Space-Time Adaptive Signal Processing for Sea Surveillance Radars

Transcription

1 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 Telecom Bretagne and the Military University of Technology Domain : Signal Processing and Telecommunications Mention : 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 Sea Surveillance Radars Defence December 9, 2008 before the examination board : Reporters : Examiners : Marc Acheroy, Professor at Royal Military Academy in Brussels Richard Klemm, Doctor at FGAN Jean Marc Le-Caillec, Professor at Telecom Bretagne Adam Kawalec, Professor at the Military University of Technology in Warsaw Laurent Ferro-Famil, Doctor with accreditation to supervise reasearch at the University of Rennes 1 Ali Khenchaf, Professor at ENSIETA

2 Contents 1 Introduction. 1 2 Radar Basics, Space-Time Adaptive Processing and Target Detection Radar principles Overview of STAP Problem Statement Radar System Airborne Clutter Adaptive MTI STAP Processing Assumptions and Limitations Detection Principles: Neyman-Pearson Test Notation Neyman-Pearson Lemma Generalized Likelihood Ratio Test Alternative Hypothesis of the Form θ > θ H Alternative Hypothesis of the Form θ θ H Examples Detection of Known Narrowband Signals in Narrowband Noise Detection of Known Narrowband Signals with Random Phase Angles Spherically Invariant Random Process (SIRP) Likelihood Ratio Test and Generalized Likelihood Ratio Test applied to the Spherically Invariant Random Process Detection of Known Narrowband Signals - Likelihood Ratio Test Detection of Known Narrowband Signals with Random Phase Angles and Random Amplitude - GLRT Detector Conclusions Sea Clutter. 31

3 CONTENTS ii 3.1 Sea clutter characterization in X band Sea clutter characterization in HF band Conclusions Two Dirac delta detector Resolving GLRT Two Dirac Deltas approximation First approach Refined approach Simulations Simulation parameters Target Simulations Additive Noise Results Classical STAP detection performance evaluation Numerical simplifications for TDD STAP Comparison of classical STAP and TDD STAP for fixed parameter Results for TDD STAP detector with automatic finding Conclusions HF radar signals experiments and STAP technique modifications WERA radar system Implementation of Adaptive MTI and STAP - covariance matrix estimation problem Adaptive MTI implementation STAP implementation Comparisons between the results of AMTI and STAP Data file and target description Detection of the tug ship from Garchine radar site Detection of the tug ship from Brezzelec radar site Detection of the fishery ship from Garchine radar site Detection of the fishery ship from Brezzelec radar site Thresholding and detections presentation Conclusions Conclusions and perspectives 104 A Gaussian complex process. 106

4 CONTENTS iii B Data generation. 108 C Space Time Adaptive Processing based on Frequency Modulated Continuous Wave system. 110 C.1 Introduction C.2 Preliminaries C.3 Antenna array with FMCW C.4 STAP system using FMCW C.5 FMCW HF system - practical example C.6 FMCW X-band system - practical example C.7 FMCW L-band system - practical example D List of symbols and abbreviations. 125 Bibliography 127

5 CHAPTER1 Introduction. Present radar systems for sea surveillance have several limitations. One group of limitations is related to strong clutter from sea waves (especially during heavy seas periods). Another group is related to range limitations of present microwave systems. These are big obstacles hindering to provide reliable surveillance data that cover Exclusive Economic Zone (200 nautical miles) 1 24 hours a day, 365 days a year. Therefore a big challenge is to find new techniques for this application. This work covers signal processing for this purpose. Space- Time Adaptive Processing (STAP) technique has a relatively long history. The theory of STAP was first published by Lawrence E. Brennan and Irving S. Reed in 1970 s [6], whereas complete 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 clutter that has Gaussian statistical properties. In this work it is proposed to apply this technique to target detection under non-gaussian conditions. The thesis of this work can be formulated as follows: STAP can be an effective technique for the Sea Surveillance Radars. Surveillance can be understood as clutter (and interferences) suppression, target matching and threshold decision. To this end there are three problems 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 with implementing them and to try to find solutions to these problems. As a result this work serves as 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 introduction to Space-Time Adaptive Processing. First section is devoted to basic radar concepts. Next section is an introduction to Adaptive MTI (AMTI) and STAP. It will be shown how STAP was introduced for airborne radar, and what was rationale standing behind this. Generally we can say, that the origin of STAP was the observation, that clutter spectrum depends on the look angle of radar system. Assumptions and limitations of STAP will be shown in the same chapter. In the next sections reader can find some elements of detection theory and Neyman-Pearson Lemma and Test. This will build a base to derive more general detectors than STAP in chapter 4. In the same chapter theory of Spherically Invariant Random Process 1 EEZ zone was defined in 1982 by United Nations Convention in Montego Bay.

6 CHAPTER 1. INTRODUCTION. 2 (SIRP) is introduced. This theory is very useful when dealing with non-gaussian clutter. It will be shown in the next chapter, that sea clutter very often has non-gaussian properties and in this case we can employ theory of SIRP. Last section will be devoted to derivation of Neymann-Pearson tests in the case of SIRP. It will be shown, that classical STAP algorithm can be derived from this more general form. The second chapter is devoted entirely to sea clutter characterization. Two radar bands will be considered: X-band and HF-band. In the first section, X-band clutter Dopplerspectrum and its properties will be presented. It will be shown, that clutter Doppler shift and statistics are related to geometry of the scene as well as many ocean parameters [9]. This property is exploited in STAP algorithm, therefore it is worth considering STAP technique to deal with this kind of clutter. Unfortunately X-band sea clutter (especially for low grazing angles) has non-gaussian properties [8], whereas STAP was derived under assumption of Gaussianity. Therefore it is possible to improve classical STAP algorithm to deal with sea clutter. This problem will be treated in chapter 4. For HF-band, sea clutter properties are different. The main contribution to the clutter is Bragg scattering. Its Doppler spectrum remains the same across different look angles. Moreover, for HF-band it is very likely to have strong radio interferences. Author s own calculations illustrating Bragg clutter and interferences will be presented. Results were obtained using real data from WERA radar system. Again we can see, that clutter and interferences have two-dimensional, space and time structure, and therefore it is reasonably to use STAP algorithm. Chapter 4 addresses the problem of the derivation of detectors under non-gaussianity that was raised in chapter 3. This is done in the framework of Spherically Invariant Random Process. A new detector will be presented. It can deal with non-gaussian clutter and noise. To evaluate performances of classical and new STAP detector in non-gaussian clutter, I performed some simulations. Receiver Operation Curves (ROC) are presented based on simulations made by the author. A discussion of the performances of usual STAP and the proposed detector under different kind of clutter (Gaussian, non-gaussian) is included. I will show, that the new developed detector can give some improvement in comparison to classical STAP algorithm in the presence of non-gaussian clutter. In chapter 5 experiments from High Frequency (HF) radar system will be presented. HF radar systems, which operate in frequency range between 3 and 30 MHz, have a potential to detect targets which are located beyond optical horizon on the sea surface (Over The Horizon visibility - OTH). In this context new problems have to be faced. An exhaustive review of these problems can be found in [3]. In this chapter two techniques will be considered. First one is AMTI, and the second is STAP. Because of practical problems, classical algorithms must be 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 processing part and to much less extent on detection problems. Additional problems are addressed in Appendices. Among them, the most important is the problem of application of STAP algorithm to Continuous Wave (CW) systems. This problem is treated in Appendix C.3. This work can be viewed as an attempt to find how to apply STAP technique to the problem of detecting targets on the sea surface. This PhD thesis is a result of cooperation between Military University of Technology in Warsaw (Poland) and Telecom-Bretagne in Brest (France).

7 CHAPTER2 Radar Basics, Space-Time Adaptive Processing and Target Detection. Present radar systems operate in very difficult environment, where target echo must compete against ground or sea clutter, noise, interferences and jamming. Therefore, apart from solving strictly technical problems such as increasing peak power, improving range resolution, there is a need of effective signal processing techniques to detect targets while maintaining reasonably low level of false alarms. In the era of non-coherent radars, the solution was to reduce resolution cell to limit the clutter power. After deployment of pulse-coherent radars it became possible to apply more advanced techniques based on Doppler effect. This is how Moving Target Indication (MTI) and pulsed-doppler radars were invented. We can generally 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 description of such systems can be found in [53]. However, these techniques are based only on time-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 directional interference and other directional distortions. More information on this topic can be found in [44]. The problem however persisted, how to combine these two techniques - one operating in time domain and the other in space domain. The solution was introduction of Space-Time Adaptive Processing (STAP), that will be described in this chapter. STAP is a technique that performs adaptation in two domains at the same time (whereas Adaptive MTI works only in time domain). This raises possibilities of suppressing distortions that have two-dimensional structure. 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 was introduced for airborne radar, and what was rationale standing behind this. Assumptions and limitations will be shown in the same section. Section 2.3 will be devoted to detection theory and to the Neyman-Pearson Lemma and Test. This will build a base to derive more general detectors than STAP. Some examples of Neyman-Pearson tests will be presented in section 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 employ theory of SIRP. Section 2.6 will be devoted to derivation of Neymann-Pearson tests in the case of SIRP. It will be shown, that classical STAP algorithm can be derived from this

8 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET 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 modulated pulses of electromagnetic wave. Electromagnetic wave is then partially reflected by an object and returns toward radar receiver as a target echo (see Fig. 2.1). Radar can measure the time between transmition of the pulse and reception of the echo, and calculate distance to the target using formula: R = ct (2.1) 2 where R is a measured distance to the target, c is a speed of light and t is elapsed time between transmission and reception of the echo. Fig. 2.2 presents basic time relations that are important in radar systems. Pulse Repetition Interval (PRI) is related to the fact that usually radars transmit signals that are periodical. PRI is a time between two successive pulses transmitted by the radar. τ denotes pulse duration. t is the time that is passed between transmission of the pulse and reception of the target echo. Usually, information about distance to the target is presented in the discreet form, therefore instead of the range it is better to use the term range gate or range cell - Fig This means nothing more but the discretization of the range value. This principle allows to obtain information about the distance and direction

9 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 5 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 necessary to introduce pulse-coherent radar. This means the use of quadrature receiver which extracts amplitude together with phase information on pulse to pulse basis. Structure of such a receiver is presented in Fig After this process, for each range gate we have a series of complex samples. 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 complex samples). If the target is moving between pulses, then the angles of complex samples are changing from pulse to pulse. This is the way to extract Doppler, and therefore velocity information. In practice, to extract Doppler information, a finite sequence of pulses is used. Therefore it is worth to remind some basic relations: f D = 2V r λ (2.2) Figure 2.4 Quadrature Receiver [41]. f c denotes radar carrier frequency, f s denotes sampling frequency.

10 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 6 Figure 2.5 Environmental diagram [53]. where f D denotes Doppler frequency of the target, λ is a radar pulse wavelength and V r is a target radial velocity relative to the radar. Doppler resolution is related to the dwelling time, which in practice means the time of coherent processing: D r = 1 T DWELL (2.3) where D r denotes Doppler resolution expressed in Hz. This can be expressed in terms of number of pulses - n. Let PRF=1/PRI (Pulse Repetition Frequency). Then: D r = PRF n (2.4) It can be seen, that to improve Doppler resolution it is necessary to increase dwelling time which in practice means to use more pulses in processing. The PRI multiplied by the number of processed pulses is often called Coherent Processing Interval (CPI) and is exactly the same as dwelling time. The other important relation is an unambiguous velocity condition: PRF > 2f D (2.5) This condition is a radar equivalent of Nyquist-Shannon-Kotelnikov theorem. It means, that to be able to unambiguously determine the target velocity it is necessary to have PRF greater than twice the Doppler shift of this target. In practice, however, very often this condition isn t fulfilled and we have situation, when velocity is ambigiuous. Operator of the radar, in such a case, must be aware that velocity of the target can be different than the one indicated by the system. Unfortunately in real scenarios, radar receives echoes not only from targets, but also other signals. This can include interferences from other sources (eg. jamming), as well as 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 sense when it means only passive distortions (excluding interferences). In Fig. 2.5 it is shown an environmental 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 signal processing techniques such as MTI, Adaptive Beamforming (AB) or STAP. MTI technique is based on Doppler principle. For example, if only ground clutter is present for not moving radar system, then clutter has approximately zero Doppler shift. Therefore

11 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 7 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 zero Doppler frequency (see Fig. 2.6). Situation is more complicated when the clutter is moving in relation to the radar. Then adaptive techniques can be useful in this case. Adaptivity means, that filters adapt themselves to the clutter properties in Doppler domain. Another group of techniques are phased or adaptive arrays. The concept of adaptive array is based on the idea of changing antenna pattern in order to steer antenna null toward interference source as presented in Fig This can be realized using two-antennas and introducing signal phase shift between antennas (Fig. 2.8). Modification of this technique is an application of quadrature receivers (the same as for MTI technique). In this case each antenna has its own quadrature receiver. Then null steering is 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 multiple nulls. This can be done using more antennas in array antenna (Fig. 2.10). Example of a complete system is presented in Fig For simplicity reasons quadrature receivers are not presented in the picture. Next step in radar technology evolution was to introduce combination of a MTI and adaptive arrays in a single technique. This concept is named Space-Time Adaptive Processing and is presented in the next section.

12 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 8 Figure 2.8 Null steering using phase shifters [33]. d denotes distance between antennas. Figure 2.9 Two element adaptive antenna [33].

13 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 9 Figure 2.10 Four antennas array [33]. Figure 2.11 Complete adaptive array system [44].

14 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 10 Figure 2.12 Airborne clutter [53]. 2.2 Overview of STAP. STAP is a modern signal processing technique, that can improve target detectability in the presence of a strong clutter. In this section only short review on this topic is presented, more exhaustive analysis of STAP can be found in a book written by Richard Klemm [37] Problem Statement. If we consider Moving Target Indication for airborne radar, then we face the problem that echo coming from non-moving ground objects (ground clutter) possess non-zero Doppler bandwidth [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 echo can fall within the clutter bandwidth and may be hidden under the clutter (Fig shows simpler case, where target echo is outside the clutter band). In this case, clutter rejection will also reject target echo. STAP objective is to filter-out ground clutter, while preserving echo coming from moving target Radar System. In order to achieve its objective, STAP employs antenna array, which allows angle of arrival filtering. In Fig we can see the geometry of airborne antenna arrays. Axis V p denotes flight 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 forwardlooking configuration, receiving elements are placed along an axis perpendicular to the flight axis, but parallel to the ground plane. Side-looking configuration is simpler to analyze, therefore we assume this configuration only. Also for simplicity reasons, we assume that receiving elements are equally-spaced, with element spacing equal to λ/2 (λ denotes wavelength). Antenna 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 P is arriving at antenna receiving elements at different moments in time. In other words: there is a phase difference among array channels. This phase difference is related to cone angle α. For other configurations (with slant angle), analysis is more complicated, but qualitative results are similar.

15 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 11 Figure 2.13 Geometry of airborne antenna arrays [37]. Figure 2.14 Moving radar system [58] Airborne Clutter. If we consider moving radar system (Fig. 2.14), it can be shown that Doppler shift of echo coming from non-moving environment depends on cone angle [37]. Each point in space, seen by the radar under angle α is approaching radar at the same speed. More precisely, this speed is 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 cos α). The cone is, therefore, surface of constant Doppler shift. But usually, airborne radar experiences reflections not from the whole space but from the ground plane only (excluding potential target). Therefore, to obtain the set of points with the same Doppler shift we should intersect cone surface with the ground plane as in Fig 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 of constant Doppler shift are named isodops. In our case a single hyperbola is exactly an isodop.

16 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 12 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 in Fig In the same figure it can be seen that zero-doppler isodop (f r = 0) is a straight line perpendicular to the flight path. On the right side of zero-doppler isodop, are placed positive-doppler isodops. The last of them is a maximum-doppler isodop (f r = 1) which is in fact 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 the range to the target. Isorange line is a line of constant range to the radar. It is a circle on the ground, which is an intersection of a ground plane and a sphere with the radius which corresponds to our range of interest(fig and 2.17). Taking into account both isodops and isorange lines, we see that for the specific range and for the specific Doppler shift, we receive echo from specific areas on the ground. This can

17 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 13 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 arriving from specific angles only. This relation is exploited in STAP. In Fig we can see space-time clutter structure in Doppler frequency-cosinus of cone angle (α) coordinate system. Clutter occupies only part of the coordinate plane. Moreover for the simple case of a side-looking antenna configuration, clutter lays on a straight line (in Fig. 2.18) as a result of the fact, that Doppler shift is proportional to the cos α, as was shown before. For other antenna configurations, clutter has a different pattern, but always occupies only small part of that plane. In the picture (Fig. 2.18) two possible targets are depicted. One so-called slow target and the other so-called fast target. Slow target (dashed line) is a target whose radial speed V r to the radar is small in comparison with the speed of the radar platform relative to the ground V p. Movement of the radar platform generates clutter whose Doppler spectrum extends from

18 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 14 Figure 2.19 Data cube [43]. 2V p /λ to +2V p /λ, where λ is a radar working wavelength (see Fig. 2.12). As a result, clutter Doppler spectrum generated by the movement of the platform can be broader than Doppler shift of the target. Therefore echo from this target is likely to be suppressed by the temporal MTI filter (see Fig. 2.6). On the other hand, fast target is a target whose radial speed to the radar platform is high in comparison to the radar platform speed. In this case there is only low risk, that echo from this target will be suppressed. Since we are interested in detecting both slow and fast targets, it can be seen, that separate Doppler (inverse temporal clutter filter in Fig. 2.18) or separate angle filtering (inverse spatial clutter filter in Fig. 2.18) may filter-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 clutter echo and preserve target echo Adaptive MTI. Before introducing STAP algorithm it is worth to recall Adaptive MTI (AMTI). This algorithm was used for example in [25]. AMTI (purely temporal adaptive filtering) utilizes echoes from coherent pulse trains received by an antenna array. Pulse trains allows temporal (Doppler) filtering, whereas different channels of antenna array allow spatial (angle) filtering. AMTI is operating on so-called data cube (Fig. 2.19). Data cube consists of complex samples taken from N pulses by M antenna elements for range cells from 1 to L. We are assuming, that pulse train consists of N coherent pulses. Echoes are coherently sampled by a 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 and 2.21). The rest of the processing is performed independently for each beam. For each range-beam cell, adaptive filter is calculated using auxiliary range cells of the same beam. Formula for the test statistic is given by [53]: η = vh ˆR 1 k x k 2 vh 1 ˆR k v (2.6)

19 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 15 Figure 2.20 Beamforming. Figure 2.21 Data snapshot. where v is a steering vector for tested velocity, x k is a data vector (N pulses) from the range cell and the beam of interest, and ˆR 1 k is an inverse of the sample covariance matrix. This test statistics has a property of Constant False Alarm Ratio (CFAR), as long as clutter follows Gaussian distribution [53]. Sample covariance matrix is calculated using auxiliary snapshots of the same beam. Usually auxiliary snapshots are taken from the same beam but from different range cells. It is worth to mention here, that above test statistics is of the same form as for STAP (which will be shown in the next section). The difference is the interpretation of vectors and matrices present in the equation (2.6). For AMTI, vectors and matrices are related to a single beam and multiple pulses, whereas for STAP it will be vectors related to two-dimensional, space and time structure. Underlying assumption of both AMTI and STAP is that clutter follows Gaussian process. After such a processing, results available in the form of the test statistics cube (range velocity angle).

20 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION STAP Processing. To make profit of angle dependency of Doppler shift, STAP is operating on the raw data cube (Fig. 2.19). STAP is processing one slice of the data cube, for the range cell of interest: [x k ] 1,1 [x k ] 1,2... [x k ] 1,N [x k ] 2,1 [x k ] 2,2... [x k ] 2,N X k =..... (2.7). [x k ] M,1 [x k ] M,2... [x k ] M,N where k denotes range cell, M is the number of antennas, N is the number of pulses processed, and in expression [x k ] a,b, a denotes antenna and b denotes pulse. Then X k is the backscatter signal from one range cell (k) but for all pulses and antennas. For further processing we need to vectorize slice (2.7) by stacking each succeeding column one beneath the other. This operation yields the space-time snapshot for the k-th range: x T k = [ [x k ] 1,1, [x k ] 2,1... [x k ] M,1, [x k ] 1,2, [x k ] 2,2... [x k ] M,2, [x k ] 1,3, [x k ] 2, [x k ] M,N ] (2.8) T denotes transposition. Let us consider a single reflecting point. Assuming that first antenna element is a reference point, received space-time snapshot should be of the form [37]: x k = a s s t (f sp, f d ) (2.9) where a denotes a random complex amplitude and s s t (f sp, f d ) is the following steering vector: 1 1 exp(j2π f sp ) 1 exp(j2π 2f sp ) 1 exp(j2π 3f sp ) 1. exp(j2π (M 1)f sp ) 1 1 exp(j2π f d ) s s t (f sp, f d ) = exp(j2π f sp ) exp(j2π f d ) (2.10). exp(j2π (M 1)f sp ) exp(j2π f d ) 1 exp(j2π 2f d ) exp(j2π f sp ) exp(j2π 2f d ).. exp(j2π (M 1)f sp ) exp(j2π (N 1)f d ) where f sp = d λ cos(α), f d = 2 ν r λ T, α denotes the cone angle (Fig. 2.13), v r - radial velocity between reflecting point and antenna, λ - wavelength and T is a PRI. The steering vector can be written as s s t (f sp, f d ) = s sp (f sp ) s t (f d ), where s sp (f sp ) is a vector performing space processing and s t (f d ) is the Doppler processing vector. When multiplying a vector by the above steering vector, we perform in fact a 2-D Fourier Transform in the space and the time domain.

21 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 17 Registered space-time snapshot is a result of coherent summation of waves from many sources: x k = s k + c k + j k + n k (2.11) s k denotes a target reflection, c k stands for clutter return, j k is jammer signal and n k represents 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]: y k = w H k x k (2.12) where y k is a resulting scalar, w k - weight vector, and superscript H denotes conjugate transpose. Formula for the optimum weight vector 1 is [37]: w k = β R 1 k s s t (f sp, f d ) (2.13) β denotes scalar, R 1 k is an inverse of covariance matrix of x k : R k = E{x k x H k } assuming no target signal present in the data. s s t (f sp, f d ) is a steering vector (2.10) for the possible target we 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 s s t (f sp, f d ) are unknown. Instead of R 1 k, an inverse of the sample covariance matrix ˆR 1 k is often used. s s t (f sp, f d ) is steering vector for, only one, possible target. To construct this vector it is necessary to possess perfect knowledge about the target velocity and angle α relative to the antenna (see Fig. 2.13). In practice neither velocity nor angle is known. Instead, we can test for targets at a series of discrete points covering angle-velocity space (test grid). Test grid must be dense enough, to sufficiently precisely approximate velocity and angle of any possible target. Then each point from velocity-angle grid is transformed into a surrogate steering vector v s t, that will be used in (2.13) as a replacement of s s t (f sp, f d ). Mismatch between real s s t (f sp, f d ) and surrogate v s t leads to some performance loss of detector. For MTI purpose, decision function has to discriminate between two hypotheses: H 0 : x k = c k + j k + n k H 1 : x k = s k + c k + j k + n k where H 0 denotes null hypothesis that there is no target within the range cell of interest, H 1 denotes an alternative hypothesis that target is present. Using optimum weight vector and taking [43]: β = v H s t 1 1 ˆR k v s t, gives test statistics [43]: η = vh 1 s t ˆR k x k 2 v H s t ˆR 1 k v s t (2.14) 1 This is true under different optimization criteria - Maximum Likelihood estimation(ml), Signal to Noise(plus Interference) Ratio maximization(snr), Minimum Noise Variance(MV) estimation and Least Mean Square Error(LMSE) estimation, but only under assumption of proper complex Gaussian process (multivariate complex Gaussian process with vanishing pseudo-covariance [46]) of the clutter. This assumption will be examined more deeply in subsequent parts of this work.

22 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 18 This test statistics has a Constant False Alarm Ratio (CFAR) property [51] (from [43]). As a result, a two-dimensional real test statistics map is obtained as an output of the STAP filter (for any range cell of interest). One dimension of this map is an angle, and the other is a velocity of a potential target. This map represents a grid of surrogates v s t for a true steering s s t, which is unknown. Decision function is constructed by comparing η to a fixed threshold γ. If η < γ then the verdict is: target is absent. If η > γ the verdict is: target is present Assumptions and Limitations. Gaussian Assumption. STAP is a special case of optimum processing in the presence of Gaussian distortions. Formulation for optimum processing can be found, among others, in [53] and [44]. STAP underlying assumption 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 Gaussian process can be represented as [44] [53]: c t = x t cos(ω c t) y t sin(ω c t) (2.15) where x t, and y t, are independently, identically distributed from normal distribution of a zero mean value and the same variance. Then: c t = v t cos(ω c t + θ t ), (2.16) θ t is uniformly distributed, and voltage envelope v t is Rayleigh distributed. We can now treat x t and y t as real and imaginary parts of complex process. For multidimensional case (e.g. multiple M sensors and coherent N pulses), x t and y t become vectors x t and y t. x t is an In-phase vector: ] x T t = [[x t ] 1, [x t ] 2, [x t ] 3... [x t ] M N and y t is an Quadrature vector: ] yt T = [[y t ] 1, [y t ] 2, [y t ] 3... [y t ] M N Similarly to one-dimensional case, we assume that: E{x t } = 0, E{y t } = 0 This implies that the process is entirely characterized by its covariance matrix: {[ x E y ] } [x T y T ] = M For narrowband processes, covariance matrix can be presented in the specific form (see Appendix A) [48] [56] [44] [53]: {[ x E y ] } [x T y T ] = 1 2 [ ] V W W V

23 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 19 Therefore it is possible to represent this process in complex notation. If we assume complex vector: [x t ] 1 + j[y t ] 1 [x t ] 2 + j[y t ] 2 z t =. [x t ] M N + j[y t ] M N then the covariance matrix is: E{zz H } = V + jw Using this notation, we describe multidimensional narrowband Gaussian process using M N x MN complex covariance matrix. This result is then used in STAP formula in (2.13) as R k. The rest of (2.13) - s s t (f sp, f d ) - can be viewed as a matched filtering. For other distributions of the clutter, covariance matrix does not describe distribution entirely, so test statistics must be reformulated. In this case, optimum detector will not necessary be a linear filter as discussed in section 2.3. Estimate of R 1 k. As it is stated on page 17, in practice we do not know real covariance matrix. Therefore we must use an estimator. Usually Sample Covariance Matrix (SM) is used [37]: ˆR k = 1 P P x m x H m (2.17) m=1 where vectors x m are usually taken from neighboring cells of space-time snapshot of data cube (see Fig. 2.19). This estimator is Minimum Likelihood Estimator if x m are IID (Independent and Identically Distributerd). Moreover they should come from the same distribution as the 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. Therefore we 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 M N - see Fig.2.19 and (2.8). However, for good performance, P should be greater than 2MN [50]. For practical applications, for example M = 22, N = 128 and P > 5632, it may be difficult 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 there is a possibility to adopt other approach like Bayesian reasoning or non-parametric tests. In this work, however, author concentrated on classical Neyman-Pearson theory. First, it will be shown LRT for simple hypothesis testing. After it will be shown GLRT that it is often used for composite hypothesis testing. Finally, it will be presented Locally Optimum (LO) test, that can be used under certain constraints for composite alternative hypothesis. Two examples of LO test will be shown.

24 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION Notation. Let us assume random vector X = (X 1, X 2, X 3,..., X n ) of observations with joint probability density function (PDF) f X (x θ), where θ is a parameter of the density function. Let x = (x 1, x 2, x 3,..., x n ) be a vector of observations (specific realization of X ) in R n. Let us further assume that θ Θ = Θ H0 Θ H1, Θ H0 and Θ H1 are disjoint. Hypothesis testing can be viewed now as deciding between: H 0 : X has PDF f X (x θ) with θ Θ H0 H 1 : X has PDF f X (x θ) with θ Θ H1 (2.18) Decision should be based on observations x. Let us use δ(x) symbol for the decision function. δ(x) = 0 means adoption of hypothesis H 0, δ(x) = 1 means adoption of hypothesis H 1. If set Θ H0 consists of a single element θ H0 only, we say that hypothesis zero is simple, otherwise we say that hypothesis zero is composite. Similar relates to alternative hypothesis H 1. The power function p(θ δ) of a test based on a test function δ is defined for θ Θ H0 Θ H1 as: p(θ δ) = E {δ(x) θ} (2.19) In target detection problems, power function can be viewed as a function that gives probability of detection as a function of unknown parameter θ. Usually we want this function to be high for θ Θ H1. The size of a test is the quantity: α = sup θ ΘH0 p(θ δ) (2.20) Size of a test in radar target detection problem is translated to the probability of a false alarm. The aim is to have power function (probability of detection) high, meanwhile having size of a test (probability of false alarm) reasonably low Neyman-Pearson Lemma. Let us consider simple hypothesis H 0 (θ = θ H0 ) and simple hypothesis H 1 (θ = θ H1 ). General structure of a most powerful test 2 may be described as one comparing the likelihood ratio to a constant threshold [35], f X (x θ H1 ) f X (x θ H0 ) > t (2.21) If the likelihood ratio on the left-hand side of (2.21) is greater than threshold t, than we decide to accept hypothesis H 1. Constant t may be evaluated to obtain desired test size α Generalized Likelihood Ratio Test. Let us consider again general test of the form (2.18) - composite hypothesis testing. To use Neyman-Pearson Lemma and likelihood ratio test (2.21) we can adopt maximum likelihood estimates ˆθ H0 and ˆθ H1 of the parameter θ, obtained under the constraints θ Θ H0 and θ Θ H1 respectively [35]. These estimates can be used in place of θ H0 and θ H1 in (2.21). Test size. 2 Most powerful test is a hypothesis test which has the greatest power among all possible tests of a given

25 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 21 derived in this procedure is named General Likelihood Ratio Test. It is possible to consider a special case of this test, when null hypothesis is composite and alternative hypothesis is simple. This is often the case in radar signal processing, when we want to detect a target echo. H 0 : X has PDF f X (x θ) with θ = θ H0 against H 1 : X has PDF f X (x θ) with θ θ H0 (2.22) Then ˆθ H0 can be replaced by exactly known θ H0, and GLRT looks as follows: f X (x ˆθ H1 ) f X (x θ H0 ) > t (2.23) denumerator is of the same as in LRT, and numerator comes from GLRT. This is form, that will be later used in derivation of target detectors Alternative Hypothesis of the Form θ > θ H0. Sometimes it is possible to derive other detectors based on LRT. This time we assume a special form of the test: H 0 : X has PDF f X (x θ) with θ = θ H0 against H 1 : X has PDF f X (x θ) with θ > θ H0 (2.24) We now say that H 0 is a simple hypothesis, but H 1 is composite. In this case we can derive Locally Optimum (LO) test. This means that, for assumed test size α, decision function will be most powerful in vicinity of θ H0 in deciding against θ > θ H1. Test structure is as follows [35]: d dθ f X(x θ) θ=θh0 f X (x θ H0 ) > t (2.25) where t is again threshold appropriately chosen to achieve desired test size. It can be seen that denominator of test statistics is the same as in (2.23). Numerator is however of an arbitrary form adjusted to be optimal in vicinity of θ H0. This detector will behave properly for θs close to θ H0. This test can be also viewed as [35]: d ( ) dθ ln f X (x θ) θ=θ H0 > t (2.26) The problem with its practical use is the necessity of knowing derivative of likelihood function Alternative Hypothesis of the Form θ θ H0. There are also many situations when we need to test two sided alternative hypothesis - the same as GLRT in (2.23): H 0 : X has PDF f X (x θ) with θ = θ H0 against H 1 : X has PDF f X (x θ) with θ θ H0 (2.27) H 0 is again simple hypothesis, H 1 is composite. In this case we can also derive LO test. In order to achieve this, we need to impose some restrictions to the test function. Test function

26 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 22 must be unbiased. Which means that test satisfies [35]: p(θ δ) α, for all θ Θ H0 p(θ δ) α, for all θ Θ H1 (2.28) For the two-sided alternative hypothesis, Locally Optimum(LO) looks as follows [35]: d 2 f dθ 2 X (x θ) θ=θh0 > t (2.29) f X (x θ H0 ) where t is, again, threshold appropriately chosen to achieve desired test size. The same as before, it can be seen that denominator of test statistics is the same as in (2.23). Numerator is again of an arbitrary form adjusted to be optimal in vicinity of θ H0. This detector will behave properly for θ s close to θ H0. This time it is necessary to know second derivative of a likelihood function. Therefore, though it may be interesting to apply these detectors for non-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 to derive and analyse. 2.4 Examples In this section, two examples will be presented as an illustration of the tests presented in section 2.3. These two examples are close to the problems faced in radar target detection Detection of Known Narrowband Signals in Narrowband Noise. Beneath will be presented an example of deriving LO detector in the case, where we can obtain a simple form of derivatives for uncorrelated noise. We will use model: X(t) = θυ(t) cos[ω 0 t + 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 with a constant power spectral density N 0 /2 over the band of interest (and zero outside). We are interested in testing θ = 0 against θ > 0. After quadrature detection with carrier frequency ω 0 and sampling with Nyquist frequency, we receive in-phase and quadrature samples x Ii and x Qi, where i = 1... n. Samples can include signal part and noise part: x Ii = θs Ii + w Ii x Qi = θs Qi + w Qi (2.31) Following the (2.30) we will assume that w Ii form a sequence of i.i.d. random variables, governed by a common univariate density function f L. Similarly we assume the same for w Qi with the same PDF f L. We also assume that s Ii and s Ii are completely known. Components w Ii and w Qi are uncorrelated for each i [56](from [35]), but independent only for the Gaussian case. Therefore we must adopt bivariate probability density function f IQ, for which marginal density functions are f l. To summarize, we assume two dimensional noise samples (W Ii, W Qi ) that are i.i.d. random variables with common bivariate PDF f IQ.

27 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 23 Moreover we assume that f IQ is a circularly symmetric 3 bivariate density function, which means that f IQ (u, υ) can be written as a function of u 2 + υ 2 : f IQ (u, υ) = h(r) (2.32) r= u 2 +υ 2 In this situation LO statistics can be derived using (2.25), and takes on the form [35]: λ LO (X I, X Q ) = n i=1 h (r i ) r i h(r i ) [s Iix Ii + s Qi x Qi ] (2.33) where r i = x 2 Ii + x2 Qi. In (2.33) we can distinguish two elements. Nonlinear envelope modification h (r i ) r i h(r i ) and linear operation on I and Q components: s Iix Ii +s Qi x Qi (matched filtering). For the special case of Gaussian noise components, h (r i ) r i h(r i ) becomes constant and LO filter becomes the usual matched filter [35]: λ LO (X I, X Q ) = n [s Ii x Ii + s Qi x Qi ] (2.34) i= Detection of Known Narrowband Signals with Random Phase Angles. Next example illustrates also LO detector. Model in this situation is different from (2.30) in the sense that we introduce additional random starting phase ψ of our signal of interest: X(t) = θυ(t) cos[ω 0 t + 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: x Ii = θ[s Ii cos ψ + s Qi sin ψ] + w Ii x Qi = θ[s Ii sin ψ + s Qi cos ψ] + w Qi (2.36) To simplify formulation let us introduce additional notation: X i = (x Ii, x Qi ) Y i = ( x Qi, x Ii ) s i = (s Ii, s Qi ) υi 2 = s 2 Ii + s2 Qi r i = x 2 Ii + x2 Qi (2.37) LO detector test statistics can be derived using (2.29), and becomes [35]: { } λ LO (X I, X Q ) = 1 n υi 2 h (r i ) [ h 2 h(r i=1 i ) (r i ) ] 2 h (r i ) + h(r i ) r i h(r i ) [ n ] 2 [ + 1 h n ] 2 (r i ) 2 r i h(r i ) s ixi T + 1 h (r i ) 2 r i h(r i ) s iyi T (2.38) i=1 3 More general multivariate case of spherically invariant random processes will be discussed later i=1

28 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 24 This form is very complicated and includes matched filtering together with nonlinear envelope modification similarly to the previous example. Even in Gaussian case, this detector remains non-linear. Generally it is quite difficult to derive LO detectors. For calculations to be feasible it is desirable to have uncorrelated noise. On the other hand, GLRT test is based directly on Neyman-Pearson test and in many cases is easier to calculate. Therefore in the rest of this work 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 SIRP can model a large number of distortions that have been used for sea clutter. This is a very usefull 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 with assumed probability density function (PDF). SIRP allows to relax assumption of Gaussianity, while keeping many of its useful characteristics [2]. PDF of a SIRV vector can be presented in the following form [49]: f(z) = 1 1 Z T M 1 Z (2π) n/2 M 1/2 0 σ n e 2σ 2 dg σ (σ) (2.39) G σ (σ) denotes cumulative distribution function of σ If E(σ 2 ) = 1 then M = E[ZZ T ]. For further analysis we introduce complex notation using transformation: X = z 1 + jz n/2+1. z n/2 + jz n (2.40) let us denote dimension of X by m = n/2. PDF of a vector X can be presented in the following form [10]: f(x) = 1 1 X H Φ 1 X π m e σ Φ 0 σ2m 2 dg σ (σ) (2.41) Similarily if E(σ 2 ) = 1 then Φ = E[XX H ]. If we assume continuous case for σ distribution then we have: f(x) = 1 1 X H Φ 1 X π m e σ Φ 0 σ2m 2 g(σ)dσ (2.42) and g(σ) is called characteristic PDF of a SIRP. It will be useful to introduce one more representation of a SIRP. Let τ = σ 2, then σ = τ, dτ = 2σdσ and dσ = dτ 2 τ. Therefore f(x) = 1 π m Φ 0 1 τ m e X H Φ 1 X τ g( τ) 2 dτ (2.43) τ

29 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 25 where g( τ) 2 τ is a PDF of a τ. Let us denote this PDF as f(τ) so: f(x) = 1 1 π m Φ 0 τ m e X H Φ 1 X τ f(τ)dτ (2.44) or in more general form f(x) = 1 1 π m Φ 0 τ m e X H Φ 1 X τ df τ (τ) (2.45) where F τ (τ) denotes cumulative distribution function for τ. We can assume restriction for τ: E(τ) = 1. Under this restriction we have Φ = E{X H X}, and this is a covariance matrix of X [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 df τ (τ) = δ(τ 0 ) in (2.45), then f(x) = 1 π m Φ 1 τ m 0 e XH Φ 1 X τ 0 (2.46) In above expression we recognize multivariate complex Gaussian random process. We can therefore say, that SIRP is a generalization of a Gaussian process. 2.6 Likelihood Ratio Test and Generalized Likelihood Ratio Test 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 exactly known signal we can use LRT, whereas in the case of not known amplitude and phase we need to employ GLRT. It will be shown, that STAP is a special case of GLRT 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 signal vector, Y is the m-dimensional complex vector of observations. Let us consider following hypothesis test: H 0 : Y = X H 1 : Y = X + S Using (2.45), it is possible to find expressions for PDF of Y under both hypotheses [52]: H 0 : f(y ) = H 1 : f(y ) = 1 π m Φ 1 π m Φ τ m e Y H Φ 1 Y τ df τ (τ) 1 τ m e (Y S) H Φ 1 (Y S) τ df τ (τ)

30 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 26 From general result of (2.21), Likelihood Ratio statistics is given by [52]: η = 0 1 τ e (Y S) H Φ 1 (Y S) m τ 0 df τ (τ) 1 τ e Y H Φ 1 Y m τ df τ (τ) If distribution function of τ is continuous then we have: (2.47) η = 0 1 τ e (Y S) H Φ 1 (Y S) m τ 0 f(τ)dτ 1 τ e Y H Φ 1 Y m τ 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 1 τ e (Y S) H Φ 1 (Y S) m τ 0 df τ (τ) 1 τ e Y H Φ 1 Y m τ df τ (τ) H 1 H 0 t, (2.49) we see that it can be rearranged: 0 1 (Y S) H Φ 1 (Y S) τ m e τ df τ (τ) H 1 t H τ m e Y H Φ 1 Y τ df τ (τ) 0 1 (Y S) H Φ 1 (Y S) τ m e τ df τ (τ) t 0 0 Now if we take threshold to be equal 1: τ m e Y H Φ 1 Y τ df τ (τ) [ 1 (Y S) H Φ 1 (Y S) τ m e τ t 1 τ m e Y H Φ 1 Y ] τ df τ (τ) 1 [ ] τ m e (Y S)H Φ 1 (Y S) τ te Y H Φ 1 Y τ df τ (τ) 1 [ ] τ m e (Y S)H Φ 1 (Y S) τ e Y H Φ 1 Y τ df τ (τ) H 1 H 0 0 H 1 H 0 0 H 1 H 0 0 (2.50) H 1 H 0 0 (2.51) Keeping in mind, that τ is positive and the fact that function e x is monotonous, it can be concluded that the sign of the expression: e (Y S)H Φ 1 (Y S) τ e Y H Φ 1 Y τ depends only on the difference between (Y S) H Φ 1 (Y S) and Y H Φ 1 Y. This is true for every τ. Recalling again that τ is positive it can be concluded, that the sign of the function under integral in (2.51) depends only on the difference (Y S) H Φ 1 (Y S) Y H Φ 1 Y, and

31 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 27 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: further simplifications: Y H Φ 1 Y H 1 H 0 (Y S) H Φ 1 (Y S) (2.52) Y H Φ 1 Y Y H Φ 1 Y S H Φ 1 Y + Y H Φ 1 S H 1 H 0 H 1 H 0 H 1 H 0 (Y S) H Φ 1 Y (Y S) H Φ 1 S Y H Φ 1 Y S H Φ 1 Y Y H Φ 1 S + S H Φ 1 S S H Φ 1 S result is: S H Φ 1 Y H 1 H 0 S H Φ 1 S/2 S H Φ 1 Y S H Φ 1 S H H 0 (2.53) the expression on the left side is possibly complex. Therefore it is necessary to take module value of the numerator: S H Φ 1 H Y 1 1 S H Φ 1 (2.54) S 2 H 0 This result shows, that in case of SIRP as a model for the clutter, optimum detector is a matched filter, but only under condition that threshold in (2.49) is equal 1. This is not true for other thresholds, because integral (2.51) cannot be simplified. Therefore it can be concluded, that this is not an interesting simplification, since it gives no possibility to adjust probability 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 Detection of Known Narrowband Signals with Random Phase Angles and Random Amplitude - GLRT Detector. This is more realistic scenario in radar applications. Our target signal is known precisely but its initial amplitude and phase are not known. In such a case we cannot employ LRT. Instead we need to use GLRT. It will be shown how to derive such a test in the case of Spherically Invariant Random Process. After, it will be shown the same simplification as for LRT. In this case however, this simplification is useless. At the end of this section it will be shown, that in fact, classical STAP is a special case of GLRT for SIRP. Let now S = as 0, where S 0 is the completely known m-dimensional signal vector with magnitude 1 (S H 0 S 0 = 1) and a is the unknown complex amplitude. It is now possible to derive GLRT (see section on page 20).

32 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 28 Taking into account (2.45), GLRT statistics becomes [52]: η = where denotes magnitude, and 1 0 τ exp( Y H Φ 1 Y (1 ρ 2 ) m τ 1 0 τ exp( Y H Φ 1 Y m τ ρ 2 = In the case when distribution of τ is continuous: η = )df τ (τ) )df τ (τ) S H 0 Φ 1 Y 2 (Y H Φ 1 Y )(S H 0 Φ 1 S 0 ) 1 0 τ exp( Y H Φ 1 Y (1 ρ 2 ) m τ 1 0 τ exp( Y H Φ 1 Y m τ where f(τ) is called characteristic PDF of SIRV. )f(τ)dτ )f(τ)dτ (2.55) (2.56) (2.57) Simplified Result. We want to perform similar reasoning to that in section Similar result to (2.50) is given by: Let t be equal 1: 0 this is equivalent to (assuming τ > 0): further: after simplification: 0 Y H Φ 1 Y 1 [ ] τ m e Y H Φ 1 Y (1 ρ 2 ) τ te Y H Φ 1 Y τ df τ (τ) 1 [ ] τ m e Y H Φ 1 Y (1 ρ 2 ) τ e Y H Φ 1 Y τ df τ (τ) ( 1 H 1 H 0 0 (2.58) H 1 H 0 0 (2.59) ) S0 HΦ 1 Y 2 H1 (Y H Φ 1 Y )(S0 H Y H Φ 1 Y (2.60) Φ 1 S 0 ) H 0 Y H Φ 1 Y + (Y H Φ 1 Y ) S H 0 Φ 1 Y 2 (Y H Φ 1 Y )(S H 0 Φ 1 S 0 ) so finally our detector turns out to be: H 1 Y H Φ 1 Y (2.61) H 0 S0 HΦ 1 Y 2 H 1 S0 H 0 (2.62) Φ 1 S 0 H 0 S H 0 Φ 1 Y 2 H 1 H 0 0 (2.63) Of course left side will be greater than zero always, so in this case detector will always report target. Although simplification is possible, however it does not give reasonable results and it will not be considered in the rest of this work.

33 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION. 29 GLRT for Gaussian Clutter - a STAP Formula. In section a classical STAP filter 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 us assume equal to σ). This means that Stieltjes integrals vanish, and we have formula for test statistics: 1 σ exp( Y H Φ 1 Y (1 ρ 2 ) η = m σ ) (2.64) 1 σ exp( Y H Φ 1 Y m σ ) after simplification: η = exp( Y HΦ 1 Y (1 ρ 2 ) σ ) (2.65) exp( Y H Φ 1 Y σ ) and taking natural logarithm: constant σ can be included in the threshold: after rearrangement: further: and substituting expression for ρ 2 from (2.56): η = [ Y H Φ 1 Y (1 ρ 2 ) ] [ Y H Φ 1 Y ] (2.66) σ σ η = [ Y H Φ 1 Y (1 ρ 2 )] [ Y H Φ 1 Y ] (2.67) η = Y H Φ 1 Y [Y H Φ 1 Y (1 ρ 2 )] (2.68) η = Y H Φ 1 Y ρ 2 (2.69) η = Y H Φ 1 Y S H 0 Φ 1 Y 2 (Y H Φ 1 Y )(S H 0 Φ 1 S 0 ) (2.70) which finally gives formula equivalent to (2.14): η = SH 0 Φ 1 Y 2 (S H 0 Φ 1 S 0 ). (2.71) Variance Estimation. In (2.71) we assume, that covariance matrix (and its inverse) is known. This is not true in general. In (2.14) inverse of the sample covariance matrix ˆΦ 1 is used as an inverse covariance matrix estimator in place of true inverse covariance matrix Φ 1. But this detector (2.14) is not GLRT detector anymore. It can be found in [51] that GLRT under assumption that covariance is unknown is given by: S0 H ˆΦ 1 Y 2 S0 H ˆΦ 1 S 0 (1 + 1 P Y H ˆΦ ) 1 Y H 1 H 0 t (2.72) where t is a threshold and ˆΦ is given by: ˆΦ = 1 P as in (2.17). P x m x H m (2.73) m=1 It can be seen that for large P (2.72) becomes (2.14).

34 CHAPTER 2. RADAR BASICS, SPACE-TIME ADAPTIVE PROCESSING AND TARGET DETECTION Conclusions This chapter introduced basic radar systems concepts. After general introduction to radar principles in section 1, a STAP technique was described in section 2. Rationale standing behind introduction of STAP was presented. STAP processing algorithm together with assumptions and limitations was shown. Next section presented more general view to the problem of detection. This is necessary in the context of non-gaussian signals. In the same section Generalized Likelihood Ratio Test was presented. For completeness Locally Optimum detector was also shown. To resolve the problem of non-gaussianity, at certain stage there is a need to introduce the model of distortions. This was done in the next section, where Spherically Invariant Random Process (SIRP) was introduced. Using this concept and GLRT it was possible to present general form of a test adjusted to the SIRP for the signal with unknown phase. As it will be shown in the next chapter, problem of non-gaussianity is a real problem in radar detection in the case of sea clutter, where reflections come from the sea surface which has a very complicated, non-linear physics. In this context GLRT for SIRP will be a basic tool for improving detection performance. Having validated the need of target detection in non-gaussian clutter, practical implementation of a GLRT for SIRP will be presented in the chapter 4.

35 CHAPTER3 Sea Clutter. In this chapter sea clutter properties will be described. After short introduction, sea clutter properties will be presented for X-band and HF-band radar systems. HF-band results will be based on author s experiments with real data. Sea clutter signal properties are related to sea roughness. It is common assumption to decompose sea roughness to capillary waves and to gravity waves (swell) [28] [31]. The electromagnetic 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 restoring force is the surface tension. Gravity waves have wavelengths ranging from few hundred meters to less than a meter [31]. Swells are produced by stable winds. Their restoring force is 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 waves with larger amplitudes and wavelengths. Sudden cessation of wind causes short waves to decay rapidly, whereas the longer waves can last several days. In reality sea state is a complex mixture of shorter and longer waves coming from different directions. Waves can be characterized by their height, length and period. Wavelength and period are in a relation, whereas waveheight fluctuates considerably [31]. Because of different length scales of different sea-wave types, sea clutter for different radar bands 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, they will have an important influence. However some general comments can be made. Doppler spectrum of sea clutter results from two main processes: the spread about the mean Doppler frequency is a manifestation of the random motion of the unresolved scatters, while the displacement 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 at which radar beam illuminates the surface with respect to the local horizontal. For HF radar systems, grazing angles are almost always small, since these radars are usually installed at the sea coast or onboard of ships. For X-band and S-band radar systems, grazing angles can vary from 0 to 90 degrees. For high grazing angles, statistical properties are usually considered Gaussian, whereas for low grazing angles for X-band and S-band radars, statistical properties of the clutter can not be anymore considered Gaussian. For this reason we have considered the influence of this departure in the next chapter (chapter 4), which deals with the detection step of the STAP.

36 CHAPTER 3. SEA CLUTTER. 32 Figure 3.1 Typical X-band, sea-clutter spectra in the upswell and near cross-swell directions 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 is therefore between 2.5 cm to 3.75 cm. This is very important, since structures of this size will influence statistical properties of the clutter in this case. At the beginning let us consider spectral properties of the clutter. In Figure 3.1 it is shown clutter power as a function of Doppler frequency for different look angles of radar and for different polarizations. It is clear, that sea clutter has broad Doppler spectrum and its maximum is related to look angle. Obviously, this spectrum Doppler hides the target Doppler and strongly disturbs usual methods such as MTI. This relation cannot be explained in the terms of wind direction in a simple way [9]. Some of other factors that play an important role are sea currents and variability of wind direction. Generally, for horizontal polarization, clutter has narrower Doppler spectrum. For both polarizations clutter power is substantially higher in directions of positive Doppler shift. Generally, power decreases as the look direction moves 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 Doppler peak spectral shift can be found in [57], where models for Doppler shift are presented.

37 CHAPTER 3. SEA CLUTTER. 33 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 to conclusion, that in this case the application of STAP technique is worth to be considered, as discussed in section 2.2. Relative dependence between the look angle and the Doppler shift of the clutter was a rationale standing behind STAP application for airborne radar as described in 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 it will be shown later, statistical properties of the clutter will be changing with grazing angle as well. Next important property of the clutter is its amplitude distribution. In the table 3.1 we see different distributions used to describe clutter properties. Depending on situation, different models 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 higher resolution radar (especially for low grazing angles) it turns out that statistical properties are different. The reason behind this is the law of large numbers. In the case of low resolution radar, many independent scattering centers contribute to overall radar cross section of a single resolution cell. In this case the law of large numbers states, that complex samples should 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 single resolution cell, and therefore their statistical properties have direct influence on radar clutter properties. This is additionally augmented by the effect of low grazing angle. This is due to the fact that for low grazing angles main contribution to clutter RCS are strong reflections from 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 probability of false alarm regions (ie. tails of the distribution that are well described by the third order moment of PDF). Therefore chi-square goodness of fit test is of limited use [9], since this kind of test mainly compares general shapes (theoretical and estimated) and is not focused on the tail of the distribution. In Fig. 3.4 and 3.5 it is shown Probability of False Alarms (P fa ) as a function of threshold. In these figures we can see probability of false alarm for a given threshold. A choice of the

38 CHAPTER 3. SEA CLUTTER. 34 Rayleigh Table 3.1 Different clutter distributions [9]. CDF : P (X) = 1 exp { x2 α } 0 x PDF : p(x) = 2x α exp { x2 α } 0 x first moment: x = (πα)1/2 2 third moment: µ 3 = 3(α/2) 3/2 π/2 Weibull CDF : P (X) = 1 exp { xn α } 0 x PDF : p(x) = n xn 1 α x2 exp { α } 0 x first moment: x = α 1/n Γ(1 + 1 n ) third moment: µ 3 = α 3/n Γ(1 + 3/n) Lognormal CDF : P (X) = 1 1 ln x m 2erfc( ) 0 x α 1/2 where erfc is the complementary error function. 1 (ln x m)2 PDF : p(x) = exp{ x(πα) 1/2 α } 0 x first moment: x = exp(m + α 4 ) third moment: µ 3 = exp(3m α) K-distribution ) νkν CDF : P (X) = 1 Γ(ν)( 2 bx 2 (bx) 0 x PDF : p(x) = 2b Γ(ν)( bx 2 ) νkν 1 (bx) 0 x where K ν 1 (z) is the modified Bessel function of the third kind of order ν first moment: x = π bγ(ν) Γ(ν ) third moment: µ 3 = Γ(ν+3/2)Γ(1+3/2) b 3 Γ(ν)

39 CHAPTER 3. SEA CLUTTER. 35 Figure 3.3 Sea clutter cross-section against grazing angle profiles of X-band in the upswell 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 we are reducing probability of detection. From figures 3.4 and 3.5 it can be seen that the data fit the K-distribution best in the amplitude region for which the P fa is 0.1 [9]. This is true for resolutions of order of 15 to 150 m. The same conclusion is confirmed in [8]. This observation implies, that detectors developed for Rayleigh clutter will not be appropriate in the case of low grazing angle radars. For higher resolutions, we can even resolve single wave crest, as it is visible in Fig Images show power range-time diagrams for X-band 50 cm resolution radar. Wave crests are traveling toward radar. In such a case, to detect targets it may be reasonable to apply image processing algorithms rather than classical radar detection methods. Moreover several furtive superevents are observed by an X band radar. These superevents are due to wave breaking and generate strong scatterer with high velocity, so they are fast scatterers (see for example Fig. 3.6). These superevents generate a lot of false alarms (which may lead to errors in tracking algorithms) and should be processed adaptively. To deal with non-gaussianity, it is essential to find a mathematical model for this type of clutter. It can be found (see for example [52]) that theory of Spherically Invariant Random Process (SIRP) in many situations is an appropriate model for sea clutter. Theory of SIRP

40 CHAPTER 3. SEA CLUTTER. 36 Figure 3.4 Comparison of P fa characteristics of an X-band, V-polarized sea clutter data set 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 have been exposed in section 2.5. Therefore a new detector presented in chapter 4 is based on SIRP model. 3.2 Sea clutter characterization in HF band. In HF radar band, main clutter component is related to Bragg scattering phenomenon. First description of this type of the clutter can be found in [11]. In the mentioned article, it is shown that for frequency MHz first-order Bragg lines have Doppler shifts 0.38 Hz and 0.38 Hz. General formula for Doppler shift is: f = g 1 π λ (3.1) where g = 9.81m/s 2 and λ is a radio wavelength. This formula means that main clutter components 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 wave length) and the speed of the sea wave is related to its length by the formula v = g 2π L (3.2)

41 CHAPTER 3. SEA CLUTTER. 37 Figure 3.5 Comparison of P fa characteristics of an X-band, V-polarized sea clutter data set with those predicted by Rayleigh, Weibull, lognormal and K-models [9]. Range resolution m. Figure 3.6 Power range-time diagrams for VV polarization (a) and HH polarization (b). Range resolution 50 cm [42].

42 CHAPTER 3. SEA CLUTTER. 38 This mechanizm is called first order Bragg scattering and is responsible for constant Doppler shift of Bragg lines. However for higher Doppler resolutions other factors play role, and in fact Bragg lines are changing slightly accross angles and ranges. This is caused by complicated sea wave dynamics. These effects make target detection more difficult and are the reason to use STAP. Bragg scattering mechanism is commonly used for sea parameters extraction such as wave height, currents velocity and direction [5]. More detailed description of such possibilities can be found in [30] and [29]. Registered signals from WERA system working in Brittany in France were used to present clutter properties. System description is presented in section 5.1. Results are obtained by beamforming followed by Doppler processing. At the day of data collection radar was working at frequency around 12 MHz. In Fig. 3.7 are presented range-doppler maps for an exemplary beam. Maps are obtained for different integration times and therefore for different Doppler resolutions. First map was obtained for 48 chirps. Single chirp duration is s, therefore integration time in this case 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 Hz ans Hz as predicted by theory (for 12 MHz carrier frequency). It is visible, that Bragg lines are present in all range cells until approximately 90 km, where noise level becomes higher than sea clutter. This is a result of a very low power transmitted by radar, which is at most 30 W. Bragg lines frequencies are constant across all range cells. In the short distance from radar we can see also land clutter. Comparing images in Fig. 3.7 we can see, that for higher integration time Bragg lines become very thin. This observation has two consequences. First one is that first-order ocean wave scattering is coherent even for 2 minutes (512 chirps). The second is that since Bragg lines are very thin in Doppler, having long integration time, may give us better opportunities to reveal targets whose Doppler frequency is very close to Bragg line. In Fig. 3.8 we can see Doppler spectra for range cell number 50, this is 20 km. These images 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 cell number 50, 100 and 150 respectively). Bragg lines are clearly visible for all ranges but there is a 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 illustrate that structure of the first order ocean waves clutter does not change significantly across different beams. Fig illustrates again how clutter is changing across beams. It is clear that though general clutter structure is not changing, Bragg lines Doppler and power is changing across different 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 can see 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 can have different sources and properties. For example in Fig we can see an interference that is relatively weak. It can be seen as a pattern of almost horizontal lines, better visible in black and white in Fig Since interference has usually a point source, it is present only in some of beams whereas

43 CHAPTER 3. SEA CLUTTER. 39 Figure 3.7 Bragg clutter for different integration time (number of chirps processed - from 48 to 512).

44 CHAPTER 3. SEA CLUTTER. 40 Figure 3.8 Bragg clutter for different integration time (number of chirps processed - from 48 to 512), for the selected, 50-th range cell.

45 CHAPTER 3. SEA CLUTTER. 41 Figure 3.9 Clutter spectra for different ranges (512 chirps integration).

46 CHAPTER 3. SEA CLUTTER. 42 Figure 3.10 Range-Doppler maps for different beams (512 chirps integration).

47 CHAPTER 3. SEA CLUTTER. 43 Figure 3.11 Beam-Doppler images for different ranges (512 chirps integration).

48 CHAPTER 3. SEA CLUTTER. 44 Figure 3.12 Clutter spectra for different starting time of the processing (512 chirps integration).

49 CHAPTER 3. SEA CLUTTER. 45 Figure 3.13 Clutter spectra for different starting time of the processing for Range cell number 50 (512 chirps integration).

50 CHAPTER 3. SEA CLUTTER. 46 Figure 3.14 Bragg clutter plus interference (256 chirps integration). Interference is visible as horizontal lines. Figure 3.15 Black and white image of Bragg clutter plus interference (256 chirps integration). Interference is visible as horizontal lines.

51 CHAPTER 3. SEA CLUTTER. 47 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-azimuth map of mean received power. Interference is present for angles between -20 deg to -40 deg. In Fig we can see range-doppler map in the beam with interference. See clutter is completely obscured by interference. Next figure (3.18) shows time-range structure of the interference. We can see that this interference has impulsive nature, which means that its power is changing considerably as a function 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 (X and HF radar bands) were presented. It was shown, that for both bands, sea clutter can have two dimensional structure in space-time domain. In X-band this was caused by Doppler shift of clutter related to wind, currents and waves direction as well as possible interference present in the data. In HF-band it was caused by joint presence of the first and second order 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 using STAP technique to improve detection possibilities. This will be verified later by performing experiments on real, recorded data. For X-band it turns out additionally, that clutter does not follow classical Rayleigh distribution, and therefore classical adaptive techniques may lead to increased number of false alarms. This will also be investigated in the next two chapters. To

52 CHAPTER 3. SEA CLUTTER. 48 Figure 3.17 Range-Doppler map oriented toward interference source. Figure 3.18 Time-range structure of the interference.

53 CHAPTER 3. SEA CLUTTER. 49 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, although there are also some other possibilities too. These other methods were not considered in this work. Among them there is, for example, a method of using passive mode of radar to estimate interferences [30].

54 CHAPTER4 Two Dirac delta detector. 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 from Gaussian (Rayleigh) case. On the other hand in chapter 2 was presented the GLRT test in the case of Spherically Invariant Random Process. This detector is capable of dealing with wide 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 is discussed below. In this chapter, I will develop a practical implementation of this detector named Two Delta Dirac STAP (TDD STAP) detector. First two sections will be devoted to development of TDD STAP detector. In particular, I discuss the estimation of parameters used in this nonlinear detector in section 4.2. To evaluate performances of classical STAP detectors in non-gaussian clutter, I performed some simulations. Description of simulation technique is presented in section 4.3. Finally, section 4.4 is devoted to performance evaluation of different STAP algorithms. Receiver Operation Curves (ROC) are presented. In particular, we discuss the performances of usual STAP and TDD STAP detector under different kinds of noise (Gaussian, non-gaussian). I will show, that new developed TDD STAP detector can give some improvements in comparison to classical STAP algorithm in the presence of non-gaussian clutter. It is also worth to pay attention to the work done by Frédéric 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): where η = 0 1 τ exp( Y H Φ 1 Y (1 ρ 2 ) m τ 1 0 τ exp( Y H Φ 1 Y m τ ρ 2 = )df τ (τ) )df τ (τ) S H 0 Φ 1 Y 2 (Y H Φ 1 Y )(S H 0 Φ 1 S 0 ) (4.1) (4.2) S 0 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 modulating scalar and H denotes conjugate transpose.

55 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 calculate the 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 in section (see Fig. 4.2). For simplicity we prefer to assume τ = 1 that results in E(τ) = 1 and Φ = E{X H X} i.e. classical STAP detector for Gaussian clutter [49]. 2. We can employ some kind of approximation as in Fig Approximating distribution may be some arbitrary distribution (for example Gamma). We may also employ some expansion approximations, for example Edgeworth. The problem with Edgeworth approximation is that it is based on Gaussian distribution, which spreads from minus infinity to plus infinity, whereas τ is a positive scalar (pure Gaussian and original distribution are shown in Fig. 4.4). In this framework, I postulate to apply some other approximation, that will be simple enough to implement, and at the same time, it will fit well enough real clutter, to give performance 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 I will present a first approach in order to introduce the idea behind the new detector. In the subsection some improvements are introduced in order to be more realistic in the noise pdf approximation 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 more degree 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

56 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 52 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.

57 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 53 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 the point b (4.5). Selection of a and b is discussed below. By rewriting (4.1) according to two Dirac delta simplification, we obtain: ] 1/2[ 1 a exp( Y H Φ 1 Y (1 ρ 2 ) η = m a ) + 1 b exp( Y H Φ 1 Y (1 ρ 2 ) m b ) ] 1/2[ 1 a exp( Y H Φ 1 Y m a ) + 1 b exp( Y H Φ 1 Y m b ) 1 a exp( Y H Φ 1 Y (1 ρ 2 ) η = m a ) + 1 b exp( Y H Φ 1 Y (1 ρ 2 ) m b ) (4.3) 1 a exp( Y H Φ 1 Y m a ) + 1 b exp( Y H Φ 1 Y m b ) From section 2.5 we have: E(τ) = 1 (4.4) having in mind that τ 0, we can rewrite: E(τ) = = = Therefore using (4.4), we can write: 0 τf(τ)dτ [ 1 τ 2 δ(τ a) + 1 ] 2 δ(τ b) τδ(τ a)dτ dτ τδ(τ b)dτ = 1 2 a b (4.5) 1 2 a b = 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(τ 2 ) E 2 (τ) = 1 2 a b2 1 (4.6) 1 2 (2 b) b2 1 V AR(τ) = 0

58 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 54 Figure 4.6 a, b as a function of V AR(τ). Figure 4.7 parameter. b 2 2b + 1 V AR(τ) = 0 this equation has two solutions if V AR(τ) is greater than zero (which is always true): b 1,2 = 1 ± V AR(τ) Keeping in mind, that both a and b must be greater than zero we have restriction for the variance V AR(τ): 1 V AR(τ) > 0 In Fig. 4.6 we see how a and b depend on V AR(τ). V AR(τ) < 1 (4.7) To simplify analysis and accordingly (4.5), I introduced additional parameter in place of a,b. This parameter will be called, and now a = 1 and b = 1 + as in Fig Since now we have one additional degree of freedom by introducing parameter, we need a procedure to estimate this additional parameter. I proposed a closed loop procedure shown in Fig. 4.8 [24].

59 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 55 Figure 4.8 estimation procedure. Figure 4.9 Unequal strength Dirac Deltas approximation. This procedure tries to minimize mean square error between theoretical histogram and histogram derived from secondary-training data. In the first step, from secondary data, covariance matrix is estimated. Having covariance matrix, it is possible to introduce f(τ) (texture), by the integral in the central part of Fig This integral allows us to obtain theoretical histogram 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 true histogram is used to update parameter. is then updated to generate texture in integral, and theoretical histogram is recalculated. Procedure is repeated until difference between theoretical and true histogram changes insignificantly. To avoid the problem of local minima, exhaustive search is used 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 devote one Dirac delta to describe the tail of the CPDF which is of great importance as discussed in section 2.5 and 3.1. Appropriate automatic procedure to find q,a and b is presented in Fig This time a,b and q are related to each other. Therefore it is necessary to find only two of them. This problem can be transformed to the problem of two dimensional optimization

60 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 56 Figure 4.10 q, a and b estimation procedure. problem. I proposed Downhill Simplex Method to cope with this problem. This is the main difference between Fig. 4.8 and Fig In order to evaluate behavior of both classical STAP and TDD STAP detectors I performed simulations. Their results are presented in the sections and Simulations. To validate algorithms and to evaluate its performances, I generated simulated data. In this section, simulation technique is described. First subsection describes parameters chosen for simulations. In subsection reader can find a description of target simulation. Subsection is devoted to noise simulation Simulation parameters. In order to perform simulations, it is necessary to choose some parameter set. This parameters set should be appropriate to the state of the art in radar technology and have been introduced in section 2.1. I have chosen to simulate airborne system, for which non-gaussianity may occur when detecting targets on the sea surface. Airborne radar systems are supposed to be the primary application of STAP technique [38]. This subsection describes process of choosing the parameter set. Some of the parameters where taken from a well known DLR system E-SAR : Carrier center frequency f c = 9.6 GHz Transmit power P = 2500 W Pulse bandwidth B = 100 M Hz Azimuth beamwidth Φ = 17 o Elevation beamwidth Θ = 30 o Platform Velocity V p = 90 m s Geometry of the system is presented in Fig and Fig Having carrier frequency it is possible to calculate wavelength: λ = c f c = 3 cm

61 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 57 Figure 4.11 Vertical situation. Figure 4.12 Earth plane geometry. Minimum range can be calculated from the formula (approximately, assuming azimuth angle=0): ( R min = H tan ξ Θ ) 2 ( R max = H tan ξ + Θ ) 2 Range resolution R resolution is determined by the pulse bandwidth B: R resolution = c 2B = 1.5 m where c is the electromagnetic wave speed. Azimuth beamwidth is related to the antenna size L a by equation: λ = Φ = 17o L a 180 o π 0.3

62 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 58 Figure 4.13 Antenna array. Figure 4.14 Azimuth angle ambiguity. so: L a 10 cm For STAP a linear antenna array parallel to the flight axis as in Fig is often assumed. In classical configuration for STAP, array elements spacing d is equal λ 2 [37], in this case 1.5 cm. This is technically difficult to achieve, because single antenna is approximately 10 cm long. Therefore here, spacing d = 10 cm is assumed. In order to avoid angle ambiguity, it is necessary to calculate ambiguous angle β. This can be done using geometry presented in Fig Ambiguous angle β is defined by the situation where a is equal λ. Therefore: so: and: cos α = λ L a = 3 10 = 0.3 α 73 o β = 90 o α 17 o Next parameter to be set is PRF. Let PRF be 4 khz (for E-SAR it was 1.2 khz). Now it is necessary to check range and velocity ambiguity. For range ambiguity, let the maximum

63 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 59 Figure 4.15 Triangle of velocities. range R max = 30 km. PRF has to be set in accordance with the condition: PRF < c 2R max = 5 khz Second condition concerns velocity ambiguity. First it is necessary to calculate maximum radial velocity from the ground. This will occur at the maximum range at the edge of the beam (see Fig. 4.15). Resolving triangle of velocities: V rmax = V p cos(90 o Φ 2 ) = = 13.3m s Additionally some moving targets can be expected. Let the additional radial movement of the target be limited to 10 m s. Overall radial velocity is therefore V r max sum = 24 m s. It is now possible to calculate Doppler shift: f D = 2V r max sum λ = = 1.6kHz PRF should be twice the Doppler shift, therefore PRF min = 3.2 khz. Last parameter to be set is dwelling time. Dwelling time determines Doppler resolution which determines velocity resolution. Let us assume velocity resolution equal 0.1 m s. Therefore Doppler resolution f D resolution should be: f D resolution = = 0.2 λ Hz Dwelling time is given by the formula: T DW ELL = 1 f D resolution = 1/6 s Finally, range migration problem should be faced. T DW ELL = 1/6 s gives (assuming 24 m s radial velocity) 4 m of range migration which is more than range resolution cell (1.5m). To

64 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 60 Figure 4.16 Radar and target geometry. resolve this problem, dwelling time reduction can be applied. Considering only 0.5 m s resolution gives: f D resolution = = 1 λ Hz therefore: 1 T DW ELL = = 0.03 s f D resolution Now range migration becomes: which is less than range resolution cell. R migration = m velocity Target Simulations. To evaluate probability of detection versus probability of false alarm, I injected target echo into 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 radar platform and the target is shown in Fig 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: where Data(n, l, k) = KP (n) σ n e jφ f(k X n ) AZ(n) Ant(l, n) (4.8) Rresolution

65 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 61 KP (n) is radar equation, σ n is a RCS, e jφ is a random unknown phase, f(k X n Rresolution ) 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 preliminary geometrical calculations. After this step, vectors describing positions and relative angles 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 be adopted. Let radar platform be flying along Y axis at the speed V p. Let us assume that, in the moment t=0 (reception of the first pulse), radar platform position be Y=0. Let us assume following model for the target movement: X(t) = X 0 + V t t (4.9) Y (t) = Y 0 (4.10) Z(t) = Z 0 = 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) V p t) 2 + H 2 We can also simply calculate angles using formulas tan(ϕ(t)) = Y (t) V p t X(t) tan(α(t)) = X(t) H θ(t) = π 2 χ(t) where (see Fig. 4.17): tan χ(t) = Y (t) V p t H 2 + X(t) 2 We should evaluate above formulas in moments defined by the reception of subsequent backscattered pulses from the target. Assuming that t 0 = 0, these moments can be calculated from PRF: t n = n PRF where n is a pulse number starting from 0. Having all this information, we can now calculate data cube evaluating each term in the equation (4.8). KP (n) - In the simplest case is constant from pulse to pulse (which means, that R changes only slightly in radar equation during dwelling time). KP (n) also reflects antenna elevation and azimuth pattern. In simulations, we adopted KP (n) = G(ϕ n, α n ), where pattern G(ϕ, α) in azimuth and elevation is shown in Fig

66 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 62 Figure 4.17 θ and χ. Figure 4.18 Single antenna azimuth G(ϕ) (left) and elevation G(α) (right) pattern. σ n - 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: The phase is however coherent within CPI. p(σ n ) = 1 σ e σn σ e jφ - This is an unknown initial phase of the target signal. We can assume e jφ = 1 without loss of generality. R f(k n Rresolution ) is function giving range cell response for the target at a distance R n. In the simplest case we can assume rectangular function as shown in Fig AZ(n) - is a modulating function of the form AZ(n) = e j 4πR n λ Ant(l, n) is additional phase advance as a result of the antenna elements displacement. Ant(l, n) = e j2πl d λ cos(θ n)

67 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 63 Figure 4.19 f function of range cell response 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 Appendix B. This allows us to add Gaussian or non-gaussian, white or colored noise according to our needs. 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 viewed as 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 a characteristic PDF (two Dirac deltas in Fig. 4.7). This is an academic example, that will show how TDD STAP will perform with the same kind of (TDD) of noise. Finally the most interesting is the case (among these tested in this work) where characteristic PDF is chosen to obtain K-distributed noise. This kind of noise can occur for high resolution radars on the 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)2 ν (bτ)2ν 1 exp ( b2 τ 2 2 ) u(τ) (4.12) where Γ(ν) is a gamma function, b and ν are parameters of K-distribution, and u(τ) is a unit step function. Covariance marix can also be randomized. Problem of Covariance matrix structure 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. Examples of results after performing STAP processing are shown in Fig In figures white covariance matrix is adopted for the purpose of technique demonstration. Last two figures present results for the same SNR 1. Results were generated under the following assumptions: 1 SNR is a Signal to Noise Ratio defined as SNR = log 10 ( P signal P noise ), where P signal is a signal power, and P noise is noise power. In practice, after quadrature receiver, if s denotes signal column vector, and x denotes noise column vector, we use formula SNR = sh s x H x

68 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 64 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 intensity image).

69 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 65 Figure 4.22 STAP result on target signal with non-gaussian, K-distributed (test statistics intensity image). X0 = 7000 m Y 0 = 500 m V t = 4 m/s P RF = 4000 Hz V p = 90 m/s L = 16 N = 120 STAP test was performed for discrete velocities [ 15 m/s,15 m/s,0.5 m/s ] and azimuth angles [ 8.5 o,8.5 o,0.5 o ]. 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 has to be slightly modified, which will be presented in section In section I will present comparison between classical STAP and TDD STAP for a priori chosen, and in section 4.4.4, similar comparison for automatic finding procedure Classical STAP detection performance evaluation. In order to evaluate influence of non-gaussianity, I performed some tests. Single target signal was generated using procedure described in section 4.3. This signal was used in all tests. For

70 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 66 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 sometimes number of repetitions was considerably higher. For each noise trial, classical STAP test was performed using different thresholds. Classical STAP test (2.14) was performed independently on Gaussian noise, signal plus Gaussian noise, non-gaussian noise, signal plus non-gaussian noise. Non-Gaussianity was introduced by K-distribution (ν = 2, b = 2 in 4.12) for real and imaginary samples with unit covariance matrix. False alarms were counted for each trial based on number of target reports in pure noise (without target signal). Therefore we can calculate 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 reported in signal plus noise was greater than number of target reported in noise only, we assumed successful detection. This procedure allows us to obtain probability of detection for different SNR s and thresholds for Gaussian and non-gaussian cases. Other parameters of simulations are presented below: X0 = 7000 m Y 0 = 500 m V t = 4 m/s P RF = 4000 Hz V p = 90 m/s L = 8 N = 30 Classical STAP test was performed for discrete velocities [ 15 m/s,15 m/s,1 m/s ] and azimuth angles [ 8.5 o,8.5 o,1 o ]. Too low resolution in tests (test grid too sparse) may lead to some performance deterioration as a result of a mismatch between the true steering vector and its surrogate from the grid (see section 2.2.5). In Fig we can see Receiver Operating Curves. Vertical axis denotes Probability of Detection (PD) and horizontal axis denotes mean number of False Alarms (FA) per single range cell. For the same PD the aim is to have the lowest possible mean number of FA. We can see that mean number of FA is higher for non-gaussian case (having the same PD) in which the characteristics are shifted to the right (higher mean number of FA). This is true for any SNR. In Fig we can see, that SNR does not influence number of false alarms (CFAR property), although number of false alarms increases for non-gaussian case (characteristics shifted up, toward higher mean number of FA). Pattern is similar for different thresholds as a result of using the same target and noise signals Numerical simplifications for TDD STAP. As a result of a finite computer calculation precision, Two Dirac Deltas (TDD STAP) detector presented in section 4.2 needs to be reformulated for practical implementation. In this subsection, I will show modifications that will be used to implement TDD STAP detector in Matlab environment. In section 4.2 Two Dirac Deltas (TDD STAP) detector (4.3) was presented in the form : η = 1 a exp( Y H Φ 1 Y (1 ρ 2 ) m a ) + 1 b exp( Y H Φ 1 Y (1 ρ 2 ) m 1 a exp( Y H Φ 1 Y m a ) + 1 b exp( Y H Φ 1 Y m b ) b ) (4.13)

71 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 67 Figure 4.23 Probability of detection (PD) as a function of mean number of false alarms per STAP picture (FA number) for different SNR s for Gaussian and non-gaussian case. where ρ 2 = S H 0 Φ 1 Y 2 (Y H Φ 1 Y )(S H 0 Φ 1 S 0 ) (4.14) I will use the following transformation: 1 a exp( X m a ) + 1 b exp( X m b ) ( ) = 1 a exp( X m a ) 1 + am exp( X b m b ) exp( X a ) = 1 a m exp( X a ) ( 1 + ( a b )m exp( X b + X a ) ) = 1 a m exp( X a ) ( 1 + ( a b )m exp ( X( 1 b 1 a )) ) Let me rewrite test statistics according to the above transformation: η = ) 1 a exp( Y H Φ 1 Y (1 ρ 2 ) m a ) (1 + ( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 a ) )) (4.15) 1 a exp( Y H Φ 1 Y m a ) (1 + ( a b )m exp ( Y H Φ 1 Y ( 1 b 1 a ))

72 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 68 Figure 4.24 Number of false alarms (FA number) as a function of SNR for different thresholds (Thr). canceling 1 a m : η = ) exp( Y H Φ 1 Y (1 ρ 2 ) a ) (1 + ( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 a )) exp( Y H Φ 1 Y a ) (1 + ( a b )m exp ( Y H Φ 1 Y ( 1 b 1 a )) ) (4.16) The logarithm of expression (4.16) is given by: ln(η) = = [ ] Y H Φ 1 Y (1 ρ 2 ) a + ln [1 + ( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 ] a )) [ ] Y H Φ 1 Y a ln [1 + ( a b )m exp ( Y H Φ 1 Y ( 1 b 1 ] a )) [ ] Y H Φ 1 Y Y H Φ 1 Y + Y H Φ 1 Y ρ 2 a + ln [1 + ( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 ] a )) (4.17)

73 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 69 ln [1 + ( a b )m exp ( Y H Φ 1 Y ( 1 b 1 ] a )) = SH Φ 1 Y 2 a(s H Φ 1 S) + ln [1 + ( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 ] a )) ln [1 + ( a b )m exp ( Y H Φ 1 Y ( 1 b 1 ] a )) If Y H Φ 1 Y (1 ρ 2 )( 1 b 1 a ) in the nominator of (4.16) is much higher than some value, depending on the numerical precision (for example 400 in Matlab), we should use approximation neglecting 1 in: ln [1 + ( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 ] a )) ln [( a b )m exp ( Y H Φ 1 Y (1 ρ 2 )( 1 b 1 ] a )) (4.18) = ln ( a b )m Y H Φ 1 Y (1 ρ 2 )( 1 b 1 a ) Similarly with the denominator. ln [1 + ( a b )m exp ( Y H Φ 1 Y ( 1 b 1 ] a )) ln [( a b )m exp ( Y H Φ 1 Y ( 1 b 1 ] a )) (4.19) = ln ( a b )m Y H Φ 1 Y ( 1 b 1 a ) Remark Below I will show, that if we apply two approximations at the same time (for nominator and denominator), we obtain classical STAP filter. ln(η) = SH Φ 1 Y 2 a(s H Φ 1 S) + ln ( a b )m Y H Φ 1 Y (1 ρ 2 )( 1 b 1 a ) ln ( a b )m + Y H Φ 1 Y ( 1 b 1 a ) ln(η) = SH Φ 1 Y 2 a(s H Φ 1 S) Y H Φ 1 Y (1 ρ 2 )( 1 b 1 a ) +Y H Φ 1 Y ( 1 b 1 a ) = SH Φ 1 Y 2 a(s H Φ 1 S) + Y H Φ 1 Y ρ 2 ( 1 b 1 a )

74 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 70 = 1 S H Φ 1 Y 2 a (S H Φ 1 S) + 1 S H Φ 1 Y 2 b (S H Φ 1 S) 1 S H Φ 1 Y 2 a (S H Φ 1 S) = 1 S H Φ 1 Y 2 b (S H Φ 1 S) In this case, there is no advantage using TDD STAP detector in comparison with classical STAP (TDD STAP is equal to classical STAP). Therefore I postulate to use approximations according to the needs (appropriate procedure should chose simplification only when necessary). This procedure was used in practical implementation of TDD STAP detector. I used approximation when exponential of nominator or denominator in 4.16 was higher than Comparison of classical STAP and TDD STAP for fixed parameter. I performed comparisons using method described in section In the first result, the value of was not estimated but chosen a priori in order to estimate the discrepancies when is poorly estimated. These results were already presented in [22]. In first experiments, Non-Gaussianity was introduced by TDD noise. This means that CPDF (for SRIP) adopted in noise generation algorithm follows TDD distribution. More precisely (see section 4.3.3): f(τ) = 1 2 δ(t a) + 1 δ(t b) 2 In Fig we can see Probability of Detection (PD) as a function of mean Number of False Alarms (NFA) for different choice of spread parameter. If approaches 0 or 1, then TDD STAP detector behaves exactly as classical STAP detector. This is not surprising since for = 0 or = 1 expressions for TDD STAP reduce to classical STAP formula. For some values of parameters (eg. 0.4 or 0.5), newly developed TDD STAP detector has a better performance than classical STAP. Unfortunately for badly chosen (eg. 0.7) TDD STAP performance is even worse than classical STAP. Similar experiments were performed for K-distributed (ν = 2 and b = 2) noise. This time in noise generation scheme (see section 4.3.3) the function f(τ) takes on the form: f(τ) = 2b Γ(ν)2 ν (bτ)2ν 1 exp ( b2 τ 2 2 ) u(τ) where Γ(ν) is a gamma function, b and ν are parameters of K-distribution, and u(τ) is a unit step function. Results are presented in Fig Again we can see, that there is some performance improvement in comparison to classical STAP. If approaches 0 or 1, then TDD STAP detector behaves exactly like classical STAP detector. When is appropriately chosen (eg. 0.3 or 0.4), TDD STAP detector has better performance than classical STAP.

75 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 71 Figure 4.25 Probability of Detection (PD) as a function of mean Number of False Alarms (NFA) for TDD STAP detector in the presence of Gaussian and TDD noise for different values of.

76 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 72 Figure 4.26 Probability of Detection (PD) as a function of mean Number of False Alarms (NFA) for TDD STAP detector in the presence of Gaussian and K-distributed noise for different values of.

77 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 73 If is badly chosen ( = 0.7) than TDD STAP performance is worse than classical STAP. To conclude it can be seen, that improvement is possible, but automatic procedures are necessary to find parameter, as discussed in section 4.2. It the parameter is poorly estimated results are much worse Results for TDD STAP detector with automatic finding. As it was shown previously, it is necessary to appropriately choose parameter. In order to do this I postulate to employ method presented in Fig. 4.8 and Results of this approach were already published in [23] and [24]. In Fig we can see results of automatic finding under assumption, that deltas strength 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 there is an improvement in detector performance. In this case detector is appropriately chosen for the noise in the framework of SIRP theory. Real CPDF is exactly the same as CPDF assumed in 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 is only an approximation of a real GLRT detector. In Fig similar results are presented for the K-distributed noise. We can see again some slight improvement of TDD STAP detector in comparison to classical STAP. This time however we can expect, that there exists better detector 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, to be better than classical STAP, and there is a possibility for further improvement. This can be achieved by better approximating CPDF. The last figure (4.29) presents results of automatic a, b and q finding for K-distributed noise and unequal Dirac deltas as described in section 4.2 and depicted in Fig We can see that in comparison to Fig there is a slight improvement. This is due to the better approximation of real CPDF of SIRP by unequal-strength deltas than by equal deltas, as a result of additional degree of freedom. Having two Dirac deltas with unequal strength we can better approximate distributions with skewness. In such a case, Dirac delta placed in point a (see Fig. 4.9) should be stronger than the other (in point b). This way it is possible to better approximate skewed distributions. In fact, because τ is greater than 0 and potentially continuous and unlimited, all considered distributions (CPDFs) will be skewed. Following this path we can expect further improvement when using for example more Dirac-deltas. Advantage of having additional degree of freedom can be outweighted by the necessity of estimating more parameters. This can be difficult to perform, and poorly estimated parameters can lead to detection losses. For high probabilities of detection we can see, that behavior of TDD STAP is slightly worse than classical STAP. Nevertheless this is in the area of high probabilities of false alarms (mean number of false alarms per image more than 1). Usually radars are working in areas with lower levels of false alarms. As a conclusion we can say that automatic procedure, presented in section 4.2, works well.

78 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 74 Figure 4.27 Automatic finding for TDD noise. Figure 4.28 Automatic finding for K-distributed noise.

79 CHAPTER 4. TWO DIRAC DELTA DETECTOR. 75 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 better fit sea clutter than traditional STAP algorithm. In chapter 3 we have seen, that sea clutter has 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 practical use in section 4.2 (this chapter). In next sections, simulation method was presented, that was later used in evaluating TDD STAP detector performance. In section it was shown, that there is a performance decrease when classical STAP algorithm is working in non-gaussian environment. The use of TDD STAP detector can bring some improvements, which was shown in section 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 automated procedures can deal with the problem, and there is a performance improvement clearly visible in characteristics presented in figures. The future improvement in this area seems possible. We can imagine more Dirac-deltas as a better approximation of characteristic PDF (under the limitations discussed in section 4.4.4). This can lead to better detectors, able to deal with a larger class of non-gaussianity.

80 CHAPTER5 HF radar signals experiments and STAP technique modifications. High Frequency (HF) radar systems, which operate in frequency range between 3 and 30 MHz, have a potential to detect targets which are located beyond optical horizon on the sea surface (Over The Horizon visibility - OTH). Therefore, in recent years, more and more attention has been paid to such a systems in the context of the ship traffic surveillance in the Exclusive Economic Zone (EEZ), that was established by United Nations Convention on the Law of the Sea in 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 from HF 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 has been theoretically described in section 2.2. The aim of the chapter is to present the problem and the solution to non-stationarity. Comparison between AMTI and STAP is of secondary importance. In the case of HF radar, there is no problem with non-gaussianity. On the other hand there is a problem with strong clutter and interference as was emphasized in section 3.2. It will be shown what new problems are to be faced in real-data implementation. Because of the difficulties with covariance matrix estimation (see sec 2.2.6), sometimes it is possible to obtain better results using Adaptive MTI. Because of these practical problems, classical algorithms must be adapted, which will be also shown in this chapter. In the section 5.1 I will introduce WERA radar system that was used in experiments. WERA system was designed to operate as an oceanographics system, therefore this application will also be described in the same section. Sections 5.2 will show how AMTI and STAP algorithm must be adapted to WERA 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 used in experiments is a property of the French Service Hydrographique et Océanographique de la Marine (SHOM) and is operated by the company SAS ActiMar. System s primary task is to provide real-time, continuous data collection about currents and waves in the area of Brest Gulf (Atlantic, west coast of France). In order to have unambiguous vectorial information of the 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

81 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 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 and storage for off-line processing. Transmitted waveform is FMCW, therefore it is necessary to perform preprocessing in range to obtain data equivalent to pulsed systems described in section This is necessary to apply algorithm like AMTI or STAP (see section 2.2). To achieve this goal, signal must be demodulated. The next step is to apply appropriate low pass filter. Signal must be then sampled with Nyquist frequency. Such raw data have to be range

82 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 78 Figure 5.3 Transmitting antenna array. processed. In the case of frequency modulated signals this means simply to perform Fast Fourier Transform (FFT). After this operation the data can be presented as a complex data cube (Fig in section 2.2.5) appropriate for AMTI or STAP processing. More detailed explanation can be found in Appendix C. WERA chirp duration can vary from s to 0.26 s. Carrier frequency of the system is around 12 MHz. Bandwidth is set independently to achieve desired range resolution (from 0.4 km to 1.5 km). Data file consists of complex samples taken from 2048 coherent chirps ( s) for 120 to 256 range cells by 16 separate channels of array antenna. The number of range cells depends on the range resolution. For example for 0.4 km of range resolution, data file consists of 240 range cells which gives maximum range 96 km. For 1.2 km of range resolution, data are taken for 120 range cells, which gives maximum range 144 km. This configuration allows to cover area presented in Fig. 5.4 (for the 96 km maximum range mode). Data used in experiments for target detection were collected in July Additionally, it was possible to obtain Automatic Identification System (AIS) data, concerning ship traffic around Brest Gulf. From AIS data it was possible to obtain information about trajectories of certain ships together with additional information e.g. MMSI number, IMO number, ship length, draft etc (see Fig. 5.5). AIS data allowed to compare HF detections against the true data. HF radar systems have a long history as an ocean monitoring systems. They proved to be capable 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 for example in [5] [40] [30] [29]. WERA radar system was designed for the same purpose. In the first stage of its processing it 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 13 spectra are calculated, each using overlapping sets of 512 samples (chirps). Chirps used to calculate each spectrum are accordingly: number 1-512, , ,...,

83 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 79 Figure 5.4 Coverage area of radars. Figure 5.5 AIS trajectories.

84 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 80 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 radar for a single Doppler band is presented in Fig 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 - covariance matrix estimation problem. In this section, the application of Adaptive MTI and STAP processing to WERA radar data will be described. The main problem of both techniques is to estimate covariance matrix. Usually covariance matrix is calculated using auxiliary range cells. Unfortunately for WERA radar system overall number of range cells is at most 256. The first step for both techniques is data trim. From the 2048 chirps in data file, only J chirps of interest are chosen for further processing (see Fig. 5.10). Number of chirps J is chosen in order to operate on data more effectively and is not important from theoretical point of view. From this point data for AMTI and STAP are processed differently Adaptive MTI implementation The theoretical description of AMTI can be found in section 2.2.4). The first step of AMTI is beamforming (Fig in section 2.2.4). Beamforming is performed using 16 antennas for discrete angles [ 60 o,+60 o,2 o ] (Fig. 5.11). After this step data are available as a cube with 61 beams instead of 16 antennas (Fig.

85 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 81 Figure 5.7 Current map (ActiMar).

86 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 82 Figure 5.8 Waves map (ActiMar).

87 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 83 Figure 5.9 Wind map (ActiMar).

88 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 84 Figure 5.10 Data trim. Figure 5.11 Angle Coverage.

89 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 85 Figure 5.12 Data snapshot. Figure 5.13 Test snapshot and training area and 2.21). For each range-beam cell, adaptive filter is calculated as described in section 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 data to estimate covariance matrix. In Fig we can see test snapshot and area that auxiliary snapshots were taken from. Auxiliary snapshots were taken by sliding snapshot window over chirps and ranges in the test area. Training areas were separated in range from test snapshot by guard cells. This prevents target self-nulling. Steering was performed for discrete velocities [ 10 m/s,+10 m/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 across one dimension. In Fig 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 is moving away from the radar, this is why echo is present in different range cells. Processing was performed using 48 chirps (N=48).

90 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 86 Figure 5.14 Example of a target visible after AMTI processing.

91 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 87 Figure 5.15 Test snapshot and training area for STAP processing STAP implementation. Fully adaptive STAP is operating on data before beamforming (suboptimum techniques were not considered in this work), but after data trim (Fig. 5.10). STAP algorithm is applied to each range cell separately to obtain test statistics as in section see equation (2.14). Covariance matrix must be calculated from secondary (training) data. Clutter covariance matrix was estimated using similar schema as for AMTI. In comparison to AMTI, STAP has a potential to suppress clutter and interference at the same time. When interference cannot be estimated independently from the clutter (because of coupling azimuth-doppler), adaptive single-domain methods may perform poorly and joint domain (space and time) adaptive processing may be advantageous [18]. However this effect can be outweighted by the problem of covariance matrix estimation. Classical schema for estimating covariance matrix fails to work because in WERA system there are at most 256 range cells whereas in the case of STAP processing it is necessary to estimate covariance matrix of a very high dimension (eg in the case of STAP for 16 antennas and 100 chirps). Therefore it is necessary to find more auxiliary snapshots. I decided (similarly as for AMTI) to use time spread of the data to obtain more snapshots. I use overlapping sliding snapshots. In Fig we can see test snapshot, training snapshot and training areas in the data cube (compare with Fig. 5.13). After STAP processing, the test statistics is available in the same form as for AMTI (test statistics cube: range velocity angle). For both techniques there is possibility to chose certain parameters of covariance matrix estimation. First one is the number of guard cells. Guard cells are necessary to avoid target self-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 increasing number 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 clutter covariance estimation area. The last parameter is time spread. Extending time spread allows us to estimate covariance matrix in the situation, when there is not enough range cells to do this. Unfortunately covariance matrix estimate converges slowly with extending timespread. Moreover there is a risk of clutter non-stationarity that will degrade covariance matrix

92 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 88 Figure 5.16 Targets localization. estimation. We can see that playing with these three parameters may be a crucial problem in application both AMTI and STAP. Results of such experiments will be shown in next sections of this 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 algorithms can cope with the interference and clutter in order to reveal targets Data file and target description. In experiments two data files from 19 July 2007 were used. One file contains recording from Brezzelec site and the other from Garchine. Recording started at 05:20 UTC and its duration was almost 8 min (2048 chirps, single chirp duration s). Bandwidth was set 375 MHz, which gave range resolution 0.4 km. Transmitted power was 30 W. Files contain samples for 240 range cells, which gave maximum range 96 km. Significant wave height at the moment of data collection was approximately 1.3 m. From AIS data I chose two interesting potential targets shown in Fig EERLAND 26 is 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 we can see Garchine data file after classical beamforming. We can see a strong radio interference from the same direction as fishery boat. This will definitely have a negative influence on detection possibilities.

93 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 89 Figure 5.17 Tug ship. Figure 5.18 Fishery boat 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 using data from Garchine localization. From AIS data it is possible to deduce, that target should be present in the range cell number 158 (distance 63 km) and in the beam number 25 (-10 degrees left to the normal to the antenna). Indeed, in Fig target is well visible. This figure presents test statistics (in decibels) for range cell number 158 and beam 25. Fig presents behavior of AMTI and STAP algorithms for different integration times and for different training samples support. Upper-left graph presents comparison between lower (100 chirps) and higher (200 chirps) integration time AMTI with low training samples support. 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 the figure denotes the reference level of detected target. We can see, that the highest non-target peak for 200 chirps AMTI is 6 db lower than for 100 chirps AMTI. We can say, that in this case 200 chirps AMTI outperforms 100 chirps AMTI by 6 db. The lower-left graph presents

94 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 90 Figure 5.19 Interference. similar results for high training samples support. In this case 200 slides and 60 range cells on both sides of test cell are used. This gives samples used to estimate covariance matrix. We can see that improvement is even better and 200 chirps AMTI outperforms 100 chirps AMTI 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 STAP fails to detect the target. Situation is different for high training samples support (lower-right in Fig. 5.20). This time we can see, that 200 chirps STAP outperforms 100 chirps STAP by 5 db. In Fig we can see comparison between AMTI and STAP. Upper-left graph presents comparison between AMTI ans STAP for shorter integration time (100 chirps) and low number 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 outperforms AMTI by 5 db. For longer integration time (200 chirps) and low number of training samples we can see, that STAP fails, whereas AMTI can reveal the target (see upper-right graph 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 on detection possibilities. General pattern is clear that more training samples gives better chances of target detection. This is valid for extending training area in time (100 vs 200 slides) as well as in range (from 15 to 60 range cells on both sides of test cell). Generally, it is possible to conclude that: AMTI works well even for relatively low number of training samples (100 slides, 15 range cells on both sides of test cell). STAP needs more training samples (200 slides and 60 range cells on both sides of test cell). Using both techniques it is possible to detect tug ship. Because of interferences present in the data file, STAP techniques gives slightly better results.

95 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 91 Figure 5.20 Tug ship detection from Garchine site - influence of integration time. Upperleft: AMTI for low training samples support, lower-left: AMTI for high training samples support, upper-right: STAP for low training samples support, lower-right: STAP for high training samples support Detection of the tug ship from Brezzelec radar site. The same target is visible from the second radar site (Brezzelec). From AIS data we can expect 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 target is visible. In the case of Bezzelec file, there are almost no interferences present. This should mean, that main advantage of STAP will not play a role here. First, let us compare the influence of integration time as before. We can see, that pattern is 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.

96 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 92 Figure 5.21 Tug ship detection from Garchine site - comparison between AMTI and STAP. For shorter integration time (100 chirps) AMTI better reveals the target (I could not explain this anomaly) Detection of the fishery ship from Garchine radar site. The second target to detect was the fishery ship L Ar Voaleden. Unfortunately strong interference was present in the same beam as the target. In figure 5.28 we can see a PPI image after 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 to detect this target. Exemplary image is presented in figure This is PPI image (rangeazimuth). PPI image is produced by taking maximum value of test statistics from all test velocities for range-azimuth cell of interest.

97 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 93 Figure 5.22 Tug ship detection from Garchine site - different range support for AMTI with 100 chirps.

98 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 94 Figure 5.23 Tug ship detection from Garchine site - different range support for AMTI with 200 chirps.

99 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 95 Figure 5.24 Tug ship detection from Garchine site - different range support for STAP with 100 chirps.

100 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 96 Figure 5.25 Tug ship detection from Garchine site - different range support for STAP with 200 chirps.

101 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 97 Figure 5.26 Tug ship detection from Brezzelec site - influence of integration time. Upperleft: AMTI for low training samples support, lower-left: AMTI for high training samples support, upper-right: STAP for low training samples support, lower-right: STAP for high training samples support Detection of the fishery ship from Brezzelec radar site. The same fishery ship should be visible from the second radar site (Brezzelec). Similar experiments were performed to detect this target. From AIS data it was possible to extract position and velocity information of the target. Fishery boat was going almost directly toward the Brezzelec radar site at the speed of about 10 knots (this is 5 m/s). To understand possibilities of this detection it is necessary to calculate Bragg line Doppler speed. From chapter 3.2 we know, 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 MHz carrier 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 integration times 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

102 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 98 Figure 5.27 Tug ship detection from Brezzelec site - comparison between AMTI and STAP. example it is hard to achieve STAP for longer than 200 chirps and AMTI for longer than 500 chirps. Moreover, having very long integration time we can face the situation, that target is no longer time-coherent. For example, target can change range speed, orientation etc. In Fig and 5.31 we can see a PPI image (range-azimuth) that is produced by taking maximum value of test statistics from all test velocities for range-azimuth cell of interest. It seems that fishery boat is not revealed by STAP, but is revealed by AMTI. This is not true. In fact neither AMTI nor STAP can reveal this target. In the case of AMTI this is false alarm, that disappears with time. Moreover this seeming detection for AMTI indicates radial velocity which 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 Doppler velocity, detection possibilities are very poor.

103 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 99 Figure 5.28 Strong interference present in the target beam. Figure 5.29 PPI image after AMTI processing for 200 chirps.

104 CHAPTER 5. HF RADAR SIGNALS EXPERIMENTS AND STAP TECHNIQUE MODIFICATIONS. 100 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 in the terms of test statistics. In a complete system, the last step it to apply thresholding for the detections (see section 2.2.5, 2.3 and chapter 4). Choosing appropriate threshold is a very important issue, that is not treated in this work. Instead, I will present some detection results for a priori chosen threshold. In figures 5.32 and 5.33 we can see PPI images after thresholding. 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 was reduced to test statistics image by taking (for each range-beam cell) maximum across all velocities. 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 mostly related to the presence of islands and rocks as well as to the sidelobes of strong echo from target. Processing after thresholding is also very important, that is not treated in this work.