)colormap(jet(256))imagesc([1,nt],[0,nr*rangeres],20*log10(abs(data)+eps))xlabel(‘Pulses’)ylabel(‘Range (m)’)title(‘Range Profiles of Bird Flapping Wings’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-20 10]);colorbardrawnow % micro-Doppler signature x = sum(data);np = nt; dT = T/length(ts);F = 1/dT;dF = 1/T; wd = 512;wdd2 = wd/2;wdd8 = wd/8;ns = np/wd; % calculate time-frequency micro-Doppler signaturedisp(‘Calculating segments of TF distribution ...’)for k = 1:ns disp(strcat(‘ segment progress: ‘,num2str(k),’/’,num2str(round(ns)))) sig(1:wd,1) = x(1,(k-1)*wd+1:(k-1)*wd+wd); TMP = stft(sig,16); TF2(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);endTF = TF2;disp(‘Calculating shifted segments of TF distribution ...’)TF1 = zeros(size(TF));for k = 1:ns-1


disp(strcat(‘ shift progress: ‘,num2str(k),’/’,num2str(round(ns-1)))) sig(1:wd,1) = x(1,(k-1)*wd+1+wdd2:(k-1)*wd+wd+wdd2); TMP = stft(sig,16); TF1(:,(k-1)*wdd8+1:(k-1)*wdd8+wdd8) = TMP(:,1:8:wd);enddisp(‘Removing edge effects ...’)for k = 1:ns-1 TF(:,k*wdd8-8:k*wdd8+8) = ... TF1(:,(k-1)*wdd8+wdd8/2-8:(k-1)*wdd8+wdd8/2+8);end % display final time-frequency signaturefigure(3)colormap(jet(256))imagesc([0,T],[-F/2,F/2],20*log10(fftshift(abs(TF),1)+eps))xlabel(‘Time (s)’)ylabel(‘Doppler (Hz)’)title(‘Micro-Doppler Signature of Flapping Wings’)axis xyclim = get(gca,’CLim’);set(gca,’CLim’,clim(2) + [-30 5]);colorbardrawnow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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5Analysis and Interpretation of Micro-Doppler SignaturesMicro-Doppler signatures are generated by the motion of objects. Human and animal motion (called the biological motion) contains a rich source of informa-tion on body movements, actions, and intentions. Human visual systems can easily retrieve information from biological motion and identify a body from its motion pattern. Why the human visual system can easily identify an object through its motion is not quite understood yet. How exactly the biologically and psychologically relevant information is encoded in the motion pattern is also not known.

Human motion is a major biological motion, and therefore, numerous studies on human motion perception have been ongoing for a long time. Many experiments have demonstrated how human observers recognize human figures through their motion and identify types of motion patterns. Tremendous ex-periments in human motion perception certainly help for selecting meaningful features from micro-Doppler signatures to identify types of motion patterns and recognize a person through its motion.

Because the micro-Doppler signature is a signature of object’s motion and the Doppler directly relates to the motion velocity, the method of extracting motion kinematic features from micro-Doppler signatures becomes an impor-tant task. If the kinematic features of an object can be extracted, the results of studies in human motion perception can be applied directly and micro-Dop-pler signatures can immediately be used to reconstruct the object’s movement.

In this chapter, the useful results on visual perception of biological motion and identification through visual motion patterns will be briefly introduced. Then, based on the knowledge of biological motion perception, the methods


of how to analyze and interpret micro-Doppler signatures, how to decompose a micro-Doppler signature into such components that are associated with body structural parts, and how to select features from micro-Doppler components will be described.

5.1 Biological Motion Perception

The pioneering experimental work on biological motion perception by Johans-son in 1973 and 1976 proved that visual motion perception of point-light dis-plays (PLD) attached to a limited number of major joints in a walking person can help observers to immediately recognize the structure of the human body [1, 2]. They showed that 0.2 second of a time interval was sufficient for an ob-server to recognize the figure of the walkers, and 0.4 second of a time interval was sufficient to identify the types of human motion patterns.

Following Johansson’s work, many researchers studied how the structural movements of a walker and the kinematic information of the movement help for identifying types of the movement, recognizing the person, and even iden-tifying the gender of the person [3–8].

It is well known that people can recognize a friend based on familiarity cues, such as face, hairstyle, or even gaiting. Face recognition is the most com-mon method for recognizing a person. However, whether and how a person can be recognized by his or her motion pattern is questionable. Many researchers have investigated this topic for many years [4, 9, 10]. Cutting and Kozlowski conducted the first experiment on person recognition from biological motion through the PLDs [4]. They recorded the gait patterns of a number of persons who were familiar to each other, but not familiar with their figures of PLDs. In this experiment, these persons repeatedly watched the PLDs from individual persons. This experiment showed that the PLDs not only are sufficient to in-dicate the presence of a human structure such as Johansson’s demonstration [1, 2], but also contain sufficient information for identifying individual person through motion patterns. This experiment demonstrated that, from learning PLDs, people can differentiate previously unknown persons by the way they walk. However, the experiment was still unclear whether there were specific fac-tors or parameters that were intrinsic to the walkers used for the identification. Researchers have also studied, from the visual motion perception point of view, which parts of structural information and kinematic information were relevant for identifying the types of movement and the identity of a person.

Troje et al. have made significant contributions to many research topics on biological motion perception and motion decomposition [7, 8, 11]. They showed that the perception of biological motion depends on the links between

Analysis and Interpretation of Micro-Doppler Signatures 249

joints (i.e., the point-light pairs), the trajectories of the individual PLDs (called the local motion information), and the information on the entire figure of the PLDs across a larger spatiotemporal interval (called the global motion informa-tion). It was found that recognizing biological motion is not solely based on the local motion information, but it is also based on the perception of the global figure. Figure 5.1 illustrates a running person represented by a limited number of point displays.

Troje et al. also proposed methods based on the principal component analysis (PCA) [7] and the Fourier analysis [8] to decompose human walking data into structural and kinematic information. The human walking pattern is an averaged posture represented by the local motion of the PLDs along with the period of gaiting. From the kinematic data, after extracting the averaged posture of a walking person in PLDs and then subtracting the posture from the kinematic data, the residual kinematic data was decomposed by the PCA algorithm [7]. It was found that the first four principal components are the major components representing the walker’s posture. It was also found that the observers are more dependent on the kinematic information for recognizing individual walkers. Troje et al. also applied the Fourier decomposition method, which can be used to separately manipulate various attributes (such as size, shape, and gaiting frequency) and to examine the differential influence of these parameters on the identification of individuals.

Researchers found that the visual kinematics of PLDs can carry informa-tion on actions [2, 3, 12, 13], emotions [14–18], and even the gender of the walker [6–8, 19–21]. From the movements of the PLDs (such as knocking,

Figure 5.1 A running person represented by a limited number of point displays.


lifting, and waving), human observers can discriminate between the neutral affect and the angry affect. Experiments also showed that the ability of distin-guishing between different emotions from the PLDs was based on the informa-tion of motion kinematics (such as velocity, acceleration, and jerk).

Knowing biological motion perception from the PLDs, the next step is how to extract these necessary kinematic and structural features from micro-Doppler signatures. The micro-Doppler signature is a characteristic of the intri-cate frequency modulations generated from each component part of an object and represented in the joint time and Doppler frequency domain. Based on the results of human motion perception from the PLDs of body component parts, it can be inferred that motion kinematics of human body components in the micro-Doppler signatures are also carry information on human actions and emotions.

However, the micro-Doppler signature of a human movement is a su-perposition of micro-Doppler components from individual body component parts. Thus, it is necessary to decompose a micro-Doppler signature into com-ponents that correspond to the motion from human body component parts. The decomposition of human micro-Doppler signatures therefore becomes a challenging problem that may lead to the identification of human actions and emotions through micro-Doppler signatures.

5.2 Decomposition of Biological Motion

The decomposition of biological motion is the decomposition of motion data into structural information components and kinematic information compo-nents. The method is based on a transformation that can transfer the biologi-cal motion data into a representation that allows the analysis using statistics and pattern recognition techniques. The transformation can be a Fourier-based transform or the PCA. In the Fourier-based method, the direct current (DC) component of the transform encodes the structural information of the moving body’s geometry, and the dynamic alternating current (AC) components of the transform encode the kinematic motion information. In the PCA method, the residual kinematic data, after subtracting the averaged posture, is used for the decomposition. The PCA is a commonly used tool in the statistic data analy-sis [22], which calculates the eigenvalue decomposition of the data covariance matrix. It can reduce the dimensionality of the data and reveal the internal structure of the data through the data variance. For biological locomotion data, the PCA method actually results in a discrete Fourier decomposition, which is optimal in the sense of maximizing variance with a minimum number of components [8].


However, decomposing biological motion is not decomposing micro-Doppler signatures. The method of how to decompose a complex micro-Dop-pler signature into components corresponding to the motion of human body components is still a challenging task.

5.2.1 Statistics-Based Decomposition

The statistically independent decomposition includes PCA or singular value decomposition (SVD) and independent component analysis (ICA) [22–25]. The PCA is a tool in statistical data analysis and uses eigenvectors with the larg-est eigenvalues to obtain a set of function bases so that the original function can be represented by a linear combination of these bases. The function bases found by the PCA are uncorrelated (i.e., they cannot be linearly predicted from each other). However, higher-order dependencies still exist in the PCA and these bases are not optimally separated. The SVD is a generalization of the PCA and can decompose a nonsquared matrix, which is possible to directly decompose the time-frequency distribution without using a covariance matrix.

The ICA was originally used for separating mixed signals into indepen-dent components, called the blind source separation (BSS). The ICA minimizes the statistical dependence between basis feature vectors and searches for a linear transformation to express a set of features as a linear combination of statistically independent function bases. As is well known, independent events must be uncorrelated, but uncorrelated events may not be independent. The PCA only requires the components to be uncorrelated. The ICA is independent and ac-counts for higher-order statistics. Thus, it is a more powerful feature representa-tion than the PCA. In fact, the PCA, just like the Fourier analysis, is basically a global component analysis, whereas the ICA, like the time-frequency analysis, is basically a localized component analysis.

Certainly, the statistically decomposed components are not these mono-components that correspond to micro-Doppler signatures generated from indi-vidual body parts. The desired decomposition method for micro-Doppler sig-natures should be the one that is based on the physical structural components of human body parts.

5.2.2 Decomposition of Micro-Doppler Signatures in the Joint Time-Frequency Domain

The micro-Doppler signature is represented in the joint time and Doppler fre-quency domain, and is a superposition of monocomponent signatures of indi-vidual body parts. There are many methods based on selecting basis (kernel) functions to decompose a time-frequency distribution into independently con-trollable components that are used to reconstruct the original time-frequency


distribution. However, these kernel functions are elementary basis functions localized in the joint time and frequency domain, but are not associated with monocomponent signatures of individual body parts.

Another type of decomposition is the statistically independent decompo-sition using PCA or SVD and ICA. These decomposed components are uncor-related or independent, but not necessarily associated with monocomponent micro-Doppler signatures of individual body parts because these monocompo-nent signatures are often correlated or dependent due to synchronized locomo-tion of the human body parts.

The EMD introduced in Chapter 1 is a signal decomposition method that decomposes an original signal into component waveforms, called the intrinsic mode functions (IMF), that are modulated in their amplitude and frequency by searching all of the oscillatory modes in the signal [26]. One property of the IMFs is that different IMFs do not have the same instantaneous frequency at the same time instant. The EMD has been applied to micro-Doppler signatures for extracting radar signal components generated by rotating or vibrating body structures [27]. Although the EMD can extract informative harmonic compo-nents from human motion data, there is no connection to structures of human body parts. Therefore, the most useful but difficult decomposition method for micro-Doppler signatures should be the one that is based on the physical struc-tural components of human body parts.

5.2.3 Physical Component–Based Decomposition

Although the micro-Doppler signature of a human motion is not a visual per-ception of the motion, it is directly related to the kinematic information about the moving structures of the human body. Human observers can easily track the component signatures from the micro-Doppler signature. The question is what algorithm can be used to track these component signatures. In general, it is not simple to track and isolate any individual component in the entire micro-Doppler signature. An effective algorithm of the physical component–based decomposition is still an open issue.

In [28, 29], a framework was proposed for decomposing a micro-Doppler signature into components that are associated with the physical parts of a hu-man body. A micro-Doppler signature of a walking person and the decomposed signatures, called motion curves, that correspond to the different physical parts of the human body are shown in Figure 5.2. The decomposition algorithm used is described in the follows. First, the class of the motion must be predetermined from the given micro-Doppler signature by examining the structure of varia-tions of the upper and lower envelopes in the micro-Doppler signature being analyzed. This was accomplished by means of a Markov chain–based inference algorithm [28]. The advantage of focusing on the upper and lower envelopes


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is that their micro-Doppler structure remains relatively invariant to the distor-tions inherent in the acquisition process. Inferring the motion class yields valu-able prior knowledge about the internal time-frequency structure of the signal to be analyzed; such knowledge is obtained by prior simulation studies in [30]. Conditioned on this knowledge, the motion class of the remaining body parts is determined by the following process.

The micro-Doppler signature being analyzed is segmented into consecu-tive different half-cycles such that each half-cycle corresponds to roughly half the period of motion, as shown in Figure 5.2(b). This segmentation is accom-plished by means of a nonlinear optimization program that is solved via dynam-ic programming. Thus, the motion curves extracted for each half-cycle can then be concatenated to form the overall motion sequence of the human. In order to extract the motion curves for each half-cycle, the significant local maxima correspond to the maximal time cell of the half-cycle. The maximal time cell is at the time coordinate where the upper/lower envelope attains global maxima. These significant local maxima are determined by means of the same nonlinear optimization algorithm described in [28]. Given this, the local corresponding maxima at the remaining time cells are determined by means of a partial track-ing algorithm described in [28, 29]. The concatenation of the corresponding local maxima at different time cells yields the motion curves for each of the body parts. The number of local maxima to be computed (i.e., the number of body parts for determining the motion curves) is known by prior knowledge obtained by simulation studies and associated with the inferred motion class. Then the initial set of motion curve estimates obtained are refined by means of a Gaussian g-Snakes–based quality measure as described in [28, 29], where Gaussian g-Snakes is a model of the micro-Doppler structure of the human gait motion that enables a blind assessment of the quality of the motion curves estimates in Figure 5.2(c) [28, 29].

Figure 5.3(a) shows the micro-Doppler signature of a running person. Figure 5.3(b) shows the corresponding decomposed motion curves extracted for this running motion.

This proposed framework was applied to the decomposition of micro-Doppler signatures of human walking and running, and demonstrated motion curves extracted from several human body parts. However, a more generalized algorithm that decomposes a more general micro-Doppler signature into mo-tion curves of physical body parts is expected for future work. To achieve this goal, it is necessary to understand in quantitative detail the nonlinear interac-tions between the various motion curves that comprise the motion, together with the need to understand, at a more basic level, the dynamics of the human gait motion including the efficient representation of complex motion dynamics and their effect on the micro-Doppler signature.


Figure 5.3 (a) The micro-Doppler signature of a running person, and (b) the decomposed motion curves from the micro-Doppler signature of the running person in (a).


5.3 Extraction of Features from Micro-Doppler Signatures

Because the most often performed human motion is walking, extracting motion features from micro-Doppler signatures of a walking human is a basic method for extracting motion features from micro-Doppler signatures.

In this section, the global human walking model [30] is used to create a human walking model and generate the micro-Doppler signature of a walking human. Because of the given human walking model, each human body part can be isolated from other parts and the micro-Doppler component of each body part can be generated and extracted. Thus, structural information and kinematic information of each human body part may be extracted.

Using the global human walking model in [30], a radar with a wavelength of 0.02m is assumed at (X

1 = 10m, Y

1 = 0m, Z

1 = 2m), a walking human is

started from the human base point located at (X0 = 0m, Y

0 = 0m, Z

0 = 0m), the

relative velocity of the walking person is VR = 1.0 m/s, and the height of the

person is assumed to be H = 1.8m, as illustrated in Figure 5.4. As mentioned in Chapter 4, because the global human walking model is

based on averaging parameters from experimental measurements, it is just an averaging human walking model without information about personalized mo-tion features.

Figure 5.5 shows the micro-Doppler signature of the walking person. The micro-Doppler components of the feet, tibias, clavicles, and torso are marked in

Figure 5.4 Geometry of a walking person and the radar at (X1 = 10m, Y1 = 0m, Z1 = 2m).


Figure 5.5. The average of the torso Doppler frequency shift is about 133 Hz. Figure 5.6 shows the corresponding micro-Doppler components of the feet, tibias, radius, and torso, respectively.

Running is the next most characteristic movement of humans. Using the human running model derived from the captured human movement database [31], assume that the radar is located at (X

1 = 10m, Y

1 = 0m, Z

1 = 2m) with a

wavelength of 0.02m and that the starting point of the running person is from (X

0 = 0m, Y

0 = 0m, Z

0 = 0m). Figure 5.6 shows the micro-Doppler signature

of the running person. The micro-Doppler components of the right foot, right tibia, and torso are marked in Figure 5.7. The average of the Doppler frequency shift of the torso is about 350 Hz.

Figure 5.8 shows the corresponding micro-Doppler components of the feet, tibias, and torso. Figures 5.8(b–d) are extracted micro-Doppler compo-nents of the feet, tibias, and torso, respectively, from the micro-Doppler signa-ture of a running person [Figure 5.8(a)].

5.4 Estimation of Kinematic Parameters from Micro-Doppler Signatures

The micro-Doppler signature of human movement shows strong reflections from human torso due to its larger RCS. From the micro-Doppler component of the torso, the average torso velocity, the cycle of torso oscillation, and the amplitude of the torso Doppler oscillation can be measured. From the simula-tion result using the global human walking model, for a Ku-band radar operat-ing at 15 GHz, the torso motion velocity is oscillating from 1.0 m/s and going

Figure 5.5 The micro-Doppler signature of a walking person using the global human walking model.


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up to 1.67 m/s, as shown in Figure 5.9. The mean value of the torso velocity is v

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0.5 second, or the cycle of the torso oscillation is 2 Hz. The corresponding lower leg (tibia) motion parameters are shown in Fig-

ure 5.10. The average tibia velocity is vtibia

= 129 × 0.02/2 = 1.29 m/s, the cycle of the tibia oscillation is 1 Hz, and the maximum amplitude of the tibia oscil-lation velocity is about 3.2 m/s.

The corresponding foot motion parameters are shown in Figure 5.11. The average foot velocity is v

foot = 127 × 0.02/2 = 1.27 m/s, the cycle of the foot

oscillation is 1 Hz, and the maximum amplitude of the foot oscillation velocity is about 5.9 m/s. Half the foot oscillation cycle is the foot forward motion, and the other half of the cycle is the foot in contact with the ground, which makes the average velocity lower. The foot motion has the highest velocity, which is approximately four to five times the average foot velocity. The oscillation frequency of the torso is two times the tibia or the foot oscillation frequency because the torso accelerates while either foot swings.

Figure 5.7 The micro-Doppler signature of a running person.


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From the simulation of a running person in the captured human move-ment database, when the radar operates at 15 GHz, the torso motion velocity is oscillating from 3.0 m/s and going up to 3.39 m/s, as shown in Figure 5.12. The mean value of the torso velocity is v

torso = f

Dλ /2 = 320 × 0.02/2 = 3.20 m/s.

The oscillating period of the torso is 0.43 to 0.50 second, or the cycle of the torso oscillation is 2.0–2.3 Hz.

The corresponding lower leg (tibia) motion parameters are shown in Fig-ure 5.13. The average tibia velocity is v

tibia = 380 × 0.02/2 = 3.80 m/s, the cycle

of the tibia oscillation is 1.2 Hz, and the maximum amplitude of the tibia oscil-lation velocity is about 5.85 m/s.

The corresponding foot motion parameters are shown in Figure 5.14. The average foot velocity is v

foot = 415 × 0.02/2 = 4.15 m/s, the cycle of the foot

oscillation is about 1.2 Hz, and the maximum amplitude of the foot oscillation velocity is about 7.66 m/s. The foot motion has the highest velocity.

Figure 5.9 The torso velocity of a walking person.


5.5 Identifying Human Movements

The purpose of the human movement identification is to identify the type or model of the movement, such as walking, running, or jumping, after classifying the movements as being made by humans, animals, or vehicles. Existing ap-proaches to identify human movements in computer vision may be divided into two different types: the exploitation of structural information, and the use of motion information [33, 34]. These motion-based algorithms use human body part moments, eigenvectors, and hidden Markov models (HMMs) [35, 36]. Although these motion-based algorithms have been successful in many scenar-ios, because of the lack of structural information, in some other scenarios they could be worse than the methods that mainly use structural information [37].

A human motion pattern is a characteristic of an observation from human movements. Different types of movements generate different types of motion patterns. The micro-Doppler signature can be seen as a type of motion pattern. From a motion pattern, the extracted structural characteristic is a feature of the movement. It can be a distinctive measurement, a transformation, or a struc-

Figure 5.10 The tibia velocity of a walking person.


tural component. The features extracted from a motion pattern are the key to the identification of a human movement.

5.5.1 Features Used for Identifying Human Movements

It has been demonstrated that human movement could be identified through the point-light displays [1]. The experiment showed that by placing point lights on the joints of human body segments and filming human movements in a dark room, the point-light displays of the human movement can still provide a vivid impression of human movements. This means that the hierarchical structure of body segments and motion constraints of the segments and joints are charac-teristic features of human movements. These characteristic features are the key features for the identification of human movements.

To identify a complete human movement, the first task is to determine the class of the human movement, such as walking, running, jumping, or crawl-ing; the second task is to identify the phase of the motion, such as stance phase or swing phase when walking; and the third task is to predict possible motion continuation. With available decomposed human motion components, such as

Figure 5.11 The foot velocity of a walking person.


components that correspond to the motion of individual human body parts, it is possible to completely identify the human movement and the motion phase and to even predict the possible intention.

5.5.2 Anomalous Human Behavior

Anomaly detection technology is to identify abnormalities and learn what nor-mal behavior looks like by defining aberrant behavior and unusual occurrences. Knowing how normal behavior functions and what it usually looks like is key to detecting possible threats. As the number of threats grows and diversifies, an anomaly detection system becomes a required element of system security.

Detection of anomalous behavior is a difficult problem. Although visual observation of human actions has opened a window to understanding human mental activities, it is still difficult to understand how human movement is decoded into intentions, what properties of human movement make it special, and how these properties are organized to make cognitive representations.

Anomaly detection is an alarm for strange behavior. An activity profile of normal behavior over an interval of time is necessary for the anomaly detec-tion. This activity profile is used to compare with current events. Anything that

Figure 5.12 The torso velocity of a running person.


deviates from normal behavior is classified as anomalous. The methods of how to interpret human movements, track these movements, select appropriate fea-tures, and detect events of interest are still open issues.

A human activity is accomplished by a sequence of a complicated motion of human body parts. Any complicated motion can be described by a rotation about an axis through the center of mass and a translation of the center of mass. Thus, the kinematic information of a body part is the sum of kinematics of rotation and the translation of the body part. This kinematic information is the discriminative features that may have higher discriminatory power for rec-ognizing human activities. The kinematic information may be obtained from decomposed micro-Doppler signatures of human body components.

To detect human anomalous behavior, it is necessary to model human be-havior. An anomalous behavior may be detected if the human behavior pattern deviates from the typical learned prototypes. HMM is popular in computer vision as an activity recognition algorithm. HMMs have been successfully used to recognize hand gestures in sign language and facial expressions and to clas-sify activities in visual surveillance systems. An approach to automated visual surveillance is to classify the normal activities using a set of discrete HMMs,

Figure 5.13 The tibia velocity of a running person.


each trained to recognize one activity, and label the unrecognized activities as unusual.

The Markov model is a probabilistic technique for learning and match-ing activity patterns. Each type of activity for people or vehicle events may be characterized by a family of event trajectories passing through the image. Each family can be represented as an HMM in which states represent regions in the image, the prior probabilities measure the likelihood of an event starting in a particular region, and the transitional probabilities capture the likelihood of progression from one state to another across the image.

5.5.3 Summary

In this chapter, based on biological motion perception, the methods of de-composing micro-Doppler signatures into component signatures of body struc-tural parts have been discussed. The physical component–based decomposi-tion method opened a window for seeing the feasibility of extracting kinematic features from the micro-Doppler signatures. More effective algorithms for the signature decomposition are needed.

Figure 5.14 The foot velocity of a running person.


Using motion kinematic features to predict possible human actions and using them to identify anomalous human behaviors are still difficult tasks in computer vision. However, if kinematic parameters of each human body part can be extracted from the micro-Doppler signatures, just like those animated human models in computer graphics using sensor motion captured data, ani-mated human body parts may be generated from the captured micro-Doppler signatures. From here, it is possible to identify human actions and even anoma-lous behaviors from micro-Doppler signatures.

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271

6 Summary, Challenges, and PerspectivesMicro-Doppler effect was originally introduced in coherent LADAR systems. Because the micro-Doppler effect is sensitive to the transmitted frequency band of the system, for radar systems operating at microwave frequency bands, the micro-Doppler effect is not as strong as in the LADAR systems, but it is still observable in many cases.

The micro-Doppler signature of an object is the characteristics of the in-tricate frequency modulations made by the structural components of the object and it is represented in the joint time and Doppler frequency domain. An ob-ject or a structure on the object may have oscillatory motion, such as an oscil-lating pendulum, the rotating rotor blades of a helicopter, a spinning top, wind turbines, a walking person with swinging arms and legs, flying birds with flap-ping wings, and quadrupedal animal motion as studied in Chapters 3 and 4.

The primary purpose of this book is to introduce the principle of the micro-Doppler effect in radar and provide a simple and easy tool for generat-ing micro-Doppler signatures of received radar signals from targets of interest. The simulation of radar returns from a target with movements is an important method for studying micro-Doppler effect in radar. Instead of using collected real radar data, this book is an introduction to the simulation of the micro-Doppler effect in radar. Based on examples provided in this book, readers may modify and extend these examples to applications of their interest.

6.1 Summary

In the simulation of the micro-Doppler effect in radar, a suitable target model and a more accurate model of the target’s motion description are necessary. Ex-amples given in Chapters 3 and 4 provided commonly used models for targets,


methods for modeling the target’s motion equation, and simple RCS modeling for the simulation of radar returns. Micro-Doppler signatures of some typical rigid body and nonrigid body motions were introduced in Chapters 3 and 4. To generate the micro-Doppler signature of a target, the first thing is to select a model suitable for describing the target and a motion model for describing the nonlinear motion of the target. Several examples of modeling different types of rigid body and nonrigid body motions were given and corresponding MAT-LAB source codes were provided. Then, to simulate radar returned signals from targets, suitable RCS prediction models are needed. The simple point-scatterer model and a more accurate physical optics model were used in these examples. The POFacet is a simple RCS prediction software package using MATLAB code; it is based on the PO model [1, 2]. It can be downloaded and used for the calculation of monostatic and bistatic RCS of a static object and it is not included the source codes provided in this book.

Many researchers have studied the micro-Doppler effect in radar and ap-plied micro-Doppler signatures for target feature extraction and identification. However, not many of them tried to extract target features that are associated with the motion of individual parts of the target of interest. Any action of a target always comes from the movement of the target’s body parts. Thus, the extraction of target movement features that closely associate with the motions of structural parts of the target is extremely useful.

In Chapter 5, some useful results from biological motion perception re-search were introduced, which may help radar engineers to consider how to use these results in radar micro-Doppler signature analysis and in the identification of targets of interest from their micro-Doppler signatures. Similar to human motion perception through the PLDs, the kinematic information about targets motion is also encoded in the components of micro-Doppler signatures. This information should be appropriately extracted from the signatures in order to detect and identify targets of interest.

6.2 Challenges

Micro-Doppler signature analysis has been applied with certain success to ex-tracting target kinematic features and identifying targets of interest. However, how to effectively interpret these extracted features and associate them with structural parts of a target of interest is challenging. The success of solving the meaningful features of a target’s structural parts may lead to improving the performance of the identification and to further identify the intention and be-havior of a biological motion.

Summary, Challenges, and Perspectives 273

6.2.1 Decomposing Micro-Doppler Signatures

The micro-Doppler signature is directly related to the kinematic information of the structural parts of a target and is the superposition of monocomponent signatures associated with individual structural parts of a target and represented in the joint time-frequency domain.

Several time-frequency decomposition methods are available. Some de-composition methods, such as matching pursuit method and adaptive Gabor method [3, 4], can only decompose a time-frequency distribution into inde-pendently controllable localized components that can reconstruct the original time-frequency distribution. However, these decomposition methods do not consider the physical phenomenon that created the monocomponent signa-tures in the joint time-frequency representation. Thus, they are not the kind of decompositions that can associate with monocomponent signatures of individ-ual structural parts of a physical target. As discussed in Section 5.2.1, although the EMD can decompose micro-Doppler signatures into monocomponents, they have no connection to target body parts. The most useful decomposition is the one that is based on the physical components of a target.

The challenge is to find an effective automatic method that tracks mono-component signatures in micro-Doppler signatures and thus decomposes into those monocomponents associated with the physical structural parts of a target of interest. Human observers can easily track monocomponent signatures from the superposed micro-Doppler signature. How to effectively decompose micro-Doppler signatures into monocomponents that relate to the physical structural parts of a target and how to measure the embedded kinematic/structural infor-mation from monocomponent signatures are still open issues.

6.2.2 Feature Extraction and Target Identification Based on Micro-Doppler Signatures

The possible features and kinematic parameters that can be extracted from micro-Doppler signatures are the time information, the frequency or period of micro motion, the magnitude and the sign of Doppler frequency shifts, the position and moving direction, the linear velocity and acceleration, and the angular velocity and acceleration.

Because the micro-Doppler signature is generated from the motion of a target, and in the monostatic radar case, the measured Doppler frequency is directly related to the radial (LOS) velocity of a target, or in the bistatic ra-dar case, the measured Doppler frequency is related to the component of the velocity projected onto the line of bisector. Therefore, to accurately locate the position and measure the true moving direction and the velocity of the target, at least two monostatic radars or one monostatic radar and one bistatic radar are needed.


Figure 6.1 depicts how to measure the true velocity of a moving target using two radars. In a two-dimensional case, the moving target is located at (X

0, Y

0), and the two radars are located at (X

1, Y

1) and (X

2, Y

2), respectively.

The cross-point of line 1 and line 2 determines the true velocity of the moving target, where line 1 is the line that is perpendicular to the radial velocity V

R1

and line 2 is the line that is perpendicular to the radial velocity VR2

. If the initial position of a physical part of a target can be measured and the micro-Doppler signature of the target can be decomposed into monocomponents associated with physical parts of the target, then the basic kinematic parameters (positions and true velocities) of each physical part of the target can be estimated by two monostatic radars.

To completely describe a target motion in a 3-D Cartesian coordinate system, linear kinematic parameters (linear position, linear velocity, and linear acceleration) are the most important motion kinematic parameters, which de-fine the manner in which the position of any point in the human body changes over time. Linear velocity describes the rate of the position change with respect to time, and linear acceleration describes the rate of velocity change with time. These three kinematic parameters will be used to determine the characteristics of a target movement. Other three kinematic parameters are the angular kine-matics, which include the angular position (orientation) of a target body part, angular velocity, and angular acceleration of the body part. Because a target may consist of a number of parts, the measurement of joint angles between two parts are useful for describing target movement. These three angular kinematic

Figure 6.1 Estimation of the true velocity V of a moving object Q based on two radial veloci-ties VR1 and VR2.


parameters will be used jointly with three linear kinematic parameters to com-pletely describe the motion of the parts of the target. By carefully handling rotation and translation, the 3-D trajectories of these joint points between body parts can be obtained. These linear and angular kinematic parameters of the target movement can be used for classification, recognition, and identification of targets of interest.

6.3 Perspectives

Chapter 2 introduced the theoretical basis of micro-Doppler effect in radar. Micro-Doppler signatures have been investigated in a certain level for the ex-traction of target features and classification of targets of interest. Since the his-tory of the research on micro-Doppler effect in radar is relatively short, many aspects of research topics are still opened and need to be exploited. These top-ics include bistatic and multistatic micro-Doppler analysis, target classification with micro-Doppler features, through-the-wall micro-Doppler analysis, and polarimetric micro-Doppler analysis.

6.3.1 Multistatic Micro-Doppler Analysis

Multistatic radar has multiple transmitter/receiver nodes over distributed loca-tions [5]. Each node in the multistatic system can have only one transmitter or receiver. Multistatic radar is considered a combination of multiple bistatic radars that observes targets from different aspects. Thus, the informative data acquired from targets is increased because of the multiple aspects viewing. Mul-tistatic radar overcomes the target’s self-occlusion and the LOS dependency in monostatic radars. Thus, it is able to observe more complete Doppler and micro-Doppler frequency shifts. Multistatic radar fuses the received data in the receivers, and the performance of the fusion depends on the degree of spatial coherency between channels, the topology of the system, the number of targets and their spatial locations, and the complexity of the targets.

Multistatic micro-Doppler signature depends on the topology of the sys-tem and the location and motion of the target. The information encoded in the micro-Doppler signature may not be linearly increased with the number of channels used in the system because of the possible correlations between channels. Reference [6] analyzed the data collected in multistatic field trials on human and vehicles. If the cross-correlation between channels is 1 with no time delay, this indicates that the received signals in the two channels are the same and, thus, the second channel contains no additional information. If the two channels are not correlated, the second channel contains different information and is an informative channel. The value of Doppler shifts detected in each channel depends on the topology of the system, the target location, and its


moving direction. By fusing information captured from multiple channels, the location, moving direction, and velocity of the target can be measured. With the increased information, the radar performance on target recognition is ex-pected to be improved.

6.3.2 Micro-Doppler Signature-Based Classification

Automatic classification based on micro-Doppler signatures is still challenging. Stove and Sykes reported an operational radar system that uses Doppler spec-trum for target classification [7, 8]. Humans, vehicles, helicopters, and ships were successfully classified with multiple Fisher linear discriminators on their Doppler spectra. The target classification based on Doppler spectrum in this operational radar system indicated the possibility of using micro-Doppler sig-natures for target classification. Micro-Doppler signature–based methods for target classification have been investigated [6, 9, 10]. Smith discussed micro-Doppler signature–based classification using an experimental multistatic system [6]. Anderson described the classification using the support vector machine and the Gaussian mixture model (GMM) classifiers from micro-Doppler data [9]. Bilik et al. also reported a GMM-based classifier using features of spectral peri-odicity extracted by the cepstrum coefficients of micro-Doppler data [10]. For classifying human activities, because different movements have various micro-Doppler signatures, by exploiting these differences, a human activity classifier can be developed. Features used in the human activity classifier may include the torso signature curve, the maximal Doppler shift of the signature, the offset of the signature, the maximum Doppler variation of the torso curve, the oscilla-tion frequency or period of the human locomotion, the kinematic parameters of limbs, and other available features. However, despite the successful tracking and interpretation of human movements reported in computer vision literature, the selection of suitable descriptive features for prediction of human actions and in-tention is still a challenge. Most classifiers are not based on the micro-Doppler features that are associated with human body parts and limbs. According to a biological motion perception study, if a kinematic information-based classifier is used based on the features of human body parts and limbs extracted from micro-Doppler signatures, this will lead to a more impressive performance for identifying human actions, emotions, and even gender information.

6.3.3 Aural Methods for Micro-Doppler–Based Discrimination

An audio sound of a micro-Doppler embedded signal, called an aural signal, may help a human listener to distinguish between different movements of a target of interest (human walking, running, or jumping) or distinguish between different targets (human and animal).


The function of the human auditory classification is based on speech pho-nemics. A phoneme is a specific sound pattern that can be recognizable by hu-man brains. It is reasonable to generalize this ability of human auditory system for listening to micro-Doppler signals of different movements and to classify target movements using movement phonemes. The human brain neural struc-tures and learned behaviors used on a daily basis are ready to perform micro-Doppler signal processing aurally.

One advantage of the auditory classification systems is that the human auditory classification process is particularly robust in the presence of noise. Thus, the human auditory system could be an effective alternative in a signal classification system based on movement phonemes. However, the aural micro-Doppler signal is not a conventional speech signal. The human auditory clas-sification system has been optimized for speech signals, but not optimized for aural micro-Doppler signals.

Aural signal classification already used in sonar signal classification [11, 12]. The potential application of the aural classification to micro-Doppler sig-natures is also possible to classify a target’s different movements by directly converting the baseband micro-Doppler signal into an audio signal for training listeners [13]. However, aural classifiers may not easily classify human actions, emotions, and intentions. They can only serve as a supplementary classifier.

6.3.4 Through-the-Wall Micro-Doppler Signatures

The ability of radar to detect human beings and their movements offers through-the-wall radar applications, including locating living humans after an earthquake or in explosion scenarios, monitoring human activities behind walls, and many other uses. Like micro-Doppler signatures of target captures in an open, free space, micro-Doppler signatures of targets behind walls can also be used to detect and identify the targets behind walls [14]. The effect of targets behind a wall undergoing a micro motion has been studied, and the impact of the wall on the micro-Doppler effect has been formulated [15]. It was found that the micro-Doppler effect in the presence of a wall has a similar form as that in free space. However, the measured aspect angle of a target behind a wall is different to that observed in free space. The measured angle depends on the thickness and dielectric constant of the wall. The change of the instantaneous aspect angle due to the wall will affect the radar imaging of the target. However, the presence of a wall does not change the pattern of the micro-Doppler signa-ture of the target; the wall only changes the absolute value of the micro-Doppler signature, depending on the wall properties. Therefore, radar micro-Doppler signatures can be used to detect the presence of human beings and their move-ments behind the wall.


Radar returns modulated by biometrical signals may be used for detecting living objects. Specific modulations in reflected radar signals from a human in-clude heartbeats, thorax motion by breathing, and even vibration in the larynx. The heartbeat is a periodic process in the frequency range from 0.8 to 2.5 Hz. The breathing process is in a lower frequency range from 0.2 to 0.5 Hz for slow breathing [16–19].

Because the micro-Doppler frequency shift is sensitive to the radar operat-ing frequency, for a relatively lower frequency used in through-the-wall radars, the micro-Doppler shifts can be very low. Even so, radar signals returned from a human behind a wall can sense human body motion, heartbeat, and even breathing to detect individuals and monitor human movements. Radar detec-tion of human objects is usually performed in a complex background clutter. The intensity of the radar signal reflected from the clutter may exceed the in-tensity of the radar signal components returned from a human object. To reject the radar signal components reflected from the clutter, the different time and Doppler frequency characteristics between the radar returns from the human object and that from the clutter must be utilized. By applying a suitable range gating and a notched frequency filtering, the signal components reflected by the human object may be largely enhanced.

Other perspectives of micro-Doppler research also include the simulation of micro-Doppler signatures of quadrupedal animals and polarimetric micro-Doppler signatures.

References

[1] Chatzigeorgiadis, F., and D. Jenn, “A MATLAB Physical-Optics RCS Prediction Code,” IEEE Antennas & Propagation Magazine, Vol. 46, No. 4, August 2004, pp. 137–139.

[2] Chatzigeorgiadis, F., “Development of Code for Physical Optics Radar Cross Section Pre-diction and Analysis Application,” Master’s Thesis, Naval Postgraduate School, Monterey, CA, September 2004.

[3] Mallat, S., and Z. Zhang, “Matching Pursuit with Time-Frequency Dictionaries,” IEEE Transactions on Signal Processing, Vol. 40, No. 12, 1993, pp. 3397–3415.

[4] Qian, S., and D. Chen, “Signal Representation Using Adaptive Normalized Gaussian Functions,” Signal Processing, Vol. 36, No. 1, 1994, pp. 1–11.

[5] Chernyak, V. S., Fundamentals of Multisite Radar Systems, Amsterdam, the Netherlands: Gordon and Breach Scientific Publishers, 1998.

[6] Smith, G. E., “Radar Target Micro-Doppler Signature Classification,” Ph.D. Dissertation, Department of Electronic and Electrical Engineering, University College London, 2008.

[7] Stove, A. G., and S. R. Sykes, “A Doppler-Based Automatic Target Classifier for a Battle-field Surveillance Radar,” 2002 International Radar Conference, Edinburgh, U.K., 2002, pp. 419–423.


[8] Stove, A. G., and S. R. Sykes, “A Doppler-Based Target Classifier Using Linear Discrimi-nants and Principal Components,” Proceedings of the 2003 International Radar Conference, Adelaide, Australia, September 2003, pp. 171–176.

[9] Anderson, M. G., “Design of Multiple Frequency Continuous Wave Radar Hardware and Micro-Doppler Based Detection and Classification Algorithms,” Ph.D. Dissertation, University of Texas at Austin, 2008.

[10] Bilik, I., J. Tabrikian, and A. Cohen, “MM-Based Target Classifi cation for Ground Surveillance Doppler Radar,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 42, No. 1, 2006, pp. 267–278.

[11] Hines, P. C., and C. M. Ward, “Classifi cation of Marine Mammal Vocalizations Using an Automatic Aural Classifi er,” J. Acoust. Soc. Am., Vol. 127, No. 1970, 2010, doi: 10.1121/1.3385036.

[12] Allen, N., et al., “Study on the Human Ability to Aurally Discriminate Between Target Echoes and Environmental Clutter in Recordings of Incoherent Broadband Sonar,” J. Acoust. Soc. Am., Vol. 119, No. 3395, 2006.

[13] Chen, V. C., W. J. Miceli, and B. Himed, “Micro-Doppler Analysis in ISAR: Review and Perspectives,” IEEE 2009 International Radar Conference, October 12–16, 2009.

[14] Chen, V. C., et al., “Radar Micro-Doppler Signatures for Characterization of Human Motion,” Chapter 15 in Through-the-Wall Radar Imaging, M. Amin, (ed.), Boca Raton, FL: CRC Press, 2010.

[15] Liu, X., H. Leung, and G. A. Lampropoulos, “Effects of Non-Uniform Motion in Through-the-Wall SAR Imaging,” IEEE Transactions on Antennas and Propagation, Vol. 57, No. 11, 2009, pp. 3539–3548.

[16] Bugaev, A. S., et al., “Radar Methods of Detection of Human Breathing and Heartbeat,” Journal of Communications Technology and Electronics, Vol. 51, No. 10, 2006, pp. 1154–1168.

[17] Lubecke, V. M., et al., “Through-the-Wall Radar Life Detection and Monitoring,” Proceedings of the 2007 IEEE Microwave Theory and Techniques Symposium, Honolulu, HI, 2007, pp. 769–772.

[18] Chia, M. Y. W., et al., “Through-Wall UWB Radar Operating Within FCC’s Mask for Sensing Heart Beat and Breathing Rate,” Proceedings of the 2005 European Radar Conference, Paris, France, 2005, pp. 267–270.

[19] Bugaev, A. S., et al., “Through Wall Sensing of Human Breathing and Heart Beating by Monochromatic Radar,” Proceedings of the 10th International Conference on Ground Penetrating Radar, Delft, the Netherlands, 2004, pp. 291–294.

281

About the AuthorVictor C. Chen is internationally recognized for his work on radar micro-Doppler signatures and time-frequency analysis. Dr. Chen received a Ph.D. in electrical engineering from Case Western Reserve University, Cleveland, Ohio. Since 1982, he has worked with several companies in the United States and in the Radar Division at the U.S. Naval Research Laboratory in Washington, D.C. He has worked on inverse synthetic aperture radar imaging, time-frequency analysis for radar signal and imaging, noncooperative target recognition, and radar micro-Doppler signature analysis. In 1995, Dr. Chen started research on the micro-Doppler effect in radar and its applications. He has published more than 140 papers in journals and proceedings, and chapters in books. He has coauthored the book Time-Frequency Transforms for Radar Imaging and Signal Analysis (Artech House, 2002). Dr. Chen is also a Fellow of the Institute of Electrical and Electronics Engineers.

283

IndexAcceleration angular, 49–50, 274 linear, 274 tangential acceleration, 160, 161 translational, 49–50ALCM model, 26–28, 32–34Aliasing phenomenon, 15, 16Angle-cyclogram pattern, 184Angular acceleration, 49–50, 274Angular kinematics, 274Angular momentum vector, 46Angular velocity, 49, 59, 274 damping pendulum, 100 pendulum, 97 spinning symmetric top, 133Anomalous human behavior, 264–66Aural methods, 276–77Aural signal classification, 277Autoregressive (AR) modeling, 13Azimuth angle, 189

Bessel function, 62Biological motion decomposition of, 250–55 human motion as, 247 perception, 248–50Bird wing flapping, 194–202 amplitude, 199 angle, 195 defined, 195 Doppler observations of, 198–99 flapping angle, 197–98 geometry, 202 kinematic model, 199, 200, 201

kinematics, 195–98 linear velocity of tip, 196, 198 locomotion calculation, 195 MATLAB source codes, 238–46 micro-Doppler signatures, 203 position vector of tip, 197 range profiles, 203 simulation of, 199–202 sweeping angle, 197–98 velocity vector, 197 vertical translation, 195 wing flap matrix, 196 wing structure, 195 wing sweep matrix, 196 wing twist matrix, 196 See also Nonrigid body motionBistatic micro-Doppler effect, 71–77 characteristics, 76 Doppler resolution, 76 Doppler shift parts, 74 induced by rotation motion, 77 maximum Doppler shift, 75–76 range, 75 range resolution, 76 of rotating target, 77 translational Doppler shift, 75Bistatic radar, 7–10 configuration illustration, 8 Doppler resolution, 10 Doppler shift, 9–10 phase change, 9 phase term, 73–74 receiver look angles, 73


Bistatic radar (continued) target location determination, 72 three-dimensional, 72 transmitter and receiver separation, 72 two-dimensional, 7, 8 See also RadarBistatic RCS, 54Bistatic triangulation factor, 75–76Blind source separation (BSS), 251

Challenges, 272–75 feature extraction, 273–75 micro-Doppler signatures, 273 target identification, 273–75Clutter suppression, 187–89 illustrated, 191 time-frequency analysis of data, 189,

192Computer-aided design (CAD), 185Coning motion defined, 66 target illustration, 67Coning motion-induced micro-Doppler

shift, 66–71 coning matrix, 69 precession matrix, 68 skew symmetric matrix, 68, 69 spinning matrix, 69 See also Micro-Doppler frequency shiftsContinuous wave (CW) radar, 14Cramer-Rao bound of Doppler frequency estimation,

17–18 of micro-Doppler estimation, 79Cramer-Rao lower bound (CRLB), 18Cross product, 81

Damping pendulum, 99–100 angular velocity, 100 micro-Doppler signatures, 106 swinging angle, 100Decomposition, 250–55 challenges, 273 in joint time-frequency domain,

251–52 physical component-based, 252–55 statistics-based, 251 See also Biological motionDegrees of freedom (DOF), 162Denavit-Hartenberg (D-H) convention, 163Diffraction, 53

Dog hierarchical model, 205Doppler, Christian, 3Doppler dilemma, 17Doppler effect, 2–3 both source and observer move, 6 experimental proving, 2–3 observed by radar, 7–10 in radial velocity measurement, 10 relativistic, 4–7Doppler frequency distributions, 158Doppler frequency estimation, 17–18Doppler frequency shifts bistatic radar, 9–10 estimation and analysis, 10–17 estimation from Doppler signal, 12 extracted by quadrature detector, 11 helicopter rotor blades, 112 induced by target motion, 58 Lorentz transformation and, 4 measurement of, 1 micro, 20–26 as proportional to emitted frequency, 7 width of, 1 wind turbines, 140Doppler resolution, 76 bistatic radar, 10 monostatic radar, 10Doppler spectrogram, 198–99Doppler spectrum, time-varying, 199

Electric field integral equation (EFIE), 53Electromagnetic scattering, 50–56 backscattering, 55 basics, 50 with body in motion, 54–56 from facets, 115 geometry of translational object, 55 helicopter rotor blades, 108 isolation, 50 occurrence, 50 point scatterer model, 94 RCS of target, 50–53 RCS prediction methods, 53–54Empirical mode decomposition (EMD), 22Equations of motion, 44–47Euler angles, 36–42 defined, 37 derivatives vector, 129 illustrated, 37 initial, 66

Index 285

inverse, transform matrix, 129 quaternion converted to, 44 spinning symmetric top, 133Euler rotation theorem, 36–37

Facets collection of, 116 PO computation, 115 PO prediction model, 114–16 triangular, 116Fast Fourier Transform (FFT), 12, 13Feature extraction, 256–57Feet micro-Doppler component, 258 motion parameters, 259, 261 of running person, 260 velocity of running person, 266 velocity of walking person, 263FFT. See Fast Fourier TransformFinite difference method (FDM), 47Finite-difference time domain (FDTD), 53Finite element method (FEM), 47Force-free rotation, 130–31, 132

Gabor representation, 25Gaussian mixture model (GMM), 276Geometric theory of diffraction (GTD), 53Gimbal lock phenomenon, 39, 41, 42Global human walking model. See Human

walking modelGyroscopes, 181

Helicopter rotor blades, 105–27 airfoil, 105 airfoil illustration, 107 backscattering, 107 blade length, 110–11 composition, 105 Doppler frequency shifts, 112 EM scattering, 108 geometry, 108 main rotor features, 128 mathematical model of, 107–12 micro-Doppler signature analysis and

interpretation, 123–27 micro-Doppler signatures, 119, 120–23 number of blades, 117 periodic modulations, 112 PO facet prediction model, 114–16 radar backscattering from, 116–20 range profiles, 119

RCS mode of, 112–14 rectangular, representation by facets,

120 required minimum PRF, 123 return, 111 rotation feature, 112 rotation geometry, 118 time-domain signature, 110 See also Rigid body motionHidden Markov models (HMMs), 265, 266Hierarchical quadrupedal models, 204–5High-speed cinematographic technique, 207Hilbert-Huang transform (HHT), 22Hilbert transform, 21Horses micro-Doppler signatures, 206 walking, 204Human auditory classification, 276–77Human body articulated motion, 159–94 two2-D position trajectory, 183 three3-D, 177–80 three3-D kinematic data collection,

182–84 angle-cyclogram pattern and, 184 angular rotation, 160 arm and leg movements micro-Doppler

signatures, 194 capture methods, 163 coordinate systems, 162 degrees of freedom (DOF), 162 displacement and velocity, 180 geometry, 190 gyroscopes, 181 hip and knee joint angles, 186 hip-knee cyclogram, 187 kinematic parameters, 160, 163,

177–81 MATLAB source codes, 209–37 model from empirical mathematical

parameterizations, 164–77 movement in one cycle, 161 movement simulation, 162 optical motion-caption system, 181 periodic motion description, 161 radar backscattering from, 184–87 segment parameters, 162–63 tangential acceleration, 161 tangential velocity, 160 walking, 159–61


Human gait, 157, 159Human motion, 19Human movement clutter suppression, 187–89 data processing, 187–89 induced radar micro-Doppler signa-

tures, 189–94 time-frequency analysis of data, 189,

192Human movement identification, 262–67 anomalous human behavior and,

264–66 class determination, 263–64 features used for, 263–64 purpose of, 262 summary, 266–67Human walking model, 164–77 three3-D orientation, 174, 175 angle trajectory calculation, 176 animation, 177 basis, 164 body reference points, 174 flexing at the ankle, 169, 171 flexing at the elbow, 172, 173 flexing at the hip, 169, 170 flexing at the knee, 169, 171 flexing at the shoulder, 172, 173 flexing/torsion trajectories, 169–72 lateral translation, 165, 166 motion of the thorax, 169, 172 in radar, 256 rotational trajectories, 167–69 rotation forward/backward, 167, 168 rotation left/right, 167, 169 segment lengths, 176 torsion rotation, 169, 170 translational trajectories, 164–67 translation forward/backward, 167, 168 vertical translation, 165, 166

Incident aspect angle, 189Independent component analysis (ICA),

251, 252Instantaneous frequency analysis, 21–22 defined, 20–21 by time-derivative operation, 21 values, 22Intrinsic mode functions (IMFs), 22, 252

Joint time-frequency analysis, 23–26Joint-time frequency domain, 251–52

Kinematics three3-D data collection, 182–84 angular, 274 bird wing flapping, 195–98 linear parameters, 274 parameter estimation, 257–62 parameters, 160, 163, 177–81 of PLDs, 249

Laser detection and ranging (LADAR), 271 coherent, 19 defined, 18Linear acceleration, 274Linear position, 274Linear velocity, 274Line of sight (LOS) velocity, 273Locomotion, 158

MacLaurin series, 6Markov chain-based inference algorithm,

252–54MATLAB source codes, 272 ALCM model, 32–34 bird wing flapping, 238–46 human walking, 209–37 radar backscattering, 143–56 radar blade returns, 148–51 radar pendulum returns, 143–48 radar top returns, 151–56 spinning theory, 83–91MATLAB time-frequency toolbox, 26Maximum likelihood estimation, 12Micro-Doppler effect, 18–20 bistatic, 71–77 concept, 28 defined, 18–19 electromagnetic scattering, 50–56 frequency spectrum, 62 induced by target with micro motion,

56–60 in LADAR systems, 271 mathematics for calculating, 56–71 modulation, 56 multistatic, 77–79 nonrigid body motion, 47–50,

157–207 in radar, 20, 35–79

Index 287

rigid body motion, 35–37, 93–141 use of, 2Micro-Doppler frequency shifts, 20–26 coning motion-induced, 66–71 Cramer-Rao bound and, 79 defined, 20 estimation and analysis, 20–26 instantaneous frequency analysis, 21–22 joint time-frequency analysis, 23–26 rotation-induced, 63–66 sensitivity, 278 vibration-induced, 60–63Micro-Doppler signature-based

classification, 276Micro-Doppler signatures analysis and interpretation, 247–67 bird wing flapping, 203 damping pendulum, 106 decomposition challenges, 273 decomposition in joint time-frequency

domain, 251–2 feature extraction, 256–57 feature extraction challenges, 273–75 feet, 258 generation of, 247, 272 helicopter rotor blades, 119, 120–23 human gait, 159 human movement, 189–94 kinematic parameter estimation from,

257–62 of objects, 26–28 of pendulum oscillation, 104, 105 person with arm and leg movements,

194 precession target, 71 precession top, 137, 138 quadrupedal animal motion, 205 radius, 258 of rigid bodies, 28 of rotating ALCM, 26–28 rotating three-blade rotor, 113, 121 rotating two-blade rotor, 122 running person, 255, 259 spinning, precession, and nutation top,

139 spinning symmetric top, 136 spinning target, 70 target application, 28 target identification challenges, 273–75

tibias, 258 torso, 258 of walking person, 193, 253, 257 wind turbines, 140Micro motion, 19–20Monostatic radar, 10Monostatic RCS, 54Multiple signal classification (MUSIC), 13Multistatic micro-Doppler analysis, 275–76Multistatic micro-Doppler effect as combination of bistatic systems,

77–78 derivation, 79 illustrated, 78 nodes, 79 processing, 78 target range determination, 77 transmitters and receivers, 77

Newton’s second law of motion, 95–96Nonlinear motion dynamics, 95–101Nonrigid body motion, 47–50 bird wing flapping, 194–202 defined, 47 human body articulated motion,

159–94 micro-Doppler effect, 157–207 multibody system, 47 quadrupedal animal motion, 202–7 slider-crank mechanism, 47, 48Nyquist interval, 15

Optical motion-caption system, 181Ordinary differential equation (ODE), 82,

96

Pendulum oscillation, 94–105 angular frequency, 96, 99 angular velocity, 97 damping, 99–100 defined, 94 differential equation, 96 illustrated, 95 initial amplitude, 97–98, 99 initial phase, 99 with linear friction, 99 micro-Doppler signatures of, 104, 105 net torque, 96 nonlinear motion dynamics, 95–101 pendulum length and, 98


Pendulum oscillation (continued) physical pendulum equation, 98–99 radar backscattering from, 102–4 range profiles of, 104 RCS modeling, 101–2 See also Rigid body motionPerspectives, 275–78 aural methods, 276–77 micro-Doppler signature-based clas-

sification, 276 multistatic micro-Doppler analysis,

275–76 through-the-wall micro-Doppler signa-

tures, 277–78Physical optics (PO), 53, 94 facet computation, 115 facet prediction model, 114–16Physical theory of diffraction (PTD), 53Point-light displays (PLDs), 248, 250 biological motion perception from, 250 kinematics of, 249 running person representation, 249 trajectories, 249Point scatterer model, 94Position, linear, 274Precession micro-Doppler signature, 71Precession top micro-Doppler signatures, 136, 137,

138 micro-Doppler signatures analysis and

interpretation, 136–39 range profiles, 137Principal component analysis (PCA), 249,

250, 251Pulse repetition frequency (PRF) defined, 16 proportional to maximum unambigu-

ous velocity, 17 required minimum, 123

Quadrature detectors Doppler shifts extracted by, 11 I and Q outputs, 11–12, 14Quadrupedal animal motion, 202–7 defined, 202 dog, 205 high-speed cinematographic technique,

207 horse, 204 illustrated, 204

micro-Doppler signatures, 205 modeling, 204–5 summary, 205–7 See also Nonrigid body motionQuaternions, 42–44 conjugate of, 43 converted to Euler angles, 44 four-component vector, 42 multiplication of, 43 rotation matrix derivation, 43–44

Radar bistatic, 7–10 coherent Doppler, 15 continuous wave (CW), 14 Doppler effect observed in, 7–10 frequency modulated continuous wave

(FM-CW), 14 micro-Doppler effect in, 20, 35–79 monostatic, 10 return modulation, 278Radar backscattering, 55 calculation, 102 ellipsoid, 101–2 helicopter rotor blades, 107, 116–20 MATLAB source codes, 105, 117,

143–56 nonnormal incidence of, 102 oscillating pendulum, 102–4 spinning symmetric top, 135–36 from walking human, 184–87Radar cross section (RCS) accuracy, 53, 117 bistatic, 54 coordinate system used in, 54 defined, 50 determination, 28 of ellipsoid backscattering, 101–2 formulation, 52 helicopter rotor blades, 112–14 high-frequency prediction models, 101 modeling methods, 94 monostatic, 54 of pendulum, 101–2 of perfectly conducting sphere, 101 prediction methods, 53–54 of rigid body, 94 spiky features and lobes, 53 symmetric top, 133–35 target characteristics and, 52

Index 289

total, 52–53 triangular facet, 116 unit of square meters, 52Radius, micro-Doppler component, 258Range profiles bird wing flapping, 203 helicopter rotor blades, 119 pendulum oscillation, 104 precession top, 137 rotating three-blade rotor, 121 rotating two-blade rotor, 122Range resolution, monostatic radar, 10RAPPORT code, 53Reflection, 53Relativistic Doppler effect, 4–7Rigid body movement, 93 RCS, 94 as solid body idealization, 93Rigid body motion, 35–37 component angular momentum, 45–46 coordinate systems, 35–36 defined, 93 dynamics of, 44–45 in EM simulation, 94 equations of motion, 44–47 Euler angles, 36–42 helicopter rotor blades, 105–27 micro-Doppler effect, 93–141 pendulum oscillation, 94–105 quaternion, 42–44 rotation in, 45 spinning symmetric top, 127–39 wind turbines, 139–41Rodrigues formula, 83Roll-pitch-way convention defined, 37 illustrated, 38 rotation matrix, 39Rotating three-blade rotor micro-Doppler features, 126 micro-Doppler signatures, 121 range profiles, 121Rotating two-blade rotor micro-Doppler features, 126 micro-Doppler signatures, 122, 125 range profiles, 122Rotation-induced micro-Doppler shift,

63–66 initial rotation matrix, 63–64

skew symmetric matrix, 65Rotation matrix three3-D, 58 components, 39, 40 defined, 39 quaternions and, 43–44 three-column vector of, 41Running, human feet, 260 foot velocity of, 266 micro-Doppler signatures, 255, 259 tibias, 260 tibias velocity of, 265 torso, 260 torso velocity of, 264

Short-term Fourier transform (STFT), 23–24

Simulation of bird wing flapping, 199–202 of human movements, 162Singular value decomposition, 251, 252Skew symmetric matrix, 59 of coning motion, 69 defined, 58, 68, 81 in Rodrigues formula, 65 of spinning motion, 69Slider-crank mechanism, 47, 48Smoothed pseudo-Wigner-Ville, 24Spinning symmetric top, 127–39 about symmetric axis, 129 angular velocity, 133 center of mass (CM), 133, 134 Euler angles, 133 force-free rotation of, 130–31, 132 micro-Doppler signatures, 136 micro-Doppler signatures analysis and

interpretation, 136–39 motion dynamics, 128 precession of, 128 principal moment, 130 radar backscattering, 135–36 RCS model, 133–35 torque, 129–30 torque-induced rotation, 132–33 See also Rigid body motionStatistics-based decomposition, 251Symmetric top. See Spinning symmetric top

Tangential acceleration, 161Tangential velocity, 160


3-D kinematic data collection, 182–843-D rotation matrix, 58Through-the-wall micro-Doppler signatures,

277–78Tibias micro-Doppler component, 258 motion parameters, 259, 261 of running person, 260 velocity of running person, 265 velocity of walking person, 262Time dilation, 4–7Time windowing, 12–13Torque-induced rotation, 132–33Torso micro-Doppler component, 258 motion parameters, 261 of running person, 260 velocity of running person, 264Translational acceleration, 49–50Translational velocity, 49True velocity, estimation of, 274

Velocity angular, 49, 59, 274 human walking, 164 linear, 274 line of sight (LOS), 273 Nyquist, 16 tangential, 160 translational, 49 true, measuring, 274Vibration-induced micro-Doppler shift,

60–63 azimuth angle, 61, 63 Bessel function, 62 elevation angle, 61, 63 geometry, 60 point scatterer, 62Vibration rotation, extraction of, 158

Walking, human, 159–61 angular rotation, 160 animating, 177 arm and leg movements micro-Doppler

signatures, 194 foot velocity, 263 geometry, 190 hip and knee joint angles, 186 hip-knee cyclogram, 187 kinematic parameters, 160 micro-Doppler signatures, 193, 253,

257 model from empirical mathematical

parameterizations, 164–77 movement in one cycle, 161 periodic motion description, 161 radar backscattering from, 184–87 tangential acceleration, 160 tangential velocity, 160 tibia velocity, 262 velocity, 164 See also Human body articulated

motionWalking horses, 204Wigner-Ville distribution (WVD), 24Wind turbines, 139–41 components, 139 Doppler components of multiple

bounces, 141 Doppler frequency shifts, 140 micro-Doppler signature illustration,

141 micro-Doppler signatures, 140 micro-Doppler signatures analysis and

interpretation, 140–41 observed by radar systems, 140 See also Rigid body motion

X-convention, 37XPATCH code, 53

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