Incorporating Pulse-to-Pulse Motion Effects into Side-Looking Array Radar Models

Justin W. Green, Todd B. Hale, Michael A. Temple, and John T. Buckreis

Air Force Institute of Technology Department of Electrical and Computer Engineering

2950 Hobson Way WPAFB Ohio, 45433 USA

Abstract. A technique is presented for incorporating pulse-to-pulse (inter-pulse) motion effects into side-looking array radar data models yielding a motion-sensitive space-time snapshot. Low flying, highly-maneuverable Unmanned Aerial Vehicles (UAV) represent a potential worst case application scenario given their roll, yaw, and pitch rates are primarily limited by structural integrity. High degrees of maneuverability during the Coherent Processing Interval (CPI) allow clutter and target returns to change significantly. The technique presented uses Mcoordinate transformations to describe platform attitude variations throughout the CPI. Ward’s model then is extended to incorporate maneuver-induced changes in spatial frequency and Doppler. The new motion-sensitive space-time snapshot is used to characterize Space Time Adaptive Processing (STAP) performance without motion-compensation applied. Results clearly show motion-induced clutter-null broadening with measurable degra-dation of STAP algorithm minimal discernable velocity.

I. INTRODUCTION

A fundamental assumption in the space-time snapshot development of [1] for a pulsed-doppler, 2D linear array radar is fixed platform-to-target geometry as shown in Fig 1. During the Coherent Processing Interval (CPI), the unit vector ˆ ,k from the array reference element to a target located at

, remains constant and is given by [1]

ˆ ˆ ˆ ˆ, cos sin cos cos sink x y z , (1)

where , , and x y z are the 3D Cartesian axis unit vectors, platform motion is along the positive x-axis (side-looking array with zero crab).

av

Fig 1 Legacy platform-to-target geometry

Consequently, a constant ˆ ,k for every pulse within a CPI implies a constant wavefront time-of-arrival delay, n ,describing propagation delay for the nth element. Thus, the spatial frequency parameter describing element-to-element spatial phase delays is constant during the CPI. The spatial frequency , a function of array-target geometry, is given by

cos sin

o

d , (2)

where is the uniform inter-element array spacing along the x-axis, and

do is the carrier wavelength.

Having uniform element spacing and constant spatial frequency increments across the array face simplified (somewhat) the linear algebra framework required to integrate the collective matched-filter responses (voltages) for each element, and for each of the M CPI pulses. For the nth

element, the mth pulse, and a given range cell of interest, the down-converted matched-filter response mnx is given by,

2 2j n j mmn tx e e , (3)

1-4244-0309-X/06/$20.00 ©2006 IEEE 580

where t describes the target (single-point scatterer) echo’s complex amplitude and is normalized target Doppler. Using constant azimuth and temporal steering vectors,

2 12( ) 1Tj Nje ea , (4)

2 12( ) 1Tj Mje eb (5)

a space-time snapshot t of dimension was formed, containing N returns from each of the M pulses in the CPI for the range cell of interest where

1NM

( ) ( )t tb a . (6)

With the space-time snapshot t for a single scatterer defined, [1] subsequently derived interference power (covariance) statistics for internal receiver thermal noise nR , interference jamming jR , and ground clutter cR . STAP techniques use the combined interference matrix or its

estimate n jR = R R Rc

R to operate on received data.

Incorporating pulse-to-pulse motion (i.e., platform maneuver) necessitates the development of a pulse-dependent framework capturing the unique geometry presented for each pulse in the CPI. The developed framework defines pulse-dependent target unit vectors ˆ ,m m mk using Cartesian

coordinate transformations operating on initial platform-target geometry. These pulse-dependent unit vectors to the target are subsequently used to define pulse dependent time delays across the array face, nm , pulse dependent spatial frequency

m , together with linear algebra manipulations to express a modified, motion-sensitive space-time snapshot t .

II. MODIFIED RADAR DATA MODEL

A. UAV Maneuvering Scenario

For the maneuvering UAV scenario, the side-looking array may experience 3D variations in attitude (yaw, roll, and/or pitch) during the CPI. Pulse-indexed unit vectors to the target ˆ ,m m mk account for attitude variations for each pulse .

The baseline UAV position

m

0 0 0ˆ ,k is defined as the

orientation for . A simple platform dynamics model using maneuver rates in yaw, roll, and pitch, designated

0m, ,

and respectively, describe attitude rotations during the CPI. Second order acceleration rates , , are zero, simplifying

framework development. Attitude position for the mth pulse in a given dimension is readily solved for by the relations

0 0

0 0

0 0

m m

m m

m m

m T

m T

m T

r

r

r

(7)

where Tr is the pulse repetition interval, mTr defines the CPI time elapsed, and m , m , and m describe the pulse-indexed incremental change in yaw, roll, and pitch respectively. Geometry is also fixed during discrete pulse intervals .

A sign convention consistent with the development in [1] is defined. Recall the elevation angle is measured off boresight in the y-z plane with 0 for targets below the y-axis. The azimuth angle is measured off boresight in the x-y plane with 0 for targets toward the positive x-axis. Consider the case in Fig. 2 where a 45o clockwise (CW) yaw (from pilot’s perspective) occurred in time extent . The m

rmTth yaw angle m is measured off the current rotated boresight

(positive my -axis) in the -m mx y plane. In order for (7) to hold, the sign of m must be negative. This assignment matches intuition as m becomes negative as the rotating boresight sweeps (in CW direction) past the original target angular location. Subsequently, the sign of m is positive for a counter-clockwise (CCW) yaw.

Next, consider a CW roll. The mth elevation angle m is measured off the current rotated boresight (positive my -axis) in the m-zm my plane. Again in order for (7) to hold, m must be negative. This assignment also matches intuition as the depression angle m becomes more negative as the rotating boresight sweeps up in the y-z plane. In similar fashion, a pitch-up maneuver results in a reduction in pitch m ,measured off the current rotated positive mx -axis in m-zm mxplane, hence m must be negative. Summarizing the sign assignments yields

0, CW yaw>0, CCW yawm , (8)

0, CW roll>0, CCW rollm , (9)

0, pitch up( )

>0, pitch downm . (10)

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Fig 2: Yaw angle 0 is measured off initial boresight (top) while

m is measured off the rotated boresight (bottom). Note motion-

compensated beam steering is off.

B. Cartesian Coordinate Transformation

Assuming INS information is available and the initial attitude is known, the UAV maneuver during each pulse interval mTr, results in a deterministic rotation of the Cartesian coordinate system. Consider the geometry shown in Fig. 3 illustrating again 45o of clockwise yaw in time extent .

Trigonometric ratios are used to determine rotated unit-vector components

rmT

xmk , , and ymk zmk in terms of the baseline

components 0xk , , and such that 0yk zmk

0 0cos( ) sin( )xm x m y mk k k , (11)

0 0sin( ) cos( )ym x m y mk k k , (12)

0zm zk k , (13)

where m is negative due to CW rotation. Equations (11) - (13) are recast in matrix form,

0

0

0

xm x

ym m y

zm z

k kkk k

= Y k

1

, (14)

where the pulse-indexed transform for the mmY th interval is given by

m

cos( ) sin( ) 0sin( ) cos( ) 0

0 0

m m

m mY . (15)

Following the same development steps for a CCW yaw yields an identical transform. A total of M transforms are required, one for each pulse interval.

mY

A similar process is used to determine Cartesian coordinate system transformations for roll and pitch. This yields the roll transform mR ,

1 0 00 cos( ) sin( )0 sin( ) cos( )

m m

m m

mR , (16)

for either a CW or a CCW roll maneuver, and the pitch transform mP

m

cos( ) 0 sin( )0 1 0

sin( ) 0 cos( )

m m

m m

P , (17)

for either a CW or a CCW pitch maneuver. An important feature of the (15)-(17) is each collapses to the identity matrix when the respective maneuver rate is zero.

This framework can be applied to the multidimensional case only if multidimensional maneuvers are discretized into a series of one-dimensional rotations occurring over the CPI, However, the order of maneuver events must be preserved and the platform orientation at the conclusion of one maneuver becomes the baseline position for the subsequent maneuver.

Fig 3. Geometry used to generate coordinate transformation

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C. Pulse-Indexed Parameters

The next step in the model development requires calculation of M unique time delays nm across the array-face

using the M unique unit vectors ˆ ,m m mk . This is done by defining nm as

ˆm

nm( , )

ck nd

, (18)

where is the vector from the reference element (nd 0n ) to to nth element. Assuming inter-element spacing is uniform,

. Pulse-indexed time delays n ndd nm yield pulse-indexed phase delays o nm across the array. In [1] phase delays were expressed in terms of spatial frequency by the relation

2o n n (19)

with defined in (2). Substitution of (18) into (19) yields a pulse-indexed spatial frequency m given by

= xmm

o

dk. (20)

As the platform attitude changes during the CPI, the projection of relative velocity onto the target unit vector

also changes. Thus it is necessary to pulse-index the normalized Doppler terms

rvˆ

mkm as

2 rmm

o p

vf

, (21)

where pf is the pulse repetition frequency, and is the

pulse-indexed relative velocity between the platform and the target. Application of (21) yields a unique Doppler term for each attitude position in the CPI.

rmv

D. Receiver Matched Filtering

The notional Matched Filter (MF) receiver implementation remains unchanged from [1]. Using complex envelope notation the transmitted signal ( )s t is given by

1

0( ) oM j t

t p rms t a u t mT e (22)

where t is the transmitted signal’s amplitude, M is the number of pulses in the CPI, (

2

2 2

( )

o tm

o tm R o tm nm

j f f tnm r p R r

j f f T j f f j

s t a u t T mT e

e e e (23)

where RT is the round trip time, of is the carrier frequency in Hz, tmf is the target Doppler, and nm is the pulse-indexed delay across the array face. Following the framework in [1], the nth receiver output, after down-conversion and MF operations, is expressed as

2 2 ,m mj n j mnm t m tmx e e t f , (24)

where the Time-Frequency Autocorrelation Function (TFAF) ,m tmt f for the mth pulse, given by

2*, ( ) ( ) tmj fm tm p p R rt f u u T mT t e d , (25)

scales the MF response in accordance with the transmitted waveform’s range and doppler tolerance [2]. For the range gate of interest, the development assumes the MF output is sampled at R rt T mT for the mth pulse (i.e. a range matched condition). The development also assumes the transmit waveform is doppler tolerant, i.e. 0, 1m tmf for all expected values of tmf . These assumptions, along with waveform energy normalization, allow a unity ,m tmt f

condition, hence the ,m tmt f term is suppressed from this

point forward in the development.

The filter response in (24) reduces to the legacy expression in (3) when there is no UAV maneuver. The spatial frequency

m in the first exponent drops the pulse index. The normalized Doppler term m in the second exponential becomes a constant, allowing combination of the second exponential with the complex amplitude t . Hence graceful degradation to the fixed attitude case is sustained through MF operations.

E. Legacy Framework Extensions

In the legacy model the space-time snapshot t arranged the MF output of all N elements for all M CPI pulses coherently into a column vector. Spatial coherence required accurate accounting of phase progression from element-to-element using the constant azimuth steering vector ( )a in (4). Temporal coherence required accurate accounting of phase progression from pulse-to-pulse due to Doppler using

)p ru t mT is a unit amplitude pulse (duration ) delayed by pulse intervals (duration

),m

rT o is the carrier radian frequency, and is the random starting phase. The received signal at the nth array element for the mth pulse is

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the implicit temporal progression term in each vector element

rmT2j me of ( )b in (5).

In the motion case, framework extensions are required to maintain spatial and temporal coherence. Consider first the need for spatial coherence, given spatial frequency m is now a function of pulse number m. For every pulse interval m and corresponding spatial frequency m , a unique pulse-indexed azimuth steering vector ( )mma , defined as

2 12( ) 1 mmTj Nj

m e ema , (26)

is required to accurate account for the element-to-element phase progression across the array. Assembling these Mazimuth steering vectors in matrix form yields a azimuth steering matrix mA of dimension , defined as N M

0 10 1 1 Mx x M xxA a a a1

, (27)

where the mth column represents the element-to-element phase progressions for the m

Nth platform orientation.

Consider next the need for temporal coherence. In (5), the normalized Doppler was constant, corresponding to a fixed platform-target geometry or constant relative velocity throughout the CPI. Here in the motion case, normalized Doppler is recalculated for every pulse interval using (21). A motion-sensitive temporal steering vector mb is required

that accounts for variation in Doppler while remaining coherent with the fixed starting phase je . The result is a modified temporal vector mb defined by

11 2 11

222( ) 1M

iijjj

m e e eb . (28)

To formulate the motion sensitive space-time snapshot t a Kronecker operation similar to (6) is used to begin the matching process of pulse-indexed terms. The Kronecker operation

Mt t m xmb A (29)

results in the matrix t of dimension MN M . Unlike (6), the Kronecker here produced mismatch terms

(i.e. elements containing dissimilar pulse indices). A necessary next step is the introduction of an indexing structure such that the indices of the

1MN M

= 1 nm pulsei n m offset (30)

where corresponds to the element of interest for n0,1, 1n N and corresponds to the pulse of interest

for m

0,1, 1m M . The pulse offset pulseoffset is defined by

pulseoffset MN N . (31)

Using (30), a vector of dimension nmi 1MN is constructed containing the only the indices for pulse-matched terms in

t . Application of these indices yields the desired result

( )t nmX i , (32)

i.e the formation of a motion-sensitive space-time snapshot that accurately accounts for maneuver-induced changes in spatial frequency and normalized Doppler over the CPI.

F. Incorporation within Clutter Model

It is desirable to utilize a motion-sensitive space-time snapshot t within the clutter model of [1] when inter-pulse motion exists. Application of t does not change all clutter model parameters. For example, grazing angle, horizon range, clutter patch reflectivity, effective radar cross section, and the number of ambiguous range rings did not change. However, clutter patch spatial frequency, Doppler, clutter to noise ratio and echo amplitude in-turn were all affected by platform maneuver. The effect of platform maneuver on the STAP sample matrix inversion technique was simulated where motion-compensated beam-steering was turned off. Table 1 indicates simulation parameters. The results are highlighted in Fig 4 and Fig 5.

Simulation results reveal clutter-null widening when motion is introduced. For comparison, Fig 4 shows the azimuth-Doppler power spectrum of where no motion effects were incorporated (top) and the same spectra given a

change in yaw during the CPI. It is evident that the UAV motion is corrupting the pulse-to-pulse temporal correlation of individual clutter-patch returns. This agrees with intuition. With UAV motion, clutter returns from the same patch, but different pulses are returning through different (rotated) locations of the antenna’s spatial power pattern, yielding temporal variations in the echo amplitude and phase. Examining SINR plots provides an alternative, yet more instructive illustration of motion effects. In Fig 5, the 1.43

R

1.43o

o

CCW yaw ( 0m ) shifts clutter ridge in the positive direction (matching intuition as ~ sin ) yielding a 22% degradation in minimal discernable velocity (MDV). A CW yaw ( 0m ) shifts the clutter ridge in the negative direction yielding a similar MDV degradation.

MN pulse-matched elements in t are defined, allowing formation of the motion-sensitive

space-time snapshot t of correct dimension, 1MN . Let be the matrix index of the pulse-matched term for the nnmi th

element and the mth pulse. It is determined that is given by nmi

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Table 1: Clutter simulation parameters

Variable ValueM (pulses in CPI) 64N (number of azimuth elements) 32

(Azimuth Transmit Direction) 0 (Elevation Transmit Direction) 0 (yaw rate) 4 rad/sec

t (target Doppler) 0.25

of (carrier frequency) 1240 MHz

pf (pulse repetition frequency) 1984 Hz (pulse width) 0.8 s

Pt (transmit power) 200 kW B (bandwidth) 800 kHz Fn (noise figure) 3 dB Nc (number of clutter patches) 2NMha (aircraft altitude) 1500 m

(clutter ridge slope) 1R (range to the target) 66km

(clutter gamma) -3 dB Array Transmit gain 22 dB Element Gain 4 dB Element Backlobe Level -30 dB d (inter-element spacing, x-axis) 0.1092 m Ls (system losses) 3 dB

III. CONCLUDING REMARKS

This paper presented a technique for incorporating inter-pulse maneuvering effects into a motion-sensitive space-time snapshot. The development of Cartesian coordinate transformations facilitated the extension of the framework in [1], allowing coherent accounting of changing spatial frequency and Doppler. Incorporating the motion-sensitive space-time snapshot within the clutter model of [1], enabled analysis of maneuvering effects on STAP performance when motion compensated beam-steering is not applied. Simulation results were consistent with analytical models, clearly showing motion-induced broadening of the clutter null with significant degradation of minimal discernable velocity.

REFERENCES[1] J. Ward, “Space--time adaptive processing for airborne radar,”

Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts, Contract F19628-95-C-0002, December 1994

[2] M.I. Skolnik, Introduction to Radar Systems 3rd ed. 1221 Avenue of the Americas, New York, NY 10020: McGraw-Hill, Inc., 2001, iSBN: 0072909803

“The views expressed in this article are those of the authors and do not reflect official policy of the United States Air Force, Department of Defense or the U.S. Government.”

Fig 4. Azimuth-Doppler clutter power spectrum of without UAV maneuver (top) and with 1.43

Ro counter -clockwise yaw (bottom) over the CPI

Fig 5. SINR plots for a 1.43o CW and a 1.43o CCW yaw over the CPI

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