Knowledge-Based Adaptive Radar Detection

Transcription

1 UNIVERSITÀ DEGLI STUDI DI CASSINO SCUOLA DI DOTTORATO IN INGEGNERIA Knowledge-Based Adaptive Radar Detection Goffredo Foglia In Partial Fulfillment of the Requirements for the Degree of PHILOSOPHIAE DOCTOR in Electrical and Information Engineering November 2006 TUTOR Prof. Ernesto Conte, Prof. Marco Lops COORDINATOR Prof. Giovanni Busatto

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3 UNIVERSITÀ DEGLI STUDI DI CASSINO SCUOLA DI DOTTORATO IN INGEGNERIA Date: November 2006 Author: Title: Degree: Goffredo Foglia Knowledge-Based Adaptive Radar Detection PHILOSOPHIAE DOCTOR Permission is herewith granted to university to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR S WRITTEN PERMISSION. THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED.

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5 Acknowledgements I would like to convey my heartfelt thanks to Prof. Antonio De Maio, with whom I was privileged to work. I express him gratitude for providing much invaluable technical support and advice. I deeply appreciate the time and effort he spent to help me toward the improvement of my research and writing abilities. I am grateful to him for giving me continuous guidance during the research process. I am also indebted to Prof. S. Haykin and Dr. B. Currie of the McMaster University (Canada) who have kindly provided the IPIX data. Goffredo Foglia v

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7 Contents Acknowledgements v Summary 1 1 Knowledge Based CFAR Processing Introduction Classical CFAR detection schemes KB-CFAR detection schemes Static KB Selector Dynamic KB Selector Performance Analysis Real Data and GIS map Number of FAs Detection Performance Conclusions Design of a Knowledge-Based Radar Detector Introduction Design of the GIS-aided Training Selector An Illustrative Example Design and Analysis of the Data-Adaptive Training Selectors One-Step GLF Procedure Two-Step GLF Procedure vii

8 CONTENTS Performance Analysis Discussion Design Issues Performance Analysis Simulated Data Real Data Conclusions Knowledge-Based Clutter Suppression Introduction Problem Formulation and Design Issues RLS Covariance Update Normalized RLS (NRLS) Covariance Update Knowledge-Based RLS (KB-RLS) Covariance Update Performance Analysis Simulated Data Real Data Conclusions Conclusions 93 Bibliography 95 viii

9 Summary To mitigate the deleterious effects of clutter and jammer, modern radar have adopted adaptive processing techniques such as Constant False Alarm Rate (CFAR) detectors, adaptive arrays, and Space Time Adaptive Processing (STAP). In the real world, however, still sub-optimal performance might occur (high false alarm rate and/or low detection probability) as a consequence of heterogeneous clutter (rapidly varying terrain, i.e. mountainous with rapid elevation/reflectivity variation, rapid land cover variations, littorals); dense target backgrounds (moving clutter, military/civilian vehicles); large discrete and spiky clutter (urban clutter, power lines, towers, step mountainous terrain). In fact the aforementioned adaptive techniques are very restrictive because they require the environment to remain stationary and homogeneous during adaptation. Bad training data selection in such adaptive filters can produce bad output signals. A possible way to circumvent this drawback is the real-time exploitation of a-priori knowledge concerning the environment surrounding the radar [1, 2, 3, 4]. In fact the environmental context is the key to efficient adaptation: sensors like humans might benefit from the context. Examples of a-priori knowledge are Digital Terrain Elevation Models (DTEM), previous look data, Geographical Information System (GIS), roadway maps (to highlight sectors of surveillance where moving cars or vehicles might be present), background of air/surface traffic, system calibration information, etc. The ultimate goal is to make the radar an intelligent device, such that it is capable of developing cognition of the surrounding environment. Otherwise stated the environment in which the radar system operates acts as a teacher and the radar can become more expert with time by learning from the environment. This is basically the concept of Knowledge-Based (KB) or cognitive radar, known to the radar community since the pioneering papers of Vannicola [5, 6] and Haykin [7]. Recent advances in environmental measurements, DTEM, future information quality and accessibility, digital processing, mass and random-access memory technologies, has opened many possibilities, unthinkable in the past, for radar systems 1

10 Summary to improve their on-line performance. New real-time processing techniques are required, to take advantages of these new opportunities to bring radar performance back to optimum under difficult operation conditions such as littorals that include mixed sea and variable terrain. The great interest on the application of KB techniques to adaptive radar signal and data processing is testified by many facts, such as the following. The Defense Advanced Research Projects Agency (DARPA) has been pioneering the development of the first ever real-time, Knowledge-Aided (KA) adaptive radar architecture. In particular, the Knowledge-Aided Sensor Signal Processing and Expert Reasoning (KASSPER) program has as its aim the development and application of a revolutionary new approach demanding multidimensional adaptive sensor systems, with a near-term focus on military applications of Ground Moving Target Indicator (GMTI) radar and Synthetic Aperture Radar (SAR). Annual KASSPER workshop started in 2002 to allow the exchange of ideas across the spectrum of research and development activities, including KB-STAP, environmental KB generation and maintenance, and real-time KB embedded computing [8]. The U.S. Air Force (USAF) Reaserch Laboratory s Sensors Directorate has been pursuing some of the most progressive work in employing KB techniques in the radar signal processing chain, specifically in the CFAR portion of the chain [9]. The USAF has an outgoing project called Autonomous Intelligent Radar System (AIRS) that is performing research in applying KB techniques to radar signal processing. The AIRS architecture design leverages advanced technologies developed by the World Wide Web Consortium (W3C) and the DARPA agent markup language (DAML) program to define the next generation Internet, also called the Semantic Web [10]. A series of lectures were devoted to KB radar signal and data processing [3]. They were sponsored by the NATO Research and Technology Organization (RTO) with the scope of promoting cooperative research and information exchange to support the development and effective use of national defense research and technology to meet the military needs of the alliance, maintaining a technological lead, and providing advice to NATO decision makers. This lecture series was held in Sweden, Hungary, and Italy in 2003; Poland and Spain in 2004; Czech Republic, Belgium and the United Kingdom in A special section of the IEEE Transactions on Aerospace and Electronic Systems devoted to KA sensor signal and data processing will appear in Winter 2

11 Summary 2006, coedited by William Melvin and Joseph Guerci. The aim of this special issue is to showcase recent research in KB systems and radar signal and data processing together in the same forum to present a range of perspectives and innovative results with potential to enable practical adaptive radar systems. The articles of this special issue review the current developments in the area and present examples of improved radar performance for augmented and upgraded systems and project the impact of KB technology on future systems. In this work we consider different applications of the KB radar signal processing. Precisely we focus on the application of KB techniques to the CFAR processing, on the design of an adaptive KB detector for doppler processing, and on the synthesis of KB filters for heterogeneous clutter suppression. We also assess the performance of the new techniques showing the potential performance improvement which can be achieved resorting to a KB processing. The performance of the new schemes have been analysed both on simulated and on measured clutter data collected by the McMaster IPIX radar in November The report is organized as follows. Chapter 1 deals with the design and the analysis of CFAR detectors exploiting KB processing techniques. In Chapter 2 we consider the synthesis and the analysis of a KB radar detector for doppler processing while in Chapter 3 we deal with the design of adaptive filters for heterogeneous clutter suppression. 3

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13 Chapter 1 Knowledge Based CFAR Processing This chapter deals with the design and the analysis of CFAR detectors exploiting KB processing techniques. The proposed algorithms are composed of two stages. The former is a KB data selector which, exploiting the a-priori information provided by a GIS, chooses the training samples for threshold adaptation. The latter stage is a conventional CFAR processor. The performance of the new schemes are analysed in the presence of real radar data, collected by the McMaster IPIX radar, also in comparison with other common CFAR detectors. The results show that noticeable performance improvements can be obtained suitably exploiting the a-priori information available about the sensed environment. 1.1 Introduction The detection of a radar target is impaired by the presence of clutter returns due to reflections from buildings, trees, ground, sea etc. Since the clutter power is usually unknown, detection schemes with fixed threshold may result in an excessive number of false alarms (FAs) and/or in poor target detectability. A possible way to circumvent this drawback relies on the use of processing devices with adaptive threshold capable of ensuring the CFAR property. Toward this goal several strategies have been proposed in open literature. Among them we mention the classical Cell Averaging CFAR (CA-CFAR) detector [11, 12, 13] that resorts to secondary (or training) data from range cells in close proximity to the Cell Under Test (CUT) in order to perform the threshold adaptation. However training data are often contaminated by power variations over range (in addition to radar range equation effect), clutter discretes, and other outliers. Moreover the strength of the clutter also fluctuates with terrain type, 5

14 CHAPTER 1. Knowledge Based CFAR Processing elevation, ground cover and the presence of man-made structures. In these situations training data may not be representative of the disturbance in the CUT and the CA-CFAR exhibits strong degradations both in the detection performance as well as in the CFAR behaviour [14]. This is especially true in regions containing varying ground cover such as regions with land and sea. Several modifications of the CA-CFAR detection scheme have been proposed during the last three decades in order to reduce the impact of a nonhomogeneous secondary data window. The Greatest Of CFAR (GO-CFAR) algorithm, devised in [15], tries to mitigate the impact of clutter discontinuities by appropriately choosing the reference window. As a result the algorithm shows a CFAR behaviour stronger than the CA-CFAR, but a detection probability (P d ) worse than the counterpart when interfering targets, and in general outliers, are present in the training window (masking effect). In [16] the Smallest Of CFAR (SO-CFAR) processor is introduced; it reduces the masking effect but sacrifices the CFAR behaviour in nonhomogeneous clutter environments. A strong robustness can be obtained exploiting Order Statistic CFAR (OS-CFAR) schemes [17], which rely on the power ranking of the reference window samples. Nevertheless the OS-CFAR processor is unable to prevent an excessive FA rate in clutter transition regions [18]. Other algorithms, based on the excision of a predetermined number of reference cells and on clutter maps have also been proposed and assessed [19, 20, 21, 22] 1. A different philosophy which can aid the selection of training samples might be the real-time exploitation of a-priori knowledge concerning the environment surrounding the radar [1, 2, 3, 4, 8, 23]. In fact the environmental context is the key to efficient adaptation: sensors like humans might benefit from the context. Examples of a-priori knowledge are DTEM s, previous look data, GIS s, roadway maps (to highlight sectors of surveillance where moving cars or vehicles might be present), background of air/surface traffic, system calibration information, etc. In this chapter we design CFAR processors capable of exploiting a-priori information, provided by a GIS, about the observed radar scene. The proposed schemes are composed of two stages. The former is a KB data selector which suitably chooses the reference samples exploiting a-priori information. The latter stage is a standard CFAR processor (in the sequel we use the CA-CFAR but another system could be also exploited). At the analysis stage we study the performance gain achievable with the aid of KB techniques. Precisely we compare, on real radar data, the performance of the new KB algorithms with the classical CFAR receivers (CA-CFAR, GO-CFAR, SO-CFAR and OS-CFAR). The chapter is organized as follows. In Section 1.2 we shortly revisit the classical CFAR schemes. In Section 1.3 we devise two GIS-based training selectors, which represent the first stage of a KB-CFAR processor. In Section 1.4 we conduct a performance analysis on measured radar data, highlighting the benefits arising 1 We explicitly point out that this reference list is not exhaustive 6

15 1.2 Classical CFAR detection schemes r K r 4 r 2 r cut r 1 r 3 r K 1 Figure 1.1: Diagram showing the reference window including the CUT. from the use of a KB pre-processing. Finally conclusions are given in Section Classical CFAR detection schemes The aim of this section is to quickly revisit some classical CFAR detection schemes, proposed during the last three decades in open literature, which will be used in the subsequent analysis for comparison purposes. With reference to a radar sweep, let r cut denote the sample from the CUT and r 1,...,r K the samples form the training cells (see Figure 1.1), namely returns from K adjacent range cells 2, symmetrically located with respect to the CUT. CA-CFAR. The CA-CFAR exploits the samples of the reference window for estimating, through the arithmetic mean of their squared modulus, the local clutter power. Then it evaluates the decision statistic normalizing the squared modulus of the return from the CUT to the aforementioned power estimate. Otherwise stated it implements the following decision procedure r cut 2 K r i 2 i=1 H 1 > < H 0 T CA, (1.1) where denotes the modulus of a complex number and T CA is the detection threshold, set according to the desired false alarm Probability (P fa ). GO-CFAR and SO-CFAR. A major assumption of the CA-CFAR is the equality of the interference statistics in the K reference cells as well as in the CUT. Unfortunately this assumption is often violated in many situations. For example, when the reference window contains a clutter edge or when returns from undesired targets are present in the training samples. A possible solution to this problem relies on dividing the reference window into two parts: the left window (data characterised by even indexes, see Figure 1.1) and the right one (data characterised by odd indexes). From this partition, two estimates of the local clutter power can be constructed, i.e. σ L 2 = 2 r i 2 σ R 2 K = 2 r i 2. (1.2) K odd i 2 For simplicity K is assumed an even integer even i 7

16 CHAPTER 1. Knowledge Based CFAR Processing The GO-CFAR [15] normalizes the squared modulus of the primary sample to the maximum of (1.2), namely it operates as follows r cut 2 K ( σ2 2 max L, σ ) U 2 H 1 > < H 0 T GO, (1.3) where T GO is the detection threshold, set according to the desired P fa. The SO-CFAR [16] normalizes the squared modulus of the primary sample to the minimum of the two local power estimates, namely it operates as follows r cut 2 K ( σ2 2 min L, σ ) U 2 H 1 > < H 0 T SO, (1.4) where T SO is the detection threshold, set according to the desired P fa. OS-CFAR. Rather than estimating the local clutter power through an arithmetic average, the OS-CFAR processor [17] orders the squared modulus of the reference samples in increasing order. Then, denoting by r (1) 2 r (2) 2... r (K) 2, the ordered statistics, the detector chooses the M-th one as an estimate of the local clutter power. In other words it works as follows r cut 2 H 1 > r (M) 2 < T OS, (1.5) H 0 with T OS the detection threshold, set according to the desired P fa. In the performance analysis of Section 1.4 the OS-CFAR has been used with M = 0.75K. 1.3 KB-CFAR detection schemes This section is devoted to the design of two data selectors which, exploiting the a- priori information provided by a GIS, are capable of discarding nonhomogeneous samples from the available secondary data set. We just recall that a GIS is a software technology which represents a geographic site through layers, namely geometrically and semantically uniform sets, characterized by an attribute describing their content. The higher the number of attributes the more detailed the representation. From a practical viewpoint there exist two different ways of representing a given layer usually referred to as the raster and the vectorial models. The former provides a regular quantization of a given area through rectangular cells (raster cells) whose greater side coincides with the spatial resolution of the representation and whose topographical content is described by an attribute. The latter, instead, provides a finer quantization of the scene, employing points, lines, and 8

17 1.3 KB-CFAR detection schemes polygons in place of the raster cells. In the sequel we focus on the raster model and highlight that the idea we introduce can be also extended to case of a vectorial representation. According to the raster model a given scene is represented by an array of rectangular cells each of them containing a numerical value which labels the corresponding attribute. For example in Figure 1.2a we have a scene composed of two lakes and homogeneous terrain. The corresponding raster representation is displayed in Figure 1.2b where the number 0 indicates the attribute terrain and the number 1 denotes the attribute lake. The training selection algorithms we propose, assume that the size of the radar cell is equal to the size of the raster cell, and resort to the GIS information to select the most homogeneous reference samples. The basic assumption is that homogeneous secondary samples come from cells which have the same terrain as the CUT. 9

18 CHAPTER 1. Knowledge Based CFAR Processing Figure 1.2: Example of GIS raster representation. Attribute 0 denotes terrain. Attribute 1 denotes lake. 10

19 1.3 KB-CFAR detection schemes Static KB Selector The first training selection algorithm devised in this chapter, referred to as Static KB (SKB) selector, excises a fixed number K S of data from the training set and works as follows. Exploiting the information provided by the GIS assign to the CUT and to the K reference cells the corresponding attributes X cut and X i, i = 1,...,K. Moreover for all i {1,...,K} evaluate the homogeneity indicator 0 if X i X cut D(i) = (1.6) 1 if X i = X cut The set Ω containing the indexes of the selected secondary data can be obtained as follows. if K i=1 D(i) K K S, then Ω is the set of the K K S smallest indexes such that D(i) = 1. This is tantamount to select as training data the K K S returns from the homogeneous cells spatially closest to the CUT. if K i=1 D(i) < K K S, then Ω is composed of the indexes such that D(i) = 1 (indexes of the homogeneous cells) and the K K S K i=1 D(i) indices of the nonhomogeneous cells closest to the CUT. After the SKB training selection procedure, a CA-CFAR detector is employed to perform the final decision about the target presence. The cascade of the SKB training selection algorithm and the CA-CFAR detector will be denoted as SKB- CFAR Dynamic KB Selector The main difference between the static training selection algorithm of the previous subsection and the dynamic one relies on the fact that the former excises a fixed number of cells while the latter chooses dynamically the quoted number. In other words the a-priori information is further used to select the number of cells to be censored. Let K MAX and K min be the maximum and minimum number of data exploited for the threshold adaptation process. The set Ω containing the indexes of the selected secondary data can be obtained as follows. if K i=1 D(i) K MAX, then Ω is the set of the K MAX smallest indexes such that D(i) = 1. This is tantamount to select as training data the K MAX returns from the homogeneous cells spatially closest to the CUT. if K min K i=1 D(i) K MAX, then Ω is the set of all indexes such that D(i) = 1. Otherwise stated all the returns from training cells homogeneous to the CUT are exploited for setting the threshold. 11

20 CHAPTER 1. Knowledge Based CFAR Processing Static KB Selector (K SKB = K K S ) or {}}{ r K 1... K 2 r 2 K 2 KB selector {}}{ r cut... r 1 r K 1 Dynamic KB Selector (K DKB [K min, K MAX]) CA-CFAR r cut Figure 1.3: Block diagram showing the proposed two-stages receivers. if K i=1 D(i) < K min, then Ω is composed of the indexes such that D(i) = 1 (indexes of the homogeneous cells) and the K min K i=1 D(i) indexes of the nonhomogeneous cells closest to the CUT. The two-stage KB detectors, which exploit one of the proposed data selectors as the first stage and the CA-CFAR as the second stage, are represented in Figure 1.3 and will be referred to SKB-CFAR and DKB-CFAR processors. 1.4 Performance Analysis This section is devoted to the performance assessment of the new algorithms in the presence of real data (X-band sea clutter), collected by the McMaster IPIX radar in November First of all the description of the real dataset and the a- priori information employed for the analysis are described. Then the performance, in terms of CFAR behaviour and P d, is evaluated also in comparison with the classical CFAR schemes of Section Real Data and GIS map Radar measurements were collected in November 1993 using the McMaster IPIX radar from a site in Dartmouth [24], Nova Scotia, on the East Coast of Canada (see Figure 1.4a). The radar was mounted on a cliff facing the Atlantic Ocean, at a height of 100 feet above the mean sea level, and scans the site over 370 Deg in 10 seconds in a continuous azimuth scan mode. More details on the experiment can be found in [24, 25, 26]. The illuminated area ranges from the Atlantic Ocean, to the shoreline of Cow Bay, and to the lands near Halifax. The specifications of the considered dataset, containing mixed land and sea clutter, are reported in 12

21 1.4 Performance Analysis Table 1.1. The data are previously preprocessed in order to remove the DC offset Dataset13 Date November 6, 1993 Time Number of Pulses N t 8000 Number of Cells N s 184 Polarizations HH RF Frequency 9.39 Ghz Pulse Length 200 ns Pulse Repetition Frequency 800 Hz Radar Scan Mode Continuous Azimuth Scan Radar Azimuth Angles (in Deg) , ,..., Grazing Angles 3.8 to 0.32 Deg Range m Radar Beam Width 0.9 Deg Range Resolution 30 m Quantization Bit 8 bit Mean/Max Wind Speed 20 / 30 Km/h Sea State: Mean/Max wave height 3 / 5 m Table 1.1: Specifications of the considered dataset. (both of I and Q channels) and the phase imbalance due to hardware imperfections. For the analysis we select the first N s = 100 range cells and N t = 1800 temporal/azimuth samples ranging from 220 Deg to 303 Deg, namely an open view of 83 Deg over the region that includes both sea, land and the heterogeneous clutter composed by the shoreline of Cow Bay. We assess the performance of the algorithms in correspondence of the most heterogeneous range cells of the dataset, in order to analyze the worst conditions for the CFAR detectors. The 2-D clutter intensity field is plotted in Figure 1.4b: the strongest return are plotted in red while the weaker returns are plotted in blue. We do not possess a GIS representation for the region of Figure 1.4a. Nevertheless, in order to prove the effectiveness of the proposed KB receivers on real data, we have constructed with the aid of the geographic map of Figure 1.4a, a GIS database assuming that the size of the radar cell coincides with the size of the raster cell. By doing so we come up with the representation depicted in Figure 1.4c where there are three different attributes: the sea, marked by the blue raster cells, the homogeneous land, plotted in orange, and the heterogeneous clutter (red-colored, corresponding to the shoreline of Cow Bay). From the synthetic GIS map of Figure 1.4c the distribution of the different attributes along the analysed region can be observed 13

22 CHAPTER 1. Knowledge Based CFAR Processing Land: for all the analysed azimuths there are areas of homogenous land, especially in correspondence of range cells far from the radar site. Sea: there is the presence of a sea area in correspondence of range cells close to the radar site. This region is delimited by the Cow Bay beach which divides the sea from the Cow Bay lake, as shown in Figure 1.4a. Mixed Land and Sea: between the Cow Bay sea and the homogeneous land region, there is the presence of a highly nonhomogeneous area including the Cow Bay lake, the Mceas island, and all the areas between the shoreline of Cow Bay and the lands near Halifax Number of FAs This subsection is devoted to a CFAR analysis of the considered detectors. It is assumed that the size of the reference window is K = 32 and that the index of the CUT varies in the set [17,84]. The overall data window is slid in space from range bin to range bin and in azimuth from 220 Deg to 303 Deg. By doing so the total number of trials available for estimating the actual number of FAs is N trials = N t (N s K) = (1.7) The SKB-CFAR detector assumes K S = 16 whereas the DKB-CFAR processor considers K min = 4 and K MAX = 16. In the sequel P fa is fixed to 10 6 and, exploiting the theoretical CFAR property, the thresholds of the receivers are set under the hypothesis of spatially homogeneous Gaussian clutter. This is tantamount to assuming that the K + 1 returns of the complete data window (CUT plus training cells) are independent and identically distributed, zero-mean, complex, Gaussian random variables. Under this simulation setup, the actual Number of FAs N fa, for each analysed system, is evaluated. The results can be summarized as follows. In Figure 1.5 the Plan Position Indicator (PPI) of the CA-CFAR detector is shown. The black areas correspond to regions where the decision statistic is below the threshold, whereas the dots correspond to range and azimuth positions where the decision statistic crosses the threshold. For the quoted receiver N fa = 621. In Figure 1.6 the PPI of the GO-CFAR is shown. In this case the total number of FAs is N fa = 387, which, as expected, is lower than the number exhibited by the CA-CFAR. In Figure 1.7 the PPI of the SO-CFAR is shown. The total number of FAs is This number is greater than the one corresponding to the CA-CFAR 14

23 1.4 Performance Analysis b) c) Range Cells Range Cells Figure 1.4: Subplot a) Geographic map of the experiment site: the red line delimitates the selected area. Subplot b) 2-D intensity field of the mixed land and sea clutter live data: the strongest return are plotted in red while the weaker returns are in blue. Subplot c) GIS representation of the considered dataset: blue raster cells represent sea, orange raster cells represent homogeneous land and red raster cells represent mixed land and sea. 15

24 CHAPTER 1. Knowledge Based CFAR Processing and the GO-CFAR due to the lower robustness of the Smallest Of training strategy with respect to clutter power mismatches. In Figure 1.8 the PPI of the OS-CFAR is shown. The total number of FAs is 4078, namely the OS-CFAR performs (in terms of FAs) better than the SO-CFAR but worse than CA-CFAR and the OS-CFAR. In Figure 1.9 the PPI of the SKB-CFAR is shown. It indicates that, resorting to the SKB selection strategy, the total number of FAs can be reduced. Indeed, in this case, N fa = 203 highlighting the effectiveness of the KB pre-processing. In Figure 1.10 the PPI of the DKB-CFAR is shown. This receiver exhibits the lowest number of FAs, i.e. N fa = 82. In other words, in the presence of strong nonhomogeneous scenarios, adaptivity on the number of cells to be censored significantly helps the adaptive threshold setting. The previous results, summarized in Table 1.2, leads to the conclusion that all the considered systems are not able to maintain rigorously the theoretical FA rate when operate in the presence of heterogeneous measured clutter data. Nevertheless Detector N fa CA-CFAR 621 GO-CFAR 387 SO-CFAR 5957 OS-CFAR 4078 SKB-CFAR 203 DKB-CFAR 82 Table 1.2: N fa over N trials = for the analyzed CFAR detectors. the use of a-priori knowledge can noticeably improve the performance. In fact the new processors exhibit a number of FAs smaller than the classical CFAR schemes. 16

25 1.4 Performance Analysis Range CUT Figure 1.5: PPI of the CA-CFAR detector for azimuth samples ranging from 220 Deg to 303 Deg and range cells from 17 to 84: the total number of FA is

26 CHAPTER 1. Knowledge Based CFAR Processing Range CUT Figure 1.6: PPI of the GO-CFAR detector for azimuth samples ranging from 220 Deg to 303 Deg and range cells from 17 to 84: the total number of FA is

27 1.4 Performance Analysis Range CUT Figure 1.7: PPI of the SO-CFAR detector for azimuth samples ranging from 220 Deg to 303 Deg and range cells from 17 to 84: the total number of FA is

28 CHAPTER 1. Knowledge Based CFAR Processing Range CUT Figure 1.8: PPI of the OS-CFAR detector for azimuth samples ranging from 220 Deg to 303 Deg and range cells from 17 to 84: the total number of FA is

29 1.4 Performance Analysis Range CUT Figure 1.9: PPI of the SKB-CFAR detector for azimuth samples ranging from 220 Deg to 303 Deg and range cells from 17 to 84: the total number of FA is

30 CHAPTER 1. Knowledge Based CFAR Processing Range CUT Figure 1.10: PPI of the DKB-CFAR detector for azimuth samples ranging from 220 Deg to 303 Deg and range cells from 17 to 84: the total number of FA is

31 1.4 Performance Analysis Detection Performance This subsection is devoted to the analysis of the considered detection schemes in the presence of synthetic targets injected into the real dataset. To this end the cell 65 is chosen as the CUT and the surrounding cells ([49,64] [66,81]) as training data (i.e. K = 32). As to the KB processors they assume respectively K S = 16, K min = 4, and K MAX = 16. Under this simulation setup, the decision statistic of the analysed systems is plotted versus the azimuth together with the theoretical detection threshold ensuring P fa = By doing so the total number of tests is 1800 referring to the azimuth positions between 220 Deg and 303 Deg. Finally three synthetic non-fluctuating targets, with a Signal to Clutter power Ratio (SCR) equal to 15 db, are injected at the azimuth positions 247 Deg (target 1), 266 Deg (target 2) and 290 Deg (target 3). The results of the analysis can be summarized as follows In Figure 1.11 the decision statistic of the CA-CFAR is plotted versus the azimuth. The stars indicate the azimuth positions where the useful targets are present. The plot shows that only target 3 is over the threshold and is thus correctly detected. Moreover, around 280 Deg, there is a great concentration of FAs. Finally, in correspondence of the azimuth interval centered at 260 Deg, the decision statistic assumes very low values (about 40 db); this is due to the bad training which causes an over-nulling effect of the clutter in the CUT. In Figure 1.12 the decision statistic of the GO-CFAR is plotted versus the azimuth showing that, even in this case, only the target 3 crosses the threshold. Again, around 280 Deg, the decision statistic is often over the detection threshold. Finally the over-nulling effect of the clutter in the CUT is still markedly present. In Figure 1.13 the decision statistic of the SO-CFAR is plotted versus the azimuth showing all the targets are over the threshold. However, as observed in the previous subsection, the main drawback of the quoted detector is the number of FAs. Actually a significant number of threshold crossings are present in absence of useful target echos. In Figure 1.14 the decision statistic of the OS-CFAR is plotted versus the azimuth: target 2 and target 3 are detected but the decision statistic is over the threshold several times also in absence of useful target, especially around 280 Deg and 293 Deg. In Figure 1.15 the decision statistic of the SKB-CFAR detector is plotted versus the azimuth: target 1 is still below the threshold, while target 2 and target 3 are correctly detected. The figure also indicates that number of FAs 23

32 CHAPTER 1. Knowledge Based CFAR Processing around the azimuth 280 Deg significantly reduces as well as the over-nulling effect around 260 Deg. In Figure 1.16 the decision statistic of the DKB-CFAR detector is plotted versus the azimuth. In this case, due to the adaptive number of censored returns, the detection threshold is no longer constant. Actually the threshold value is ruled by the expression T CA = P 1 K fa 1, which clearly indicates that the higher the number of training data, exploited for evaluating the normalization factor in (1.1), the smaller the threshold. For the case at hand, it ranges between 1.3 db (corresponding to homogeneous regions, i.e. K MAX = 16) and 14.9 db (in correspondence of very nonhomogeneous regions, where the DKB algorithm decides to use K min = 4 training cells). The plot indicates that all the targets are over the threshold and thus correctly detected. Moreover the FAs around 280 Deg are now absent as the detection threshold increases in correspondence of the quoted azimuth position. Finally the very flat behaviour of the decision statistic indicates that the over-nulling effect of the clutter within the CUT is significantly reduced. In the last part of this subsection we evaluate the P d versus the SCR, setting the detection threshold of the receivers in order to guarantee the same N fa = 82. The total number of trials available for evaluating P d, for a given SCR value, is N trials = In Figure 1.17 the P d of the analysed detectors is compared. The plots highlight that the OS-CFAR and SO-CFAR exhibit the worst performance. For low/medium values of P d. the CA-CFAR and GO-CFAR achieve almost the same performance, but for high P d s, the former has a slight performance advantage over the latter. The DKB-CFAR achieves a better performance than the SKB- CFAR and both uniformly outperform the classical CFAR detection schemes. For P d = 0.5 the gain of the DKB-CFAR is 5.6 db over the SKB-CFAR, 11.6 db over the CA-CFAR, 12.0 db over the GO-CFAR, 14.2 db over the OS-CFAR, and 14.3 db over the SO-CFAR. In conclusion the analysis has clearly highlighted the advantages of KB training selection strategies, which not only keep as low as possible the number of FAs, but also improve the detection performance of classical CFAR processors. 24

33 1.4 Performance Analysis Test Statistic (db) Azimuth (Deg) Figure 1.11: Test Statistic versus the azimuth for the CA-CFAR (solid line). Starmarked values are in correspondence of the positions where the useful targets, with SCR = 15 db, are present. The dotted line represents the theoretical detection threshold for P fa =

34 CHAPTER 1. Knowledge Based CFAR Processing Test Statistic (db) Azimuth (Deg) Figure 1.12: Test Statistic versus the azimuth for the GO-CFAR (solid line). Starmarked values are in correspondence of the positions where the useful targets, with SCR = 15 db, are present. The dotted line represents the theoretical detection threshold for P fa =

35 1.4 Performance Analysis Test Statistic (db) Azimuth (Deg) Figure 1.13: Test Statistic versus the azimuth for the SO-CFAR (solid line). Starmarked values are in correspondence of the positions where the useful targets, with SCR = 15 db, are present. The dotted line represents the theoretical detection threshold for P fa =

36 CHAPTER 1. Knowledge Based CFAR Processing Test Statistic (db) Azimuth (Deg) Figure 1.14: Test Statistic versus the azimuth for the OS-CFAR (solid line). Starmarked values are in correspondence of the positions where the useful targets, with SCR = 15 db, are present. The dotted line represents the theoretical detection threshold for P fa =

37 1.4 Performance Analysis Test Statistic (db) Azimuth (Deg) Figure 1.15: Test Statistic versus the azimuth for the SKB-CFAR (solid line). Star-marked values are in correspondence of the positions where the useful targets, with SCR = 15 db, are present. The dotted line represents the theoretical detection threshold for P fa =

38 CHAPTER 1. Knowledge Based CFAR Processing Test Statistic (db) Azimuth (Deg) Figure 1.16: Test Statistic versus the azimuth for the DKB-CFAR detector (solid line). Star-marked values are in correspondence of the positions where the useful targets, with SCR = 15 db, are present. The dotted line represents the theoretical detection threshold for P fa =

39 1.4 Performance Analysis P d SCR (db) Figure 1.17: P d versus SCR of the CA-CFAR (solid curve), GO-CFAR (dash-dot curve), SO-CFAR (dotted curve), OS-CFAR (dashed curve), SKB-CFAR (dotmarked curve), and DKB-CFAR (plus-marked curve). The thresholds of all the receivers are set in order to guarantee N fa =

40 CHAPTER 1. Knowledge Based CFAR Processing 1.5 Conclusions In this chapter we have introduced and assessed two KB-CFAR detectors which exploit the a-priori information provided by a GIS about the topology of the illuminated environment. They are composed of two stages. The former is a KB data selector which chooses the most homogeneous returns exploiting the GIS information. The latter is a classical CFAR detector. At the analysis stage we have compared, on real radar data, the performance of the new detectors with that of conventional CFAR schemes. The results have highlighted that, even with few attributes and a low resolution GIS map, the a- priori knowledge can lead to significant performance improvements. Actually the introduced detectors ensure a CFAR behaviour and a detection performance better than the classical CFAR schemes. 32

41 Chapter 2 Design of a Knowledge-Based Radar Detector In this chapter we introduce and assess a KB radar detector composed of three elements: a GIS-based data selector, which eliminates static outliers from the secondary data exploiting the a-priori information concerning the topography of the observed scene, a data-adaptive training selector which removes dynamic outliers from the training data, and an adaptive radar detector which performs the final decision about the target presence. The performance of the new algorithm is analyzed both on simulated as well as on real radar data collected by the McMaster University IPIX radar. The results show that the new KB system achieves a satisfactory performance level and can outperform some previously proposed adaptive detection schemes. 2.1 Introduction In a typical adaptive radar detector [27] the disturbance covariance matrix is estimated using secondary data namely returns from range cells spatially close to the cell under test and sharing the same spectral properties. However training data are often contaminated by power variations over range (in addition to that induced by the radar equation), clutter discretes, and undesired or unwanted outlier signals of different types. For instance, outlier sources are [28] unintentional temporally sporadic electromagnetic interference, intentional blanking interference in the main channel of a sidelobe canceller, other desired target returns located in range cells surrounding the one under 33

42 CHAPTER 2. Design of a Knowledge-Based Radar Detector test (this situation may arise when a formation of aircrafts or maritime targets is present), signals at doppler (with reference to a doppler processing), angle (with reference to a spatial processing), or doppler-angle (for the case of STAP) locations different than that specified by the actual steering vector. In these situations the training data are no longer representative of the disturbance in the cell under test. As a consequence covariance matrix estimates from heterogeneous/outlier dense 1 environments result in a significant under-nulled interference which leads to a severe degradation of the detection performance as well as of the CFAR behaviour of adaptive radar detectors. It is thus of primary concern the design of selection procedures which excise outliers from the training data. Previous solutions to this challenging problem can be found in [29] where the Power Selected Training and the Power Selected Deemphasis, which use measurements of the interference environment to select training data, are described and analyzed also on recorded radar data. In [30], the Nonhomogeneity Detector (NHD) is introduced and assessed. Moreover, in [31], it is further analyzed with emphasis on its use as a preprocessing step for STAP algorithms, while, in [32], it is generalized to the case of small sample support introducing the concept of Innovation Inner Product (IIP). In [28], approximate Maximum Likelihood (ML) procedures, also re-iterative, are proposed and analyzed. Finally, in [33], the problem of secondary data selection in non-gaussian disturbance modeled as a Spherically Invariant Random Process (SIRP) is considered. The quoted algorithms are usually referred to as data-adaptive algorithms as the decision concerning the cells to be excised are driven by the data. Nevertheless data-independent methodologies such as the Sliding Window, Exponential Window, and Range Segmentation, have been also proposed and analyzed (see [34, 35] for a survey) showing that they usually result in inadequately nulled clutter [29]. Finally the use of a KB data selection procedure and its application to post-doppler radar processing is discussed in [36, 37, 38]. Since optimality cannot be claimed for all of the aforementioned procedures, in this report we still deal with the problem of secondary data selection. We design and assess algorithms capable of choosing the training vectors which better represent the disturbance from the cell under test. Precisely we propose a data-independent KB training selector which exploits the a-priori information provided by a GIS in order to choose the most homogeneous secondary cells; 1 With the term heterogeneous/outlier dense environment we indicate a radar scenario accounting for both the spatial variations of the clutter power as well as the presence of outlier signals. 34

43 2.2 Design of the GIS-aided Training Selector two data-adaptive selectors which estimate the most homogeneous secondary data subset resorting to the one-step and the two-step Generalized Likelihood Function (GLF). Otherwise stated, resorting to the GLF, we develop approximate ML estimates of the most homogeneous secondary data subset. Even if optimality cannot be claimed in any sense also for the proposed selection rules, the performance analysis shows that they are very effective in rejecting outliers. The benefits of employing both the selection strategies as a pre-processing step of an adaptive detection algorithm are highlighted. More precisely, in the last part of the chapter, we devise a KB detector exploiting the GIS-based selector, for the elimination of static outliers due to the environmental characteristics, one of the proposed data-adaptive selectors for the elimination of dynamic outliers (i.e. moving objects such as vehicles), and the Recursive Persymmetric Adaptive Normalized Matched Filter (RP-ANMF) [39] as the final stage performing the decision concerning the target presence. The chapter is organized as follows. In Section 2.2 we devise the GIS-aided data selector. In Section 2.3 we propose the aforementioned data-adaptive selection algorithms and analyze their performance on simulated radar data. In Section 2.4 we introduce the KB adaptive detector and, in Section 2.5, we assess its performance, also on measured radar data, highlighting the benefits arising from the joint use of both KB and data-adaptive pre-processing. Finally conclusions are given in Section Design of the GIS-aided Training Selector This section is devoted to the design of a data selector which exploits the a-priori information provided by a GIS and is capable of removing static outliers from the available secondary data set. The training selection algorithm we propose, assumes that the size of the radar cell is greater than or equal to the size of the raster cell and exploits the GIS information in the following way. Acquisition Step 1. The GIS provides the raster grids of the cell under test and of the K available secondary data. Acquisition Step 2. With reference to the cell under test, the selection algorithm counts the number of attributes and their Number of Raster Cells (NRC). Hence it builds a table which associates to each attribute the corresponding NRC. Moreover there exists an indifference threshold X a (expressed in percentage of NRC and depending on the specific attribute a) which allows to discard attributes whose NRC is smaller than X a NRC tot, where denotes the smallest integer greater than or equal to the argument 35

44 CHAPTER 2. Design of a Knowledge-Based Radar Detector and NRC tot is the NRC constituting a radar cell. Otherwise stated attributes with a significantly small area occupation with respect to the radar cell area are considered not significant. Finally, the attributes are sorted in descending order with reference to the NRC and L denotes the number of significant attributes. Acquisition Step 3. For each of the available secondary cells the algorithm constructs a table which associates the N RC to each attribute. A default zero value is written in correspondence of attributes which are present in the primary cell but are not in the considered secondary gate. Finally the secondary cells attributes are sorted according to the order induced by those of the primary cell. Selection Step 1. With reference to the first ordered attribute of the cell under test, the algorithm considers homogeneous all the secondary cells which exhibit an NRC (associated with that specific attribute, say NRC 1,k, k = 1,...,K), satisfying the following condition NRC 1,cut NRC1,cut NRC 1,k NRC 1,cut + NRC1,cut, (2.1) where NRC1,cut = X 1 NRC 1,cut, NRC 1,cut and X 1 are respectively the N RC and the indifference threshold associated with the first ordered attribute of the cell under test. The cells which do not comply with the quoted condition are considered non-homogeneous to that under test and, as a consequence, are discarded. Selection Steps 2-L. For all i = 2,...,L, the cells which pass the homogeneity test at the step (i 1) are processed with reference to the i-th primary cell ordered attribute. Precisely, the algorithm considers homogeneous all the cells which exhibit an N RC (associated with that specific attribute, say NRC i,k, k = 1,...,K), satisfying the following condition NRC i,cut NRCi,cut NRC i,k NRC i,cut + NRCi,cut, (2.2) where NRCi,cut = X i NRC i,cut, NRC i,cut and X i are respectively the N RC and the indifference threshold associated with the i-th ordered attribute of the cell under test. Selection Step L+1. For all the cells which are declared homogeneous at the step L, the algorithm focuses on those attributes which are present in the considered secondary cell but are not in the primary one. Precisely, for each of such attributes, if the characteristic NRC is greater than or equal to X a NRC tot, then the corresponding secondary cell is discarded. 36

45 2.2 Design of the GIS-aided Training Selector We explicitly highlight that simplified versions of the proposed GIS-aided selection algorithm can be also conceived. For instance one can think to implement only L s < L + 1 steps of the aforementioned procedure which is tantamount to considering only the most relevant L s attributes of the primary cell. In the limiting case a selection can be performed only based on the most significant attribute of the cell under test. Clearly the higher the number of considered attributes the higher the accuracy of the selection procedure. Finally one can also think to stop the algorithm just after that a given number of cells M GIS are classified as nonhomogeneous, providing also a top priority of excision to some specific attributes such as strong clutter discretes An Illustrative Example Consider the scenario of Figure 2.1 where we assume a total number of training cells equal to 4, hence five cells are available including the one under test. In the same figure we also display the raster grids provided during the acquisition step 1 by the GIS (each radar cell is represented by 30 raster cells), the attributes, their labels, and their indifference thresholds. According to the acquisition step 2, in Table 2.1 we report, the attributes and the corresponding NRC of the primary cell. These last values are compared with the number reported in the last column Cell Under Test Attribute NRC X a NRC tot Table 2.1: NRC and X a NRC tot for all the primary cell attributes. of the table namely with the smallest integer greater than the product between the indifference threshold and the total NRC constituting the radar cell. It follows that the attributes 0,1,2 and 3 are significant, 0 is the most significant, and the total steps of the selection algorithm are 5. Finally they are sorted in descending order with reference to the NRC. The attributes of the secondary cells and their NRC are given in Table 2.2 where the default zero value is introduced in correspondence of the attributes which are present in the primary cell but are not in that specific secondary gate (acquisition step 3). Selection Step 1. The attribute 0 is the first ordered attribute of the cell under test and thus NRC 1,cut = 12. Moreover, since X 1 = 24% 37

46 CHAPTER 2. Design of a Knowledge-Based Radar Detector Figure 2.1: Illustrative example showing the working principle of the KB GISaided data selector. 38

47 2.3 Design and Analysis of the Data-Adaptive Training Selectors NRC Attribute Secondary Secondary Secondary Secondary Cell 1 Cell 2 Cell 3 Cell Table 2.2: NRC for all the attributes of the secondary cells. then NRC1,cut = 3 and the algorithm considers homogeneous all the secondary cells which exhibit NRC 1,k, k = 1,...,4, satisfying the condition 9 NRC 1,k 15. It follows that the cells 1, 2, and 4 are classified as homogeneous to that under test. Selection Step 2. The attribute 1 is the second ordered attribute of the cell under test as NRC 2,cut = 7. Moreover, since X 2 = 20% then NRC2,cut = 2 and the algorithm considers homogeneous all the secondary cells (survived at the previous step) which exhibit NRC 2,k, k = 1,...,4, complying with the condition 5 NRC 1,k 9. It follows that the cells 1 and 4 are classified as homogeneous to that under test. Selection Steps 3 and 4. Repeating the procedure with reference to the third and fourth ordered primary cell attributes we have that the cells 1 and 4 can be still considered homogeneous to that under test. Selection Step 5. For the case at hand this step is not necessary as there aren t attributes which are present in a given secondary cell but are not in the primary one. In conclusion the GIS-aided algorithm classifies the cells 1 and 4 as homogeneous to the cell under test. 2.3 Design and Analysis of the Data-Adaptive Training Selectors The aim of this section is the design of data-adaptive selection techniques capable of detecting the presence of outliers within the secondary data. To this end, denote by r i, i = 1,...,K, the N-dimensional complex vector of the samples from 39

48 CHAPTER 2. Design of a Knowledge-Based Radar Detector baseband equivalent of the received signal from the i-th range cell and assume that r i = c i i Ω Ω 0 (2.3) r i = p i + c i i Ω 0 where Ω = {1,...,K} is a set of size K and Ω 0 = {i 1,...,i M } is a subset of Ω with distinct elements and of size M (see Figure 2.2). c 1,..., c K, are independent, zero-mean, complex, circular, Gaussian, random vectors sharing the same covariance structure M assumed to be positive definite, i.e. E[ c i c i ] = σ2 i M i = 1,...,K, (2.4) where, in turn, E[ ] denotes statistical expectation, ( ) conjugate transpose, and σi 2 the power level (possibly random) of the i-th random vector. p i s are unknown, possibly random, N-dimensional complex vectors, representing outliers, namely discretes, non-homogeneities, clutter edges, or multiple targets. i Ω 0 they contaminate the c i s, producing vectors r i s with a covariance structure very different from M. Otherwise stated, assuming that the p i s are statistically independent of the c i s the covariance of r i can be written as E[ r i r i ] = σ2 i M + E[ p i p i ] = σ2 i M i, i Ω 0, (2.5) and the corresponding structure M i = M + E[ p i p i ] σ 2 i (2.6) may deeply deviate from M. We consider the problem of selecting the vectors sharing the same covariance structure, namely whose indexes belong the set Ω Ω 0. To this end, in the sequel, we devise two distinct procedures One-Step GLF Procedure In this case we model M, σ1,...,σ 2 K 2, p i 1,..., p im as unknown quantities and observe that the ML estimate of Ω 0 is the solution to the problem [ Ω 0 = arg max Ω0 max M max σ 2 1,...,σ 2 max p K i1,..., p im f( r 1,..., r K M,σ1,...,σ 2 K 2, p i 1,..., p im ) ] (2.7) 40

49 2.3 Design and Analysis of the Data-Adaptive Training Selectors Ω 1 K M }{{} Ω 0 Figure 2.2: Diagram showing Ω and Ω 0. 41

50 CHAPTER 2. Design of a Knowledge-Based Radar Detector where f( r 1,..., r K M,σ1,...,σ 2 K 2, p i 1,..., p im ) is the multivariate probability density function (pdf) of r 1,..., r K and arg max Ω0 ( ) denotes the set between K the subsets of Ω with distinct elements and of size M which maximizes M the argument. Thus we remove from the set r 1,..., r K the outliers namely the vectors whose indexes belong to Ω 0. The remaining K M vectors represent the ML estimate of data sharing the same covariance structure. Unfortunately the GLF max M max σ 2 1,...,σ 2 K max p i1,..., p im f( r 1,..., r K M, σ 2 1,...,σ 2 K, p i 1,..., p im ) (2.8) is unbounded and hence the procedure does not lead to any significant solution. In order to circumvent this drawback it is necessary to introduce constraints on the parameter space which avoid unbounded GLF. To this end we notice that a physically grounded condition relies on the assumption that the disturbance power levels σ 2 i s cannot take values smaller than the noise floor of the receiver σ2. This last condition implies that the maximum over the generic σ 2 i must be performed within the set Ξ = [σ 2,+ ) and leads to the following criterion for selecting Ω 0 Ω 0 = arg max Ω0 [ max M max (σ 2 1,...,σ 2 K ) ΞK max p i1,..., p im f( r 1,..., r K M,σ 2 1,...,σ 2 K, p i 1,..., p im ) ] (2.9) The solution of (2.9) requires specifying the conditional multivariate pdf of r 1,..., r K. To this end, exploiting the statistical independence of c 1,..., c K, in conjunction with the circular property, we can write f( r 1,..., r K M,σ 2 1,...,σ 2 K, p i1,..., p im ) = 1 π N σ i Ω i 2N 0 1 π N σ i Ω Ω i 2N 0 [ ( det( M) exp r i p i ) M 1 [ det( M) exp r i M 1 σ 2 i σ 2 i r i ], ] ( r i p i ) (2.10) where det( ) denotes the determinant of a square matrix. Maximizing (2.10) over p i, i Ω 0, yields max f( r p i1,..., p 1,..., r K M,σ1,...,σ 2 K, 2 p i1,..., p im ) = im 1 π N σ 2N i Ω 0 i det( M) 1 π N σ i Ω Ω i 2N 0 [ det( M) exp 42 r i M 1 σ 2 i r i ], (2.11)

51 2.3 Design and Analysis of the Data-Adaptive Training Selectors Maximizing, i Ω 0, (2.11) over σ 2 i Ξ leads to max max f( r (σi 2,...,σ 2 ) Ξ 1 i M p i1,..., p 1,..., r K M,σ1,...,σ 2 K, 2 p i1,..., p im ) = im M 1 π NM σ 2NM det M ( M) 1 π N σ i Ω Ω i 2N 0 [ det( M) exp r i M 1 σ 2 i r i ] = g( r 1,..., r K M,σi 2,i Ω Ω 0 ). (2.12) It remains to maximize (2.12) over σi 2 Ξ, i Ω Ω 0, and M. Unfortunately it seems that the quoted problem cannot be solved in closed form. Thus we propose to approximate the maximization through a recursive algorithm capable of increasing the value that the function (2.12) attains at pre-assigned (initial) value of M. To this end, assume K M N and denote by σi 2(t), i Ω Ω 0, and M (t) the values of σi 2 and M, respectively, at the t-th recursion. Then, calculate σi 2(t+1) and M (t+1) at the step t + 1 as σ 2(t+1) i = arg max σ 2 i Ξ g( r 1,..., r K M (t),σ 2 i,i Ω Ω 0 ) = ( max σ 2, 1 ( N r i M (t)) ) (2.13) 1 ri where max (, ) denotes the maximum of the arguments, and M (t+1) =arg max M g( r 1,..., r K M,σ 2 (t+1) i,i Ω Ω 0 ) = 1 K i Ω Ω 0 r i ri. σi 2(t+1) (2.14) In other words, i Ω Ω 0, σi 2(t+1) is obtained maximizing the function g( ) with respect to σi 2 when M is equal to M(t), as stated by (2.13). Similarly, M (t+1) is obtained maximizing the function g( ) over M, assuming that i Ω Ω 0 σi 2 = σ2(t+1) i, according to (2.14). Working as in [40], it can be shown that the above introduced procedure increases the function g( ). Otherwise stated the sequence g( r 1,..., r K M (t),σ 2(t) i,i Ω Ω 0 ) (2.15) increases as t increases. In conclusion the devised selection algorithm can be summarized as follows. One-Step Data Selection Procedure (1S-DSP). For each element Ω 0 43

52 CHAPTER 2. Design of a Knowledge-Based Radar Detector which belongs to the set composed of the elements and of size M K M subsets of Ω with distinct 1. Select the data vectors whose indexes don t belong to Ω Starting from the selected vectors construct the initial estimate of the covariance structure through a normalized sample covariance matrix [41], i.e. 2 N K M i Ω Ω 0 where denotes the Euclidean norm. r i ri r i 2. (2.16) 3. Apply the recursive maximization procedure to the function g( ) and denote by g Nit ( ) its value after N it iterations. The K M dimensional set of vectors ensuring the highest value of g Nit ( ) is chosen as the most homogeneous (in the sense of sharing the same covariance structure). The main drawback of the proposed selection method is the computational K complexity because it requires applying times the recursive maximization M procedure which involves O(N it KN 2 ) floating point operations (flops) 3. It is thus of interest envisaging sub-optimum implementations of the selection algorithm which permit a more affordable computational burden. A possible solution toward this goal is to apply repeatedly the procedure for the removal of a single outlier. Precisely the modified procedure can be implemented as follows. One-Step Modified Data Selection Procedure (1S-MDSP). Let Ψ = { r 1,..., r K }. For j = 1 to M 1. Apply the DSP for the removal of a single outlier to the set Ψ and denote by i j the index of the data vector discarded. 2. Set Ψ = Ψ { r ij }. By doing so we come up with a set Ψ composed of K M data vectors which are chosen, with reference to the covariance structure, as the most homogeneous. 2 Notice that we exploit this covariance estimator as it is functionally independent of the power levels σi 2. Indeed they factor out between the numerator and the denominator of the estimator. The plain sample covariance matrix does not exhibit this property. 3 Herein we use the usual Landau notation O(n); hence, an algorithm is O(n) if its implementation requires a number of flops proportional to n [42]. 44

53 2.3 Design and Analysis of the Data-Adaptive Training Selectors We explicitly point out that the sub-optimum nature of the modified algorithm stems from the fact that it does not account for the alteration in the statistics of Ψ after the removal of an outlier. A further simplification arises if ( ( Pr r i M (t)) ) 1 ri Nσ 2 0 i = 1,...,K, where Pr( ) denotes the probability of the argument. In fact, in this case, we can approximate the maximum in (2.13) with the second argument. Hence the procedure for eliminating a single outlier reduces to the construction of the quantities δ i = det j=1,j i r j r j σ 2(t) j K N K j=1,j i 1 σ 2(t) j (2.17) i = 1,...,K, and the excision of the cell ensuring the highest value of δ i. The simplified algorithm allows for a significant reduction of the computational complexity, namely it requires O(N it MK 2 N 2 ) because it involves the evaluation of (2.17) M times. Finally simulations show that N it = 1 represents a good compromise between complexity and performance (this value will be adopted in the sequel for the performance analysis) Two-Step GLF Procedure This design technique is based on two steps. In the former we assume that M is known and derive the ML estimate of Ω 0 modeling σ 2 1,...,σ 2 K, p i 1,..., p im as unknown parameters. Then, in the second step, a suitable estimate of M is substituted in place of its exact value. Step 1. For known M the ML estimate of Ω 0 is the solution to the problem Ω 0 = arg max Ω0 [max σ 2 1,...,σ 2 K max p i1,..., p im f( r 1,..., r K M,σ 2 1,...,σ 2 K, p i 1,..., p im ) ]. (2.18) Maximizing over p i i Ω 0, yields max f( r p i1,..., p 1,..., r K M,σ1,...,σ 2 K, 2 p i1,..., p im ) = im 1 π N σ 2N i Ω 0 i det( M) 1 π N σ i Ω Ω i 2N 0 [ det( M) exp r i M 1 σ 2 i r i ]. (2.19) 45

54 CHAPTER 2. Design of a Knowledge-Based Radar Detector Moreover the maximization, i Ω 0, of (2.19) over σ 2 i Ξ leads to max max f( r (σi 2,...,σ 2 ) Ξ 1 i M p i1,..., p 1,..., r K M,σ1,...,σ 2 K, 2 p i1,..., p im ) = im M 1 π NM σ 2NM det M ( M) 1 π N σ i Ω Ω i 2N 0 [ det( M) exp r i M 1 σ 2 i r i ]. (2.20) It still remains to maximize (2.20) over σ 2 i Ξ, i Ω Ω 0. After some algebraic manipulations this last maximization yields 1 π NM σ 2NM det M ( M) [ 1 i Ω Ω 0 π N σ 2N i det( M) exp r i M 1 σ i 2 r i ], (2.21) where σ 2 i = max (σ 2, 1N ) r i M 1 r i. (2.22) As a consequence Ω 0 = arg max Ω 0 { 1 2N i Ω Ω 0 σ exp i [ r i M 1 σ i 2 r i ] }. (2.23) Step 2. The second step is tantamount to substituting a suitable estimate of M in place of its exact value. To this end we exploit the normalized covariance matrix estimator [41], i.e. N M = K i Ω r i r i r i 2. (2.24) As a consequence the devised procedure can be implemented as follows Two-Step Data Selection Procedure (2S-DSP) 1. i Ω construct the quadratic forms ξ i = r i M 1 r i (Generalized Inner Product (GIP) with normalized sample covariance matrix) and sort them in decreasing order. 2. The set Ψ, composed of the K M most homogeneous vectors, can be obtained censoring from Ω the M vectors characterized by the highest values of ξ i. The computational complexity of the 2S-DSP is O(KN 2 ) which is significantly smaller than that connected with the implementation of the 1S-MDSP. We also highlight that the proposed procedure resembles that devised in [28] with the main and fundamental difference that the normalized sample covariance matrix is employed in place of the conventional sample covariance. 46

55 2.3 Design and Analysis of the Data-Adaptive Training Selectors Before concluding this section we highlight that the proposed algorithms assume the knowledge of M, namely the number of cells to be censored. This might stem from KB criteria [2], which depending on the type of operational environment, choose how many cells are to be excised Performance Analysis This section is devoted to the performance assessment of the new algorithms in the presence of simulated data also in comparison with both the NHD of [30] and the Reiterative NHD (R-NHD) introduced in [28]. The performance is evaluated in terms of the probability P r of correctly rejecting the outliers. At the analysis stage we model the σi 2 s as statistically independent Gamma random variables with the same shape parameter ν and mean square value σ 2. This is tantamount to modeling c i as Spherically Invariant Random Vectors with K-distributed amplitude pdf (apdf) [43]. As to M we assume an exponentially shaped covariance with 1-lag correlation coefficient ρ = Moreover we inject equal power (non-fluctuating) outliers in N out range cells. Each outlier shares the following temporal steering vector p i = α N [1,exp(j2πf d ),...,exp(j2π(n 1)f d )] T, (2.25) i = 1,...,N out, where ( ) T denotes transpose, j = 1, f d is the normalized doppler frequency, and α accounts for the outlier amplitude. Then we apply the four considered algorithms to the input data vectors, using M N out. Due to the lack of a closed form expression for the P r, the performance analysis is conducted resorting to Monte Carlo simulations. Nevertheless it can be shown that K N out lim P r = OCR 0 M N out K M where OCR denotes the Outlier to Clutter power Ratio, given by (2.26) OCR = α 2 σ 2, (2.27) and represents the modulus of a complex number. In Figure 2.3 the P r is plotted versus OCR for N = 8, K = 20, M = 4, N out = 1, and ν = 0.5. As to f d two situations are considered: The former assumes f d = 0.05 namely an outlier embedded in deep clutter. The latter refers to f d = 0.15, namely the outlier normalized doppler frequency is in the tail of the 47

56 CHAPTER 2. Design of a Knowledge-Based Radar Detector clutter Power Spectral Density (PSD). The curves highlight that for f d = 0.15 the 1S-MDSP achieves the best performance, and 2S-DSP is outperformed by the R-NHD and the NHD. On the contrary when f d = 0.05 the 2S-MDSP ensures the best performance and both the new selection rules outperform the R-NHD and the NHD for P r values in the medium high range. In Figure 2.4 the same simulation setup of Figure 2.3 is assumed with the only difference that N out = 2. In this case the new algorithms achieve an effective outliers rejection while the R-NHD and the NHD suffer a saturation effect both for f d = 0.05 and f d = Moreover the saturation becomes more and more pronounced when N out increases. This is shown in Figure 2.5 where N out = M = 4. It is an evidence that the performance of the R-NHD and the NHD are not very satisfactory in correspondence of both the considered values of f d. Finally the simulation results reported in Figure 2.6 have highlighted that the smaller ν (i.e. the higher the spiky nature of the clutter) the smaller the P r value where the saturation occurs. 48

57 2.3 Design and Analysis of the Data-Adaptive Training Selectors f d =0.15 f d = P r OCR (db) Figure 2.3: P r versus OCR of the 1S-MDSP (solid curve), 2S-DSP (dot-marked curve), R-NHD (dashed curve) and NHD (dotted curve) for N = 8, K = 20, M = 4, N out = 1, ν = 0.5, and two values of f d. 49

58 CHAPTER 2. Design of a Knowledge-Based Radar Detector P r f d =0.15 f d = OCR (db) Figure 2.4: P r versus OCR of the 1S-MDSP (solid curve), 2S-DSP (dot-marked curve), R-NHD (dashed curve) and NHD (dotted curve) for N = 8, K = 20, M = 4, N out = 2, ν = 0.5, and two values of f d. 50

59 2.3 Design and Analysis of the Data-Adaptive Training Selectors P r f d = f d = OCR (db) Figure 2.5: P r versus OCR of the 1S-MDSP (solid curve), 2S-DSP (dot-marked curve), R-NHD (dashed curve) and NHD (dotted curve) for N = 8, K = 20, M = 4, N out = 4, ν = 0.5, and two values of f d. 51

60 CHAPTER 2. Design of a Knowledge-Based Radar Detector a) P r f d = f d = OCR (db) b) 0.8 f d = f d =0.05 P r OCR (db) Figure 2.6: P r versus OCR of the 1S-MDSP (solid curve), 2S-DSP (dot-marked curve), R-NHD (dashed curve) and NHD (dotted curve) for N = 8, K = 20, M = 4, N out = 2, and two values of f d. Subplot a) ν = Subplot b) ν = 5. 52

61 2.4 Design Issues Discussion In this section we have devised and assessed two data-adaptive selection rules based on the theory of the GLF. They achieve an effective outlier rejection and, usually, the 1S-MDSP algorithm performs better than the 2S-DSP procedure when the outlier doppler is located in the tail of the clutter PSD. Moreover the 2S-DSP requires a smaller computational complexity than the 1S-MDSP for its implementation. Finally both the proposed algorithms outperform the R-NHD and the NHD in the presence of operational scenarios with more than two outliers. 2.4 Design Issues The system we consider is depicted in Figure 2.7 and is composed of three distinct blocks the KB GIS-based data selector, introduced in Section 2.2, exploiting the raster representation of the GIS data for the excision of static outliers; the data-adaptive selection algorithm 2S-DSP, devised in Section 2.3, whose main goal is the excision of secondary data containing dynamic outliers; the detector RP-ANMF, devised in [39], whose expression is also reported in the sequel for completeness. Precisely, let r e k = 1 2 ( r k + J r k) and r o k = 1 2 ( r k J r k), where ( ) denotes complex conjugate and J is the permutation matrix, i.e J = Define the recursive persymmetric covariance structure estimator at the (t + 1)-step as follows Σ (t+1) = N K K k=1 r e k r e k ) 1 (2.28) re k r e k ( Σ(t) 53

62 CHAPTER 2. Design of a Knowledge-Based Radar Detector removes static removes dynamic outliers outliers MAP Selector 2S-DSP Selector K 2 {}}{ r 1 K 2 {}}{... r... r K K M MAP M 2S secondary data RP-ANMF r primary data Figure 2.7: Block scheme of the proposed KB radar detector. initializing the recursion with 4 Σ (0) = 1 K K r e k r ( e k T r o k r o k T ) {i} {i} k=1 (2.29) with T = (I + J) + j(i J) and I the identity matrix. The RP-ANMF is obtained substituting (2.28) in place of Σ in the following decision rule p Σ 1 r 2 ( r Σ 1 r) ( p Σ 1 p) H 1 > < H 0 T (2.30) where r is the primary data (namely the data vector where the presence of the useful target is sought), p the target temporal steering vector, and T is the detection threshold. 2.5 Performance Analysis This section is devoted to the performance assessment of the new algorithm in the presence of both simulated and real radar data (X-band mixed sea and land clutter), collected by the McMaster University (Canada) IPIX radar in November Precisely we focus on highly heterogeneous scenarios (accounting for the presence of spiky/heterogeneous clutter, static and dynamic outliers) and assess the performance of the introduced KB processor also in comparison with the plain RP-ANMF and the RP-ANMF exploiting the 2S-DSP algorithm only. For all the 4 A {i} {i}, i = 1,..., N, denotes the (i, i)-th element of the N N matrix A. 54

63 2.5 Performance Analysis considered system configurations we set the detection threshold resorting to Monte Carlo simulations assuming homogeneous white Gaussian disturbance. Finally, in order to limit the computational burden, we consider a nominal P fa = Simulated Data In the simulated scenario the clutter powers from different range cells are still modeled as statistically independent Gamma random variables with shape parameter ν = 0.5. The clutter covariance matrix is exponentially shaped with 1-lag correlation coefficient ρ = Equal power (non-fluctuating) outliers, whose steering vector is given by (2.25) are injected in N out = 4 range cells. Two of them are static, namely with f d = 0, and the remaining two are dynamic with f d = 0.25 and f d = 0.35 respectively. Finally the OCR has been chosen equal to 15 db. In Figure 2.9 the RP-ANMF detector is used without any training selection strategy (block scheme of Figure 2.8). By doing so, both the static as well as the dynamic outliers contaminate the training set. The number of available secondary data is K = 38 and, since no cells are excised, the size of the actual training set is still equal to 38. In subplot a) of Figure 2.9 the actual P fa is plotted versus the normalized doppler frequency f d. The curve shows that there is a significant mismatch between the nominal P fa (equal to 10 4 ) and its actual value, especially for the doppler frequencies around f d = 0 and f d = 0.3. This is due to the wrong covariance matrix estimate that leads to over-null the doppler frequencies where the outliers are present. A similar behavior can be seen in subplot b) where the P d is plotted versus the normalized doppler frequency f d for three different values of the SCR, i.e. SCR = α tg 2 p M 1 p, (2.31) with α tg a complex scalar accounting for the target reflectivity. Otherwise stated the presence of the outliers strongly affects the detection performance, around f d = 0 and especially around f d = 0.3, where the P d is close to 0 for all the considered SCR values. In Figure 2.11 the performance of the RP-ANMF detector exploiting the 2S-DSP data selector (block scheme of Figure 2.10) is studied. Therein we still assume K = 38 and that the 2S-DSP selector removes 4 training cells, namely 5 M 2S = 4. By doing so the size of the actual training set exploited for covariance matrix estimation is K M 2S = 34. Subplot a) shows that the actual P fa is equal to 10 4 but for the doppler frequencies around f d = 0. This behavior can be explained observing that the adopted data-adaptive 5 In the sequel we denote by M 2S the number of secondary cells excised by the 2S-DSP selector. 55

64 CHAPTER 2. Design of a Knowledge-Based Radar Detector selector with high probability rejects the dynamic outliers but the static outliers are not effectively eliminated. Similar considerations can be done with reference to subplot b) where the detection performance degrades in the region around f d = 0 due to the presence of the static outliers. In Figure 2.12 the performance of the proposed KB system (block scheme of Figure 2.7) is analyzed assuming K = 38, M 2S = 4 and M GIS = 2 (M GIS is the number of cells excised by the GIS-based data selector). This setup implies that the size of the actual training set used for covariance matrix estimation is K M 2S M GIS = 32. Subplot a) highlights that the actual P fa is practically equal to the nominal value 10 4 for all the doppler frequencies. This behavior is confirmed by the subplot b) showing that the P d exhibits a flat behavior too. Otherwise stated both the static and the dynamic outliers are successfully censored. In conclusion the conducted analysis (with simulated data) clearly shows the benefits of the proposed KB strategy. 56

65 2.5 Performance Analysis K 2 K 2 K secondary data {}}{{}}{ r 1... r... r K RP-ANMF r primary data Figure 2.8: Block scheme of the plain RP-ANMF. a) 10 2 P fa P d b) f d SCR=20dB SCR=15dB 0.2 SCR=10dB f d Figure 2.9: P fa and P d versus the normalized doppler frequency f d for the plain RP-ANMF detector with N = 8, K = 38, N out = 4, and ν =

66 CHAPTER 2. Design of a Knowledge-Based Radar Detector K 2 {}}{ r 1 K 2 {}}{ removes dynamic outliers 2S-DSP Selector K M 2S secondary data... r... r K RP-ANMF r primary data Figure 2.10: Block scheme of the RP-ANMF exploiting the 2S-DSP training selector. a) 10 2 P fa b) f d SCR=20dB P d SCR=10dB SCR=15dB f d Figure 2.11: P fa and P d versus normalized doppler frequency f d for the RP- ANMF detector exploiting the 2S-DSP selection algorithm. N = 8, K = 38, N out = 4, M 2S = 4, and ν =

67 2.5 Performance Analysis a) 10 2 P fa b) f d 0.8 SCR=20dB P d SCR=10dB SCR=15dB f d Figure 2.12: P fa and P d versus normalized doppler frequency f d for the proposed KB detector. N = 8, K = 38, N out = 4, M 2S = 4, M GIS = 2, and ν =

68 CHAPTER 2. Design of a Knowledge-Based Radar Detector Real Data Radar measurements were collected in November 1993 using the McMaster IPIX radar from a site in Dartmouth [24], Nova Scotia, on the East Coast of Canada (see Figure 2.13a). The radar was mounted on a cliff facing the Atlantic Ocean, at a height of 100 feet above the mean sea level, and scans the site over 370 Deg in 10 seconds in a continuous azimuth scan mode. More details on the experiment can be found in [24, 25, 26]. The illuminated area ranges from the Atlantic Ocean, to the shoreline of Cow Bay, and to the lands near Halifax. The specifications of the considered dataset, containing mixed land and sea clutter, are reported in Table 1.1. The data are previously preprocessed in order to remove the DC offset (both of I and Q channels) and the phase imbalance due to hardware imperfections. The 2-D clutter intensity field is plotted in Figure 2.13b showing the presence of two different sectors From 71 to 220 Deg there is an open view of the Atlantic Ocean. From 221 to 303 Deg there is a highly inhomogeneous region including the shoreline of Cow Bay and the lands near Halifax. In this environment we inject equal power (non-fluctuating) outliers, with OCR=15 db and whose range and azimuth positions are randomly chosen. Precisely we introduce 30 static outliers (f d = 0) representing strong clutter discretes, 15 dynamic outliers with f d = 0.25, and 15 dynamic outlier with f d = 0.35 accounting for the presence of moving objects. For the analysis we select N t = 5000 temporal/azimuth samples ranging from 71 Deg to 303 Deg namely an open view of 232 Deg. Then we assess the performance of the algorithms in correspondence of the most heterogeneous range cells of the dataset in order to analyze the worst conditions for the training selection algorithms. More precisely we choose the range cell 65 as that under test and exploit K = 38 surrounding cells ([46,64] [66,84]) as training data. We apply the statistical tests for all the available azimuth positions using N = 8 temporal/azimuth samples. By doing so the total number of tests is 625. We do not possess a GIS representation for the region of Figure 2.13a. Nevertheless in order to prove the effectiveness of the proposed algorithm on real data we have constructed with the aid of the geographical map of Figure 2.13a and the 2-D clutter intensity field of Figure 2.13b a low resolution GIS map assuming that the size of the radar cell coincides with the size of the raster cell. Otherwise stated each radar cell is represented by only one attribute. By doing so we come up with the representation depicted in Figure 2.14 where there are four different attributes: the sea, marked by the blue raster cells, the homogeneous land, plotted in orange, the heterogeneous land and sea (red-colored, corresponding to the shoreline of Cowbay) and the strong clutter discretes (static outliers) represented by the white raster cells. As to the GIS-based selection algorithm we suppose that 60

69 2.5 Performance Analysis it removes from the training only M GIS = 2 cells giving a top priority of excision to the cells containing strong clutter discretes (white raster cells). We inject three synthetic targets with SCR = 15 db and zero-doppler frequency at the azimuths 137 Deg (target 1, embedded in sea clutter), 223 Deg (target 2, embedded in mixed land and sea clutter) and 252 Deg (target 3, embedded in land clutter). In Figure 2.15 the decision statistic is plotted versus the azimuth. The straight line for each subplot is in correspondence of the threshold set in order to achieve P fa = The stars indicate the azimuth positions where the useful targets are present. In the subplot a) the RP-ANMF detector is employed without any training selection procedure. In this situation only the target 3 is over the threshold and is thus detected. In the subplot b) the K = 38 secondary data are pre-processed by the 2S-DSP selector block, with M 2S = 4, leading to K M 2S = 34 available cells: in this situation the target 1 is also detected, while the target 2, embedded in the highly heterogeneous clutter, is still below the threshold. In the subplot c) the K tot = 38 secondary data are pre-processed by both the 2S-DSP selector block (with M 2S = 4) as well as by the GIS-aided data selector (with M GIS = 2 and exploiting the information of the Figure 2.14), providing K M 2S M GIS = 32 available cells. In this last case all the targets are over the threshold. 61

70 CHAPTER 2. Design of a Knowledge-Based Radar Detector Figure 2.13: Subplot a) Geographic map of the experiment site. Subplot b) 2-D intensity field of the mixed land and sea clutter live data. The strongest returns, colored in red, represent the outliers (both static and dynamic) range cells time/azimuth samples Figure 2.14: GIS representation of the considered dataset: blue raster cells represent sea, orange raster cells represent homogeneous land, red raster cells represent mixed land and sea, and white raster cells represent static outliers. 62

71 2.5 Performance Analysis 0 a) 10 Test Statistic (db) b) c) Azimuth (Deg) Figure 2.15: Test Statistic versus the azimuth for N = 8 and K = 38. Starmarked values are in correspondence of the positions where the useful targets with SCR = 15 db are present. Subplot a) RP-ANMF detector without training selection. Subplot b) RP-ANMF detector exploiting the 2S-DSP selection algorithm (with M 2S = 4). Subplot c) RP-ANMF detector exploiting the 2S-DSP selection algorithm (with M 2S = 4) and with the GIS-aided data selector (with M GIS = 2). 63

72 CHAPTER 2. Design of a Knowledge-Based Radar Detector 2.6 Conclusions In this chapter we have designed a KB signal processor for coherent radar detection of targets embedded in heterogeneous/outlier dense environments. To this end we have first introduced a GIS-aided data selector which exploits the a-priori information provided by a GIS in order to select the most homogeneous training data from a topographical point of view, thus removing strong environmental non-homogeneities. Moreover we have devised and assessed two data-adaptive secondary data selectors which choose the most homogeneous secondary data according to the statistical ML criterion. Hence we have synthesized a three stage KB radar detector which resorts to the GIS-aided data selector for the excision of static outliers and to one of the proposed data-adaptive algorithm for the elimination of dynamic outliers. The RP-ANMF detector is employed after this pre-processing as a final stage for the decision about the signal presence. The performance of the new signal processor has been analyzed both on simulated as well as on measured radar data, collected by the McMaster IPIX radar in The results have confirmed that the joint use of the a-priori information concerning the topography of the observed scene in conjunction with a data-adaptive selection algorithm is a suitable mean to deal with scenarios including environmental non-homogeneities and strong outlier signals. 64

73 Chapter 3 Knowledge-Based Clutter Suppression In this chapter we deal with the design of KB adaptive algorithms for the cancellation of heterogeneous clutter. To this end we revisit the application of the Recursive Least Squares (RLS) technique for the rejection of unwanted clutter and devise modified RLS filtering procedures accounting for the spatial variation of the clutter power as well as of the disturbance covariance persymmetry property. Then we introduce the concept of KB RLS and explain how the a-priori knowledge about the radar operating environment can be adopted for improving the system performance. Finally we assess the benefits resulting from the use KB processing both on simulated and on measured clutter data collected by the McMaster IPIX radar in November Introduction Adaptive filters for clutter suppression are sub-optimal implementations of the optimum linear processor which, assuming the a-priori knowledge of the disturbance spectral properties, maximizes the output Signal to Interference plus Noise power Ratio (SINR) [44]. In order to estimate the clutter covariance matrix they exploit training data, namely clutter returns collected from range cells spatially closed to the one under test and assumed free of useful signal component. The estimation procedure is usually performed through the sample covariance matrix and then the adaptive filter weights are derived [45]. It is clear that the imperfect estimate of the clutter PSD leads to sub-optimal performances even in the presence of a spatially homogeneous environment, namely secondary data independent and identically distributed (iid). With reference to this last scenario the loss with respect to the optimum processor, in terms of Improvement Factor (IF), has been quantified analytically and a closed form ex- 65

74 CHAPTER 3. Knowledge-Based Clutter Suppression pression can be found in [45]. Nevertheless the assumption of homogeneous clutter over the extent of the reference window is restrictive and quite often violated. Clutter heterogeneities [46] are usually present yielding severe losses in the performance of adaptive filters [47, 48]. In the sequel we report the most common forms of clutter dishomogeneities which cause the departure from the homogeneous assumption [48]. Clutter edges which lead to abrupt changes in the clutter intensity profiles along the range. This could typically represent either transitions between land and sea clutter or between shadowed and illuminated surface clutter. Clutter in which the power from distinct range bins varies randomly (this is usually the case of sea clutter); Clutter discretes, namely returns from range bins containing point-clutter sources or extraneous targets characterized by significantly different powers and/or spectral properties from the surrounding environment. Their effects on the performance of adaptive doppler filters have been studied thoroughly in [48] and a technique based on the incorporation of a standard Moving Target Indicator (MTI) filter before the adaptive processor is also proposed. This solution leads to performance benefits especially in situations where a few stationary clutter discretes are embedded into an homogeneous scenario or in the presence of clutter edges where one side of the edge has a dominant stationary component. An alternative approach to circumvent the severe performance loss caused by clutter heterogeneities is to resort to KB techniques which should be valuable in using a-priori information to restore the radar performance. They are based on the real-time exploitation of a-priori knowledge concerning the environment surrounding the radar [1, 2]. As already pointed out, examples of a-priori knowledge are DTEM, previous look data, GIS s, roadways (to highlight sectors of surveillance where moving cars or vehicles might be present), background of air/surface traffic, system calibration information, et cetera. In this chapter we show how KB techniques can be exploited for the adaptive implementation of the optimum doppler processor. To this end, we first revisit the RLS algorithm [49, 50, 51], which permits a reduced complexity adaptive implementation of the optimum filter, designed under the assumption of homogeneous environment. Then we introduce three more recursive procedures exploiting either the persymmetry property of the clutter covariance matrix [52] or suitable data normalizations, or both the concepts. However, due to the strong clutter variability, none of the algorithms uniformly outperforms the others and the problem of choosing the most appropriate procedure arises. In this context we propose to use the environmental information for the selection of the most suitable adaptation technique and to mitigate the deleterious effects of clutter heterogeneity. 66

75 3.2 Problem Formulation and Design Issues The chapter is organized as follows. In Section 3.2 we deal with the problem formulation and the design issues while in Section 3.3 we assess the performance of the algorithms in the presence of both simulated and measured data, collected by the McMaster IPIX radar in November Finally conclusions are given in Section Problem Formulation and Design Issues Denote by r the N-dimensional complex vector of the samples from the base-band equivalent of the received signal from the cell under test. Under the hypothesis H 0, namely target absence, r contains disturbance only, i.e. H 0 : r = d, where the disturbance vector d accounts for both clutter and thermal noise. Under H 1, instead, r contains also a target component, i.e. H 1 : r = α p + d, with α the complex amplitude accounting for both the target as well as the channel propagation effects and p the target temporal steering vector. A doppler processor performs the inner product between a suitable weight vector w and the vector r of the returns from the cell under test. Ideally it provides coherent gain on target while forming doppler response nulls to suppress the disturbance components. Specifically the optimum doppler filter, which maximizes the output SINR [53], is given to within a scale factor by w = M 1 p, (3.1) where M denotes the disturbance covariance matrix. Its performance is usually evaluated in terms of the IF and the frequency response with respect to specified operational conditions. Equation (3.1) highlights that the design of the optimum filter requires the a- priori knowledge of the clutter covariance matrix. From a practical point of view this knowledge is not available and thus it is necessary to develop a processor which online estimates M and automatically adapts its filtering action in response to a changing environment. This self-adjusting capability makes the operation of such system more flexible and reliable than non-adaptive processors based on a fixed filter structure with constant parameters. In order to perform the covariance estimate it is customary to resort to training data, namely returns from range cells spatially closed to the one under test and free of useful signal component. Moreover it is assumed that they share the same 67

76 CHAPTER 3. Knowledge-Based Clutter Suppression covariance matrix of the disturbance component from the cell under test (homogeneous assumption). Denoting by r 1,..., r K the N-dimensional vectors of the training set (K N) the ML estimator of M is the sample covariance matrix M = 1 K K k=1 r k r k, (3.2) where ( ) denotes conjugate transpose. A fundamental trade-off exists between the speed of adaptation and the steady state performance; indeed a common rule of thumb suggests that K = 2N training data are required for satisfactory performance [45]. Any adaptive system design should therefore aim to optimize the quoted trade off. Better accuracy and fast adaptation can be obtained exploiting some structural properties of the clutter covariance matrix thus reducing the uncertainty in learning. For instance the clutter covariance might exhibit a doubly symmetric form namely it is hermitian about its principal diagonal and persymmetric about its cross diagonal (the Toeplitz structure which results for the case of a uniformly spaced pulse train is a special case of persymmetry. Moreover this structure is not appropriate for the case of staggered pulses). In this case the ML estimate of M is [52] M p = 1 2K K ( k=1 ) r k r k + J r k r T k J, (3.3) where ( ) T denotes transpose, ( ) the complex conjugate operator, and J is the permutation matrix, i.e J = Under the persymmetry assumption the adaptive filter which exploits (3.3) requires K 2 training data to achieve the same performance of the processor which resorts to the plain sample covariance matrix (3.2) with K secondary vectors [52]. Other structural information might be also employed for the design of covariance estimators. Nevertheless it is worth to point out that severe performance losses might be experienced if the actual environmental condition deviates from the nominal one, namely if the exploited structural property is not fully respected. Both the adaptive filter which uses the sample covariance matrix (3.2) and the one exploiting the persymmetric estimator (3.3) require the computation of M or M p and its online inversion. It is well known that the computational burden connected with this last operation is O(N 3 ) 1. Hence, in order to save computation 1 Notice that computational savings can be obtained if the matrix is Toeplitz. 68

77 3.2 Problem Formulation and Design Issues time, recursive algorithms which estimate the inverse covariance directly from the input samples can be conceived RLS Covariance Update The RLS algorithm was first conceived for beamforming applications in [54] and then adopted for clutter suppression purposes in [55] and [56]. It is relies on a recursive procedure which estimates the inverse covariance matrix M 1 directly from the input samples. By doing so the online matrix inversion is no longer required and a significant saving in computation time can be achieved. The recursive equation which defines the RLS can be obtained applying the matrix inversion lemma [57] to the equation M k+1 = λ M k + r k+1 r k+1, 0 < λ 1, (3.4) where the weighting coefficient λ, referred to in the sequel as the forgetting factor, determines the relative confidence of the input data with respect to the current estimate. The aforementioned procedure leads to M 1 k+1 = λ 1 1 M k λ λ 1 r k+1 M 1 k 1 M k r k+1 r 1 k+1 M k, (3.5) r k+1 which defines the estimate of the inverse covariance at the step k +1 as a function of the estimate at the previous step, the new input data r k+1 vector, and the forgetting factor. This last parameter rules the speed of adaptation of the algorithm. In fact the higher λ the more accurate the estimate. This, however implies a longer reaction time due to the longer memory of the algorithm. A faster adaptation is achieved by means of a smaller forgetting factor at the expense of a poorer accuracy, resulting in a loss of the steady state SINR. As a consequence a trade off between speed and accuracy must be achieved. The recursion is usually initialized assuming M 0 = δi (I is the identity matrix and δ is a scalar called loading factor) which is tantamount to introducing in the covariance estimate an exponentially decaying diagonal loading [51], namely the actual covariance estimate after K training data is M K = λ K δi + K λ K k r k r k. (3.6) k=1 Exploiting (3.5) we can obtain the recursive equation for the filter coefficients update, i.e ( ) w k+1 = λ 1 I k k+1 r k+1 w k, (3.7) 69

78 CHAPTER 3. Knowledge-Based Clutter Suppression where w k is filter vector at the k-the step and λ 1 M 1 k r k+1 k k+1 = 1 + λ 1 r k+1 M 1 k r k+1. (3.8) Notice that the filter at the K-th instance of the recursion can be interpreted (but for a scaling factor) as the solution to the optimization problem [ ( ) ] K w K = argmin w K w K λ K δi + λ K k r k r k w K w K p = 1 k=1 (3.9) where argmin w K ( ) denotes the value of w K which minimizes the argument. The computational complexity connected with the implementation of the RLS is O ( N 2) flops which indicates a significant saving in computation time with respect to the online inversion of the sample covariance matrix, which, as already pointed out, involves O(N 3 ) flops. A recursive algorithm which exploits the covariance persymmetry can be also developed. To this end the covariance update equation is M k+1,p = λ M k,p r k+1 r k J r k+1 r T k+1j, (3.10) and the recursion which defines the Persymmetric RLS (PRLS) can be obtained applying twice the matrix inversion lemma to (3.10) and is given by Φ k+1 = λ 1 M 1 k,p M 1 k+1,p = Φ k+1 λ λ 1 r k+1 1 M k,p r k+1 r k+1 k,p r k+1 M 1 M 1 k,p (3.11) r T k+1 JΦ k+1j r Φ k+1 J r k+1 r T k+1jφ k+1 k+1 It follows that the algorithm for the recursive computation of the filter weights is λ 1 M 1 k,p r k+1 h k+1,p = 2 + λ 1 r k+1 M 1 k,p r k+1 Φ k+1 = λ 1 1 M k,p λ 1 h k+1,p r 1 k+1 M k,p Φ k+1 J r k+1 k k+1,p = 2 + r k+1 JΦ k+1j r k+1 M 1 k+1,p = Φ k+1 k k+1,p r T k+1jφ k+1 70

79 3.2 Problem Formulation and Design Issues w k+1,p = λ 1 (I h k+1,p r k+1 k k+1,p r T k+1 J+ + k k+1,p r T k+1 Jh k+1,p r k+1 ) w k,p (3.12) Notice that, assuming M 0,p = δi, the filter at the K-th instance of the recursion can be still deemed (but for a scaling factor) as the solution of a constrained optimization problem, i.e. { w K,p = argmin w K,p w [ K,p λ K δi w K,p p = 1 K k=1 ( ) ] } λ K k r k r k + J r k+1 r T k+1j w K,p (3.13) 1 M k+1 or 1 As a final remark we point out that the same M k+1,p can be used to implement a bank of parallel doppler filters, by merely changing the useful vector p Normalized RLS (NRLS) Covariance Update In the presence of an heterogeneous environment, where the clutter power varies from cell to cell (this is for instance the case of sea cutter [58] and high resolution land clutter [59]), the filter (3.7) suffers a severe performance loss with respect to the optimum processor [48]. This behavior is due to the fact that in this situation the sample covariance matrix is no longer the ML covariance estimator. A possible way to restore the system performance in such a case is to employ a normalized sample covariance matrix [41] in place of the usual one, i.e. M n = N K K k=1 r k r k r k 2, (3.14) where denotes the Euclidean norm of a complex vector. The resulting performance improvement stems from the observation that, in the presence of a clutter dominated environment, the normalized covariance estimator is invariant with respect to scaling of the secondary data. As a consequence it gets rid of the local clutter powers since they simplify between the numerator and the denominator of the generic normalized outer product r k r k r k 2 /N k = 1,...,K. 71

80 CHAPTER 3. Knowledge-Based Clutter Suppression A normalized version of the RLS algorithm can be also conceived. Precisely denoting by r k r k,n = r k / N the k-th normalized data vector, we get the following equation for the recursive update of the inverse covariance matrix M 1 k+1,n = λ 1 M 1 k,n λ 2 1 M k,n r k+1,n r k+1,n k,n r k+1,n M 1 M 1 k,n, 1 + λ 1 r k+1,n (3.15) 1 where M k,n denotes the estimate at the step k. If the recursion is initialized with M 0,n = δi then the actual covariance estimate after K training data is given by M K,n = λ K δi + K k=1 λ K k Moreover 2, taking the statistical average of (3.16) we get r k r k r k 2 /N. (3.16) [ ] [ ] E MK,n = λ K r k r K k δi + E r k 2 λ K k, (3.17) /N which for K and N tends to [ ] 1 1 λ lim E r k r k N r k 2 /N k=1 = 1 Σ, (3.18) 1 λ showing that (3.16) is an asymptotically unbiased estimator of the covariance structure but for a scalar deterministic factor. The equation which defines the filter coefficients update can be obtained analogously to (3.7), but for r k,n in place of r k, and the filter vector is still proportional to the solution of a constrained optimization problem. The computational complexity connected with the implementation of the NRLS is O ( N 2) flops. Finally we notice that a Persymmetric NRLS (PNRLS) can be also developed applying twice the matrix inversion lemma to M k+1,p,n = λ M k,p,n r k+1,n r k+1,n J r k+1,n r T k+1,nj. (3.19) 2 We assume that the training data r k are zero mean complex circular Gaussian vectors with the same covariance matrix but for a scaling factor, i.e. E[ r k r k ] = σ2 kσ, where E[ ] denotes statistical expectation, σ k the local disturbance power level, and Σ the common covariance structure. 72

81 3.2 Problem Formulation and Design Issues This procedure leads to the following recursive equations ( ) Φ k+1,n = λ 1 1 λ 2 1 M k,p,n r k+1,n r 1 k+1,n M k,p,n M k,p,n 2 + λ 1 r 1 k+1,n M k,p,n r k+1,n (3.20) M 1 k+1,p,n = Φ k+1,n Φ k+1,nj r k+1,n rt k+1,n JΦ k+1,n 2 + r T k+1,n JΦ k+1,nj r k+1,n from which a procedure for the filter coefficients updating can be derived analogously to the non-normalized case but for r k,n in place of r k Knowledge-Based RLS (KB-RLS) Covariance Update The previous subsections highlight that two different families of updating algorithms for the filter coefficients can be conceived. The former exploits a conventional recursive covariance estimator (3.5) (or (3.11) for the persymmetric case) whereas the latter resorts to a normalized one (3.15) (or (3.20) for the persymmetric case). It is clear that in the presence of an homogeneous environment the RLS provides a better performance than the NRLS since in this situation the conventional sample covariance matrix is the ML estimate of the clutter covariance. On the contrary if the scene is such that the clutter power varies from one cell to another then the NRLS usually outperforms the classic RLS procedure (3.7). It follows that the problem of selecting the most suitable filtering procedure arises. To this end it is quite natural to exploit KB techniques which might be very valuable in using a-priori information to select the most appropriate filtering algorithm. In this context, exploiting the geographical information provided by a GIS, it is possible to know the exact location of transition regions where the updating algorithm must be changed 3. Otherwise stated, if we refer to an interface between homogeneous land (region 1) and sea (region 2), resorting to the GIS information, we can use in region 1 the RLS procedure while in region 2 the NRLS algorithm. Another relevant source of a-priori information which can be exploited for the selection of the algorithm is the wind data. Precisely, if the filter must operate into an homogeneous region covered by vegetation, it is reasonable to employ the plain RLS algorithm in the absence or for weak wind. However, for very windy conditions (for instance in the presence of wind gusts), it is known that the windblown vegetation causes strong fluctuations of the clutter power from cell to cell [60] and thus the NRLS must be adopted. We also point out that KB techniques might be employed for the choice of the forgetting factor λ. In other words we suggest to adopt a space-varying forgetting 3 A simpler way for the clutter edge identification might also be the use of a clutter map constructed from previous radar scans. 73

82 CHAPTER 3. Knowledge-Based Clutter Suppression factor whose value at the step k must be a function of the environment, i.e. λ = λ k (environment). Several a-priori information sources might be employed toward the selection of the best value; for instance wind and meteorological information, geographical map, DTEM, et cetera. The aforementioned idea may represent a very powerful technique to mitigate the effects of clutter edges. Indeed, exploiting a GIS for locating the presence of transition regions, we can think to lower (according to a certain mathematical law) the forgetting factor when we are approaching the first side of the interface and then to gradually increase λ right after the transition. By doing so, as it will be shown in the performance analysis, the a-priori knowledge helps the algorithm to rapidly forget the clutter conditions within the first region. Otherwise stated the filter will exhibit a shorter reaction time, namely a faster adaptation, than the plain RLS. KB GIS-aided strategies might also be helpful to deal with the presence of sporadic point clutter sources embedded into an homogeneous environment (for instance several man-made structures within an open country). It is not convenient to update the inverse covariance in correspondence of range bins containing clutter discretes and, to this end, the a-priori knowledge helps to excise contaminated clutter returns from the training set. As a final remark we provide some considerations as to the adoption of the persymmetric algorithms. The application of the quoted procedures can be decided according to two different criteria. The former relies on the use of data-dependent tests, such as that developed in [60], which permit to establish online if the persymmetry holds. The latter relies on the adoption of persymmetry maps, namely databases constructed from data collected in the region of interest (under several distinct operational conditions), which allow to establish whether or not, in that specific situation, the persymmetry assumption can be met. 3.3 Performance Analysis This section is devoted to the performance assessment of the previously presented algorithms both on simulated and on real radar data (X-band mixed sea and land clutter), collected by the McMaster University IPIX radar in November In Subsection we consider simulated data accounting for either homogeneous or heterogeneous environments with variable statistical properties (i.e clutter apdf, clutter PSD parameters, and Clutter to Noise Power Ratio (CNR)) in order to evaluate the performance achievable under different situations like land clutter, sea clutter, and heterogeneous environments (i.e. rapid land cover variations, littorals, et cetera) including clutter edges and/or discretes. In Subsection we 74

83 3.3 Performance Analysis focus on real radar data and assess the performance benefits which can be achieved resorting to a KB processing. All the experiments assume that the number of integrated pulses N is equal to 16; the structure of the useful signal component is p = 1 N [1,exp(j2πf d ),...,exp(j2π(n 1)f d )], where f d = 0.01 (slow moving target) is the normalized doppler frequency; the loading factor δ for the initialization of the algorithms is equal to 10 3 which is tantamount to assume w 0 = 1 δ p Simulated Data In the first experiment (Figure 3.1a) we assume an homogeneous Gaussian environment, namely we model the vectors r k, k = 1,...,K, as zero-mean iid complex circular Gaussian vectors with covariance matrix M = M c + σ 2 I, where M c is the clutter covariance and σ 2 is the thermal noise level. The matrix M c is assumed exponentially shaped, namely its (i,j)-th entry is given by M c (i,j) = P c ρ i j exp(j2π(i j)f c ) (3.21) where ρ is the one-lag correlation coefficient, P c and f c denote respectively the clutter power level and doppler frequency [61]. We also suppose ρ = 0.999, f c = 0, and CNR = P c σ 2 = 30dB, which is the typical situation of an homogeneous land environment. The performance is evaluated in terms of SINR, i.e. SINR = w k p 2 w k M w, (3.22) k at the k-th instance of the recursion. Precisely, in Figure 3.1a, the normalized SINR of the RLS and NRLS (both with forgetting factor 0.99), averaged over 400 independent trials, is plotted versus the number of range samples N R. As expected, since the simulated environment is homogeneous, after a transient of about 30 cells, the plain RLS outperforms the normalized algorithm. 75

84 CHAPTER 3. Knowledge-Based Clutter Suppression An opposite behavior is shown by the plots of Figure 3.1b where an heterogeneous scenario accounting for the spatial variations of the clutter power is considered. Therein the disturbance vectors r k are modeled as the sum of a clutter contribution c k and the receiver thermal noise n k (statistically independent of c k ), i.e. r k = c k + n k. The n k are iid zero-mean complex circular Gaussian vectors with covariance matrix σ 2 I. The clutter vectors c k are iid complex circular SIRV s [43] with covariance matrix given by (3.21) and Gamma distributed squared textures whose shape parameter, which rules the impulsive nature of the clutter, is denoted by ν. The plots also assume ρ = 0.8, f c = 0.2, ν = 0.5, and CNR = 20 db, which is a typical scenario of sea clutter. Finally the forgetting factor is again λ = It is clear that the NRLS outperforms the plain RLS and this behavior stems from the observation that in the simulated environment the sample covariance matrix is no longer the ML estimator of the disturbance covariance. The previous analyses have shown that no algorithm uniformly outperforms the others and the superiority of a specific filtering procedure is strictly dependent on the actual disturbance environment. Thus it appears quite natural to exploit the a-priori knowledge, provided by a GIS about the observed geographical site, in order to select the most appropriate clutter suppression algorithm. In the next example the performance improvements achievable resorting to KB processing are illustrated. Therein an heterogeneous scenario composed of two different regions is simulated. The former, composed of the range cells from k = 1 to 400, contains homogeneous land clutter plus noise (simulated according to model used in Figure 3.1a with ρ = 0.999, f c = 0, and CNR= 30 db). The latter, namely range cells from k = 400 to 800, includes sea clutter plus thermal noise (simulated according to the model used in Figure 3.1b with ρ = 0.8, f c = 0.2, and CNR= 20 db). The RLS and NRLS with forgetting factor λ = 0.99 are compared with the KB-RLS that exploits the a-priori information concerning the location of the clutter edge. Precisely the KB-RLS coincides with the plain RLS algorithm in the first region. Moreover after the clutter edge it switches into the NRLS. As to the forgetting factor the KB-NRLS algorithm assumes a space-varying λ in order to provide a fast adaptation to the new clutter conditions. Otherwise stated the forgetting factor is ruled by the following linear piecewise function 0.99 k = 1,...,400 λ k (environment) = (k 401) k = 401,...,420 (3.23) 0.99 k = 421,...,800 It is not excluded the existence of other functions, as for instance parabolic piecewise laws, capable of ensuring better performances than the linear one. Neverthe- 76

85 3.3 Performance Analysis less the difficult problem of devising the optimum function is still opened. When, in correspondence of the clutter edge, the KB-RLS commutes into 1 the NRLS the inverse covariance estimate is normalized, i.e. M 401 = (1 λ)tr( M ) M 400, with tr( ) denoting the trace of a square matrix. Extensive simulation results have shown that this normalization ensures a fast convergence even if it adds a small increase in the algorithm computational complexity since a further scalar recursive equation for the trace updating, i.e. tr( M k+1 ) = λtr( M k ) + r k+1 r k+1, is required. In Figure 3.2a the normalized averaged SINR of the RLS, NRLS, and KB-RLS is plotted versus N R. The curves highlight an abrupt transition in correspondence of the clutter edge. The RLS performs better than NRLS in the first region while the opposite behavior is observed in the second region. The KB-RLS algorithm achieves the best performance especially after the transition, requiring a shorter adaptation time than the NRLS to reach the steady state SINR. This behavior is confirmed by the curves of Figure 3.2b where the amplitude of the filter frequency response at the range cell 800 is plotted versus the normalized frequency. The plots show that the RLS does not yet forget the land clutter environment; indeed the filter response exhibits a deep null in correspondence of f c = 0. The quoted null is less pronounced with reference to the curve of the NRLS (Figure 3.2c) and vanishes if the KB-NRLS (Figure 3.2d) is employed. In this last case, only one null is present in correspondence of f c = 0.2, namely the clutter doppler frequency in the second region. The performance benefits achievable exploiting persymmetry are shown in Figure 3.3a and Figure 3.3b. In the former the RLS is compared with the PRLS assuming the same simulation setup of Figure 3.1a. In the latter the NRLS is compared with the PNRLS under the same disturbance conditions of Figure 3.1b. In both the cases the use of persymmetry leads to a performance gain implying a shorter adaptation time than the conventional algorithms. In Figure 3.4, which assumes the same simulation parameters of Figure 3.2, the performance of the KB-RLS which exploits also the persymmetry is compared with that of the other algorithms. In this case the KB-RLS coincides with the PRLS in the first region. Then after the clutter edge it switches into the PNRLS. Moreover, the same law of Figure 3.2 for the spatial variation of the forgetting factor is exploited. The curves confirm that the correct use of a-priori information can lead to significant performance benefits. In Figure 3.5, the effects of a strong clutter discrete on the performance of the RLS is assessed also in comparison with the KB-RLS which a-priori knows the location of the outlier and hence doesn t exploit data from that region in order to update the filter weights. Precisely we simulate land clutter, according to the 77

86 CHAPTER 3. Knowledge-Based Clutter Suppression model of Figure 3.1a, for all the range cells 1 k 800; moreover in the range bins 401 k 410 we inject a discrete narrowband outlier with f d = and with the same power of the clutter. The plots show that the RLS experiences a significant performance loss in the presence of training data contaminated by discrete outliers. Moreover, due to the long memory, more that 200 cleaned range cells are necessary in order for the system forgets the contaminated region. On the contrary the KB-RLS, which knows the location of the discretes, does not update the filter coefficients in correspondence of the contaminated data and as a consequence the SINR loss is only confined to a small fractions of range cells. We finally highlight that it might be also possible to reduce the loss in the contaminated area if one decides to adopt in the quoted region a non-adaptive filter, as for instance that proposed in [62]. 78

87 3.3 Performance Analysis a) 0 SINR (db) b) N R SINR (db) N R Figure 3.1: Normalized SINR (db) versus the number of range samples N R for the RLS (solid line) and the NRLS (dashed line). a) Simulated land clutter. b) Simulated sea clutter. 79

88 CHAPTER 3. Knowledge-Based Clutter Suppression a) 0 5 SINR (db) N R Filter Renspose Amplitude (db) b) c) d) f d Filter Renspose Amplitude (db) f d Filter Renspose Amplitude (db) f d Figure 3.2: a) Normalized SINR (db) versus the number of range samples N R for the RLS (solid line), the NRLS (dashed line), and the KB-RLS (bold line). Simulated land clutter for 1 N R 400. Simulated sea clutter for N R > 400. Amplitude of the filter frequency response (db) versus the normalized doppler frequency f d for the RLS (b), the NRLS (c), and the KB-RLS (d). 80

89 3.3 Performance Analysis 0 a) SINR (db) b) 0 N R SINR (db) N R Figure 3.3: Normalized SINR (db) versus the number of range samples N R. a) Simulated land clutter: RLS (solid line) and PRLS (dotted line). b) Simulated sea clutter: NRLS (dashed line) and PNRLS (dashed-dotted line). 81

90 CHAPTER 3. Knowledge-Based Clutter Suppression 0 5 SINR (db) N R Figure 3.4: Normalized SINR (db) versus the number of range samples N R for the RLS (solid line), the NRLS (dashed line), the PRLS (dotted line), the PNRLS (dashed-dotted line), and the KB-RLS (bold line). Simulated land clutter for 1 N R 400. Simulated sea clutter for N R >

91 3.3 Performance Analysis SINR (db) N R Figure 3.5: Normalized SINR (db) versus the number of range samples N R for the RLS (solid line) and the KB-RLS (bold line). Simulated land clutter for 1 N R 400 and N R > 410, Simulated land clutter with the presence of outliers for 401 N R

92 CHAPTER 3. Knowledge-Based Clutter Suppression Real Data Radar measurements were collected in November 1993 using the McMaster IPIX radar from a site in Dartmouth [24], Nova Scotia, on the East Coast of Canada (see Figure 3.6). The radar was mounted on a cliff facing the Atlantic Ocean, at a height of 100 feet above the mean sea level, and scans the site over 370 Deg in 10 seconds in a continuous azimuth scan mode. More details on the experiment can be found in [24, 25, 26]. The illuminated area ranges from the Atlantic Ocean, to the shoreline of Cow Bay, and to the lands near Halifax. The specifications of the considered dataset, containing mixed land and sea clutter, are reported in Table 1.1. The data are previously preprocessed in order to remove the DC offset (both of I and Q channels) and the phase imbalance due to hardware imperfections. The 2-D clutter intensity field is plotted in Figure 3.6a showing the presence of two different sectors containing respectively returns from land and sea. In order to perform the analysis in the presence of clutter edge we select two different angular sectors and put them near as pictorially shown in Figure 3.6b. Precisely the selected areas are Region 1, from 250 to 270 Deg, where there is a land clutter region including the lands near Halifax (the normalized PSD of the returns from this region is plotted in Figure 3.7a). Region 2, from 150 to 170 Deg, where there is an open view of the Atlantic Ocean (the normalized PSD of the returns from this region is plotted in Figure 3.7b). We employ N = 16 azimuth returns and a total of 400 trials for estimating the SINR in a given range position. The RLS, NRLS, PRLS and PNRLS exploit a forgetting factor equal to The KB-RLS coincides with RLS algorithm in Region 1, and with PRLS procedure in Region 2. It also exploits a space-varying forgetting factor ruled by the following linear piecewise function 0.99 k = 1,...,184 λ k (environment) = (k 185) k = 185,...,204 (3.24) 0.99 k = 295,...,368 In Figure 3.8 the normalized SINR is plotted versus N R showing that the PRLS and PNRLS are outperformed by the RLS and NRLS respectively; this is probably due to the lack of a persymmetric covariance matrix with reference to the land clutter region of the analyzed dataset. Moreover the PRLS, NRLS, and PNRLS provide a faster convergence rate than the RLS in the first 50 cells but are outperformed by the KB-RLS, until the end of the Region 1. 84

93 3.3 Performance Analysis After the clutter edge the PRLS and PNRLS exhibit a better performance than the RLS and NRLS; an indirect consequence of the persymmetry property of the sea clutter covariance matrix. Moreover both the NRLS and PNRLS perform better than the non-normalized algorithm. As to the KB-RLS, it requires a short adaptation time after the clutter edge, achieving the best performance from the cell 215 to the end of Region 2. A further evidence of this behavior is highlighted in Figure 3.9, where the amplitude of the filter frequency response at the end of Region 2 is plotted versus the normalized doppler frequency. Figure 3.9a refers to the RLS filter and shows a null around zero doppler frequency. It is due to the filter memory which has not yet forgotten the clutter condition of the first region and its effect is the degradation of the performance achievable for slow moving target. On the contrary the shape of the KB-RLS filter (displayed in Figure 3.9b) is correctly adapted to the sea clutter PSD of Figure 3.7b. Finally, in Figure 3.10, two different KB-RLS configurations, referred to in the sequel as KB1-RLS and KB2-RLS respectively, exploiting two distinct functions for the spatial variation of λ, are compared under the same clutter environment of Figure 3.8. The former, whose performance is plotted in (Figure 3.10a), adopts a space-varying forgetting factor, ruled by the linear piecewise function (3.24) displayed in Figure 3.10b. The latter, whose performance is plotted in (Figure 3.10c), resorts to the following forgetting function 0.99 k = 1,...,144 λ k (environment) = (k 145) k = 145,..., (k 185) k = 185,..., k = 295,...,368 (3.25) which is also displayed in Figure 3.10d. Otherwise stated, in this last case, the system starts to forget a few cells (40 in the case at hand) before the clutter edge. The figure highlights that this second strategy leads to a more gradual SINR transition at the clutter interface. However this improvement is accompanied by a small performance degradation in correspondence of the cells where the forgetting process is started. 85

94 CHAPTER 3. Knowledge-Based Clutter Suppression a) 184 range cells Region Region b) Region 1 Region Figure 3.6: a) 2-D intensity field of the mixed land and sea clutter live data (red are the strongest returns, blue are the weaker returns). The straight red lines delimitate the selected regions, namely Region 1 ([250, 270], land clutter) and Region 2 ([150, 170], sea clutter). b) Schematic representation of the exploited clutter regions. 86