introduction to radar part i

1

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Introduction to RadarPart I

Scriptum of a lecture at theRuhr-Universitat Bochum

Dr.-Ing. Joachim H.G. Ender

Honorary Professor of the Ruhr-University BochumHead of the Fraunhofer-Institute for High Frequency

Physics and Radar Techniques FHR

Neuenahrer Str. 20, 53343 Wachtberg, Phone 0228-9435-226, Fax 0228-9435-627E-Mail: [email protected]

Copyright remarks This document is dedicated exclusively to purposes of edu-cation at the Ruhr-University Bochum, the University of Siegen, the RWTH Aachenand the Fraunhofer-Institute FHR. No part of this document may be reproduced ordistributed in any form or by any means, or stored in a database or retrieval system,without the prior written consent of the author.

Contents

1 Radar fundamentals 9

1.1 The nature of radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Coherent radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Quadrature modulation and demodulation . . . . . . . . . . . 12

1.2.2 A generic coherent radar . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Optimum receive filter . . . . . . . . . . . . . . . . . . . . . . 15

1.2.4 The point spread function . . . . . . . . . . . . . . . . . . . . 19

1.2.5 Two time scales for pulse radar . . . . . . . . . . . . . . . . . 20

1.3 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Definitions of resolution . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.1 The idea of pulse compression . . . . . . . . . . . . . . . . . . 26

1.5.2 The chirp waveform, anatomy of a chirp I . . . . . . . . . . . 27

1.5.3 Spatial interpretation of the radar signal and the receive filter 28

1.5.4 Inverse and robust filtering . . . . . . . . . . . . . . . . . . . . 30

1.5.5 Range processing and the normal form . . . . . . . . . . . . . 32

1.5.6 The de-ramping technique . . . . . . . . . . . . . . . . . . . . 33

1.6 Range-Doppler processing . . . . . . . . . . . . . . . . . . . . . . . . 34

1.6.1 Separated processing . . . . . . . . . . . . . . . . . . . . . . . 35

1.6.2 Range-velocity processing . . . . . . . . . . . . . . . . . . . . 35

1.6.3 Ambiguity function . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6.4 Towards 1D imaging . . . . . . . . . . . . . . . . . . . . . . . 41

1.7 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.7.1 Estimation problem for a standard signal model . . . . . . . . 41

1.7.2 Maximum likelihood estimator . . . . . . . . . . . . . . . . . . 42

3

4 CONTENTS

1.7.3 Cramer-Rao Bounds . . . . . . . . . . . . . . . . . . . . . . . 44

1.8 Target detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.8.1 The target detection problem . . . . . . . . . . . . . . . . . . 45

1.8.2 Simple alternatives . . . . . . . . . . . . . . . . . . . . . . . . 46

1.8.3 Composed alternatives . . . . . . . . . . . . . . . . . . . . . . 50

1.8.4 Probability of detection for basic detectors . . . . . . . . . . . 53

1.9 MTI for ground based radar . . . . . . . . . . . . . . . . . . . . . . . 57

1.9.1 Temporal stable clutter . . . . . . . . . . . . . . . . . . . . . . 57

1.9.2 Temporal variation of clutter . . . . . . . . . . . . . . . . . . 58

1.10 Radar equation, SNR, CNR, SCNR . . . . . . . . . . . . . . . . . . . 60

1.10.1 Power budget for the deterministic signal . . . . . . . . . . . . 60

1.10.2 Spectral power density of the noise . . . . . . . . . . . . . . . 62

1.10.3 Signal-to-noise ratio and maximum range . . . . . . . . . . . . 62

2 Signal processing for real arrays 65

2.1 Beamforming for array antennas . . . . . . . . . . . . . . . . . . . . . 65

2.1.1 The SNR-optimum beamformer . . . . . . . . . . . . . . . . . 65

2.1.2 Characteristics of the array . . . . . . . . . . . . . . . . . . . 69

2.1.3 Ambiguities of the array characteristics . . . . . . . . . . . . . 71

2.2 Interference suppression . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.2.1 Test for the detection of the useful signal . . . . . . . . . . . . 73

2.2.2 Optimum beamformer for colored interference . . . . . . . . . 74

2.2.3 SNIR before and after optimum filtering . . . . . . . . . . . . 76

2.2.4 The case of a single source of interference . . . . . . . . . . . . 76

2.2.5 Several sources of interference . . . . . . . . . . . . . . . . . . 80

2.2.6 Signal subspace, interference subspace, noise subspace . . . . . 81

2.2.7 A basic data reduction retaining optimal detection- and esti-mation properties for signals in interference . . . . . . . . . . 82

2.2.8 An example for continuous distributed interference . . . . . . 82

2.3 Eigenspaces and sample matrix . . . . . . . . . . . . . . . . . . . . . 83

2.3.1 Determination of the weights . . . . . . . . . . . . . . . . . . 85

2.4 DOA-estimation and Cramer-Rao Bounds . . . . . . . . . . . . . . . 86

2.4.1 Estimation problem and signal model . . . . . . . . . . . . . . 86

2.4.2 Cramer-Rao Bounds for the general problem . . . . . . . . . . 86

CONTENTS 5

2.4.3 The basic estimation problem . . . . . . . . . . . . . . . . . . 88

2.4.4 Transformation of the general problem to the basic problem . 89

2.5 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.5.1 Gain loss by linear transformations . . . . . . . . . . . . . . . 91

2.5.2 Special transformations . . . . . . . . . . . . . . . . . . . . . . 92

3 Air- and spaceborne radar 93

3.1 Definition of basic angles . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2 Airborne radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.2.1 Energy distribution in range and Doppler . . . . . . . . . . . . 95

3.2.2 Three dimensional geometry for horizontal flight . . . . . . . . 96

3.2.3 ISO-Range and ISO-Doppler contours . . . . . . . . . . . . . . 97

3.2.4 Mapping between range - radial velocity and earth surface . . 99

3.3 Real and synthetic arrays . . . . . . . . . . . . . . . . . . . . . . . . 100

3.3.1 The Doppler spectrum of clutter . . . . . . . . . . . . . . . . . 104

3.4 Ambiguities and the critical antenna size . . . . . . . . . . . . . . . . 106

3.4.1 Range ambiguities . . . . . . . . . . . . . . . . . . . . . . . . 106

3.4.2 Azimuth ambiguities . . . . . . . . . . . . . . . . . . . . . . . 107

3.4.3 Full description of ambiguities . . . . . . . . . . . . . . . . . . 107

3.4.4 Minimum antenna size . . . . . . . . . . . . . . . . . . . . . . 109

3.5 Spacebased geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.5.1 Coordinate systems and transformations . . . . . . . . . . . . 111

3.5.2 Coordinate systems for earth observation . . . . . . . . . . . . 112

3.5.3 Range and radial velocity of clutter points . . . . . . . . . . . 113

4 Synthetic Aperture Radar 117

4.1 SAR history and applications . . . . . . . . . . . . . . . . . . . . . . 117

4.1.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2 Introduction to imaging Radar . . . . . . . . . . . . . . . . . . . . . . 120

4.2.1 Imaging small scenes and objects . . . . . . . . . . . . . . . . 120

4.3 SAR Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.4 Heuristical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.5 The azimuth signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.5.1 Cylinder coordinates . . . . . . . . . . . . . . . . . . . . . . . 126

6 CONTENTS

4.5.2 Range and direction history . . . . . . . . . . . . . . . . . . . 126

4.5.3 The azimuth chirp . . . . . . . . . . . . . . . . . . . . . . . . 128

4.5.4 Temporal parameters of the azimuth signal . . . . . . . . . . . 129

4.5.5 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . 131

4.6 The principle of stationary phase . . . . . . . . . . . . . . . . . . . . 132

4.6.1 The basic statement . . . . . . . . . . . . . . . . . . . . . . . 132

4.6.2 Application to the Fourier transform . . . . . . . . . . . . . . 132

4.6.3 Application to a SAR-relevant phase term . . . . . . . . . . . 133

4.7 Azimuth compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.7.1 The flying radar as a time-invariant linear operator . . . . . . 133

4.7.2 The spectrum of the azimuth chirp . . . . . . . . . . . . . . . 134

4.7.3 Filters for azimuth compression . . . . . . . . . . . . . . . . . 136

4.7.4 Azimuth resolution . . . . . . . . . . . . . . . . . . . . . . . . 137

4.7.5 Spotlight and sliding mode . . . . . . . . . . . . . . . . . . . . 137

4.8 The signal in two dimensions . . . . . . . . . . . . . . . . . . . . . . . 138

4.8.1 The signal in the ξ − r domain . . . . . . . . . . . . . . . . . 139

4.8.2 The signal in the ξ − kr domain . . . . . . . . . . . . . . . . . 140

4.8.3 The signal in the kx − kr domain . . . . . . . . . . . . . . . . 141

4.8.4 k-set and point spread function . . . . . . . . . . . . . . . . . 144

4.9 SAR-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.9.1 Generic SAR processor . . . . . . . . . . . . . . . . . . . . . . 146

4.9.2 Range-Doppler processor . . . . . . . . . . . . . . . . . . . . . 147

4.9.3 Range-migration processor . . . . . . . . . . . . . . . . . . . . 147

4.9.4 Back-projection processor . . . . . . . . . . . . . . . . . . . . 148

4.10 The radar equation for SAR imaging . . . . . . . . . . . . . . . . . . 149

A Mathematical Annex 155

A.1 Complex linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.1.1 Products of matrices . . . . . . . . . . . . . . . . . . . . . . . 155

A.1.2 Scalar product and the inequality of Cauchy-Schwarz . . . . . 156

A.1.3 Linear Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 156

A.1.4 Properties of Matrices . . . . . . . . . . . . . . . . . . . . . . 157

A.1.5 Order of matrices . . . . . . . . . . . . . . . . . . . . . . . . . 160

CONTENTS 7

A.2 Energy signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.3 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.4 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . 162

A.4.1 Random variables, realizations, probability distributions anddensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.5 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.6 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.6.1 The general estimation problem . . . . . . . . . . . . . . . . . 166

A.6.2 Approaches to estimators . . . . . . . . . . . . . . . . . . . . . 166

A.6.3 Cramer-Rao Bounds . . . . . . . . . . . . . . . . . . . . . . . 168

A.7 Power of filtered noise . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B Exercises 175

B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

B.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

B.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

B.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

B.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

B.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

B.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

B.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

B.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

B.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8 CONTENTS

Preface

This lecture addresses the basics of modern radar with an emphasis on signal mod-elling and processing. It is intended to convey the technical and physical backgroundof radar to the students to enable them to understand the mechanisms in depth andso to apply well-founded statistical signal models.

The lecture requires as prerequisites the bases of communications engineering(usual signals, convolution, Fourier transformation, linear systems and filters, sam-pling theorem, ...). Elementary knowledge of high-frequency technology like wavepropagation and antennas is communicated during the lectures.

The emphasis is set on the one hand on the statistical modelling of sensor signals,on the other hand on the multidimensional application of the Fourier transformationin the temporal and spatial domain. Based on these investigations, modern radartechniques are investigated like radar with phased array antennas and syntheticaperture radar (SAR). The lecture is illustrated with many graphics explaininggeometrical situations, processing architectures and principles. Also real examplesof radar signals are included.

In chapter 1 on radar fundamentals complex valued radar signals for coherentpulse radar are introduced including the Doppler effect caused by target motion,followed by pulse compression techniques. Ambiguities in range and Doppler arestudied as well as the design of radar waveforms based on the ambiguity function.Since radar is an ever-lasting fight against noise, statistical parameter estimationis explained in the application to radar; the Cramer-Rao bounds are introduced asmeans for measuring the information content of statistical signals with respect tothe estimation of parameters like direction or Doppler frequency. Radar detectorsare derived based on the theory of statistical test, receiver-operating characteristicsare calculated as performance measures.

Chapter 2 concerns the processing of vector-valued signals originated from an-tenna arrays. Beamformers are derived as spatial filters maximizing the signal-to-interference ratio. The monopulse estimator for the direction of a target is deducedusing the maximum-likelihood technique. The meaning of the spatial covariancematrix is demonstrated in the context of adaptive beamforming.

The chapter 3 on air and space borne radar platforms describes the geometry offlight and gives insight into the nature of clutter in spatial and frequency domains asa pre-requisite to chapter 4 on synthetic aperture radar (SAR). Here, model signalsof point scatterers in two dimensions are presented, ambiguities are derived and themost important SAR processors are investigated. Moreover, this chapter aims toobtaining insights about the meaning of the space of wave number vectors (’k-space’)for imaging.

This document is marked as ’part I’ since a second part covering also space-time-adaptive processing (STAP) and multi-dimensional SAR is planned.

Remagen, April 2011 Joachim H. G. Ender

Chapter 1

Radar fundamentals

In this chapter we will describe the basics of a coherent radar system with the abilityto emit an arbitrary wave form and to receive and digitize the backscattered echoes.Afterwards the elementary signal processing units will be built up step by step. Themost content of this chapter can be found in books on signal theory or in radar books[1, 2, 3, 5, 6, 7, 8]; nevertheless, we believe that it is very useful for the reader tohave all the needed tools together with a unified notation. The concept of matchedfilter and other matters are dealt in many publications only in the real domain. Wedescribe the signals in the complex domain from the beginning.

1.1 The nature of radar

The main advantages of radar compared to other sensors are its weather and day-light independence, its range-independent resolution and the sensitivity to smalldisplacements and motions, which have led to a triumphal procession especially forremote sensing. The core components of a radar system are: a waveform generator,the transmitter bringing the radio frequency signals to the transmit antenna whichradiates the electromagnetic waves into the space, the receiving antenna collectingthe waves reflected by objects, the receiver amplifying the weak signals and makingthem available to the analogue and - above all - digital processing.

A radar system measures the distance and direction to the object, its velocityand some signatures for the purpose of classification. There are two sources for thedistance measurement: A coarse but unambiguous information by measurement ofthe wave travelling time and a fine but ambiguous information by phase measure-ment. The accuracy is in the order of meters down to decimeters for the first kindand in the order of fractions of the wavelength (millimeters!) for the second. Byusing the phase modulation over time (Doppler frequency) the radar can measurealso the radial velocity with a high accuracy.

There are two different basic concepts of radar: The first aims at the detectionand localization of targets; here usually the resolution cell size (range, direction,Doppler) is greater or equal to the extensions generated by the object. Targetclassification can be achieved via the signal strength (radar cross section, RCS),

9

10 CHAPTER 1. RADAR FUNDAMENTALS

Glossary for this chapterα Frequency modulation rate of a chirpα Time stretch factor (Doppler effect)α Level for a statistical testβ = v/c0 Ratio of target velocity to velocity of lightη Thresholdχ(τ, ν) Ambiguity functionϕ Signal parameter (e.g. Doppler phase shift)Φ Set of signal parametersΨ Random phaseγ Signal-to-noise ratioPU Projector to the subspace Uλ Wavelengthλ0 Wavelength related to the reference frequencyΛ(z) Likelihood ratioψ(z) Statistical testρ(s1, s2) Correlation coefficientσ2 Noise varianceϑ Parameter vector

ϑ Estimator of parameter vector ϑΘ Set of parameter vectorsτ Time delaya Complex Amplitudeb Bandwidthc0 Velocity of lightC(x) Fresnel integralC(T ) Clutter processE[X] Expectation of the random variable Xf Radio frequencyf0 Reference frequencyfs Sampling frequency fast-timeF Doppler frequencyFs = PRF Sampling frequency slow-time = Pulse repetition frequencyFϑ(x) Distribution functionGrx Gain of the receive antennaGtx Gain of the transmit antennah(t) Pulse responseH(f) Transfer functionHη(k) Transfer function of robustified filterH Hypothesis (for a statistical test)In n× n unit matrixI0(x) Modified Bessel function of order 0= Imaginary partJ Fisher’s information matrixkr = 2π/λ Wavenumberk0 = 2π/λ0 Wavenumber related to the reference frequencyK Wave number set, k-setK Alternative (for a statistical test)k = 1.38 · 10−23 [Ws/K] Boltzmann constantkT = 4 · 10−21 [Ws] Noise power density at room temparature

1.1. THE NATURE OF RADAR 11

LZ(z) Log-likelihood functionLtx Power density effected by the transmit antennaLrx Power density at the receive antennam(ϑ) Parametrized model for the deterministic signal partN(t) Noise processN Noise vectorNCn(µ,R) Normal (Gaussian) distribution of a complex random vectornf Noise figure of the receive channelp(t), P (f) Point spread function and its Fourier transformp(R), P (k) Point spread scaled to range and its FTpZ(z) probability density function (pdf) of ZPZ Distribution of ZP PowerPtx Transmit powerPD Probability of detectionPF Probability of false alarmPr(A) Probability for the event Aq Radar cross section (RCS) of the targetr Distance to a scattererδr Range resolutionR = 2r Travelling way of the waveδR Travelling way resolutionr(t) Ramp signalRZ Covariance matrix of ZrZZ(τ) Correlation function of the process Z(t)RZZ(f) Spectral power density of the process Z(t)< Real parts(t), S(f) Transmit wave form and its Fourier transforms(R), S(k) Transmit wave form scaled to range and its FTsα(t), Sα(f) Chirp and its Fourier transformsRF Transmit signal in the RF-domains(t) Deterministic receive signals Signal vectorS(x) Fresnel integralt Fast timeδt temporal resolutionT Slow timeT Absolute temperature in degree Kelvin∆T = 1/∆F Pulse repetition intervalT (z) Test statisticsVar[X] Variance of the random variable Xw Weight vectorx(t), X(f) Input signal and its Fourier transformy(t), Y (f) Filter output and its Fourier transformZ(t) Received signalZ Random vector as model for the measured vectorz Realization of Z, measurement


Doppler modulations by moving parts of the object, polarization, and the dynamicsof motion. The second concept is that of radar imaging. The aim is to generate aquasi optical image (SAR, ISAR). Here the resolution cells have to be much smallerthan the objects extension. Information about the target can be extracted from thethe two- or three-dimensional images or even one-dimensional images (range profile,micro-Doppler).

1.2 Coherent radar

In this section we will discuss the principle of a coherent pulse radar system includingthe optimum receive filter. Coherent in this context means that the stability of theimplemented frequency sources allows to use the phase of the received signals frompulse to pulse.

1.2.1 Quadrature modulation and demodulation

Since the generation of sophisticated waveforms and the adequate processing of thereceived signals are practically impossible in the radio frequency domain, the partsrelevant to signal processing are mostly handled in the baseband. The signals in thebaseband are not only mathematically described in the complex number domain;real and imaginary components of the signals are realized in the hardware. Thetransformation of complex valued baseband signals to real valued radio frequency(RF) signals and vice versa is performed by the quadrature modulator (QM) and thequadrature demodulator (QDM), respectively.

Let us proceed from an arbitrary complex valued waveform s(t) generated by adevice like an arbitrary waveform generator (AWG) or a direct digital synthesizer(DDS)1. This waveform is fed into an QM transforming it into the real valued RFsignal sRF (t) via

sRF (t) = <s(t)ej2πf0t

(1.1)

= <s(t) cos(2πf0t)−=s(t) sin(2πf0t).

s(t) is called the complex envelope of the RF-signal sRF (t) with respect to thereference frequency f0. The technical realization of this transformation is depictedin Fig. 1.1 (left), using mixers with local frequency (LO) f0. The LO-signalscos(2πf0t) and − sin(2πf0t) are outputs of a stable frequency source, where one isthe time delayed version of the other.

The Fourier-transforms of s(t) and sRF (t), denoted by S(f) and SRF (f), respec-tively, are related by

SRF (f) =1

2(S(f + f0) + S∗(−f − f0)) . (1.2)

1We will use sanserif characters like s to indicate transmit waveforms

1.2. COHERENT RADAR 13

Q D M

s ( t )

s R F ( t )

- s i n ( 2 p f 0 t )

R e I ms ( t )

9 0 o

* 2

* 2

c o s ( 2 p f 0 t )

f 0

s R F ( t )

Q M

s ( t )

s R F ( t )

- s i n ( 2 p f 0 t )

R e I ms ( t )

9 0 o

c o s ( 2 p f 0 t )

f 0

s R F ( t )

Figure 1.1: Left: quadrature modulator. Right: quadrature demodulator

Let the signal s(t) be bandlimited to the frequency interval [−b/2, b/2] and letf0 > b/2. In this case SRF (f) is non-zero only over two disjunct intervals [f0 −b/2, f0 + b/2] and [−f0 − b/2,−f0 + b/2].

For signals with this property the complex envelope s(t) is uniquely determinedby the RF-signal sRF (t) [4]. Mathematically, the baseband signal s(t) can be recov-ered by deleting the negative frequency band [−f0−b/2,−f0+b/2] of SRF (f), shiftingthe spectrum to the left by f0, multiplying with the factor 2 and transforming itback to the time domain.

Technically, the recovery of s(t) is done by the QDM (see Fig. 1.1 right). The RF-signal sRF (t) is given to two mixers and mixed down with the LO signals 2 cos(2πf0t)and −2 sin(2πf0t). The outputs of the mixers are filtered by low pass filters witha bandwidth larger than b and lower than 2f0 − b which remove the higher bandscentered at ±2f0.

If the output of a QM is given directly into a QDM (’bypass’), the resultingoutput signal of the QDM is identical to the input signal of the QM, see Fig. 1.2,so the QDM performs the inverse operation to that of the QM, the real valuedRF-signal may be regarded as a carrier of the base band-signal s(t), able to betransmitted as RF waves over long ranges.

Q M

s ( t )

f 0Q D M

s R F ( t ) s R F ( t )

s ( t )Figure 1.2: Bypass of quadrature modulator and demodulator

In real systems, mostly there are several mixing stages with individual local fre-quencies. Nevertheless, the mathematics remains the same, if all LO-frequencies


are summed up to the reference frequency f0. f0 is not necessarily the center fre-quency; if the spectrum of the base band-signal s(t) is not centered at zero, thecenter frequency of the RF-signal and the reference frequency are dislocated by thesame amount.

1.2.2 A generic coherent radar

Now we extend the structure shown in Fig. 1.2 to a radar system. The output ofthe QM is amplified, led over a transmit/receive switch to the antenna and emittedinto the free space, see Fig. 1.3. A non moving point target at distance r reflectsa part of the wave, the reflected wave reaches the antenna delayed by the time τ ,is amplified and put into the QDM. The delay is τ = R/c0 where R = 2r is thecomplete travelling way of the wave forward to the scatterer and backward to thereceive antenna2 and c0 is the velocity of light in the free space.

During the amplification stages noise is added to the received signal which can betaken into account by accumulating all noise contributions to one resulting complexnoise signal N(t) superposed to the output of the QDM.

P o i n t t a r g e t

A n t e n n a

T / R S w i t c h

r*

Q M

s ( t )

f 0Q D M

a s ( t ; r )N ( t )

Z ( t )Figure 1.3: Radar system with baseband signals

Consequently, the receive signal in the baseband domain Z(t) can be written as

Z(t) = as(t; r) + N(t) (1.3)

with s(t; r) being the normalized deterministic part of the receive signal and a acomplex constant reflecting the system gain, the attenuation over the propagation

2The use of the variable R allows to include also bistatic radar systems with the completetravelling way R = r(tx) + r(rx) with different distances from the transmit antenna to the scattererr(tx) and from the scatterer to the receive antenna r(rx)!


path and the reflectivity of the point target. Following a large part of the SARliterature, we discriminate the parameters referring to the radar system (in thissituation the receiver time t) and the parameters related to the scene (here: thedistance to the scatterer r) by a colon.

The normalized noise-free received RF-signal sRF (t; r) is the delayed transmittedRF-signal:

sRF (t; r) = sRF (t− τ) (1.4)

= <s(t− τ)ej2πf0(t−τ)

= <s(t− τ)e−j2πf0τej2πf0t

.

Since the QDM recovers the complex envelope, i.e. the term before the harmonictime function ej2πf0t, the output of it is

s(t; r) = s(t− τ)e−j2πf0τ . (1.5)

Applying a Fourier transform in the baseband, we get

S(fb; r) = S(fb)e−j2π(f0+fb)τ = S(fb)e

−j2πfτ (1.6)

with f = f0 + fb being the true RF frequency of the transmitted wave componentrelated to the signal component S(fb).

If we denote the wavenumber by k0 = f0/c0 = 2π/λ0, where λ0 is the wavelengthwith respect to the reference frequency f0, we get 2πf0τ = 2πf0R/c0 = 2πR/λ0 =k0R and Eq. (1.5) can be re-written as

s(t; r) = s(t−R/c0)e−jk0R with R = 2r. (1.7)

S(fb; r) = S(fb)e−jkrR with kr = 2π(f0 + fb)/c0 = k0 + kb. (1.8)

kr is the wave number for the electromagnetic wave corresponding to the har-monic decomposition of the transmit signal at baseband frequency fb.

Apart from the expected time delay in the base band-signal in Eq. (1.7), theimportant term e−jkrR in Eq. (1.8) shows the sensitivity of the phase against targetmotion. If the distance to the scatterer is increased by a half wavelength, the phaserotates by 2π; since also much lower phase changes can be measured, range variationsin the order of millimeters are detectable in principle independent of the range whichmay be 1000 km or more!

1.2.3 Optimum receive filter

Since radar is a continuous fight against noise (and clutter, i.e. echoes of unwantedobjects), we will optimize the signal-to-noise-ratio (SNR) by filtering the data. First,two related results for the time-discrete case are presented:


Optimum weight vector

a s

Y

N 1

w 1 *

N n

w n *

S

Figure 1.4: Signal vector superposed by noise and combined by a weight vector

Problem: Let s be a complex n-dimensional signal vector, a a complexamplitude and N a noise vector with expectation zero. The componentsof N are assumed to be statistically independent with variance σ2. Tothe observer only the noisy data

Z = as + N (1.9)

are available. This simple model will come across several times through-out this scriptum. We call this standard model I. Now we search for aweighting vector w for which the scalar product Y = w∗Z reaches themaximum SNR (see Fig. 1.4).

Solution: The covariance matrix of the noise is given by RN = E [NN∗] = σ2In whereIn denotes the n-dimensional unit matrix. The SNR of Y is calculated as a function of wby

γ(w) =|aw∗s|2

E |w∗N|2 (1.10)

and wopt = argmax γ(w) : w ∈ Cn. The denominator of Eq. (1.10) can be developed as

E |w∗N|2 = E [w∗NN∗w] = w∗E [NN∗]w = w∗σ2Inw = σ2‖w‖2. (1.11)

It follows

γ(w) =|a|2 |w∗s|2σ2‖w‖2

=|a|2|s‖2

σ2

|w∗s|2‖w‖2‖s‖2

=|a|2|s‖2

σ2|ρ(w, s)|2 . (1.12)

ρ(w, s) = w∗s‖w‖‖s‖ is the correlation coefficient of w and s whose absolute value is always

lower or equal to 1, and equal to 1 if w and s are collinear (inequality of Cauchy andSchwarz, see A.1.2). So, the maximum SNR for the above problem is achieved for theweight vector

wopt = ws (1.13)

with an arbitrary complex constant w 6= 0 and takes the value γ(wopt) = |a|2‖s‖2σ2 .


Essence 1.1 The weight vector optimizing the SNR of its scalar product with Z isthe signal vector itself or a complex multiple of it. The obtained SNR is equal to theenergy of the deterministic part divided by the noise variance.

We will meet this property throughout the scriptumfor several times.

Time discrete matched filter

Now it is easy to derive from this solution an optimum filter response to a delayedsignal (see Fig. 1.5).

D e l a y e d s i g n a l sT i m e s e r i e s Z 0 , Z 1 , . . .

L i n e a r f i l t e r

h n - 1 h 0

F i l t e r e d s i g n a l Y 0 , Y 1 , . . .

Figure 1.5: Discrete matched filter

Problem: Let a be a complex constant,s ∈ Cn a signal vector andZ0, Z1, . . . a complex time series with the following property:

Zν =

Nν for ν = 0, . . . , k − 1Nν + asν−k for ν = k, . . . , k + n− 1Nν for ν = k + n, . . .

. (1.14)

N0, N1, . . . here is a sequence of statistically independent complex noisevariables with expectation zero and variance σ2.This time series is filtered by a linear time invariant filter with the pulseresponse hν . Which pulse response maximizes the SNR of the filter out-put at the time index k + n?

Solution: Obviously, the signal vector is ’hidden’ in the time series and shiftedby the amount k. The length of the optimum filter cannot be larger than n, sinceotherwise only noise will be added without growing signal energy. The filtered


output is the convolution with the pulse response:

Yν =n−1∑µ=0

Zν−µhµ (1.15)

Yk+n =n−1∑µ=0

(asn−µ + Nk+n−µ)hµ (1.16)

Following the essence 1.1, Eq.(1.13), the optimum filter coefficients are hµ =ws∗n−µ with a non-zero constant w.

Essence 1.2 The pulse response of the optimum filter is equal to the time-inverted,complex conjugated signal:

hµ = ws∗n−µ for µ = 0, . . . , n− 1 (1.17)

The maximum SNR is γ(hopt) = |a|2‖s‖2σ2 . This filter is called matched filter.

Time continuous matched filter

This time discrete optimization task can easily be transferred to the time con-tinuous case. Let s(t) be a bandlimited complex signal with Fourier transformS(f) and N(t) be a wide sense stationary noise process3 with expectation zero.Let rNN(τ) = E [N(t + τ)N∗(t)] be the correlation function of N(t) (= covari-ance function, since the expectation is zero) and its Fourier transform RNN(f) =∫

exp−j2πfτrNN(τ)dτ the spectral noise density.

Problem: We assume that the noise process is white over the signalbandwidth, i.e. RNN(f) = σ2 for all f with S(f) 6= 0. Z(t) is thesuperposition of the delayed signal s(t − τ) and the noise process N(t).Z(t) is given into a filter with pulse response h(t), the output of the filteris Y (t).

Z(t) = as(t− τ) + N(t) (1.18)

Y (t) = (h ? Z)(t) (1.19)

Which h(t) optimizes the SNR γ(h) at time t = τ?

Solution: The noise related part of the output signal is again a wide sense stochasticprocess. The SNR of Z(τ) is given by

γ(h) =

∣∣a ∫s(t− τ)h(τ − t)dt

∣∣2E|(h ? N)(0)|2 . (1.20)

3A stochastical process is called stationary, if its statistical properties are invariant against timeshifts; it is called wide sense stationary, if its correlation properties are invariant against time shifts.


If we define a new signal by q(t) := s∗(−t), the nominator is easily calculated to

|a|2∣∣∣∣∫

s(t− τ)h(τ − t)dt

∣∣∣∣2

= |a|2∣∣∣∣∫

q(t)h∗(t)dt

∣∣∣∣2

(1.21)

= |a|2 |< q, h >|2 . (1.22)

The denominator is given by (see appendix A.7)

E |(h ? N)(0)|2 = σ2‖h‖2. (1.23)

Since ‖s‖2 = ‖q‖2, we get

γ(h) =|a|2 |< q, h >|2

σ2‖h‖2(1.24)

=|a|2‖s‖2

σ2|ρ(h, q)|2 (1.25)

where ρ(h, q) denotes the correlation coefficient of h and q. Following Cauchy-Schwarz’s in-equality for continuous signals (see appendix A.2), the optimum filter is given by hopt(t) =wq(t) = ws∗(−t) with any non-zero constant w.

Essence 1.3 The pulse response of the optimum filter is equal to the time-inverted,complex conjugated signal

hopt(t) = ws∗(−t). (1.26)

The maximum SNR is γ(hopt) = |a|2‖s‖2σ2 . This filter is called matched filter.

In the foregoing, we have derived the optimum way to filter the output signal ofthe QDM in the generic radar system sketched in Fig. 1.3. There are two possibleways to implement it: Either an analogue filter is attached to the receive chain witha pulse response close to Eq.(1.26) and the filter output is converted to the digitaldomain, or the signal is first sampled by an analogue-to-digital converter (ADC)and digitally filtered according to Eq.(1.17). The latter is the more modern methodand will be followed throughout the scriptum.

1.2.4 The point spread function

If any signal x(t) is given into the filter matched to s(t), the output y(t) is

y(t) = (hopt ? x)(t) (1.27)

=

∫hopt(τ)x(t− τ)dτ (1.28)

=

∫s∗(−τ)x(t− τ)dτ (1.29)

=

∫s∗(τ)x(t + τ)dτ. (1.30)

Obviously the matched filtering is identical to a correlation with the transmitsignal. In the frequency domain, we get

Y (f) = S∗(f)X(f). (1.31)


If the transmit signal itself is given into an arbitrary filter with pulse responseh(t), the output is called point spread function

p(t) = (h ? s)(t), P (f) = H(f) · S(f). (1.32)

The radar system plus receive filter reacts on a point scatterer at distance r withthe deterministic part y(t) = ap(t− 2r/c0) exp−j2k0r.

For the matched filter, the output is given by

p(t) = (hopt ? s)(t) (1.33)

=

∫s∗(τ)s(t + τ)dτ (1.34)

= rss(t) (1.35)

and in the frequency region we get

P (f) = |S(f)|2. (1.36)

We summarize:

Essence 1.4 Matched filtering means correlation with the transmit signal. Thepoint spread function is the reaction of the receive filter to the transmit signal.The point spread function is equal to the autocorrelation of the transmit signal, if amatched filter is used. In this case it is the Fourier back transform of the magnitude-squared of the signal spectrum.

1.2.5 Two time scales for pulse radar

The analogue signal outputs of the receiver unit have to be sampled and A/D-converted as input to digital signal processing. After transmission of a pulse thesystem is switched to receive. A receive window is opened starting at the timecorresponding to the minimum desired range. Now the sampling is performed at asampling rate fs = 1/∆t a bit larger than the signal bandwidth. Sampling stopsat the time of desired maximum range. The samples fill a rangeline which can bearranged as a column in a two-dimensional array (sample field), as illustrated in Fig.1.6. The next pulse is emitted after the pulse repetition interval ∆T , the samples ofthe echoes build a new rangeline which is inserted in the two-dimensional field andso on. The frequency Fs = 1/∆T is called pulse repetition frequency (PRF).

In this way, the sample field is filled from pulse to pulse and is the starting pointfor further signal processing. This two-dimensional raw-data structure is used for themost modern radar systems. If there are more than one receive channel in parallel,for each of the channels a sample field is generated effecting a three-dimensional rawdata structure.

Both of the two dimensions of the sample field have the original dimension ’time’;the vertical axis (t) is called fast-time controlling the interval between the pulses andthe horizontal axis (T ) slow-time related to the pulse-to-pulse clock.

1.3. DOPPLER EFFECT 21

D TD t

Sample No. ( mDt)

T r a n s m i t t e d s i g n a l

F i l t e r e d e c h o o f p o i n t t a r g e t

S a m p l i n g

P u l s e - N o . ( n D T )

Figure 1.6: Two time scales for pulse radar

In the fast-time domain the sampling frequency fs = 1/∆t ranges in generalfrom some MHz to a few GHz, related to a range sampling interval c0/(2fs) ofabout 150 m (1 MHz) downto 15 cm (1GHz). The sampling frequency in the slow-time domain Fs = 1/∆T normally is in the order of hundreds of Hz to a MHz leadingto an unambiguous range interval c0/(2Fs) of some hundred kilometers downto somehundred meters.

1.3 Doppler effect

Moving targets change the frequency of the backscattered waveform due to theDoppler effect. Let us assume that the distance to the target r(t) is time varyingand vr(t) = d

dtr(t) denotes the radial velocity.

Due to the motion of the target, the delay of the wave τ(t) becomes time-dependent where t is defined at the time at the receiver. The RF receive signalwill be the time-delayed transmit signal according to Eq. (1.4) from which thecomplex baseband signal is determined:

sRF (t; r(.)) = sRF (t− τ(t)) (1.37)

= <s(t− τ(t))e−j2πf0τ(t)ej2πf0t


s(t; r(.)) = s(t− τ(t))e−j2πf0τ(t). (1.38)

The rotating phase in the exponential term constitutes the basic component ofthe Doppler frequency:

F0(t) =1

2π

d

dt[−2πf0τ(t)] = −f0

d

dtτ(t). (1.39)

Since the varying time delay in the base band signal introduces additional phaserotations, we resolve the waveform in its spectral components and get

S(fb; r(.)) = S(fb)e−j2πfbτ(t)e−j2πf0τ(t) = S(fb)e

−j2π(f0+fb)τ(t). (1.40)

So, the Doppler frequency related to the baseband frequency fb is

F (t, fb) = −(f0 + fb)d

dtτ(t) = F0(t)− fb

d

dtτ(t). (1.41)

Object motion negligible during the wave’s travelling time If the radialvelocity of the target is so low that its position at the time point of transmissionand its position at the time point of receive are small against the wavelength, themotion of the target during the wave’s travelling time can be neglected. Often thissituation is called stop and go approximation. The idea behind is that the targetstops for a moment where a pulse is transmitted, reflected and received and thenproceeds to the next position according to its radial velocity.

In this case, τ(t) ≈ 2r(t)/c0, neglecting that the range of the target at the time ofreflection is slightly different to r(t). Under this approximation, d

dtτ(t) = 2vr(t)/c0

and from Eq.(1.39) the Doppler frequency is

F (t, fb) = −2(f0 + fb)

c0

vr(t) = −2vr(t)

λwith λ =

c0

f0 + fb

. (1.42)

Exact expression For targets with a large radial velocity this approximationmay not be sufficient. We are looking for an exact solution. At time t the echo isreceived from a pulse which was emitted at time t− τ(t) and hit the target at timet′ = t − τ(t)/2. The distance at this point was r(t′), so the time for forward andbackward travelling is τ(t) = 2r(t′)/c0. We get the following equation

τ(t) =2r

(t− τ(t)

2

)

c0

, (1.43)

which implicitly determines the time delay τ(t).

For a linear time dependence r(t) = r0 + vrt the delay τ(t) and its temporalderivative are easily calculated from Eq. (1.43):

τ(t) =2(r0 + vrt)

c0 + vr

=1

1 + β

2(r0 + vrt)

c0

(1.44)

d

dtτ(t) =

1

1 + β

2vr

c0

(1.45)

1.3. DOPPLER EFFECT 23

with β = vr/c0. The resulting Doppler frequency is calculated to

F (t, fb) = −(f0 + fb)d

dtτ(t) = − 1

1 + β

2(f0 + fb)vr

c0

= − 1

1 + β

2vr

λ. (1.46)

This result differs from the stop-and-go approximation by the factor 11+β

whichis normally very close to 1.

Essence 1.5 The Doppler frequency of a moving target is given by F = − 11+β

2vr

λ

with β = vr/c0. For the stop-and-go approximation this is simplified to F = −2vr/λ.

——————————————-

We go back to the received signal Eq. (1.38). With g(t) = t− τ(t) we get

s(t) = s(t− τ(t))e−j2πf0τ(t) (1.47)

= s(g(t)) exp −j2πf0(t− g(t)) (1.48)

For constant radial velocity g(t) is computed from Eq. (1.44) to

g(t) = t− 1

1 + β

2(r0 + vrt)

c0

(1.49)

=1− β

1 + βt− 1

1 + β

2r0

c0

(1.50)

= αt− τ ′ (1.51)

with α = 1−β1+β

and τ ′ = 11+β

2r0

c0and by inserting this expression into Eq. (1.47) the

received signal is

s(t) = s(αt− τ ′) exp −j2πf0(τ′ + (1− α)t) . (1.52)

We draw the following conclusions:

• The time scale of the receive signal has been stretched by the factor α withrespect to the transmitted signal.

• The signal has been modulated by the Doppler frequency F0 = −f0(1− α).

• The initial time delay and initial phase have been modified to τ ′ and −2πf0τ′.

The effects of the Doppler modulation on a sequence of rectangular pulses areillustrated in Fig. 1.7. The time-continuous receive signal can be split into the fastand slow-time domains and sorted into the two dimensional array of samples. Thereare three main effects: Firstly the phase modulation during one received pulse,secondly the phase shift from pulse to pulse and thirdly the increasing delay frompulse to pulse. The latter is called range migration or range walk . The effects areof different importance. Most significant is the phase change from pulse to pulsewhich is the first to be exploited in signal processing. The Doppler modulationwithin the received single pulse often can be neglected, if the pulse duration is nottoo long. The range migration effect is negligible for short pulse trains and coarserange resolution, but in general not for imaging radars.


1.4 Definitions of resolution

Let us return to the point spread function p(t), Eq. 1.32. It has a maximum att = 0 and a certain width of the main beam which expresses how the response toa point like target is smeared. Obviously this width is related to the ability of theradar system to resolve point like targets which are close together. So, the value’resolution’ should be a measure how far two point targets have to be separatedto be recognized as two. A general mathematical definition does not exist. Thesimplest way is to measure the width of the main beam. Since this is not uniquelydefined, we mention different approaches:

tI m

R e

tI m

R eP h a s e r o t a t i o nw i t h i n o n e p u l s e

T i m e e x p a n s i o n

T r a n s m i t t e d s i g n a l

Im

t

Re

TP h a s e r o t a t i o n f r o m p u l s e t o p u l s e

t h e r a n g e c e l l s !

R e c e i v e d s i g n a l

M i g r a t i o n t h r o u g h

Figure 1.7: Doppler effect for a pulse train; bottom: After transformation to theT − t plane

1.4. DEFINITIONS OF RESOLUTION 25

3dB-resolution: The temporal resolution δt is defined by t+−t− where t+ and t−are that points of time closest to t = 0 where the power of the point spread functionhas decreased to the half of the maximum value:

|p (t−)|2 = |p (t+)|2 =1

2|p (0)|2 . (1.53)

The 3dB-resolution is a more technical value which for example is given in datasheets to systems and units. More feasible in our context is the

Distance of the maximum to the first zero: For many wave forms the mainbeam of the point spread function is bounded by a clear zero. The distance from themaximum to this zero is a measure of width, too. If a rectangular pulse of length tsis used, the response of the matched filter is a triangular pulse with the first zero att = ts. So the temporal resolution in this sense is δt = ts.

Rayleigh resolution: If the spectrum of the pulse is represented by a rectangularfunction with bandwidth b, the point spread function for matched filtering is a sinx/x function with first zero at t = 1/b. Following the preceding definition, theresolution in this case is δt = 1/b. This definition can be used also for spectra whichare not rectangular, if the bandwidth is measured in another way, for instance as a3dB-bandwidth.

t t

d t 3 d B

d t R a y l e i g h = 1 / b

fb

| S ( f ) | 2

| p ( t ) | 2| p ( t ) | 2

Figure 1.8: Definitions of resolution

From the temporal resolution δt immediately the range resolution results in δR =c0δt, where R is the complete travelling way of the wave, and consequently

δr =c0

2δt (1.54)

for monostatic radars with R = 2r.

If the range-axis is partitioned into intervals of length δr, these are called rangeresolution cells.


Similar conditions can be found in other dimensions of the radar like Dopplerfrequency or angle. In general a radar resolution cell is a multidimensional bodywhose extensions along the axes are the range resolution, Doppler resolution, angularresolution and so on.

1.5 Pulse compression

1.5.1 The idea of pulse compression

The range resolution achieved with a rectangular pulse of length ts is given byδr = c0ts/2. The energy ‖s‖2 of the pulse, which determines the maximum range,is ‖s‖2 = tsP where P denotes the power. This is - especially if semiconductors areused for the high power amplifier - limited.

Problem: How can a finer range resolution be achieved without loss ofpulse energy?

The solution is to expand the bandwidth by modulation of the pulse. From Eq.1.54 the Rayleigh range resolution of a waveform with a rectangular spectrum S(f)of bandwidth b is given by

δr =c0

2b(1.55)

without direct dependence on the pulse length.

If the original waveform s(t) with Fourier transform S(f) is passed through a filterwith transfer function H(f) = expjϕ(f) the spectrum of the filtered waveformsRF (t) is given by SRF (f) = H(f)S(f) and |SRF (f)|2 = |S(f)|2. Since the pointspread function for matched filtering is the Fourier back transform of the squaredmagnitude of the signal spectrum, the point spread functions corresponding to s(t)and sRF (t) are the same. If for instance the primary waveform is a rectangular pulsewith a C times shorter pulse length ts/C is passed through a filter with transferfunction H(f) = exp−j2πτ(f)f, the delay τ(f) is frequency dependent causinga time stretch of the original signal. Assume that a time stretch by a factor Cis achieved. Then the resulting wave form can again be amplified to the power Presulting in the same energy content as the rectangular pulse with duration ts. Butthe spectrum is stretched by the factor C effecting a C times better range resolution.C is called compression factor . The time-bandwidth product of the rectangular pulseis ts · 1

ts= 1, that of the expanded pulse ts · C

ts= C, so the time-bandwidth product

is equal to the compression factor.

Essence 1.6 For the range resolution the bandwidth of the transmitted waveform isdecisive: δr = c0

2b. The gain in range resolution with respect to a rectangular pulse of

same duration is called compression rate, which is equal to the time-bandwidth prod-uct. Two different waveforms s(t) and sRF (t) effect the same point spread functionfor matched filtering, if sRF (t) is generated by s(t) by passing it through a filter witha transfer function with magnitude 1.

1.5. PULSE COMPRESSION 27

1.5.2 The chirp waveform, anatomy of a chirp I

t

R e s ( t )

t s / 2- t s / 2

t

f

f = a t

t s / 2- t s / 2

Figure 1.9: Chirp

An outstanding role in coded waveforms plays the linearly frequency modulatedpulse with constant amplitude, in radar jargon called ’chirp’ , defined by

sα(t) = expjπαt2w(t) (1.56)

We call α frequency modulation rate or simply ’rate’. If the window w(t) isrectangular (w(t) = rect(t/ts)), the instantaneous frequency f(t) = αt runs throughthe interval [−αts/2, αts/2] (see Fig. 1.9) effecting a bandwidth which is close to theinstantaneous frequency span αts. So the time-bandwidth product (=compressionrate!) is approximately equal to αt2s.

Exactly, the Fourier transform of this rectangular chirp is given by

Sα(f) =

∫ ts/2

−ts/2

expjπαt2 − j2πftdt (1.57)

=

√1

2αe−jπ 1

αf2

[C(x1) + C(x2) + j(S(x1) + S(x2))] (1.58)

with

x1 =

√α

2ts

(1 +

2f

αts

), x2 =

√α

2ts

(1− 2f

αts

)

and the Fresnel-Integrals

S(x) =

x∫

0

sin(π

2t2

)dt, C(x) =

x∫

0

cos(π

2t2

)dt. (1.59)

If ts is small, the phase modulation is negligible and the Fourier transform tendsto a sin x/x function whose main beam becomes narrower if ts grows, see Fig. 1.10.At a point where the time-bandwidth product exceeds 1, the main beam splits andbroadens more and more and takes approximately the form of a rectangular functionwith a bandwidth close to the frequency span αts. Anticipating the chapter on SAR,we observe exactly this behavior of the point spread function in azimuth if the motioncompensation is not perfect leaving a quadratic phase error.


When ts tends to infinity, x1 and x2 also tend to infinity and integral cosine andintegral sine tend to 1/2, so the limiting form of the Fourier transform is

Sα(f) → 1 + j√2α

e−jπ 1α

f2

=1 + j√

2αs−1/α(f). (1.60)

Essence 1.7 The magnitude of the Fourier transform of a rectangular chirp withrate α has the approximate shape of a rectangular function with bandwidth close tothe frequency span αts. For infinite duration the Fourier transform is again a chirpwith rate −1/α.

Figure 1.10: Magnitude of the Fourier transforms of chirps with growing time basis

Fig. 1.11 shows the steps of the compression of a chirp in the time-domain andthe frequency domain.

1.5.3 Spatial interpretation of the radar signal and the re-ceive filter

In this section, we pass to the spatial domain. From the viewpoint of focussing toimages (SAR), the spatial domain is the primary one.

We transform the temporal transmit signal, the pulse response of the receivefilter and the point spread function into spatial signals, dependent on the spatial


t

f

t

f

C h i r p

F F - 1

a ct

t | . | 2

F o u r i e r t r a n s f o r mP o w e r

s p e c t r u m

C o m p r e s s i o nr e s u l t

Figure 1.11: Compression of a chirp signal

variable R = 2r via t → R = c0t. In this section we will use the symbols s, h, p asfunctions of R.

The Fourier transforms are referring now to spatial frequencies, the wave-numberreplaces the circular frequency:

S(k) =

∫s(R)e−jkRdR (1.61)

H(k) =

∫h(R)e−jkRdR (1.62)

P (k) =

∫p(R)e−jkRdR (1.63)

Note that these relations are formulated in the baseband, so k = 2πfb/c0 is thatpart to the physical effective wavenumber kr = k0 + k = 2π

λcontributed by the base

band frequencies.

p(R) is the point spread function of the signal, now in the spatial domain. Itswidth determines the range resolution in the variable R, for monostatic radars therange resolution in r is half of that in R.

For matched filtering, we get:

p(R) =1

2π

∫|S(k)|2ejkRdk. (1.64)


The power of the signal spectrumW(k) := |S(k)|2 can be interpreted as a weight-ing in the k-domain, and we conclude:

Essence 1.8 The point spread function for matched filtering in the range domainis given by the inverse Fourier transform of the power of the signal spectrum in thewave number domain.

1.5.4 Inverse and robust filtering

The matched filter achieves the optimum signal-to-noise ratio. On the other hand,the sidelobes of the matched filter may be unacceptable. For an arbitrary receivefilter h(R) we introduce the following quantities:

Ls(h) = |p(0)|2

Ln(h) =

∫|h(R)|2dR

Lp(h) =

∫|p(R)|2dR. (1.65)

Ls(h) is the power of the point spread function at its maximum, Ln(h) the noisepower density at the output of the filter for normalized white noise at its input, andLp(h) is the total energy of the point spread function, containing also the powerspread to the sidelobes. The maximization of the ratio Ls(h)/Ln(h) leads to thematched filter, the maximization of Ls(h)/Lp(h) to the inverse filter, as we will seein the following. To get any state in between, we define a weighted quantity ofunwanted power by Li(h) := ηLn(h) + Lp(h) and solve the optimization task

hη = argmax

Li(h)

Ln(h): h ∈ L2(C)

(1.66)

The result in the wave number domain is4

Hη(k) = wS∗(k)

η + |S(k)|2 (1.67)

with an arbitrary non-zero constant w.

The Fourier transform of the related point spread function takes the form

4Derivation: In the wavenumber domain we have Ls(h) =∣∣∫ H(k)S(k)dk/(2π)

∣∣2, Ln(h) =∫ |H(k)|2dk/(2π), Lp(h) =

∫ |H(k)|2|S(k)|2dk/(2π). By defining H ′(k) := H(k)√

η + |S(k)|2

and S′(k) := S(k)/√

η + |S(k)|2 we get Ls(h) =∣∣∫ H ′(k)S′(k)dk/(2π)

∣∣2 and ηLn(h) + Lp(h) =∫ |H ′(k)|2dk. From the inequality of Cauchy-Schwarz follows for the maximization of the ratioH ′

opt(k) = wS′∗(k) and from that immediately the statement.


Pη(k) = Hη(k)S(k) = w|S(k)|2

η + |S(k)|2 (1.68)

For η = 0 (noise neglected) and w = 1 we get H0(k) = 1/S(k) which defines theinverse filter. For w = η and η tending to infinity (only noise regarded) Hη(k) → S∗

results in the matched filter.

Let K :=k : |S(k)|2 > 0

be the carrier of the baseband signal spectrum which

we will call in the following ’k-set’. Here, the k-set is placed at the baseband. Laterwe will consider also the related k-set in the physical wave number domain.

The application of the inverse filter is defined on K. Outside this set the transferfunction H0(k) may be set equal to zero.

We get for the point spread function p0(k) in this case:

P0(k) =

S(k)S(k)

= 1 for k ∈ K0 else

(1.69)

Obviously, P0(k) is the indicator function of the set K where the indicator func-tion IM(x) of an arbitrary set M is defined as 1, if x ∈ M and 0 else. From Eq.(1.69) it follows

p0(R) =1

2π

∫IKejkRdk =

1

2π

∫

KejkRdk. (1.70)

Essence 1.9 For the application of the inverse filter, the point spread function isequal to the Fourier back transform of the indicator function of the carrier of thesignal spectrum (k-set)

If the signal power spectrum |S(k)|2 itself is 1 for k ∈ K and zero else, i.e. auniform power distribution over the carrier, matched filter and inverse filter coincide.In this case, there is no SNR loss.

For the arbitrary case we get a normalized SNR of

SNR =

∣∣∣∫K

|S(k)|2η+|S(k)|2 dk

∣∣∣2

2π∫K

|S(k)|2

(η+|S(k)|2)2 dk(1.71)

=

12π

∫K |S(k)|2dk for η →∞

(∫K dk)

2

2π∫K|S(k)|−2dk

for η = 0(1.72)

If |S(k)|2 takes in K values close to zero, the denominator in the second case maybe very large or even infinity effecting a high loss in SNR. If the robustified filter fora suitable choice of η is used, this can be avoided. The robustified filter behaves likethe matched filter for small values of the spectral signal power and like the inversefilter for large values. It is able to correct variations of the spectral signal power asillustrated in Fig. 1.12.


S i g n a l s p e c t r u m

R o b u s t i f i e di n v e r s e f i l t e r

R e s u l t

K

Figure 1.12: Robustified filter

1.5.5 Range processing and the normal form

For radar engineers it is convenient to think in processing schemes. Starting point isalways the array of complex samples as introduced in section 1.2.5. We assume thatthe analogue filtering before sampling reduces to a low path filter limiting the noisebandwidth to a value smaller than the sampling frequency according to the Nyquistcondition. Clearly the sampling frequency should be larger than the bandwidth ofthe useful signal. Under these conditions, range compression can be done in thedigital domain.

Fig. 1.13 shows the processing scheme in the temporal and frequency domain.For each range line the compression filter has either to be applied as convolution inthe temporal domain or as multiplication with the transfer function in the frequencydomain.

h ( t ) = s * ( - t )C o m p r e s s i o nf i l t e r

Tt

R a n g e c o m -p r e s s e d d a t a

F o r e a c h r a n g e l i n e

R a n g e F F T

Tt R a w d a t a

Tf

F o r e a c h r a n g e l i n eT

t R a w d a t a

H ( f ) = S * ( f )

Tf

R a n g e I F F T

Tt

R a n g e c o m -p r e s s e d d a t a

Figure 1.13: Range compression with the matched filter. Left: direct convolution,right: processing in the frequency domain

Regarding the transformation to the wavenumber domain and the inverse filter-ing we can come to an elegant signal representation, called normal form :


First, the fast-time variable is substituted as range variable. Let the measuredsignal along one range line be z(R). The first step of range compression is to applythe Fourier transform yielding Z(k) =

∫z(R) exp−jkRdR. Now the inverse filter

is applied resulting in y(k) = Z(k)/S(k). We call the output pre-processed data.

A following inverse Fourier transform (IFFT) would yield the range compresseddata. Nevertheless, in some applications, the next step is not the IFFT but aprocessing along the slow-time variable. If z(R) is the noise-free echo of a pointscatterer at R = R0, we get

z(R) = as(R−R0) exp−jk0R0 (1.73)

Z(k) = aS(k) exp−j(k0 + k)R0. (1.74)

After application of the inverse filter, the signal is

y(k; R0) = Z(k)/S(k) (1.75)

= a exp−j(k0 + k)R0 (1.76)

for all k in the measured baseband k-set. Obviously, the dependance on the waveform has been eliminated by the inverse filtering.

Regarding kr = k0 + k we can define the model signal as function of kr bys(kr; R0) := y(kr − k0; R0) and get the following normal form for a point scattererat range R:

s(kr; R) = exp−jkrR. (1.77)

R a n g e F F T

TR

Tk

F o r e a c h r a n g e l i n e

1 / S ( k )

Tt R a w d a t a

v a r i a b l es u b s t i t u t i o n

TkP r e - p r o c e s s e d

d a t a

R a n g e I F F T

Tt

R a n g e c o m - p r e s s e d d a t a

o p t i o n a l

Figure 1.14: Pre-processing to the k domain

1.5.6 The de-ramping technique

We return to the representation in time. If the received echoes are sampled with theNyquist-rate according to the signal bandwidth the A/D conversion technique is soon


R e c e i v e d s i g n a l B a n d w i d t h b

L o c a l = ' r a m p '

B a n d p a s s

A / D

R e d u c e d s a m p l i n g r a t e f s

F F T

t

f

t

f

- fInt

ensity

r a m p

O r i g i n a l R F s i g n a l sb a n d p a s s

b

A f t e r d o w n - c o n v e r s i o n

t h i s e c h o i s s u p p r e s s e d

A f t e r F o u r i e r t r a n s f o r m a t i o n

Figure 1.15: De-Ramping

at its limits, if the bandwidth advances into the GHz area. An elegant possibility toavoid this bottle neck is the de-ramping-method, see Fig. 1.15. A linearly frequencymodulated waveform with the baseband representation s(t) = expjπαt2rect(t/ts)is used as transmit pulse. The returning echoes s(t − τ) are down-converted by a’ramp’ r(t) = expjπα(t− t0)

2 resulting in

r∗(t)s(t− τ) = expjπα(t20 + τ 2) expjπα2(τ − t0)trect((t− τ)/ts). (1.78)

This mixing product has the constant frequency α(τ − t0). This frequency cor-responds to the range of the scatterer f(r) = α(2r/c0 − t0). Now a bandpass isapplied letting only the frequencies in the interval [f1, f2] pass. This is equivalentto cut the range of scatterers which are not suppressed by the filter to a range win-dow [r1, r2]. Subsequently the filtered echoes can be sampled at a lower samplingfrequency obeying the Nyquist condition with respect to the pass band. A followingFourier transform effects a range compression; the frequency axis can be re-scaledto range. A simple calculation shows that the original range resolution δr = c0/(2b)is preserved. The price payed for the reduced sampling rate is the abandonment ofranges outside the ’zoom’ interval.

If the duration of the chirp is large compared to the used range-interval, thedown-converted sampled signals can be used directly as entries in the normal formEq. (1.77), if the axis is re-scaled from f to kr.

1.6 Range-Doppler processing

The relative radial motion of a target (i.e. either the target or the radar is movingor both of them) is described by a time-dependent range r(T ) to the antennas phase

1.6. RANGE-DOPPLER PROCESSING 35

center. r(T ) is called range history. The complete travelling way of the wave thenis R(T ) = 2r(T ) when we use the stop-and-go approximation.

Under this approximation, the signal of a target moving relatively to the radarin the (k, T )-domain can be idealized in the normal form as

s(kr,T )(kr, T ; R(.)) = exp−jkrR(T ). (1.79)

If the range history is time-linear or can be approximated as time-linear during theduration of a sequence of pulses: r(T ) = r0+vrT , the double way is R(T ) = R0+vRTwith R0 = 2r0, vR = 2vr and the signal in two dimensions is given by

s(kr,T )(k, T ; R0, vR) = exp−jkr(R0 + vRT ) (1.80)

= exp−jkrR0 exp−jkrvRT (1.81)

= exp−jkrR0 expj2πF (kr)T (1.82)

where F (kr) = − 12π

krvR = − 1λvR is the Doppler frequency which is proportionally

to kr.

1.6.1 Separated processing

Conventionally, range-Doppler processing is done in two separated steps. First, therange-compression of the pre-processed data z(k,T )(k, T ) is performed by the Fourierback transformation:

z(R,T )(R, T ) =1

2π

∫z(k,T )(k, T ) exp−jkRdk. (1.83)

Then follows a Fourier transform into the Doppler domain

z(R,F )(R, F ) =

∫z(R,T )(R, T ) exp−j2πFTdT. (1.84)

While the radial motion of the target mostly can be neglected for the first step, itoften is a problem for the second, since for the Fourier transformation along the rowsof the sample field it has to be assumed that the target remains in the same rangeresolution cell during the time of coherent integration. This is often not the caseespecially for long pulse sequences and/or fine range resolution (range migration).In this case the output of separated processing will show a response smeared alongrange corresponding to the change of range during the slow-time interval. Also theDoppler resolution will suffer since the effective time basis decreases. Additionallya variation of the target’s radial velocity during the time of integration leads to asmear in the Doppler domain.

1.6.2 Range-velocity processing

The effect of range migration can be compensated in its linear term by Range-velocityprocessing . For a fixed radial velocity vr the Doppler frequency is proportional to the


RF frequency, or the wave number: F (kr) = −krvr/π. At a reference wavenumber k0

we will observe the Doppler frequency F (k0) leading to a periodicity ∆T0 = 1/F (k0)in the slow-time domain. If the time axis for the other wavenumbers is re-scaledby a factor kr/k0, the periodicity at all wavenumbers will also be T0 instead of∆T = 1/F (kr). If this re-scaling is done before the last step of range compression,the inverse Fourier transformation along k, the range walk will be compensatedsince the final data are no longer arranged in wavenumber/Doppler but in fact inwavenumber/radial velocity. The re-scaling can be done by re-sampling to a newraster (’re-formatting’) with an interpolator or by aids of the chirp scaling algorithm,see [14]. This approach is state-of the art in SAR-systems but can be applied alsoto other systems like airspace surveillance radars, if range walk is a problem. In Fig.1.16 both approaches are illustrated.

Tk rP r e - p r o c e s s e d

d a t a

S l o w - t i m e F F T

Fk r


d a t a

R a n g e I F F T

TR


FR

R a n g e -D o p p l e r d a t a

D o u b l e f r e -q u e n c y d a t a

2 D I F F T

FR

R a n g e -D o p p l e r d a t a


d a t a


Fk r

D o u b l e f r e -q u e n c y d a t a

R e - f o r m a t t i n g

v r

W a v e n u m b e r -v e l o c i t y d a t ak r 2 D I F F T

v r

R a n g e -v e l o c i t y d a t aR

Figure 1.16: Range Doppler processing and Range velocity processing

1.6.3 Ambiguity function

The choice of a specific waveform s(t)is not only decisive for the range resolution butalso for the response to Doppler shifts. To judge the performance of a waveform inboth dimensions, the ambiguity function was introduced in the 1950th as the squaremagnitude of the function

χ(τ, ν) :=

∫s(t)s∗(t− τ)ej2πνtdt. (1.85)

Obviously, χ(τ, ν) represents the reaction of a filter matched to Doppler frequencyF0 to a signal with Doppler frequency F0 + ν in a simplified form. The maximumof the magnitude is at the origin. A peak at another place in the (τ, ν)-plane(ambiguity plane) indicates that at the corresponding temporal-frequency shift the


T

k r

k r e f

T '

k r

r rr 0

Figure 1.17: Re-formatting for the range velocity processing. For each wavenum-ber, the time axis is scaled to yield the same slow-time frequency as at a referencewavenumber. At the same time, the range frequency becomes constant along T lead-ing to a fixed range, the linear range walk is removed.

signal is difficult to discriminate from the signal at τ = ν = 0: obviously there is anambiguity.

Transformed to the frequency domain, we get

χ(τ, ν) :=

∫S(f − ν)S∗(f)e2πjfτdf. (1.86)

The one-dimensional cuts along the time and Doppler axes are given by

χ(τ, 0) = rss(τ) (1.87)

χ(0, ν) = rSS(−ν). (1.88)

Along the τ axis we recognize the autocorrelation function of the temporal signal,along the ν axis the mirrored autocorrelation function of the frequency representa-tion.

If |χ(τ, ν)|2 is integrated over the whole ambiguity plane the following rule aboutthe volume invariance of the ambiguity function is obtained:

∫ ∫|χ(τ, ν)|2dτdν = ‖s‖4. (1.89)

Essence 1.10 The volume under the ambiguity function does not depend on theshape of the waveform but only on the signal energy.


This means that it is not possible to reduce the sidelobe energy completely overthe whole ambiguity plane without change to the main lobe, but that only a shift ofthe energy to other regions is possible, for instance to places in which ambiguitiesare not severe, since they are e.g. outside of the practically to expect range-Dopplerarea, an aim of wave form design.

t / t s

n t s n t s

t / t s

t / t s

n t s

Figure 1.18: Ambiguity functions of three waveforms in level plot representation.Top left: rectangular pulse, top right: chirp with Gaussian envelope, bottom: trainof five rectangular pulses

Ambiguity function of important waveforms

Rectangular pulse: If s(t) = rect(t/ts) the ambiguity function is given by χ(τ, ν) =ejπντ (ts − |τ |)si(πν(ts − |τ |)) for |τ | ≤ ts and zero else. In the temporal variablewe recognize the triangular function as autocorrelation function of the rectangularpulse, in the Doppler frequency the expected si function.

Chirp with Gaussian envelope: For a chirp with Gaussian envelope:

s(t) = exp

− t2

2σ2+ jπαt2

, (1.90)


the ambiguity function is

χ(τ, ν) =√

πσ2ejπντ exp

− τ 2

4σ2− (πσ)2(ατ + ν)2

. (1.91)

For fixed τ the magnitude of the ambiguity function is at ν = −ατ , so the ’ridgeof the ambiguity mountain’ is tilted against the axes. This can be heuristicallyexplained as follows (see Fig. 1.19):

t i m e o f b e s t f i t

D o p p l e r s h i f t e d s i g n a l

M a t c h e d f i l t e r f o r f 0

t 0t

f 0 t

n

f

A r e a o f p h a s e m a t c h

f 0 + n

t 0 - t t i m e s h i f t

n

Figure 1.19: TimeShift

The matched filter for a mean Doppler frequency f0, which is represented by astraight line in the time-frequency plain, moves along the time axis. If it is congruentwith a signal at the same mean Doppler, at the output of the filter there will arisea peak at the time t = t0, if the echo signal is centered at t0. But if the same echosignal is shifted in Doppler by the amount ν, the filter will match a part of the signalat the time t = t0 − τ with τ = ν/α. So the peak will occur at the time shift τ andwith a widened and lowered main beam due to the shorter time base.

On the one hand, this is a disadvantage, since the position of the range responseis changed due to a Doppler shift effecting a wrong range information. On the otherhand, the chirp waveform matches the Doppler shifted signal nevertheless over apart of the pulse duration, accumulating SNR and making detection possible. Thisproperty is called Doppler tolerance. Other waveforms, e.g. a pseudo-noise codedpulse, don’t have this property, slight Doppler shifts cause a complete mismatchalong the whole time axis, detection is prevented. In this case, pulse compression hasto be performed with a bank of filters matched to a selection of Doppler frequencies.

Obviously, for the chirp waveform there is an ambiguity between Doppler andrange. If one of the two is known, the other variable can be measured with high


accuracy. The Doppler frequency may be measured over a sequence of pulses andused for a correction of range.

Essence 1.11 The chirp wave form is Doppler tolerant, i.e. a Doppler shift of theecho with respect to the reference chirp leads only to a moderate SNR loss corre-sponding to the non-overlapping part of the signal spectra. A time shift is effectedwhich is proportional to the Doppler shift.

Train of rectangular pulses If a train of N equal pulses s(t) is transmitted witha temporal period ∆T , the resulting wave form is q(t) =

∑N−1n=0 s(t− n∆T ) and the

ambiguity function χq of q is evaluated from the ambiguity function χs of s by

χq (τ, ν) =1

N

N−1∑n=0

N−1∑m=0

ej2πνn∆T χs (τ − (n−m) ∆T, ν) (1.92)

=1

N

N−1∑

p=−(N−1)

sin (πν (N − |p|) ∆T )

sin (πν∆T )ejπν∆T (N−1+p)χs (τ − p∆T, ν).

For a train of rectangular pulses the ambiguity function is shown in Fig. 1.18.The range resolution is the same as for the single pulse, while the Doppler resolutionis decreased to 1/tq, where tq is the duration of the whole pulse train. In theambiguity plane we recognize a temporal ambiguity with period ∆T leading to arange ambiguity c

2∆T and a Doppler ambiguity with a period PRF = 1/∆T . If we

define a ambiguity rectangle with the edges ∆T and 1/∆T , its area is for any choiceof the PRF equal to 1. This leads to the following

Essence 1.12 For a pulse train repeated with ∆T , the product of temporal andfrequency ambiguity is equal to 1. The product of range ambiguity and Doppler am-biguity is equal to c0/2. The product of range ambiguity and radial velocity ambiguityis c0/λ = f .

So the choice of the PRF will changes the ambiguities, but not their product. Forthe most radar designs, it is not possible to place both ambiguities simultaneouslyinto areas where they are harmless. Three cases can be discriminated:

• Low-PRF mode: After the transmission of a pulse, the next pulse is trans-mitted not before all echoes of the first pulse have arrived, also from themaximum range rmax with non-negligible echo power. So the PRF for range-unambiguous operation is limited by PRFmax = c0/(2rmax).

• High-PRF mode: To evaluate also the highest expected radial velocity±vmax unambiguously from the Doppler frequency, this has to be at leastPRFmin = 4vmax/λ.

• Medium-PRF mode: The PRF lies in between, there are ambiguities inrange as well as in Doppler

1.7. ESTIMATION 41

1.6.4 Towards 1D imaging

a ( t ) ^

F F

A ( f )

- 1

r s s ( t )

. | S ( f ) | 2

a ( t )

A ( f )^

a ( r ) ^

F F

A ( k )

- 1

r s s ( r )

. | S ( k ) | 2

a ( r )

A ( k )^

Figure 1.20: 1DImaging

Pulse compression can be regarded as one-dimensional imaging, see Fig. 1.20. Ifthe reflectivity in range is given by a(r), the result of transmission of a waveform s(r)scaled to the range domain and matched filtering of the echoes is the convolution ofa(r) with the autocorrelation function of s(r). This can be regarded as the imperfect’reconstruction’ a(r) of the range reflectivity, the range profile. In the wavenumberdomain A(k) - the Fourier transform of a(r) - is multiplied by the squared magnitudeof the signal spectrum resulting in A(k) whose back transform is again a(r). Since|S(k)|2 is unequal to zero only over an interval determined by the RF band of thetransmitted signal, the multiplication effects a cutting of the reflectivity spectrum,which limits the bandwidth of a(r), resulting in an ’unsharp’ image.

1.7 Estimation

1.7.1 Estimation problem for a standard signal model

If the model for the distribution of a random variable X contains an unknownparameter vector ϑ ∈ Θ, the basic problem is to estimate this parameter vectorwith an estimator ϑ(x) for a given realization x of X. For each ϑ the distributionPX

ϑ of X and the probability distribution function (pdf) pXϑ (x) are parametrized by

this parameter vector.

General basics of the estimation theory proceeding from this problem are de-scribed in the mathematical appendix A.6. Here, we start with an estimation prob-lem which often arises in radar signal processing (we call this standard model II):


Example : Let the n-dimensional random vector Z be modelled by

Z = as(ϕ) + N,N ∼ NCn(0, σ2In). (1.93)

The complex amplitude a and the parameter ϕ are unknown and haveto be estimated. The signal vector is dependant on ϕ. For instance,the coefficients of s(ϕ) could describe the signal along slow-time, whileϕ = 2πF∆T is the phase increment from pulse to pulse due to Dopplermodulation. Or s(ϕ) denotes the signals at an antenna array, while ϕdescribes the angle of arrival of an impinging wave.For this example, the parameter vector is ϑ = (a, ϕ)t with values inΘ = C× Φ, where Φ is e.g. the interval [0, 2π]. The related probabilitydistribution of Z is PZ

a,ϕ = NCn(as(ϕ), σ2In).

1.7.2 Maximum likelihood estimator

This problem is of the formZ = m(ϑ) + N (1.94)

with N ∼ NCn(0, σ2In). m(ϑ) describes the deterministic part in dependance onthe parameter vector ϑ.

The related set of pdf’s is

pZϑ(z) =

1

πnσ2nexp

− 1

σ2‖z−m(ϑ)‖2

(1.95)

and their logarithm (log-likelihood function)

LZϑ(z) = − ln(πnσ2n)− 1

σ2‖z−m(ϑ)‖2 (1.96)

xx

p d f ' s f o r d i f f e r e n t J ' s

s e l e c t e d J ( x )^

Figure 1.21: Principle of Maximum Likelihood estimation

The maximum likelihood estimator (ML estimator) for this problem is given asthe maximum of the pdf with respect to ϑ as illustrated in Fig. 1.21 - or its logarithm(see appendix A.6.2) and results in the minimization task

1.7. ESTIMATION 43

ϑ(z) = argmin‖z−m(ϑ)‖2 : ϑ ∈ Θ

. (1.97)

Obviously, for the assumption of a Gaussian probability distribution, the MLestimate is equivalent to the minimum mean square estimator.

If we return to the signal model in Eq.(1.93), we have to solve

(aϕ

)(z) = argmin

‖z− as(ϕ)‖2 : a ∈ C, ϕ ∈ Φ

. (1.98)

The solution can be done in two steps. First, we minimize with respect to a andget

aϕ(z) =s∗(ϕ)z

‖s(ϕ)‖2(1.99)

‖z− aϕ(z)s(ϕ)‖2 = ‖z‖2 − |s∗(ϕ)z|2‖s(ϕ)‖2

. (1.100)

If we define the normalized signal by s(ϕ) = s(ϕ)/‖s(ϕ)‖, the ML estimation of ϕand a are obtained by

ϕ(z) = argmax|s∗(ϕ)z|2 : ϕ ∈ Φ

(1.101)

a(z) = s∗(ϕ(z))z (1.102)

Essence 1.13 For estimating the unknown variables in the signal model Eq.(1.93),first the maximum magnitudes of the matched filter outputs with respect to the nor-malized signal is to be determined, then the amplitude estimate is given as the resultof the related matched filter.

( ) .~1

* js ( ) .~2

* js

z

| . | 2M a x !

| . | 2 | . | 2| . | 2( ) .~ *

kjs

Figure 1.22: Maximum likelihood estimator for a signal with unknown parameter(signal model Eq.(1.93))

If the set Φ = ϕ1, . . . , ϕk is finite, this can be accomplished by a filter bank,as illustrated in Fig. 1.22.


For the signal model of unknown Doppler frequency s(ϕ) =

exp·jϕ. . .

expn · jϕ

and Φ = 0 · 2π/n, . . . , (n − 1) · 2π/n the filter bank is equivalent to the discreteFourier transform and the ML estimator searches for the maximum of the magnitudeof the Doppler spectrum.

Maximum likelihood estimator for colored noise

Now, let the noise in the more general model

Z = m(ϑ) + N (1.103)

be ’colored’, i.e. not white. Let the noise have the distribution N ∼ NCn(0,R) witha general non-singular covariance matrix R. Now the pdf is

pZϑ(z) =

1

πn detRexp

−(z−m(ϑ))∗R−1(z−m(ϑ))

(1.104)

and we get analogue to the preceding

ϑ(z) = argmin(z−m(ϑ))∗R−1 (z−m(ϑ)) : ϑ ∈ Θ

(1.105)

= argmin∥∥R−1/2 (z−m(ϑ))

∥∥2: ϑ ∈ Θ

. (1.106)

The application of R−1/2 has ’whitened’ the random vector, as can seen by

E[(

R−1/2N) (

R−1/2N)∗]

= R−1/2RR−1/2 = In. (1.107)

1.7.3 Cramer-Rao Bounds

A widely method used for the analysis of possible performance of estimators are theCramer-Rao Bounds (CRB) which are treated in the appendix A.6.3. At this point,we will present the CRB for the model Eq.( 1.94). Fisher’s information matrix isdefined by

J(ϑ) = Eϑ

[(∂

∂ϑln pZ

ϑ(Z)

)(∂

∂ϑln pZ

ϑ(Z)

)t]

. (1.108)

In Eq.(1.96) the logarithm of the pdf was presented, from which we can compute itspartial derivatives

∂

∂ϑLZ

ϑ(z) = − 1

σ2

∂

∂ϑ‖z−m(ϑ)‖2 (1.109)

=2

σ2<m∗

ϑ(ϑ) (z−m(ϑ)) (1.110)

1.8. TARGET DETECTION 45

where mϑ =(

∂∂ϑ1

m, . . . , ∂∂ϑk

m)

denotes the partial derivatives of m. Since the

second factor in the brackets is the noise contribution, Fisher’s information matrixis easily calculated:

J(ϑ) =4

σ4Eϑ

[<m∗ϑ(ϑ)N (<m∗

ϑ(ϑ)N)t]

(1.111)

=2

σ4<Eϑ [m∗

ϑ(ϑ)NN∗mϑ(ϑ)] (1.112)

=2

σ2<m∗

ϑ(ϑ)mϑ(ϑ) . (1.113)

So under the assumptions of the signal model Eq.( 1.94) the partial derivativesof the deterministic part of the signal constitute the information matrix. Underslight conditions, the covariance matrix of any unbiased estimator is bounded bythe inverse of Fisher’s information matrix. If there is only one real parameter ϑ,this is simplified to

Varϑ

[ϑ(Z)

]≥ J−1(ϑ) =

σ2

2‖mϑ(ϑ)‖2(1.114)

for any unbiased estimator ϑ(z) of ϑ.

If, for example, m(ϕ) = a(expj(ν − ν)ϕ)ν=1,...,n models a signal with Dopplerangular increment ϕ and a is assumed to be known, we get

J−1(ϕ) =σ2

2|a|2 ∑nν=1(ν − ν)2

(1.115)

ν has to be chosen as the mean value of all subscripts ν = (n + 1)/2 to achieve thephase of a being independent on ϕ. Otherwise the assumed knowledge of a wouldalso include an information on ϕ. The resulting CRB is

J−1(ϕ) =6σ2

|a|2n(n2 − 1)=

6

(n2 − 1)SNR(1.116)

where SNR is the accumulated signal-to-noise ratio.

Recall, that this result is obtained without specifying a concrete estimator. Itreflects the information content of the data with respect to the parameter to beestimated. This analysis can often be used for the design of system parameterswithout going into details of estimation.

1.8 Target detection

1.8.1 The target detection problem

We regard the radar measurement z which can be interpreted as a realization of therandom vector Z. It is not known from the beginning, if the measurement includes a


K : T a r g e t p r e s e n tH : O n l y n o i s ez D e t e c t o r

y ( . )1

0

Figure 1.23: A target detector

target signal, or if it is produced by noise only. A device would be of great interest,which automatically decides if a target is supposed to be present or not. Such adevice (as illustrated in Fig. 1.23) is called detector in this context and can bedescribed by a function ψ(z) taking values in 0, 1 with the result 1, if the detectordecides for ’target present’ and 0 else. In the general mathematical background, thisdetector is called test.

The two possibilities for the origin of data are called hypothesis H for the ’nor-mal’ or ’uninteresting’ case that there was only noise, and alternative K for the’important’ and ’interesting’ case that a target echo is hidden in the measurement.In mathematical language, it has to be tested the alternative K (target present)against the hypothesis H (only noise).

Since Z is a random vector, the decision cannot be made with absolute certainty,wrong decisions are possible with certain probabilities. If the detector decides for’K: target present’ though there was only noise (H), this is called false alarm, if itdecides for ’H: only noise’ though there is indeed a target (K), we call this targetmiss.

The probability for the first kind of error is called false alarm probability PF ,that for the second kind of error target miss probability. More usual as the latteris the expression detection probability PD for the opposite case, that the detectordecides correctly for K, if a target is present. Clearly, PD is equal to 1 minus theprobability of target miss.

To compute these probabilities we need models for the distribution of Z.

Example 1: detector for a signal with known phase As an ex-ample we proceed from the standard model I with Gaussian noise:

Z = as + N, N ∼ NCn

(0, σ2In

), a ∈ C (1.117)

For a fixed a ∈ C the distribution of Z is PZ(a) = NCn (as, σ2In) withthe pdf

pZ(a; z) =1

(πσ2)nexp

− 1

σ2‖z− as‖2

(1.118)

1.8.2 Simple alternatives

A stringent solution for an optimum detector is possible, if there are exactly twoprobability distributions for the two alternatives (simple alternatives). For the noise-only case we have for this model a simple hypothesis P0 = PZ(0) = PN. For the


alternative the possible values for the amplitude a have - for the present - to belimited to a fixed value a = a0 6= 0 leading to the distribution P1 = PZ(a0). Thecorresponding pdf’s are p0(z) and p1(z).

So the detection problem reduces to test the alternative K : Z ∼ P1 against thehypothesis H : Z ∼ P0.

If a certain detector ψ is used, the two basic probabilities are

PF = Prdecision for K|H = P0(ψ(Z) = 1) (1.119)

PD = Prdecision for K|K = P1(ψ(Z) = 1) (1.120)

The approach of Neyman and Pearson

Radar detection follows the approach of Neyman and Pearson[12]: The two alter-natives are not symmetrical. The error of the first kind (here: a target is indicated,though only noise is present) possibly induces unwanted activities. Its probabilityPF has to be controlled, it shall not be larger than a pre-given level α. Among alltests fulfilling this demand, a test ψopt is called best test for the level α , if it has themaximum detection probability PD.

Likelihood ratio test

Neyman and Pearson have not only proofed that such a best test exists, but theyhave constructively given how it has to be performed. For continuous distributionslike in the example, it is given by

ψ(z) =

1 if Λ(z) > η0 if Λ(z) ≤ η

(1.121)

with Λ(z) =p1(z)

p0(z). (1.122)

Λ(z) is called likelihood ratio, and the threshold η has to be determined to achievePF = α. This test (detector) is called likelihood ratio test (likelihood ratio detector).

If f : R→ R is a strictly monotonic increasing function, then the comparison ofT (z) := f(Λ(z)) with the new threshold η′ = f(η) defines a best test, too. For theregarded example, we use the logarithm as such a monotonic function, resulting inthe log likelihood ratio


ln Λ(z) = ln

pZ(a0; z)

pZ(0; z)

(1.123)

= ln

1

(πσ2)n exp− 1

σ2 |z− a0s|2

1(πσ2)n exp

− 1σ2 |z|2

(1.124)

=1

σ2

(−‖a0s‖2 + 2Re a∗0s∗z)

(1.125)

Let a0 = |a0|ejφ. Applying the second strictly monotonic increasing function

g(x) = σ2

2|a0|x+ |a0|2‖s‖2 we finally get the function T (z) = Re

e−jφs∗z

and the best

test

ψ(z) =

1 if <

e−jφs∗z

> η′

0 else(1.126)

(1.127)

Obviously the best test is achieved if the filter matched to s is applied followedby a phase compensation of the phase of a0. Then the deterministic part of themeasurement is rotated to the real part. Only this is compared to a threshold, seeFig. 1.24 top. See Fig. 1.27.

| . |z < >T a r g e t p r e s e n tO n l y n o i s ehs * .

R e a l p a r tz < >

T a r g e t p r e s e n tO n l y n o i s ehs * . . e - j j I m a g i n a r y

p a r t

Figure 1.24: Top: detector for known phase. Bottom: detector for unknown phase


Example 2: detector for a signal with random phase In general,the phase of the signal will not be known in the radar application, sinceit depends on the exact range of the scatterer in the accuracy of fractionsof a wavelength. So in this example we model the phase Ψ as rectangulardistributed over (0, 2π):

Z = γejΨs + N, N ∼ NCn (0, σ2In) , γ ∈ RΨ ∼ R (0, 2π) ,N and Ψ statistical independent.

H : γ = 0K : γ > 0

(1.128)

We search for a best test for K against H for the level α.

Solution: For arbitrary γ we find the pdf pZγ (z) by integrating over the conditioned

distribution for Ψ = ψ:

pZγ (z) =

2π∫

0

pZ|Ψ=ψγ (z)pΨ(ψ)dψ (1.129)

=(πσ2

)−n 1

2π

2π∫

0

exp

−

∥∥z− γejψs∥∥2

σ2

dψ (1.130)

=(πσ2

)−ne−(‖z‖2+‖γs‖2)/σ2 1

2π

2π∫

0

exp

2<γs∗ze−jψ

σ2

dψ (1.131)

=(πσ2

)−ne−(‖z‖2+‖γs‖2)/σ2 1

2π

2π∫

0

exp

2|γs∗z|

σ2cos ψ

dψ (1.132)

=(πσ2

)−ne−(‖z‖2+‖γs‖2)/σ2

I0

(2|γs∗z|

σ2

). (1.133)

I0(x) here denotes the modified Bessel function of order 0, see appendix Eq.(A.60). The distribution described by the pdf evaluated in Eq. (1.128) is calledRician distribution, see appendix Eq. (A.94) and Rayleigh distribution, if γ = 0.

The likelihood ratio now is computed to

Λ(z) =pZ

γ (z)

pZ0 (z)

= e−‖γs‖2/σ2

I0

(2|γs∗z|

σ2

). (1.134)

Since I0(x) is a strictly monotonic increasing function, we can replace Λ(z) by

T (z) = |s∗z|. (1.135)


The parameter γ has disappeared, so this test function is optimum for arbitraryγ > 0, it leads to an uniform best test which simply compares the amplitude of thematched filter to the threshold, see Fig. 1.24 bottom.

The dependance of the probability of detection on the SNR is depicted in Fig.1.27. Compared to the detection with known phase, a loss of about 1 dB at repre-sentative parameters can be stated.

1.8.3 Composed alternatives

Unfortunately, uniform best tests don’t exist in general. Often a test problem ex-tends to two sets of distributions instead of only two single distributions. Like inthe estimation problem section 1.7 the model for a random variable X contains anunknown parameter vector ϑ ∈ Θ for the distribution PX

ϑ and the pdf pXϑ (x). Let

Θ0 and Θ1 be two disjunct subsets of Θ. Now, the test problem is to decide for arealization x of X, to which of the two classes the parameter vector belongs:

H : ϑ ∈ Θ0 (1.136)

K : ϑ ∈ Θ1 (1.137)

Maximum likelihood ratio tests

A practical way is to insert the maximum likelihood estimators for the two alter-natives ϑi(x) = argmax

pX

ϑ (x) : ϑ ∈ Θi

for i = 1, 2 and form with these the

maximum likelihood ratio:

Λ(x) : =pX

ϑ1(x)(x)

pXϑ0(x)

(x)(1.138)

=max

pX

ϑ (x) : ϑ ∈ Θ1

max pXϑ (x) : ϑ ∈ Θ0 . (1.139)

Example 3: Detector for a signal in colored noise Throughoutthis scriptumthis test problem is of great importance.

Z = as + N, N ∼ NCn (0,R) , a ∈ CH : a = 0K : a 6= 0.

(1.140)

We search for a maximum likelihood ratio test.

Solution: For arbitrary a the pdf. is given by


pZa (z) =

1

πn detRexp

− (z− as)∗R−1 (z− as)

(1.141)

The maximum likelihood ratio is given by

Λ(z) = maxa

pZa (z)

pZ0 (z)

(1.142)

= expz∗R−1z−min

a(z− as)∗R−1 (z− as)

(1.143)

The amplitude a leading to the extremum is a maximum likelihood estimator ofa for the conditions of the alternative, and is computed to

a =s∗R−1z

s∗R−1s. (1.144)

The logarithm of the maximum likelihood ratio now is

ln Λ(z) = z∗R−1z− (z− as)∗R−1 (z− as) (1.145)

= z∗R−1z− z∗(In − R−1ss∗

s∗R−1s

)R−1

(In − ss∗R−1

s∗R−1s

)z (1.146)

= z∗R−1z− z∗(R−1 − R−1ss∗R−1

s∗R−1s

)z (1.147)

=|z∗R−1s|2s∗R−1s

(1.148)

Equivalent to this is the test function

T (z) =∣∣z∗R−1s

∣∣2 . (1.149)

Example 4: Detector for a signal with unknown parameter Anexample for the maximum likelihood ratio test is the detection of a signalwith unknown parameter according to standard model II:

Z = as(ϕ) + N, N ∼ NCn (0, σ2In) , a ∈ Cϕ ∈ Φ unknown parameter

‖s (ϕ)‖2 > 0 for all ϕ ∈ ΦH : a = 0K : a 6= 0.

(1.150)

We search for a maximum likelihood ratio test.


Solution: The test problem is parametrized by ϑ = (a, ϕ), Θ0 = (0, ϕ) : ϕ ∈ Φ,Θ1 = (a, ϕ) : a ∈ C\ 0 , ϕ ∈ Φ.

For a = 0 the pdf does not depend on ϕ:

pZ0 (z) = (πσ2)−n exp

−‖z‖2/σ2

, (1.151)

while for the alternative the maximum likelihood estimator has to be inserted:

pZa,ϕ(z) = (πσ2)−n exp

−‖z− as(ϕ)‖2

σ2

(1.152)

= (πσ2)−n exp

−‖z‖

2 − |s∗(ϕ)z|2σ2

, (1.153)

where s is the signal s normalized to norm equal to one, and ϕ(z) = argmax|s∗(ϕ)z|2 : ϕ ∈ Φ

according to Eq. (1.99).

Consequently, the maximum likelihood ratio is given by

Λ(x) =exp

−‖z‖2−|s∗(ϕ)z|2

σ2

exp−‖z‖2

σ2

(1.154)

= exp

max

|s∗(ϕ)z|2 : ϕ ∈ Φ

σ2

(1.155)

Equivalent to this, since monotonic increasing depending, is the test function

T (z) = max|s∗(ϕ)z|2 : ϕ ∈ Φ

. (1.156)

For a finite set Φ = ϕ1, . . . , ϕk the detector is builded as illustrated in Fig.1.25.

| . |s * ( j 1 ) z-Maximum

| . |s * ( j k ) z-

| . |-

z > T a r g e t p r e s e n tO n l y n o i s eh<

s * ( j K ) z

Figure 1.25: Maximum likelihood ratio test

As in the section on maximum likelihood estimators, for the signal model of un-

known Doppler frequency s(ϕ) =

exp0 · jϕ. . .

exp(n− 1) · jϕ

and Φ = 0·2π/n, . . . , (n−

1) · 2π/n the filter bank is equivalent to the discrete Fourier transform. The maxi-mum amplitude of the Fourier spectrum now has to be compared to a threshold.

For the case of orthogonal signals (as in the discrete Fourier transformation), theprobability of detection is plotted against the SNR in Fig. 1.27.


1.8.4 Probability of detection for basic detectors

Probability of detection versus Signal-to-noise-ratio

Following the approach of Neyman and Pearson, a fixed upper bound α for the falsealarm probability PF is specified.

P r o b a b i l i t y d e n s i t i e s o f T ( X ) f o r H a n d K

bh at

K : P X = P 1H : P X = P 0

Figure 1.26: False alarm probability and detection probability as areas below theprobability density functions of the test statistics

If an arbitrary test function T (X) is used for the general test problem Eq. (1.136)the threshold η has to be determined by

η := min

t : P(X)ϑ (T (X) > t) ≤ α for all ϑ ∈ Θ0

. (1.157)

If Fϑ(t) := P(X)ϑ (T (X) ≤ t) denotes the distribution function of T (X) for an

arbitrary ϑ, η can be found by

η := F−10 (1− α) with F0(t) := min Fϑ(t) : ϑ ∈ Θ0 (1.158)

if F0 is bijective. Then the detection probability can be derived by

PD(ϑ) = Fϑ(η). (1.159)

The dependance of the detection probability on the SNR is depicted in Fig. 1.27for different conditions.


5 . 0 1 0 . 0 1 5 . 0 2 0 . 0S i g n a l - t o - n o i s e - r a t i o ( d B )

0.00

0.20

0.40

0.60

0.80

1.00

Probabil

ity of

detec

tion

P h a s e , D o p p l e r , S i g m a k n o w nD o p p l e r , S i g m a k n o w nP h a s e , S i g m a k n o w nS i g m a k n o w nD o p p l e r k n o w nI n c o h e r e n t d e t e c t i o n

n = 3 2P F = 1 0 - 5

Figure 1.27: Detection probability in dependence on the SNR. The false alarm prob-ability is fixed at 10−5 and the number of samples is 32.

Receiver operating characteristics

For a fixed SNR the dependence PD(PF ) is called receiver operating characteristics,plotted in Fig. 1.28 for the test problem example 2. If the SNR is zero, PD = PF andwe get the diagonal. For growing SNR the curves are vaulted upwards but retainingthe points (0,0) and (1,1). A best test is characterized by the property that its ROCcurve is the upper bound for the ROC of any test for this problem at each value ofPF .

0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0F a l s e a l a r m p r o b a b i l i t y

0 . 00

0 . 20

0 . 40

0 . 60

0 . 80

1 . 00

P r ob ab i l

i t y o f

d et e c

t i on

s / n = 0 s / n = 1 0

Figure 1.28: Receiver operating characteristics (ROC)


Constant false alarm rate detectors

All detectors regarded up to now assume that the noise variance is known. If athreshold was computed for a certain σ2 and the real noise variance becomes larger- for instance because of variations of the amplification factor - the false alarmprobability can increase dramatically. To anticipate this, a test can be constructedwith a PF independent on σ2. For instance, example 2 can be expanded to thissituation:

Example 5: detector for unknown noise variance

Z = γejΨs + N, N ∼ NCn (0, σ2In) , γ ∈ RΨ ∼ R (0, 2π) ,N and Ψ statistical independent,σ2 unknown.

H : γ = 0K : γ > 0

(1.160)

The statistical model is invariant against a multiplication with a scalar. In themathematical test theory a method was developed to construct best invariant tests.We relinquish going into detail, but give directly the solution:

. .

.2

2

zs * z 2<

> h .

M a t c h e d f i l t e r V a r i a b l e t h r e s h h o l d

P r o j e c t i o n t o t h e s i g n a l f r e e s u b s p a c e

E s t i m a t i o n o f t h en o i s e p o w e r

T a r g e t d e c i s i o n

P S z

Figure 1.29: CFAR

Without loss of generality we can assume that the signal vector s is normalizedto ‖s‖2 = 1. Then,

T (z) =|s∗z|2

‖z‖2 − |s∗z|2 =‖P<s>z‖2

‖P<s>⊥z‖2 (1.161)

is a best invariant test (without proof). P<s> = ss∗ and P<s>⊥ = Inss∗ de-

note the projections into the signal subspace and the subspace orthogonal to it,respectively. P<s>⊥Z blends out the deterministic part of the signal, and onlythe noise part is left: P<s>⊥Z = P<s>⊥N. The expectation of its square normis E[P<s>⊥N] = (n − 1)σ2, so ‖P<s>⊥z‖2 /(n− 1) can be regarded as a bias-freeestimate of the noise variance. Instead of dividing by this, the threshold can beobtained by multiplication of a fixed value η with this estimator, to ’adjust’ thethreshold to the varying noise power. This is illustrated in Fig. 1.29.


Besides example 4 there are many other CFAR detectors, which are invariantagainst special transformations, e.g. nonparameteric tests like sign tests or ranktests.

Incoherent detection

If there is no phase information available - as it is the case for very old or very cheapsystems - only the amplitude values Aν , ν = 1..n can be observed.

Example 6: Incoherent detector

Z = γs + N, N ∼ NCn (0, σ2In) , γ ∈ RAν = |Zν | , ν = 1..n,

H : γ = 0K : γ = γ0 > 0

(1.162)

For a realization a of the random vector A containing this amplitudes, the like-lihood ratio is given by the product of the likelihood ratios for the individual am-plitudes, which are Rician distributed for the alternative and Rayleigh distributedfor the hypothesis. With the densities given in the appendix App. A.92 and A.94it is evaluated as

Λ(a) =n∏

ν=1

2aνe−(a2

ν−γ20 |sν |2)/σ2

I0 (2γ0|sν |aν/σ)

2aνe−a2ν/σ2 (1.163)

= eγ20 |sν |2/σ2

n∏ν=1

I0 (2γ0|sν |aν/σ) . (1.164)

Equivalent is the test statistic

T (a) =n∑

ν=1

ln (I0 (2γ0|sν |aν/σ)) . (1.165)

In this case, the test statistics really depends on the construction parameter γ0.Only for γ = γ0 it will result in the maximum detection probability. For smallargument the logarithm of the modified Bessel function can be approximated by itsabsolute and quadratic Taylor expansion resulting in the simple equivalent form

T ′(a) =n∑

ν=1

|sν |2a2ν . (1.166)

Compared to the coherent detectors mentioned before, this has a much worseperformance which can be recognized in Fig. 1.27. While for n → ∞ the SNR of

1.9. MTI FOR GROUND BASED RADAR 57

a single component tends to zero like 1/n to achieve a fixed detection probabilitywith a coherent detector, it behaves like 1/

√n for the incoherent detector. So the

performance difference grows with the number of samples.

1.9 MTI for ground based radar

One of the most important goals of a radar system is to detect a moving targetagainst clutter. The expression ’clutter’ comprises all echoes of unwanted reflectorson ground like woods and fields and in the air (clouds, rain, ...).

As we know from the previous, radar is an excellent instrument to sense alsosmall changes of the distance from pulse to pulse independent on range. The radialmotion of a target induces the Doppler modulation, as described in section 1.3,resulting in a phase change from pulse to pulse.

To detect the moving target against the clutter background, it is necessary tosuppress or cancel the clutter returns with as small suppression of the target signalas possible.

1.9.1 Temporal stable clutter

If the clutter echoes don’t change from pulse to pulse, as it is the case e.g. forbuildings and rocks, and for calm wind, the Doppler frequency of the clutter isequal to zero and it is easy to cancel the clutter. Heuristical methods use recursivefilters with a zero at Doppler=0, e.g. the two-pulse canceller Yν = Zν −Zν−1. For apulse train of N pulses, we introduce the following standard model III:

Z = as + c1 + N, N ∼ NCN

(0, σ2IN

). (1.167)

Here, 1 is the vector with all components equal to 1, and c is an unknownconstant reflecting the clutter amplitude. To find an optimum method we couldderive a best test based on this model, or we may search for a maximum likelihoodestimate of a and c, given a realization z of Z. The latter is given by the best fitminimizing the mean square error

(ac

)(z) =

(s∗s s∗11∗s 1∗1

)−1 (s∗

1∗

)z (1.168)

It follows

a(z) =s∗P<1>⊥z

s∗P<1>⊥s, c(z) =

1∗P<s>⊥z

1∗P<s>⊥1. (1.169)

P<1>⊥ = IN − 11∗1∗1 is the projector to the subspace orthogonal to 1 , similar for

P<s>⊥ . Note that the application of P<1>⊥ means to subtract the mean value ofall components from each component. Clearly this operation cancels the clutterand produces a zero for a signal at Doppler=0. After clutter cancellation the target


match is performed. The result for a can directly be used for detection by comparingits magnitude to a threshold.

Now, we further investigate the relevant expression T (z) = s∗P<1>⊥z. Letd(F ) = (expj2πF∆T, . . . , expjN2πF∆T) denote the signal vector effected byDoppler frequency F (Note that d(0) = 1). Further let s = d(F0) be the signal fora chosen Doppler frequency F0.

Then D(F ; F0) = |d∗(F0)P<1>⊥d(F )|2 describes the filter gain if the filter ismatched to frequency F0 and a signal at frequency F comes in. The resulting gaincurves for a bank of filters for varying F0 is shown in Fig. 1.30. The gain D(F0; F0)at matching frequency shows a notch at F0 = 0.

- 0 . 5 0 - 0 . 3 0 - 0 . 1 0 0 . 1 0 0 . 3 0 0 . 5 0- 0 . 5 0 - 0 . 3 0 - 0 . 1 0 0 . 1 0 0 . 3 0 0 . 5 0

1 0. 0

2 0. 0

3 0. 0

4 0. 0

F i lt e r

g ai n

[ d B]

F D T

Figure 1.30: Red: Doppler filter gains for varying center frequencies; green: gain atmatching frequency

1.9.2 Temporal variation of clutter

If the wind is moving the trees, or if clouds are passing the sky, the clutter is nolonger temporal stable. In this case, the clutter echoes have to be described by astochastic process C(T ). The samples of this process are collected in the cluttervector C = (C(1 ·∆T ), . . . , C(N ·∆T )). The model for the random vector of themeasured data is now

Z = as + C + N, N ∼ NCN

(0, σ2IN

)(1.170)

C,N stoch. independent , E[C] = 0, E[CC∗] = RC . (1.171)

The distribution of C is left open intentional. It can be a normal distribution or amore sophisticated distribution with a better fit to the reality. The linear processing

1.9. MTI FOR GROUND BASED RADAR 59

of the data Y = w∗Z with the optimum resulting signal-to-clutter-plus-noise ratioSCNR is obtained with the weight vector

wopt = R−1C s. (1.172)

This will be shown in the context of array processing, see Eq. (2.27).

Often C(T ) can be modelled as a stationary stochastic process, i.e. the statisticalproperties don’t change with time. A weaker condition is the property to be a widesense stationary stochastic process, that means that the correlation properties don’tchange with time.

Since the phase of C(T ) will be uniformly distributed over (0, 2π), its expectationis zero. Then, its correlation properties are described by the correlation function(which is in this case equal to the covariance function)

rCC(τ) = E[C(T + τ)C∗(τ)]. (1.173)

If the common distribution of each sample vector of C(T ) is Gaussian, the covariancefunction completely determines the statistical properties of C(T ).

The Fourier transform of rCC(T ) is called power spectrum or spectral power den-sity:

RCC(F ) =

∫exp−j2πFTrCC(T )dT. (1.174)

The power spectrum is always larger or equal to zero and represents the spectraldistribution of the clutter power.

If we regard an infinite sequence of samples Cν = C(ν∆T ), they form a timediscrete stationary process and have the correlations E[Cν+µC

∗ν = RC(µ∆T ). The

power spectrum for the discrete process is defined by the discrete Fourier transform

P(d)C (F ) =

∑µ

exp−j2πµF∆TRC(µ∆T ). (1.175)

Let ∆F = 1/∆T be the pulse repetition frequency. If PC(F ) = 0 for F /∈(−∆F

2, ∆F

2) the sampling fulfills the Nyquist condition and we get

P(d)C (F ) = PC(F )∆F for F ∈

[−∆F

2,∆F

2

]. (1.176)

Now we regard the normalized Fourier transform of the clutter echoes:

C(F )(F ) =1√N

N∑n=1

Cν exp−j2πnF∆T (1.177)

The magnitude squared of C(F )(F ) is called periodogram and is often used forthe estimation of the power spectrum.


Its expectation is

E[∣∣C(F )(F )

∣∣2]

=1

N

N∑ν=1

N∑µ=1

E[CνC

∗µ

]exp−j2π(ν − µ)F∆T (1.178)

=1

N

N−1∑

κ=−(N−1)

minN,N+κ∑

ν=max1,1+κRC(κ∆T ) exp−j2πκF∆T(1.179)

=N−1∑

κ=−(N−1)

(1− |κ|

N

)RC(κ∆T ) exp−j2πκF∆T. (1.180)

This is a windowed version of the discrete power spectrum. If the correlationvanishes for |τ | > τ0 or, equivalently for |κ| > κ0, then for n > κ0

E[∣∣C(F )(F )

∣∣2]

=

κ0∑κ=−κ0

(1− |κ|

N

)RC(κ∆T ) exp−j2πκF∆T (1.181)

→κ0∑

κ=−κ0

RC(κ∆T ) exp−j2πκF∆T for N →∞ (1.182)

= P(d)C (F ) (1.183)

The covariance matrix of C takes the form

RC = E [CC∗] (1.184)

=

R0 R1 . . . RN−2 RN−1

R−1 R0 . . . RN−3 RN−2

. . . . . . . . . . . . . . .R−(N−2) R−(N−1) . . . R0 R1

R−(N−1) R−(N−2) . . . R−1 R0

(1.185)

with Rν = RC(−ν∆T ). (1.186)

A matrix of that structure is called Toeplitz matrix; the rows are shifted by oneposition from row to row.

1.10 Radar equation, SNR, CNR, SCNR

Now we tie the results from section 1.8 together with physical facts to calculate themaximum range of a radar.

1.10.1 Power budget for the deterministic signal

Power density generated by the transmit antenna Let the power of thetransmit signal at the input of the transmit antenna be Ptx [W ]. If the complete

1.10. RADAR EQUATION, SNR, CNR, SCNR 61

power is transferred to the emitted wave, the integral of the power density overthe surface of each sphere enclosing the transmit antenna will be Ptx, too, if noabsorption in the atmosphere is present. For an omnidirectional lossless antennathis leads in the range r to the power density Ltx = Ptx/(4πr2) [W/m2].

If a directed antenna with gain Gtx (antenna losses taken into account) in thedirection to the scatterer is used, the power density has to be multiplied by thefactor Gtx, so we get Ltx = GtxPtx/(4πr2).

Power of the reflected wave The radar cross section (RCS) of the reflectedobject is defined by q = Prefl/Ltx where Prefl is the virtual power of an omni-directional radiator generating the same power density in direction of the receiveantenna as the wave reflected by the object. So q is the virtual area capturing thepower Prefl = qLtx which replaces the object in order to generate the reflected powerdensity5.

We get Prefl = qGtxPtx/(4πr2) and the power density at the receive antennaLrx = Prefl/(4πr2), combined in

Lrx = qGtxPtx/(4πr2)2 (1.187)

Power of the receive signal Following antenna theory ([9]), a perfectly matchedomnidirectional receive antenna has an effective cross section qrx = λ2/(4π) andwould yield the power Lrxλ

2/(4π) at its output, if the power density of the incomingwave is Lrx. If we instead of this we use a directed antenna with gain Grx in thedirection to the scatterer the effective cross section is enlarged to qrx = Grxλ

2/(4π)and the power at the output is Prx = LrxGrxλ

2/(4π). If we insert this into Eq.(1.177) we get

Prx = S21Ptx with S21 =GtxGrxλ

2q

(4π)3r4(1.188)

S21 is the transmission coefficient between transmit and receive antenna via theway to the reflector and back which is normally extremely small. If the RF transmitsignal sRF (t) is normalized in that way that |sRF (t)|2 is the effective power feededinto the transmit antenna at time t and sRF (t) is the received signal from thebackscattering object at the antenna output, we get the same proportionality forthe energies

5A metallic sphere with radius d À λ has an RCS of approximately its geometrical area q = πd2

in the projection to a plane. Radar cross sections of targets range from about 10−4m2 (fly) to105m2 (large ship) for usual radar frequencies, see [6]. If the extension of the target is smallerthan a wavelength, its backscattering is sited in the Rayleigh region, the RCS is approximatelyproportional to the forth power of the frequency. For instance, the leaves in a forest which mayhave an extension in the order of a decimeter, reflect strongly at frequencies of 10 and more GHzand are more or less transparent for frequencies lower than 1 GHz (foliage penetration).


∫|sRF (t)|2dt = S21

∫|sRF (t)|2dt (1.189)

and the spectral power densities

|SRF (f)|2 = S21|SRF (f)|2. (1.190)

1.10.2 Spectral power density of the noise

The spectral power density of the natural noise at the absolute temperature T ofthe receiver (in Kelvin degrees [K]) is given by Cnoise = kT, where k = 1.38065 ·10−23 [Ws/K] is the Boltzmann constant, see [6]. For room temperature this valueis about kT ≈ 4 · 10−21 [Ws]. Each receiving system has additional noise sourcesincreasing the spectral power density by a factor fn (noise figure) resulting in

Cnoise = kTnf [Ws]. (1.191)

Depending on the frequency and the bandwidth, a noise figure of below 1 dB(X-band) or 3 dB (W-band) can be realized.

If the white noise is given into a filter with transfer function H(f), the resultingspectral power density of the output noise is

Coutnoise(f) = |H(f)|2Cnoise. (1.192)

In this context the filter is called form filter. The total power of the noise output is

σ2n = Cnoise

∫|H(f)|2df (1.193)

and for a filter matched to the wave form we get

σ2n = Cnoise

∫|SRF (f)|2df (1.194)

For a rectangular signal spectrum of bandwidth b we get a noise power of

σ2n = kTbnf [W ]. (1.195)

1.10.3 Signal-to-noise ratio and maximum range

Let γmin be the minimum SNR to obtain a detection probability PF for a targetwith fixed RCS q at a false alarm probability PF . Representative is the coherentdetection of a target with unknown phase (see problem Eq. (1.128)). Here we needfor PF = 10−5 and PE = 0.9 an SNR of about γmin ≈ 20 (13 dB).

From Eq.(1.188) together with Eq.(1.195) we get an SNR of

1.10. RADAR EQUATION, SNR, CNR, SCNR 63

γ =PtxGtxGrxλ

2q

(4π)3r4kTbnf

η. (1.196)

Here the new inserted constant η subsumes all the remaining losses on the wayto the detector. The total loss can be distributed to the units in different ways.We prefer the way that the antenna losses including electromagnetic mismatch isincluded in the antenna gain, and that the additional noise in the RF part, decreasingthe SNR, is contained in the noise figure nf of the receiver. The remaining lossescan be found in the mismatch of the receive filter, sampling losses, suboptimumdetectors, and so on.

To obtain the minimum SNR, the inequality γ ≥ γmin has to be fulfilled and canbe resolved to get the maximum range according to Eq. (1.196):

r ≤ rmax = 4

√PtxGtxGrxλ2qη

(4π)3kTbnfγmin

(1.197)

Eq. (1.197) is known in this or a similar form as the radar equation.

For coherent integration of N pulses the SNR is increased by a factor N . Thiscan be taken into account, if for b the bandwidth according to the whole coherentintegration time is inserted:

b =1

Tint

=1

Nts=

1

dTcompl

(1.198)

where d = ts/∆T is the duty factor and Tcompl the whole observation time.

Essence 1.14 The maximum range to obtain a minimum SNR is given by Eq.(1.197) which is called the ’radar equation’. It is proportional to the forth rootof the transmit power, the gain of the antennas, and the RCS of the target.

In this form, the maximum range is proportional to the square root of the wave-length for fixed antenna gain. To keep the gain constant, the antenna dimensionshave to be scaled proportional to the wave length. On the other hand, if the an-tenna’s size is fixed, the gain for transmit and receive will be proportional to 1/λ2,resulting in the maximum range being proportional to 1/

√λ. Further, if the time

to scan a given spatial sector is fixed, the necessary number of beams grows pro-portionally to 1/λ2, so the illumination time for each beam will decrees at the samerate. Under these conditions, λ cancels out and the maximum range is independenton the frequency.

Instead of the forth root for SAR in the stripmap mode we will get the thirdroot, since the illumination time here is proportional to the range, see Eq.(4.113).

Chapter 2

Signal processing for real arrays

In this chapter we summarize the facts and tools for array processing as pre-requisitesfor the air- and spaceborne clutter suppression. We start with beamforming forarray antennas, deal the spatial interference suppression, regard the estimation ofthe target direction and present the Cramer-Rao Bounds (CRB) for this estimationtask.

2.1 Beamforming for array antennas

2.1.1 The SNR-optimum beamformer

Linear arrays

We regard a linear array antenna with M omnidirectional single elements at thepositions xm (see Fig. 2.1).

P h a s e f r o n t

x m c o s a

x

a

x m

Figure 2.1: Travelling way differences for a linear array

65

66 CHAPTER 2. SIGNAL PROCESSING FOR REAL ARRAYS

Glossary for this chapterα Direction angle with respect to the array axisα Magnitude of the signal’s amplitudeδu Resolution of directional cosineδu Deviation of focusing directionδ(x) Difference beam∆u Ambiguity of directional cosine∆x Element spacingη Interference-to-noise-ratioγ Signal-to-noise ratioλ Wavelengthλ Eigen value∆ψ Phase increment from element to elementρ(s1, s2) Correlation coefficient of the two signals s1 and s2

σ2 Variance of noiseσ(x) Sum beama Complex amplitudea(x) Complex amplitude along a linear aperturea(~x) Element distribution function

A(~k) Fourier transform of the element distribution functionA(u) Complex amplitude of a wave from direction ub Bandwidthb Beamformer vectorB Random vector of interference amplitudesd(~u) Direction of arrival (DOA) vectorD(~u, ~u0) Characteristics for the array focused to ~u0

Di(u,M) Dirichlet functione(u) Normalized DOA vectorF (~u, ~u0) Array factor for focusing to ~u0

G~u0(.) Physical characteristics for focusing direction ~u0

H Interference suppression operatorIn Identity matrix of dimension nJ External interference at array outputJ Amplitude of external interference sourceJ(ϑ) Fisher’s information matrixkr = 2π/λ Wavenumber~k = kr~u Wavenumber vectorlx Length of a linear arraym(ϑ) Signal including amplitude dependent on parameter vector ϑM Number of single elementsPU Projector to the subspace UQ External interference plus noiseR Covariance matrix of interference plus noiseRN Covariance matrix of noiseRJ Covariance matrix of interference

R Sample matrix or empirical covariance matrixS Matrix of signal column vectors

2.1. BEAMFORMING FOR ARRAY ANTENNAS 67

u Directional cosineumax Maximum scan~u Unit vector in line-of-sight~u0 Unit vector in focusing directionu = cos α Directional cosine with respect to the array axisu0 Directional cosine for focusing directionxm x-Position of mth array element~xm 3D-Position of mth array elementZ Random vector of array outputs

A plane wave coming in from a direction which encloses an angle α with thearray-axis, has to travel a way to the mth single element which is shorter by theamount uxm than the way to the origin, where u = cos α denotes the directionalcosine .

The vector collecting the voltage outputs at the elements obtains the form

d(u) = c

ejkrux1

...ejkruxM

. (2.1)

This signal vector depending on the direction of the incoming wave, is calleddirection of arrival vector or shortly DOA-vector. c is a constant which can bechosen by convenience. 1

Note that the DOA vector only depends on the angle α or - equivalent - thedirectional cosine u. In the three-dimensional space, all directions with the sameangle with respect to the x-axis define a cone surface. We will call this cone lateiso-DOA cone.

For each component we may assume a superposition of statistically independentnoise with variance σ2 leading to the following model of the spatial sample vector:

Z = ad(u) + N. (2.2)

A linear combination Y = b∗Z with an arbitrary complex vector b will lead toa certain antenna pattern, for this reason the vector b is called beamformer. Fig.1.4 may serve as an illustration. Now we derive the SNR-optimum beamformer toa selected direction u0.

Analogue to Eq.(1.12) for an arbitrary b the output of the array will have theSNR

γ(b) =|a|2 ‖d (u0)‖2

σ2|ρ (b,d (u0))|2 . (2.3)

Like in the section 1.2.3 the optimum solution turns out to be the beamformerbopt(u0) = wd(u0) with any non-zero complex constant w. For convenience, we takew = 1.

1If the antenna characteristics Dm(u) of the single elements are not equal, they have to beincorporated into the DOA-vector: d(u) = c (D1(u) expjkrux1, . . . , DM (u) expjkruxM)t.


Essence 2.1 For white noise, the maximum SNR for a source in direction u0 isobtained by the beamformer bopt(u0) = d(u0) equal to the DOA-vector to u0. Themaximum obtains the value |a|2 ‖d (u0)‖2/σ2, i.e. the ratio of the received poweraccumulated over all elements and the noise variance at a single element.

So, by adding elements, the SNR can be increased proportional to the numberof elements.

In the transmit case the same beamformer (with c = 1) produces the maxi-mum power density in direction u0, if all single elements have the same antennacharacteristics and radiate with the same power.

Two- and three-dimensional arrays

It is easy to transfer this to the two- and three-dimensional case.

x

y

z

u v

w

A n t e n n a e l e m e n t

P h a s e p l a ne

U n i t s p h e r e

< u , x m >

x m u

Figure 2.2: Three dimensional array

Let the element positions be ~xm,m = 1, . . . , M in a Cartesian three dimensionalcoordinate system. A plane wave comes in from a direction defined by the unitvector ~u = (u, v, w)t. The variables u, v, w are directional cosines with respect tothe x, y, z axes, see Fig. 2.2. Further, the travelling distance to the element position~xm, normalized to that to the origin, is given by 〈~u, ~xm〉. Consequently, the followingDOA vector is generated:

d(~u) = c

ejkr〈~u,~x1〉...

ejkr〈~u,~xM 〉

= c

ej〈~k,~x1〉...

ej〈~k,~xM〉

. (2.4)


Here ~k = kr~u is a wave number vector (but with the opposite sign as the physicalwave number vector of the incoming wave). Again, SNR-optimum beamforming ina selected direction ~u0 is achieved with the beamformer

bopt(~u0) = d(~u0). (2.5)

2.1.2 Characteristics of the array

In the following we will use as normalization constant c = 1/√

M . In this case,DOA vector and optimum beamformer have unit norm.

Let the array be focused to direction ~u0. An incoming wave with normalizedpower from an arbitrary direction ~u will produce at the output of the array a signalD(~u, ~u0). For fixed focusing direction the array has the (physical) characteristics

G~u0(.) =D(., ~u0)

C~u0

(2.6)

where C~u0 is the normalization constant C~u0 =√

14πη

∫Ω|D(~u′, ~u0)|2dO(~u′), the

integration area is the surface Ω of the unit sphere, dO(~u′) denotes the infinitesimalsurface element, and η the efficiency of the antenna array (which might also bedependent on the focusing direction).

On the other hand, the direction to the source can be fixed, and only the beam-former focusing direction is varied. This leads to another kind of characteristics, letus call this scan characteristics D(~u0, .).

While the physical characteristics Eq.( 2.6) obeys the physical laws like energypreservation, the scan characteristics is a mathematical tool happening exclusivelyin the computer, if the array is fully digital, and in the phase shifters for a phasedarray.

If the array is composed of identical identically orientated single elements withthe common characteristics D0(~u), we get from the direction ~u the signal vectorD0(~u)d(~u). If the array is focused to the direction ~u0 with the beamformer b(~u0) =d(~u0), we get the array output

D(~u, ~u0) = D0(~u)d∗(~u0)d(~u) (2.7)

= D0(~u)F (~u, ~u0). (2.8)

with

F (~u, ~u0) = d∗(~u0)d(~u) (2.9)

=1

M

M∑m=1

ejkr〈~u−~u0,~xm〉. (2.10)


F (~u, ~u0) is called array factor and depends only on the geometry of the phasecenter positions; the influence on the single elements has been factored out. Forthe performed choice c = 1/

√M the magnitude of the array factor is for running ~u

maximum at ~u = ~u0 and assumes there the value 1.

Essence 2.2 The characteristics of an array antenna with identical identically ori-entated single elements is the product of the array factor with the single elementcharacteristics.

The array characteristics as spatial Fourier transform

With ~k0 = kr~u0, ~k = kr~u it follows

F (~u, ~u0) =

∫e−jkr〈~u0−~u,~x〉 1

M

M∑m=1

δ (~x− ~xm) d~x (2.11)

= A(~k0 − ~k), (2.12)

where A(~k) is the Fourier transform of the element distribution functiona(~x) = 1

M

∑Mm=1 δ (~x− ~xm).

Essence 2.3 The array factor is the Fourier transform of the spatial element dis-tribution.

Now we regard a linear array with equidistant spacing ∆x, the positions on thex-axis are xm = m∆x. For the case of a linear array, only the first component u ofa general direction vector ~u is necessary, so the DOA-vectors and array-factors willbe written as functions of the directional cosine u. The array factor is given by

F (u, u0) =1

M

M∑m=1

ejm∆ψ (2.13)

= ejm∆ψ sin(M ∆ψ

2

)

M sin(

∆ψ2

) (2.14)

= ejm∆ψDi

(∆ψ

π,M

)(2.15)

with ∆ψ = kr∆x(u−u0) and m = 1M

∑Mm=1 m and the Dirichlet function Di, defined

by Di(u,M) :=sin(M π

2u)

M sin(π2u)

.

Resolution and beamwidth

The mainbeam with its maximum at u = u0 decreases to zero at ∆ψ = 2π/M . Thisis equivalent to a directional cosine difference of

δu =∆ψ

(kr∆x)=

λ

M∆x=

λ

lx(2.16)


where lx is the length of the array. It is a Rayleigh measure for the resolution in thedirectional cosine, or the beamwidth, respectively.

Essence 2.4 The beamwidth of an SNR-optimum focused linear antenna array inthe directional cosine domain is given by λ/lx.

2.1.3 Ambiguities of the array characteristics

Secondary mainlobes for a linar array

We proceed from a linear array with equidistant elements with inter-element distance∆x. The scan characteristics 1

MFs(u, z) =

∑e−jkrum∆xzm obviously is periodical in

the directional cosine with the period ∆u determined by kr∆u∆x = 2π leading to

∆u = λ/∆x. (2.17)

The same behavior can be observed for an arbitrary beamformer b at the char-acteristics Fp(b, u) = b∗d(u). The periodicity is illustrated in Fig. 2.3. A mainbeam focused to direction u1 is repeated at the direction u2 = u1 + ∆u. If u2 fallsinto the visible area u ∈ [−1, 1] a secondary main beam arises at u = u2.

To avoid secondary main beams for any main beam direction u1 ∈ [−1, 1], thecondition

λ/∆x ≥ max |u2 − u1| : u1, u2 ∈ [−1, 1] = 2 (2.18)

has to be fulfilled, i. e. the inter-element distance ∆x has to be smaller or equal toλ/2.

If the array shall be steered only in the restricted interval [−umax, umax], thespatial sampling condition is given by

∆x ≤ λ/(1 + umax). (2.19)

Nyquist criterium for unambiguous sampling

The condition ∆x ≤ λ/2 for unambiguous array characteristics corresponds to an-other important result: If waves with the complex amplitudes A(u) impinge fromall directions u ∈ [−1, 1] on an infinite aperture along the x-axis, they superimposeto the complex amplitude function

a(x) =

∫ 1

−1

A(u)ejkruxdu =1

kr

∫ kr

−kr

A(kx/kr)ejkxxdkx. (2.20)

Obviously, a(x) is expressed as the Fourier transform of a signal with the variable’directional cosine’ over a finite interval, i. e. it is a bandlimited signal where


D u = l / D x

M a i n b e a m

u = - 1 u = 1

v i s u a l a r e a

S c a n i n t e r v a l

S e c o n d a r ym a i n b e a m

u = u m a xu = - u m a x

u

S e c o n d a r ym a i n b e a m

Figure 2.3: Ambiguity of the array factor, secondary main lobes

the spatial bandwidth is given by b = 2kr/(2π) = 2/λ. Following the Nyquist-Shannon sampling theorem, the signal has to be sampled at a sampling interval∆x ≤ 1/b = λ/2 to reconstruct the signal A(u) perfectly.

Essence 2.5 To avoid secondary main beams, the spatial sampling interval of alinear array has to be smaller or equal to the half wavelength, if the whole visibilityarea shall be scanned, and equal or smaller to λ/(1 + umax), if only the interval[−umax, umax] has to be covered. For an infinite linear aperture, the spatial samplinginterval has to be smaller or equal to λ/2 to reconstruct the complex amplitudes ofthe impinging waves unambiguously.

Secondary mainlobes for a planar array

The mostly used phased array antenna is a planar phased array. Here the twodirectional cosines u and v corresponding to the x and y dimensions of the arraydetermine uniquely the direction ~u, if the half space backwards of the antenna can beneglected. A rectangular raster ∆x×∆y in the aperture plane leads to a rectangularambiguity grid ∆u = λ/∆x, ∆v = λ/∆y in the directional cosine plane, see Fig. 2.4.Unambiguity in the visible area can be forced by a quadratic grid with spacing λ/2.For other grid shapes (e.g. triangular) already a slightly minor element density leadsto the disappearance of secondary sidelobes in the visible area, see Fig. 2.4.

2.2 Interference suppression

If apart from the noise there are external interferences like clutter returns (or activeexternal emitters of electromagnetic waves), the model of the received signal vectortakes the form

Z = ad(~u) + J + N (2.21)

2.2. INTERFERENCE SUPPRESSION 73

x

y

D y

D xu

vV i s i b l e a r e a

M a i n b e a m

S c a n a r e a

D u = l / D x

x

y

D au

v

Dv= l

/ Dy

2 l / ( 31 / 2 D a )

V i s i b l e a r e aM a i n b e a m

S c a n a r e a

Figure 2.4: Secondary mainlobes of a planar array. Left: rectangular grid, right:triangular grid

where d(~u) is the DOA-vector from a source in direction ~u and J denotes the re-ceived external interference which will be modelled as a random vector statisticallyindependent on the noise and with expectation 0. The DOA-vector can be arbitrary,also including different amplitudes due to non identical elements.

Interference and noise are summarized to one random vector Q = J + N withthe covariance matrix

R = RQ = E [(J + N)(J + N)∗] = E[JJ∗] + E[NN∗] = RJ + σ2IM . (2.22)

This is a situation with ’colored’ interference.

2.2.1 Test for the detection of the useful signal

First we regard the problem to detect the deterministic signal d(~u) and formulatethe


Test problem: Let Z = ad(~u) + Q be Gaussian distributed: Z ∼NCM (ad(~u),R). Decide between

H: a = 0 (only interference)and

K: |a| > 0 (Useful signal superposed by interference)

Solution: We know the solution from the test example in section 1.8, Eq. (1.149).The maximum likelihood ratio test for this problem is given by comparing the teststatistic

T (z) =∣∣z∗R−1s

∣∣2 (2.23)

to a threshold.

2.2.2 Optimum beamformer for colored interference

Alternative, we can search for an SNR-optimum beamformer. In contrary to thepreceding test problem, no explicit probability distribution is demanded. We onlyneed the properties that the expectation of the interference plus noise is zero andthat the covariance matrix is given as before.

Problem: For a selected direction ~u0, find a beamformer b(~u0) forwhich Y = b∗(~u0)Z achieves a maximum signal-to-noise-plus-interferenceratio (SNIR)!

.

Solution: Since R is hermitian and positive definite, we can form the matrixR−1/2. The covariance matrix of Q = R−1/2Q is the M ×M unit matrix, since

E[R−1/2Q

(R−1/2Q

)∗]= R−1/2E [QQ∗]R−1/2 = R−1/2RR−1/2 = IM . (2.24)

Obviously, R−1/2 ’whitens’ the interference. If this ’whitener’ is applied to therandom vector Z, we get

Z = R−1/2Z = ad(~u0) + Q (2.25)

with the deterministic component d(~u0) = R−1/2d(~u0) and the stochastic com-

ponent Q = R−1/2Q. The random vector Z bears the same information as theoriginal vector, since the transformation is reversible by multiplying with R1/2.

Now the optimum beamformer for Z is easily found using Essence 2.1 which isvalid since Z contains only white interference: bopt(~u0) = d(~u0). If this beamformer

is applied to Z, the result is


Y = d∗(~u0)Z =(d∗(~u0)R

−1/2) (

R−1/2Z)

= d∗(~u0)R−1Z =

(R−1d(~u0)

)∗Z. (2.26)

Obviously, the optimum beamformer to be applied to Z is

bopt(~u0) = R−1d(~u0). (2.27)

This is in good agreement with the result for the optimum detector given inEq.(2.23).

Essence 2.6 The SNIR-optimum beamformer for colored interference can be foundby the principle of ’pre-whiten and match’. It is given by the inverse noise+interferencecovariance matrix applied to the DOA-vector in the desired direction. For Gaussiandistribution, also the maximum likelihood ratio test is based on the output of thisbeamformer.

The ’pre-whiten and match principle’ for adaptive arrays is illustrated in Fig.2.5

z

m a t c h

R - 1 / 2

p r e - w h i t e n

( R - 1 / 2 d ( u 0 ) ) * y

Figure 2.5: Pre-whiten and match principle

b 1 ( u 0 )

Z 1 Z m

b m ( u 0 )** d 1 ( u 0 ) d m ( u 0 )**

Y = d * ( u 0 ) ( R - 1 Z )

R - 1

Z 1 Z m

Y = ( R - 1 d ( u 0 ) ) * Z

Figure 2.6: Two realizations of an adaptive array

The operation Y = d∗(~u0)R−1Z can be put into brackets in different ways which

are mathematical equivalent but not the same in the technical realization, see Fig.


2.6. If we put together the two first factors, we get the optimum beamformerbopt(~u0) = R−1d(~u0) leading to the realization shown on the left side of Fig. 2.6.For each new direction the beamformer has to be calculated again. If the righttwo terms are bracketed, we get the situation depicted at the right side of thefigure. First the element outputs are transformed by the matrix R−1 (suppressionof interference) and then the classical beamformer is applied to the resulting vectorfor any direction.

2.2.3 SNIR before and after optimum filtering

For an arbitrary beamformer b the SNIR is calculated to

γ(b) = |a|2 |b∗d(~u0)|2

E |b∗Q|2 (2.28)

= |a|2 |b∗d(~u0)|2b∗Rb

(2.29)

For the optimum beamformer, bopt = R−1d(~u0), and it follows

γ(bopt) = |a|2 |d∗(~u0)R−1d(~u0)|2

d∗(~u0)R−1RR−1d(~u0)(2.30)

= |a|2d∗(~u0)R−1d(~u0) (2.31)

If the conventional beamformer is used, we get

γ(d(~u0)) = |a|2 ‖d(~u0)‖4

d∗(~u0)Rd(~u0). (2.32)

The gain of the optimum beamformer compared to the conventional beamformer isexpressed by

γ(bopt(~u0))

γ(d(~u0))=

d∗(~u0)R−1d(~u0)

‖d(~u0)‖2

d∗(~u0)Rd(~u0)

‖d(~u0)‖2 . (2.33)

Moreover, it is usual to compare the SNIR to the SNR obtained without externalinterference with the conventional beamformer. This is expressed by

SNIR

SNR=

σ2d∗(~u0)R−1d(~u0)

‖d(~u0)‖2. (2.34)

2.2.4 The case of a single source of interference

The general form of interference suppression as treated in the foregoing section isvalid for any kind of interference which can be modelled as a random vector with


fixed covariance matrix. We regard an array with identical identically orientatedelements and an interference coming from a distinct direction ~v.

Let the complex amplitude J of the interference be random with expectationzero and variance σ2

J . Then

Z = ad(~u) + J + N = ad(~u) + Jd(~v) + N. (2.35)

The covariance matrices of interference J and interference plus noise Q = J+Nare given by

RJ = E[JJ∗] = σ2Jd(~v)d∗(~v) (2.36)

R = E[QQ∗] = σ2IM + σ2Jd(~v)d∗(~v). (2.37)

A normalization to the noise variance yields

1

σ2R = IM + η

d(~v)d∗(~v)

‖d(~v)‖2(2.38)

= IM + ηd(~v)d∗(~v) (2.39)

with η = σ2J‖d(~v)‖2/σ2 being the interference-to-noise-ratio (INR) and using the

normalized DOA-vector d(~v) = d(~v)/‖d(~v)‖.

Shape of the inverse covariance matrix: Using the matrix equality

(I + ηvv∗)−1 = I− η

1 + ηv∗vvv∗ (2.40)

(see Annex Eq.(A.28)) 1σ2R can be inverted to

σ2R−1 = IM − ζd(~v)d∗(~v) with ζ =η

1 + η. (2.41)

Now the application of the interference suppression operator H = σ2R−1 to themeasured array output z can be written as

Hz = z− ζd(~v)d∗(~v)z = z− ζJ(z)d(~v) = z− ζJ(z) (2.42)

with J(z) = d∗(~v)z and J(z) = J(z)d(~v).

It is illuminating to interpret this formula. J(z) = d∗(~v)z means to form a beam

into the direction ~v of the interference. Since ‖d(~v)‖ = 1 the output J(z) is anunbiased estimator of the complex interference amplitude. (see Annex Eq.(A.31))

J(z) = J(z)d(~v) is a reconstruction of the interference vector. 2

Obviously a complex multiple w of the signal d(~v) expected from the interferencedirection v is subtracted from the original sample vector z. To determine the factor,a beam is formed into this direction: y = d∗(~v)z and the output is scaled withw = ηy

1+ηd∗(~v)d(~v).

2To be worked over from here on


Adaptive characteristics The characteristics of the array for fixed beamformerb(~u0) = R−1d(~u0) is calculated to D(~u, ~u0) = βd∗(~u0)R

−1d(~u). Here, β is a normal-ization factor which is chosen in that way that D(~u0, ~u0) = 1, if only noise is present.We assume further that the single elements are omnidirectional and that the DOAvectors are normalized to unit vectors. Under these circumstances it follows β = σ2.We get

D(~u, ~u0) = βd∗(~u0)R−1d(~u) (2.43)

= d∗(~u0) (IM − γd(~v)d∗(~v)d(~u)) (2.44)

= d∗(~u0)d(~u)− γd∗(~u0)d(~v)d∗(~v)d(~u) (2.45)

= F (~u, ~u0)− γF ∗(~v, ~u)F (~v, ~u0). (2.46)

F (~u, ~u0) is the array factor as defined in Eq. (2.9) with F (~u, ~u) ≡ 1,

In direction of the interfering source we get

D(~v, ~u0) = F (~v, ~u0)− γF ∗(~v, ~u)F (~v, ~u0) (2.47)

= F (~v, ~u0)

(ηF (~v,~v)

1 + ηF (~v,~v)

). (2.48)

D(~v,~v) = F (~v,~v)

(ηF (~v,~v)

1 + ηF (~v,~v)

)(2.49)

=η

1 + η. (2.50)

Figure 2.7: Adaptive Null

Obviously a minimum is obtained in the direction of interference, that for η →∞tends to a null, see Fig. 2.7. The optimum beamformer is a linear combination fromthe two classical beamformers in target- and interference direction:


b(~u0) = σ2R−1d(~u0) (2.51)

= (IM − γd∗(~v)d(~v))d(~u0) (2.52)

= d(~u0)− γd∗(~v)d(~u0)d(~v) (2.53)

(The factor σ2 is allowed since the SNIR is invariant against multiplication of thebeamformer by a scalar).

The signal to noise plus interference ratio (SNIR) in dependence on thetarget direction The signal to noise plus interference ratio (SNIR) at the outputof the beamformer focused to the target direction ~u0 = ~u is given by

SNIR(~u) =|ad∗(~u)R−1d(~u)|2

E |d∗(~u)R−1 (J + N)|2 (2.54)

= |a|2 d∗(~u)R−1d(~u) (2.55)

=|a|2σ2

d∗(~u) (IM − γd(~v)d∗(~v)d(~u)) (2.56)

=|a|2σ2

(1− γ

M|F (~v, ~u)|2

). (2.57)

From the value without outer interference |a|2/σ2 the characteristics with focus-ing to the interference direction is subtracted, see Fig. 2.8.3

Figure 2.8: SNIR curve for one interference direction

Limit for large INR For a large η tends γ to γ = 1/(d∗(~v)d(~v)), the limitingform of R−1 is

3A further illustration could show the subtsction of the pattern


σ2R−1 → IM − d(~v)d∗(~v)

d∗(~v)d(~v)= IM −P<d(~v)>, (2.58)

where P<d(~v)> denotes the projector to the linear subspace spanned by d(~v).The characteristics tends to null in direction of the interference.

2.2.5 Several sources of interference

Again, we regard the model Z = ad(~u0)+Q,Q = J+N with the covariance matrixR of the sum Q of internal and external interference.

Let in the directions ~v1, . . . , ~vk be interfering sources with the complex randomamplitudes B1, . . . , Bk, collected in the vector B = (B1, . . . , Bk)

t with zero expecta-tions and covariances described by the covariance matrix RB = E[BB∗].

The related DOA-vectors are the columns of the M×k - Matrix D = (d(~v1), · · · ,d(~vk)).For D we presume regularity, i.e. the DOA-vectors to the interfering sources arelinearly independent.

Then we get

J =k∑

κ=1

Bκd(~vκ) = DB (2.59)

andZ = ad(~u0) + DB + N. (2.60)

Let UJ be the linear subspace spanned by d(~v1), · · · ,d(~vk) and U⊥J the subspace

orthogonal to UJ . Then

RJ = DE [BB∗]D∗ = DRBD∗ (2.61)

R = σ2IM + DRBD∗. (2.62)

If the number of external interference sources k is smaller than the number ofchannels M , the matrix RJ is not regular, its rank is maximal equal to k. Theinverse of the M ×M dimensional matrix R can be reduced to an equivalent matrixoperation in the k-dimensional subspace UJ according to 4

σ2R−1 = IM −DED∗. (2.63)

The k × k dimensional matrix E is given by

E =(σ2R−1

B + D∗D)−1

. (2.64)

So we get

4This identity is known as the ’matrix inversion lemma’, see Appendix


σ2R−1 = IM −D(σ2R−1

B + D∗D)−1

D∗. (2.65)

Limit for large INR If the noise is negligible with respect to the interference,i.e. the INR in all channels tends to infinity, σ2R−1

B will approach 0 resulting in

σ2R−1 → IM −D (D∗D)−1 D∗ = I−PUJ= P⊥

UJ. (2.66)

Essence 2.7 For the limiting case of negligible noise, the optimum interference sup-pression results in the projection to the subspace orthogonal to the subspace spannedby the DOA vectors to the sources of interference (interference space).

2.2.6 Signal subspace, interference subspace, noise subspace

As long as linearity can be assumed, for the case of simultaneous useful signals,interference, and noise the following general model can be applied:

Z = Sa + DB + N, (2.67)

with the useful signals S = (s1, · · · , sl), the interfering signals D = (d1, · · · ,dk)and the noise N. The complex amplitudes of the useful signals are collected in thedeterministic vector a, and those of the interfering signals in the random vector B.We assume that the signal vectors as well as the interference vectors are linearlyindependent, i. e. the matrix S has the full rank l and the matrix D has the rankk.

Useful signals and noise If Z consists of only useful signals and noise, we haveseen in the preceding that the projection PUS

:= S(S∗S)−1S∗ to the space of usefulsignals US = Lin s1, · · · , sl reconstructs the signals in terms of minimum meansquare optimally. The noise looses l degrees of freedom due to the projection, sothe remaining complete noise power will be equal to lσ2. The useful signals are notaffected by the projection.

Useful signals, interfering signals, and noise The interfering signals are con-tained in the interference subspace UJ = Lin d1, · · · ,dl. To remove the wholeinterference (necessary, if the noise is negligible with respect to the interference), theprojector PU⊥J

:= IM −D(D∗D)−1D∗ has to be applied. This projector eliminatescompletely the interference, but also a part of the useful signals. The remaining

parts of the useful signals are(PU⊥J

s1, · · · ,PU⊥Jsl

), the optimal reconstruction is

obtained by the projection PUSP⊥

C = PU⊥JPUS

. This is the projector to the subspace

US ∩ U⊥J .


Essence 2.8 For the model of linearly superposed useful signals, and interferingsignals the related subspaces can be identified. The projection to the signal subspacedoes not decrease the performance of detection and estimation, it is a sufficientstatistics. To remove the whole interference a subsequent projector to the subspaceorthogonal to the interference subspace may be applied, which may decrease the signalpower. The two projections - which can be subsumed to only one projection - alsoreduce the noise power.

2.2.7 A basic data reduction retaining optimal detection-and estimation properties for signals in interference

2.2.8 An example for continuous distributed interference

As an example we examine a linear array composed of omnidirectional elements withan external interference coming continuously and uniformly from all directions.

2.3. EIGENSPACES AND SAMPLE MATRIX 83

Problem: We regard a linear array of M elements with inter-elementdistance d, arranged along the z-axis of a Cartesian coordinate system.We assume external interference coming from all three-dimensional di-rections uniformly and independent (Interference being white over theset of directions on the unit sphere). Determine the covariance matrix ofthe vector of the element outputs!Solution: Let the z-coordinates of the phase centers be zm =m∆z, m = 1, . . . , M , the other components are equal to zero.The normalized DOA vector to an arbitrary direction ~u =(sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ)t characterized in spherical coordinates isgiven by

d(~u) =

expjkr∆z cos ϑ...

expjkrM∆z cos ϑ

, (2.68)

the received signal is C =∫Ω

A(~u)d(~u)dO(~u) and the covariance matrixis calculated to

RJ = E

[∫

Ω

∫

Ω

A(~u)A∗(~u′)d(~u)d∗(~u′)dO(~u)dO(~u′)]

(2.69)

= σ2A

∫

Ω

d(~u)d∗(~u)dO(~u) (2.70)

= σ2A

(∫ 2π

0

∫ π

0

expjkr(m− µ)∆z cos ϑ sin ϑdϑdϕ

)M

m,µ=1

(2.71)

= σ2A

(2π

∫ 1

−1

expjkr(m− µ)∆zudu

)M

m,µ=1

(2.72)

= πσ2A (si(kr(m− µ)∆z))M

m,µ=1 . (2.73)

If ∆z = λ/2, then kr∆z = π and

RJ = πσ2AIM . (2.74)

Essence 2.9 The outputs of a linear array with spacing λ/2 are uncor-related for spatial white interference uniformly distributed over the unitsphere.

2.3 Eigenspaces and sample matrix

Eigenvalues of a covariance matrix If u is a normalized eigenvector of thecovariance matrix RZ of an arbitrary random vector Z with expectation 0 withrespect to the eigenvalue λ, then


RZu = λu (2.75)

u∗RZu = λu∗u = λ, (2.76)

it follows

E[‖u∗Z‖2] = λ. (2.77)

Now we return to the interfering part of Eq. (2.60). The vector combiningexternal and internal interference is Q = DB + N with the covariance matrix R,the subspace of external interference is UJ = 〈d1, . . . ,dk〉. For each u ∈ U⊥

J wehave:

Ru = (DRBD∗ + σ2I)u = σ2u, (2.78)

so u is an eigenvector of R to the eigenvalue σ2. Since U⊥J has the dimension

M −k, M −k eigenvectors uk+1, · · · ,um can be found spanning U⊥J . The remaining

k eigenvectors to the eigenvalues λ1, · · · , λk are orthogonal to U⊥J , are consequently

elements of UJ . For these the following equation is valid

u∗κRuκ = u∗κ(DRBD∗ + σ2I)uκ ≥ σ2. (2.79)

If the columns of D are linearly independent is RB regular, even the greater signis valid.

Essence 2.10 The eigenvectors to the k largest eigenvalues span the interferencesubspace, while the remaining eigenvectors are related to the remaining noise sub-space and have the eigenvalues σ2.

E i g e n v a l u e n o .

l

s 2

I n t e r f e r e n c e s u b s p a c e

k M

l

s 2

k M

h

E s t i m a t e d i n t e r f e r e n c es u b s p a c e

E i g e n v a l u e n o .

Figure 2.9: The eigenvalues of the covariance matrix (’eigen spectrum’)

2.3. EIGENSPACES AND SAMPLE MATRIX 85

2.3.1 Determination of the weights

In the most applications the covariance matrix is unknown. An adaptive determina-tion of the coefficients of the beamformer is based on a learning sequence q1, . . . ,qn

which are ideally independent realizations of interference plus noise Q. In this con-text, the terminus ’snapshot’ is often used.

In the literature, many methods to determine the weights are described; we onlymention a few of them. Generally, we apply an M × M filter matrix H to thedata. To maximize the SNIR, H has to be chosen as a multiple of R. Since thetrue covariance matrix is not available other matrices have to be used. The loss iscalculated to

loss =γ(H)

γ(R−1)(2.80)

=|d∗(~u0)Hd(~u0)|2

d∗(~u0)H∗RHd(~u0)d∗(~u0)R−1d(~u0). (2.81)

The sample matrix: The matrix

R =1

n

n∑ν=1

qνq∗ν (2.82)

is called sample matrix or empirical covariance matrix. Its expectation is equal tothe wanted covariance matrix R, so R is an unbiased estimate of R.

Sample matrix inversion: Instead of R the estimate R is inverted. This methodis called sample matrix inversion (SMI).

Diagonal loading: Since R is a random variable, H = R−1 fluctuates especially ifthe number n of snapshots is low. The robustness of this technique can be increasedby using diagonal loading: To the diagonal of R a small constant ε is added beforeinversion: H = (R + εIM)−1.

Projection method: If UJ is an estimate of the interference subspace, the pro-jector H = P⊥

UJis applied. The estimation UJ can be extracted from the sample

matrix: The eigenvalues λ1, . . . , λM of the sample matrix are ordered. Accordingto the behavior of the eigenvalues of the true covariance matrix, we expect k largeeigenvalues related to the interference subspace and M − k eigenvalues related tothe remaining noise subspace. A comparison with a suitable threshold yields thek largest eigenvalues to the eigenvectors u1, . . . ,uk. The span of these vectors isthe estimation of the interference subspace. The remaining eigenvectors span thesubspace U⊥

J which is asymptotically free from interference.


2.4 DOA-estimation and Cramer-Rao Bounds

If the parallel signals of an antenna array or subapertures of an array are available,the possibility to estimate the direction of arrival with a high accuracy is opened. Incontrary to finding the maximum of the main beam by scanning this is possible inprinciple by processing the echoes of only one pulse. For this reason, these methodsare called monopulse estimators. In the following, we will have a closer look atthe general angular estimation problem, derive its Cramer-Rao Bounds (CRB) as ameans to judge the estimation accuracy, and present an algorithm for a generalizedmonopulse estimator for colored interference.

2.4.1 Estimation problem and signal model

We start by formulating the general signal model and the estimation problem. Thebasic model is the same as in Eq.(2.21), except for restricting it to the case of alinear array.

Problem: Based on a realization z of

Z = ad(u) + Q (model A) (2.83)

with unknown complex constant a and known covariance matrix R, uhas to be estimated.

This is a well known problem in array processing theory (see e.g. [13] or [15]).Nevertheless, for the application on moving platforms, this has rarely been ad-dressed, e.g. in [10], [11].

2.4.2 Cramer-Rao Bounds for the general problem

The Cramer-Rao Bounds offer the possibility to calculate a bound for the expectedquadratic deviation of an arbitrary estimator of the true value. They are basedon Fisher’s information matrix, see Annex section A.6.3. In section 1.7.3 we havealready derived the Cramer-Rao Bounds for the estimation of the direction, if onlynoise is present.

If we use the statistical model

Z = m(ϑ) + Q (2.84)

with a deterministic complex vector m(ϑ) and interference Q ∼ NCn(0,R), Fisher’sinformation matrix is given by

J(ϑ) =2

σ2Re

m∗

ϑ(ϑ)R−1mϑ(ϑ)

(2.85)

where mϑ(ϑ) means the gradient of mϑ with respect to ϑ written as n× k matrix,see appendix ??.

2.4. DOA-ESTIMATION AND CRAMER-RAO BOUNDS 87

If the unknowns in Eq. (2.83) are combined into the real-valued parameter vectorϑ = (u, α, ϕ) with a = α expjϕ, the vector m is expressed as

m(ϑ) = αejϕd(u). (2.86)

The Fisher information matrix is calculated starting from Eq.(A.111) to

J = 2<

α2d∗uR−1du αd∗uR

−1d jα2d∗uR−1d

αd∗R−1du d∗R−1d 0−jα2d∗R−1du 0 α2d∗R−1d

(2.87)

For simplicity, the dependence on u was omitted, and the subscript u denotes thederivation with respect to u. The variance of the estimation error of the directionalcosine is limited by the first component of the inverse Fisher matrix, which can beanalytically calculated from Eq. (2.87) using the representation of sub-determinants:

(J−1)11 =1

2|a|2p

q(2.88)

with

p = d∗R−1d

q = d∗R−1dd∗uR−1du −

∣∣d∗uR−1d∣∣2 . (2.89)

Fig. 2.10 shows the SNIR and CRB curves for two sources of interference.

−1 −0.5 0 0.5 1−14

−12

−10

−8

−6

−4

−2

0

u

SN

IR (

dB)

−1 −0.5 0 0.5 10.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

u

Sqr

t(C

RB

)

Figure 2.10: SNIR- and CRB plot for two external sources of interference

Essence 2.11 For colored interference, the expected quadratic error of any estima-tor of the directional cosine for the signal model Eq.(2.83) is bounded by the firstcoefficient of the inverse Fisher matrix given in Eq.(2.88).


2.4.3 The basic estimation problem

As we will see, the general problem can be reduced to a much simpler basic problem:

X = be(u) + N (model B) (2.90)

with the properties

‖e(u)‖2 ≡ 1 (2.91)

e∗u(u)e(u) = 0 (2.92)

RN = IM . (2.93)

Again, the complex amplitude b and the parameter u are unknown, the task is toestimate u based on a realisation x of X.

Cramer-Rao-Bounds

From Eq. (2.83) we see that under these conditions the Fisher information matrixis diagonal, i.e. the estimation of the three parameters is decoupled. The inverse ofthe element of J dedicated to u is simply given by

J−111 =

1

2|b|2‖eu(u)‖2. (2.94)

Maximum-Likelihood estimator

Under the assumption that N is Gaussian distributed, the ML estimator of u isgiven to

u = argmaxu

|e∗(u)x|2 : u ∈ [−1, 1]

, (2.95)

while b = e∗(u)x is the corresponding ML estimate of b. The first derivative ofEq. (2.95) with respect to u yields the necessary condition:

<(e∗u(u)x) (e∗(u)x)∗ = 0. (2.96)

Monopulse estimator

In order to reduce computational complexity, we may approximate the ML equation(2.96). Let u0 be a direction close to the unknown direction u, e.g. the main lookdirection of the phased array, and ∆u = u− u0. By Taylor-expansion

e(u) ≈ e(u0) + ∆ueu(u0) (2.97)

eu(u) ≈ eu(u0) + ∆ueuu(u0). (2.98)

around u0 we get as the approximation of the ML equation:

<(

σ(x) + ∆uδ(x))∗ (

δ(x) + ∆uγ(x))

= 0, (2.99)

2.4. DOA-ESTIMATION AND CRAMER-RAO BOUNDS 89

with the abbreviations

σ(x) = e∗(u0)x, δ(x) = e∗u(u0)x, γ(x) = e∗uu(u0)x. (2.100)

Omitting terms of the order ∆u2

yields

<

σ∗(x)δ(x) + ∆u(|δ(x)|2 + σ∗(x)γ(x)

)= 0. (2.101)

For sufficient SNR and small ∆u2

the terms in the brackets right from ∆u can bereplaced by |δ(x)|2 ≈ 0, γ(x) ≈ σ(x)e∗uu(u0)e(u0) = −σ(x)‖eu(u0)‖2 resulting in

∆u ≈ 1

‖eu(u0)‖2<

δ(x)

σ(x)

. (2.102)

This is the well-known monopulse estimator for white noise interference; σ(x) is thesum channel, δ(x) the difference channel.

Essence 2.12 For white Gaussian noise under the assumptions of the basic esti-mation problem of model B Eq.(2.90 the approximate maximum likelihood estimatoris given in Eq. (2.102). In principle, only one pulse is necessary to estimate thedeviation ∆u from the focusing direction, if two beamformers are used: The sumchannel σ(x) and the difference channel δ(x).

Unfortunately, this estimator possesses an infinite variance, since the probabilitythat the denominator takes values in the area around 0 is extremely low for sufficientSNR but not zero. Nevertheless, this can be avoided by blending out these valueswith a clipping function g(x) tending to 1 for x →∞ and g(x)/x bounded for x → 0.

∆u :=1

‖eu(u0)‖2<

δ(x)

σ(x)

g(|σ(x)|). (2.103)

It can be shown that for this version the estimator is efficient in the limit |b| → ∞and ∆u → 0, that means the expectation of the quadratic error approaches theCRB.

2.4.4 Transformation of the general problem to the basicproblem

Reduction equations

Model A Eq. (2.83) can be transformed into model B Eq. (2.90) in the followingway: First, the random vector is multiplied by the ’whitening filter’ R−1/2:

X = R−1/2Z (2.104)

resulting in a identity covariance matrix for X. During this operation also the DOAvector is distorted. We can further normalize the DOA vector by changing theamplitude a appropriately:

X = be(u) + N (2.105)


with

N = R−1/2Q (2.106)

e(u) =R−1/2d(u)

‖R−1/2d(u)‖ (2.107)

b = a‖R−1/2d(u)‖. (2.108)

To force the condition e∗u(u)e(u) = 0 in Eq. (2.91), the phase of the signal vector isnormalised to zero derivation at u = u0:

e(u) :=e∗(u)e(u0)

|e∗(u)e(u0)| e(u) (2.109)

This phase shift can be compensated by an opposite phase shift of b to the finalamplitude b. Hence the transformations in Eq. (2.104)-(2.109) yield X = be(u) + Nwith the desired properties that is identical to model B.

Generalised Monopulse estimator

The monopulse estimator Eq. (2.102) can be expanded to a general monopulse esti-mator for the model A. At u = u0, we get with the abbreviations d00 = d∗(u)R−1d(u),du0 = d∗u(u)R−1d(u), duu = d∗u(u)R−1du(u) and omitting the dependance on u:

|b|2 = |a|2d00 (2.110)

e = d−1/200 R−1/2d (2.111)

eu = d−1/200 R−1/2du − d

−3/200 <du0R−1/2d (2.112)

eu = eu + (e∗ue)e

= d−1/200 R−1/2du − d

−3/200 d∗u0R

−1/2d (2.113)

‖eu‖2 = ‖eu‖2 − |e∗ue|2 = d−100 duu − d−2

00 |du0|2 (2.114)

As verification of the consistency of the CRB for the basic form (Eq. 2.94) with thegeneral form, we may compute

1

2|b|2‖eu‖2=

d00

2|a|2(d00duu − |du0|2) , (2.115)

which in fact is consistent with Eq. (2.88).

We are now left with the derivation of the general forms of the sum and differencechannels:

σ(x) = e∗(u0)x (2.116)

= d−1/200

(d∗(u0)R

−1/2)R−1/2z = d

−1/200 Σ(z)

δ(x) = e∗u(u0)x

=(d−1/200 d∗uR

−1/2 − d−3/200 du0d

∗R−1/2)

R−1/2z

= d−1/200 ∆(z)− d

−3/200 du0Σ(z) (2.117)

2.5. LINEAR TRANSFORMATIONS 91

with Σ(z) := d∗(u0)R−1z and ∆(z) := d∗u(u0)R

−1z. Using (Eq. 2.102) we canderive the generalized monopulse estimator to

∆u =1

‖eu(u0)‖2<

δ(x)

σ(x)

=d2

00

d00duu − |du0|2<

∆(z)

Σ(z)− du0

d00

. (2.118)

Finally, by using the beamformer vectors bΣ = R−1d(u0) and b∆ = R−1du(u0), theestimator can be expressed as

∆u =|b∗Σd(u0)|2

b∗Σd(u0)b∗∆du(u0)− |b∗∆d(u0)|2 (2.119)

×<

b∗∆z

b∗Σz− b∗∆d(u0)

b∗Σd(u0)

. (2.120)

This form is exactly identical to that given in [13], where it was called ’correctedadaptive monopulse’. However, therein it was derived from another viewpoint,namely to linearly correct a monopulse quotient Q = <b∗∆z/b∗Σz to eliminatebias and slope of the interference-less quotient. In our derivations we have shown,that the generalized monopulse estimator Eq. (2.119) is a natural expansion of thesimple monopulse estimator Eq. (2.102), therefore clearly possessing the same prop-erty of asymptotic efficiency. For u close to u0, large SNR and a clipping prohibitingthat the denominator comes close to zero, the variance of the generalized monopulseestimator approaches the CRB in Eq. (2.89) and (2.115).

2.5 Linear transformations

To reduce the technical and computational effort, transformations into a lower-dimensional data space is of interest. Starting from the standard model Z = as+Qwe regard a linear transformation described by the K ×M dimensional matrix Ttransforming the data into a new K-dimensional data space via Ztr = TZ. Theresulting deterministic part is str = Ts and the interference is transformed by Jtr =TJ leading to the new model Ztr = astr + Qtr. The transformed covariance matrixis given by Rtr = TRT∗.

2.5.1 Gain loss by linear transformations

The SNIR before transformation is γ = |a|2s∗R−1s. Now we calculate the new SNIRγT after transformation:

γT = |a|2s∗trR−1tr str (2.121)

= |a|2s∗T∗ (TRT∗)−1 Ts. (2.122)


The loss induced by the transformation is

LossT =γT

γ=

s∗T∗ (TRT∗)−1 Ts

s∗R−1s. (2.123)

2.5.2 Special transformations

Invertible transformation If K = M and T is invertible, T and T∗ can beshifted out of the brackets as their inverse yielding after cancellation with T and T∗

the same expression as without transformation, there is no loss.

Transformation using the optimum beamformer If K = 1 and T = (R−1s)∗

we get no loss, too, which is easily varified. This is an extreme reduction of dimen-sionality, nevertheless it does not really spare computation since the inverse of thefull covariance matrix has to be calculated.

Transformation into the subspace spanned by interference plus signal Ifthe vectors v1, . . . ,vk span the interference subspace, the transformation matrix

T = (v1, . . . ,vK , s)∗ (2.124)

also will not lead to a loss 5.

5Proof follows

Chapter 3

Air- and spaceborne radar

The aim of this chapter is to introduce the special situation of a radar mounted onan airplane or satellite. The main specific problem is that the radar itself is moving.All ground scatterers are moving relative to the radar, different radial velocities offixed scatterers are the consequence. We will concentrate on the properties of theechoes from the temporal stable ground surface for an airborne radar - where theearth surface can be approximated by a plane - and for space based radar with anappropriate model of ta curved surface and a satellite orbit. The results are usefulfor synthetic aperture imaging as well as for moving target recognition. Throughoutthis chapter, the regarded time interval is short enough to justify a time-linearapproximation of the phase.

3.1 Definition of basic angles

Before treating the ground echoes of an air- or spaceborne radar, let us define someangles which apply for flat as well as curved earth surface, see Fig. 3.1. The angle εbetween the horizontal plane at the radar and the line-of-sight (LOS) to a point onthe ground, characterized by the unit vector ~u is called depression angle. The angleϑ = ε+π/2 is used for a mathematical description in a spherical coordinate system.On the earth surface the wave comes in at the incidence angle β with respect to thevertical axis at this point. It is also usual to measure the direction of the incomingwave relative to the horizontal plane. This angle γ = π/2−β is called grazing angle.In the literature, there are different definitions of the elevation angle. We will usethis expression only in the relation to the antenna surface. For the approximationof a flat earth - which is usual for airborne radar with medium range - the grazingangle and the depression angle are equal γ = ε and for the incidence angle we getβ = π − ϑ.

Let the radar be moving in direction of the x-axis, see Fig. 3.1 right. The angleα between x-axis and the LOS is called coneangle, the angle ϕ between the x-axisand the projection of the LOS to the (x, y)-plane is called azimuthangle. Cone- andazimuth angle are related by

93

94 CHAPTER 3. AIR- AND SPACEBORNE RADAR

Glossary for this chapter

α Cone angleβ Incidence angleδr, δv, δF Resolutions in range, velocity and Dopplerδu Resolution in directional cosine∆T Pulse repetition period∆x Spacing of synthetic array~δr, ~δv Edges of a resolution patch on groundε Depression angleϕ Azimuth angleγ Grazing angleλ WavelengthωE Rotation velocity of the earthΩBA Angular velocity matrixσ0 Terrain scattering coefficientϑ Angle between z axis and look directionA Area of a ground resolution patchBmain, Btot Main beam and total Doppler bandwidthsD(u), D(~u) Two-way characteristics of antennaF Doppler frequencyFs Sampling frequency in slow time = PRFFmax = 2V/λ Maximum clutter Doppler frequencyG,H Mappings from the earth surface to the range-velocity planeh Flight altitude over groundlx Length of antenna in flight directionLx Length of synthetic arrayLBA Rotation matrix from the system A to the system B~p Position of a scattererPC(F ) Doppler power spectrum of clutterPRF = Fs Pulse repetition frequencyr Range to scattererrg Ground range~r, ~rg Vector from the antenna to a scatterer and its ground projection~R, ~Rg Position of the antennas phase center and its ground projectionu = cos α Cosine of cone angle~u = (u, v, w)t Unit vector pointing from the antenna to a scatterer~V Platform velocity vectorV Platform velocityvr Radial velocityW (sin γ) Weighting function for reflectivity(xA, yA, zA) Antenna coordinate system(xe, ye, ze) Earth fixed coordinate system with origin at the surface(xE, yE, zE) Earth fixed coordinate system with origin at the center(xI , yI , zI) Inertial coordinate system(xP , yP , zP ) Platform coordinate system

3.2. AIRBORNE RADAR 95

J

gb

e R a d a r

G r o u n d s c a t t e r e r

x

yz

R a d a r

J

ae

j

G r o u n d t r a c k

z

uj

u

Figure 3.1: Left: Definition of angles in the vertical plane: β: Incidence angle, ε:Depression angle, γ: Grazing angle, ϑ: Angle used for spherical coordinate system.Right: ϕ: Azimuth angle, α: Cone angle

cos α = cos ϕ cos ε. (3.1)

3.2 Airborne radar

3.2.1 Energy distribution in range and Doppler

To get a feeling of the expected ground returns, in Fig. 3.2 a coarse distributionof the energy of the ground-returns is depicted for a forward looking configuration.If the flight altitude is h, the first return in range comes from the surface patchdirectly under the airplane containing the nadir , i.e. the projection of the antennasphase center to the ground plane. Since the range is minimum, and the area of thispatch is relatively large, and since the wave comes in perpendicular to the surfaceproducing a specular reflection, this echo is very strong and appears normally alsoin the data, if the nadir direction is outside the main beam of the antenna. In thearea of illumination of the ground by the main beam, the mean echo power is -apart from the modulation by the antenna pattern - decreasing proportional to r−4

according to the range law Eq.(1.188).

In the Doppler domain also a strong nadir return is observed at the Dopplerfrequency zero, since the radial velocity of a scatterer directly under the platformvanishes according to the LOS being perpendicular to the flight path. In the mainbeam illumination area the radial velocities of the ground scatterers with respectto the platform vary between v

(1)r = −V cos α1 and v

(2)r = −V cos α2 where α1

and α2 are the cone angles at the boundaries of the main beam. This leads to


a Doppler spread over the interval [Fmax cos α1, Fmax cos α2] with Fmax = 2V/λ.Doppler frequencies with magnitudes larger then Fmax can never appear for nonmoving scatterers. Also the power of the spectrum is governed by the elevationpattern of the antenna and the r−4 law.

R a n g e

G r o u n d r e t u r nr - 4 l a w

D o p p l e r

h

h

0

v r ( 2 )

V

V F m a x F m a x

a 1 a 2

v r ( 1 )

v r ( 1 )V F m a xv r ( 2 )

0

x

Figure 3.2: Energy distribution of the ground-returns in range and Doppler

3.2.2 Three dimensional geometry for horizontal flight

For airborne radar, we may assume the earth surface to be flat. Let the platform- with the antenna’s phase center as the reference point - move along a straightflight path with constant velocity ~V . Fig. 3.3 shows the geometrical situation forhorizontal flight at the altitude h above ground. At a fixed point of time the positionof the antenna’s phase center is ~R.

We discriminate three identically orientated cartesian coordinate systems, onefixed to the platform (xP , yP , zP ) with the antennas phase center at its origin andxP arranged to the direction of motion and one to the earth (xe, ye, ze) where ze = 0at the earth’s surface and the last (xe′ , ye′ , ze′) centered to the ground point directlybeneath the platform, the nadir. The vector ~r from the antenna’s phase center to aground scatterer at the position ~p is given by ~r = ~p− ~R. The unit vector ~u = ~r/‖~r‖represents the LOS-vector.

The ground range vector ~Rg is the projection of the platform vector ~R to theearth plane, ~rg the projection of the vector ~r. The lengths of the latter vectors aredenoted by r and rg =

√r2 − h2, respectively.


x P

y Pz P

h

E a r t h s u r f a c e

J

a

j g

u

ry e

z eR

R gr g

( x , y )

x e

e

x ' e

y ' e

R a d a r

p

V

Figure 3.3: Three dimensional geometry for horizontal flight

The LOS-vector ~u can be expressed by spherical polar coordinates with respectto the platform coordinate system:

~u =

sin ϑ cos ϕsin ϑ sin ϕ

cos ϑ

=

cos ε cos ϕcos ε sin ϕ− sin ε

. (3.2)

Since the earth surface is assumed to be flat, the grazing angle γ is equal to thedepression angle ϕ and related to the range via sin γ = sin ε = h/r. The radialvelocity of any point scatter with respect to the antenna’s phase center is given byvr =< ~V , ~u >= V cos α where α is the cone angle.

3.2.3 ISO-Range and ISO-Doppler contours

In Fig. 3.4 the correspondence of the radar parameters range and Doppler in thethree dimensional space and on the earth surface is illustrated. Clearly, the pointsof equal range r to the antenna’s phase center are placed on the surface of a spherewith radius r centered at ~R and the points of equal radial velocity of non moving


scatterers on the surface of a cone with the cone angle α and its corner at ~R. Thelines in the three-dimensional space with equal range and radial velocity are cuts ofthat spheres with that cones: They are circles orientated perpendicular to the flightpath.

The lines on the earth surface with the same distance to the phase center (ISO-range lines) are the cuts of the range sphere with the earth planes, i.e. circles withits center at the nadir. The lines with the same radial velocity are the cuts of theabove mentioned cone surfaces with the earth plane, i.e. parabulae. At a fixed RFthe radial velocity is proportional to the Doppler frequency, so the cone can also becalled Doppler cone and the cone cuts with the earth surface ISO-Doppler lines .

Also in the case of non-horizontal flight the ISO-Doppler lines are cone-cuts: forascending flight directions they are hyperbolae, for descending flight they also canbe ellipses.

D o p p l er c o n e

Range s

phere

I s o - R a n g e I s o - D o p p l e r

N a d i r

F l i g h t p a t h

G r o u n d t r a c k

V

Figure 3.4: Range sphere and Doppler cone

Fig. 3.5 shows a projection of the ISO-range and ISO-Doppler contours to theearth plane for a side-looking radar. In this figure, also the antenna pattern asit illuminates the earth surface is indicated. Already by inspecting this figure itbecomes clear that the Doppler spectrum of a homogeneous area is related to theantenna pattern. We will deepen this fact in the following.

Essence 3.1 The ISO-range lines on the surface are circles centered at the nadir,the ISO-Doppler lines are cuts of the Doppler-cone with the surface. The Doppler-cone has its axis in flight direction and its corner at the antennas phase center, thecone angle α is related to the radial velocity by vr = −V cos α.


I s o - R a n g e

I s o - D o p p l e r

G + ( r , v r )- 1

G - ( r , v r )- 1

x P

Figure 3.5: ISO-range and ISO-Doppler contours

3.2.4 Mapping between range - radial velocity and earthsurface

In the following we will investigate the mapping from a position on the groundto the radar parameters range/radial velocity (range/Doppler, respectively), andvice versa. For each point (x, y)t on the ground surface measured in nadir-centeredcoordinates (xe′ , ye′ , ze′) , see Fig. 3.3, range and radial velocity are determined bythe mapping

G(x, y) =

(rvr

)(x, y) =

√x2 + y2 + h2

−V x√x2+y2+h2

. (3.3)

The mapping is symmetric in y, i.e. G(x,−y) = G(x, y). It is unique on thehalf space y > 0. The inverses of the restrictions to the positive and the negativehalf space, respectively, are given by the functions G+ and G−, defined by

G−1± (r, vr) =

(xy±

)(r, vr) =

−r vr

V

±√

r2(1− (

vr

V

)2)− h2

. (3.4)

ISO-range and ISO-Doppler lines in (x, y)-coordinates are easily derived by


Eq. (3.4), considering the unique relation between radial velocity and DopplerF = −vr

VFmax, as long as no Doppler ambiguities arise. A range-Doppler pair

(r, F ) is related to the two points G−1+ (r, vr) and G−1

− (r, vr) on the earth withvr = −V F/Fmax, see Fig. 3.5.

Consequently, the range-Doppler map generated in the radar processor accordingto Fig. 1.16 can be regarded as a first coarse radar image of the earth surface whichis left-right ambiguous with respect to the flight path. If a side-looking antenna isused as in Fig. 3.5 where the antenna gain to the back plane can be neglected, thisambiguity vanishes and a one-to-one correspondence between range-Doppler mapand the half space on the earth surface is established.

Analogue to Eq. (3.3) we may express these relations also using polar coordinates(rg, ϕ) on ground with (x, y) = (rg cos ϕ, rg sin ϕ) defining a referring mapping H by

H(rg, ϕ) =

(rvr

)(rg, ϕ) =

√r2g + h2

−V rg√r2g+h2

cos ϕ

. (3.5)

Again, H is unambiguous in the half spaces, and the inverses are given by

H−1± (r, vr) =

(rg

ϕ±

)(r, vr) =

√r2 − h2

± arccos(

vrrV√

r2−h2

)

. (3.6)

Essence 3.2 For each range-velocity pair (r, vr) there are two points on the earthsurface generating echo signals at (r, vr). Their positions are symmetric to the flightpath projected on the ground.

3.3 Real and synthetic arrays

There are many correspondences between a real array and sampling during the flight:The sampling interval ∆T = 1/Fs in slow time is transferred by the platform’smotion into a spatial sampling ∆x = V ∆T along the flight path. In this way, asynthetic array is generated whose properties are quite similar to a real array, seeTab. 3.3.

The signal model for a sequence of N pulses assuming a scatterer in direction uis given by

s(u) =

ej2kru∆x

...ej2kruN∆x

. (3.7)

This is mathematically identical to the model of a DOA-vector for a real array,Eq. (2.1) apart from an additional factor 2 in the exponent regarding the two-way

3.3. REAL AND SYNTHETIC ARRAYS 101

Real array Synthetic array Temporalinterpretation

Spatial sampling ∆xa ∆x = V ∆TArray length lx Lx = V Tint

Directional cosine resolution δua = λ/lx δu = λ/(2Lx)Maximum spatial frequency kxmax = 2π/λ kxmax = 4π/λMaximum Doppler frequency Fmax = 2V/λSpatial frequency kx = ukxmax kx = ukxmax

Doppler frequency F = −uFmax

k-ambiguity ∆kx = 2π/∆xa ∆kx = 2π/∆xDoppler ambiguity ∆F = 1/∆TDirectional cosine ambiguity ∆u = λ/∆x ∆u = λ/(2∆x) ∆u = ∆F/Fmax

Nyquist condition to cover[u1, u2] ∆x ≤ λ

u2−u1∆x ≤ λ

2(u2−u1)∆T ≤ λ

2V (u2−u1)

Table 3.1: Relations for real and synthetic arrays

travelling of the wave. So the properties of a real array can be transferred to thesynthetic array analogue.

Spatial frequency A Doppler frequency F can also be regarded as a spatialfrequency along the flight path with kx = (2πF/V )F . Clutter Doppler frequenciesF (u) = −2uV/λ are transferred to clutter spatial frequencies via kx(u) = −4πu/λ =−2ukr.

Beamforming Doppler processing can be interpreted as the formation of a syn-thetic beam for each Doppler cell with Doppler frequency F to that direction u(F ) =−F/Fmax corresponding to the clutter Doppler frequency F . If the Doppler filter isapplied over a time interval of length Tint, the platform has moved over a distanceLx = V Tint. This is the length of the synthetic array, the synthetic aperture. Thoughthis expression is used mainly related to the SAR technique, it can be applied withmeaning also for air- or space borne MTI. In this chapter, we presume that Lx issmall enough to approximate the wavefronts to be planar, i.e. the phase propa-gation along the synthetic aperture can be assumed to be linear. In other words:The synthetic array is small enough to accept the far field condition given for realantennas (Eq. (??)) and considering a factor 2: Lx ≤

√rλ with an allowed phase

deviation from a linear function not larger than ±π/4.

Ambiguities Due to the finite sampling along the aperture with period ∆x, am-biguities arise in the directional cosine according to ∆u = λ/(2∆x) like that of a realarray (see Eq. (2.17)). These are directly related to the Doppler ambiguity ∆F = Fs

via ∆u = ∆F/Fmax. Therefore, to cover a directional cosine interval [u1, u2] unam-biguously the spatial sampling period according to the Nyquist condition must notbe larger than ∆x = λ/(2(u2 − u1)).


Interaction between real and synthetic characteristics The Doppler ambi-guities can be suppressed, if the main beam of the real array is smaller than theambiguities in the directional cosine caused by Doppler ambiguities. This means,the beamwidth of the real array δua has to be smaller than the ambiguity of thesynthetic beam ∆u. It follows

δua ≤ ∆u (3.8)

λ

lx≤ ∆F

Fmax

(3.9)

lx ≥ λFmax

∆F= 2V ∆T. (3.10)

This means that the antenna length has to be larger or equal to twice the waythe platform moves from pulse to pulse.

The output of a Doppler filter for frequency F0 forms a beam to the directionu0 = −F0/Fmax. So the response h(F, F0) of this Doppler filter to other frequenciesF can also be regarded as a spatial characteristics Ddop(u, u0) = h(−uFmax,−u0Fmax

of the synthetic array. If D(u) is the two-way characteristics of the real antenna, thesensitivity of the Doppler filter output to clutter echoes from direction u is given bythe product of the two characteristics:

H(u, u0) = D(u)Ddop(u, u0). (3.11)

This is illustrated in Fig. 3.6.

F i g _ A i r A n d S p a c e _ 0 2 5 _ C h a r a c t e r i s t i c s _ D o p p l e r

Figure 3.6: The combination of Dopper filter transfer function and spatial charac-teristics for clutter


Range-Doppler resolution patch on the earth surface

By the radar processor range-Doppler cells are generated with the resolutions δr andδF . Range - radial velocity resolution cells are derived from this with the resolutionsδr and δv = δFλ/2. We now look for the area on the earth surface correspondingto δr and δv? These are areas with the form of grid cells shown in Fig. 3.5 if thespacing between the ISO-range and the ISO-Doppler lines are chosen as the relatingresolutions.

Inspecting Eq. (3.4) the derivations with respect to r and vr are given by

~gr :=∂

∂rG−1

+ (r, vr) =

−vr

V

r(1−( vr

V )2)

√r2

(1−( vr

V )2)−h2

=

− cos α

rysin2 α

(3.12)

~gv :=∂

∂vr

G−1+ (r, vr) =

− rV

−r2 vrV 2√

r2(1−( vr

V )2)−h2

= − r

V

1

rycos α

. (3.13)

So, as long as linearity is justified, the resolution cell in range and radial veloc-ity on the earth surface is approximately given by a parallelogram with the edgesdetermined by the vectors ~δr = δr~gr and ~δv = δv~gv, see Fig. 3.7.

d r dv

Figure 3.7: A range Doppler resolution patch

The area of this patch is easily determined from Eq. (3.12):

Ares = ‖~δr × ~δv‖ =r2

|y|V δrδv =r2

|y|V ares (3.14)

with the resolution area ares = δrδv = c02b

λ2Tint

in the original (r, vr) plane. Obviously,the grid degenerates in the vicinity of y = 0. The reason is that the slope of the


radial velocity (Doppler) in the y-direction is zero for y = 0. We will find thisproperty also in the application to SAR imaging.

3.3.1 The Doppler spectrum of clutter

Coarse consideration in two dimensions

F

- F m a x

B m a i n = d u a F m a x

P R F / 2- P R F / 2

F m a x

Figure 3.8: Doppler Spectrum of clutter

Let’s go back to Fig. 3.5 and assume that the antenna pattern is zero on thehalf space y < 0. Let the two-way antenna characteristics D(u) be dependent inthe first place only on the directional cosine u. In direction u the radial velocity ofthe stationary ground scatterers is given by the linear relation F (u) = −uFmax oru(F ) = −F/Fmax, vice versa. So in this direction the ground echoes at Doppler Fare weighted by the antenna characteristics D(u(F )). From this, we can derive thecoarse form of the Doppler spectrum of the ground echoes:

If the reflectivity of the ground is homogeneous, or if the Doppler power spectrumis averaged over a long time, we get

PC(F ) ≈ const

∣∣∣∣D(

F

Fmax

)∣∣∣∣2

. (3.15)

In other words, the Doppler power spectrum is a scaled version of the two-way antenna pattern. This fact is illustrated in Fig. 3.8. In the application ofmoving target indication, the ground echoes are regarded as clutter and their Dopplerspectrum are called clutter spectrum. It can be divided into three areas:

• The main beam clutter. It’s bandwidth is given by Bmain = δuaFmax, whereδua denotes the beamwidth of the two-way characteristics.

• The sidelobe clutter. The spectrum is sharply bounded by the condition −1 ≤u ≤ 1. So we have for the bandwidth of the whole clutter spectrum: Btot =2Fmax.


• The clutter free region. This is outside the bandwidth Btot.

For most airborne radar systems, it is impossible to cover all of the sidelobeclutter or even parts of the clutter free region unambiguously by the sampling fre-quency Fs = PRF. This can be achieved only if ∆T > 1/Btot = λ/(4V ), i.e. thepulse repetition interval has to be shorter than the time needed by the platform tofly the distance of λ/4. For GHz frequencies and normal airspeed this would requirea PRF larger than allowed to avoid range ambiguities. As a consequence, normallyat least the sidelobe part of the clutter will be aliased, so there will hardly be anycompletely clutter-free region! Common SAR systems apply an azimuth samplingfrequency equal to or a little above Bmain, MTI-systems should use a considerablyhigher PRF!

To sample the main beam clutter according to the Nyquist criterium Fs has tobe larger or equal to Bmain. For the way the airplane is allowed to fly betweentwo pulses follows V ∆T ≤ V/Bmain = λ/(2δua). If we insert the main beam widthδua = λ/lx of an antenna of length lx, we get

Bmain =V

lx/2, V ∆T ≤ lx

2, (3.16)

so the airplane should not move more than a half antenna length between two pulses.

Essence 3.3 The clutter spectrum can be approximated as the two way antennapattern scaled via u = −F/Fmax with Fmax = 2V/λ. Three areas of the clutterspectrum can be discriminated: the main beam clutter, the sidelobe clutter and theclutter free region |F | > Fmax.

Clutter spectrum - three dimensional derivation

Let us have a closer look to the clutter spectrum taking all three dimensions intoaccount! The two-way antenna characteristics D(~u) now is expressed as functionof the LOS-vector ~u. For a certain fixed range r there are two patches on theearth surface effecting the Doppler frequency F , namely at G−1

+ (r,−V F/Fmax) andG−1− (r,−V F/Fmax), see Eq. (3.4). There are two LOS-vectors ~u±(r, F ) pointing to

these points which can be evaluated to

~u±(r, F ) =

u

±√1− u2 − w2

−w

(3.17)

with u = cos α = − F

Fmax

(3.18)

w = sin ε = sin γ = h/r. (3.19)

The areas of the patches have been calculated to A = r2

|y|V δrδv, see Eq. (3.12),which can be further expressed as


A(r, F ) =r2

V

√1−

(F

Fmax

)2

− (hr

)2

δrδv. (3.20)

For homogeneous ’spatial white’ clutter the energy coming from the directions~u±(r, F ) will be proportional to the patch area. Summarizing, the power of theclutter spectrum at range r can be approximated by

PC(r, F ) ≈ const(|D(~u+(r, F ))|2 + |D(~u−(r, F ))|2) A(r, F )r−4W (h/r)σ0. (3.21)

Here, σ0 - sometimes written in texts simply as ’sigma zero’ - denotes the terrainscattering coefficient, i. e. the reflectivity of the surface, which is defined by theproperty that an area a of that type of surface has the RCS q = aσ0. Tables of σ0

for different kinds of vegetation depending on the frequency and the grazing anglehave been published. W (sin γ) is a weighting function reflecting the influence ofthe grazing angle. In the simplest case, set W (w) = w to take into account thatthe effective area perpendicular to the LOS is equal to sin γ times the ground area.Remind that the approximation for A(r, F ) is not valid in the vicinity of y = 0!

3.4 Ambiguities and the critical antenna size

In this section we will have a closer look to the ambiguities in range and radialvelocity. To suppress ambiguities, the antenna’s main beam must not cover morethan one ambiguity rectangle. This leads to a condition to the minimum size of thereal aperture.

3.4.1 Range ambiguities

Let’s first study range ambiguities ∆r = c02∆T effected by a low pulse repetition

interval ∆T = 1/Fs. That means that in a range resolution cell dedicated to theprimary range r also the echoes from all the ranges rν = r + ν∆r with ν ∈ Z andrν ≥ h are superposed. From Eq. (3.6) follows for the ambiguous ground ranges:

r(ν)g =

√r2ν − h2. (3.22)

This is illustrated in Fig. 3.9. The desired range r might be in a secondaryrange ambiguity interval like indicated in the illustration. The range ambiguitiescause elevation angle ambiguities according to

εν = arcsin

(h

rν

). (3.23)

3.4. AMBIGUITIES AND THE CRITICAL ANTENNA SIZE 107

D r

r ' g = r g ( 0 )r g ( - 1 ) r g ( 1 )

e - 1e 0 e 1

r ' = r ( 0 )r ( - 1 ) r ( 1 )

g - 1 g 0 g 1

Figure 3.9: Range ambiguities

3.4.2 Azimuth ambiguities

Due to low PRF sampling, ambiguities in the Doppler frequency are introducedwith the period ∆F = 1/∆T resulting in a velocity ambiguity with the period∆v = λ/(2∆T ). So the Doppler filter for the radial velocity vr also receives signalsfrom scatterers with the radial velocities vµ = vr + µ∆v, µ ∈ Z. That means thatwe receive not only echoes from points on the Doppler cone with the cone angle αwith cos α = vr/V , but also from all those cones with the cone angles cos αµ = vµ/Vfor which |vµ| ≤ V , see Fig. 3.10.

Since cos α = cos ϕ cos ε, there follows for fixed depression angle ε, i.e. for a fixedrange, an ambiguity of the azimuth angle:

ϕµ = ± arccos( vµ

V cos ε

)= ± arccos

(vµr

V√

r2 − h2

). (3.24)

3.4.3 Full description of ambiguities

We can fit together the two types of ambiguities by inserting rν = r + ν∆r andvµ = vr + µ∆v into Eq. (3.6) and we get

(rg

ϕ±

)

νµ

=

√r2ν − h2

± arccos

(vµrν

V√

r2ν−h2

)

. (3.25)

The result is a grid on the earth surface, which is principally the same as in Fig.3.5, if the ISO-range and ISO-Doppler contours are drawn in the raster of ∆r and∆F .

An area bounded by the lines rν ≤ r ≤ rν+1 and vµ ≤ vr ≤ vµ+1 forms anambiguity facet on the earth surface. In the range-Doppler map generated by theprocessor, the contributions of all ambiguity facets are superposed, weighted by theantenna characteristics and the r−4-law.


D uD u

a 1

a 2

a 3

u 1u 2u 3- 1 + 1

x P

Figure 3.10: Azimuth ambiguities

For a side-looking configuration we regard the main ambiguity facet around theline perpendicular to the flight path, see Fig. 3.11. A rectangular in the (v, r)plane with the lateral lengths ∆v and ∆r corresponds to a circular ring segmentbounded by parabulae on the earth surface. The extensions in x- and y-directionare approximately given by

∆ξ ≈ r∆v

Vand ∆y ≈ ∆r

cos ε(3.26)

where ε is the depression angle at the reference point (√

r2 − h2, 0). From thiswe get the area of the ambiguity facet

Aamb ≈ ∆ξ∆y ≈ r

V cos ε∆v∆r =

r

V cos ε

λc0

4. (3.27)

Note that this expression is independent on the PRF and is given only by a fewbasic parameters.

As a numerical example we regard the following figures: r = 700 km, V = 7600m/s, λ = 3 cm, ε = 45 deg. It follows an ambiguity area of 293 km2. For earthobservation satellites this number can only be changed significantly by choosinganother wavelength since the other parameters are more or less given.

3.4. AMBIGUITIES AND THE CRITICAL ANTENNA SIZE 109

r y

xv

D rD v

D y

D xFigure 3.11: Ambiguity facet in the sidelooking situation

This ambiguity area covers a spatial angle as seen by the antenna of

Ωamb =Aamb

r2sin ε ≈ λc0

4rVtan ε. (3.28)

3.4.4 Minimum antenna size

For SAR as well as for MTI it is desired that the main beam illuminates only oneambiguity facet in order to exclude the superposition of secondary returns within themain beam. The spatial angle of the main beam of an antenna with the effective areaaant perpendicular to the look direction is approximately given by Ωant = λ2/aant.To fulfill the condition that the main beam has illuminate an area smaller or equalto the ambiguity facet From Eq. (3.28) we conclude directly

Ωant ≤ Ωamb (3.29)

λ2

aant

≤ λc0

4rVtan ε (3.30)

aant ≥ 4λrV

c0 tan ε(3.31)

For the before mentioned numerical example we get a minimum antenna areaof about aant=2.1 m2. It should be noted, that this area just covers the ambiguityfacet in the sense of Rayleigh’s resolution. In order to fit the whole main beam fromzero to zero into this area, the dimensions of the antenna have to be increased by afactor two, i. e. the antenna area has to be four times as large; for the example weget a minimum area of 8.5 m2.

This result is independent on the PRF. To get a feeling about minimum lengthand height for a given PRF, we recapitulate Eq. (3.8), where the minimum antennalength was derived to lx,min = 2V ∆T .


y

zJ 1

J 2l z

r 2r 1

e

l p

D qD r

e

Figure 3.12: Minimum antenna height

The minimum antenna extension perpendicular to LOS and flight path is easilyevaluated, see Fig. 3.12: The length ∆q between two ambiguous range radialsperpendicular to the LOS is ∆q = tan ε∆r. On the other hand, if lp is the projectedantenna length perpendicular to the LOS, in range r the extension of the main beamperpendicular to the LOS is δq = rλ/lp. It follows from the condition δq ≤ ∆q forthe minimum projected antenna length: lp,min = λr/(∆r tan ε). From these twominimum lengths we get the minimum projected antenna area:

aant ≥ lx,minlp,min =2V ∆Tλr

∆r tan ε(3.32)

=4λrV

c0 tan ε(3.33)

which is the same result as in Eq. (3.29). Of course, there are additional condi-tions for the antenna size, for example to get a sufficient antenna gain for the powerbudget, but also with respect to mechanical constraints.

Essence 3.4 If range and azimuth ambiguities shall be avoided by the illuminationby the antenna main beam, the antenna area has to fulfill the inequation Eq. (3.32).

3.5. SPACEBASED GEOMETRY 111

3.5 Spacebased geometry

The most of the studied properties of a radar moving over ground can be transferredto the spacebased situation. The most important difference is that the curvature ofthe earth has to be taken into account. In this section the geometry describing theflight path of the radar platform in relation to the curved earth surface is considered,followed by the analysis of ambiguous clutter directions due to discrete sampling inrange and azimuth.

3.5.1 Coordinate systems and transformations

1

If (xA, yA, zA) and (xB, yB, zB) are two cartesian coordinate systems with iden-

tical origin, any vector ~RA can be transformed into the B-system according to

~RB = LBA~RA, (3.34)

where LBA is an orthonormal rotational matrix, i.e. LtBALBA = Id and LAB =

L−1BA = Lt

BA. Two subsequent transformations LBA and LCB can be combined to asingle via LCA = LCBLBA.

LBA can be described by the three Euler angles:

LBA = L1(Φ)L2(Θ)L3(Ψ)

=

1 0 00 cosΦ sin Φ0 − sinΦ cosΦ

cosΘ 0 − sinΘ0 1 0

sinΘ 0 cosΘ

cosΨ sinΨ 0− sinΨ cosΨ 0

0 0 1

=

cosΘ cos Ψ cosΘ sinΨ − sinΘsinΦ sinΘ cos Ψ− cosΦ sinΨ sinΦ sinΘ sinΨ + cosΦ cosΨ sinΦ cos ΘcosΦ sinΘ cos Ψ + sin Φ sinΨ cos Φ sinΘ sinΨ− sinΦ cosΨ cos Φ cosΘ

(3.35)

where three simple rotations (Euler rotations) are performed subsequently: First,a rotation about the z-axis by the angle Ψ, for the second, a rotation about y-axisby the angle Θ, and for the third, a rotation about x-axis by the angle Φ2. Thesigns for the rotation about the y-axis seem to be inverted compared to the othertwo rotations. The reason is that all rotations are counted positive for left-turningif we look along the rotation axis to the origin.

If the vector ~rA(t) and the transformation matrix LBA(t) are time-dependent,we obtain

1Could be shifted to annex2In the literature, different variations of the Euler rotations can be found. We use a type

common in the field of aero-space engineering


~vB(t) =d

dt~rB(t)

=

(d

dtLBA(t)

)~rA(t) + LBA(t)

d

dt~rA(t)

= ΩBA(t)~rA(t) + LBA(t)~vA(t) (3.36)

Since LtBA(t)LBA(t) = Id, the time derivative of this expression is zero, resulting

in

ΩtBA(t)LBA(t) + Lt

BA(t)ΩBA(t) = 0 (3.37)

LtBA(t)ΩBA(t) = − (

LtBA(t)ΩBA(t)

)t(3.38)

that means the matrix LtBA(t)ΩBA(t) is antisymmetrical and ΩBA(t) can be writ-

ten in the form

ΩBA(t) = LBA(t)

0 −ωz(t) ωy(t)ωz(t) 0 −ωx(t)−ωy(t) ωx(t) 0

(3.39)

3.5.2 Coordinate systems for earth observation

Let the path of the satellite with respect to an Cartesian inertial coordinate system(xI , yI , zI) with its origin at the earth’s center and the zI-axis directed along the

earth rotational axis be described by a vector valued time function ~RI(T ) with time

derivative ~VI(T ), i.e. the history of the velocity vector.

An earth fixed coordinate system (xE, yE, zE) with origin and zE-axis identicalto the inertial system rotates with the angular velocity ωE with respect to the latter,which can be described by a time dependent transformation matrix LEI(T ) fromthe I-system to the E-system.

According to Eq. 3.35 the path of the satellite is transformed into the E-systemby ~RE(T ) = LEI(T )~RI(T ), the satellite velocity vector with respect to the E-systemis given by

~VE(T ) = ΩEI(T )~RI(T ) + LEI(T )~VI(T ) (3.40)

with the angular velocity matrix ΩEI(T ) being the time derivative of LEI(T ).

As illustrated in Fig. 3.13, we additionally introduce an antenna fixed Cartesiancoordinate system (xA, yA, zA) with its origin at the position ~RE(T ). An arbitrarylook direction is denoted in antenna coordinates by the unit vector ~u = (u, v, w)t


w E T

x I

y I

z I

x E

y E

z Ex Az A y A

p ER E ( T )

r E ( T )

Figure 3.13: Geometry for space based radar

originating in the center of the antenna coordinate system. For phased array sys-tems, the main beam direction can be steered by the phase shifters correspondingto a unit vector ~u0.

Let LAE(T ) and ΩAE(T ) be the rotational matrix and its time derivative de-scribing the orientation of the antenna with respect to the earth, and the angularvelocities, respectively.

3.5.3 Range and radial velocity of clutter points

For any point on the earth surface, denoted by a vector ~pE with length equal to theradius ρE of the earth, the look direction in the E-system is given by

~uE(T ) =~pE − ~RE(T )

‖~pE − ~RE(T )‖(3.41)

(3.42)

and its path, the look direction and its velocity vector in the A-system are


~rA(T ) = LAE(T )(~pE − ~RE(T )

)(3.43)

~uA(T ) =~rA(T )

‖~rA(T )‖ = LAE(T )~uE(T ) (3.44)

~vA(T ) = ΩAE(T )~rA(T )− LAE(T )~VE(T ) (3.45)

The radial velocity of this point is

vr(T ) =d

dT

∥∥∥~pE − ~RE(T )∥∥∥ (3.46)

= −⟨

~VE(T ), ~uE(T )⟩

(3.47)

= 〈~vA(T ), ~uA(T )〉 . (3.48)

In the following, we regard the geometrical situation at a fixed point of timeand don’t express the dependency on t explicitly. VE = ‖~VE‖ denotes the scalar

velocity of the platform in earth coordinates, RE = ‖~RE‖ the distance from theearth center to the satellite. Inverse to the previous derivations, we can determinethe point ~pE(~uA) where the LOS-vector ~uA hits the earth surface. With the LOS-vector ~uE = LEA~uA in the E-system the necessary condition is formulated as thequadratic equation

∥∥∥~RE + η~uE

∥∥∥2

= ρ2E (3.49)

which has to be solved with respect to the real parameter η. This equationis solvable only if the LOS-vector points into the silhouette of the earth, which issatisfied when

⟨~uE, ~RE

⟩≤ −

√R2

E − ρ2E (3.50)

which can be seen in Fig. ??. Otherwise the LOS-vector misses the earth. Let theset of all ’admissible directions’ characterized by Eq.(3.50)be denoted by U.

For an admissible direction ~uE we get from Eq.(3.49) the solutions

R2E + 2η

⟨~uE, ~RE

⟩+ η2 = ρ2

E (3.51)

η± = −⟨~uE, ~RE

⟩±

√⟨~uE, ~RE

⟩2

− (R2E − ρ2

E) (3.52)

Since η+ ≥ η− only the second solution η− is valid, since η+ is related to thepoint where the LOS hits the surface on the earth’s backside. The range r(~uA) to


the hit point on the front side is simply given by η−, i.e.

r(~uA) = −⟨~uA, ~RA

⟩−

√⟨~uA, ~RA

⟩2

− (R2E − ρ2

E) (3.53)

since the rotation of a vector pair doesn’t change their scalar product.

The radial velocity at this point can be deduced from Eq. (3.46):

vr(~uA) = −⟨

~VE,LEA ~uA

⟩(3.54)

Chapter 4

Synthetic Aperture Radar

In this chapter we will deal the basics of SAR as a pre-requisite for SAR movingtarget recognition. Starting from a tight expos’e of the SAR principles we willintroduce a general model of SAR signals gathered with a SAR system and deducedata processing techniques.

4.1 SAR history and applications

The technique of Synthetic Aperture Radar (SAR) permits the imaging of the earth’surface with high resolution performed from a moving platform (aircraft, satellite).In contrast to other remote sensing methods, for instance optical systems, advan-tages of SAR comprise on one hand the independency from weather and day light,and on the other hand the provision of additional features such as polarimetry andmulti-frequency. The imaging geometry of SAR is hereby different to the opticalgeometry, as the imaging plane is rather spanned by azimuth and range instead ofazimuth and elevation. By means of SAR-Interferometry for instance it is possibleto determine the third dimension and hence to chart the terrain height for eachresolution cell.

The SAR principle and its first demonstration date back into the fifties. C.A.Wiley of Goodyear Ltd exhibited the first system in 1952 and applied for a patent in1954. In the early days the data recording and imaging was performed with opticallenses due to a lack of digital signal processors. Only recently prevailed the digitalprocessor owing to achievable computation power and its palpable advantages. In1987 the first SAR-satellite called ‘SeaSAT’ was launched; today numerous satelliteswith outstanding imaging performances exist for civilian remote sensing as well asmilitary surveillance. It has been a long journey from the rather ‘prosaic’ beginningsof SAR towards today’s possible. Its evolution, however, is still rousingly advancingand is definitely not at the end!

117

118 CHAPTER 4. SYNTHETIC APERTURE RADAR

Glossary for this chapterα Chirp rateβ Doppler slope of azimuth chirpδua Beamwidth of real antennaϕ Phase functionλ Wavelengthλ0 Wavelength related to the reference frequencyσ2 Noise varianceω Angular velocity of aspect angleξ xPosition of the platform along motion axisξ0 Position at stationary phasea Complex Amplitudea(x) Reflectivitya(T ) Reflectivity scaled to slow timeA(K) Fourier transform of reflectivity

A(T ) Fourier transform of reflectivity as time functionb Bandwidth fast timeB Bandwidth slow timec0 Velocity of lightD(u) Two-way characteristicsf Radio frequencyf0 Reference frequencyF Doppler frequencyF0 Instantaneous Doppler frequency of the model signalFs = PRF Sampling frequency slow-time = Pulse repetition frequencyh(T ) Pulse response of azimuth compression filterH(F ) Transfer function of azimuth compression filterIM Indicator function of the set Mk Spatial frequency in the baseband domaink0 = 2π/λ0 Wavenumber related to the reference frequencykr = 2π/λ Wavenumber in the RF domainkx Spatial frequency along the apertureK Spatial frequencyK = (Kx, Ky, Kz)

t Vector of spatial frequenciesK Wave number set, k-setLa Azimuth extension of the antenna footprintlx Length of the real antennap(T ) Azimuth point spread functionP (F ) Fourier transform of p(T )r(ξ; x, ρ) Range historyr0(ξ; ρ) Range history of model signalrs Spatial pulse length

4.1. SAR HISTORY AND APPLICATIONS 119

s(.) Transmitted waveforms(kr, ξ; x, ρ) Signal of a scatterers0(kr, ξ; ρ) Model signals0(T ) Azimuth signalS0(F ) Fourier transform of azimuth signalsd1,d2(.) Signal in the (d1, d2)-domainTs Time duration of azimuth chirpT0 Time point of stationary phaseu(ξ; x, ρ) Direction historyu0(ξ; ρ) Direction history of model signalV Platform velocitywT (T ) Window in the time domainwF (F ) Window in the frequency domain(x, ρ) Coordinates of a scattererx(T ) Measured azimuth signal without noiseX(F ) Fourier transform of x(T )z(T ) Measured azimuth signal including noise

4.1.1 Application

Today, applications are mainly found in various areas of remote sensing:

• Geology, Cartography: Chartering of areas which are permanently under densecloud cover, surveying of geological structures, worldwide surveying of humansettlings and cultivated land.

• Oceanography: Analysis of ocean waves, conclusions on wind fields, researchon tsunamis.

• Hydrology: Surveying of floodings and soil moisture.

• Glaciology: Monitoring of ice in the arctic regions, icebergs, and snow coverage.

• Nautical shipping and economy: Recording of the sea states, monitoring ofshipping lanes, chartering of seaweed appearances.

• Agriculture and Forestry: Identification of cultivation types, control of fallow-ing schemes, monitoring of planting states, observation of grasshopper swarms.

• Emergency management: Recognition of surface changes, e.g. cost erosion,Detection of earth- and seaquakes, search and rescue of missing people.

• Environmental monitoring: In addition to the applications mentioned above:Recognition and imaging of pollution, e.g. oil films on water surfaces, charter-ing of damage to forests.


Figure 4.1: SAR-image of the Elbe river near Domitz at times of a disastrous flood-ing. Left: 29 August 2002, Right: 22 October 2003. c©Fraunhofer FHR

4.2 Introduction to imaging Radar

4.2.1 Imaging small scenes and objects

To understand imaging radar, it is favorable to start with techniques to image smallscenes and objects, like turntable imaging and inverse synthetic aperture radar(ISAR) . As a pre-requisite we will address the properties of the spatial Fouriertransform.

Spatial Fourier transform

If a(x) is a complex function of a spatial variable x, so

A(Kx) :=

∫a(x)e−jKxxdx (4.1)

is defined as the spatial Fourier transform of a(x). Kx is a spatial frequency measuredin the units rad/m stating the change of phase with the spatial progress. Eq.(4.1) represents the function a(x) as the superposition of spatial harmonic functionswKx(x) = ejKxx.

A function a(~r) in d spatial variables (d ∈ 1, 2, 3) given by the vector ~r istransformed by the multi-dimensional Fourier transform

4.2. INTRODUCTION TO IMAGING RADAR 121

A( ~K) :=

∫a(~r)e−j〈 ~K,~r〉d~r (4.2)

into the domain of spatial frequency vectors ~K. The back-transform is performedby

a(~r) =1

(2π)d

∫A( ~K)ej〈 ~K,~r〉d ~K. (4.3)

The spatial Fourier transform of a spatial Dirac function a(~r) = δ(~r − ~r0) leads

to the harmonic function e−j〈 ~K,~r0〉. The phase of this function changes most rapidlyin the direction of ~r0.

Vice versa, if A( ~K) = δ( ~K − ~K0) is a Dirac distribution in the wave-numberdomain, the back-transform represents the time harmonics of a wave propagating inthe direction of ~K0 and a wavelength λ = 2π/| ~K0|.

K xxK yy

Figure 4.2: Spatial Fourier correspondence

Fig. 4.2 illustrates this correspondence. All properties known from the Fourierpartners time - frequency can be transferred to the Fourier partners space / wavenum-ber. Spatial functions a(~r) vanishing outside a cube [~r0, ~r0 + ∆~r] with ∆~r = (∆x, ∆y, ∆z)t

lead to ’band limited’ functions A( ~K), which are - following the Nyquist/Shannontheorem - uniquely determined by sampling on a grid

~Kνµκ = ~K0 +

ν∆Kx

µ∆Ky

κ∆Kz

, (4.4)

if ∆Kx ≤ 2π/∆x, ∆Ky ≤ 2π/∆y, ∆Kz ≤ 2π/∆z.


Incomplete back-transform If the back transform is performed only over a sub-set K

aK(~r) =1

(2π)d

∫

KA( ~K)ej〈 ~K,~r〉d ~K, (4.5)

so aK(~r) is an - though unsharp - reconstruction of a(~r). Nevertheless, it is afamous property of the Fourier transform that cutting the transform for any subsetalways lead to the same image! This property is used for radar imaging with themulti look technique.

4.3 SAR Terminology

Figure 4.3: Synthetic Aperture Radar in the stripmap mode

We will explain some usual expressions used in the SAR terminology by means ofFig. 4.3. Ideally, the platform moves along a straight line. The antenna is mountedto achieve a side looking geometry , i.e. the beam points orthogonal to the motiondirection - or with a certain squint angle - to the earth surface. The look directionis tilted by the depression angle from the horizontal plane downwards. The area onground covered by the antenna main beam is called antenna footprint .

The aim of a SAR mission in the stripmap mode is to image a strip of a certainswath width slipped by the antenna footprint. Besides this mode the imaging of a

4.4. HEURISTICAL APPROACH 123

smaller area with finer resolution can be obtained by the spotlight mode where thebeam center is fixed to a point on the earth surface or the sliding mode where theantenna footprint moves with a lower (sometimes higher) velocity than the platform.

High resolution in range is obtained by modulated transmit signals and pulsecompression. We have to discriminate the range in line-of-sight, which is calledslant range and the range projected to the earth surface, the ground range. Theground range resolution of course is coarser than the slant range resolution.

To obtain sufficient resolution parallel to the flight direction the beamwidth ofthe antenna for real aperture imaging normally is not narrow enough, since in themicrowave region the antenna length would have to be some hundreds or thousandsof meters; to overcome this handicap the principle of synthetic aperture is used: dueto the motion of the platform it is possible to generate a synthetic aperture. By theplatform’s movement time is transformed into space - a synthetic array is spanned,the recorded echoes are processed similar as for a real array summing up the returnsafter compensating the phases. In this way a synthetic beam is generated effectinga fine resolution in azimuth direction.

4.4 Heuristical approach

We consider a synthetic array, as in section 3.3, generated by the motion of theplatform with velocity V at a pulse repetition interval ∆T . The positions of thissynthetic array are ξn = n∆x, n = −∞, . . . ,∞ with the spacing ∆x = V ∆T . InFig. 4.4 the principle of a synthetic array is illustrated.

Let a point scatterer be placed at the x-coordinate x = 0 at a distance ρ tothe flight path. By compensation of the phases effected by the varying distancesfrom the scatterer to the elements of the synthetic array, the received signals canbe focused to this point. In contrary to the situation in chapter 3 we don’t restrictthe length of the synthetic array according to the far field condition. So we have totake into account phase terms of higher orders than linear.

The distance of an element of this synthetic array at position ξ to the pointscatterer is given by r0(ξ; ρ) =

√ρ2 + ξ2. To focus a beam to the scatterer, the

phase ϕ(ξ; ρ) = −2krr0(ξ; ρ) has to be compensated by a ’phase shifter’. Since thearray is synthetic, this phase shift has to be performed in the processor.

If we take into account all elements of the synthetic array along a syntheticaperture of size LX , a beam is generated with a beamwidth in terms of directionalcosine of size δu = λ/(2Lx). The achieved resolution of the scene in direction of theflight (azimuth resolution) will be δx = ρδu = ρλ/(2Lx).

Clearly, this requires that the scatterer is illuminated during the whole timewhere the platform is moving along the synthetic aperture of length Lx. If theantenna has a fixed look direction - as it is the case for stripmap SAR - and has thereal beamwidth δua, only a restricted azimuth length La of the scene is illuminated,i.e. the azimuth length of the antenna footprint. This length is roughly given byLa = ρδua.


T r a v e l l i n g d i s t a n c e sc o m p e n s a t e d b y p h a s e s h i f t sr

F o c u se l l i p s e

d u a

L a = r d u a

d u a

xx

r 0 ( x ; r )

Figure 4.4: Focused SAR as synthetic antenna

For stripmap SAR, the distance the platform is flying from a scatterer enteringinto the mainbeam until its disappearance is La, too. So for this mode, the lengthof the synthetic array Lx is bounded by Lx ≤ La. Consequently, the achievableazimuth resolution for stripmap SAR is

δxmin = ρλ

2La

= ρλ

2ρδua

=λ

2δua

. (4.6)

Obviously, the azimuth resolution is independent on the range, since the length ofthe available synthetic aperture increases proportional to ρ. Because of the azimuthresolution being proportional to ρ and inverse proportional to Lx, the range ρ cancelsout.

The last interesting step is done by inserting the equality δua = λ/lx for thebeamwidth of an antenna with real aperture lx:

δxmin =λ

2δua

=λ

2λ/lx=

lx2

. (4.7)

Essence 4.1 The minimum obtainable azimuth resolution of a SAR in stripmapmode with an antenna of length lx is independent on the range and the wavelength

4.5. THE AZIMUTH SIGNAL 125

equal to the half length of the antenna.

There are some differences compared to a real array:

• The phase terms always are related to two-ways travelling. Compared to areal array always the double phase has to be inserted. As a consequence, forthe same aperture the half beamwidth is obtained. On the other hand, theminimum spatial sampling interval is reduced by the factor 2. Omnidirectionalantennas would have to be spaced at the raster λ/4. The phase centers ofdirected antennas with the real aperture lx are not allowed to move more thanlx/2 from pulse to pulse, see Eq.(3.15).

• The pattern of the synthetic antenna shows only to a one-way characteristics,since the way forward and backward had been regarded in the phase terms.Unfortunately, this leads to a higher side lobe level compared to the two-waypattern of a real array!

• Because of the long synthetic aperture a far field focusing is no longer possible.The quadratic term of the phase along the synthetic aperture is related to adistinct range. Like for an optical lens a high quality focusing is obtained onlyin a certain range interval, the depth of focus.

The depth of focus can be roughly estimated by the following consideration: The rangevariation for a scatterer at the slant range ρ is approximately r0(ξ; ρ)− ρ ≈ ξ2/(2ρ), thatof a scatterer at slant range ρ± d/2, where d will be the depth of focus, is approximatedin an analogue way. To keep the difference between the variations below γλ, where γ is afraction, e.g. γ = 1/8, the following inequation has to be fulfilled:

∣∣∣∣ξ2

2ρ− ξ2

2(ρ± d/2)

∣∣∣∣ ≤ γλ for all ξ ∈[−Lx

2,Lx

2

]. (4.8)

This leads to the condition

γλ ≥(

Lx

2

)2 ∣∣∣∣12ρ− 1

2(ρ− d/2)

∣∣∣∣ ≈(

Lx

2

)2 d

4ρ2(4.9)

and finally to the depth of focus:

dmax =16γλρ2

L2x

. (4.10)

4.5 The azimuth signal

In this section, we will investigate the signal in the slow time domain effected byan azimuthal reflectivity distribution along a line parallel to the flight path, thefocusing by a temporal filter and the resulting point spread function. We will startat a three-dimensional geometry which can be reduced to two dimensions, if theflight path is exactly a straight line.


4.5.1 Cylinder coordinates

x '

y

z

S e n s o r c o o r d i n a t e s

px

y

z

S c e n e c o o r d i n a t e s

r

J

E a r t h p l a n eh

V

r

R a d a r x = V T

Figure 4.5: Cylinder coordinates (x, ρ, ϑ) for the description of signals from pointscatterers

The three dimensional geometry is illustrated in Fig. 4.5. The phase center ofthe antenna is placed in the origin of the sensor coordinate system (x′, y, z), which isidentically orientated as the scene coordinate system (x, y, z). The platform moves

at an altitude h with the constant velocity ~V in direction of the x-axis, so the phasecenter is at the time T at the position ~R(T ) = (V T, 0, 0)t measured in the scenecoordinate system. We denote the x-component with ξ = V T . A scatterer at theposition ~p in sensor coordinates is seen by the radar at the vector ~r(ξ) whose lengthdepends only on the x-position and the slant range ρ =

√y2 + z2. Therefore it

is convenient to introduce a cylindric coordinate system (x, ρ, ϑ), where the thirdparameter, the elevation angle does not influence the range function. It merelymodulates the signal amplitude by the antenna characteristics.

So we can reduce the geometry to two dimensions (x, ρ) and (x′, ρ), see Fig. 4.6.

4.5.2 Range and direction history

Let the position of a fixed scatterer be at (x, ρ). Range and directional cosine tothis scatterer as functions of ξ are given by

r(ξ; x, ρ) =√

(x− ξ)2 + ρ2, u(ξ; x, ρ) =x− ξ

r(ξ; x, ρ). (4.11)

where u(ξ; x, ρ) = cos α(ξ; x, ρ) is the x-component of the LOS vector ~u(ξ; x, ρ)


M o t i o n o f S A R s e n s o r

P o i n t s c a t t e r e r

r

x ( T ) = V T

( x , r )

r ( x ; x , r )

u ( x ; x , r )

V

S e n s o r c o o r d i n a t e sS c e n e c o o r d i n a t e sx

a ( x ; x , r )

r

x '

Figure 4.6: SAR geometry in two dimensions

pointing to the scatterer. α is again the angle between x-axis and scatterer, the cone-angle. We will call the range- and directional cosine functions of ξ range history anddirectional cosine history.

If we regard a scatterer at x = 0, its histories are representative for any scattererwith arbitrary x-coordinate, since they only have to be shifted in the ξ-variableby x to get the histories of this scatterer. We define the representative historiesby r0(ξ; ρ) := r(ξ; 0, ρ) and u0(ξ; ρ) := u(ξ; 0, ρ), so r(ξ; x, ρ) = r0(ξ − x; ρ) andu(ξ; x, ρ) = u0(ξ − x; ρ). We get

r0(ξ; ρ) =√

ξ2 + ρ2, u0(ξ; ρ) = − ξ

r0(ξ; ρ). (4.12)

The later needed derivatives of these histories with respect to ξ are

r′0(ξ; ρ) = −u0(ξ; ρ), u′0(ξ; ρ) = − ρ2

r30(ξ; ρ)

(4.13)

r′′0(ξ; ρ) = −u′0(ξ; ρ) (4.14)

and they take following values at ξ = 0:

r′0(0; ρ) = 0, u′0(0; ρ) = −1

ρ, r′′0(0; ρ) =

1

ρ. (4.15)

Evaluation of the range history

For some rough estimates it is practical to use the second order Taylor approximationof the range history with respect to ξ:


r0(ξ; ρ) ≈ ρ +1

2r′′0(0; ρ)ξ2 = ρ +

1

2ρξ2 (4.16)

r′0(ξ; ρ) ≈ ξ

ρ. (4.17)

Note that these approximations are only valid for small angular deviations fromthe normal direction! The area of validity is constrained by the condition that thesinus (ξ/r0) and the tangens (ξ/ρ) of that angle can be regarded as equal - or thatthe exact range hyperbola Eq.(4.12) can be sufficiently approximated by the rangeparabola Eq.(4.16).

Essence 4.2 The azimuth signal of any point scatterer at (ξ, ρ) is a shifted versionof the model signal of a point scatterer at (0, ρ). The range hyperbola can be ap-proximated by the range parabola for sufficiently small angular deviations from thenormal direction. In this region, the directional cosine can be regarded as a linearfunction of the position ξ.

4.5.3 The azimuth chirp

The signal for this scatterer in dependence on the slow time (azimuth signal) fora fixed wave number kr is given by the range history and the modulation by thetwo-way antenna characteristics D(u):

s0(T ) = exp −j2krr0(V T ; ρ)D (u0(V T ; ρ)) . (4.18)

If the antenna is steered during the flight, as it is the case for example for the spot-light mode, the antenna characteristics has to be replaced by D(u0(V T ; ρ), us(V T ))where us(ξ) means the steering direction at platform position ξ. But, for the time,we remain at the stripmap mode, where the antenna look direction is fixed.

If the quadratic approximation is used, one gets the form

s0(T ) ≈ e−j2krρ exp

−jkr

(V T )2

ρ

D(u0(V T ; ρ)). (4.19)

We state that not only the transmit signal can have a chirp shape but also in theslow-time domain we get a chirp the so called azimuth chirp which is nearly lineararound the direction perpendicular to the flight direction.

The histories of range (= negative phase), Doppler frequency (= directionalcosine) and the real part of the model signal itself are depicted in Fig. 4.7. We willsee that analogue to the compression of the transmit signal this can be compressed,when the received azimuth signal is correlated with the azimuth chirp.


R a n g e h i s t o r y

D o p p l e r f r e q u e n c y

A z i m u t h c h i r p

T

T

T

Figure 4.7: Time histories of the range (phase), the angle, the Doppler frequencyand the azimuth chirp

4.5.4 Temporal parameters of the azimuth signal

Phase and Doppler history

From Eq.(4.18) the phase and Doppler histories of the model signal can be derived:

ϕ(ξ; ρ) = −2krr0(ξ; ρ) (4.20)

F0(ξ; ρ) = − 1

2π2krV r′0(ξ; ρ) (4.21)

= −2V

λr′0(ξ; ρ) (4.22)

= Fmaxu0(ξ; ρ) (4.23)

with Fmax = 2V/λ. In the second order approximation, this is

ϕ(ξ; ρ) ≈ −2kr(ρ +1

2ρξ2) (4.24)

F0(ξ; ρ) ≈ −Fmaxξ

ρ. (4.25)


Summary of the interdependencies between the variables

For fixed ρ and kr each of the four variables ξ,r0,u0 and F0 determines the state. Fora better survey we summarize all of the interdependencies in the following table.

Variable Position Range

ξ ≡ r0(ξ) =√

ρ2 + ξ2

r0 ξ(r0) = ±√

r20 − ρ2 ≡

u0 ξ(u0) = −ρ u0√1−u2

0

r0(u0) = ρ 1√1−u2

0

F0 ξ(F0) = ρ F0√F 2

max−F 20

r0(F0) = ρ Fmax√F 2

max−F 20

Direction Doppler

ξ u0(ξ) = − ξ√ρ2+ξ2

F0(ξ) = Fmaxξ√

ρ2+ξ2

r0 u0(r0) = ±√

r20−ρ2

r0F0(r0) = ∓Fmax

√r20−ρ2

r0

u0 ≡ F0(u0) = Fmaxu0

F0 u0(F0) = F0

Fmax≡

(4.26)

Duration of illumination Let u1 and u2 be the directional cosines boundingthe main beam. Then by Eq. (4.26) the illuminated ξ-interval - i.e. the azimuth-extension of the antenna footprint - at slant range ρ is

La = ρ(u2/

√1− u2

2 − u1/√

1− u21

). We get a temporal duration of

Ts = La/V. (4.27)

For narrow beams of width δua directed to the broadside, La can be approximatedby ρδua = ρλ/lx resulting in

Ts ≈ δuaρ

V=

ρλ

lxV. (4.28)

Note that ρV

is the inverse to the aspect angular velocity ω = V/ρ at the point ofclosest approach.

Doppler modulation rate From Eq. (4.20) the time derivative of the Doppler-history is

∂

∂TF0(V T ; ρ) = FmaxV u′0(V T ; ρ) (4.29)

= −FmaxVρ2

r30(V T ; ρ)

(4.30)

At time T = 0 the Doppler slope reaches the maximum magnitude and takes thevalue

β =∂

∂TF0(V T ; ρ)|T=0 = −Fmax

V

ρ= −2V 2

λρ; (4.31)


Doppler bandwidth The Doppler varies between Fmaxu1 and Fmaxu2, so thebandwidth is - in agreement with Eq. (3.15) - given by

B = Fmax(u2 − u1) = Fmaxδua =2V

λδua. (4.32)

For a real aperture of length lx we get

B ≈ 2V

λ

λ

lx=

V

lx/2(4.33)

This again shows that the spatial sampling interval in slow time must not belarger than lx/2. Since the bandwidth given in Eq. (4.33) is independent on thedirection of the main beam, the sampling condition is valid also for squinted beams.

Time-bandwidth product From Eq.(4.27) and (4.32) the time-bandwidth prod-uct is calculated to

BTs =La

V

2V

λδua =

2Laδua

λ=

La

lx/2. (4.34)

Since the time-bandwidth product equals the compression ratio, we can readthat the maximum compression ratio of the azimuth chirp can be expressed as thenumber of half real apertures fitting into the synthetic aperture. For narrow beamsdirected to broadside we get approximately

BTs ≈ 2L2

λρ=

2λρ

l2x. (4.35)

4.5.5 Numerical example

The following numerical example should give us a feeling for the order of magnitude ofthe parameters:

given deductedV 150 m/s δua 0,01 radλ 3 cm La 1,5 kmlx 3 m Ts 10 sρ 150 km β 10 Hertz/s

B 100 HertzBTs 1000

Essence 4.3 The azimuth signal has the form of a chirp over slow-time with ap-proximately linear frequency modulation around the direction perpendicular to theflight direction, multiplied with the two-way characteristics of the antenna in theinstantaneous direction to the scatterer passing by. It is called ’azimuth chirp’.


4.6 The principle of stationary phase

In the following treatment of SAR we will make use of the principle of stationaryphase. Because of the importance of this tool this section will be dedicated to thismethod.

4.6.1 The basic statement

The principle of stationary phase This states that for a sufficientlysmooth signal s(t) and a rapidly changing two times differentiable phaseϕ(t) with exactly one local extremum at the point t = t0 the integral∫

expjϕ(t)s(t)dt can be approximated by

∫expjϕ(t)s(t)dt ≈ C expjϕ(t0)s(t0) (4.36)

with C = 1+j√2α

and α = 12π

d2

dt2ϕ(t)|t=t0 .

———————————————————————————

If the phase function is strictly convex, α is positive and the above formula canbe applied directly. If it is strictly concave, α is negative and the expression for Cmay be replaced by C = 1−j√

2|α| . If there are more than one points of stationary phase

all contributions have to be summed up to replace the integral in Eq. 4.36.

4.6.2 Application to the Fourier transform

If a signal z(t) is the product of a function expjϕ(t) with a rapidly changing phaseterm with the above stated properties and a slowly varying complex signal s(t), theFourier transform Z(ω) of z(t) can be approximated by

Z(ω) =

∫exp−jωt expjϕ(t)s(t)dt (4.37)

=

∫expjΦ(t)s(t)dt (4.38)

≈ C(ω) expjΦ(t0)s(t0) (4.39)

with

Φ(t) = −ωt + ϕ(t) (4.40)

d

dtϕ(t)|t=t0 = ω (4.41)

C(ω) =1 + j√

2 12π

d2

dt2ϕ(t)|t=t0

. (4.42)

4.7. AZIMUTH COMPRESSION 133

4.6.3 Application to a SAR-relevant phase term

This result is often applied to SAR where the phase term ϕ(t) is of the form

ϕ(t) = −K√

ρ2 + βt2 (4.43)

where the square root represents the range history and K = 2kr for the monostaticcase. The related signal is

f(t; K, ρ, β) = exp−jK

√ρ2 + βt2

s(t). (4.44)

The application of Eq.(4.37) yields the Fourier transformation F (ω):

F (ω; K, ρ, β) ≈ C(ω) expjΦ(t0)s(t0). (4.45)

In the Fourier transform of the azimuth chirp Eq. (4.18)

S0(F ) =

∫e−j2πFT s0(T )dT (4.46)

=

∫e−j2πFT exp −j2krr0(V T ; ρ)D (u0(V T ; ρ)) dT (4.47)

the accumulated phase term - neglecting phase variations of the antenna pattern -is given by

φ(T ) = −2πFT + ϕ(T ) with ϕ(T ) = −2krr0(V T ; ρ). (4.48)

ϕ(T ) is the phase of the model signal.

Problem Determine the point of stationary phase T0, the phase φ(T0), andits second derivative at this point!

4.7 Azimuth compression

4.7.1 The flying radar as a time-invariant linear operator

The signal s0(T ) is a normalized model signal for a scatterer placed at x = 0. If a(x)denotes the reflectivity along a line parallel to the flight path at a distance ρ, and a(T ) =a(V T ) is the same function scaled to slow time, the measured azimuth signal of this linewill be of the form

z(T ) =∫

s0(T − T ′)a(T ′)dT ′ + n(T ) = x(T ) + n(T ) (4.49)

where x(T ) = (s0 ? a)(T ) is the deterministic part of z(T ) and n(T ) is the realization ofa white-noise process N(T ). Obviously the flying radar acts on the azimuth reflectivity


signal like a time-invariant linear filter, whose pulse response is s0(T ). The correspondingtransfer function is its Fourier-transform S0(F ). If the Fourier transforms of the signalsx(T ) and a(T ) are denoted by the corresponding capital letters, we get

X(F ) = S0(F )A(F ). (4.50)

Essence 4.4 The moving radar responds to the azimuth reflectivity like a time-invariantlinear filter with pulse response s0(T ) and transfer function S0(T ).

We know from the investigation of pulse compression, that a way to compress theazimuth signal will be to apply a time-invariant compression filter with the pulse responseh(T ) to the data according to y(T ) = (h ? z)(T ). The noise-free output of the filter isy(T ) = (h ? x)(T ), in the Doppler frequency region

Y (F ) = H(F )X(F ) = H(F )S0(F )A(F ). (4.51)

Analogue to the pulse compression in fast time, the point spread function in azimuth isthe response of the compression filter to the azimuth model signal:

p(T ) = (h ? s0)(T ), P (F ) = H(F )S0(F ). (4.52)

where the capital letters again denote the Fourier transforms.

4.7.2 The spectrum of the azimuth chirp

Rough estimate of the spectral power

A rough estimation of the square magnitude of S0(F ) can be obtained in the followingway: In the survey equations Eq.(4.26) the direction coupled to the Doppler frequency Fwas given by u0(F ) = F/Fmax. Here the power effected by the two-way antenna patternis P (u0) = |D(u0)|2. So we draw the simple conclusion

|S0(F )|2 ∼ |D(u0(F ))|2 =∣∣∣∣D

(u0

(F

Fmax

))∣∣∣∣2

. (4.53)

Again, the Doppler power spectrum is a scaled version of the square magnitude of the two-ways antenna pattern, as similar stated in Eq. (3.15). In those investigations we regardedthe clutter spectrum generated by many ground based scatterers at different azimuth po-sitions, observed over a short time. Here, we assumed a single scatterer observed for thewhole time of illumination. Obviously, the simultaneous echoes of the whole illuminatedground and the long term observation of the echo of one scatterer varying its directionover the whole beam width lead to the same result.

The usable bandwidth again corresponds to the main beam of the real antenna, leadingto the Doppler bandwidth given in Eq.(4.32 )


Azimuth point target reference spectrum

Up to now, we only presented statements about the power spectrum. For the compressionfilter the phase is much more important. Since it is difficult or impossible to evaluate theFourier transform of s0(T ) analytically, we apply the ’trick’ of the stationary phase.

dfafadadffaadfafsdsdf

Solution The necessary condition ddT φ(T )|T=T0 = 0 is equivalent to

0 = −2πF +d

dTϕ(T )|T=T0 (4.54)

F = F0(V T0; ρ). (4.55)

So the principle of stationary phase demands that the point of time T = T0 is foundwhere the instantaneous Doppler frequency F0(V T0; ρ) is equal to the frequency F of theFourier cell.

Using the survey Eq. (4.26) at this point the time, we get by inserting 2πF =d

dT ϕ(T )|T=T0 = 2krV u0 (the dependencies on the variables are omitted for better clar-ity):

φ(T0) = −2πFT0 + ϕ(T0) = −2krV T0u0 + ϕ(T0)= −2krξ0u0 − 2krr0 = 2krr0u

20 − 2krr0

= −2krr0(1− u20) = −2krρ

√1− u2

0

= −2krρ

√1−

(F

Fmax

)2

(4.56)

Further

d2

dT 2φ(T )|T=T0 = 2krV

2u′0 = −2krV2 ρ2

r30

= −2krV2(1− u2

0)3/2 1

ρ

= −2krV2

ρ

(1−

(F

Fmax

)2)3/2

. (4.57)

From these derivations follows the azimuth point target reference spectrum asstationary-phase approximation:

S0(F ) ≈ C(F ) exp

−j2krρ

√1−

(F

Fmax

)2

D

(F

Fmax

)(4.58)

with


C(F ) =1 + j√2α(F )

(4.59)

α(F ) =1

2π

2krV2

ρ

(1−

(F

Fmax

)2)3/2

(4.60)

Essence 4.5 The power spectrum of the azimuth chirp is a scaled version of thesquare magnitude of the two-way antenna characteristics Eq.( 4.53). The complexazimuth point target reference spectrum S0(F ) can be approximated by the principleof the stationary phase by Eq. (4.58) and (4.59). This approximation can be usedto perform the azimuth compression in the Doppler domain.

4.7.3 Filters for azimuth compression

Matched filter On the basis of the model Eq.(4.49) the SNR-optimum filter is -in analogy to the pulse compression in fast time - given by the time inverted complexconjugated azimuth model signal: hmf (T ) = s∗0(−T ). In this case, Hmf (F ) = S∗0(F ),and the point spread function is derived by

pmf (T ) = (s0♦s0)(T ), Pmf (F ) = |S0(F )|2, (4.61)

where ’♦’ means correlation. Eq.(4.51) shows that the response of this compression

filter to an arbitrary azimuth reflectivity with Fourier transform A(F ) is in the

Doppler frequency domain given by Ymf (F ) = |S0(F )|2A(F ).

To decrease the numerical effort, the matched filter mostly is applied in thefrequency domain. In this case, the transfer function S0(F ) has to be calculatedfor each ρ or for a variety of range strips each within the depth of focus zones.A fast method to evaluate S0(F ) without Fourier transformation is the method ofstationary phase which will be explained later.

Windowed matched filter In the time domain as well as in the Doppler fre-quency domain window functions can be applied to lower the sidelobes or to getindependent subbands for multilook processing. The corresponding pulse responseor transfer function, are

hw(T ) = s∗0(T )wT (T ), Hw(F ) = S∗0(F )wF (F ). (4.62)

Inverse filter Similar as for pulse compression, an inverse filter can be appliedaccording to

Hinv(F ) =

1S0(F )

if F ∈ F,

0 else.(4.63)

Here F is a set of frequency values with |S0(F )| > 0. The Fourier transform ofthe point spread function for the inverse filter is


Pinv(F ) = IF(F )1

S0(F )S0(F )(F ) = IF(F ), (4.64)

and the response to the azimuth reflectivity spectrum

Yinv(F ) = Pinv(F )A(F ) = IF(F )A(F ), (4.65)

where IF(F ) is the indicator function of F. If Yinv(F ) is regarded as the recon-

struction of A(F ) the inverse filter is obviously the most accurate one with the bestresults if F is as large as possible.

Essence 4.6 If the azimuth compression filter is chosen as the matched filter, theazimuth point spread function is the autocorrelation function of the azimuth chirp, itis the Fourier back transform of the square magnitude of the azimuth chirp spectrum.If the compression filter is chosen as the inverse filter, the point spread function isthe Fourier back transform of the Indicator function of the Doppler frequency band.

4.7.4 Azimuth resolution

In Eq. (4.6) it was already shown in a more or less heuristical manner that theachievable azimuth resolution in the stripmap mode is equal to δx = λ/δua.

This can be confirmed by inspecting the point spread function generated by theinverse filter. The usable frequency band F for non vanishing signal power willcorrespond to the main beam of the antenna, so it will be a frequency interval oflength B given in Eq. (4.32) and (4.33). So the Fourier back transform will yield ansi-function as point spread function with the Rayleigh width δT = 1/B. This leadsto the azimuth resolution

δx = δTV =V

B= V

λ

2V δua

=λ

2δua

= lx/2. (4.66)

This result is independent on the focusing direction of the real antenna.

4.7.5 Spotlight and sliding mode

As we have seen in section 4.7.4, for the stripmap mode the antenna must not belonger then twice the desired azimuth resolution. That means that a very shortantenna has to be used to obtain a fine resolution. But this is a problem in severalaspects: The antenna gain will be low leading to a poor SNR. At the same timethe Nyquist condition for azimuth sampling (at least one pulse per movement of thehalf antenna length) requires a high PRF leading to a high data rate and perhapsrange ambiguities.


Spotlight mode

For this problems, the spotlight mode affords relief: The antenna beam is steeredto a fixed point at the earth. In this way the synthetic aperture can be elongatednearly arbitrarily to achieve fine resolution. The instantaneous Doppler bandwidthis still given by B = 2V/lx and the PRF has to be adopted to this bandwidth, whilethe center Doppler frequency changes according to the varying look direction. Adisadvantage of the spotlight mode is the restriction of the azimuth length of thescene to the width of the antenna footprint.

Sliding mode

A continuous intermediate state between stripmap and spotlight mode can be ob-tained by the sliding mode. Here the center point of the antenna footprint movesalong the earth surface with a velocity which is normally between zero and the plat-form velocity, but sometimes it is also desirable to let the footprint slide over theearth with a higher velocity or even opposite to the flight direction. In all these cases,a feasible balance between azimuth extension of the scene and azimuth resolutioncan be achieved.

4.8 The signal in two dimensions

L x

x

r

d j

r 0 ( x ; r )

x m a xx

2 k r

P o i n t s c a t t e r e r l / 2

L a

k x- x m a x

Figure 4.8: Azimuth chirp and synthetic aperture

At this stage of the chapter we have regarded only one-dimensional signal pro-cessing, we have seen the similarity of pulse compression in the fast-time domain andthe compression of the azimuth chirp in the slow-time domain. The two dimensions

4.8. THE SIGNAL IN TWO DIMENSIONS 139

cannot be separated: e.g. after range compression the echoes of a point scatterer arenot aligned in a straight row of the data matrix, but will migrate through the rangelines due to the hyperbolic variation of range along slow time (range curvature). Soit is necessary to analyze the signal simultaneously in two dimensions.

Since there are several distinct two-dimensional domains, it is not feasible tocreate for each of these domains a new symbol for the signal. For this reason wewill use superscripts indicating the concerned domain, like sξ,kr indicating that thesignal is regarded in the (ξ, kr)-domain.

4.8.1 The signal in the ξ − r domain

We go back to the original raw data before pre-processing to the normal form. Ifthe transmit signal s(r) - written as function of the spatial variable r - is used, thefollowing raw data signal in the (ξ, r)-plane from a point scatterer at the position(x, ρ) is got:

sξ,r(ξ, r; x, ρ) = s (r − r0(ξ − x; ρ)) (4.67)

× exp −j2k0r0(ξ − x; ρ)D (u0(ξ − x; ρ)) (4.68)

where k0 again is the wave number for the reference frequency.

Fresnel figures If the transmit signal is a chirp, written in the spatial variable as

s(r) = rect

(r

rs

)exp

jαπ

(r

rs

)2

, (4.69)

where the temporal duration of the signal ts is transferred to a spatial extensionrs = tsc0/2, the signal takes the form

sξ,r(ξ, r; x, ρ) = rect

(r − r0(ξ − x; ρ)

rs

)exp

jαπ

(r − r0(ξ − x; ρ)

rs

)2

× exp −j2k0r0(ξ − x; ρ)D (u0(ξ − x; ρ)) (4.70)

If the antenna characteristics is assumed to be real, the phase of this expressionis

ϕ(ξ, r) = απ

(r − r0(ξ − x; ρ)

rs

)2

− 2k0r0(ξ − x; ρ)

≈ απ

(r − ρ

rs

)2

− 2k0

(ρ +

1

2ρ(ξ − x)2

)(4.71)

= −2k0ρ +απ

r2s

(r − ρ)2 − 2k0

2ρ(ξ − x)2 (4.72)


where the approximation is valid in the vicinity of (x, ρ). In the two-dimensional rep-resentation, the range shifted transmitted chirp is multiplied with the azimuth chirpin the ξ dimension. Since the exponents, which are both approximately quadratic intheir respective variables add, the lines of equal phases are approximately ellipses, ifthe signs of the quadratic terms coincide, otherwise they are hyperbolas. Since theazimuth chirp is always a down-chirp, ellipses appear if the transmit signal is alsoa down chirp, and hyperbolas, otherwise. In Fig. 4.9 the real part of a simulatedsignal at x = 0 is presented.

ξ (m)

r (m

)

−100 −80 −60 −40 −20 0 20 40 60 80 100200

220

240

260

280

300

320

340

360

380

400

Figure 4.9: Model signal in the ξ − r-plane (real part). Note the deformation alongthe azimuth coordinate due to the range hyperbola.

Fig. 4.10 shows the raw data of a real SAR measurement, exhibiting these ellipsepattern accompanying every dominant point scatterer.

Around a fixed point targets we observe in the first case a sequence of concen-tric ellipses, which have the same sequence of radius as the fringes of a Fresnel zoneplate which are the cuts of concentric sphere shells with equidistantly growing radius.These interference fringes are known from holography. Since such Fresnel interfer-ence patterns can be focused using coherent light, can SAR raw data, exposed on aphotographic film, be focused to an image using optical methods, as it was done inthe early period of SAR processing.

4.8.2 The signal in the ξ − kr domain

After the application of a Fourier transform from the r- to the kr-dimension followedby an inverse filtering, the signal assumes the form


Figure 4.10: Real raw data in the azimuth-range domain

sξ,kr(ξ, kr; x, ρ) = exp −j2krr0(ξ; ρ)D (u0(ξ; ρ)) . (4.73)

This equation expresses the same as Eq.(4.18) apart from the use of the spatialvariable ξ instead of the slow time variable T . Fig. 4.11 shows the model signal inthe kr − ξ domain.

4.8.3 The signal in the kx − kr domain

A spatial Fourier transform along ξ yields

skx,kr(kx, kr; x, ρ) = e−jkxxskr,kx(kr, kx; 0, ρ)

= e−jkxx

∫e−jkxξ exp −j2krr0(ξ; ρ)D (u0(ξ; ρ)) dξ(4.74)

(4.75)

Again, the principle of stationary phase can be applied. Not to repeat the deriva-tions of section 4.7.2 we follow a little different way using geometrical interpretations.

The complete phase is given by

φ(ξ) = −kxξ − 2krr0(ξ; ρ). (4.76)

For each kx, a position ξ0 has to be determined where the derivation of φ(ξ) iszero, resulting in


ξ (m)

k r (ra

d/m

)

−100 −80 −60 −40 −20 0 20 40 60 80 100

16

17

18

19

20

21

22

23

24

25

26

Figure 4.11: Model signal in the ξ − kr-plane (real part).

0 = −kx − 2krr′0(ξ0; ρ) (4.77)

=⇒ kx = 2kru0(ξ0; ρ) (4.78)

(4.79)

The right side of this equation is the phase gradient along the synthetic aperture.Obviously, that position ξ0 is searched for which this phase gradient is equal to kx.

Since the absolute value of the phase gradient is always lower or equal to 2kr,we can characterize kx as the abscissa component of a vector k with length 2kr inthe k-domain (kx, kρ), see Fig. 4.12 top. In this plane, kx = 2kr cos β where β is theangle from the kx-axis to the vector.

According to Eq.(4.76) the phase φ(ξ) may be written as φ(ξ) = −2krg(ξ) whereg(ξ) = ξ cos β + r0(ξ; ρ) can be interpreted as a way length, see Fig. 4.13. Forarbitrary ξ the part r0(ξ; ρ) is the distance between the points A and B in thedrawing. The distance between B and C is equal to |ξ| cos β. Since we regard theleft upper quadrant, the ξ-values are negative, so g(ξ) will be the difference between

the length of the lines ~AB and ~BC, i.e. the length of the line ~BC ′. For varying ξthe point C ′ travels along the marked path, the minimum distance to the point A isreached at C = C0, where the line of sight and the line through the origin and thepoints C defined by the ratio kx/(2kr) are perpendicular.

This is easily proven by inspecting Eq.(4.77). From this follows

cos β = u0(ξ0; ρ) = cos α (4.80)


k x

k r

k x

2 k r

b x

r

b

rr 0

x 0

Figure 4.12: Correspondence from the spatial domain to the domain of spatial fre-quencies.

where α is the cone-angle corresponding to the point of the stationary phase.So the direction of the defined k-vector and the look direction at the point of thestationary phase coincide. From this we get immediately the values for ξ0 andr0(ξ0; ρ):

ξ0 = −ρ cot β, r0(ξ0; ρ) = ρ/ sin β. (4.81)

Also the value of g(ξ0) is easily obtained:

g(ξ0) = ρ sin β

= ρ√

1− cos2 β

= ρ

√1− k2

x

4k2r

(4.82)

and the corresponding phase is

φ(ξ0) = −2krg(ξ0) = −ρ√

4k2r − k2

x. (4.83)

Fig. 4.13 bottom shows the behavior of the distance g(ξ0) in dependence on thevariable kx: In the right bottom quadrant the coordinate system (kx, kρ) is sketched.A specific value of kx defines the angle β in the circle with radius 2kr. In the leftupper quadrant we see in the spatial coordinate system (x, ρ) the way of the pointC0 corresponding to the stationary phase for the actual value of kx. Because of the


orthogonal angle to the line of sight this point moves on the half circle of Thalesbetween the origin and the point A. The length of the cathetus emanating from Adefines the stationary phase.

Essence 4.7 Using the principle of the stationary phase, the phase of the Fouriertransform of the azimuth chirp in dependence on the spatial frequency along theazimuth can be deduced to the cathetus length in the half circle of Thales as illustratedin Fig. 4.13.

For the final formula we still need the second derivative of the phase:

φ′′(ξ0) = −2krr′′0(ξ0; ρ)

= −2krρ2

r30(ξ0; ρ)

= −2krρ2

(ρ/ sin α)3

= −2krsin3 α

ρ

= −2kr

ρ

(1− k2

x

4k2r

)3/2

= − 1

ρ4k2r

(4k2

r − k2x

)3/2. (4.84)

The final result for the signal in the kx − kr domain now is

skx,kr(kx, kr; x, ρ) ≈ C(kx) exp−j

(kxx + ρ

√4k2

r − k2x

)D

(kx

2kr

)(4.85)

with

C(kx) = (1 + j)

√πρ4k2

r

(4k2r − k2

x)3/2

. (4.86)

Essence 4.8 Using the principle of the stationary phase, the signal in the kx − kr-domain can be approximated by Eq.(4.85) and (4.86).

Fig. 4.14 shows the model signal in the double-wavenumber domain.

4.8.4 k-set and point spread function

We return to the three dimensional model and regard a reference point, which isat the coordinates ~p0, further a scattering center at the coordinates ~p = ~p0 + ~p ′ in


the neighborhood of the reference point. The platform motion is described by thevector ~R(ξ). The range history to this scatterer is now

r(ξ; ~p) = ‖~R(ξ)− ~p‖ (4.87)

= ‖~R(ξ)− ~p0 − ~p ′‖ (4.88)

≈ r(ξ; ~p0) + 〈~u0(ξ), ~p′〉 (4.89)

with the direction vector ~u0(ξ) =~R(ξ)−~p0

|~R(ξ)−~p0| pointing from the radar to the position

~p0. The signal is given by

s(ξ, kr; ~p′) = e−j2krr(ξ;~p0) exp−j2kr 〈~u0(ξ), ~p

′〉D(2kr~u0(ξ)), (4.90)

with the two-ways characteristics D(2kr~u) written as function of the double k-vectorand the measurements (without noise) by

z(ξ, kr) =

∫a(~p ′)s(ξ, kr; ~p

′)d~p ′, (4.91)

where a(~p ′) describes the reflectivity in the vicinity of the reference point. The firstexponential function in Eq. (4.90) is equal for all ~p ′, the phase can be motion-corrected according to r(ξ; ~p0) by

zmc(ξ, kr) := ej2krr(ξ;~p0)z(ξ, kr) (4.92)

=

∫a(~p ′)smc(ξ, kr; ~p

′)d~p ′ (4.93)

with smc(ξ, kr; ~p′) = ej2krr(ξ;~p0)s(ξ, kr;~r

′) (4.94)

= exp−j2kr 〈~u0(ξ), ~p′〉D(2kr~u0(ξ))). (4.95)

The SNR optimum processor correlates with the model signal:

a(~p ′) =

∫ ∫zmc(ξ, kr) expj2kr 〈~u0(ξ), ~p

′〉D(2kr~u0(ξ)))dξdkr∫ ∫ |D(2kr~u0(ξ)))|2dξdkr

. (4.96)

By variation of the wavenumber in the interval [kr1, kr2] and by variation of theantenna position ξ ∈ [ξ1, ξ2] a k-set

K = 2kr~u0(ξ) : kr ∈ [kr1, kr2], ξ ∈ [ξ1, ξ2] (4.97)

is generated and Eq. (4.96) using the variable ~K := 2kr~u0 can be written as:

a(~p ′) =

∫K ej〈 ~K,~p ′〉zmc( ~K)D( ~K)d ~K∫

K |D( ~K)|2d ~K. (4.98)

The k-set is a circular ring segment as illustrated in Fig. 4.15. If the antennacharacteristics is assumed to be rectangular and a ξ-area is admitted which covers the


bounds of the main beam, the k-set can be restricted to the ring segment referringto the main beam. If this restricted k-set is denoted by K0, the equation Eq. (4.98)becomes

a(~p ′) =

∫K0

ej〈 ~K,~p ′〉zmc( ~K)d ~K∫K0

d ~K. (4.99)

Obviously the reconstruction is obtained by the Fourier transformation of themotion compensated data over the k-set.

If for zmc the motion compensated signal of a point scatterer at ~p = ~p0 is inserted,we get the point spread function

p(~p ′) =

∫K0

ej〈 ~K,~p ′〉d ~K∫K0

d ~K. (4.100)

Essence 4.9 The point spread function is the Fourier transform of the indicatorfunction of the k-set.

4.9 SAR-Processing

4.9.1 Generic SAR processor

The measured signal z(ξ, r) consists of the superposition of the echoes of many pointscatterers with the reflectivity at a(x, ρ):

z(ξ, r) =

∫ ∫a(x, ρ)s0(ξ, r; x, ρ)dxdρ, (4.101)

or, after transformation into the (ξ, kr)-domain,

zξ,kr(kr, ξ) =

∫ ∫a(x, ρ)sξ,kr(ξ, kr; x, ρ)dxdρ. (4.102)

Principally, the raw data could be processed to an image optimally, if for eachpossible position of a point scatterer the two-dimensional normalized matched filterfor the model signal in Eq. (4.67), and (4.73), respectively, would be applied:

a(x, ρ) =

∫ ∫s∗(ξ, r; x, ρ)z(ξ, r)drdξ∫ ∫ |s(ξ, r; x, ρ)|2drdξ

(4.103)

or

a(x, ρ) =

∫ ∫sξ,kr∗(ξ, kr; x, ρ)zξ,kr(ξ, kr)drdξ∫ ∫ |sξ,kr(ξ, kr; x, ρ)|2drdξ

. (4.104)

The computational effort would be much to high. The first approach is to applyseparated processing, i.e. range compression and azimuth compression are appliedone after the other.

4.9. SAR-PROCESSING 147

In Fig. 4.17 the essential building blocks of such a separated processor areillustrated.

4.9.2 Range-Doppler processor

The range-Doppler processor starts with the classical range compression as a rule as’fast correlation’ performed in the range frequency domain. Now the effect that thesignal runs along a hyperbolic curve through the range cells - the range curvatureproblem - has to be solved.

Curvature correction in the kx-domain

There is a possibility to reduce the numerical effort of the described interpolationalong the range hyperbolas: First the data are transformed by an azimuth FFT intothe kx domain. The signals of scatterers with the same ρ but varying x-coordinatesare shifted in slow time; but by the Fourier transform the pathes in the kx-domainare overlayed1. For this reason, the range curvature correction can be performed inthe kx-domain simultaneously.

From Eq.(4.83) the phase for a certain kx can be transferred to the range:

r(ξ0, ρ) = − 1

2kr

φ(ξ0) =ρ√

4k2r − k2

x

2kr

. (4.105)

From this, we get the range variation over kx. All echoes are shifted back tothe value ρ, see Fig. 4.19. This can be performed either by interpolation or by anapproximate range curvature relating to the mean distance which can be executedin the range frequency domain. The result for the last possibility is depicted in Fig.4.19. Only for the reference-ρ the correction is perfect. For smaller or larger slantranges errors remain which can be neglected for small swathwidths. In the othercase, a procedure called differential range correction has to be performed. Finally,the azimuth compression is completed by fast convolution along the rows.

4.9.3 Range-migration processor

The range-Doppler processing is based on a classical primary range compression.Since all transformations are linear, the order of application can be altered. Let usstart with a pre-processing to the normal form (range FFT and inverse filtering)and an azimuth Fourier transform.

The signal of a point scatterer at (x, ρ) in the kx, kr-domain was calculated inEq.(4.85) using the principle of stationary phase to

1Shift in time means multiplication by an exponential in the frequency domain!


skx,kr(kx, kr; x, ρ) ≈ C(kx) exp−j

(kxx + ρ

√4k2

r − k2x

)D

(kx

2kr

),(4.106)

so the data will be the superposition of these signals with the azimuth reflectivity:

zkx,kr(kx, kr) =

∫ ∫skx,kr(kx, kr; x, ρ)a(x, ρ)dxdρ (4.107)

≈ C(kx)

∫ ∫exp

−j

(kxx + ρ

√4k2

r − k2x

)D

(kx

2kr

)a(x, ρ)dxdρ.

Stolt-interpolation

If we set kρ :=√

4k2r − k2

x and perform the variable substitution

zkx,kρ(kx, kρ) := zkx,kr(kx, 2√

k2x + k2

ρ) (4.108)

we get in the new (kx, kρ)-domain:

zkx,kρ(kx, kρ) ≈ C(kx)D

(kx√

k2x + k2

ρ

)∫ ∫exp −j (kxx + kρρ)) a(x, ρ)dxdρ.

= C(kx)D

(kx√

k2x + k2

ρ

)A(kx, kρ) (4.109)

with A(kx, kρ) being the two-dimensional Fourier transformation of the reflec-tivity. This substitution has changed the data in the (kx, kr)-domain into the 2DFourier transformation of the reflectivity distribution, modulated by the antennacharacteristics.

Similar as for the polar reformatting the grid in (kx, kr) has to be transferredinto a grid in the (kx, kρ)-domain. For this the Stolt-interpolation is applied. At theend only a two-dimensional inverse Fourier transformation has to be performed toget the focused SAR image.

4.9.4 Back-projection processor

The back projection processor starts with a usual range compression. A two-dimensionalarray relating to the later image in the x, ρ-domain is initially filled with zeros. Foreach pulse the data are distributed along the related range rings and accumulated,see Fig. 4.21).

4.10. THE RADAR EQUATION FOR SAR IMAGING 149

4.10 The radar equation for SAR imaging

In the power budget Eq. (1.196) we got an SNR of

γ =PtxGtxGrxλ

2q

(4π)3r4kTbnf

η (4.110)

where Ptx is the transmit power, Gtx and Grx are the gains of the transmit andreceive antennas, λ the wavelength, q the radar cross section of the target, r therange, k the Boltzman constant, T the absolute temperature of the receiver, b thebandwidth, nf the noise figure of the receiver chain and η the accumulated efficiencyof the whole system. This forms the basis for completion of the power budget forSAR imaging.

Instead of a radar cross section q for SAR purposes we take the reflectivitydensity σ0, defined by q = σ0A being the cross section of an area A with constantσ0. For SAR we chose the area A = δx × δr of a resolution cell, so q = σ0δx × δrcan be inserted into Eq. (1.196).

The bandwidth b is the inverse illumination time Tcompld of the patch where d isthe duty factor and Tcompl is the observation time which is related to the syntheticaperture via Tcompl = Lx/V . On the other hand, the azimuth resolution is given byδx = rλ/(2Lx) resulting in

b =1

Tcompld=

V

Lxd=

2V δx

rλd. (4.111)

By inserting also this expression into Eq. (1.196) we get the final result, theradar equation for SAR:

γ =PtxGtxGrxλ

3σ0δrd

(4π)3r3kTnf2Vη. (4.112)

The noise-equivalent sigma-zero (NESZ) is defined as that σ0 which leads tothe same power density as the noise. From this we can solve Eq. (4.112) to themaximum range where a specified NESZ is obtained:

rmax = 3

√PtxGtxGrxλ3σ0δrd

NESZ(4π)3kTnf2Vη (4.113)


x

r

bx 0

A

B

C

C '

C 0

x

r

bx 0

A

k x

2 k r

k r

k x

b

Figure 4.13: Geometrical interpretation of the stationary phase


kx (rad/m)

k r (ra

d/m

)

−15 −10 −5 0 5 10 15

16

17

18

19

20

21

22

23

24

25

26

Figure 4.14: Model signal in the kr − kx-plane (real part).

k - s e t k m i n

k m a x

Figure 4.15: The k-set in three dimensions generated by a stripmap SAR at a refer-ence point at the surface


A z i m u t h

Entfernung

R e f e r e n c e f u n c t i o n r a n g e

A z i m u t h

Range

R e f e r e n c e f u n c t i o n a z i m u t h

S

Slant range

A z i m u t h

P u l s e c o m p r e s s i o n

A z i m u t h c o m p r e s s i o n

C o m p l e x S A R i m a g e

Figure 4.16: Principle range Doppler processing

Figure 4.17: Principal steps in SAR processing


R a n g e F F T

xr

xk r

xk r

R a n g e R e f e r e n c e

A z i m u t h I F F T

xA z i m u t hF F T

k x

r

C u r v a t u r eC o r r e c t i o n

k x

A z i m u t h R e f e r e n z

r

R a n g e I F F T

xr

k x

r r

Figure 4.18: Range Doppler processor

k x

R R c o rr

r

k x k x

Figure 4.19: Range curvature over kx; Center: After correction; right: perfect cor-rection

R a n g e F F T

xr

xk r

xk r

R a n g e r e f e r e n c e

x

k x

S t o l t -I n t e r p o l

A z i m u t h F F T

k r k xk r

2 D I F F T

r

Figure 4.20: Range migration (omega-k) processor


R a n g e F F T

xr

xk r

xk r

R a n g e R e f e r e n c e

R a n g e I F F T

xr

xy

A c c u m u l a t i o n t of i n a l i m a g e

F o r e a c h c o l u m n

c o n t r i b u t i o n t o i m a g e

b y i n t e r p o l a t i o n

Figure 4.21: Back projection processor

Appendix A

Mathematical Annex

A.1 Complex linear algebra

A.1.1 Products of matrices

The vector space Cn is defined as the set of n-dimensional with elements in C, where

Cn :=

a1

· · ·an

: a1, · · · , an ∈ C

. (A.1)

Cn×m describes the set of all n × m matrices with complex coefficients. Thematrix product is defined as common.

The superscript ∗ denotes complex conjugate, such as

a11 · · · a1m

· · · · · · · · ·an1 · · · anm

∗

=

a∗11 · · · a∗n1

· · · · · · · · ·a∗1m · · · a∗nm

. (A.2)

For matrices A and B yields

(AB)∗ = B∗A∗ (A.3)

and for invertible quadratic matrices

(AB)−1 = B−1A−1. (A.4)

Please note that for vectors a,b ∈ Cn the expression a∗b represents a complexnumber and ab∗ a complex matrix with rank one (a so-called dyad).

The trace of matrices is defined as the sum of all diagonal elements. It yields:

trace(AB) = trace(BA). (A.5)

155

156 APPENDIX A. MATHEMATICAL ANNEX

A.1.2 Scalar product and the inequality of Cauchy-Schwarz

The scalar product (inner product, dot product) a,b ∈ Cn is defined as

< a,b >:= b∗a =n∑

ν=1

b∗νaν , (A.6)

and the L2-norm as

‖a‖ :=√

< a, a > =

√√√√n∑

ν=1

|aν |2. (A.7)

A vector a ∈ Cn is called a unit vector when ‖a‖ = 1; Two vectors a,b ∈ Cn

are called orthogonal when < a,b >= 0. Two orthogonal unit vectors are denotedorthonormal.

The correlation coefficient of two non-zero vectors a,b ∈ Cn is given by

ρ(a,b) :=< a,b >

‖a‖‖b‖ . (A.8)

Using Cauchy-Schwarz’ inequality it follows that |ρ(a,b)| ≤ 1, with ’=’ if bothvectors are linearly dependent.

A.1.3 Linear Subspaces

A linear subspace (LSS) U in Cn is a non-zero subset which is closed against thesummation and multiplication with complex scalars.

If s1, · · · , sK are vectors in Cn, the linear hull of these vectors is defined by theset of all linear combinations:

Lin (s1, · · · , sK) := a1s1 + · · ·+ aKsK : a1, · · · , aK ∈ C . (A.9)

The dimension od an LSS is the minimum number of vectors which span this space:

dim(U) := min K : It exist s1, · · · , sK ∈ Cn mit Lin (s1, · · · , sK) = U . (A.10)

s1, · · · , sK are linear independent, if dim(Lin (s1, · · · , sK)) = K. In this case, thevectors constitute a basis of this space.

A system u1, · · · ,uK is called an orthogonal system if these vectors are mutuallyorthogonal, and called orthonormal system if they are additionally unit vectors:

u∗kul = δkl. (A.11)

A orthonormal system u1, · · · ,uK , combined in the n×K-matrix U = (u1, · · · ,uK),consists the property

U∗U = IK , (A.12)

where IK denotes the K ×K identity matrix. Every orthonormal system is linearlyindependent and the spanned space constitutes a basis.

Let U be A LSS of Cn. Then U⊥ denotes the LSS of all vectors, which areorthogonal to each vector of U .

A.1. COMPLEX LINEAR ALGEBRA 157

A.1.4 Properties of Matrices

Hermitian Matrices

A matrix A ∈ Cn×m is called Hermitian if and only if A∗ = A.

It is called positive semidefinite if each quadratic form x∗Ax is non-negative:

x∗Ax ≥ 0 for each x ∈ Cn. (A.13)

It is called positive definite if and only if each of those quadratic form is greaterthan zero for arbitrary vectors x ∈ Cn with x 6= 0.

Unitary Matrices

A n× n-matrix U of n vectors of a orthonormal system is called unitary. It yields:

U∗U = U∗ = In, (A.14)

U∗ = U−1. (A.15)

Projectors

A n × n-matrix PU is called projector or projection matrix if PUPU = PU . Let Uthe subspace spanned by the columns of P . Then it yields:

PUs = s, if s ∈ U, (A.16)

PUs = 0, if s ∈ U⊥. (A.17)

PU⊥ := I−PU is also a projector, namely onto the orthogonal space to U . It yields:

PU⊥PU = 0. (A.18)

Futher, it yields that for each s ∈ Cn:

s = PUs + PU⊥s. (A.19)

The two vectors sU := PUs and sU⊥ := PU⊥s are orthogonal to each other and withthe generalized Pythagoras’ theorem:

‖s‖2 = ‖sU‖2 + ‖sU⊥‖2. (A.20)

Eigen-Decomposition

A vector v is called eigenvector to the eigenvalue λ of the matrix A, if

Av = λv. (A.21)


Eigenvectors to different eigenvalues are orthogonal. Each Hermitian matrix A canbe diagonalized:

A = UΛU∗, (A.22)

where U is a unitary matrix and Λ is a diagonal matrix with real diagonal elementsλν .

It is easy to see that the column vectors uν of U constitute a complete system ofeigenvectors to the eigenvalues λν . Eq. A.22 can also be written as a sum of dyads:

A =n∑

ν=1

λνuνu∗ν . (A.23)

A projector PU onto a K-dimensional subspace U can be written to

PU =K∑

ν=1

uνu∗ν , (A.24)

where uν , ν = 1, · · · , K is a orthonormal basis of U . Consequently, PU has Keigenvalues = 1 and n−K eigenvalues = 0.

On a Hermitian matrix A functions can be applied such as potentiate, logarith-mal etc., if the values of the diagonal matrix are used:

f(A) = U

f(λ1) · · · 0· · · · · · · · ·0 · · · f(λn)

U∗, (A.25)

e. g.

Ax = U

λx1 · · · 0

· · · · · · · · ·0 · · · λx

n

U∗. (A.26)

Matrix inversion

A positive definite Hermitian matrix A is invertible. Using the eigen decompositionA = UΛU∗ it yields

A−1 = U

1λ1

· · · 0

· · · · · · · · ·0 · · · 1

λn

U∗. (A.27)

A matrix of the form A = In + ss∗ is positive definite and invertible. The inverse iscomputed to

A−1 = In − 1

s∗s + 1ss∗ (A.28)

If S is a n× k-matrix with k ≤ n and D a invertible k × k-matrix, then is a matrixof the form A = In + SDS∗ invertible and it yields:

A−1 = In − S (Ik + DS∗S)−1 DS∗. (A.29)

Please note, that in the so called (Matrix-Inversion-Lemma) (A.29) the inversion inthe n-dimensional space has been transformed into the k-dimensional space.

A.1. COMPLEX LINEAR ALGEBRA 159

Differentiating of complex vectors

Let x be a real variable, and F(x) a complex vector depending on x. Then

d

dx‖F(x)‖2 = 2<

(d

dxF(x)

)∗F(x)

. (A.30)

Best linear approximations

If z and s are n-dim. column vectors and a is a complex constant, then

a(z) := argmin‖z− as‖2 : a ∈ C =s∗

s∗sz. (A.31)

The best approximation of z by a scalar multiple of s in the minimum squareerror sense is

a(z)s =ss∗

s∗sz. (A.32)

P<s> := ss∗s∗s is the projector to the subspace spanned by s. Using the generalized

Pythagoras’ equation

‖z‖2 = ‖P<s>‖2 + ‖P<s>⊥‖2. (A.33)

shows that ‖P<s>⊥‖2 is the remaining quadratic error.

If s1, . . . , sk are linear independent n-dim. column vectors we search for the bestapproximation of z by a linear combination of s1, . . . , sk with complex coefficientsa1, . . . , ak:

‖z− (a1s1 + · · ·+ aksk) ‖2 = Min! (A.34)

Let S = (s1, . . . , sk) be the n × k-matrix composed of the signal vectors anda = (a1, . . . , ak)

t the k-dim. vector of coefficients. Then Eq. (A.34) can be writtenin the form

a(z) := argmin‖z− Sa‖2 : a ∈ Ck. (A.35)

The solution is

a(z) = (S∗S)−1 S∗z. (A.36)

The best approximation of z can be written in the form

a(z) = S (S∗S)−1 S∗z (A.37)

= P<s1,...,sk>z (A.38)

= PUSz (A.39)


if US :=< s1, . . . , sk > denotes the subspace spanned by the columns of S.PUS

= S (S∗S)−1 S∗ is the projector to this space.

A.1.5 Order of matrices

For a Hermitean matrix A ∈ CN×N it is defined A ≥ 0 if for all x ∈ CN theinequality x∗Ax ≥ 0 is valid. This is identical to the property that all eigenvaluesof A are greater or equal to 0, i.e. A is positive semidefinite.

For two Hermitean matrices A,B the half order A ≥ B is defined by A−B ≥ 0.

A.2 Energy signals

The space L2(C) is defined as the set of all square-integrable signals s : R → (C).For s, s1, s2 ∈ L2(C) we define

< s1, s2 > :=

∫s1 (t) s∗2 (t) dt (dotproduct),

‖s‖ :=√

< s, s > =

√∫|s (t)|2 dt (‖s‖2 = signalenergy .) (A.40)

It yields

< as1 + bs2, s > = a < s1, s > +b < s2, s > (A.41)

< s, as1 + bs2 > = a∗ < s, s1 > +b∗ < s, s2 > for all a, b ∈ C, (A.42)

‖s‖ = 0 if and only ifs(t) = 0 (almost everywhere) . (A.43)

Possess s1, s2 non-vanishing signal energy, then

ρ (s1, s2) :=< s1, s2 >

‖s1‖ ‖s2‖ (correlationcoefficient, generalized cosine). (A.44)

It yields|ρ (s1, s2)| ≤ 1 (A.45)

with equality if and only if an a ∈ C exists with s1(t) = as2(t) almost everywhere(Cauchy-Schwarz’ inequality).

The convolution of s1 with s2 is defined to

(s1 ∗ s2) (t) :=

∫s1 (τ) s2 (t− τ) dτ =

∫s1 (t− τ) s2 (τ) dτ, (A.46)

the cross-correlation function to

rs1s2 (t) :=

∫s1 (t + τ) s∗2 (τ) dτ =

∫s1 (τ) s∗2 (τ − t) dτ, (A.47)

and the auto-correlation function to

rss (t) :=

∫s1 (t + τ) s∗ (τ) dτ . (A.48)

A.3. SPECIAL FUNCTIONS 161

It yields

rs1s2 = s∗1s2 with s2 (t) := s∗2 (−t) (A.49)

rs1s2 (t) = < s1, s(t)2 > with s

(t)2 (τ) := s2 (τ − t) (A.50)

The Fourier transform of a signal is defined to

S (ω) :=

∫s (t) e−jωtdt. (A.51)

with the back transformation

s (t) =1

2π

∫S (ω)ejωtdω. (A.52)

The application of the Fourier transform is written as S = F(s), s = F−1(S) andS(ω) = F(s(t)), s(t) = F−1(S(ω)), respectively.

For the Fourier pairs S = F (s), S1 = F (s1), and S2 = F (s2) yields:

‖s‖2 =1

2π‖S‖2 (Parzeval’s Lemma), (A.53)

< s1, s2 > =1

2π< S1, S2 >, (A.54)

F (s1 ∗ s2) = S1S2 F (s1s2) = S1 ∗ S2, (A.55)

F (rs1s2) = S1S2 F (rss) = |S|2 . (A.56)

A.3 Special Functions

Rectangular function:

rect (t) :=

1 for |t| ≤ 1

2

0 else(A.57)

sin x/x-function:

si (x) :=

1 for x = 0sin (x) /x else

(A.58)

Integral sine:

Si(x) :=

x∫

0

si(t)dt (A.59)

Modified Bessel function:

Iν(x) :=1

2π

2π∫

0

cosν (ϕ) ex cos(ϕ)dϕ (A.60)

Fresnel-Integrals:

S (x) :=

x∫

0

sin(π

2t2

)dt (A.61)

C (x) :=

x∫

0

cos(π

2t2

)dt (A.62)


A.4 Probability distributions

A.4.1 Random variables, realizations, probability distribu-tions and densities

Let X be a random variable with values in ΩX . If x is the result of a ’randomexperiment’ performing X, x is called a realization of X. If the random experimentis repeated n times, we observe the realizations x1, · · · , xn.

The probability to get a result x is characterized by the probability distributionPX . The probability that X falls into a (measurable) subset A ⊆ ΩX is given byPX(A). If X takes continuous values, the probability density function (pdf)pX(x)defines the distribution via

PX(X ∈ M) =

∫

M

pX(x)dx. (A.63)

Expectation and Variance of a random variable X with values in R and the pdfpX are defined by

E[X] =

∫xpX(x)dx (A.64)

Var[X] =

∫(x− E[X])2pX(x)dx (A.65)

=

∫x2pX(x)dx−

(∫xpX(x)dx

)2

. (A.66)

For each function f(x) we get

E[f(X)] =

∫f(x)pX(x)dx. (A.67)

Random variables X and Y are (statistically, stochastically) independent if

P (X,Y )(X ∈ MX , Y ∈ MY ) = PX(X ∈ MX)P Y (Y ∈ MY ) (A.68)

for all (measurable) subsets MX and MY . If there exists the common pdf of (X,Y )this is equivalent to

p(X,Y )(x, y) = pX(x)pY (y) (A.69)

(almost everywhere).

If X and Y are independent, also functions of these f(X) and g(Y ) are indepen-dent. Further independent random variables have the following properties:

E[XY ] = E[X]E[Y ] (A.70)

Var[X + Y ] = Var[X] + Var[Y ] (A.71)

For random variables X and Y with values in R Jensen’s Inequality is valid:

E[XY ] ≤√

E[X2]E[Y 2]. (A.72)

A.4. PROBABILITY DISTRIBUTIONS 163

Complex random variables

For random variables Z,W with values in C expectation and variance are definedby analogy:

E[Z] := E[<Z] + jE[=Z] (A.73)

Var[Z] := E|Z − EZ|2 = E|Z|2 − |EZ|2 (A.74)

Cov[Z, W ] := E[Z − EZ]E[W − EW ]∗ = E[ZW ∗]− E[Z]E[W ]∗ (A.75)

Variance and Covariance have the properties Var[Z] ∈ R, Var[Z] ≥ 0 with 0 ifand only if Z = 0 almost everywhere. Cov[W,Z] = Cov[Z, W ]∗ and Cov[Z, Z] =Var[Z] ∈ R.

If Cov[W,Z] = 0 the random variables are called uncorrelated. If W and Z areindependent, they are also uncorrelated.

The covariance properties of real- and imaginary part of a complex randomvariable are uniquely determined under the condition

Cov[<Z,=Z] = 0, Var[<Z] = Var[=Z]. (A.76)

It follows Var[Z] = 2Var[<Z] = 2Var[=Z].

Covariance matrix of complex random vectors Let Z be an n-dimensionalcomplex random column vector with expectation µ := EZ. The covariance matrixof Z is defined by

RZ := E [(Z− µ) (Z− µ)∗] (A.77)

=

E [(Z1 − µ1) (Z1 − µ1)∗] . . . E [(Z1 − µ1) (Zn − µn)∗]

. . . . . . . . .E [(Zn − µn) (Z1 − µ1)

∗] . . . E [(Zn − µn) (Z1 − µ1)∗]

(A.78)

=

Var[Z1] . . . Cov[Z1, Zn]. . . . . . . . .

Cov[Zn, Z1] . . . Var[Zn]

(A.79)

A covariance matrix of a complex random vector is always Hermitean since

R∗Z = (E [(Z− µ) (Z− µ)∗])∗ (A.80)

= E [(Z− µ) (Z− µ)∗] (A.81)

= RZ (A.82)

Further it is positive semidefinit since


x∗RZx = x∗E [(Z− µ) (Z− µ)∗]x (A.83)

= E [x∗ (Z− µ) (Z− µ)∗ x] (A.84)

= E |(Z− µ)∗ x|2 (A.85)

≥ 0 for all x ∈ Cn. (A.86)

Special distributions

Gaussian distribution, normal distribution of a real valued random vari-able: A real valued random variable is called normal distributed or Gaussian dis-tributed with expectation µ and variance σ2

X if it has the pdf

pX(x) =1√

2πσ2X

exp

−(x− µ)2

2σ2X

(A.87)

We write X ∼ NR(µ, σ2X).

Gaussian distribution, normal distribution of a real valued random vector:A real valued random vector is called normal distributed or Gaussian distributed withexpectation µ and covariance matrix RX if it has the pdf

pX(x) =1√

(2π)n detRX

exp

−1

2(x− µ)t R−1

X (x− µ)

(A.88)

We write X ∼ NRn(µ,RX).

Gaussian distribution, normal distribution of a complex valued randomvariable: The random variable Z is called complex Gaussian distributed with ex-pectation µ and variance σ2, if <Z and =Z are uncorrelated and have the samevariances σ2/2. The density pZ is given by

pZ (z) =1

πσ2exp

−|z − µ|2

σ2

. (A.89)

We write Z ∼ NC(µ, σ2) .

Gaussian distribution, normal distribution of a complex valued randomvector: A complex valued random vector Z is called normal distributed or Gaus-sian distributed with expectation µ and covariance matrix RX if it has the pdf

pZ(z) =1

πn detRZ

exp− (z− µ)∗R−1

Z (z− µ)

(A.90)

We write Z ∼ NCn(µ,RZ).

A.5. STOCHASTIC PROCESSES 165

Uncorrelated and stochastically independent Gaussian random variablesIf two real- or complex valued random variables are uncorrelated, they are inde-pendent. The phase of a NC(0, σ2)-distributed random variable is rectangular dis-tributed on [0, 2π] and independent on the amplitude.

Rectangular distribution: A real valued random variable R is distributed ac-cording to a rectangular distribution (or uniformly distribution) on the interval [a, b],if the pdf is given by

pR(r) =

1

b−a, if r ∈ [a, b]

0 else(A.91)

We write R ∼ R(a, b).

Rayleigh distribution: Distribution of the amplitude of a NC(0, 1) distributedrandom variable X = |Z| :

FRayleigh (x) = Pr (|Z| ≤ x)

=1

π

∫e−|z|

2

dz =1

π

x∫

0

2π∫

0

e−r2

rdϕdr |z| ≤ x

= 2

x∫

0

e−r2

rdr = 1− e−x2

(A.92)

Density: pX (x) = 2xe−x2

(A.93)

Rician distribution: Distribution of X = |a + Z| with a ∈ C and Z ∼ NC(0, 1).

FRicea (x) =

1

π

∫e−|z−a|2dz |z| ≤ x

=1

π

x∫

0

2π∫

0

e−r2|a|2+2|a|r cos ϕrdϕdr

= 2e−|a|2

x∫

0

e−r2

I0 (2 |a| r) rdr (A.94)

Density: pX (x) = 2x e−x2−|a|2I0 (2 |a|x) (A.95)

A.5 Stochastic processes

For complex valued processes Z(t) the definitions are analogue to the real valuedcase:


µZ(t) := E[Z(t)] deterministic partZ(t)− µZ(t) stochastic partσ2

ZZ(t) := Var[Z(t)] variance functionrZZ(t1, t2) := E[Z(t1)Z

∗(t2)] correlation functioncZZ(t1, t2) := Cov[Z(t1), Z(t2)] covariance function

ρZZ(t1, t2) := cZZ(t1,t2)√σ2

ZZ(t1)σ2ZZ(t2)

normalized covariance function

(A.96)

For wide sense stationary complex valued stochastic processes the following def-initions are given:

rZZ(τ) := E[Z(t + τ)Z∗(t)] (A.97)

cZZ(τ) := Cov[Z(t + τ), Z(t)] (A.98)

ρZZ(τ) :=cZZ(τ)

σ2ZZ

(A.99)

CZZ(ω) :=

∫CZZ(τ)e−2πjωτdτ spectral power density (A.100)

A.6 Estimation

A.6.1 The general estimation problem

If the model for the distribution of the random variable X contains an unknownparameter vector ϑ ∈ Θ, the basic problem is to estimate this parameter vectorwith an estimator ϑ(x) for a given realization x of X.

For each ϑ the distribution PXϑ of X and the probability distribution function

(pdf) pXϑ (x) are parametrized by this parameter vector.

A.6.2 Approaches to estimators

Performance of estimators

If ϑ(x) is an estimator of ϑ, the bias is defined by

Bϑ(ϑ) = Eϑ

[ϑ(X)

]− ϑ. (A.101)

If the bias is zero for all ϑ ∈ Θ, we call this estimator bias-free.

As a performance measure of the estimator we introduce the expectation of thequadratic deviation from the true value:

Qϑ(ϑ) := Eϑ

[(ϑ(X)− ϑ

)(ϑ(X)− ϑ

)t]

(A.102)

= Rϑ(ϑ) + Bϑ(ϑ)Btϑ(ϑ) (A.103)

A.6. ESTIMATION 167

where Rϑ is the covariance matrix of the estimator ϑ(x) as a function of ϑ.

Best unbiased estimator. Weights of estimation. Order of matrices.

Best unbiased estimators

A desired property of an estimator is that its expectation is equal to the parametervector itself. The bias is defined by

Bϑ(ϑ) = Eϑ

[ϑ(X)

]− ϑ. (A.104)

The subscript ϑ at the expectation operator means that the probability distributionaccording to this parameter vector is assumed. An estimator is called bias free ifBϑ(ϑ) = 0 for all ϑ ∈ Θ.

The second desired property of an one-dimensional real valued estimator is thatits variance is minimum. If the parameter vector has a dimension larger than 1,the variance has to be replaced by the covariance matrix, and the half order ofcovariance matrices is defined according to section A.1.5.

So an estimator ϑ is called best unbiased estimator, if it is unbiased and has theminimum covariance matrix among all unbiased estimators for all ϑ ∈ Θ.

Maximum likelihood estimators

Unfortunately, for the most estimation problems in radar signal processing, bestunbiased estimators don’t exist. A more practical way is the maximum likelihoodestimator which mostly leads to a feasible near optimum algorithm. This approachchooses as estimator the parameter vector for which the realization has a ’maximumlikelihood’ (illustrated in Fig. 1.21):

ϑ(x) = argmaxpX

ϑ (x) : ϑ ∈ Θ

. (A.105)

pXϑ (x) is called likelihood function (as function of ϑ), ϑ(x) is the maximum-

likelihood estimator (ML estimator). Equivalent with the maximization of the likeli-hood function is the maximization of the logarithm of the pdf: LX

ϑ (x) := ln(pX

ϑ (x)),

called log-likelihood function.

ϑ(x) = argmaxLX

ϑ (x) : ϑ ∈ Θ

. (A.106)

If the partial derivatives of the log-likelihood function with respect to ϑ exist,the necessary condition for an extremum is

∂

∂ϑLX

ϑ (x)|ϑ=ϑ(x) = 0. (A.107)


A.6.3 Cramer-Rao Bounds

The Cramer-Rao Bounds offer the possibility to obtain a lower bound for the ex-pected quadratic error of an estimator without specifying an estimation algorithm.The quality of any estimator is dependent on the information implicitly containedin the probability distribution as function of the parameter to be estimated. Thisinformation is measurable: The Fisher-information relates to a single real param-eter, the Fisher-information matrix represents the information about a parametervector. For a family of pdf’s pX

ϑ , ϑ ∈ Θ the Fisher-information matrix is defined by:

J(ϑ) = E

[(∂

∂ϑln pX

ϑ (X)

)(∂

∂ϑln pX

ϑ (X)

)t]

. (A.108)

This matrix has remarkable properties appropriate to an information measure;so invertible transformations of a statistic do not change the Fisher-informationmatrix, the information of an independent pair of distributions is the sum of theindividual information measures and so on.

For any unbiased estimator ϑ for ϑ the following inequation is valid, if Fisher’sinformation matrix is invertible:

Rϑ(ϑ) ≥ J−1(ϑ). (A.109)

If the equality sign holds, the estimator is called efficient.

If we use the statistical model ??

Z = m(ϑ) + Q (A.110)

with a deterministic complex vector m(ϑ) and interference Q ∼ NCn(0,R), Fisher’sinformation matrix is given by

J(ϑ) =2

σ2Re

m∗

ϑ(ϑ)R−1mϑ(ϑ)

(A.111)

where mϑ(ϑ) means the gradient of mϑ with respect to ϑ written as n× k matrix.

Proof: The log-likelihood Function is

LZϑ(z) = − ln(πnσ2n)− 1

σ2‖z−m(ϑ)‖2 (A.112)

and the Fisher-information matrix is calculated using mϑ(ϑ) := ∂∂ϑ

m(ϑ) :=

A.7. POWER OF FILTERED NOISE 169

(∂

∂ϑ1m(ϑ), · · · , ∂

∂ϑKm(ϑ)

)to

J(ϑ) = Eϑ

[(∂

∂ϑLZ

ϑ(Z)

)(∂

∂ϑLZ

ϑ(Z)

)t]

(A.113)

=1

σ4Eϑ

[(∂

∂ϑ‖Z−m(ϑ)‖2

)(∂

∂ϑ‖Z−m(ϑ)‖2

)t]

(A.114)

=1

σ4Eϑ

[2Re N∗mϑ(ϑ)t 2Re N∗mϑ(ϑ)] (A.115)

=1

σ4Eϑ

[(N∗mϑ(ϑ) + m∗

ϑ(ϑ)N)t (N∗mϑ(ϑ) + m∗ϑ(ϑ)N)

]. (A.116)

We get

J(ϑ) =2

σ2Re (mϑ(ϑ))∗mϑ(ϑ) (A.117)

A.7 Power of filtered noise

Let N be white noise, t.i. CNN(f) = σ2 for all frequencies f , which is passed througha filter with pulse response h ∈ L2(C). The expected power of the filter output isevaluated over the spectral density:

E |(h ? N)(0)|2 = E

∣∣∣∣∫

h(−t)N(t)dt

∣∣∣∣2

(A.118)

= E

∫ ∫h∗(−t)N∗(t)h(−t′)N(t′)dtdt′ (A.119)

= E

∫ ∫h∗(−t)N∗(t)h(−t− τ)N(t + τ)dtdτ (A.120)

=

∫ (∫h∗(t)h(t− τ)dt

)rNN(τ)dτ (A.121)

=

∫ (∫|H(f)|2 exp−j2πfτdf

)rNN(τ)dτ (A.122)

=

∫|H(f)|2CNN(f)df (A.123)

= σ2

∫|H(f)|2df (A.124)

= σ2‖h‖2 (A.125)

Index

Adaptive array, 75Alternative, 46Ambiguity function, 36Ambiguity plane, 36Ambiguity rectangle, 40Antenna footprint, 122, 123, 130Arbitrary waveform generator, 12Aspect angular velocity, 130Azimuth angle, 93Azimuth chirp, 128Azimuth resolution, 123Azimuth signal, 128

Back projection processor, 148Baseband signal, 13Basis, 156Beamformer, 67Beamwidth, 71Best invariant test, 55Best unbiased estimator, 167Bias free estimator, 167Bias of an estimator, 166, 167Bias-free, 166Boltzmann constant, 62

Cauchy and Schwarz, inequality, 16Cauchy-Schwarz’ inequality, 156, 160CFAR, 55Chirp scaling algorithm, 36Chirp waveform, 27, 38Clutter, 57Clutter spectrum, 104, 134Complex envelope, 12Compression factor, 26Cone angle, 93Constant false alarm rate, 55Correlation coefficient, 16, 156Covariance matrix, 163Cramer-Rao Bounds, 168Cramer-Rao Bounds, 86

Data, pre-processed, 33De-ramping, 33Depression angle, 93Depth of focus, 125Detection probability, 46Detector, 46Diagonal loading, 85Difference channel, 89Dimension, 156Direct digital synthesizer, 12Direction of arrival vector, 67Directional cosine, 67Directional cosine history, 127DOA-vector, 67Doppler cone, 98Doppler slope, 130Doppler tolerance, 39Dot product, 156Duty factor, 63Dyad, 155

Elevation angle, 93, 126Empirical covariance matrix, 85Euler angles, 111Euler rotations, 111

False alarm, 46False alarm probability, 46Fast time, 20Filter, optimum, 18Fisher information matrix, 44Fisher-information, 168Fisher-information matrix, 86, 168Foliage penetration, 61Form filter, 62Frank code, 175Fresnel zone plate, 140

Grazing angle, 93Ground range, 123

170

INDEX 171

Hypothesis, 46

Incidence angle, 93Incoherent detection, 56, 175Indicator function, 31Inner product, 156INR, 77Interference space, 81Interference-to-noise-ration, 77Inverse filter, 30, 31Inverse synthetic aperture radar, 120ISAR, 120Iso-DOA cone, 67ISO-Doppler lines, 98ISO-range lines, 98

Likelihood function, 167Likelihood ratio, 47Likelihood ratio test, 47Linear array antenna, 65Linear combinations, 156Linear hull, 156Linear subspace, 156LO-signal, 12Local frequency signal, 12Log likelihood ratio, 47Log-likelihood function, 42, 167

Mainbeam clutter, 104Matched filter, continuous, 19Matched filter, discrete, 18Maximum likelihood estimator, 42, 167Maximum-likelihood estimator, 167minimum mean square estimator, 43Modified Bessel function, 49Multi channel synthetic aperture radar,

117Multi-look, 122Multilook processing, 136

Nadir, 95Neyman and Pearson, approach, 47Noise figure, 62Noise-equivalent sigma-zero, 149Normal form, 32, 33

Orthogonal, 156Orthogonal system, 156Orthonormal, 156

Orthonormal system, 156

Periodogram, 59Planar phased array, 72Point of stationary phase, 132Point spread function, 20Positive semidefinit, 163Power spectrum, 59Pre-processing in range, 33Principle of stationary phase, 132Products of matrices, 155Pulse repetition interval, 20

Quadrature demodulator, 12Quadrature modulator, 12

Radar cross section, 61Radar equation, 63Radar equation for SAR, 149Radio frequency signal, 13Range curvature, 139Range curvature correction, 147Range history, 35, 127Range hyperbola, 128Range migration, 23Range parabola, 128Range profile, 41Range resolution, 25Range walk, 23Range-Doppler processor, 147Range-velocity processing, 35Rangeline, 20Rayleigh distribution, 49Rayleigh region, 61Rayleigh resolution, 25RCS, 61Re-formatting, 36Real aperture imaging, 123Receiver operating characteristics, 54Reference frequency, 12, 14Reflectivity of surface, 106Resolution, 24Resolution cell, 25RF-signal, 12Rician distribution, 49, 165

Sample matrix, 85Sample matrix inversion, 85

172 INDEX

SAR, 117SAR applications, 117SAR history, 117Scalar product, 156scan characteristics, 69SeaSAT, 117Separated processing, 146Side looking geometry, 122Sidelobe clutter, 104Signal-to-clutter-plus-noise ratio, 59Signal-to-noise-plus-interference ratio,

74Signal-to-noise-ratio, 15Simple alternatives, 46Slant range, 123, 126Sliding mode, 123, 138Slow time, 20SMI, 85Snapshot, 85SNIR, 74SNR, 15Spatial interpretation of radar signals,

28Spectral power density, 59Specular reflection, 95Spotlight mode, 123, 128, 138Squint angle, 122Standard model I, 16Standard model II, 41Stationary phase, 135Stationary stochastic process, 59Statistically independent, 162Stolt interpolation, 148Stop and go approximation, 22Stripmap mode, 122Sum channel, 89Swath width, 122Synthetic aperture, 101, 123Synthetic aperture radar, 117Synthetic array, 100Synthetic beam, 101Synthetical array, 123

Target miss, 46Target miss probability, 46Terrain scattering coefficient, 106Test, 46

Threshold, 47Time-bandwidth product, 26Toeplitz matrix, 60Trace, 155Turntable imaging, 120Two-pulse canceller, 57

Uncorrelated random variables, 163Uniform best test, 50Unit vector, 156

Volume invariance, 37

Wave form design, 38Wave number domain, 30Wavenumber, 15Weights, optimum, 16Wide sense stochastic process, 59

Bibliography

[1] F. Le Chevalier: ’Principles of Radar and Sonar Signal Processing’, ArtechHouse 2002, ISBN 1-58053-338-8

[2] A. Farina: ’Antenna-Based Signal Processing Techniques for Radar Systems’,Artech House 1992, ISBN 0-89006-396-6

[3] N. Fourikis: ’Phased Array-Based Systems and Applications’, Wiley series inmicrowave and optical engineering, 1997, ISBN 0-471-01212-2

[4] A. Papoulis: ’Signal analysis’, McGraw-Hill, New York, 1977

[5] A. W. Rihaczek: ’Principles of High-Resolution Radar’, Artech House 1996,ISBN 0-89006-900-X

[6] M. I. Skolnik: ’Introduction to Radar Systems’, third edition 2001, McGraw-Hill, ISBN 0-07-290980-3

[7] D. R. Wehner: ’High-Resolution Radar’, second edition, Artech House, 1995,ISBN 0-89006-727-9

[8] W. D. Wirth: ’Radar techniques using array antennas’, IEE Radar, Sonar,Navigation and Avionics Series 10, London 2001, ISBN 0 85296 798 5

[9] R. E. Collin, F. J. Zucker: ’Antenna Theory’, McGraw Hill 1969

[10] J. Ender, D. Cerutti-Maori: Position Estimation of Moving Vehicles for Space-Based Multi-Channel SAR/MTI Systems, European Conference on SyntheticAperture Radar, European Conference on Synthetic Aperture Radar, EUSAR2006, Dresden, May 2006

[11] J. Ender, C. Gierull, D. Cerutti-Maori, ”Improved Space-Based Moving TargetIndication via Alternate Transmission and Receiver Switching,” Geoscience andRemote Sensing, IEEE Transactions on , vol. 46, no.12, pp.3960-3974, Dec.2008.

[12] J. Neyman, E. Pearson: ’On the Problem of the Most Efficient Tests of Statis-tical Hypotheses’, Philosophical Transactions of the Royal Society of London.Series A, Containing Papers of a Mathematical or Physical Character 231, pp.289-337, 1933

173

174 BIBLIOGRAPHY

[13] U. Nickel: An Overview of Generalized Monopulse Estimation, IEEE A&ESystem Magazine Vol. 21, No.6 June 2006, Part 2:Tutorials, pp. 27-55

[14] R. K. Raney, H.Runge, R. Bamler, I. G. Cumming, F. H. Wong: ’PrecisionSAR Processing using Chirp Scaling’, IEEE Transactions on Geoscience andRemote Sensing, vol. 32, No. 4, pp. 786-799, July 1994

[15] J. Ward, G. F. Hatke: An Efficient Rooting Algorithm for Simultaneous Angleand Doppler Estimation with Space-Time Adaptive Processing Radar ASILO-MAR97, Nov. 1997, Pacific Grove, CA

Appendix B

Exercises

B.1

How large is the maximum signal-to-noise-ratio for a matched filter in the RF domainif a narrow band signal with fixed carrier frequency and triangular envelope is used?The additive noise is assumed to be white with the spectral power density N0/2.

B.2

Calculate the point spread function for a Gaussian base band signal

s(t) = 1√2πt2s

exp− t2

2t2s

for matched filtering!

B.3

Let Z = as + N be a vector composed by N complex samples with a known deter-ministic signal s and NCN (0, σ2IN)-distributed noise N. Derive with the methodof minimum mean squares an estimator a(z) of a and calculate its expectation andvariance!

B.4

As exercise B.3, but as superposition of known signals with unknown complex am-plitudes:

Z =K∑

k=1

aksk + N with K ≤ N . In matrix notation: Z = Sa + N with

S = (s1, . . . , sK) , a = (a1, . . . , aK)t.Determine

1) a (z) = argmin‖z− Sa‖2 : a ∈ CK

175

176 APPENDIX B. EXERCISES

2) z− Sa (z)

3) ‖z− Sa (z) ‖2

and the expectations of these quantities!

B.5

For the base band signal s(t) = rect(t/ts) with subsequent matched filtering andsampling with ∆t = ts determine the maximum-likelihood estimate of the time delayτ for unknown complex amplitude a. It is assumed to be known that

τ/∆t ∈ [ν0 − 1/2, ν0 + 1/2] .

B.6

Let a coherent pulse radar with RF frequency (f0 = 3GHz) be operated with rect-angular pulses of the width ts = 1µs at a PRF of 1 kHz. How large is the un-ambiguous range interval if the transmit-receive switch causes a ’dead time’ of 2µs? How many range cells with full integration are obtainable at a sampling with∆t = ts? At which radial velocity rotates the phase from pulse to pulse by 360

(’blind velocity’)? Which shape has the spectrum of the received base band signalif a pulse train of ten pulses is transmitted? How does this change if the object ismoving at a radial velocity vr = 10m/s? (A sketch with declaration of the relevantfrequencies is sufficient!).

B.7

At the output of a quadrature modulator the band limited signal sRX with band-width b is superposed by noise with power density CNN(f) = N0rect(f/bn) andbs < bn. Let this signal be sampled an filtered by a digital filter with transferfunction H(f) = S∗(f). Discuss the deterministic signal spectrum and the noisepower spectral density of the output signal for the cases bs < fs < bn and bn < fs!What does this mean for the signal-to-noise ratio of the filtered signal (heuristicsand sketches are sufficient!).

B.8

Let a continuous-wave (CW) radar transmit a signal s which is repeated with the

period ∆T : q (t) =∞∑

ν=∞s (t− ν∆T ).

At the receiver baseband output a filter is applied matched to one cycle. Whichpoint spread function can be stated for a phase coded base band signal of the

B.9. 177

form s (t) = ejψµ for t ∈ [µ∆t, (µ + 1) ∆t] , µ = 0, . . . , m − 1 with m = k2 andk ∈ N, ∆t = ∆T/m? ψµ is defined by ψµ = (µ mod k) (µ div k) 2π/k (Frank code).

B.9

Evaluate the ambiguity function for a chirp with rectangular envelopes (t) = t

−1/2s rect (t/ts) ejπαt2 .

a) How large is the range resolution, defined by the first null, for ν = 0 (noDoppler shift) as a function of α? Regard also the limiting cases α = 0 and α →∞.

b) For which τ decreases χ (τ, ν (τ)) to the half power? Here, ν(τ) is defined asthe Doppler shift for which the power ambiguity is maximum for fixed τ .

B.10

Design a detector for the model

Zν = aejφν + Nν for ν = 1 . . . n, φν , Nν stoch. independent for ν = 1 . . . n,φν ∼ R (0, 2π),Nν ∼ NC (0, σ2) for ν = 1 . . . n. a is assumed to be non-negative realvalued, and the test problem is to decide between H: a = 0 and K: a > 0 (incoherentdetection).

Approximate the distribution of the obtained test by a normal distribution andderive the approximated threshold and detection probability for n tending to ∞.For fixed PE and PF a signal-to-noise ratio γn = (a/σ)2 is necessary. Determine therate of γn tending to zero and compare this with the analogue result for coherentdetection!

B.11

Let a Doppler filter bank for a data vector of the length N consist of 2N filtersfor a uniform distribution of the Doppler angle increment over [0, 2π]. Calculatethe covariance matrix of the 2N dimensional output vector, if the input is given bya deterministic signal plus noise with covariance matrix σ2IN ! Further, derive theeigenvalues of this covariance matrix!

B.12

Let Z be NCN (0, σ2IN)-distributed. Develop a test for H: σ2 = σ20 against K:σ2 = σ2

1

with σ1 > σ0!

178 APPENDIX B. EXERCISES

B.13

Develop a maximum-likelihood-ratio test for the modelZ = Sa + N, N ∼ NCN (0, σ2IN) with a known N ×K matrix S and an unknownamplitude vector a ∈ CK for H: a = 0 against K: a 6= 0! Which distribution hasthe test statistic under the hypothesis?

B.14

Often the temporal behavior of clutter echoes can be modelled by an autoregressivestochastic process. Let the process Cν := C(ν∆T ), ν ∈ Z be modelled as autore-gressive of order one:

Cν = ρCν−1 + Mν for ν ∈ Z with identical independent excitations Mν ∼NC(0, σ2) and |ρ| < 1.

a) Which transfer function has the linear system that transforms the time series(Mν)ν∈Z into the process (Cν)ν∈Z?

b) Calculate the Doppler spectrum of the clutter!

c) Determine a sequential clutter filter which transforms (Cν)ν∈Z into white noiseand calculate its transfer function!

d) Which SCR gain obtains this filter?

B.15

Show that the de-ramping technique does not change the range resolution comparedto direct matched filtering according to the transmitted chirp!

B.16

Recall that the point spread function for matched filtering is equal to the Fourierback transform of the square magnitude of the signal spectrum (P (f) = |S(f)|2). Asyou know, the application of a ’well-shaped’ window in the frequency domain beforethe Fourier back transform effects low side lobes of the point spread function. Butthis is connected to a loss of SNR. It would be preferable to design the transmittedwaveform in order to get a ’well-shaped’ magnitude-square spectrum |S(.)|2 from thebeginning. On the other hand, the amplitude of the waveform should be constant inthe time domain to drive the last amplifier to maximum output power. How can youconstruct a waveform s(t) = expjϕ(t), t = −ts/2, · · · , ts/2 with phase modulationonly to obtain approximately a given shape W (.) of |S(.)|2?

B.17. 179

B.17

Let the sampling of a signal from pulse to pulse modelled by an N dimensionalrandom vector Z = as(F ) + N,N ∼ NCN (0, σ2IN), where

s(F ) =

expj2π (1− N+1

2

)F∆T

...expj2π (

N − N+12

)F∆T

.

The complex amplitude a = αejψ is unknown as well as the Doppler frequencyF . Derive the maximum-likelihood estimator for F and calculate the Cramer-RaoBounds for the estimation of F at unknown a!

B.18

For a simple monopulse radar with the aperture divided into two equal parts the

two receive signals can be modelled by a vector Z = a

(ejkrdu0/2

e−jkrdu0/2

)+ N with

N ∼ NC2(0, σ2I2).

The estimation of the directional cosine is performed by

u(z) = 2krd=Q( z) with Q(z) = ∆

Σ= z1−z2

z1+z2.

Calculate the variance of u, if the real direction is u0 = 0 in the approximationfor SNR →∞. Compare the result with the Cramer-Rao Bounds from exercise B.17with N = 2 and equivalent replacement of the Doppler frequency by the directionalcosine!

B.19

We regard an object in the object coordinate system (x, y) which is rotated on aturntable in clock sense while a radar measures this object from a large (far field)distance R0 to the origin. The direction of the transmitted wave is described by theunit vector

u(ϕ) =

(cos ϕsin ϕ

). (B.1)

in the object-fixed coordinate system. Now we assume a point scatterer at position~r = (x, y)t. For the azimuth angle ϕ the distance to the point scatterer is given by

R(ϕ; x, y) =√

(R0 cos ϕ + x)2 + (R0 sin ϕ + y)2 (B.2)

≈ R0 + (x cos ϕ + y sin ϕ)

= R0 + 〈~u(ϕ), ~r〉 .Now a Fourier transform is performed over ϕ:S(kϕ, kr; x, y) := 1

2π

∫ 2π

0e−jkrϕs(ϕ, kr; x, y)dϕ. Using the principle of stationary phase,

approximate this signal!

introduction to radar part i

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