warsaw mti

reg. number : 2008telb0075

Thesis

presented at the

Military University of Technology in Warsaw

with the authorisation of the University of Rennes 1

to obtain the degree of

Doctor of Philosophy in association with TelecomBretagne and the Military University of Technology

Domain : Signal Processing and TelecommunicationsMention : Traitement du Signal et Télécommunication

by

Tomasz Górski

Universities : Telecom Bretagne and the Military University of Technology in Warsaw

Space-Time Adaptive Signal Processing for SeaSurveillance Radars

Defence December 9, 2008 before the examination board :

Reporters : Marc Acheroy, Professor at Royal Military Academy in BrusselsRichard Klemm, Doctor at FGAN

Examiners : Jean Marc Le-Caillec, Professor at Telecom BretagneAdam Kawalec, Professor at the Military University of Technology in WarsawLaurent Ferro-Famil, Doctor with accreditation to supervise reasearch

at the University of Rennes 1Ali Khenchaf, Professor at ENSIETA

Contents

1 Introduction. 1

2 Radar Basics, Space-Time Adaptive Processing and Target Detection. 3

2.1 Radar principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Overview of STAP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Radar System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Airborne Clutter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.4 Adaptive MTI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.5 STAP Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.6 Assumptions and Limitations. . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Detection Principles: Neyman-Pearson Test. . . . . . . . . . . . . . . . . . . . 19

2.3.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Neyman-Pearson Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.3 Generalized Likelihood Ratio Test. . . . . . . . . . . . . . . . . . . . . 20

2.3.4 Alternative Hypothesis of the Form θ > θH0. . . . . . . . . . . . . . . 21

2.3.5 Alternative Hypothesis of the Form θ 6= θH0. . . . . . . . . . . . . . . 21

2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Detection of Known Narrowband Signals in Narrowband Noise. . . . . 22

2.4.2 Detection of Known Narrowband Signals with Random Phase Angles. 23

2.5 Spherically Invariant Random Process (SIRP). . . . . . . . . . . . . . . . . . 24

2.6 Likelihood Ratio Test and Generalized Likelihood Ratio Test applied to theSpherically Invariant Random Process. . . . . . . . . . . . . . . . . . . . . . . 25

2.6.1 Detection of Known Narrowband Signals - Likelihood Ratio Test. . . . 25

2.6.2 Detection of Known Narrowband Signals with Random Phase Anglesand Random Amplitude - GLRT Detector. . . . . . . . . . . . . . . . 27

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Sea Clutter. 31

CONTENTS ii

3.1 Sea clutter characterization in X band. . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Sea clutter characterization in HF band. . . . . . . . . . . . . . . . . . . . . . 36

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Two Dirac delta detector. 50

4.1 Resolving GLRT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Two Dirac Deltas approximation. . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 First approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.2 Refined approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.2 Target Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.3 Additive Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.1 Classical STAP detection performance evaluation. . . . . . . . . . . . 65

4.4.2 Numerical simplifications for TDD STAP. . . . . . . . . . . . . . . . . 66

4.4.3 Comparison of classical STAP and TDD STAP for fixed ∆ parameter. 70

4.4.4 Results for TDD STAP detector with automatic ∆ finding. . . . . . . 73

4.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 HF radar signals experiments and STAP technique modifications. 76

5.1 WERA radar system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Implementation of Adaptive MTI and STAP - covariance matrix estimationproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Adaptive MTI implementation . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 STAP implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Comparisons between the results of AMTI and STAP . . . . . . . . . . . . . 88

5.3.1 Data file and target description. . . . . . . . . . . . . . . . . . . . . . 88

5.3.2 Detection of the tug ship from Garchine radar site. . . . . . . . . . . . 89

5.3.3 Detection of the tug ship from Brezzelec radar site. . . . . . . . . . . . 91

5.3.4 Detection of the fishery ship from Garchine radar site. . . . . . . . . . 92

5.3.5 Detection of the fishery ship from Brezzelec radar site. . . . . . . . . . 97

5.4 Thresholding and detections presentation. . . . . . . . . . . . . . . . . . . . . 100

5.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Conclusions and perspectives 104

A Gaussian complex process. 106

CONTENTS iii

B Data generation. 108

C Space Time Adaptive Processing based on Frequency Modulated Contin-uous Wave system. 110

C.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C.3 Antenna array with FMCW. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

C.4 STAP system using FMCW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

C.5 FMCW HF system - practical example. . . . . . . . . . . . . . . . . . . . . . 122

C.6 FMCW X-band system - practical example. . . . . . . . . . . . . . . . . . . . 123

C.7 FMCW L-band system - practical example. . . . . . . . . . . . . . . . . . . . 124

D List of symbols and abbreviations. 125

Bibliography 127

CHAPTER1 Introduction.

Present radar systems for sea surveillance have several limitations. One group of limitationsis related to strong clutter from sea waves (especially during heavy seas periods). Anothergroup is related to range limitations of present microwave systems. These are big obstacleshindering to provide reliable surveillance data that cover Exclusive Economic Zone (200nautical miles)1 24 hours a day, 365 days a year. Therefore a big challenge is to find newtechniques for this application. This work covers signal processing for this purpose. Space-Time Adaptive Processing (STAP) technique has a relatively long history. The theory ofSTAP was first published by Lawrence E. Brennan and Irving S. Reed in 1970’s [6], whereascomplete monographies on this subject were published by Richard Klemm in 2002 and 2004[37, 38]. It is also worth to mention here contribution of the paper written by E. J. Kelly [36].Nevertheless applications are mainly related to target detection in the presence of clutterthat has Gaussian statistical properties. In this work it is proposed to apply this techniqueto target detection under non-Gaussian conditions.

The thesis of this work can be formulated as follows: STAP can be an effective tech-nique for the Sea Surveillance Radars. Surveillance can be understood as clutter (andinterferences) suppression, target matching and threshold decision. To this end there are threeproblems that can be identified with regards to the sea clutter problem. These problems are:

1. Clutter non-stationarity in space and time.

2. Clutter non-Gaussianity.

3. Clutter with spread Doppler spectrum.

The purpose of this work is to evaluate different algorithms, to find possible problems withimplementing them and to try to find solutions to these problems. As a result this work servesas a complete guide how to deal with sea clutter by modifying STAP technique.

In the first chapter, reader can find elementary radar concepts as well as an introductionto Space-Time Adaptive Processing. First section is devoted to basic radar concepts. Nextsection is an introduction to Adaptive MTI (AMTI) and STAP. It will be shown how STAPwas introduced for airborne radar, and what was rationale standing behind this. Generallywe can say, that the origin of STAP was the observation, that clutter spectrum depends onthe look angle of radar system. Assumptions and limitations of STAP will be shown in thesame chapter. In the next sections reader can find some elements of detection theory andNeyman-Pearson Lemma and Test. This will build a base to derive more general detectorsthan STAP in chapter 4. In the same chapter theory of Spherically Invariant Random Process

1EEZ zone was defined in 1982 by United Nations Convention in Montego Bay.

CHAPTER 1. INTRODUCTION. 2

(SIRP) is introduced. This theory is very useful when dealing with non-Gaussian clutter. Itwill be shown in the next chapter, that sea clutter very often has non-Gaussian propertiesand in this case we can employ theory of SIRP. Last section will be devoted to derivation ofNeymann-Pearson tests in the case of SIRP. It will be shown, that classical STAP algorithmcan be derived from this more general form.

The second chapter is devoted entirely to sea clutter characterization. Two radar bandswill be considered: X-band and HF-band. In the first section, X-band clutter Doppler-spectrum and its properties will be presented. It will be shown, that clutter Doppler shiftand statistics are related to geometry of the scene as well as many ocean parameters [9]. Thisproperty is exploited in STAP algorithm, therefore it is worth considering STAP techniqueto deal with this kind of clutter. Unfortunately X-band sea clutter (especially for low graz-ing angles) has non-Gaussian properties [8], whereas STAP was derived under assumptionof Gaussianity. Therefore it is possible to improve classical STAP algorithm to deal withsea clutter. This problem will be treated in chapter 4. For HF-band, sea clutter propertiesare different. The main contribution to the clutter is Bragg scattering. Its Doppler spec-trum remains the same across different look angles. Moreover, for HF-band it is very likelyto have strong radio interferences. Author’s own calculations illustrating Bragg clutter andinterferences will be presented. Results were obtained using real data from WERA radar sys-tem. Again we can see, that clutter and interferences have two-dimensional, space and timestructure, and therefore it is reasonably to use STAP algorithm.

Chapter 4 addresses the problem of the derivation of detectors under non-Gaussianitythat was raised in chapter 3. This is done in the framework of Spherically Invariant Ran-dom Process. A new detector will be presented. It can deal with non-Gaussian clutter andnoise. To evaluate performances of classical and new STAP detector in non-Gaussian clutter,I performed some simulations. Receiver Operation Curves (ROC) are presented based onsimulations made by the author. A discussion of the performances of usual STAP and theproposed detector under different kind of clutter (Gaussian, non-Gaussian) is included. I willshow, that the new developed detector can give some improvement in comparison to classicalSTAP algorithm in the presence of non-Gaussian clutter.

In chapter 5 experiments from High Frequency (HF) radar system will be presented. HFradar systems, which operate in frequency range between 3 and 30 MHz, have a potential todetect targets which are located beyond optical horizon on the sea surface (Over The Horizonvisibility - OTH). In this context new problems have to be faced. An exhaustive review ofthese problems can be found in [3]. In this chapter two techniques will be considered. First oneis AMTI, and the second is STAP. Because of practical problems, classical algorithms mustbe adapted, which will be also shown in this chapter. Results were obtained using real data,from the oceanographic system WERA. This chapter is concentrated on signal processingpart and to much less extent on detection problems.

Additional problems are addressed in Appendices. Among them, the most important isthe problem of application of STAP algorithm to Continuous Wave (CW) systems. Thisproblem is treated in Appendix C.3.

This work can be viewed as an attempt to find how to apply STAP technique to theproblem of detecting targets on the sea surface.

This PhD thesis is a result of cooperation between Military University of Technology inWarsaw (Poland) and Telecom-Bretagne in Brest (France).

CHAPTER2 Radar Basics,Space-Time AdaptiveProcessing and TargetDetection.

Present radar systems operate in very difficult environment, where target echo must competeagainst ground or sea clutter, noise, interferences and jamming. Therefore, apart from solv-ing strictly technical problems such as increasing peak power, improving range resolution,there is a need of effective signal processing techniques to detect targets while maintainingreasonably low level of false alarms. In the era of non-coherent radars, the solution was toreduce resolution cell to limit the clutter power. After deployment of pulse-coherent radarsit became possible to apply more advanced techniques based on Doppler effect. This is howMoving Target Indication (MTI) and pulsed-Doppler radars were invented. We can gener-ally say that these techniques are based on Fourier analysis and sometimes adaptive filtering(Adaptive MTI) and proved to be very effective in suppressing clutter. Very exhaustive de-scription of such systems can be found in [53]. However, these techniques are based only ontime-domain. The other group of signal processing techniques is based on adaptive antennas.This group is operating in space domain and employ array antennas to suppress directionalinterference and other directional distortions. More information on this topic can be foundin [44]. The problem however persisted, how to combine these two techniques - one operatingin time domain and the other in space domain. The solution was introduction of Space-TimeAdaptive Processing (STAP), that will be described in this chapter. STAP is a technique thatperforms adaptation in two domains at the same time (whereas Adaptive MTI works only intime domain). This raises possibilities of suppressing distortions that have two-dimensionalstructure.

In section 2.1 elementary radar concepts will be presented. In section 2.2 Adaptive MTI(AMTI) and classical STAP algorithm will be introduced. It will be shown how STAP wasintroduced for airborne radar, and what was rationale standing behind this. Assumptionsand limitations will be shown in the same section. Section 2.3 will be devoted to detectiontheory and to the Neyman-Pearson Lemma and Test. This will build a base to derive moregeneral detectors than STAP. Some examples of Neyman-Pearson tests will be presented insection 2.4. Section 2.5 will present theory of Spherically Invariant Random Process (SIRP).This theory is very useful when dealing with non-Gaussian clutter. It will be shown later,that sea clutter very often has non-Gaussian properties and in this case we can employtheory of SIRP. Section 2.6 will be devoted to derivation of Neymann-Pearson tests in thecase of SIRP. It will be shown, that classical STAP algorithm can be derived from this

CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING ANDTARGET DETECTION. 4

Figure 2.1 — Radar principle - electromagnetic wave reflection from a target.

more general form.

2.1 Radar principles.

Radar is a sensor that uses electromagnetic waves to obtain information about the range,direction or speed of moving or fixed objects. The simplest radar system transmits modulatedpulses of electromagnetic wave. Electromagnetic wave is then partially reflected by an objectand returns toward radar receiver as a target echo (see Fig. 2.1). Radar can measure thetime between transmition of the pulse and reception of the echo, and calculate distance tothe target using formula:

R =ct

2(2.1)

where R is a measured distance to the target, c is a speed of light and t is elapsed timebetween transmission and reception of the echo. Fig. 2.2 presents basic time relations thatare important in radar systems. Pulse Repetition Interval (PRI) is related to the fact thatusually radars transmit signals that are periodical. PRI is a time between two successivepulses transmitted by the radar. τ denotes pulse duration. t is the time that is passed betweentransmission of the pulse and reception of the target echo. Usually, information about distanceto the target is presented in the discreet form, therefore instead of the range it is better to usethe term range gate or range cell - Fig. 2.3. This means nothing more but the discretization ofthe range value. This principle allows to obtain information about the distance and direction


Figure 2.2 — Basic time relations.

Figure 2.3 — Range gates.

to the target. To be able to extract velocity information about the target it is necessaryto introduce pulse-coherent radar. This means the use of quadrature receiver which extractsamplitude together with phase information on pulse to pulse basis. Structure of such a receiveris presented in Fig. 2.4. After this process, for each range gate we have a series of complexsamples. Samples are obtained from successive pulses of the radar. If the target is not moving,relative to the radar, phase of this series of samples is constant (precisely angles of complexsamples). If the target is moving between pulses, then the angles of complex samples arechanging from pulse to pulse. This is the way to extract Doppler, and therefore velocityinformation. In practice, to extract Doppler information, a finite sequence of pulses is used.Therefore it is worth to remind some basic relations:

fD =2Vr

λ(2.2)

Figure 2.4 — Quadrature Receiver [41]. fc denotes radar carrier frequency, fs denotes sam-pling frequency.


Figure 2.5 — Environmental diagram [53].

where fD denotes Doppler frequency of the target, λ is a radar pulse wavelength and Vr is atarget radial velocity relative to the radar. Doppler resolution is related to the dwelling time,which in practice means the time of coherent processing:

Dr =1

TDWELL(2.3)

where Dr denotes Doppler resolution expressed in Hz. This can be expressed in terms ofnumber of pulses - n. Let PRF=1/PRI (Pulse Repetition Frequency). Then:

Dr =PRF

n(2.4)

It can be seen, that to improve Doppler resolution it is necessary to increase dwelling timewhich in practice means to use more pulses in processing. The PRI multiplied by the numberof processed pulses is often called Coherent Processing Interval (CPI) and is exactly the sameas dwelling time. The other important relation is an unambiguous velocity condition:

PRF > 2fD (2.5)

This condition is a radar equivalent of Nyquist-Shannon-Kotelnikov theorem. It means, thatto be able to unambiguously determine the target velocity it is necessary to have PRF greaterthan twice the Doppler shift of this target. In practice, however, very often this condition isn’tfulfilled and we have situation, when velocity is ambigiuous. Operator of the radar, in sucha case, must be aware that velocity of the target can be different than the one indicated bythe system. Unfortunately in real scenarios, radar receives echoes not only from targets, butalso other signals. This can include interferences from other sources (eg. jamming), as wellas reflection from objects that are not interesting from the point of view of radar operator.All of this signals are called clutter. Sometimes the term clutter is used in a narrower sensewhen it means only passive distortions (excluding interferences). In Fig. 2.5 it is shown anenvironmental diagram that radar engineer can face. It can be seen, that apart from targets,many other sources of signal are present. These distortions can be suppressed by signalprocessing techniques such as MTI, Adaptive Beamforming (AB) or STAP.

MTI technique is based on Doppler principle. For example, if only ground clutter is presentfor not moving radar system, then clutter has approximately zero Doppler shift. Therefore


Figure 2.6 — Principle of MTI technique [53].

Figure 2.7 — Concept of steering antenna null toward interference [33].

to reject clutter it is necessary to implement Doppler filter having stopband around zeroDoppler frequency (see Fig. 2.6).

Situation is more complicated when the clutter is moving in relation to the radar. Thenadaptive techniques can be useful in this case. Adaptivity means, that filters adapt themselvesto the clutter properties in Doppler domain.

Another group of techniques are phased or adaptive arrays. The concept of adaptivearray is based on the idea of changing antenna pattern in order to steer antenna null towardinterference source as presented in Fig. 2.7.

This can be realized using two-antennas and introducing signal phase shift between an-tennas (Fig. 2.8).

Modification of this technique is an application of quadrature receivers (the same as forMTI technique). In this case each antenna has its own quadrature receiver. Then null steeringis performed by applying appropriate complex weight to each of the antennas (see Fig. 2.9).

Finally when dealing with multiple interference sources, it is necessary to apply multiplenulls. This can be done using more antennas in array antenna (Fig. 2.10).

Example of a complete system is presented in Fig. 2.11.

For simplicity reasons quadrature receivers are not presented in the picture.

Next step in radar technology evolution was to introduce combination of a MTI andadaptive arrays in a single technique. This concept is named Space-Time Adaptive Processingand is presented in the next section.


Figure 2.8 — Null steering using phase shifters [33]. d denotes distance between antennas.

Figure 2.9 — Two element adaptive antenna [33].


Figure 2.10 — Four antennas array [33].

Figure 2.11 — Complete adaptive array system [44].


Figure 2.12 — Airborne clutter [53].

2.2 Overview of STAP.

STAP is a modern signal processing technique, that can improve target detectability in thepresence of a strong clutter. In this section only short review on this topic is presented, moreexhaustive analysis of STAP can be found in a book written by Richard Klemm [37].

2.2.1 Problem Statement.

If we consider Moving Target Indication for airborne radar, then we face the problem thatecho coming from non-moving ground objects (ground clutter) possess non-zero Dopplerbandwidth [53] (Fig. 2.12). This is a result of relative velocity between antenna platform(aircraft) and ground area illuminated by the radar system. As a consequence, target echocan fall within the clutter bandwidth and may be hidden under the clutter (Fig. 2.12 showssimpler case, where target echo is outside the clutter band). In this case, clutter rejection willalso reject target echo. STAP objective is to filter-out ground clutter, while preserving echocoming from moving target.

2.2.2 Radar System.

In order to achieve its objective, STAP employs antenna array, which allows angle of arrivalfiltering. In Fig. 2.13 we can see the geometry of airborne antenna arrays. Axis Vp denotesflight direction. There are possible two basic configurations: side-looking and forward-looking.For side-looking configuration, receiving elements are placed along the flight axis. For forward-looking configuration, receiving elements are placed along an axis perpendicular to the flightaxis, but parallel to the ground plane. Side-looking configuration is simpler to analyze, there-fore we assume this configuration only. Also for simplicity reasons, we assume that receivingelements are equally-spaced, with element spacing equal to λ/2 (λ denotes wavelength). An-tenna array allows cone-angle (α in Fig. 2.13) filtering. Echo arriving from point P (Fig.2.13) on the ground is sampled in space by array elements. Wave-front coming from point Pis arriving at antenna receiving elements at different moments in time. In other words: thereis a phase difference among array channels. This phase difference is related to cone angleα. For other configurations (with slant angle), analysis is more complicated, but qualitativeresults are similar.


Figure 2.13 — Geometry of airborne antenna arrays [37].

Figure 2.14 — Moving radar system [58].

2.2.3 Airborne Clutter.

If we consider moving radar system (Fig. 2.14), it can be shown that Doppler shift of echocoming from non-moving environment depends on cone angle [37]. Each point in space, seenby the radar under angle α is approaching radar at the same speed. More precisely, this speedis proportional to the radar platform velocity and to the cosinus of the angle α. Therefore,all such points have the same Doppler shift (proportional to cosα). The cone is, therefore,surface of constant Doppler shift. But usually, airborne radar experiences reflections not fromthe whole space but from the ground plane only (excluding potential target). Therefore, toobtain the set of points with the same Doppler shift we should intersect cone surface withthe ground plane as in Fig. 2.14. Result of intersection of a cone with a plane is a hyperbola.Hyperbola is therefore a set of all points with the same Doppler shift. Lines on the ground ofconstant Doppler shift are named isodops. In our case a single hyperbola is exactly an isodop.


Figure 2.15 — Isodops [37].

Figure 2.16 — Range sphere [58].

Each Doppler shift is related to a single isodop (hyperbola). Bunch of isodops is shown inFig. 2.15. In the same figure it can be seen that zero-Doppler isodop (fr = 0) is a straightline perpendicular to the flight path. On the right side of zero-Doppler isodop, are placedpositive-Doppler isodops. The last of them is a maximum-Doppler isodop (fr = 1) which is infact a half of a straight line. On the left side are placed negative-Doppler isodops.

We are usually interested in target detection, together with the information about therange to the target. Isorange line is a line of constant range to the radar. It is a circle onthe ground, which is an intersection of a ground plane and a sphere with the radius whichcorresponds to our range of interest(Fig. 2.16 and 2.17).

Taking into account both isodops and isorange lines, we see that for the specific rangeand for the specific Doppler shift, we receive echo from specific areas on the ground. This can


Figure 2.17 — Isorange circles and isodops [53].

Figure 2.18 — Space-time structure of clutter [37].

be paraphrased: clutter from the specific range and of the specific Doppler shift is arrivingfrom specific angles only. This relation is exploited in STAP.

In Fig. 2.18 we can see space-time clutter structure in Doppler frequency-cosinus of coneangle (α) coordinate system. Clutter occupies only part of the coordinate plane. Moreoverfor the simple case of a side-looking antenna configuration, clutter lays on a straight line (inFig. 2.18) as a result of the fact, that Doppler shift is proportional to the cosα, as was shownbefore. For other antenna configurations, clutter has a different pattern, but always occupiesonly small part of that plane.

In the picture (Fig. 2.18) two possible targets are depicted. One so-called slow target andthe other so-called fast target. Slow target (dashed line) is a target whose radial speed Vr tothe radar is small in comparison with the speed of the radar platform relative to the groundVp. Movement of the radar platform generates clutter whose Doppler spectrum extends from


Figure 2.19 — Data cube [43].

−2Vp/λ to +2Vp/λ, where λ is a radar working wavelength (see Fig. 2.12). As a result, clutterDoppler spectrum generated by the movement of the platform can be broader than Dopplershift of the target. Therefore echo from this target is likely to be suppressed by the temporalMTI filter (see Fig. 2.6). On the other hand, fast target is a target whose radial speed to theradar platform is high in comparison to the radar platform speed. In this case there is onlylow risk, that echo from this target will be suppressed. Since we are interested in detectingboth slow and fast targets, it can be seen, that separate Doppler (inverse temporal clutterfilter in Fig. 2.18) or separate angle filtering (inverse spatial clutter filter in Fig. 2.18) mayfilter-out target signal together with the clutter. Some of the targets will not be detected.Two-dimensional filtering (space-time clutter filter in Fig. 2.18) allows to filter out clutterecho and preserve target echo.

2.2.4 Adaptive MTI.

Before introducing STAP algorithm it is worth to recall Adaptive MTI (AMTI). This algo-rithm was used for example in [25].

AMTI (purely temporal adaptive filtering) utilizes echoes from coherent pulse trains re-ceived by an antenna array. Pulse trains allows temporal (Doppler) filtering, whereas differentchannels of antenna array allow spatial (angle) filtering.

AMTI is operating on so-called data cube (Fig. 2.19). Data cube consists of complexsamples taken from N pulses by M antenna elements for range cells from 1 to L. We areassuming, that pulse train consists of N coherent pulses. Echoes are coherently sampled bya quadrature receiver as described in section 2.1.

The first step of processing is beamforming (Fig. 2.20).

After this step data are available as a cube with beams instead of antennas (Fig. 2.20and 2.21). The rest of the processing is performed independently for each beam. For eachrange-beam cell, adaptive filter is calculated using auxiliary range cells of the same beam.Formula for the test statistic is given by [53]:

η =|vHR−1

k xk|2vHR−1

k v(2.6)


Figure 2.20 — Beamforming.

Figure 2.21 — Data snapshot.

where v is a steering vector for tested velocity, xk is a data vector (N pulses) from the rangecell and the beam of interest, and R−1

k is an inverse of the sample covariance matrix. This teststatistics has a property of Constant False Alarm Ratio (CFAR), as long as clutter followsGaussian distribution [53]. Sample covariance matrix is calculated using auxiliary snapshotsof the same beam. Usually auxiliary snapshots are taken from the same beam but fromdifferent range cells. It is worth to mention here, that above test statistics is of the same formas for STAP (which will be shown in the next section). The difference is the interpretationof vectors and matrices present in the equation (2.6). For AMTI, vectors and matrices arerelated to a single beam and multiple pulses, whereas for STAP it will be vectors related totwo-dimensional, space and time structure. Underlying assumption of both AMTI and STAPis that clutter follows Gaussian process.

After such a processing, results available in the form of the test statistics cube (range ×velocity × angle).


2.2.5 STAP Processing.

To make profit of angle dependency of Doppler shift, STAP is operating on the raw datacube (Fig. 2.19). STAP is processing one slice of the data cube, for the range cell of interest:

Xk =

[xk]1,1 [xk]1,2 . . . [xk]1,N

[xk]2,1 [xk]2,2 . . . [xk]2,N...

.... . .

...[xk]M,1 [xk]M,2 . . . [xk]M,N

(2.7)

where k denotes range cell, M is the number of antennas, N is the number of pulses processed,and in expression [xk]a,b , a denotes antenna and b denotes pulse. Then Xk is the backscattersignal from one range cell (k) but for all pulses and antennas. For further processing weneed to vectorize slice (2.7) by stacking each succeeding column one beneath the other. Thisoperation yields the space-time snapshot for the k -th range:

xTk =

[[xk]1,1, [xk]2,1 . . . [xk]M,1, [xk]1,2, [xk]2,2 . . . [xk]M,2, [xk]1,3, [xk]2,3 . . . . . . [xk]M,N

](2.8)

T denotes transposition.

Let us consider a single reflecting point. Assuming that first antenna element is a referencepoint, received space-time snapshot should be of the form [37]:

xk = a · ss−t(fsp, fd) (2.9)

where a denotes a random complex amplitude and ss−t(fsp, fd) is the following steering vector:

ss−t(fsp, fd) =

1 · 1exp(j2π · fsp) · 1

exp(j2π · 2fsp) · 1exp(j2π · 3fsp) · 1

...exp(j2π · (M − 1)fsp) · 1

1 · exp(j2π · fd)exp(j2π · fsp) · exp(j2π · fd)

...exp(j2π · (M − 1)fsp) · exp(j2π · fd)

1 · exp(j2π · 2fd)exp(j2π · fsp) · exp(2π · 2fd)

...

...exp(j2π · (M − 1)fsp) · exp(j2π · (N − 1)fd)

(2.10)

where fsp = dλ cos(α), fd = 2·νr

λ T , α denotes the cone angle (Fig. 2.13), vr - radial velocitybetween reflecting point and antenna, λ - wavelength and T is a PRI. The steering vectorcan be written as ss−t(fsp, fd) = ssp(fsp)⊗ st(fd), where ssp(fsp) is a vector performing spaceprocessing and st(fd) is the Doppler processing vector. When multiplying a vector by theabove steering vector, we perform in fact a 2-D Fourier Transform in the space and the timedomain.


Registered space-time snapshot is a result of coherent summation of waves from manysources:

xk = sk + ck + jk + nk (2.11)

sk denotes a target reflection, ck stands for clutter return, jk is jammer signal and nk repre-sents uncorrelated noise. Of course target and jammer signals are not always present.

Classical STAP processor is a linear filter [36] [51] of the form [37]:

yk = wHk · xk (2.12)

where yk is a resulting scalar, wk - weight vector, and superscript H denotes conjugatetranspose. Formula for the optimum weight vector1 is [37]:

wk = β ·R−1k · ss−t(fsp, fd) (2.13)

β denotes scalar, R−1k is an inverse of covariance matrix of xk: Rk = E{xkxH

k } assuming notarget signal present in the data. ss−t(fsp, fd) is a steering vector (2.10) for the possible targetwe want to detect. Above formula can be derived from Generalized Likelihood Ratio Test,which will be shown in the section 2.6. In practice both R−1

k and ss−t(fsp, fd) are unknown.Instead of R−1

k , an inverse of the sample covariance matrix R−1k is often used.

ss−t(fsp, fd) is steering vector for, only one, possible target. To construct this vector itis necessary to possess perfect knowledge about the target velocity and angle α relative tothe antenna (see Fig. 2.13). In practice neither velocity nor angle is known. Instead, we cantest for targets at a series of discrete points covering angle-velocity space (test grid). Testgrid must be dense enough, to sufficiently precisely approximate velocity and angle of anypossible target. Then each point from velocity-angle grid is transformed into a surrogatesteering vector vs−t, that will be used in (2.13) as a replacement of ss−t(fsp, fd). Mismatchbetween real ss−t(fsp, fd) and surrogate vs−t leads to some performance loss of detector.

For MTI purpose, decision function has to discriminate between two hypotheses:

H0 : xk = ck + jk + nk

H1 : xk = sk + ck + jk + nk

where H0 denotes null hypothesis that there is no target within the range cell of interest, H1

denotes an alternative hypothesis that target is present.

Using optimum weight vector and taking [43]:

β =1√

vHs−tR

−1k vs−t

,

gives test statistics [43]:

η =|vH

s−tR−1k xk|2

vHs−tR

−1k vs−t

(2.14)

1This is true under different optimization criteria - Maximum Likelihood estimation(ML), Signal toNoise(plus Interference) Ratio maximization(SNR), Minimum Noise Variance(MV) estimation and Least MeanSquare Error(LMSE) estimation, but only under assumption of proper complex Gaussian process (multivari-ate complex Gaussian process with vanishing pseudo-covariance [46]) of the clutter. This assumption will beexamined more deeply in subsequent parts of this work.


This test statistics has a Constant False Alarm Ratio (CFAR) property [51] (from [43]). Asa result, a two-dimensional real test statistics map is obtained as an output of the STAPfilter (for any range cell of interest). One dimension of this map is an angle, and the otheris a velocity of a potential target. This map represents a grid of surrogates vs−t for a truesteering ss−t, which is unknown.

Decision function is constructed by comparing η to a fixed threshold γ. If η < γ then theverdict is: target is absent. If η > γ the verdict is: target is present.

2.2.6 Assumptions and Limitations.

Gaussian Assumption.

STAP is a special case of optimum processing in the presence of Gaussian distortions. Formu-lation for optimum processing can be found, among others, in [53] and [44]. STAP underlyingassumption is: clutter can be modeled by a proper complex Gaussian (band-pass) process[37] which is explained underneath. For the simple one-dimensional case, proper Gaussianprocess can be represented as [44] [53]:

ct = xt cos(ωct)− yt sin(ωct) (2.15)

where xt, and yt, are independently, identically distributed from normal distribution of a zeromean value and the same variance. Then:

ct = vt cos(ωct + θt), (2.16)

θt is uniformly distributed, and voltage envelope vt is Rayleigh distributed. We can now treatxt and yt as real and imaginary parts of complex process.

For multidimensional case (e.g. multiple M sensors and coherent N pulses), xt and yt

become vectors xt and yt. xt is an In-phase vector:

xTt =

[[xt]1, [xt]2, [xt]3 . . . [xt]M ·N

]

and yt is an Quadrature vector:

yTt =

[[yt]1, [yt]2, [yt]3 . . . [yt]M ·N

]

Similarly to one-dimensional case, we assume that:

E{xt} = 0, E{yt} = 0

This implies that the process is entirely characterized by its covariance matrix:

E{[ x

y

][xTyT]

}= M

For narrowband processes, covariance matrix can be presented in the specific form (seeAppendix A) [48] [56] [44] [53]:

E{[ x

y

][xTyT]

}=

12

[V −WW V

]


Therefore it is possible to represent this process in complex notation. If we assume complexvector:

zt =

[xt]1 + j[yt]1[xt]2 + j[yt]2...[xt]M ·N + j[yt]M ·N

then the covariance matrix is:

E{zzH} = V + jW

Using this notation, we describe multidimensional narrowband Gaussian process using MN xMN complex covariance matrix. This result is then used in STAP formula in (2.13) as Rk. Therest of (2.13) - ss−t(fsp, fd) - can be viewed as a matched filtering. For other distributions ofthe clutter, covariance matrix does not describe distribution entirely, so test statistics must bereformulated. In this case, optimum detector will not necessary be a linear filter as discussedin section 2.3.

Estimate of R−1k .

As it is stated on page 17, in practice we do not know real covariance matrix. Therefore wemust use an estimator. Usually Sample Covariance Matrix (SM) is used [37]:

Rk =1P

P∑

m=1

xmxHm (2.17)

where vectors xm are usually taken from neighboring cells of space-time snapshot of data cube(see Fig. 2.19). This estimator is Minimum Likelihood Estimator if xm are IID (Independentand Identically Distributerd). Moreover they should come from the same distribution asthe clutter at the range cell of interest (this restriction is called homogeneity condition).Additionally we must exclude target signal from samples taken for calculating SM. Thereforewe must exclude sample from range cell of interest and samples in vicinity (guard cells).Moreover, for SM to be invertible P must be greater than dimension of covariance matrix[12], which is MN - see Fig.2.19 and (2.8). However, for good performance, P should begreater than 2MN [50].

For practical applications, for example M = 22, N = 128 and P > 5632, it may bedifficult to provide adequate homogeneous sample support.

2.3 Detection Principles: Neyman-Pearson Test.

In this section Neyman-Pearson Test will be introduced. It is worth to mention, that thereis a possibility to adopt other approach like Bayesian reasoning or non-parametric tests. Inthis work, however, author concentrated on classical Neyman-Pearson theory. First, it willbe shown LRT for simple hypothesis testing. After it will be shown GLRT that it is oftenused for composite hypothesis testing. Finally, it will be presented Locally Optimum (LO)test, that can be used under certain constraints for composite alternative hypothesis. Twoexamples of LO test will be shown.


2.3.1 Notation.

Let us assume random vector X = (X1, X2, X3, . . . , Xn) of observations with joint probabilitydensity function (PDF) fX(x|θ), where θ is a parameter of the density function. Let x =(x1, x2, x3, . . . , xn) be a vector of observations (specific realization of X ) in Rn. Let us furtherassume that θ ∈ Θ = ΘH0 ∪ΘH1 , ΘH0 and ΘH1 are disjoint. Hypothesis testing can be viewednow as deciding between:

H0 : X has PDF fX(x|θ) with θ ∈ ΘH0

H1 : X has PDF fX(x|θ) with θ ∈ ΘH1

(2.18)

Decision should be based on observations x. Let us use δ(x) symbol for the decisionfunction. δ(x) = 0 means adoption of hypothesis H0, δ(x) = 1 means adoption of hypothesisH1. If set ΘH0 consists of a single element θH0 only, we say that hypothesis zero is simple,otherwise we say that hypothesis zero is composite. Similar relates to alternative hypothesisH1.

The power function p(θ|δ) of a test based on a test function δ is defined for θ ∈ ΘH0 ∪ΘH1

as:p(θ|δ) = E {δ(x)|θ} (2.19)

In target detection problems, power function can be viewed as a function that gives probabilityof detection as a function of unknown parameter θ. Usually we want this function to be highfor θ ∈ ΘH1 .

The size of a test is the quantity:

α = supθ∈ΘH0p(θ|δ) (2.20)

Size of a test in radar target detection problem is translated to the probability of a falsealarm. The aim is to have power function (probability of detection) high, meanwhile havingsize of a test (probability of false alarm) reasonably low.

2.3.2 Neyman-Pearson Lemma.

Let us consider simple hypothesis H0 (θ = θH0) and simple hypothesis H1 (θ = θH1). Generalstructure of a most powerful test2 may be described as one comparing the likelihood ratio toa constant threshold [35],

fX(x|θH1)fX(x|θH0)

> t (2.21)

If the likelihood ratio on the left-hand side of (2.21) is greater than threshold t, than wedecide to accept hypothesis H1. Constant t may be evaluated to obtain desired test size α.

2.3.3 Generalized Likelihood Ratio Test.

Let us consider again general test of the form (2.18) - composite hypothesis testing. To useNeyman-Pearson Lemma and likelihood ratio test (2.21) we can adopt maximum likelihoodestimates θH0 and θH1 of the parameter θ, obtained under the constraints θ ∈ ΘH0 andθ ∈ ΘH1 respectively [35]. These estimates can be used in place of θH0 and θH1 in (2.21). Test

2Most powerful test is a hypothesis test which has the greatest power among all possible tests of a givensize.


derived in this procedure is named General Likelihood Ratio Test. It is possible to considera special case of this test, when null hypothesis is composite and alternative hypothesis issimple. This is often the case in radar signal processing, when we want to detect a targetecho.

H0 : XhasPDF fX(x|θ) with θ = θH0 againstH1 : XhasPDF fX(x|θ) with θ 6= θH0

(2.22)

Then θH0 can be replaced by exactly known θH0 , and GLRT looks as follows:

fX(x|θH1)fX(x|θH0)

> t (2.23)

denumerator is of the same as in LRT, and numerator comes from GLRT. This is form, thatwill be later used in derivation of target detectors.

2.3.4 Alternative Hypothesis of the Form θ > θH0.

Sometimes it is possible to derive other detectors based on LRT. This time we assume aspecial form of the test:

H0 : XhasPDF fX(x|θ) with θ = θH0 againstH1 : XhasPDF fX(x|θ) with θ > θH0

(2.24)

We now say that H0 is a simple hypothesis, but H1 is composite. In this case we can deriveLocally Optimum (LO) test. This means that, for assumed test size α, decision function willbe most powerful in vicinity of θH0 in deciding against θ > θH1 . Test structure is as follows[35]:

ddθfX(x|θ)

∣∣∣∣θ=θH0

fX(x|θH0)> t (2.25)

where t is again threshold appropriately chosen to achieve desired test size. It can be seen thatdenominator of test statistics is the same as in (2.23). Numerator is however of an arbitraryform adjusted to be optimal in vicinity of θH0 . This detector will behave properly for θs closeto θH0 . This test can be also viewed as [35]:

d

dθln

(fX(x|θ)

)∣∣∣∣∣θ=θH0

> t (2.26)

The problem with its practical use is the necessity of knowing derivative of likelihood function.

2.3.5 Alternative Hypothesis of the Form θ 6= θH0.

There are also many situations when we need to test two sided alternative hypothesis - thesame as GLRT in (2.23):

H0 : XhasPDF fX(x|θ) with θ = θH0 againstH1 : XhasPDF fX(x|θ) with θ 6= θH0

(2.27)

H0 is again simple hypothesis, H1 is composite. In this case we can also derive LO test. Inorder to achieve this, we need to impose some restrictions to the test function. Test function


must be unbiased. Which means that test satisfies [35]:

p(θ|δ) ≤ α, for all θ ∈ ΘH0

p(θ|δ) ≥ α, for all θ ∈ ΘH1

(2.28)

For the two-sided alternative hypothesis, Locally Optimum(LO) looks as follows[35]:

d2

dθ2 fX(x|θ)|θ=θH0

fX(x|θH0)> t (2.29)

where t is, again, threshold appropriately chosen to achieve desired test size. The same asbefore, it can be seen that denominator of test statistics is the same as in (2.23). Numeratoris again of an arbitrary form adjusted to be optimal in vicinity of θH0 . This detector willbehave properly for θ’s close to θH0 . This time it is necessary to know second derivative ofa likelihood function. Therefore, though it may be interesting to apply these detectors fornon-Gaussian clutter, in the rest of the work GLRT test in the form (2.23) will be used.GLRT test can be applied in more general case and in many situations GLRT is simpler toderive and analyse.

2.4 Examples

In this section, two examples will be presented as an illustration of the tests presented insection 2.3. These two examples are close to the problems faced in radar target detection.

2.4.1 Detection of Known Narrowband Signals in Narrowband Noise.

Beneath will be presented an example of deriving LO detector in the case, where we canobtain a simple form of derivatives for uncorrelated noise. We will use model:

X(t) = θυ(t) cos[ω0t + φ(t)] + W (t) (2.30)

Here υ(t) and φ(t) are known amplitude and phase modulations. θ is overall signal amplitude.Noise process W (t) will be assumed to be stationary, zero mean, band-pass white noise witha constant power spectral density N0/2 over the band of interest (and zero outside). We areinterested in testing θ = 0 against θ > 0. After quadrature detection with carrier frequencyω0 and sampling with Nyquist frequency, we receive in-phase and quadrature samples xIi andxQi, where i = 1 . . . n. Samples can include signal part and noise part:

xIi = θsIi + wIi

xQi = θsQi + wQi(2.31)

Following the (2.30) we will assume that wIi form a sequence of i.i.d. random variables,governed by a common univariate density function fL. Similarly we assume the same for wQi

with the same PDF fL. We also assume that sIi and sIi are completely known.

Components wIi and wQi are uncorrelated for each i [56](from [35]), but independent onlyfor the Gaussian case. Therefore we must adopt bivariate probability density function fIQ,for which marginal density functions are fl. To summarize, we assume two dimensional noisesamples (WIi,WQi) that are i.i.d. random variables with common bivariate PDF fIQ.


Moreover we assume that fIQ is a circularly symmetric3 bivariate density function, whichmeans that fIQ(u, υ) can be written as a function of

√u2 + υ2:

fIQ(u, υ) = h(r)∣∣∣r=√

u2+υ2(2.32)

In this situation LO statistics can be derived using (2.25), and takes on the form [35]:

λLO(XI , XQ) =n∑

i=1

−h′(ri)rih(ri)

[sIixIi + sQixQi] (2.33)

where ri =√

x2Ii + x2

Qi. In (2.33) we can distinguish two elements. Nonlinear envelope modifi-

cation −h′(ri)rih(ri)

and linear operation on I and Q components: sIixIi+sQixQi (matched filtering).

For the special case of Gaussian noise components, −h′(ri)rih(ri)

becomes constant and LO filterbecomes the usual matched filter [35]:

λLO(XI , XQ) =n∑

i=1

[sIixIi + sQixQi] (2.34)

2.4.2 Detection of Known Narrowband Signals with Random Phase An-gles.

Next example illustrates also LO detector. Model in this situation is different from (2.30) inthe sense that we introduce additional random starting phase ψ of our signal of interest:

X(t) = θυ(t) cos[ω0t + φ(t) + ψ] + W (t) (2.35)

We assume that ψ is uniformly distributed over [0, 2π]. After similar operations as for (2.30),we get samples:

xIi = θ[sIi cosψ + sQi sinψ] + wIi

xQi = θ[sIi sinψ + sQi cosψ] + wQi(2.36)

To simplify formulation let us introduce additional notation:

Xi = (xIi, xQi)Yi = (−xQi, xIi)si = (sIi, sQi)υ2

i = s2Ii + s2

Qi

ri =√

x2Ii + x2

Qi

(2.37)

LO detector test statistics can be derived using (2.29), and becomes [35]:

λLO(XI ,XQ) =12

n∑

i=1

υ2i

{h′′(ri)h(ri)

−[h′(ri)

h(ri)

]2+

h′(ri)rih(ri)

}

+12

[n∑

i=1

−h′(ri)rih(ri)

siXTi

]2

+12

[n∑

i=1

−h′(ri)rih(ri)

siYTi

]2

(2.38)

3More general multivariate case of spherically invariant random processes will be discussed later


This form is very complicated and includes matched filtering together with nonlinearenvelope modification similarly to the previous example. Even in Gaussian case, this detectorremains non-linear.

Generally it is quite difficult to derive LO detectors. For calculations to be feasible itis desirable to have uncorrelated noise. On the other hand, GLRT test is based directly onNeyman-Pearson test and in many cases is easier to calculate. Therefore in the rest of thiswork only GLRT will be considered.

2.5 Spherically Invariant Random Process (SIRP).

In this section theory of SIRP will be introduced. In fact, as we detail in chapter 3, the SIRPcan model a large number of distortions that have been used for sea clutter. This is a veryusefull tool, that will be used in other sections of this work.

Spherically Invariant Random Process can be represented as [2]:

Z = σI

where Z denotes n-dimensional Spherically Invariant Random Vector (SIRV), I is a n-dimensional multivariate Gaussian random vector and σ is a positive random scalar withassumed probability density function (PDF). SIRP allows to relax assumption of Gaussian-ity, while keeping many of its useful characteristics [2]. PDF of a SIRV vector can be presentedin the following form [49]:

f(Z) =1

(2π)n/2|M |1/2

∫ ∞

0

1σn

e−ZTM−1Z

2σ2 dGσ(σ) (2.39)

Gσ(σ) denotes cumulative distribution function of σ If E(σ2) = 1 then M = E[ZZT]. Forfurther analysis we introduce complex notation using transformation:

X =

z1 + jzn/2+1...zn/2 + jzn

(2.40)

let us denote dimension of X by m = n/2. PDF of a vector X can be presented in thefollowing form [10]:

f(X) =1

πm|Φ|∫ ∞

0

1σ2m

e−XHΦ−1X

σ2 dGσ(σ) (2.41)

Similarily if E(σ2) = 1 then Φ = E[XXH]. If we assume continuous case for σ distributionthen we have:

f(X) =1

πm|Φ|∫ ∞

0

1σ2m

e−XHΦ−1X

σ2 g(σ)dσ (2.42)

and g(σ) is called characteristic PDF of a SIRP. It will be useful to introduce one morerepresentation of a SIRP. Let τ = σ2, then σ =

√τ , dτ = 2σdσ and dσ = dτ

2√

τ. Therefore

f(X) =1

πm|Φ|∫ ∞

0

1τm

e−XHΦ−1X

τg(√

τ)2√

τdτ (2.43)


where g(√

τ)2√

τis a PDF of a τ . Let us denote this PDF as f(τ) so:

f(X) =1

πm|Φ|∫ ∞

0

1τm

e−XHΦ−1X

τ f(τ)dτ (2.44)

or in more general form

f(X) =1

πm|Φ|∫ ∞

0

1τm

e−XHΦ−1X

τ dFτ (τ) (2.45)

where Fτ (τ) denotes cumulative distribution function for τ . We can assume restriction for τ :E(τ) = 1. Under this restriction we have Φ = E{XHX}, and this is a covariance matrix ofX [49]. This form will be used in the next sections.

Gaussian distribution as a special case of SIRP.

Let δ(x) denotes Dirac-delta function. If we assume dFτ (τ) = δ(τ0) in (2.45), then

f(X) =1

πm|Φ|1

τm0

e−XHΦ−1X

τ0 (2.46)

In above expression we recognize multivariate complex Gaussian random process. We cantherefore say, that SIRP is a generalization of a Gaussian process.

2.6 Likelihood Ratio Test and Generalized Likelihood RatioTest applied to the Spherically Invariant Random Process.

In this section it will be shown how to derive LRT and GLRT for SIRP. In the case of exactlyknown signal we can use LRT, whereas in the case of not known amplitude and phase weneed to employ GLRT. It will be shown, that STAP is a special case of GLRT.

2.6.1 Detection of Known Narrowband Signals - Likelihood Ratio Test.

If we know exactly the signal we want to detect, we can employ LRT. After deriving the test,it will be shown, that for a threshold equal 1, test statistics becomes matched filter.

Let now X be an m-dimensional SIRV, S is the known m-dimensional complex signalvector, Y is the m-dimensional complex vector of observations. Let us consider followinghypothesis test:

H0 : Y = XH1 : Y = X + S

Using (2.45), it is possible to find expressions for PDF of Y under both hypotheses [52]:

H0 : f(Y ) =1

πm|Φ|∫ ∞

0

1τm

e−Y HΦ−1Y

τ dFτ (τ)

H1 : f(Y ) =1

πm|Φ|∫ ∞

0

1τm

e−(Y−S)HΦ−1(Y−S)

τ dFτ (τ)


From general result of (2.21), Likelihood Ratio statistics is given by[52]:

η =

∫∞0

1τm e−

(Y−S)HΦ−1(Y−S)τ dFτ (τ)

∫∞0

1τm e−

Y HΦ−1Yτ dFτ (τ)

(2.47)

If distribution function of τ is continuous then we have:

η =

∫∞0

1τm e−

(Y−S)HΦ−1(Y−S)τ f(τ)dτ

∫∞0

1τm e−

Y HΦ−1Yτ f(τ)dτ

(2.48)

Simplified Result. Let us investigate result (2.47) more deeply (similar derivation to [59]).If we compare test statistics to the threshold:

η =

∫∞0

1τm e−

(Y−S)HΦ−1(Y−S)τ dFτ (τ)

∫∞0

1τm e−

Y HΦ−1Yτ dFτ (τ)

H1

≷H0

t, (2.49)

we see that it can be rearranged:

∫ ∞

0

1τm


τ dFτ (τ)H1

≷H0

t

∫ ∞

0

1τm

e−Y HΦ−1Y

τ dFτ (τ)

⇔∫ ∞

0

1τm


τ dFτ (τ)− t

∫ ∞

0

1τm

e−Y HΦ−1Y

τ dFτ (τ)H1

≷H0

0

⇔∫ ∞

0

[ 1τm


τ − t1

τme−

Y HΦ−1Yτ

]dFτ (τ)

H1

≷H0

0

⇔∫ ∞

0

1τm

[e−

(Y−S)HΦ−1(Y−S)τ − te−

Y HΦ−1Yτ

]dFτ (τ)

H1

≷H0

0 (2.50)

Now if we take threshold to be equal 1:

∫ ∞

0

1τm

[e−

(Y−S)HΦ−1(Y−S)τ − e−

Y HΦ−1Yτ

]dFτ (τ)

H1

≷H0

0 (2.51)

Keeping in mind, that τ is positive and the fact that function ex is monotonous, it can beconcluded that the sign of the expression:


τ − e−Y HΦ−1Y

τ

depends only on the difference between (Y − S)HΦ−1(Y − S) and Y HΦ−1Y . This is true forevery τ . Recalling again that τ is positive it can be concluded, that the sign of the functionunder integral in (2.51) depends only on the difference (Y −S)HΦ−1(Y −S)−Y HΦ−1Y , and


therefore the sign of the result also depends on this difference. So finally, our detector from(2.51) turns out to be a linear filter:

Y HΦ−1YH1

≷H0

(Y − S)HΦ−1(Y − S) (2.52)

further simplifications:

Y HΦ−1YH1

≷H0

(Y − S)HΦ−1Y − (Y − S)HΦ−1S

⇔ Y HΦ−1YH1

≷H0

Y HΦ−1Y − SHΦ−1Y − Y HΦ−1S + SHΦ−1S

⇔ SHΦ−1Y + Y HΦ−1SH1

≷H0

SHΦ−1S

⇔ SHΦ−1YH1

≷H0

SHΦ−1S/2

result is:

SHΦ−1Y

SHΦ−1S

H1

≷H0

12

(2.53)

the expression on the left side is possibly complex. Therefore it is necessary to take modulevalue of the numerator:

|SHΦ−1Y |SHΦ−1S

H1

≷H0

12

(2.54)

This result shows, that in case of SIRP as a model for the clutter, optimum detector isa matched filter, but only under condition that threshold in (2.49) is equal 1. This is nottrue for other thresholds, because integral (2.51) cannot be simplified. Therefore it can beconcluded, that this is not an interesting simplification, since it gives no possibility to adjustprobability of false alarm or probability of detection (both of which depend on the threshold).It is necessary to have a detector independent on the threshold.

2.6.2 Detection of Known Narrowband Signals with Random Phase Anglesand Random Amplitude - GLRT Detector.

This is more realistic scenario in radar applications. Our target signal is known precisely butits initial amplitude and phase are not known. In such a case we cannot employ LRT. Insteadwe need to use GLRT. It will be shown how to derive such a test in the case of SphericallyInvariant Random Process. After, it will be shown the same simplification as for LRT. In thiscase however, this simplification is useless. At the end of this section it will be shown, thatin fact, classical STAP is a special case of GLRT for SIRP.

Let now S = aS0, where S0 is the completely known m-dimensional signal vector withmagnitude 1 (SH

0 S0 = 1) and a is the unknown complex amplitude. It is now possible toderive GLRT (see section 2.3.3 on page 20).


Taking into account (2.45), GLRT statistics becomes [52]:

η =

∫∞0

1τm exp(−Y HΦ−1Y (1−|ρ|2)

τ )dFτ (τ)∫∞0

1τm exp(−Y HΦ−1Y

τ )dFτ (τ)(2.55)

where || denotes magnitude, and

|ρ|2 =|SH

0 Φ−1Y |2(Y HΦ−1Y )(SH

0 Φ−1S0)(2.56)

In the case when distribution of τ is continuous:

η =

∫∞0


τ )f(τ)dτ∫∞0


τ )f(τ)dτ(2.57)

where f(τ) is called characteristic PDF of SIRV.

Simplified Result. We want to perform similar reasoning to that in section 2.6.1. Similarresult to (2.50) is given by:

∫ ∞

0

1τm

[e−

Y HΦ−1Y (1−|ρ|2)τ − te−

Y HΦ−1Yτ

]dFτ (τ)

H1

≷H0

0 (2.58)

Let t be equal 1:

∫ ∞

0

1τm

[e−

Y HΦ−1Y (1−|ρ|2)τ − e−

Y HΦ−1Yτ

]dFτ (τ)

H1

≷H0

0 (2.59)

this is equivalent to (assuming τ > 0):

−Y HΦ−1Y

(1− |SH

0 Φ−1Y |2(Y HΦ−1Y )(SH

0 Φ−1S0)

) H1

≷H0

− Y HΦ−1Y (2.60)

further:

−Y HΦ−1Y +(Y HΦ−1Y )|SH

0 Φ−1Y |2(Y HΦ−1Y )(SH

0 Φ−1S0)

H1

≷H0

− Y HΦ−1Y (2.61)

after simplification:

|SH0 Φ−1Y |2

SH0 Φ−1S0

H1

≷H0

0 (2.62)

so finally our detector turns out to be:

|SH0 Φ−1Y |2

H1

≷H0

0 (2.63)

Of course left side will be greater than zero always, so in this case detector will always reporttarget. Although simplification is possible, however it does not give reasonable results and itwill not be considered in the rest of this work.


GLRT for Gaussian Clutter - a STAP Formula. In section 2.2.5 a classical STAPfilter was shown. Optimum test statistics is given in (2.14) and can be derived using (2.55),which will be shown in this section. First, for the Gaussian process, τ is constant (let usassume equal to σ). This means that Stieltjes integrals vanish, and we have formula for teststatistics:

η =1

σm exp(−Y HΦ−1Y (1−|ρ|2)σ )

1σm exp(−Y HΦ−1Y

σ )(2.64)

after simplification:

η =exp(−Y HΦ−1Y (1−|ρ|2)

σ )

exp(−Y HΦ−1Yσ )

(2.65)

and taking natural logarithm:

η′ = [−Y HΦ−1Y (1− |ρ|2)σ

]− [−Y HΦ−1Y

σ] (2.66)

constant σ can be included in the threshold:

η′′ = [−Y HΦ−1Y (1− |ρ|2)]− [−Y HΦ−1Y ] (2.67)

after rearrangement:η′′ = Y HΦ−1Y − [Y HΦ−1Y (1− |ρ|2)] (2.68)

further:η′′ = Y HΦ−1Y |ρ|2 (2.69)

and substituting expression for |ρ|2 from (2.56):

η′′ = Y HΦ−1Y|SH

0 Φ−1Y |2(Y HΦ−1Y )(SH

0 Φ−1S0)(2.70)

which finally gives formula equivalent to (2.14):

η′′ =|SH

0 Φ−1Y |2(SH

0 Φ−1S0). (2.71)

Variance Estimation. In (2.71) we assume, that covariance matrix (and its inverse) isknown. This is not true in general. In (2.14) inverse of the sample covariance matrix Φ−1

is used as an inverse covariance matrix estimator in place of true inverse covariance matrixΦ−1. But this detector (2.14) is not GLRT detector anymore. It can be found in [51] thatGLRT under assumption that covariance is unknown is given by:

|SH0 Φ−1Y |2

SH0 Φ−1S0

(1 + 1

P Y HΦ−1Y)

H1

≷H0

t (2.72)

where t is a threshold and Φ is given by:

Φ =1P

P∑

m=1

xmxHm (2.73)

as in (2.17).

It can be seen that for large P (2.72) becomes (2.14).


2.7 Conclusions

This chapter introduced basic radar systems concepts. After general introduction to radarprinciples in section 1, a STAP technique was described in section 2. Rationale standing be-hind introduction of STAP was presented. STAP processing algorithm together with assump-tions and limitations was shown. Next section presented more general view to the problemof detection. This is necessary in the context of non-Gaussian signals. In the same sectionGeneralized Likelihood Ratio Test was presented. For completeness Locally Optimum detec-tor was also shown. To resolve the problem of non-Gaussianity, at certain stage there is aneed to introduce the model of distortions. This was done in the next section, where Spheri-cally Invariant Random Process (SIRP) was introduced. Using this concept and GLRT it waspossible to present general form of a test adjusted to the SIRP for the signal with unknownphase. As it will be shown in the next chapter, problem of non-Gaussianity is a real problemin radar detection in the case of sea clutter, where reflections come from the sea surface whichhas a very complicated, non-linear physics. In this context GLRT for SIRP will be a basictool for improving detection performance. Having validated the need of target detection innon-Gaussian clutter, practical implementation of a GLRT for SIRP will be presented in thechapter 4.

CHAPTER3 Sea Clutter.

In this chapter sea clutter properties will be described. After short introduction, sea clutterproperties will be presented for X-band and HF-band radar systems. HF-band results will bebased on author’s experiments with real data.

Sea clutter signal properties are related to sea roughness. It is common assumption todecompose sea roughness to capillary waves and to gravity waves (swell) [28] [31]. The electro-magnetic wave that is backscattered by capillary waves modulated by three marine features(wind, gravity waves, current) has wavelength of order of centimeters or less [28]. Their restor-ing force is the surface tension. Gravity waves have wavelengths ranging from few hundredmeters to less than a meter [31]. Swells are produced by stable winds. Their restoring forceis the force of gravity. If we consider time relations, first results of a wind are capillary waves[28]. As capillary waves build up, their energy is transfered in a nonlinear process to waveswith larger amplitudes and wavelengths. Sudden cessation of wind causes short waves to de-cay rapidly, whereas the longer waves can last several days. In reality sea state is a complexmixture of shorter and longer waves coming from different directions.

Waves can be characterized by their height, length and period. Wavelength and periodare in a relation, whereas waveheight fluctuates considerably [31].

Because of different length scales of different sea-wave types, sea clutter for different radarbands will have different properties. Note that for example for HF radar band (3-30 MHz)capillary waves will have no effect on clutter properties, whereas for X-band systems, theywill have an important influence. However some general comments can be made. Dopplerspectrum of sea clutter results from two main processes: the spread about the mean Dopplerfrequency is a manifestation of the random motion of the unresolved scatters, while thedisplacement of the mean Doppler frequency is caused by the evolution of resolved waves[31], in particular by the effects of currents.

Clutter properties are strongly related to grazing angle. Grazing angle is the angle atwhich radar beam illuminates the surface with respect to the local horizontal. For HF radarsystems, grazing angles are almost always small, since these radars are usually installed at thesea coast or onboard of ships. For X-band and S-band radar systems, grazing angles can varyfrom 0 to 90 degrees. For high grazing angles, statistical properties are usually consideredGaussian, whereas for low grazing angles for X-band and S-band radars, statistical propertiesof the clutter can not be anymore considered Gaussian. For this reason we have consideredthe influence of this departure in the next chapter (chapter 4) , which deals with the detectionstep of the STAP.

CHAPTER 3. SEA CLUTTER. 32

Figure 3.1 — Typical X-band, sea-clutter spectra in the upswell and near cross-swell di-rections for vertical (upper figure) and horizontal (lower figure) polarizations [9].

3.1 Sea clutter characterization in X band.

X-band radar systems operate at frequencies between 8 GHz and 12 GHz. Wavelength istherefore between 2.5 cm to 3.75 cm. This is very important, since structures of this size willinfluence statistical properties of the clutter in this case.

At the beginning let us consider spectral properties of the clutter. In Figure 3.1 it isshown clutter power as a function of Doppler frequency for different look angles of radar andfor different polarizations. It is clear, that sea clutter has broad Doppler spectrum and itsmaximum is related to look angle. Obviously, this spectrum Doppler hides the target Dopplerand strongly disturbs usual methods such as MTI. This relation cannot be explained in theterms of wind direction in a simple way [9]. Some of other factors that play an importantrole are sea currents and variability of wind direction. Generally, for horizontal polarization,clutter has narrower Doppler spectrum. For both polarizations clutter power is substantiallyhigher in directions of positive Doppler shift. Generally, power decreases as the look directionmoves away from the upswell direction.

In Fig. 3.2 general trend of mean Doppler frequency against azimuth angle is presented.It is seen that relation takes on the form of sinusoid. More detailed discussion on Dopplerpeak spectral shift can be found in [57], where models for Doppler shift are presented.


Figure 3.2 — Plot showing the trend of mean Doppler frequency against azimuth angle[8].

Observation that clutter power and Doppler shift depend on look angle may lead toconclusion, that in this case the application of STAP technique is worth to be considered, asdiscussed in section 2.2. Relative dependence between the look angle and the Doppler shift ofthe clutter was a rationale standing behind STAP application for airborne radar as describedin section 2.2. In Fig. 3.3 it is shown how radar cross section is changing with grazing angle.We can see general trend, that with higher grazing angle, radar cross section increases. As itwill be shown later, statistical properties of the clutter will be changing with grazing angleas well.

Next important property of the clutter is its amplitude distribution. In the table 3.1 we seedifferent distributions used to describe clutter properties. Depending on situation, differentmodels can describe clutter. Basic model is Rayleigh model. It is useful in many situations,for example for low resolution radars with a high grazing angle. If we are dealing with higherresolution radar (especially for low grazing angles) it turns out that statistical properties aredifferent. The reason behind this is the law of large numbers. In the case of low resolutionradar, many independent scattering centers contribute to overall radar cross section of asingle resolution cell. In this case the law of large numbers states, that complex samplesshould be Gaussian distributed, and therefore envelope should be Rayleigh distributed [53].On the other hand, for higher resolutions, there are only few scattering centers in a singleresolution cell, and therefore their statistical properties have direct influence on radar clutterproperties. This is additionally augmented by the effect of low grazing angle. This is due tothe fact that for low grazing angles main contribution to clutter RCS are strong reflectionsfrom wave crests which are few in a single range cell.

It is worth noticing, that goodness of fit is much more important in the low probabilityof false alarm regions (ie. tails of the distribution that are well described by the third ordermoment of PDF). Therefore chi-square goodness of fit test is of limited use [9], since thiskind of test mainly compares general shapes (theoretical and estimated) and is not focusedon the tail of the distribution.

In Fig. 3.4 and 3.5 it is shown Probability of False Alarms (Pfa) as a function of threshold.In these figures we can see probability of false alarm for a given threshold. A choice of the


Table 3.1 — Different clutter distributions [9].

Rayleigh

CDF : P (X) = 1− exp {−x2

α } 0 ≤ x ≤ ∞

PDF : p(X) = 2xα exp {−x2

α } 0 ≤ x ≤ ∞

first moment: 〈x〉 = (πα)1/2

2

third moment: µ3 = 3(α/2)3/2√

π/2

Weibull

CDF : P (X) = 1− exp {−xn

α } 0 ≤ x ≤ ∞

PDF : p(X) = nxn−1

α exp {−x2

α } 0 ≤ x ≤ ∞

first moment: 〈x〉 = α1/nΓ(1 + 1n)

third moment: µ3 = α3/nΓ(1 + 3/n)

Lognormal

CDF : P (X) = 1− 12erfc( ln x−m

α1/2 ) 0 ≤ x ≤ ∞where erfc is the complementary error function.

PDF : p(X) = 1x(πα)1/2 exp{− (ln x−m)2

α } 0 ≤ x ≤ ∞

first moment: 〈x〉 = exp(m + α4 )

third moment: µ3 = exp(3m + 94α)

K-distribution

CDF : P (X) = 1− 2Γ(ν)

(bx2

)νKν(bx) 0 ≤ x ≤ ∞

PDF : p(X) = 2bΓ(ν)

(bx2

)νKν−1(bx) 0 ≤ x ≤ ∞

where Kν−1(z) is the modified Bessel function of the third kind of order ν

first moment: 〈x〉 =√

πbΓ(ν)Γ(ν + 1

2)

third moment: µ3 = Γ(ν+3/2)Γ(1+3/2)b3Γ(ν)


Figure 3.3 — Sea clutter cross-section against grazing angle profiles of X-band in theupswell and near cross-swell directions [9].

threshold plays an important role in detection scheme (that is not considered in this work).If we increase threshold we are reducing the number of false alarms, but at the same time weare reducing probability of detection. From figures 3.4 and 3.5 it can be seen that the data fitthe K-distribution best in the amplitude region for which the Pfa is ≤ 0.1 [9]. This is true forresolutions of order of 15 to 150 m. The same conclusion is confirmed in [8]. This observationimplies, that detectors developed for Rayleigh clutter will not be appropriate in the case oflow grazing angle radars.

For higher resolutions, we can even resolve single wave crest, as it is visible in Fig. 3.6.Images show power range-time diagrams for X-band 50 cm resolution radar. Wave crestsare traveling toward radar. In such a case, to detect targets it may be reasonable to applyimage processing algorithms rather than classical radar detection methods. Moreover severalfurtive ”superevents” are observed by an X band radar. These superevents are due to wavebreaking and generate strong scatterer with high velocity, so they are ”fast scatterers” (seefor example Fig. 3.6). These superevents generate a lot of false alarms (which may lead toerrors in tracking algorithms) and should be processed adaptively.

To deal with non-Gaussianity, it is essential to find a mathematical model for this type ofclutter. It can be found (see for example [52]) that theory of Spherically Invariant RandomProcess (SIRP) in many situations is an appropriate model for sea clutter. Theory of SIRP


Figure 3.4 — Comparison of Pfa characteristics of an X-band, V-polarized sea clutter dataset with those predicted by Rayleigh, Weibull, lognormal and K-models [9]. Range resolution

- 15 m.

can be used as a model for different types of clutter (Gaussian, lognormal, K etc..) and havebeen exposed in section 2.5. Therefore a new detector presented in chapter 4 is based onSIRP model.

3.2 Sea clutter characterization in HF band.

In HF radar band, main clutter component is related to Bragg scattering phenomenon. Firstdescription of this type of the clutter can be found in [11]. In the mentioned article, it isshown that for frequency 13.56 MHz first-order Bragg lines have Doppler shifts −0.38 Hz and0.38 Hz. General formula for Doppler shift is:

∆f =

√g

π

1λ

(3.1)

where g = 9.81m/s2 and λ is a radio wavelength. This formula means that main cluttercomponents are more or less constant over look angles of the radar and over different ranges.This comes from the fact, that resonance occurs when λ = 2L (where L denotes sea wavelength) and the speed of the sea wave is related to its length by the formula

v =√

g

2πL (3.2)


Figure 3.5 — Comparison of Pfa characteristics of an X-band, V-polarized sea clutter dataset with those predicted by Rayleigh, Weibull, lognormal and K-models [9]. Range resolution

- 150 m.

Figure 3.6 — Power range-time diagrams for VV polarization (a) and HH polarization (b).Range resolution 50 cm [42].


This mechanizm is called first order Bragg scattering and is responsible for constant Dopplershift of Bragg lines. However for higher Doppler resolutions other factors play role, and in factBragg lines are changing slightly accross angles and ranges. This is caused by complicated seawave dynamics. These effects make target detection more difficult and are the reason to useSTAP. Bragg scattering mechanism is commonly used for sea parameters extraction such aswave height, currents velocity and direction [5]. More detailed description of such possibilitiescan be found in [30] and [29]. Registered signals from WERA system working in Brittanyin France were used to present clutter properties. System description is presented in section5.1. Results are obtained by beamforming followed by Doppler processing. At the day of datacollection radar was working at frequency around 12 MHz.

In Fig. 3.7 are presented range-Doppler maps for an exemplary beam. Maps are obtainedfor different integration times and therefore for different Doppler resolutions. First map wasobtained for 48 chirps. Single chirp duration is 0.2275 s, therefore integration time in thiscase is 10.9 s. Other maps were obtained using 96 chirps (22 s), 256 chirps (59 s) and 512(117 s) chirps respectively.

We can see, that indeed there are advance and recede Bragg lines at frequencies -0.35 Hzans +0.35 Hz as predicted by theory (for 12 MHz carrier frequency). It is visible, that Bragglines are present in all range cells until approximately 90 km, where noise level becomes higherthan sea clutter. This is a result of a very low power transmitted by radar, which is at most30 W. Bragg lines frequencies are constant across all range cells. In the short distance fromradar we can see also land clutter. Comparing images in Fig. 3.7 we can see, that for higherintegration time Bragg lines become very thin. This observation has two consequences. Firstone is that first-order ocean wave scattering is coherent even for 2 minutes (512 chirps). Thesecond is that since Bragg lines are very thin in Doppler, having long integration time, maygive us better opportunities to reveal targets whose Doppler frequency is very close to Braggline.

In Fig. 3.8 we can see Doppler spectra for range cell number 50, this is 20 km. Theseimages illustrate how integration time increases influence Bragg lines thickness.

In the next figure (3.9) we can see spectra for different ranges (20, 40 and 60 km - range cellnumber 50, 100 and 150 respectively). Bragg lines are clearly visible for all ranges but there isa significant noise level increase relatively to Bragg lines maximum. It is also clearly visible,that Bragg lines present constant Doppler across all ranges. Images in Fig. 3.10 illustrate thatstructure of the first order ocean waves clutter does not change significantly across differentbeams.

Fig. 3.11 illustrates again how clutter is changing across beams. It is clear that thoughgeneral clutter structure is not changing, Bragg lines Doppler and power is changing acrossdifferent azimuths. We can see that Doppler band is changing considerably accross angles.We also have a slight change in Doppler shift. These are so called second order Bragg effects.Because of them it is worth to consider using STAP.

Next two figures (3.12 and 3.13) illustrate that clutter can vary only little in time. We cansee spectra for time shifts 0 s, 11 s, 113 s, 226 s (for 0, 50, 500 and 1000 chirps respectively).

In HF radar band there is also high probability of radio interferences. Interferences canhave different sources and properties. For example in Fig. 3.14 we can see an interferencethat is relatively weak. It can be seen as a pattern of almost horizontal lines, better visiblein black and white in Fig. 3.15.

Since interference has usually a point source, it is present only in some of beams whereas


Figure 3.7 — Bragg clutter for different integration time (number of chirps processed -from 48 to 512).


Figure 3.8 — Bragg clutter for different integration time (number of chirps processed -from 48 to 512), for the selected, 50-th range cell.


Figure 3.9 — Clutter spectra for different ranges (512 chirps integration).


Figure 3.10 — Range-Doppler maps for different beams (512 chirps integration).


Figure 3.11 — Beam-Doppler images for different ranges (512 chirps integration).


Figure 3.12 — Clutter spectra for different starting time of the processing (512 chirpsintegration).


Figure 3.13 — Clutter spectra for different starting time of the processing for Range cellnumber 50 (512 chirps integration).


Figure 3.14 — Bragg clutter plus interference (256 chirps integration). Interference is vis-ible as horizontal lines.

Figure 3.15 — Black and white image of Bragg clutter plus interference (256 chirps inte-gration). Interference is visible as horizontal lines.


Figure 3.16 — Range-angle map of directional interference.

in other beams it is not visible.

In figure 3.16 we can see very strong directional interference. Image presents range-azimuthmap of mean received power. Interference is present for angles between -20 deg to -40 deg.

In Fig. 3.17 we can see range-Doppler map in the beam with interference. See clutter iscompletely obscured by interference.

Next figure (3.18) shows time-range structure of the interference. We can see that thisinterference has impulsive nature, which means that its power is changing considerably as afunction of chirp number. This property can be used to suppress the interference.

On the other hand, in HF radar systems usually there is no problem with non-Gaussianity,because range resolution is very low (a few hundred meters to a few kilometers).

3.3 Conclusions

In this chapter the characteristic properties of the sea sea clutter for two radar bands (Xand HF radar bands) were presented. It was shown, that for both bands, sea clutter canhave two dimensional structure in space-time domain. In X-band this was caused by Dopplershift of clutter related to wind, currents and waves direction as well as possible interferencepresent in the data. In HF-band it was caused by joint presence of the first and secondorder sea clutter (Bragg lines) and interference which is inevitable in this frequency area.Two dimensional structure of the clutter and interferences is a reason to consider usingSTAP technique to improve detection possibilities. This will be verified later by performingexperiments on real, recorded data. For X-band it turns out additionally, that clutter does notfollow classical Rayleigh distribution, and therefore classical adaptive techniques may lead toincreased number of false alarms. This will also be investigated in the next two chapters. To


Figure 3.17 — Range-Doppler map oriented toward interference source.

Figure 3.18 — Time-range structure of the interference.


summarize, we can see that because of:

• Correlated and strong clutter with spread Doppler,

• Clutter that may be nonstationary,

• Impossibility to estimate interference-free clutter and clutter-free interference,

• Lots of parameters to be taken into account,

it is reasonably to consider adaptive (in space and time) methods, such as STAP, althoughthere are also some other possibilities too. These other methods were not considered in thiswork. Among them there is, for example, a method of using passive mode of radar to estimateinterferences [30].

CHAPTER4 Two Dirac deltadetector.

In this chapter the problem of the detector construction under Non-Gaussianity is addressed.In section 3.1 it was shown, that sea clutter has statistical properties that deviates fromGaussian (Rayleigh) case. On the other hand in chapter 2 was presented the GLRT test inthe case of Spherically Invariant Random Process. This detector is capable of dealing withwide spectrum of non-Gaussian distortions (clutter plus noise) including Weibull, log-normal,K-distribution and others [52]. There is, however, a problem with its practical use that isdiscussed below. In this chapter, I will develop a practical implementation of this detectornamed Two Delta Dirac STAP (TDD STAP) detector. First two sections will be devoted todevelopment of TDD STAP detector. In particular, I discuss the estimation of parametersused in this nonlinear detector in section 4.2. To evaluate performances of classical STAPdetectors in non-Gaussian clutter, I performed some simulations. Description of simulationtechnique is presented in section 4.3. Finally, section 4.4 is devoted to performance evaluationof different STAP algorithms. Receiver Operation Curves (ROC) are presented. In particular,we discuss the performances of usual STAP and TDD STAP detector under different kindsof noise (Gaussian, non-Gaussian). I will show, that new developed TDD STAP detectorcan give some improvements in comparison to classical STAP algorithm in the presence ofnon-Gaussian clutter.

It is also worth to pay attention to the work done by Frederic Pascal on the same subject[47].

4.1 Resolving GLRT.

In section 2.6 it was shown, that GLRT detector for SIRP is of the form of test statistics(2.55):

η =

∫∞0


τ )dFτ (τ)∫∞0


τ )dFτ (τ)(4.1)

where

|ρ|2 =|SH

0 Φ−1Y |2(Y HΦ−1Y )(SH

0 Φ−1S0)(4.2)

S0 is the completely known m-dimensional signal vector with magnitude 1, Y is a data vector,m is a data vector dimension, Φ is a covariance matrix of the distortions, τ is a modulatingscalar and H denotes conjugate transpose.

CHAPTER 4. TWO DIRAC DELTA DETECTOR. 51

Figure 4.1 — Original distribution of τ .

The problem now arises how to estimate all necessary parameters and how to calculatethe integrals in the formula (4.1). To do this we should know the distribution of τ (Fig. 4.1).This is unrealistic, but we have still some options:

1. Assuming Gaussian clutter simplification we put constant τ to achieve solution as insection 2.6.2 (see Fig. 4.2). For simplicity we prefer to assume τ = 1 that results inE(τ) = 1 and Φ = E{XHX} i.e. classical STAP detector for Gaussian clutter [49].

2. We can employ some kind of approximation as in Fig. 4.3. Approximating distributionmay be some arbitrary distribution (for example Gamma). We may also employ some ex-pansion approximations, for example Edgeworth. The problem with Edgeworth approx-imation is that it is based on Gaussian distribution, which spreads from minus infinityto plus infinity, whereas τ is a positive scalar (pure Gaussian and original distributionare shown in Fig. 4.4).

In this framework, I postulate to apply some other approximation, that will be simpleenough to implement, and at the same time, it will fit well enough real clutter, to giveperformance improvement. This approximation is discussed in the next section.

4.2 Two Dirac Deltas approximation.

Solution, that I proposed in [22], is picturized in Fig. 4.5 and is named Two Dirac Delta(TDD STAP). In the subsection 4.2.1 I will present a first approach in order to introduce theidea behind the new detector. In the subsection 4.2.2 some improvements are introduced inorder to be more realistic in the noise pdf approximation.

4.2.1 First approach.

I approximate CPDF (Characteristic Probability Distribution Function) by two Dirac deltas.In the simplest case, both Dirac deltas have equal strength. This solution gives one moredegree of freedom comparing to simple Gaussian clutter case (compare Fig. 4.2 and Fig. 4.5).We can see, that now instead of a single Dirac delta, there are two Dirac deltas (each one of


Figure 4.2 — Original distribution of τ and Dirac delta simplification.

Figure 4.3 — Original distribution of τ and approximation.

Figure 4.4 — Original distribution of τ and pure Gaussian approximation.


Figure 4.5 — Original distribution of τ and two Dirac delta simplification.

them multiplied by 1/2). One of them is placed in the point a of the τ axis, the other in thepoint b (4.5). Selection of a and b is discussed below.

By rewriting (4.1) according to two Dirac delta simplification, we obtain:

η =1/2

[1

am exp(−Y HΦ−1Y (1−|ρ|2)a ) + 1

bm exp(−Y HΦ−1Y (1−|ρ|2)b )

]

1/2[

1am exp(−Y HΦ−1Y

a ) + 1bm exp(−Y HΦ−1Y

b )]

⇔ η =1

am exp(−Y HΦ−1Y (1−|ρ|2)a ) + 1




b )(4.3)

From section 2.5 we have:E(τ) = 1 (4.4)

having in mind that τ ≥ 0, we can rewrite:

E(τ) =∫ ∞

0τf(τ)dτ

=∫ ∞

0τ[12δ(τ − a) +

12δ(τ − b)

]dτ

=12

∫ ∞

0τδ(τ − a)dτ +

12

∫ ∞

0τδ(τ − b)dτ

=12a +

12b (4.5)

Therefore using (4.4), we can write:

12a +

12b = 1 ⇔ a + b = 2 ⇔ a = 2− b

Now if we have true value of variance V AR(τ) or at least its estimate, then:

V AR(τ) = E(τ2)−E2(τ) =12a2 +

12b2 − 1 (4.6)

⇔ 12(2− b)2 +

12b2 − 1− V AR(τ) = 0


Figure 4.6 — a, b as a function of V AR(τ).

Figure 4.7 — ∆ parameter.

⇔ b2 − 2b + 1− V AR(τ) = 0

this equation has two solutions if V AR(τ) is greater than zero (which is always true):

b1,2 = 1±√

V AR(τ)

Keeping in mind, that both a and b must be greater than zero we have restriction for thevariance V AR(τ):

1−√

V AR(τ) > 0

V AR(τ) < 1 (4.7)

In Fig. 4.6 we see how a and b depend on V AR(τ).

To simplify analysis and accordingly (4.5), I introduced additional parameter in place ofa,b. This parameter will be called ∆, and now a = 1−∆ and b = 1 + ∆ as in Fig. 4.7.

Since now we have one additional degree of freedom by introducing ∆ parameter, we needa procedure to estimate this additional parameter. I proposed a closed loop procedure shownin Fig. 4.8 [24].


Figure 4.8 — ∆ estimation procedure.

Figure 4.9 — Unequal strength Dirac Deltas approximation.

This procedure tries to minimize mean square error between theoretical histogram andhistogram derived from secondary-training data. In the first step, from secondary data, covari-ance matrix is estimated. Having covariance matrix, it is possible to introduce f(τ) (texture),by the integral in the central part of Fig. 4.8. This integral allows us to obtain theoreticalhistogram of samples in our model. This histogram is then compared to the true histogram,obtained from secondary (training) data. Mean square error between theoretical and truehistogram is used to update ∆ parameter. ∆ is then updated to generate texture in integral,and theoretical histogram is recalculated. Procedure is repeated until difference between the-oretical and true histogram changes insignificantly. To avoid the problem of local minima,exhaustive search is used.

4.2.2 Refined approach.

Next step in this reasoning is to introduce not equal strength of Dirac deltas as in Fig. 4.9[23]. In particular, this approach allows us to break the symmetry of the CPDF and to devoteone Dirac delta to describe the tail of the CPDF which is of great importance as discussedin section 2.5 and 3.1.

Appropriate automatic procedure to find q,a and b is presented in Fig. 4.10.

This time a,b and q are related to each other. Therefore it is necessary to find only twoof them. This problem can be transformed to the problem of two dimensional optimization


Figure 4.10 — q, a and b estimation procedure.

problem. I proposed Downhill Simplex Method to cope with this problem. This is the maindifference between Fig. 4.8 and Fig. 4.10 .

In order to evaluate behavior of both classical STAP and TDD STAP detectors I per-formed simulations. Their results are presented in the sections 4.4.3 and 4.4.4.

4.3 Simulations.

To validate algorithms and to evaluate its performances, I generated simulated data. In thissection, simulation technique is described. First subsection describes parameters chosen forsimulations. In subsection 4.3.2 reader can find a description of target simulation. Subsection4.3.3 is devoted to noise simulation.

4.3.1 Simulation parameters.

In order to perform simulations, it is necessary to choose some parameter set. This parametersset should be appropriate to the state of the art in radar technology and have been introducedin section 2.1. I have chosen to simulate airborne system, for which non-Gaussianity may occurwhen detecting targets on the sea surface. Airborne radar systems are supposed to be theprimary application of STAP technique [38].

This subsection describes process of choosing the parameter set. Some of the parameterswhere taken from a well known DLR system E-SAR :

Carrier center frequency fc = 9.6GHzTransmit power P = 2500WPulse bandwidth B = 100MHzAzimuth beamwidth Φ = 17o

Elevation beamwidth Θ = 30o

Platform Velocity Vp = 90 ms

Geometry of the system is presented in Fig. 4.11 and Fig. 4.12. Having carrier frequencyit is possible to calculate wavelength:

λ =c

fc= 3 cm


Figure 4.11 — Vertical situation.

Figure 4.12 — Earth plane geometry.

Minimum range can be calculated from the formula (approximately, assuming azimuth an-gle=0):

Rmin = H · tan(ξ − Θ

2

)

Rmax = H · tan(ξ +

Θ2

)

Range resolution Rresolution is determined by the pulse bandwidth B:

Rresolution =c

2B= 1.5m

where c is the electromagnetic wave speed. Azimuth beamwidth is related to the antenna sizeLa by equation:

λ

La= Φ =

17o

180oπ ≈ 0.3


Figure 4.13 — Antenna array.

Figure 4.14 — Azimuth angle ambiguity.

so:La ≈ 10 cm

For STAP a linear antenna array parallel to the flight axis as in Fig. 4.13 is often assumed.In classical configuration for STAP, array elements spacing d is equal λ

2 [37], in this case1.5 cm. This is technically difficult to achieve, because single antenna is approximately 10 cmlong. Therefore here, spacing d = 10 cm is assumed. In order to avoid angle ambiguity, it isnecessary to calculate ambiguous angle β. This can be done using geometry presented in Fig.4.14. Ambiguous angle β is defined by the situation where a is equal λ. Therefore:

cosα =λ

La=

310

= 0.3

so:α ≈ 73o

and:β = 90o − α ≈ 17o

Next parameter to be set is PRF. Let PRF be 4 kHz (for E-SAR it was 1.2 kHz). Now itis necessary to check range and velocity ambiguity. For range ambiguity, let the maximum


Figure 4.15 — Triangle of velocities.

range Rmax = 30 km. PRF has to be set in accordance with the condition:

PRF <c

2Rmax= 5 kHz

Second condition concerns velocity ambiguity. First it is necessary to calculate maximumradial velocity from the ground. This will occur at the maximum range at the edge of thebeam (see Fig. 4.15). Resolving triangle of velocities:

Vrmax = Vp · cos(90o − Φ2

) = 90 · 0.147 = 13.3m

s

Additionally some moving targets can be expected. Let the additional radial movement ofthe target be limited to 10m

s . Overall radial velocity is therefore Vrmax sum = 24ms . It is now

possible to calculate Doppler shift:

fD =2Vrmax sum

λ=

2 · 240.03

= 1.6kHz

PRF should be twice the Doppler shift, therefore PRFmin = 3.2 kHz. Last parameter to beset is dwelling time. Dwelling time determines Doppler resolution which determines veloc-ity resolution. Let us assume velocity resolution equal 0.1m

s . Therefore Doppler resolutionfD resolution should be:

fD resolution =2 · 0.1

λ=

0.20.03

≈ 6Hz

Dwelling time is given by the formula:

TDWELL =1

fD resolution= 1/6 s

Finally, range migration problem should be faced. TDWELL = 1/6 s gives (assuming 24ms

radial velocity) 4m of range migration which is more than range resolution cell (1.5m). To


Figure 4.16 — Radar and target geometry.

resolve this problem, dwelling time reduction can be applied. Considering only 0.5ms velocity

resolution gives:

fD resolution =2 · 0.5

λ=

10.03

≈ 33Hz

therefore:TDWELL =

1fD resolution

= 0.03 s

Now range migration becomes:

Rmigration = 24 · 0.03 ≈ 0.75m

which is less than range resolution cell.

4.3.2 Target Simulations.

To evaluate probability of detection versus probability of false alarm, I injected target echointo clutter data. This section describes methodology that was used to do this.

Simulations were based on the backscattered energy given in [45]. Geometry of the radarplatform and the target is shown in Fig. 4.16.

The result of simulations should be a complex data cube:

Data(n, l, k)

where n is a pulse index l is an antenna index and k is range cell index (see section 2.2.5).For a single target, formula for data cube is given by:

Data(n, l, k) = KP (n) · σn · ejφ · f(k − Xn

Rresolution) ·AZ(n) ·Ant(l, n) (4.8)

where


• KP (n) is radar equation,

• σn is a RCS,

• ejφ is a random unknown phase,

• f(k − XnRresolution) is a range gate factor,

• AZ(n) is a modulation function,

• Ant(l, n) is factor related to an antenna array element.

In order to calculate all elements of the data cube equation it is necessary to perform pre-liminary geometrical calculations. After this step, vectors describing positions and relativeangles of the target as well as the radar platform in subsequent moments defined by the PRF,will be available. In order to achieve this, some radar and target movement models must beadopted. Let radar platform be flying along Y axis at the speed Vp. Let us assume that, inthe moment t=0 (reception of the first pulse), radar platform position be Y=0. Let us assumefollowing model for the target movement:

X(t) = X0 + Vt · t (4.9)Y (t) = Y0 (4.10)Z(t) = Z0 = 0 (4.11)

Having this information it is possible to calculate slant range to the target (see Fig. 4.16):

R(t) =√

X(t)2 + (Y (t)− Vp · t)2 + H2

We can also simply calculate angles using formulas

tan(ϕ(t)) =Y (t)− Vp · t

X(t)

tan(α(t)) =X(t)H

θ(t) =π

2− χ(t)

where (see Fig. 4.17):

tanχ(t) =Y (t)− Vp · t√H2 + X(t)2

We should evaluate above formulas in moments defined by the reception of subsequentbackscattered pulses from the target. Assuming that t0 = 0, these moments can be calculatedfrom PRF:

tn =n

PRFwhere n is a pulse number starting from 0.

Having all this information, we can now calculate data cube evaluating each term in theequation (4.8).

• KP (n) - In the simplest case is constant from pulse to pulse (which means, that Rchanges only slightly in radar equation during dwelling time). KP (n) also reflects an-tenna elevation and azimuth pattern. In simulations, we adopted KP (n) = G(ϕn, αn),where pattern G(ϕ, α) in azimuth and elevation is shown in Fig. 4.18.


Figure 4.17 — θ and χ.

Figure 4.18 — Single antenna azimuth G(ϕ) (left) and elevation G(α) (right) pattern.

• σn - We can assume Swerling II model for example [39], assuming some mean RCS σ.Then for each pulse (n) we should choose a random RCS using PDF:

p(σn) =1σ

e−σnσ

The phase is however coherent within CPI.

• ejφ - This is an unknown initial phase of the target signal. We can assume ejφ = 1without loss of generality.

• f(k − RnRresolution) is function giving range cell response for the target at a distance Rn.

In the simplest case we can assume rectangular function as shown in Fig. 4.19

• AZ(n) - is a modulating function of the form

AZ(n) = e−j 4πRnλ

• Ant(l, n) is additional phase advance as a result of the antenna elements displacement.

Ant(l, n) = e−j2πl dλ

cos(θn)


Figure 4.19 — f function of range cell response.

4.3.3 Additive Noise.

In order to simulate the data close to the real one, we add noise and clutter to our target data.This is done using procedure from [49]. More detailed explanations are presented in AppendixB. This allows us to add Gaussian or non-Gaussian, white or colored noise according to ourneeds. In this work three types of noise were used. First type is a classical Gaussian noise.This noise occurs for low resolution radars, where many scatterers contribute to overall RCS.This is typical situation for airborne radars at high grazing angles. This noise can be viewedas a noise with Dirac Delta as a characteristic PDF (see section 2.5) of SIRP (as in Fig.4.2). The second type is TDD noise. This is a noise generated using two Dirac deltas as acharacteristic PDF (two Dirac deltas in Fig. 4.7). This is an academic example, that willshow how TDD STAP will perform with the same kind of (TDD) of noise. Finally the mostinteresting is the case (among these tested in this work) where characteristic PDF is chosento obtain K-distributed noise. This kind of noise can occur for high resolution radars onthe sea surface or for radars with low grazing angle. In order to achieve this kind of noise,characteristic PDF must be of the form [49]:

f(τ) =2b

Γ(ν)2ν(bτ)2ν−1 exp

(− b2τ2

2

)u(τ) (4.12)

where Γ(ν) is a gamma function, b and ν are parameters of K-distribution, and u(τ) is aunit step function. Covariance marix can also be randomized. Problem of Covariance matrixstructure is separable from the problem of non-Gaussianity as a result of employing SIRP.

After this step we have data, that can be used as an input for a STAP processor. Examplesof results after performing STAP processing are shown in Fig. 4.20-4.22. In figures 4.20-4.22white covariance matrix is adopted for the purpose of technique demonstration. Last twofigures present results for the same SNR 1.

Results were generated under the following assumptions:

1SNR is a Signal to Noise Ratio defined as SNR = log10(PsignalPnoise

), where Psignal is a signal power, and Pnoise

is noise power. In practice, after quadrature receiver, if s denotes signal column vector, and x denotes noise

column vector, we use formula SNR = sHsxHx


Figure 4.20 — STAP result on target signal without noise (test statistics intensity image).

Figure 4.21 — STAP result on target signal with Gaussian noise (test statistics intensityimage).


Figure 4.22 — STAP result on target signal with non-Gaussian, K-distributed (test statis-tics intensity image).

X0 = 7000mY 0 = 500mVt = 4 m/s

PRF = 4000HzVp = 90 m/sL = 16N = 120

STAP test was performed for discrete velocities [−15m/s,15 m/s,0.5m/s ] and azimuth angles[ −8.5o,8.5o,0.5o ].

4.4 Results.

On the basis of simulation experiments I was able to compare different processing algorithms.In the first subsection I will show, that classical STAP performance becomes worse under non-Gaussian conditions. As a result of computer precision limitations, TDD STAP detector hasto be slightly modified, which will be presented in section 4.4.2. In section 4.4.3 I will presentcomparison between classical STAP and TDD STAP for a priori chosen ∆, and in section4.4.4, similar comparison for automatic ∆ finding procedure.

4.4.1 Classical STAP detection performance evaluation.

In order to evaluate influence of non-Gaussianity, I performed some tests. Single target signalwas generated using procedure described in section 4.3. This signal was used in all tests. For


different SNRs, Gaussian and non-Gaussian Noise was generated independently 300 times.300 was chosen to achieve reasonably high (eg.10) number of false alarms. However sometimesnumber of repetitions was considerably higher. For each noise trial, classical STAP test wasperformed using different thresholds. Classical STAP test (2.14) was performed independentlyon Gaussian noise, signal plus Gaussian noise, non-Gaussian noise, signal plus non-Gaussiannoise. Non-Gaussianity was introduced by K-distribution (ν = 2, b = 2 in 4.12) for realand imaginary samples with unit covariance matrix. False alarms were counted for each trialbased on number of target reports in pure noise (without target signal). Therefore we cancalculate average number of false alarms per STAP image for different SNR’s and thresholds,independently for Gaussian noise and for non-Gaussian noise. If number of targets reportedin signal plus noise was greater than number of target reported in noise only, we assumedsuccessful detection. This procedure allows us to obtain probability of detection for differentSNR’s and thresholds for Gaussian and non-Gaussian cases. Other parameters of simulationsare presented below:

X0 = 7000mY 0 = 500mVt = 4 m/s

PRF = 4000HzVp = 90 m/sL = 8N = 30

Classical STAP test was performed for discrete velocities [−15m/s,15m/s,1m/s ] andazimuth angles [ −8.5o,8.5o,1o ]. Too low resolution in tests (test grid too sparse) may leadto some performance deterioration as a result of a mismatch between the true steering vectorand its surrogate from the grid (see section 2.2.5).

In Fig. 4.23 we can see Receiver Operating Curves. Vertical axis denotes Probability ofDetection (PD) and horizontal axis denotes mean number of False Alarms (FA) per singlerange cell. For the same PD the aim is to have the lowest possible mean number of FA. Wecan see that mean number of FA is higher for non-Gaussian case (having the same PD) inwhich the characteristics are shifted to the right (higher mean number of FA). This is true forany SNR. In Fig. 4.24 we can see, that SNR does not influence number of false alarms (CFARproperty), although number of false alarms increases for non-Gaussian case (characteristicsshifted up, toward higher mean number of FA). Pattern is similar for different thresholds asa result of using the same target and noise signals.

4.4.2 Numerical simplifications for TDD STAP.

As a result of a finite computer calculation precision, Two Dirac Deltas (TDD STAP) de-tector presented in section 4.2 needs to be reformulated for practical implementation. In thissubsection, I will show modifications that will be used to implement TDD STAP detector inMatlab environment.

In section 4.2 Two Dirac Deltas (TDD STAP) detector (4.3) was presented in the form :

η =1

am exp(−Y HΦ−1Y (1−|ρ|2)a ) + 1




b )(4.13)


Figure 4.23 — Probability of detection (PD) as a function of mean number of false alarmsper STAP picture (FA number) for different SNR’s for Gaussian and non-Gaussian case.

where

|ρ|2 =|SH

0 Φ−1Y |2(Y HΦ−1Y )(SH

0 Φ−1S0)(4.14)

I will use the following transformation:

1am exp(−X

a ) + 1bm exp(−X

b )

= 1am exp(−X

a )(1 + am

bm

exp(−Xb

)

exp(−Xa

)

)

= 1am exp(−X

a )(1 + (a

b )m exp(−Xb + X

a ))

= 1am exp(−X

a )(1 + (a

b )m exp (−X(1b − 1

a)))

Let me rewrite test statistics according to the above transformation:

η =1

am exp(−Y HΦ−1Y (1−|ρ|2)a )

(1 + (a

b )m exp (− Y HΦ−1Y (1− |ρ|2)(1b − 1

a)))


a )(1 + (a

b )m exp (− Y HΦ−1Y (1b − 1

a))) (4.15)


Figure 4.24 — Number of false alarms (FA number) as a function of SNR for differentthresholds (Thr).

canceling 1am :

η =exp(−Y HΦ−1Y (1−|ρ|2)

a )(1 + (a

b )m exp (− Y HΦ−1Y (1− |ρ|2)(1b − 1

a)))

exp(−Y HΦ−1Ya )

(1 + (a

b )m exp (− Y HΦ−1Y (1b − 1

a))) (4.16)

The logarithm of expression (4.16) is given by:

ln(η) =

[− Y HΦ−1Y (1− |ρ|2)

a

](4.17)

+ ln[1 + (

a

b)m exp (− Y HΦ−1Y (1− |ρ|2)(1

b− 1

a))

]

−[− Y HΦ−1Y

a

]

− ln[1 + (

a

b)m exp (− Y HΦ−1Y (

1b− 1

a))

]

=

[Y HΦ−1Y − Y HΦ−1Y + Y HΦ−1Y |ρ|2

a

]

+ ln[1 + (

a

b)m exp (− Y HΦ−1Y (1− |ρ|2)(1

b− 1

a))

]


=1a

|SHΦ−1Y |2(SHΦ−1S)

+1b


− 1a


=1b


In this case, there is no advantage using TDD STAP detector in comparison with classicalSTAP (TDD STAP is equal to classical STAP). Therefore I postulate to use approximationsaccording to the needs (appropriate procedure should chose simplification only when neces-sary). This procedure was used in practical implementation of TDD STAP detector. I usedapproximation when exponential of nominator or denominator in 4.16 was higher than 400.

4.4.3 Comparison of classical STAP and TDD STAP for fixed ∆ parameter.

I performed comparisons using method described in section 4.4.1. In the first result, the valueof ∆ was not estimated but chosen a priori in order to estimate the discrepancies when ∆ ispoorly estimated. These results were already presented in [22].

In first experiments, Non-Gaussianity was introduced by TDD noise. This means thatCPDF (for SRIP) adopted in noise generation algorithm follows TDD distribution. Moreprecisely (see section 4.3.3):

f(τ) =12δ(t− a) +

12δ(t− b)

In Fig. 4.25 we can see Probability of Detection (PD) as a function of mean Number ofFalse Alarms (NFA) for different choice of spread parameter ∆.

• If ∆ approaches 0 or 1, then TDD STAP detector behaves exactly as classical STAPdetector. This is not surprising since for ∆ = 0 or ∆ = 1 expressions for TDD STAPreduce to classical STAP formula.

• For some values of ∆ parameters (eg. 0.4 or 0.5), newly developed TDD STAP detectorhas a better performance than classical STAP.

• Unfortunately for badly chosen ∆ (eg. 0.7) TDD STAP performance is even worse thanclassical STAP.

Similar experiments were performed for K-distributed (ν = 2 and b = 2) noise. This timein noise generation scheme (see section 4.3.3) the function f(τ) takes on the form:

f(τ) =2b

Γ(ν)2ν(bτ)2ν−1 exp

(− b2τ2

2

)u(τ)

where Γ(ν) is a gamma function, b and ν are parameters of K-distribution, and u(τ) is a unitstep function.

Results are presented in Fig. 4.26. Again we can see, that there is some performanceimprovement in comparison to classical STAP.

• If ∆ approaches 0 or 1, then TDD STAP detector behaves exactly like classical STAPdetector.

• When ∆ is appropriately chosen (eg. 0.3 or 0.4), TDD STAP detector has better perfor-mance than classical STAP.


Figure 4.25 — Probability of Detection (PD) as a function of mean Number of FalseAlarms (NFA) for TDD STAP detector in the presence of Gaussian and TDD noise for

different values of ∆.


Figure 4.26 — Probability of Detection (PD) as a function of mean Number of FalseAlarms (NFA) for TDD STAP detector in the presence of Gaussian and K-distributed noise

for different values of ∆.


• If ∆ is badly chosen (∆ = 0.7) than TDD STAP performance is worse than classicalSTAP.

To conclude it can be seen, that improvement is possible, but automatic procedures arenecessary to find ∆ parameter, as discussed in section 4.2. It the parameter is poorly estimatedresults are much worse.

4.4.4 Results for TDD STAP detector with automatic ∆ finding.

As it was shown previously, it is necessary to appropriately choose ∆ parameter. In order todo this I postulate to employ method presented in Fig. 4.8 and 4.10. Results of this approachwere already published in [23] and [24].

In Fig. 4.27 we can see results of automatic ∆ finding under assumption, that deltasstrength is equal (see Fig. 4.7) and noise is generated using TDD as a characteristic PDF(see section 2.5 and 4.3.3). Automatic procedure properly choses the ∆ parameter, and thereis an improvement in detector performance. In this case detector is appropriately chosen forthe noise in the framework of SIRP theory. Real CPDF is exactly the same as CPDF assumedin TDD STAP detector. We cannot have better detector in this model.

Situation is different if we use other type of noise. In such a case, TDD STAP detector isonly an approximation of a real GLRT detector. In Fig. 4.28 similar results are presented forthe K-distributed noise. We can see again some slight improvement of TDD STAP detectorin comparison to classical STAP. This time however we can expect, that there exists betterdetector than TDD STAP. This is due to the fact, that in this case real CPDF is continuous,and in TDD STAP, I assume discrete CPDF. Nevertheless, TDD STAP detector proves, tobe better than classical STAP, and there is a possibility for further improvement. This canbe achieved by better approximating CPDF.

The last figure (4.29) presents results of automatic a, b and q finding for K-distributednoise and unequal Dirac deltas as described in section 4.2 and depicted in Fig. 4.10. We cansee that in comparison to Fig. 4.28 there is a slight improvement. This is due to the betterapproximation of real CPDF of SIRP by unequal-strength deltas than by equal deltas, as aresult of additional degree of freedom. Having two Dirac deltas with unequal strength we canbetter approximate distributions with skewness. In such a case, Dirac delta placed in pointa (see Fig. 4.9) should be stronger than the other (in point b). This way it is possible tobetter approximate skewed distributions. In fact, because τ is greater than 0 and potentiallycontinuous and unlimited, all considered distributions (CPDFs) will be skewed.

Following this path we can expect further improvement when using for example moreDirac-deltas. Advantage of having additional degree of freedom can be outweighted by thenecessity of estimating more parameters. This can be difficult to perform, and poorly esti-mated parameters can lead to detection losses.

For high probabilities of detection we can see, that behavior of TDD STAP is slightlyworse than classical STAP. Nevertheless this is in the area of high probabilities of falsealarms (mean number of false alarms per image more than 1). Usually radars are working inareas with lower levels of false alarms. As a conclusion we can say that automatic procedure,presented in section 4.2, works well.


Figure 4.27 — Automatic ∆ finding for TDD noise.

Figure 4.28 — Automatic ∆ finding for K-distributed noise.


Figure 4.29 — Automatic q, a and b finding for K-distributed noise.

4.5 Conclusions.

In this chapter I have presented a concept of TDD STAP detector. This detector can betterfit sea clutter than traditional STAP algorithm. In chapter 3 we have seen, that sea clutterhas different statistical properties than those assumed in classical STAP detection schema.Therefore GLRT from section 2.6 was employed. I resolved the problem with its practicaluse in section 4.2 (this chapter). In next sections, simulation method was presented, that waslater used in evaluating TDD STAP detector performance. In section 4.4.1 it was shown, thatthere is a performance decrease when classical STAP algorithm is working in non-Gaussianenvironment. The use of TDD STAP detector can bring some improvements, which was shownin section 4.4.3. However this improvement can be outweighted by badly chosen ∆ parameter.Therefore in section 4.4.4, I postulated an automatic procedure. It is clear that automatedprocedures can deal with the problem, and there is a performance improvement clearly visiblein characteristics presented in figures.

The future improvement in this area seems possible. We can imagine more Dirac-deltasas a better approximation of characteristic PDF (under the limitations discussed in section4.4.4). This can lead to better detectors, able to deal with a larger class of non-Gaussianity.

CHAPTER5 HF radar signalsexperiments and STAPtechnique modifications.

High Frequency (HF) radar systems, which operate in frequency range between 3 and 30MHz, have a potential to detect targets which are located beyond optical horizon on thesea surface (Over The Horizon visibility - OTH). Therefore, in recent years, more and moreattention has been paid to such a systems in the context of the ship traffic surveillance in theExclusive Economic Zone (EEZ), that was established by United Nations Convention on theLaw of the Sea in 1982. Such a trials were already undertaken and described for example in[1] [13] [18] [19] [15] [16] [17] [14] [26]. In this chapter experiments, concerning real data fromHF radar system WERA, will be described. In particular two techniques will be presented.First one is AMTI (purely temporal adaptive filtering), and the second is STAP that hasbeen theoretically described in section 2.2. The aim of the chapter is to present the problemand the solution to non-stationarity. Comparison between AMTI and STAP is of secondaryimportance. In the case of HF radar, there is no problem with non-Gaussianity. On the otherhand there is a problem with strong clutter and interference as was emphasized in section3.2. It will be shown what new problems are to be faced in real-data implementation. Becauseof the difficulties with covariance matrix estimation (see sec 2.2.6), sometimes it is possibleto obtain better results using Adaptive MTI. Because of these practical problems, classicalalgorithms must be adapted, which will be also shown in this chapter. In the section 5.1 I willintroduce WERA radar system that was used in experiments. WERA system was designedto operate as an oceanographics system, therefore this application will also be described inthe same section. Sections 5.2 will show how AMTI and STAP algorithm must be adapted toWERA radar system design. Finally, in section 5.3 results of traffic survey will be presented.

5.1 WERA radar system.

Data were collected from the WERA [30] [29] HF radar system. Installation that was usedin experiments is a property of the French Service Hydrographique et Oceanographique de laMarine (SHOM) and is operated by the company SAS ActiMar. System’s primary task is toprovide real-time, continuous data collection about currents and waves in the area of BrestGulf (Atlantic, west coast of France). In order to have unambiguous vectorial information ofthe currents, full system consists of two radars displaced approximately 50 km apart (see Fig.5.1).

Each radar consists of 16 line-aligned receiving antennas separated by approximately

CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUEMODIFICATIONS. 77

Figure 5.1 — Radars displacement (Google Earth).

Figure 5.2 — Receiving antenna array.

9.3 m (Fig. 5.2). Transmitting array consists of 4, square arranged, antennas (Fig. 5.3).

Signals from receiving antennas are independently sampled for digital beamforming andstorage for off-line processing. Transmitted waveform is FMCW, therefore it is necessaryto perform preprocessing in range to obtain data equivalent to pulsed systems described insection 2.2.5. This is necessary to apply algorithm like AMTI or STAP (see section 2.2). Toachieve this goal, signal must be demodulated. The next step is to apply appropriate low passfilter. Signal must be then sampled with Nyquist frequency. Such raw data have to be range


Figure 5.3 — Transmitting antenna array.

processed. In the case of frequency modulated signals this means simply to perform FastFourier Transform (FFT). After this operation the data can be presented as a complex datacube (Fig. 2.19 in section 2.2.5) appropriate for AMTI or STAP processing. More detailedexplanation can be found in Appendix C. WERA chirp duration can vary from 0.2275 s to0.26 s. Carrier frequency of the system is around 12 MHz. Bandwidth is set independently toachieve desired range resolution (from 0.4 km to 1.5 km). Data file consists of complex samplestaken from 2048 coherent chirps (532.48 s) for 120 to 256 range cells by 16 separate channelsof array antenna. The number of range cells depends on the range resolution. For example for0.4 km of range resolution, data file consists of 240 range cells which gives maximum range 96km. For 1.2 km of range resolution, data are taken for 120 range cells, which gives maximumrange 144 km.

This configuration allows to cover area presented in Fig. 5.4 (for the 96 km maximumrange mode).

Data used in experiments for target detection were collected in July 2007. Additionally,it was possible to obtain Automatic Identification System (AIS) data, concerning ship trafficaround Brest Gulf. From AIS data it was possible to obtain information about trajectoriesof certain ships together with additional information e.g. MMSI number, IMO number, shiplength, draft etc (see Fig. 5.5). AIS data allowed to compare HF detections against the truedata.

HF radar systems have a long history as an ocean monitoring systems. They proved to becapable of extracting currents speed and direction as well as sea state and wind information.Bragg scattering from ocean waves (first described in [11]) plays a crucial role in this process.Possibility of employing HF radar system to ocean monitoring is presented in more detail forexample in [5] [40] [30] [29].

WERA radar system was designed for the same purpose. In the first stage of its processingit uses classical beamforming and Doppler processing (by Fast Fourier Transform - FFT).Beamforming uses 16 signals from different antennas. The next step is Doppler processing.Doppler processing is performed in a little bit complicated manner. At the beginning 13spectra are calculated, each using overlapping sets of 512 samples (chirps). Chirps used tocalculate each spectrum are accordingly: number 1-512, 129-640, 257-768, ... , 1537-2048.


Figure 5.4 — Coverage area of radars.

Figure 5.5 — AIS trajectories.


Figure 5.6 — Single subband after beam-Doppler processing.

After this operation all 13 spectra are incoherently added (ie. summation of the magnitude)in order to obtain 512 Doppler bands. A typical X-Y image of an area surveilled by the radarfor a single Doppler band is presented in Fig. 5.6.

After appropriate analysis, it is possible to obtain oceanographical information. In Fig.5.7 we can see current map obtain from WERA system. Fig. 5.8 presents waves map and Fig.5.9 wind information.

5.2 Implementation of Adaptive MTI and STAP - covariancematrix estimation problem.

In this section, the application of Adaptive MTI and STAP processing to WERA radar datawill be described. The main problem of both techniques is to estimate covariance matrix.Usually covariance matrix is calculated using auxiliary range cells. Unfortunately for WERAradar system overall number of range cells is at most 256. The first step for both techniquesis data trim. From the 2048 chirps in data file, only J chirps of interest are chosen for furtherprocessing (see Fig. 5.10). Number of chirps J is chosen in order to operate on data moreeffectively and is not important from theoretical point of view. From this point data for AMTIand STAP are processed differently.

5.2.1 Adaptive MTI implementation

The theoretical description of AMTI can be found in section 2.2.4). The first step of AMTIis beamforming (Fig. 2.20 in section 2.2.4).

Beamforming is performed using 16 antennas for discrete angles [−60o,+60o,2o] (Fig.5.11).

After this step data are available as a cube with 61 beams instead of 16 antennas (Fig.


Figure 5.7 — Current map (ActiMar).


Figure 5.8 — Waves map (ActiMar).


Figure 5.9 — Wind map (ActiMar).


Figure 5.10 — Data trim.

Figure 5.11 — Angle Coverage.


Figure 5.12 — Data snapshot.

Figure 5.13 — Test snapshot and training area.

2.20 and 2.21). For each range-beam cell, adaptive filter is calculated as described in section2.2.4. Sample covariance matrix was calculated using auxiliary snapshots of the same beam.Since in WERA there are at most 256 range cells I decided to use time spread of the datato estimate covariance matrix. In Fig. 5.13 we can see test snapshot and area that auxiliarysnapshots were taken from. Auxiliary snapshots were taken by sliding snapshot window overchirps and ranges in the test area. Training areas were separated in range from test snapshotby guard cells. This prevents target self-nulling. Steering was performed for discrete velocities[−10 m/s,+10m/s,0.2 m/s].

After such a processing, results are available in the form of the test statistics cube (range×velocity × angle). It is not possible to present it directly. It is necessary to cut this cube acrossone dimension. In Fig. 5.14 we can see example of a target visible after AMTI processing.Test statistic is presented as an image for the specific range. 6 images present target that ismoving away from the radar, this is why echo is present in different range cells. Processingwas performed using 48 chirps (N=48).


Figure 5.14 — Example of a target visible after AMTI processing.


Figure 5.15 — Test snapshot and training area for STAP processing.

5.2.2 STAP implementation.

Fully adaptive STAP is operating on data before beamforming (suboptimum techniques werenot considered in this work), but after data trim (Fig. 5.10). STAP algorithm is applied toeach range cell separately to obtain test statistics as in section 2.2.5 - see equation (2.14).Covariance matrix must be calculated from secondary (training) data. Clutter covariancematrix was estimated using similar schema as for AMTI. In comparison to AMTI, STAP hasa potential to suppress clutter and interference at the same time. When interference cannot beestimated independently from the clutter (because of coupling azimuth-Doppler), adaptivesingle-domain methods may perform poorly and joint domain (space and time) adaptiveprocessing may be advantageous [18]. However this effect can be outweighted by the problemof covariance matrix estimation. Classical schema for estimating covariance matrix fails towork because in WERA system there are at most 256 range cells whereas in the case of STAPprocessing it is necessary to estimate covariance matrix of a very high dimension (eg. 1600 ×1600 in the case of STAP for 16 antennas and 100 chirps). Therefore it is necessary to findmore auxiliary snapshots. I decided (similarly as for AMTI) to use time spread of the datato obtain more snapshots. I use overlapping sliding snapshots.

In Fig. 5.15 we can see test snapshot, training snapshot and training areas in the datacube (compare with Fig. 5.13). After STAP processing, the test statistics is available in thesame form as for AMTI (test statistics cube: range × velocity × angle).

For both techniques there is possibility to chose certain parameters of covariance matrixestimation. First one is the number of guard cells. Guard cells are necessary to avoid targetself-nulling. Usually 2 or 3 guard cells on both sides of test cell is sufficient to avoid this effect.The next parameter to chose is range spread. Covariance matrix converges with increasingnumber of auxiliary range cells. Unfortunately there is also a risk of clutter non-homogeneity.Moreover having wide range-spread means high probability of other targets falling into cluttercovariance estimation area. The last parameter is time spread. Extending time spread allowsus to estimate covariance matrix in the situation, when there is not enough range cells todo this. Unfortunately covariance matrix estimate converges slowly with extending time-spread. Moreover there is a risk of clutter non-stationarity that will degrade covariance matrix


Figure 5.16 — Targets localization.

estimation.

We can see that playing with these three parameters may be a crucial problem in appli-cation both AMTI and STAP. Results of such experiments will be shown in next sections ofthis chapter.

5.3 Comparisons between the results of AMTI and STAP

In this section I will compare AMTI and STAP techniques. It will be shown how both algo-rithms can cope with the interference and clutter in order to reveal targets.

5.3.1 Data file and target description.

In experiments two data files from 19 July 2007 were used. One file contains recording fromBrezzelec site and the other from Garchine. Recording started at 05:20 UTC and its durationwas almost 8 min (2048 chirps, single chirp duration 0.2275 s). Bandwidth was set 375 MHz,which gave range resolution 0.4 km. Transmitted power was 30 W. Files contain samples for240 range cells, which gave maximum range 96 km. Significant wave height at the momentof data collection was approximately 1.3 m.

From AIS data I chose two interesting potential targets shown in Fig. 5.16. EERLAND 26is a small tug ship 25 m long (Fig. 5.17), L’AR VOALEDEN is a fishery boat 23 m long (Fig.5.18). We will compare AMTI and STAP techniques for both targets and both localizations(Garchine and Brezzelec).

In Fig. 5.19 we can see Garchine data file after classical beamforming. We can see a strongradio interference from the same direction as fishery boat. This will definitely have a negativeinfluence on detection possibilities.


Figure 5.17 — Tug ship.

Figure 5.18 — Fishery boat.

5.3.2 Detection of the tug ship from Garchine radar site.

In this subsection I will present results of the processing in order to reveal the tug ship usingdata from Garchine localization. From AIS data it is possible to deduce, that target shouldbe present in the range cell number 158 (distance 63 km) and in the beam number 25 (-10degrees left to the normal to the antenna). Indeed, in Fig. 5.20 target is well visible. Thisfigure presents test statistics (in decibels) for range cell number 158 and beam 25.

Fig. 5.20 presents behavior of AMTI and STAP algorithms for different integration timesand for different training samples support. Upper-left graph presents comparison betweenlower (100 chirps) and higher (200 chirps) integration time AMTI with low training samplessupport. In this case, 100 time slides and 15 range cells on both sides of test cell (see Fig.5.13) were used for training. This gives total number of 3000 training snapshots. 0 dB in thefigure denotes the reference level of detected target. We can see, that the highest non-targetpeak for 200 chirps AMTI is 6 dB lower than for 100 chirps AMTI. We can say, that in thiscase 200 chirps AMTI outperforms 100 chirps AMTI by 6 dB. The lower-left graph presents


Figure 5.19 — Interference.

similar results for high training samples support. In this case 200 slides and 60 range cells onboth sides of test cell are used. This gives 24000 samples used to estimate covariance matrix.We can see that improvement is even better and 200 chirps AMTI outperforms 100 chirpsAMTI by 8 dB. Right graphs show similar results for STAP. For low training samples support(upper-right) we can see that only 100 chirps STAP can reveal the target. 200 chirps STAPfails to detect the target. Situation is different for high training samples support (lower-rightin Fig. 5.20). This time we can see, that 200 chirps STAP outperforms 100 chirps STAP by5 dB.

In Fig. 5.21 we can see comparison between AMTI and STAP. Upper-left graph presentscomparison between AMTI ans STAP for shorter integration time (100 chirps) and low num-ber of training samples. We can see, that in this case AMTI outperforms STAP by 8 dB.Situation changes dramatically with high number of training samples. In this case, STAP out-performs AMTI by 5 dB. For longer integration time (200 chirps) and low number of trainingsamples we can see, that STAP fails, whereas AMTI can reveal the target (see upper-rightgraph of Fig. 5.21). For high number of training samples, STAP outperforms AMTI by 3 dB.

In figures from 5.22 to 5.25 we can see the influence of different training schemes ondetection possibilities. General pattern is clear that more training samples gives better chancesof target detection. This is valid for extending training area in time (100 vs 200 slides) aswell as in range (from 15 to 60 range cells on both sides of test cell). Generally, it is possibleto conclude that:

• AMTI works well even for relatively low number of training samples (100 slides, 15 rangecells on both sides of test cell).

• STAP needs more training samples (200 slides and 60 range cells on both sides of testcell).

• Using both techniques it is possible to detect tug ship.

• Because of interferences present in the data file, STAP techniques gives slightly betterresults.


Figure 5.20 — Tug ship detection from Garchine site - influence of integration time. Upper-left: AMTI for low training samples support, lower-left: AMTI for high training samplessupport, upper-right: STAP for low training samples support, lower-right: STAP for high

training samples support.

5.3.3 Detection of the tug ship from Brezzelec radar site.

The same target is visible from the second radar site (Brezzelec). From AIS data we canexpect target in the range cell number 160 (distance 64 km) and in the beam number 34(6 degrees right to the normal to the antenna). Indeed, in Fig. 5.26 target is visible. In thecase of Bezzelec file, there are almost no interferences present. This should mean, that mainadvantage of STAP will not play a role here.

First, let us compare the influence of integration time as before. We can see, that patternis similar as for Garchine file (Fig. 5.26). For longer integration time, target is better visible.

If we compare AMTI and STAP (5.27), we can see that:

• There is no advantage of STAP.

• For longer integration time (200 chirps), STAP gives the same result.


Figure 5.21 — Tug ship detection from Garchine site - comparison between AMTI andSTAP.

• For shorter integration time (100 chirps) AMTI better reveals the target (I could notexplain this anomaly).

5.3.4 Detection of the fishery ship from Garchine radar site.

The second target to detect was the fishery ship L’Ar Voaleden. Unfortunately strong inter-ference was present in the same beam as the target. In figure 5.28 we can see a PPI imageafter beamforming. It is well visible, that interference covers the area where target is present.Interference was so strong that, using similar techniques as for tug ship, it was impossible todetect this target. Exemplary image is presented in figure 5.29. This is PPI image (range-azimuth). PPI image is produced by taking maximum value of test statistics from all testvelocities for range-azimuth cell of interest.


Figure 5.22 — Tug ship detection from Garchine site - different range support for AMTIwith 100 chirps.


Figure 5.23 — Tug ship detection from Garchine site - different range support for AMTIwith 200 chirps.


Figure 5.24 — Tug ship detection from Garchine site - different range support for STAPwith 100 chirps.


Figure 5.25 — Tug ship detection from Garchine site - different range support for STAPwith 200 chirps.


Figure 5.26 — Tug ship detection from Brezzelec site - influence of integration time. Upper-left: AMTI for low training samples support, lower-left: AMTI for high training samplessupport, upper-right: STAP for low training samples support, lower-right: STAP for high

training samples support.

5.3.5 Detection of the fishery ship from Brezzelec radar site.

The same fishery ship should be visible from the second radar site (Brezzelec). Similar experi-ments were performed to detect this target. From AIS data it was possible to extract positionand velocity information of the target. Fishery boat was going almost directly toward theBrezzelec radar site at the speed of about 10 knots (this is 5 m/s). To understand possibilitiesof this detection it is necessary to calculate Bragg line Doppler speed. From chapter 3.2 weknow, that Bragg Doppler frequency (f) is around 0,38 Hz for 12 MHz radar. Let us recall,that f = 2V/λ, where V is a speed and λ is radar wavelength (around 28 m for 12 MHzcarrier frequency). From mentioned expression we can derive, that Bragg lines (ocean waves)speed is around 4.9 to 5.1 m/s. We can see, that this is exactly the radial speed of our target.This will make detection very difficult. Generally we should try to have very long integrationtimes in order to achieve Doppler resolution allowing to separate target from Bragg lines.On the other hand it is difficult to estimate covariance matrix for long integration times. For


Figure 5.27 — Tug ship detection from Brezzelec site - comparison between AMTI andSTAP.

example it is hard to achieve STAP for longer than 200 chirps and AMTI for longer than 500chirps. Moreover, having very long integration time we can face the situation, that target isno longer time-coherent. For example, target can change range speed, orientation etc. In Fig.5.30 and 5.31 we can see a PPI image (range-azimuth) that is produced by taking maximumvalue of test statistics from all test velocities for range-azimuth cell of interest. It seems thatfishery boat is not revealed by STAP, but is revealed by AMTI. This is not true. In factneither AMTI nor STAP can reveal this target. In the case of AMTI this is false alarm, thatdisappears with time. Moreover this seeming detection for AMTI indicates radial velocitywhich is negative, whereas fishery boat is in fact approaching radar.

We can conclude, that for the target with the radial velocity equal to Bragg line Dopplervelocity, detection possibilities are very poor.


Figure 5.28 — Strong interference present in the target beam.

Figure 5.29 — PPI image after AMTI processing for 200 chirps.


Figure 5.30 — PPI image after AMTI processing for 200 chirps.

Figure 5.31 — PPI image after STAP processing for 100 chirps.

5.4 Thresholding and detections presentation.

In sections before, it was shown how AMTI and STAP techniques can reveal the target inthe terms of test statistics. In a complete system, the last step it to apply thresholding forthe detections (see section 2.2.5, 2.3 and chapter 4). Choosing appropriate threshold is avery important issue, that is not treated in this work. Instead, I will present some detectionresults for a priori chosen threshold. In figures 5.32 and 5.33 we can see PPI images afterthresholding. I used AMTI algorithm for 200 chirps and low number of training samples.This was due to the computational reasons. After AMTI processing, test statistics cube wasreduced to test statistics image by taking (for each range-beam cell) maximum across allvelocities. The last step was to apply threshold. We can see that some targets were detected,but there are still some undetected targets. We can also see some false alarms, that are mostlyrelated to the presence of islands and rocks as well as to the sidelobes of strong echo fromtarget. Processing after thresholding is also very important, that is not treated in this work.


Figure 5.32 — PPI images 05:20-05:22.

Tracking can reduce number of false alarms considerably (see for example [1] and [13]).

To reduce sidelobe false detections it is worth considering to use dominant scatter removal.Knowledge aided techniques could be used to remove false detections from islands and rocks.Finally, simultaneous processing from two radars, after thresholding, can also reduce thenumber of false alarms as well as give additional vectorial information about target velocity.

Thresholding for other techniques (AMTI with longer integration time and STAP) wasnot performed from computational reasons, but it should give better results (lower numberof false alarms).

5.5 Conclusions.

In this chapter a practical implementation of AMTI and STAP techniques was shown. Wecould see that:


Figure 5.33 — PPI images 05:24-05:26.


• AMTI and STAP techniques can reveal quite small ships even 60 km away from theshore.

• Longer-distance detections are difficult because of the very low transmitting power ofWERA radar system (only 30 W).

• In the case of the presence of a strong interference, STAP technique seems to be moreeffective than AMTI.

• On the other hand, AMTI performance is good enough when very little interference ispresent. Moreover AMTI is less computational and training data demanding. This maybe an important factor.

There are however some other possibilities to improve STAP calculations, that were notconsidered in this work, for example:

• Parametric adaptive methods.

• Diagonal loading.

• Reduced-rank STAP.

Nevertheless, both techniques failed to detect target that is buried completely in interferenceor having the same speed as the Bragg waves. In this chapter, only two targets were consideredas an example of detections. In fact, during experiments many other targets were detectedsuccessfully. Generally we can conclude, that bigger ships (eg. 100 m) are well visible usingboth AMTI and STAP even at longer distances (100 km).

Tu summarize we can say that for HF radar the rationale standing behind using STAPis second order Bragg effects (azimuth-Doppler coupling of the clutter) plus interferences(jamming). In the case of a simple first order Bragg clutter plus interferences, adaptive arraysfollowed by adaptive temporal filtering may be sufficient. Second order Bragg scattering makesSTAP prevail over AMTI. Either way, STAP prevails over AMTI when interference is presenttogether with clutter.

It is worth to reconsider radar waveforms used in HF radars. WERA radar was designedto provide information about waves and currents. For target detection it may be profitable,to change some radar parameters such as transmitted power, or chirp repetition frequency.For example for low chirp repetition frequency, all fast targets fall into false Doppler cell ofa slow target. Therefore an airplane or a very fast speed boats can be detected with falsevelocity.

It seems that HF radar systems have a potential to improve target detections at longdistances that is not yet sufficiently explored. In order to achieve this, it is also necessary toperform analysis on the influence of meteorological conditions. For example sea state can bea crucial factor when detecting targets.

CHAPTER6 Conclusions andperspectives

In this work it was shown how classical STAP concept can be adapted to the sea surveillance.Two main problems were addressed. First one is non-Gaussianity of the clutter and the secondis covariance matrix estimation.

In the first chapter basic concepts were introduced. Classical STAP algorithm was shownas well as Adaptive MTI processing. Detection theory was also presented in order to havea good base to develop other detectors. In order to show potential detection difficulties, seaclutter properties were presented in chapter 3. We could sea, that for see clutter we can expectnon-Gaussianity, Doppler clutter related do Bragg scattering and clutter non-stationarity.These problems were addressed in chapters 4 and 5.

Two Dirac Delta (TDD STAP) detector was presented in chapter 4 as a solution tothe problem of non-Gaussianity. It was shown, that indeed, this detector can give someimprovement in the case of non-Gaussian clutter. There is a possibility to develop this conceptand introduce for example more Dirac deltas to further improve detections. There is, however,a limit of improvement that is related to non-Gaussian distribution itself. This can be viewedas an Cramer-Rao bound for the problem of detection.

In figure 6.1 I presented graphically three cases in the typical detection problems.

They are:

• Gaussianity

• Non-Gaussianity

• Determinism

Figure 6.1 — Triangle of detection problems.

CHAPTER 6. CONCLUSIONS AND PERSPECTIVES 105

Very often we are placed in between all of these cases. Typical detectors were developedunder assumption of Gaussian clutter and noise. This case is well studied. Its characteristicfeature is that in fact this is most random case. It means that this is a result of the lawof large numbers, where we have large number of independent contributors to overall effect.From this point of view this is the worst case, because there is only strictly random patternand as such cannot be predicted precisely. On the other hand, in the case of determinism,there is a perfect information about the process standing behind the clutter, and clutter canbe predicted (anticipated) and suppressed. This case can occur for very high resolution radars.The third case is statistical non-Gaussian distribution. This is very difficult situation. In factthis may occur in the situations that are in the middle between Gaussian and deterministiccase (middle resolution radars). We cannot apply imaging algorithms and classical (Gaussian)algorithms fail in the sense, that they produce more false alarms. This is the case, when newalgorithms must be developed, that can cope with non-Gaussianity. Even then, because ofdistribution imminent properties, usually, performance will be worse than classical algorithmsworking under Gaussian noise and clutter.

In this work it was proved, that under non-Gaussian K-distributed clutter there is apossibility to improve detection in comparison to classical STAP algorithms using a newconcept of Two Dirac Delta detector. To validate results, simulated data were used in chapter4.3.

To test adaptive methods on real data, WERA radar system was used in chapter 5. Themain problem in practical application was the way the covariance matrix should be estimated.I presented sliding window method. Because of low number of range cells in WERA radarsystem as well as clutter nonstationarity it was necessary to precisely chose certain algorithmparameters. STAP algorithm was tested together with Adaptive MTI. It was proved, thatindeed, both STAP and AMTI are effective algorithms in canceling clutter and interferences.It was shown, that having clutter covariance matrix well calculated STAP outperforms AMTIwhen clutter plus interference is present in the data. Nevertheless, in the case of underesti-mated covariance matrix, this advantage can vanish.

To conclude we can say, that STAP technique can be an effective algorithm to detecttargets on the sea surface, and this work give certain solutions to some problems with itspractical application.

The future work should concentrate on:

• Testing algorithm under different sea state and weather conditions.

• Developing concept of TDD STAP detector by introducing more Dirac deltas.

• Applying techniques like reduced-rank STAP, knowledge aided algorithms, dominatingscatterer removal and diagonal loading.

• Applying tracking algorithms after signal processing.

These improvements should reduce number of false alarms and false tracks considerably.

APPENDIXA Gaussian complexprocess.

In this appendix, it will be presented transformation that is commonly used in narrowbandsystems, that allows to employ compact complex notation instead of classical real-numbersnotation. This notation is used in the rest of this PhD thesis, and therefore it is necessary topresent it in an appendix.

Let us assume Gaussian N-dimensional process. Let us split random, N dimensional vector(N is even) into vectors x and y (both having dimension N/2 and zero mean value).

Now we have formula for a Gaussian process [44]:

p([

xy

])=

1(2Π)N/2

√|M| exp(−

[xTyT]M−1

[xy

]

2) (A.1)

where M is a covariance matrix:

M = E{[ x

y

][xTyT]

}

where || denotes determinant. For a narrowband Gaussian process, covariance matrix is of aspecial form [44]:

M =12

[V −WW V

]

where WT = −W, and T denotes transposition.

Therefore we can easily apply transformation into complex domain [44]: Let

z = x + jy (A.2)

z is a N/2 dimensional complex vector. Let also

C = V + jW (A.3)

There are some interesting properties of this transformation:

[xTyT][

V −WW V

] [xy

]= zHCz (A.4)

APPENDIX A. GAUSSIAN COMPLEX PROCESS. 107

[xTyT][

V −WW V

]−1 [xy

]= zHC−1z (A.5)

So

[xTyT]M−1

[xy

]

2= zHC−1z (A.6)

Moreoverp(z) =

1ΠN/2|C| exp(−zHC−1z) (A.7)

where H denotes transpose conjugate.

Additional properties: If z is a random complex Gaussian vector, A is a transformationmatrix and b is a vector such that:

s = Ax + b (A.8)

thenE(s) = AE(x) + b (A.9)

andCOV (s) = ACAH (A.10)

APPENDIXB Data generation.

In order to test performance of STAP, as described in chapter 4, we need to generate Gaussianand SIRP vectors. To do this we will use this representation of a PDF (for real SIRP) [49]:

f(X) =1

(2π)n/2|M |1/2

∫ ∞

0

1σn

e−XTM−1X

2σ2 dGσ(σ) (B.1)

where X is a Spherically Invariant Random Vector (SIRV) and T denotes transpose. IfE(σ2) = 1 then M = E[XXT]. We will be using continuous version:

f(X) =1

(2π)n/2|M |1/2

∫ ∞

0

1σn

e−XTM−1X

2σ2 g(σ)dσ (B.2)

SIRV has a, very useful property [49]: If X is a SIRV with characteristic PDF g(σ), then

Y = AX + b (B.3)

is also SIRV with the same characteristic PDF.It is assumed that A is a matrix such thatAAT = Σ and b is a known vector having the same dimension as X. If X is a zero meanGaussian vector with identity covariance matrix, then covariance matrix of Y = E(Y Y T) =Σ. So general algorithm looks as follows [49]:

1. Generate a white zero mean Gaussian random vector Z, having identity covariancematrix.

2. Generate a random variable v from the PDF g(v). Denote mean square value of v by a2.

3. Normalize v to obtain σ: σ = v/a.

4. Generate product X = Zσ. At this step, we have a white SIRV having zero mean andidentity covariance matrix.

5. Perform linear transformation to obtain SIRV having desired mean value andcovariance matrix.

In radar detection problems we need to have complex samples. Therefore algorithm must bemodified. In Fig. B.1 exemplary histograms are shown.

APPENDIX B. DATA GENERATION. 109

Figure B.1 — Generated SIRVs histograms.

APPENDIXC Space Time AdaptiveProcessing based onFrequency ModulatedContinuous Wavesystem.

C.1 Introduction.

The fundamental assumption that is made while space-time adaptive processing (STAP)technique is considered is that the signal to be processed takes on the form of a sequence ofcoherent pulses. Frequency modulated continuous wave (FMCW) systems are in widespreaduse because of their numerous advantages, for example low transmitting power. It seemsextremely important to look for a possibility of using STAP procedures in FMCW systemsto make them even more attractive. The appendix presents a simple analysis of a proposedapproach to achieve this goal. A structure of the signal processor is presented and assumptionsnecessary to be fulfilled are defined. Effects of the proposed approach implementation in HF,I, and X bands are evaluated on the basis of selected parameters of the appropriate FMCW-STAP systems. Derivations presented in this appendix were inspired by the work [4]. Moreprecisely, all results from section C.2 were published in [4], but without all step-by-steptransformations. Therefore, here I present a complete derivation.

C.2 Preliminaries.

Let us consider single receiving antenna system shown in Fig. C.1. In this document weassume the following standard data set:

fc = 10 MHzfdoppler ± 0.5Hz

fr = PRF = 1 Hz → PRI = Tr = 1 sDoppler resolution = 0.02Hz → TDWELL = 50 s

range resolution = 1.5 km → B = 100 kHz

(C.1)

APPENDIX C. SPACE TIME ADAPTIVE PROCESSING BASED ON FREQUENCYMODULATED CONTINUOUS WAVE SYSTEM. 111

Figure C.1 — Single receiving antenna system.

Transmitted waveform (we assume here only one period from FMCW waveform, centered attime zero):

vT(t) = cos[ωct + πBfrt2] = cos[φT (t)] (C.2)

where −Tr/2 < t < +Tr/2From above we can derive instantaneous frequency as:

fT (t) =12π

dφT (t)dt

= fc + Bfrt (C.3)

Let us consider a single target:

ν = 5 m/st = 0

R0 = 15kmR(t) = R0 + νt = 15 km + 5m

s t

td = 2R(t)c

(C.4)

Received signal is both delayed in time (due to distance from radar to target) and shiftedin frequency (due to non-zero radial velocity). Frequency (over time) of transmitted andreceived signal is shown in Fig. C.2. After mixing with transmitted signal, frequency overtime is shown in Fig. C.3. Mixing with transmitted signal is the same as subtracting thephase. Therefore two frequencies f1 and f2 in Fig. C.3 are :

f1 =12π

d

dt[φT (t− td)− φT (t)] (C.5)

f2 =12π

d

dt[φT (t− td)− φT (t + Tr)] (C.6)

If f2 À f1, than it is possible to filter out part of the signal having frequency f2 as shown inFig. C.4. Condition f2 À f1 is fulfilled if we assume that time delay td of echo from targetis much less than Tr (we keep in mind that T = Tr − td). Therefore distance to the target isbounded. This condition also imply that Tr > T À 1

2Tr. Now it is possible to recenter eachpulse as shown in Fig. C.5. Following calculations assume only re-centered part of the mixed


Figure C.2 — Frequency over time of transmitted and received signal.

Figure C.3 — Received signal after mixer and low pass filter.


Figure C.4 — Signal after filtering out high frequency factor.

Figure C.5 — Demodulated pulse.


signal. Backing off to primitive echo from the target:

vR(t) = AvT(t− td) = A cos[ωc(t− td) + πBfr(t− td)2]

=A

2

[ej[ωc(t−td)+πBfr(t−td)2] + e−j[ωc(t−td)+πBfr(t−td)2]

](C.7)

After mixing, received signal is:

vb(t) =A

2

[ej[ωc(t−td)+πBfr(t−td)2] + e−j[ωc(t−td)+πBfr(t−td)2]

]·

12

[ej[ωct+πBfrt2] + e−j[ωct+πBfrt2]

]

=A

4

[ej[ωc(t−td)+πBfr(t−td)2+ωct+πBfrt2] + e−j[ωc(t−td)+πBfr(t−td)2−ωct−πBfrt2] +

ej[ωc(t−td)+πBfr(t−td)2−ωct−πBfrt2] + e−j[ωc(t−td)+πBfr(t−td)2+ωct+πBfrt2]]

(C.8)

Then signal is passed through a low-pass filter. After this operation only underlined terms in(C.8) remain:

vl(t) =A

4

[ej[ωc(t−td)+πBfr(t−td)2−ωct−πBfrt2] + e−j[ωc(t−td)+πBfr(t−td)2−ωct−πBfrt2]

]

=A

2cos[ωc(t− td) + πBfr(t− td)2 − ωct− πBfrt

2] (C.9)

Let us introducetd = 2R

c = t0 + 2νtic , t0 = 2R0

c (C.10)

Then :

φ(ti) = 2πfc(ti − td) + πBfr(ti − td)2 − 2πfcti − πBfrt2i

= 2πfcti − 2πfctd + πBfr(t2i − 2titd + t2d)− 2πfcti − πBfrt2i

= 2πfcti − 2πfctd + πBfrt2i − 2πBfrtitd + πBfrt

2d − 2πfcti − πBfrt

2i

= −2πfctd − 2πBfrtitd + πBfrt2d

= −2πfc(t0 +2νtic

)− 2πBfrti(t0 +2νtic

) + πBfr

(t0 +

2νtic

)2

= −2πfct0 − 2πfc2νtic

− 2πBfrtit0 − 2πBfr2νt2i

c

+πBfrt20 + 2πBfrt0

2νtic

+ πBfr(2νtic

)2

= [− 2πfct0 + πBfrt20] + 2π[− 2fc

ν

c

+2Bfrt0ν

c− Bfrt0]ti − 2πB

2ν

cfr

[1− ν

c

]t2i (C.11)

For assumed set of data (eg. providing ν/c ¿ 1), during one pulse duration the quadraticterm and second part of the linear term may be neglected. Therefore we have:

φ(ti) ≈ φ0 − 2π[2ν

cfc + Bfrt0]ti (C.12)

hence within the first pulse, frequency fI is

fI =12π

dφ(t)dt

=2ν

cfc + Bfrt0 (C.13)


As can be seen, this frequency offset consists of two terms: the first is due to target velocityand the second is due to time delay (range) to the target. It is not possible to separate velocityfrom range, analysing only one pulse. In our example for the target:

2ν

cfc =

13Hz, Bfrt0 = 10 Hz (C.14)

similar analysis for the nth pulse:

td = 2Rc = t0 + 2ν

c (nTr + ti) , t0 = 2R0c (C.15)

ti− is a time variablewithin one pulse duration (C.16)

φ(ti) = 2πfc(ti − td) + πBfr(ti − td)2 − 2πfcti − πBfrt2i

= 2πfcti − 2πfctd + πBfr(t2i − 2titd + t2d)− 2πfcti − πBfrt2i

= 2πfcti − 2πfctd + πBfrt2i − 2πBfrtitd + πBfrt

2d − 2πfcti − πBfrt

2i

= −2πfctd − 2πBfrtitd + πBfrt2d

= −2πfc[t0 +2ν

c(nTr + ti)]

−2πBfrti[t0 +2ν

c(nTr + ti)]

+πBfr

[t0 +

2ν

c(nTr + ti)

]2

= −2πfct0 − 2πfc2ν

cnTr − 2πfc

2ν

cti

−2πBfrtit0 − 2πBfrti2ν

cnTr − 2πBfrti

2ν

cti

+πBfr

[t20 + 2t0

2ν

c(nTr + ti) +

4ν2

c2(nTr + ti)2

]


cnTr − 2πfc

2ν

cti


cnTr − 2πBfrti

2ν

cti

+πBfr

[t20 + 2t0

2ν

cnTr + 2t0

2ν

cti +

4ν2

c2(n2T 2

r + 2nTrti + t2i )]


cnTr − 2πfc

2ν

cti


cnTr − 2πBfrti

2ν

cti

+πBfrt20 + πBfr2t0

2ν

cnTr + πBfr2t0

2ν

cti + πBfr

4ν2

c2(n2T 2

r + 2nTrti + t2i )


cnTr − 2πfc

2ν

cti


cnTr − 2πBfrti

2ν

cti


2ν

cnTr + πBfr2t0

2ν

cti + πBfr

4ν2

c2n2T 2

r

+πBfr4ν2

c22nTrti + πBfr

4ν2

c2t2i


cnTr − 2πfc

2ν

cti



cnTr − 2πBfrti

2ν

cti


2ν

cnTr + πBfr2t0

2ν

cti + πBfr

4ν2

c2n2T 2

r

+πBfr4ν2

c22nTrti + πBfr

4ν2

c2t2i


cnTr − 2πfc

2ν

cti


cnTr − 4πBfr

ν

c

[1− ν

c

]t2i


2ν

cnTr + πBfr2t0

2ν

cti + πBfr

4ν2

c2n2T 2

r + πBfr4ν2

c22nTrti

= −2πfct0 + πBfrt20 − 2πfc

2ν

cnTr − 2πfc

2ν

cti − 2πBfrtit0 − 2πBfrti

2ν

cnTr

−4πBfrν

c

[1− ν

c

]t2i

+πBfr2t02ν

cnTr + πBfr2t0

2ν

cti + πBfr

4ν2

c2n2T 2

r + πBfr4ν2

c22nTrti


2ν

cnTr − 2π

[2ν

cfc + Bfrt0 + n

2ν

cB

]ti

−4πBfrν

c

[1− ν

c

]t2i

+πBfr2t02ν

cnTr + πBfr

4ν2

c2n2T 2

r + πBfr2t02ν

cti + πBfr

4ν2

c22nTrti


2ν

cnTr − 2π

[2ν

cfc + Bfrt0 + n

2ν

cB

]ti

−4πBfrν

c

[1− ν

c

]t2i

+πB2t02ν

cn + πB

4ν2

c2n2Tr

+πBfr2t02ν

cti + πB

4ν2

c22nti (C.17)

In order to neglect terms:

−4πBfrν

c

[1− ν

c

]t2i

+πB2t02ν

cn + πB

4ν2

c2n2Tr

+πBfr2t02ν

cti + πB

4ν2

c22nti (C.18)

certain assumptions must be made:

1.

4πBfrν

c

[1− ν

c

]t2i ¿ 2π → B

(2νmax

c

)Tr

14¿ 1

since max value of ti is 12Tr

2.

πB2t02ν

cn ¿ 2π → B

(2Rmax

c

)(2νmax

c

)N ¿ 1

where N is a number of pulses and Rmax is a max range from the radar to the target


3.

πB4ν2

c2n2Tr ¿ 2π → 1

2B

(2νmax

c

)2N2Tr ¿ 1

4.

πBfr2t02ν

cti ¿ 2π → B

(2Rmax

c

)(2νmax

c

)12¿ 1

again we have to remember that max value of ti is 12Tr and fr = 1

Tr. This condition is

weaker than (2.) since 12 ¿ N .

5.

πB4ν2

c22nti ¿ 2π → 1

2B

(2νmax

c

)2NTr ¿ 1

again max value of ti is 12Tr and max value of n is N . This condition is weaker than (3.)

because N < N2.

So after simplifications based on conditions 1. 2. 3. we have:

φIn(ti) = φ0 − 2nπfc2ν

cTr − 2π[

2ν

cfc + Bfrt0 + n

2ν

cB]ti (C.19)

so

fIn =12π

dφ(t)dt

=2ν

cfc + Bfrt0 + n

2ν

cB (C.20)

Let us perform Fourier transformation on the received (n-th) pulse (C.9) using (C.19)

VIn(f) =∫ T/2

−T/2Acos (φIn(t))e

−j2πftdt

=∫ T/2

−T/2Acos (φ0 − 2nπfc

2ν

cTr − 2πfInt)e−j2πftdt

=∫ T/2

−T/2

A2

(ej(φ0−2nπfc

2νc

Tr−2πfInt) + e−j(φ0−2nπfc2νc

Tr−2πfInt)

)e−j2πftdt

=∫ T/2

−T/2

A2

(ej(φ0−2nπfc

2νc

Tr−2πfInt−2πft)

)e−j2πftdt +

∫ T/2

−T/2

A2

(e−j(φ0−2nπfc

2νc

Tr−2πfInt+2πft)

)e−j2πftdt

=∫ T/2

−T/2

A2

ej(φ0−2nπfc2νc

Tr−2πfInt−2πft)dt +

∫ T/2

−T/2

A2

e−j(φ0−2nπfc2νc

Tr−2πfInt+2πft)dt

=A2

ej(φ0−2nπfc2νc

Tr)

∫ T/2

−T/2ej(−2πfInt−2πft)dt +

A2


Tr)

∫ T/2

−T/2e−j(−2πfInt+2πft)dt


Figure C.6 — Spectrum of the received pulse.

=A2

ej(φ0−2nπfc2νc

Tr)

∫ T/2

−T/2e−j(2πfInt+2πft)dt +

A2


Tr)

∫ T/2

−T/2e−j(2πft−2πfInt)dt

=A2

ej(φ0−2nπfc2νc

Tr) 2 sin [2π(fIn + f)T/2]2π(fIn + f)

+A2


Tr) 2 sin [2π(f − fIn)T/2]2π(f − fIn)

=AT2

ej(φ0−2nπfc2νc

Tr) 2 sin [2π(fIn + f)T/2]2π(fIn + f)T

+AT2


Tr) 2 sin [2π(f − fIn)T/2]2π(f − fIn)T

=AT2

[ej(φ0−2nπfc

2νc

Tr) 2 sin [2π(fIn + f)T/2]2π(fIn + f)T

+ e−j(φ0−2nπfc2νc

Tr) 2 sin [2π(f − fIn)T/2]2π(f − fIn)T

]

=AT2

[ej(φ0−2nπfc

2νc

Tr) sin [2π(fIn + f)T/2]2π(fIn + f)T/2

+ e−j(φ0−2nπfc2νc

Tr) sin [2π(f − fIn)T/2]2π(f − fIn)T/2

]

=AT2

[ejφ0−j2nπfc

2νc

Trsin [2π(f + fIn)T/2]

2π(f + fIn)T/2+ e−jφ0+j2nπfc

2νc

Trsin [2π(f − fIn)T/2]

2π(f − fIn)T/2

]

=AT2

[ejφ0−j2πfc

2νc

nTrsin [2π(f + fIn)T/2]

2π(f + fIn)T/2+ e−jφ0+j2πfc

2νc

nTrsin [2π(f − fIn)T/2]

2π(f − fIn)T/2

]

(C.21)

Let us investigate (C.21). Considering only positive frequencies, maximum of (C.21) is in pointf = fIn , and fIn is related to the distance to the target (see (C.20) and (C.10)). Thereforedistance to the target can be read from the spectrum of the received signal as can be seenin Fig. C.6 (this is true as long as 2ν

c fc and n2νc B are much less than Bfrt0, eg. (C.14)). We

see that from pulse to pulse, two parameters can change: fIn in sin [2π(f±fIn)T/2]2π(f±fIn)T/2 and nTr in

e±jφ0∓j2πfc2νc

nTr . For standard data set, over dwelling time (50 s) fIn changes only slightly,

hence sin [2π(f±fIn)T/2]2π(f±fIn)T/2 can be interpreted as function of f only (let say K(f)). Taking into

account only positive frequencies we can neglect term

ejφ0−j2πfc2νc


2π(f + fIn)T/2

Therefore we have the following approximation:

VIn(f) ≈ K(f)e−jφ0+j2πfc2νc

nTr (C.22)


Including e−jφ0 in K(f) leads to:

VIn(f) = K(f)ej2πfc2νc

nTr (C.23)

We can see that for the specific range (which means frequency f) from pulse to pulse onlyterm ej2πfc

2νc

nTr changes. Assumptions are summarized in the table C.1.

Table C.1 — Assumptions.

T À 12Tr → 1

2Tr À 2Rmaxc → Rmax ¿ 1

4Trc range limitation

B(

2νmaxc

)Tr

14 ¿ 1 1.

B(

2Rmaxc

)(2νmax

c

)N ¿ 1 2.

12B

(2νmax

c

)2N2Tr ¿ 1 3.

νmaxNTr ¿4R 4R− range resolution,no range migration condition

dνdt NTr ¿4ν 4ν − velocity resolution,

no velocity migration condition

2νmaxc fc ¿ Bfrt0 → νmax

fc

B Tr ¿ Rmin t0 = 2Rminc

violation of this condition leadsto range errors which can bemitigated after Dopplerprocessing (not true for STAP)

N 2νmaxc B ¿ Bfrt0 → νmaxNTr ¿ Rmin t0 = 2Rmin

cthis condition is weaker than conditionfor the range migration

C.3 Antenna array with FMCW.

Lets perform similar calculations for multi-dimensional case (see Fig. C.7). Received signal,coming from the target will be now expressed in similar way as in (C.7):

vmr(t) = Av(t− td − ta) = A cos[ωc(t− td − ta) + πBfr(t− td − ta)2]


Figure C.7 — System with antenna array.

Figure C.8 — Geometry of the target echo and antenna array.

=Aej[ωc(t−td−ta)+πBfr(t−td−ta)2] + Ae−j[ωc(t−td−ta)+πBfr(t−td−ta)2]

2

=A

2

[ej[ωc(t−td−ta)+πBfr(t−td−ta)2] + e−j[ωc(t−td−ta)+πBfr(t−td−ta)2]

]

(C.24)

where ta is an additional delay due to the distance between antenna elements. We assumethe first antenna as a reference antenna (see Fig. C.8). Subscript m denotes antenna element(m = 1 . . . M). Time delay will be equal md

c sin(α), where d is a distance between antennaelements and c is a speed of wave propagation. After low pass filter we have:

vml(t) =A

2cos[ωc(t− td − ta) + πBfr(t− td − ta)2 − ωct− πBfrt

2]

(C.25)

Let tboth = td + ta:

tboth = t0 +2νtic

+md

csin(α) = t′0 +

2νtic

(C.26)

Now:

φm(ti) = 2πfc(t− tboth) + πBfr(t− tboth)2 − 2πfcti − πBfrt2i


= −2πfctboth − 2πBfrtitboth + πBfrt2both

= [− 2πfct′0 + πBfrt

′02] + 2π[− 2fc

ν

c+ 2Bfrt

′0

ν

c− Bfrt

′0]ti − 2πB

2ν

cfr

[1− ν

c

]t2i

(C.27)

Similar simplifications as before lead us to formulas as follows. Phase for single pulse:

φm(ti) ≈ φ0 − 2πfcmd

csin(α)− 2π[

2ν

cfc + Bfrt

′0]ti (C.28)

hence within the first pulse, instantaneous frequency fmI is

fmI =12π

dφ(t)dt

=2ν

cfc + Bfrt

′0 (C.29)

And for the for the nth pulse

φmIn(ti) = φ0 − 2πfcmd

csin(α)− 2nπfc

2ν

cTr − 2π[

2ν

cfc + Bfrt

′0 + n

2ν

cB]ti

so

fmIn =12π

dφ(t)dt

=2ν

cfc + Bfrt

′0 + n

2ν

cB (C.30)

After Fourier transformation:

VmIn(f) =∫ T/2

−T/2Acos (φmIn(t))e

−j2πftdt

=AT2

[ejφ0−j2πfc

mdc

sin(α)−j2πfc2νc


2π(f + fIn)T/2+

e−jφ0+j2πfcmdc

sin(α)+j2πfc2νc

nTrsin [2π(f − fIn)T/2]

2π(f − fIn)T/2

]

(C.31)

Again for positive frequencies we have the following approximation:

VmIn(f) ≈ K(f)e−jφ0+j2πfcmdc

sin(α)+j2πfc2νc

nTr (C.32)

Including e−jφ0 in K(f) leads to:

VmIn(f) = K(f)ej2πfcmdc

sin(α)+j2πfc2νc

nTr

= K(f)ej2πfcmdc

sin(α)ej2πfc2νc

nTr (C.33)

We see that (C.33) changes (from pulse to pulse and between array elements) exactly in thesame way as it is assumed for STAP processing (see for example [37]).

C.4 STAP system using FMCW.

In this section digital processing will be explained. First we need to set cut-off frequency of thelow-pass filter. To get rid of frequency f2 (C.6) the cut-off frequency can be set B/2. Thenwe are left with the signal as in the Figure C.9 (for clarity reasons transmitted sawtoothwaveform is also shown but on a different frequency scale). Now for N pulses we have to


Figure C.9 — Sampling after filtering.

Figure C.10 — Data cube.

separately perform L point FFT. Thus the sampling rate should be L/Tr. L should be setthat condition L/TR > 2fImax is fulfilled. fImax depends on the maximum range of the radarsystem and on the range resolution since B is set to obtain desired range resolution. Aftersampling and FFT we have L/2 samples for positive frequencies and L/2 samples for negativefrequencies (from which 2 are the same frequency-zero frequency). After this operation (foreach antenna array receiving element) we can arrange our complex samples into data cube:M antenna elements * N pulses * L/2 range cells. This is the form accepted by digital STAPprocessor (Fig. C.10) see for example [37].

C.5 FMCW HF system - practical example.

In this section we summarize practical considerations


fc = 10 MHz (C.34)

which gives:λ = 30 m (C.35)

LetB = 100 kHz → range resolution = 1.5 km (C.36)

Let max Doppler shift be:

fdoppler ± 0.5Hz (Vmax < 7.5ms

) (C.37)

Which gives:fr = PRF = 2 ∗ 0.5Hz = 1Hz → PRI = Tr = 1 s (C.38)

We want to have:Doppler resolution = 0.02Hz → TDWELL = 50 s (C.39)

Now let us consider conditionf2 À f1 (C.40)

This implies that

T À 12Tr → 1

2Tr À 2R

c(C.41)

After calculations we have condition for the radar range

R ¿ 75 000 km (C.42)

Now we want to have a radar range 150 km. This means that fImax = 100 Hz. This is becausetwo way distance is 300 km which gives 1 ms time delay. Sawtooth signal changes within 100kHz band during 1 s. So after 1 ms, frequency difference is 100 Hz. Our sampling rate shouldbe twice this number. After calculations we have L=200. Since FFT operates on samplesnumber equal power of 2, we can set L=256 (we will have 256 range cells). N will be 50 sincedwelling time is 50 s.

C.6 FMCW X-band system - practical example.

In this section similar considerations for X-band system are presented.

fc = 10 GHz (C.43)

which gives:λ = 0.03m (C.44)

LetB = 1 MHz → range resolution = 150 m (C.45)


fdoppler ± 500Hz (Vmax < 7.5ms

) (C.46)

Which gives:fr = PRF = 2 ∗ 500Hz = 1 kHz → PRI = Tr = 1 ms (C.47)


We want to have:Doppler resolution = 10Hz → TDWELL = 0.1 s (C.48)


This implies that

T À 12Tr → 1

2Tr À 2R

c(C.50)


R ¿ 75 km (C.51)

Additional calculations to verify assumptions (target at the range of 1 km and at the speedof 5m

s ) :2ν

cfc =

13kHz, Bfrt0 = 6 kHz (C.52)

If we put radar range 10 km we will have L > 120 (eg. 128), and N=100.

C.7 FMCW L-band system - practical example.

In this section are presented similar considerations for L-band system.

fc = 1 GHz (C.53)

which gives:λ = 0.3m (C.54)

LetB = 1 MHz → range resolution = 150 m (C.55)


fdoppler ± 50Hz (Vmax < 7.5ms

) (C.56)

Which gives:fr = PRF = 2 ∗ 50Hz → PRI = Tr = 10 ms (C.57)

We want to have:Doppler resolution = 1 Hz → TDWELL = 1 s (C.58)


This implies that

T À 12Tr → 1

2Tr À 2R

c(C.60)


R ¿ 750 km (C.61)

Additional calculations to verify assumptions (target at the range of 10 km and at the speedof 5m

s ) :2ν

cfc = 30 Hz, Bfrt0 = 6000 Hz (C.62)

If we put radar range 150 km we will have L > 2000 (eg. 2048), and N=100.

APPENDIXD List of symbols andabbreviations.

Abbreviations have following meaning (unless otherwise stated):AB - Adaptive Beamforming

A/D - Analog-Digital converterAIS - Automatic Identification System

AMTI - purely temporal Adaptive MTIc - Speed of light

CDF - Cumulative Distribution FunctionCFAR - Constant False Alarm RatioCPDF - Characteristic PDF

CW - Continuous WaveDFT - Discrete Fourier TransformEEZ - Exclusive Economic Zone

fD - Doppler frequencyFA - False Alarm

FFT - Fast Fourier TransformFMCW - Frequency Modulated Continuous WaveGLRT - Generalized Likelihood Ratio Test

H - Conjugate transposeHF - High Frequency

IMO - International Maritime OrganizationL - Number of range cells

LMSE - Least Mean Square ErrorLO - Locally Optimum (detector)

LPF - Low Pass FilterLRT - Likelihood Ratio Test

M - Number of antennasML - Maximum Likelihood

MMSI - Maritime Mobile Service Identity

APPENDIX D. LIST OF SYMBOLS AND ABBREVIATIONS. 126

MTI - Moving Target IndicationMV - Minimum Variance

N - Number of pulses/chirpsOTH - Over The Horizon

PD - Probability of DetectionPDF - Probability Distribution FunctionPPI - Plan position indicator

PRF - Pulse Repetition FrequencyPRI - Pulse Repetition Interval

R - DistanceRCS - Radar Cross SectionROC - Receiver Operating CurveSIRP - Spherically Invariant Random ProcessSIRV - Spherically Invariant Random Vector

SM - Sample covariance MatrixSNR - Signal to Noise Ratio

STAP - Space Time Adaptive ProcessingT - Timet - Time

TDD - Two Dirac DeltasUTC - Universal time

Vr - Radial velocity relative to the radarVAR - Variance

λ - Wavelength⊗ - Hadamard product

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TELECOM BRETAGNE

MILITARY UNIVERSITY OF TECHNOLOGY

MEMOIRE DE THESE - SYNTHESE

Traitement Adaptatif Temps-Frequence Pour les Radars de

Surveillance Maritime

Tomasz Gorski

30.06.2008

Directeurs :dr hab. inz Adam Kawalec, prof. WATdr hab. inz Jean-Marc Le Caillec, prof. Telecom Bretagne

L’objet de ce travail est l’etude de l’application de la technique du traitement spatio-temporelle adaptative (en anglais STAP, Space-Time Adaptative Processing) pour la sur-veillance de la surface de la mer. Ces techniques du STAP sont ete developpees afin d’utiliserconjointement toute l’information disponible (energie/Doppler) pour la detection de cibles,et dans notre cas de cibles marines. En particulier, ce travail analyse les limitations dues aufouillis de mer. En effet, le fouillis de mer presentent des caracteristiques qui ne sont pascompatibles avec les hypotheses theoriques d’utilisation des STAP. Le fouillis presentent defortes non stationnarites aussi bien temporelles que spatiales. Il presente des statistiques nongaussiennes, ce qui influe sur la forme du detecteur optimal. De plus, il presente un spectreDoppler tres etendu englobant l’echo Doppler de la cible. Par consequent une redefinition desalgorithmes de detection est un probleme primordial pour une application en milieu marin.Afin de tenir compte de la non gaussianite des signaux retrodiffuses par la surface de la mer,nous avons modelise le fouillis de mer par un processus aleatoire spheriquement invariant(SIRP) dont l’un des cas particuliers est le processus gaussien. En effet, les differentes sous-classes des signaux SIRP se distinguent par une fonction auxiliaire d’une variable aleatoireτ (lorsque cette variable aleatoire devient constante, nous retrouvons le cas gaussien usuel).Afin de generaliser les detecteur a des cas non gaussien, une approximation de la densite deprobabilite de cette variable aleatoire par deux Dirac a ete proposee. Une optimisation concer-nant la distance entre les deux Diracs ainsi que des coefficients de ponderation de ces Diracs aete effectuee. Cette approximation sert a simplifier les calculs necessaires pour la proceduresde detection et plus particulierement pour l’estimation de la matrice de covariance. Cettematrice est indispensable pour l’estimation du rapport de vraisemblance sur lequel se baseles procedures de detection. Plusieurs experiences d’application sur des donnees simulees dutraitement spatio-temporelle ont ete faites. Des experiences similaires ont ete effectues avecdes donnees reelles (obtenues a partir du radar HF WERA.), avec un etat de mer connu ainsique deux navires dont les positions ont ete fournies par un AIS (Automatic IdentificationSystem). Les resultats (de simulation et experimentaux) obtenus confirment la possibilite detraitement STAP efficace pour la surveillance maritime. Des algorithmes plus classiques, tel” l’Adaptative Multiple Target Indicators ” AMTI, ont aussi ete developpes a des fins decomparaisons. Malgre le bruit, les interferences ou simplement le fouillis de mer en HF, lesresultats de detection des objets flottants par la technique STAP sont tres prometteurs. Unepoursuite des recherches semble cependant necessaire pour une application temps reel.

Space-Time Adaptive Signal Processing for Sea Surveillance Radars

There are several limitations of radar systems for sea surveillance. One group of limitationsis related to strong clutter from sea waves (especially during heavy seas periods). Thereforea big challenge is to find new processing techniques for this application. In this work it isproposed to apply Space-Time Adaptive Technique for target detection on the sea surface.Problems addressed in this work cover : clutter non-stationarity in space and time, clutternon-Gaussianity, clutter with spread Doppler spectrum. The purpose of this work is to resolvethese problems. As a result this work serves as a complete guide how to deal with sea clutterby modifing STAP technique. In experiments author used a simulated data as well as datafrom th real Surface Wave system WERA.

This PhD thesis is a result of cooperation between Military University of Technology inWarsaw (Poland) and Telecom-Bretagne in Brest (France).

Traitement adaptatif temps-frequence pour les radars de surveillance mari-time

Il existe plusieurs limitations dans les systemes radar de surveillance maritime. En par-ticulier, un probleme important est le fort retour electromagnetique de la surface de mer(notamment pour les mers formees). La deefinition de nouvelles techniques de traitement deces donnees radar est un domaine en pleine expansion. Dans cette these, nous proposons l’uti-lisation de traitement adapatif dans les domaines temps-frequence (STAP en englais) pourla detection d’objets sur la surface de mer. Cependant, cette methodologie est confrontee aplusieurs difficultes : Non stationnarite (dans le temps et l’espace) du fouillis de mer, Nongaussianite de ce fouillis, fouillis avec un spectre Doppler etendu. Le propos de ce travail a eted’etudier des solutions a ces divers problemes. En particulier, une modification des techniquesde STAP (traitement et decision) pour la prise en compte des problemes mentionnes ci-dessusest detaillee dans cette these. Ces modifications ont ete testees sur des donnees simulees, maisaussi sur des donnees reelles obtenues grace au radar a onde de surface WERA.

Cette these s’inscrit dans le cadre de la cooperation entre l’Universite Militaire de Varsovieet Telecom bretagne (Brest).

warsaw mti

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