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- Permanent Link:
- http://ufdc.ufl.edu/AA00024538/00001
## Material Information- Title:
- Ground moving target detection and feature extraction with airborne phased array radar
- Creator:
- Jiang, Nanzhi
- Publication Date:
- 2001
- Language:
- English
- Physical Description:
- vii, 125 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Calibration ( jstor )
Estimation methods ( jstor ) Feature extraction ( jstor ) Frequency ranges ( jstor ) Jamming ( jstor ) Phased arrays ( jstor ) Radar ( jstor ) Radar range ( jstor ) Supernova remnants ( jstor ) Vector autoregression ( jstor ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 2001.
- Bibliography:
- Includes bibliographical references (leaves 118-123).
- General Note:
- Printout.
- General Note:
- Vita.
- Statement of Responsibility:
- by Nanzhi Jiang.
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- University of Florida
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- Resource Identifier:
- 47291877 ( OCLC )
ocm47291877
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Ground Moving Target Detection and Feature Extraction with Airborne Phased Array Radar By Nanzhi Jiang A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2001 This is a dedication to my loving and supportive parents. ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my advisor, Dr. Jian Li, for her constant support, enthusiasm, and patience in guiding this research. Special thanks are due to Drs. Jose C. Principe, Fred J. Taylor, John M. Anderson, and David C. Wilson for serving on my supervisory committee and for their contribution to my graduate education at the University of Florida. I also wish to thank all the fellow graduate students in the spectral analysis laboratory with whom I had the great pleasure of interacting. Drs. Guoqing Liu, Renbiao Wu, Jianhua Liu, and Xi Li have my gratitude for sharing many interesting discussions with me. I would like to gratefully acknowledge all the people who helped me during my Ph.D. program. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . ....................... ........ iii LIST OF FIGURES ......................... ...... vi ABSTRACT .................................... ix CHAPTERS 1 INTRODUCTION ................... .......... 1 1.1 Background....... ... . ..... . . ....... 1 1.2 Scope of the Work .......................... 5 1.3 Contributions ................... ......... 6 1.4 Outline ................... ............. 7 2 LITERATURE SURVEY ......................... 10 2.1 Clutter Suppression ......................... 10 2.1.1 Displace-phase-center-antenna (DPCA) .......... 10 2.1.2 Adaptive Filtering Based Space-time Adaptive Processing (ST A P ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Vector Auto-regressive (VAR) Filtering .......... 12 2.2 Moving Target Detection ............. . ...... 13 2.3 Parameter Estimation ...................... . 14 2.3.1 The RELAX algorithm . .................. 15 2.3.2 ML methods in Radar Array Signal Processing ...... 16 3 MOVING TARGET PARAMETER ESTIMATION FOR AIRBORNE LRR RADAR ............. . ... ............. ..... 18 3.1 Introduction ......... ..... .. .......... 18 3.2 Clutter Simulation ................... ....... 19 3.3 Clutter Suppression ................... ...... 21 3.3.1 Data model ............. .. .......... 21 3.3.2 VAR Filtering ...... ........ ....... . 22 3.3.3 DPCA ....... ... ....... ....... 25 3.4 Parameter Estimation ..... .............. ..... 26 3.5 Numerical Results ................... ....... 30 3.6 Summary ................... . .......... 32 iv 4 MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR ......... .......... .. .................... 39 4.1 Introduction ................... .......... 39 4.2 Data Model and VAR Filtering ................. ......41 4.3 Target Feature Extraction ...................... 45 4.3.1 Doppler Frequency and Spatial Signature Estimation . .. 46 4.3.2 Phase History Sequence and Array Steering Vector Estimation .................... .. ........ 47 4.3.3 Target Range Feature Estimation . . .......... 51 4.4 Numerical Examples ................... ...... 52 4.5 Summary ...................... ........ 56 5 MULTIPLE MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR .......... .... ............. 65 5.1 Introduction ................... ........ . 65 5.2 Data Model and VAR Filtering .. . . . . . . . . . . . . . . 65 5.3 Feature Extraction of Multiple Moving Targets . ......... 68 5.3.1 Space-Time Parameter Estimation . ...... ..... . 69 5.3.2 Target Range Feature Estimation . ..... .... . . 75 5.4 Numerical Examples ................... ...... 76 5.5 Summary ................... .. ......... 80 6 MULTIPLE MOVING TARGET DETECTION FOR AIRBORNE HRR RADAR ............. ................... ........ 87 6.1 Introduction ............... .. ... ....... . 87 6.2 Data Model and VAR Filtering . .................. 88 6.3 Detection of a Single Moving Target ... ............... 90 6.3.1 Doppler Frequency and Spatial Signature Estimation ... 91 6.3.2 GLRT Detection Strategy . ................. 92 6.3.3 Asymptotic Statistical Analysis . .............. 93 6.3.4 Moving Target Detection Steps . .............. 94 6.4 Detection of Multiple Moving Targets . .............. 95 6.5 Numerical Examples ................... ...... 96 6.6 Summary ............. .. .. .. ........ . 99 7 SUMMARY AND CONCLUSIONS . .................. 103 APPENDIXES REFERENCES .............. . .. . ...... 115 BIOGRAPHICAL SKETCH ................. . ........ 121 v LIST OF FIGURES 1.1 Echoes from an aircraft and a moving vehicle on the ground. ..... 8 1.2 Conceptual block diagram of line array antenna. . .......... . 8 1.3 Mesh plot of (a) the power spectral density of the interference (including clutter, jamming, and noise) and (b) Fully adaptive response when the clutter ridge is along the diagonal (7 = 1) and the jamming is from 0j = 120. ...... .. ...... ................. 9 3.1 Clutter ridge for (a) 7 = 1, (b) 7 = 1.2, and (c) a, = 0.2 and , = 1. 33 3.2 Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB and there is no system mismatches , no array calibration errors, and no jamming. ................................ 34 3.3 Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB, the jammer has JNR=25 dB, and there is no system mismatches and no array calibration errors. .................. ...... 35 3.4 Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency in the presence of system mismatches (7 = 1.2) and when the clutter is non-fluctuating with CNR = 40 dB and there is no array calibration errors and no jamming. . ................... 36 3.5 Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is fluctuating with o, = 0.05 and CNR = 40 dB and there is no system mismatches, no array calibration errors, and no jamming. ....... .................... 37 3.6 Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB in the presence of antenna mutual coupling with p = 0.1 and there is no system mismatches and no jamming. .......... ........ 38 4.1 HRR range profiles are divided into LRR segments with each segment containing L HRR range bins .............. ........ 58 4.2 Flow chart of the proposed clutter suppression and target parameter estimation algorithm for HRR phased array radars. . ........ . 59 vi 4.3 (a) Mesh plot and (b) projection of the power spectral density of the interference (including clutter, jamming, and noise) and the target when the clutter ridge is along the diagonal (y = 1), the jamming comes from Oj = 450, and the target is located at DOA = 60' with wo = 0.27r. The axes 9 and w are for the spatial frequency and the normalized Doppler frequency, respectively, (see Equations (3.1) and (3.2)). 60 4.4 Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, no jammer, no array calibration errors, no system mismatch.) . ............ 61 4.5 Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 450 with JNR=30 dB, no array calibration errors , no system mismatch.) ............... ........... 62 4.6 Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, a jammer at 450 with JNR=30 dB, array calibration error covariance matrix 0.041, no system mismatch.) ............................63 4.7 Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 450 with JNR=30 dB, array calibration error covariance matrix 0.041, system mismatch with clutter ridge slope y = 1.1.) . . . 64 5.1 (a) The real HRR range profile without interference (clutter and jammer) and the estimated HRR range profile after clutter suppression and feature extraction, and (b) the cluttered HRR range profile with CNR= 40 dB, SNR= 0 dB. ....................... 82 5.2 Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and no array calibration error. ..... ................... 83 5.3 Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and array calibration error (covariance matrix = 0.041) . ......... 84 5.4 (a) Target photo taken at 450 azimuth angle ........... .. . . 85 5.5 (a) the HRR range profile corrupted by clutter and jamming and (b) the average of the normalized FFT range profiles without interference (clutter and jammer) and the estimated HRR range profile by using the algorithms presented. ................... ...... 86 6.1 Flow chart of the proposed clutter suppression and single moving target detection algorithm for HRR phased array radars. . . . . . . . 100 6.2 Multiple moving target detection probability (for Pf = 10-6) under conditions of: (1) no degradation factors except for the presence of clutter plus noise (solid line), (2) the presence of a point source jammer at 1200 (dotted line), (3) the presence of both the array calibration errors and a point source jammer at 1200 (dashed line), and (4) the presence of the system mismatch (y = 1.1) in addition to the degradation factors in (3) (dashdotted line). . ................. 101 vii 6.3 Multiple moving target detection probability (for P = 10-6) under conditions of: (1) no target in the secondary data (only clutter and noise) (solid line), (2) the presence of two targets in the seondary data with different DOAs and different Doppler frequencies from the two targets in the primary data (dotted line), (3) the presence of two targets in the secondary data with different DOAs and the same Doppler frequencies from the two targets in the primary data (dashed line), and (4) the presence of two targets in the secondary data with both the same DOAs and Doppler frequencies as the two targets in the primary data (dashdotted line). . .................. ....... 102 viii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Ground Moving Target Detection and Feature Extraction with Airborne Phased Array Radar By Nanzhi Jiang May 2001 Chairman: Dr. Jian Li Major Department: Electrical and Computer Engineering "Battlefield awareness" is critical to the success of future military operations. Enabling radar signal processing techniques for battlefield awareness includes the detection and recognition of ground moving targets. Since the Gulf War, DARPA (Defence Advanced Research Projects Agency) has initiated several programs to improve the detection and recognition capability of existing systems, including MSTAR (Moving and Stationary Target Acquisition and Recognition), MTE (Moving Target Exploitation), and SHARP (System-oriented HRR Automatic Recognition Program). Our efforts herein are partially funded by one of the DARPA programs. In this dissertation, we investigate the detection and feature extraction of ground moving targets with airborne phased array radar. The ground clutter observed by an airborne radar is spread over both the range and spatial angle. The clutter spectrum also covers a certain Doppler region due to the platform motion. Without clutter suppression, moving target detection and parameter estimation are impossible. For airborne low range resolution (LRR) phased array radar, we demonstrate that the combination of a vector auto-regressive (VAR) filtering technique and a maximum likelihood (ML) parameter estimation ix method is an effective approach for carrying out clutter suppression and parameter estimation, and being more robust against system mismatches than conventional displaced-phase-center-antenna (DPCA) processing. Compared to a conventional airborne LRR radar, an airborne High Range Resolution (HRR) radar can not only enhance the radar's capability of detecting, locating and tracking moving targets, but can also provide valuable features for applications including automatic target recognition (ATR). We study moving target feature extraction algorithms in the presence of ground clutter for airborne HRR phased array radar. The VAR filtering technique is extended for airborne HRR phased array radar for clutter suppression. We also devise effective and robust feature extraction algorithms for the radar system. Multiple moving target scenarios occur frequently in radar applications. Yet, to the best of our knowledge, little research on the topic has been reported in the literature. We present a relaxation-based algorithm for multiple moving target feature extraction, which reduces the multiple moving target feature extraction problem to a sequence of single moving target feature extraction problems. Target detection is critical for every radar system since without target detection, target feature extraction and ATR are impossible. Our final discussion is focused on multiple moving target detection for airborne HRR phased array radar. We combine the multiple moving target feature extraction methods with a single moving target detection algorithm for multiple target detection. Finally, the VAR filtering technique is demonstrated to be effective for clutter suppression via numerical examples. The proposed moving target detection and feature extraction algorithms are also shown to be both robust and accurate. X CHAPTER 1 INTRODUCTION This dissertation is concerned with clutter suppression and moving target detection and feature extraction for airborne phased array radar. This chapter serves as a general introduction to the background and scope of the work. Contributions are also highlighted in this chapter. 1.1 Background Airborne radars are used to detect the presence of moving targets and estimate their parameters in the presence of noise, ground clutter, and jammers. A typical application (see Fig. 1.1) of these radars is surveillance to detect low-flying aircrafts or vehicles moving over terrain through possible weather disturbances. The function of the airborne radar in this situation is to reject the returns from terrain (usually called ground clutter) and weather while retaining the return from the targets of interest, such as the aircrafts or vehicles, thereby allowing target detection and followed by target parameter estimation for possible automatic target recognition. Meanwhile, the ability to locate targets in angle is a key characteristic of virtually every airborne radar. The high angle (spatial) resolution can be offered by phased arrays. Phased arrays have been used in fields as diverse as radar, communications, electronic warfare and radio astronony; furthermore, researchers plan to use phased array-based systems to enhance airport safety and traffic efficiency. The phased array technique is a much newer technology than a single antenna system. The antenna array consists of many elements such as dipoles. Each element of the phased array is phase shifted to steer the beam to the desired direction. Depending 1 upon the array design, the number of array elements could be several or a few thousand. Because phase shifters and associated equipment are relatively expensive, the technique is expensive to implement. Its advantage is that the technique has significantly better performance than a parabolic reflector type of single antenna. Thus, the use of this technique requires a performance versus implementation cost trade-off. Unlike the traditional mechanically scanned radars, the phased arrays need not scan mechanically. Scanning is accomplished electronically by adjusting the phase within each transmit/receive (T/R) module (i.e., radiating element) to focus the wavefront, which is a line of equal phase radiation, in the desired direction. Key performance advantages of the phased arrays are the wide (octave) bandwidth, spectral purity, and the agile beam pointing capabilities. Fig. 1.2 shows a line array used as a receiving antenna. Each element is multiplied by a gain that phase shifts and amplifies the received signal from each array element. The array output is the sum of these weighted signals. In any usual radar environment, there are many reflectors. Some of these reflectors are of interest to the radar system and are called "targets." Some of these reflectors are of no interest to the radar system and are called "clutter." Reflectors that are of no interest to the radar system cause difficulties in two manners: (1) erroneously reporting clutter as target tends to overload the data processor's capacity and (2) clutter can prevent the detection and parameter estimation of desired targets. For airborne phased array radar, due to the strong ground clutter reflection, clutter suppression is critical for airborne phased array radar signal processing. The ground clutter seen by an airborne radar is extended in both range and angle. It also is spread over a region in Doppler due to the platform motion. A mesh plot view of the interference environment seen by an airborne phased array radar is shown in Fig. 1.3(a). The Jammer in Fig. 1.3 is localized in an angle and distributed over all Doppler frequencies. The clutter echo from a single ground patch has a Doppler 2 frequency that depends on its aspect angle with respect to the radar platform; clutter from all angles lies on the "clutter ridge" shown in Figure 1.3(a). A potential target may be obscured not only by mainlobe clutter that originates from the same angle as the target but also by sidelobe clutter that comes from different angles but has the same Doppler frequency. Therefore, clutter suppression is very challenging and important for airborne phased array radar. Clutter suppression has drawn a lot of attention during these years and many signal processing schemes have been proposed for it, which will be addressed in the following chapters. Target detection is critical for radar applications since without target detection, target feature extraction and target recognition are not possible. The radar target detection issue has been widely investigated since the beginning of the radar era. As we know, the degree of detectability of targets in a noise environment is controlled by the ratio of the reflected signal energy and the thermal noise power. Often, however, even when the target's signal-to-noise ratio (SNR) value is very large, it is difficult to detect it because the reflected energy is less than that reflected from the clutter. That is, if the radar cross section (RCS) of the clutter in the resolution volume containing the target is greater than the RCS of the target, it may be impossible to detect the presence of the target since it is obscured by the clutter. Thus, there are two related problems. The first is to prevent clutter from being reported as a target. The second is to detect targets despite the presence of clutter in the same resolution cell. Note that if a signal processor, based on a discriminant, attenuates the clutter return while not affecting the target return, both problems are alleviated. Again, this is another reason why clutter suppression is critically important for airborne phased array radar. For a radar system, resolution is defined as the ability of the radar to distinguish between two closely spaced targets in at least one of the state variables used 3 to describe the target vector. Conceptually, resolution is applicable to phased array radar in the domains of range or angle or Doppler frequency. In applications of conventional Low Range Resolution (LRR) airborne wide-area surveillance phased array radars, the moving target parameters (state variables) of interest include the target range (distance between the radar and target), Radar Cross Section (RCS), Direction-Of-Arrival (DOA), and Doppler frequency due to the relative motion between the target and the radar [13]. With the development of modern radar technology, especially with the dramatic improvement of the radar range resolution capability, the attention has been shifted from simple target detection and tracking to target recognition, mapping, and imaging capabilities [82, 42]. Future airborne radars will be required to provide increasingly high range resolution features of ground targets, which makes the signal processing needed by airborne high range resolution (HRR) phased array radars more important. Compared to a conventional airborne LRR radar [58], an airborne HRR radar can not only enhance the radar's capability of detecting, locating and tracking moving targets, but can also provide more features for applications including Automatic Target Recognition (ATR) [23, 25]. Two important technical issues associated with the signal processing of an airborne HRR radar are clutter suppression and feature extraction. It appears that few techniques have been reported for HRR clutter suppression [24, 17]. Moreover, the range migration resulted from the radial motion between the radar and the moving target will accumulate from pulse to pulse and destroy the range alignment. The range migration makes it impossible to directly use the LRR target parameter estimation approaches proposed in [72]. Therefore, the major signal processing techniques for airborne phased array radar are clutter suppression and moving target detection and parameter estimation, which are the motivation of our work. 4 1.2 Scope of the Work Our work starts with the airborne LRR phased array radar. Due to the importance of clutter suppression for airborne radar, in Section 3.2, we simulate the high fidelity ground clutter [80]. Then in Section 3.3, we introduce the Vector AutoRegressive (VAR) filtering technique [72] which whitens the clutter temporally, as well as a robust unstructured maximum likelihood (ML) method for moving target parameter estimation. We demonstrate that the VAR filtering technique and the unstructured ML estimation method work effectively and robustly for clutter suppression and parameter estimation. In Chapter 4, our discussion is extended to the airborne HRR phased array radar. To avoid the range migration problems that occur in HRR radar data, we first divide the HRR range profiles into LRR segments. Since each LRR segment contains a sequence of HRR range bins, no information is lost due to the division and hence no loss of resolution occurs. We show how to extend the VAR filtering technique to suppress the ground clutter for the HRR radar. Then a parameter estimation algorithm is proposed for target feature extraction. Numerical results are provided to demonstrate the effectiveness performances of the proposed algorithm. Since the multiple moving target scenario occurs frequently in radar applications yet little research work has been reported in the literature. In Chapter 5, our discussion is focused on multiple moving target feature extraction. The feature extraction is achieved by using a relaxation-based algorithm to minimize a nonlinear least squares fitting function by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently determined values. Thus the problem of multiple target feature extraction is reduced to the feature extraction of a single target, which is presented in Chapter 4, in a relaxation-based iteration step. Numerical examples are used to demonstrate the effectiveness of this multiple moving target feature extraction algorithm. 5 The discussion of multiple moving target detection follows the discussion of multiple moving target feature extraction. In Chapter 6, our discussion is concerned with multiple moving target detection for airborne HRR phased array radar, which is simplified to a sequence of single moving target detection problems based on a relaxation-based algorithm. Numerical examples have demonstrated the effectiveness of this multiple moving target detection algorithm even for contaminated secondary data. 1.3 Contributions The main results of this dissertation are as follows: * We introduce the VAR filtering technique and a ML parameter estimation method. We show the effectiveness of clutter suppression and parameter estimation by using these schemes. We compare the robustness of VAR filtering techniques to (Displaced-phase-center-antenna) DPCA in the presence of different system mismatches. * We present our data model of a moving target observed by an airborne HRR phased array radar in the presence of range migration. We extend the VAR filtering technique so that it can be used for airborne HRR phased array radar. We devise an effective parameter estimation method for moving target feature extraction. * We present the data model of multiple moving targets observed by an airborne HRR phased array radar in the presence of range migration. We derive a relaxation-based parameter estimation method for multiple moving target feature extraction by taking advantage of the technique we devised for a single moving target 6 * We introdue a moving target detector based on a Generalized Likelihood Ratio Test (GLRT) detection strategy and extend it to the case of multiple moving target detection. 1.4 Outline The dissertation is organized as follows. Chapter 2 gives a literature survey of topics including clutter suppression for airborne phased array radar, target parameter estimation, and target detection. In Chapter 3, we first introduce the ground clutter observed by an airborne radar, VAR filtering based clutter suppression technique, and unstructure ML target parameter estimation algorithm. We next demonstrate the effectiveness of the VAR filtering technique and ML parameter estimation method for airborne LRR phased array radar clutter suppression and parameter estimation. Chapter 4 addresses the VAR filtering technique for airborne HRR phased array radar and the target feature extraction algorithms. Chapter 5 discusses multiple moving target feature extraction for airborne HRR phased array radar. In Chapter 6, we investigate the multiple moving target detection problem. Finally, we summarize this work in Chapter 7. 7 Figure 1.1: Echoes from an aircraft and a moving vehicle on the ground. Wavefront angle of arrival Phase and amplitude control V V V V V out out sum Figure 1.2: Conceptual block diagram of line array antenna. 8 0 -25 -30, -1 -1 -0.5 -0.5 0.5 0 0.5 1 1 cos(\y) 2 fd/fr (a) -1 0 -0.8 -10 -0.6 -0.4 -20 --00 -30 0.2 -40 0.4 -50 0.6 -60 1 0.8 0.6 0,4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 2 fd/fr (b) Figure 1.3: Mesh plot of (a) the power spectral density of the interference (including clutter, jamming, and noise) and (b) Fully adaptive response when the clutter ridge is along the diagonal (' = 1) and the jamming is from Oj = 1200. 9 CHAPTER 2 LITERATURE SURVEY Historical and modern perspectives on the general topics of clutter suppression, moving target detection, and parameter estimation for airborne radar have been well documented in the literature. Among them, there are many classical articles, both theoretical and application-oriented, [74, 51, 1, 31, 32, 71, 47, 12], excellent texts are also available [45, 62, 52, 76, 4, 85, 78, 43, 3, 80, 11, 27, 72]. In this Chapter, we give a brief review of a number of subjects that are related to our work, namely clutter suppression and moving target detection and parameter estimation for airborne phased array radar. 2.1 Clutter Suppression For airborne radar, due to the strong ground clutter reflection, clutter suppression is critical for airborne radar signal processing. The ground clutter seen by an airborne radar is extended in both range and angle. It also is spread over a region in Doppler due to the platform motion. A potential target may be obscured not only by mainlobe clutter that originates from the same angle as the target but also by sidelobe clutter that comes from different angles but has the same Doppler frequency. In recent years, many signal processing schemes have been proposed for ground clutter suppression for airborne low range resolution (LRR) phased array wide-area surveillance radar. 2.1.1 Displace-phase-center-antenna (DPCA) DPCA [10, 62] processing is one of the simplest methods for clutter suppression currently used in certain airborne surveillance radar systems. DPCA is designed to 10 compensate for the phase shift of the outputs of antenna sensors due to the platform motion. For example, for the simple case of an array of two antennas with spacing ( in the direction of the flight path, DPCA compensation requires that ( = 2vT be satisfied, where v is the radar platform speed and T is the radar pulse repetition interval (PRI). Hence the airborne clutter can be cancelled by subtracting out the second antenna sensor output due to the (i + 1)st pulse from the first antenna sensor output due to the ith pulse. Since it is difficult to change the spacing among antenna sensors, the non-adaptive DPCA technique requires that the platform velocity v and PRI T be precisely controlled, which is difficult to achieve in practice. If ( = 2vT cannot be perfectly satisfied, then we have a system mismatch, to which DPCA is very sensitive. DPCA is also very sensitive to other uncertainties including array calibration errors and clutter fluctuations. 2.1.2 Adaptive Filtering Based Space-time Adaptive Processing (STAP) The adaptive filtering based STAP techniques [6, 31, 32, 12, 33, 77, 79, 80] simultaneously process the signals received from multiple elements of an antenna array and from multiple pulses. A basic illustration of space-time adaptive processing is given in Figure 1.3(b). A space-time adaptive processor may be thought of as a twodimensional (2-D) filter that represents combined receiver beamforming and target Doppler filtering. Note the high gain at the target angle and Doppler, and the deep nulls along both the jamming and clutter lines. Applying this filter to the data will suppress the interference and enable target detection. A bank of adaptive filters is then formed to cover all potential target angles and velocities. STAP is an adaptive approach and can outperform DPCA in the presence of many system mismatches and uncertainties. However, STAP may require a significant amount of computations because it needs a bank of filters and may require the inversion of matrices of large dimensions, which makes it often too complex for real-time implementation. STAP requires that the clutter is homogeneous. That is, 11 the statistical distribution of the interference in the primary data, which is usually with targets of interest, is the same as (or at least "close to") the distribution of the clutter in the secondary data, which is assumed to be target free. Unfortunately, the large number of desired secondary data often prevents the secondary data from containing truly homogeneous data, which could result in a dramatic performance degradation. Further, STAP cannot be used for jamming suppression if the secondary data bins used to obtain the second-order statistics of the ground clutter do not contain jamming interference (deceptive jammers). 2.1.3 Vector Auto-regressive (VAR) Filtering The VAR filtering technique [44, 72] is proposed to model the ground clutter observed by an airborne radar as a vector auto-regressive random process, whose coefficients are estimated adaptively from the target-free secondary data. The VAR filter is then used to suppress the clutter in the primary data temporally where a target may be present. Although the technique can be easily used for spatial whitening as well, it is not needed since the VAR-filtered interference is assumed to be spatially colored with an arbitrary unknown covariance matrix, which automatically achieves jamming suppression when the VAR filter output is used with the Maximum Likelihood (ML) methods presented in [72] to estimate the target parameters for LRR radar. Different from the adaptive filtering based STAP, in which usually clutter suppression is performed first and then the target direction of arrival (DOA) and Doppler frequency are determined by the pointing angle of the beam (or monopulse processing), the VAR-filtered primary data can be used with a robust unstructured Maximum Likelihood (ML) parameter estimation method proposed in [72] to achieve asymptotically statistically efficient results. In this dissertation, we not only demonstrate the effectiveness of the VAR filtering technique for airborne LRR phased array radar clutter suppression but also extend it to be used for airborne HRR phased array radar. 12 2.2 Moving Target Detection Target detection is critical for radar applications since without target detection, target feature extraction and automatic target recognition (ATR) are not possible. The radar target detection issue has been widely investigated since the beginning of the radar era. Most previous investigations on the detection of a radar target in clutter and noise are based on some statistical models of the clutter plus noise, which include Gaussian, Log-Normal [20], Weibull [19, 16], K-family [81, 46, 7, 56] distributions, and more recently, the so called alpha-stable (SaS) distribution [28, 60] and compound-Gaussian distribution [9, 8]. These statistical model based detection algorithms consider mainly the temporal statistics of the radar measurements of the clutter. The non-Gaussian detection schemes may suffer from complicated model parameter estimation problems. A classical radar problem is to detect targets of unknown amplitude embedded in clutter with unknown statistical properties. It is assumed that returns from other range cells are available to provide an estimate of the statistical properties of the clutter. Kelly derived a decision rule for detection of a signal of unknown amplitude masked by Gaussian interference with unknown covariance [30]. The decision rule is a generalized maximum likelihood ratio test (GLRT) in the sense that the conditional densities that constitute the likelihood ratio are maximized over the unknown parameters. In his paper, Kelly does not make any optimality claims for his detector. However, recent work by Scharf and Friedlander has shown that equivalence of the GLRT and uniformly most powerful invariant detectors, thereby conferring a sense of optimality to the GLRT [57]. Additionally, the GLRT exhibits the property of constant false alarm rate (CFAR) implying that the probability of false alarm is independent of the noise covariance matrix. More recently, Swindlehurst and Stoica develop a GLRT for target detection under an unstructured array model [72]. 13 They compare the unstructured GLRT with the standard test via simulation for various levels of array perturbation. Even in the absence of such perturbations, the unstructured GLRT may be useful in providing a more rapid "initial scan" of the environment prior to application of the structured model. In [41], Liu and Li investigate the moving target detection in the presence of ground clutter for airborne HRR phased array radar. Due to the high range resolution of the HRR radar, the scatterers of a moving target may move out of their range bins during the coherent processing interval (CPI) and hence results in the range migration problem. They divide the HRR range profiles into non-overlapping large range segments to avoid these range migration problems that occur in HRR radar data. A moving target detector based on a GLRT detection strategy is derived in [41]. The detection threshold is determined according to the desired false alarm rate, which is made possible via an asymptotic statistical analysis. After the target Doppler frequency and spatial signature vectors are estimated, a simple detection variable is computed and compared to the detection threshold to render a decision on the presence of a target. Numerical results are provided to demonstrate the performance of the proposed moving target detection algorithm. 2.3 Parameter Estimation The essence of the spectral estimation problem is [66] "From a finite record of a stationary data sequence, estimate how the total power is distributed over frequency". Spectral analysis is widely used in many diverse fields, including economics, meteorology, astronomy, speech analysis, and radar and sonar systems. Generally speaking, there are two branches of analysis methods: non-parametric and parametric methods. Just opposite to the non-parametric method, the parametric or model based methods of spectral estimation assume that the signal satisfies a generating model with known functional form, and then proceed by estimating the parameters 14 in the assumed model. The signal's spectral characteristics of interest are then derived from the estimated model. In those cases where the assumed model is a close approximation to the reality, it is no wonder that the parametric methods provide more accurate spectral estimates than the nonparametric techniques. The nonparametric approach to Power Spectral Density (PSD) estimation remains useful, though, in applications where there is little or no information about the signal in question. The parametric approach postulates a model for the data, which provides a means of parameterizing the spectrum, and thereby reduces the spectral estimation problem to that of estimating the parameters in the assumed model. Besides the Maximum Likelihood (ML) estimation method [65, 67], the parametric methods often used include, for example, autoregressive (AR) model based methods [15, 14], eigendecomposition based methods including MUSIC [59, 29, 48, 22, 50, 68] and ESPRIT [55, 54, 34, 35], and nonlinear least squares (NLS) fitting based methods as CLEAN [21, 53, 73] and RELAX [38, 39, 40]. RELAX is used in the feature extraction of our work due to its conceptual and computational simplicity and its excellent asymptotic statistical properties. 2.3.1 The RELAX algorithm The RELAX algorithm [38]. To minimize the Nonlinear Least Square (NLS) cost function, the RELAX algorithm performs a complete search for the global minimum of the cost function by letting only the parameters of one scatterer vary at a time while freezing the parameters of all other scatterers at their most recently determined values. The RELAX algorithm can also be referred to as SUPER CLEAN. If the number of reiterations in each step of the RELAX algorithm is set to be zero, then RELAX becomes the CLEAN algorithm, which was first proposed in radio astronomy [21]. CLEAN is computationally more efficient than RELAX but its resolution and estimation accuracy are inferior to RELAX. Hence RELAX is preferred in 15 many applications where high resolution and high estimation accuracy of individual scatterers are desired. 2.3.2 ML methods in Radar Array Signal Processing Standard solutions to the problem of radar array signal processing involve the use of classical space-time filters, either data independent (e.g., as with a delay-andsum beam former) or "adaptive" (e.g., using linear constraints, maximum signalto-interference-plus-noise ratio criteria, etc.) [80]. ML approaches have not been extensively considered for this problem, mainly because they are perceived to be too computationally complex. Recent work has shown that for a single target source in Gaussian interference with unknown spatial covariance, the ML solution can be obtained via a two-dimensional (2-D) search over target DOA and Doppler [75]. In [72], Swindlehurst and Stoica demonstrate that the 2-D search required for the ML solution can be replaced with two simpler one-dimensional (1-D) searches without affecting the asymptotic accuracy of the estimates. If the array is uniform and linear, then only a single 1-D search is required. The resulting computational savings have the potential for making the ML approach more feasible for radar applications. The key idea behind the simplification of the algorithm is the use of the so-called extended invariance principle (EXIP) in estimation first introduced in [70]. This technique involves reparameterizing the ML criterion in a way that admits a simple solution and then refining that solution by means of a weighted least squares (WLS) fit. In particular, they initially use an unstructured model for the array response instead of one parameterized by the DOA of the target signal. Under this model, the ML solution reduces to a search over only the target Doppler frequency. Refined estimates of the target parameters, including DOA, are then obtained via WLS, where the optimal weighting is simply the Fisher information matrix (FIM) corresponding to the unstructured ML criterion. An additional advantage of using an unstructured 16 model for the array is that it provides robustness to errors in the model for the array response. 17 CHAPTER 3 MOVING TARGET PARAMETER ESTIMATION FOR AIRBORNE LRR RADAR 3.1 Introduction The ground clutter observed by an airborne radar is Doppler spread due to the platform motion. Because of the importance of clutter suppression for airborne radar, many clutter suppression methods are presented in the literature in recent years, which include DPCA and adaptive filtering based STAP. Their advantages and disadvantages are introduced in Chapter 2. In [44, 72], a new idea was proposed to model the ground clutter observed by an airborne radar as a VAR random process, whose coefficients are estimated adaptively from the target-free secondary data. The VAR filter is then used to suppress the clutter in the primary data temporally where a target may be present. Different from adaptive filtering based STAP, in which usually clutter suppression is performed first and then the target DOA and Doppler frequency are determined by the pointing angle of the beam (or monopulse processing), the VAR-filtered primary data can be used with a robust unstructured ML parameter estimation method proposed in [72] to achieve asymptotically statistically efficient results. In this chapter, we first simulate the high fidelity ground clutter [80]. Then we quantitatively demonstrate the effectiveness of the VAR filtering technique proposed in [72] for clutter and jamming suppression, which is missing in [72]. We also compare the performances of VAR filtering and DPCA when used with the robust unstructured ML method in [72] for moving target parameter estimation in the presence of various system mismatches. 18 3.2 Clutter Simulation The ground clutter used in our simulations is generated according to the clutter model in [80]. A continuous field of clutter is modeled in [80] as a superposition of a large number of uncorrelated point scatterers which are evenly distributed in azimuth around the radar in all possible ambiguous range bins. Assume that the number of range bins is N,, each of which consists of N, patches in azimuth. The center location of the lth clutter patch in the ith range bin is specified by the slant range Ri, the elevation angle /i, and the azimuth angle 01. Assume that the airborne radar platform is moving with a velocity v. The spatial frequency and the normalized Doppler frequency of the lth clutter patch in the ith range bin are, respectively, di = -cosoicosol, (3.1) and 2vT wil = -Tcosoicos01, (3.2) where ( is the spacing between two adjacent sensors. The clutter return due to the mth spatial antenna for the nth pulse is Nr Nc Xnm= --ailexp{j27r(mi l+nwii)}, n = 0, **,N-1, m = 0, .,M-1, (3.3) i=1 l 1 where ail denotes the random complex amplitude of the lth clutter patch in the ith range bin. Let E{lai2} = a2il, (3.4) where E{.} represents the expectation, a2 is the noise power per element, and (il denotes the clutter-to-noise ratio (CNR) of the lth clutter patch in the ith range bin, which is related to the terrain type and can be determined by the radar equation [80]. 19 Define the spatial and temporal steering vectors associated with the lth clutter patch in the ith range bin to be, respectively, vs(i, 1) =[ ej2i ... ej2r(M-1)til T (3.5) and vt(i,) = [ ej2)mji ... e2x(N-l)~i ]T (3.6) Rewrite the clutter sequence Xnm, m = 0, ,M - 1, n = 0,... , N - 1, in a 1-D vector Nr Nc X = ZZ avt(i,1) 0 v,(i,1) c CMNx', (3.7) i=1 1=1 Where 0 denotes the Kronecker matrix product [18]. Then the clutter is modeled as a Gaussian random process with the space-time covariance matrix expressed as Qc = E{xXc Nr Nc = E{il2}(vt(i 1)VH(i, 1)) 0 (v'(i, 1)VH(i, 1)), (3.8) i=1 1=1 which is a Toeplitz-block-Toeplitz matrix with dimension MN x MN. Given (, v, and T, the clutter ridge slope . . . (3.9) -i 2vT is a constant. The 2-D clutter spectrum in the plane of normalized spatial frequency versus normalized Doppler frequency appears as a straight line with a slope of 7 (see Figure 3.1 (a) and (b) for the 2-D clutter spectrum with 7 = 1 and 7 = 1.2, respectively). So far, the clutter model in (3.8) is non-fluctuating, i.e., the returns from a clutter patch do not fluctuate from pulse to pulse. However, in practice, many factors, such as any pulse-to-pulse instabilities of the radar system components and the intrinsic clutter motion due to wind, may cause small pulse to pulse fluctuations 20 in the clutter returns. Clutter fluctuation will broaden the clutter ridge and make the clutter rejection more difficult. Define, asi= ail ... ail 1 , (3.10) where a' is the random complex amplitude for the lth clutter patch in the ith range bin from the nth pulse. Then (3.7) becomes N, Nc S= (i 0 vt(i, 1)) 0 vs(i, 1), (3.11) i=1 1=1 where 0 denotes the Hadamard matrix product [18]. Usually, the fluctuation is modeled as a wide-sense stationary Gaussian random process. Thus we have [80]: Ti (n) = E i+n)(a,) H _ a2ilexp - 2 4 n2 j (3.12) where a, is referred to as the velocity standard derivation. Plugging (3.12) into (3.8), we have the covariance of the fluctuating clutter: N, Nc Qc = (Fit 0 v(i, l)vt(i, 1)H) ï¿½ (v,(i, 1)v (i, (3.13) i=1 1=1 where [ it (0) Tit(1) - yil,(N - 1) Fit = E{aii(iit)H = Yil() il(O) - iz(N - 2) 7i1(N - 1) yii(N - 2) ... -yi (O) is the N x N covariance matrix describing the fluctuations for the lth patch of the ith range bin (see Figure 3.1 (c) for the 2-D clutter ridge with a = 1 and 7 = 1). 3.3 Clutter Suppression 3.3.1 Data model For an airborne surveillance radar, where at most one target is assumed to be present in any given range bin, the sampled output vector of an array of M sensors 21 due to the nth look of the N looks during a coherent processing interval (CPI) can be written as: x(n) = ba(O)ejwn + e(n) E CMxl, n = 0,. N - 1, (3.14) where b is the complex amplitude of the signal proportional to the target RCS, w is the normalized Doppler frequency due to the relative motion between the array platform and the target, a(O) is the array response due to the target from the DOA 0 relative to the flight path, and e(n) represents the interference due to clutter, jamming, and noise. Assume that the data is collected with a uniform linear array (ULA) of M sensors, i.e., a() = [1 e3\cose ... ej(M-1)?\cos IT . (3.15) The data sequence {x(n)}No'1 is referred to as the primary data. The data from adjacent range bins are assumed to be target free and are referred to as the secondary data. Note that due to the jamming free secondary data, the conventional STAP methods fail and hence are not considered hereafter. (For the suppression of highduty-cycle jammers where jammers exist in both primary and secondary data, both our method and other STAP approaches can handle it without difficulty and hence are ignored.) Rewrite (3.14) as x(n) = aejwn + e(n) E CMX1, n = 0, . . N - 1, (3.16) with a = ba(0). (3.16) and (3.14) are the so called unstructured and structured data model, respectively. 3.3.2 VAR Filtering By assuming that the statistics of the interference in the primary and secondary data are the same, the secondary data are used to adaptively estimate the 22 VAR filter coefficients [72], which are assumed to have the form: P '(z-1) = I + Hz-P, (3.17) p=1 where P is the VAR filter order, z-1 denotes the unit delay operator, and I is the identity matrix. Next we outline the estimation of the VAR filter W(z-1) using the target-free secondary data. We denote the secondary data by {e1(n)} , = 1, - *, S, where S is the number of secondary range bins used to estimate the VAR filter coefficients. The VAR filter is obtained such that [72] lt(z-')es(n) = es(n) + Hles(n- 1) +... -+ Hpes(n- P) = E(n), n= P, N ,- 1, (3.18) is temporally white for each s. This approach assumes that the statistics of the clutter is the same in all of the primary and secondary data sets. The problem of estimating the VAR filter coefficients is based on the following least-squares error criterion [72]: S N-1 P 2 I1,... ,Hp arg min e(n)+ 1 He(n -p) . (3.19) s=1 n=P p=l Let H = [H... Hp], (n) -[e (n - 1)...e T(n- )], , = [(P) .. ,(N- 1)], E, = [e,(P) ... e,(N - 1)], and E = [El... Es] 23 Then (3.19) can be rewritten as HI = argmin liE - Hll|, (3.20) H where I - ||F denotes the Frobenius norm. Provided that the matrix TFH has full rank, then H has the form [72] HI = EWiH (XITH)-1. (3.21) A necessary condition for the existence of the matrix inverse in (3.21) is that WI has at least as many columns as rows, i.e., S(N - P) > MP, a condition that can be easily satisfied by choosing S large enough. Let us take a look at the effect from the possiblely existing jammer in the secondary data on the filter coefficients. Assume the interference in the secondary data includes only jammer and noise. From (3.18), straightforwardly, we have {es(n)eH(n)+..+HLes(n-L)e H(n)} = E{E(n)eH(n)} E{e(n)eH(n(-1) + ...+HLe(n- L)e*(n - 1)} = E{(n)eH (n (3.22) E{es(n)eH(n- L) +...+ HLeS(n - L)eH(n- L)} = E{e,(n)eH(n- L)}, where E{x} denotes the expected value of x. Let E{e,(n)e,H(n-1)} = Q(1) C CMXM, then rewrite (3.23) by matrices, we have Q(0) Q(-1) ... Q(-L) Q(1) Q(0) ... Q(-L + 1) [I H1 ... HL] Q(1) Q(0) . Q(-L+1) = Q(1) 0 ... 0]. Q(L) Q(L - 1) ... Q(0) Hence, [fl1 " i L] = -[ (-1) .. (^(-L)] 24 Q(0) Q(-1) ... Q(-L + 1) Q(1) Q(0) ... Q(-L + 2) Q(L-1) Q(L-1) ... Q(0) where Q(1) denotes the estimated covariance matrix from the available secondary data. If there are only noise and jammer in the secondary data, due to the properties of noise and jammer, Q(1) = 0, for 1 # 0. So from eqn. (3.23), we will have [ ]1i ... IIL ] 0. (3.24) Therefore, the jammer in the secondary data contributes little to the VAR filter coefficients and the VAR filter has little effect on the jammer in the primary data. The filtered array output will have the form, y(n) = l(z-')x(n) = b-i(e-)a(O)eiwn + E(n) ba(0, w)e" + E(n) (3.25) = e + E(n) E CM , n=P, - - -, N - 1, (3.26) where a(0, w) = W(e-j')a(O) and a = bW(e-j")a(O). The moving target parameters {b, 0, w} can then be estimated from the VAR filter output by using the unstructured approach presented in [72]. 3.3.3 DPCA Let Xm(n) be the mth element of x(n) in (3.14). Then the DPCA processing yields zm(n) = x+1(n) - m(n + 1), n = 0, , N - 2, m = 0, , M - 2. (3.27) 25 Let z(n) = [ zo(n) . ZM-2(n) ]T. Rewriting (3.27) in a vector form, we have z(n) = ba(0)e + i(n), c &ej" + i(n) C C(M-1)x1, n = 0, ... , N - 2, (3.28) where b = b(eJ2" COs - ejw), (O) is a subvector of a(0) containing the first M - 1 elements of a(O), and i(n) is the interference vector after the DPCA processing. For the ideal case of non-fluctuating clutter and no system mismatches, z(n) is free of clutter, but the noise level is doubled. By applying the unstructured approach presented in [72] to z(n), we can estimate the moving target parameters {b, 0, w}. 3.4 Parameter Estimation We outline the robust unstructured ML parameter estimation method [72] here. The negative log-likelihood function for N samples of data from model (3.25) is shown as: VN(b, 0, w) = log|QI + Tr{Q-C(b, 0, w)}, (3.29) where N-1 C(b, 0, w) = (y(n) - ba(O, w)eJ")(y(n) - ba(0, w)ej")H. (3.30) n=O Let Qj with Qj being the (ij)th element of Q. By using some matrix computation rules, we have OlnIQI -1 I =Qi tr(Q-Qj) (3.31) and 8Q-1 1Q = -Q- Q = -Q- QQ'. (3.32) So ac O = -tr(Q-'Qj) + tr(Q-1'Q Q- C) (3.33) and tr(Q-Ql )= [Q-']i . (3.34) 26 Hence the ML estimate of Q is given by Q-T + [Q-1CQ-1] lq=' = 0. (3.35) Then Qs = C(b, 0, w). (3.36) where the subscript s corresponds to the estimates of the structured data model. Substitute (3.36) into (3.29), we have: VN(b, 0, w) = log C(b, 0, w)| + M. (3.37) As shown in [37], we have bs = ) (3.38) 2 a(9, w)(w) 2 O, ij, = arg max (3.39) o,w (1 - yH(w)R-1(w))gH(O, w)R-1(O, w)' where N-1 (w) = 1 y (n)eJn", (3.40) n=0 N-1 R= E y(n)yH(n). (3.41) n=o Re = R- y(w)yH(w), (3.42) b5, 9,, and &, are the ML estimates of b, 0, and w in (3.25), respectively. However, the estimates 0, and OcD are coupled in a 2-D search. To simplify the 2-D search algorithm, we turned to the unstructured data model (3.26). Similarly, the negative log-likelihood function for N samples of data is shown to be: VN(d, w) = logQI + Tr{Q-'C(, w)}, (3.43) where SN-1 C(a,w) = N k (y(n) - dewn)(y(n) - deijn)H. (3.44) n=O 27 N-1 (, = Z (y(n) - e")(y() - ejwn)H. (3.45) n=0 Hence, 1 N-1 Jwn) VN(a, w) = log Z(y (n) - (y(n)n) - e - e n)(H( n=O = log R- y(w)&H - H(w) + H = log (- ( ())( - y(w))H - (c)() . (3.46) From (3.46), clearly, the estimate of a is, Ce = y(w). (3.47) The remaining term in (3.46) is a function of w only, and can be written as log R( - w)y H(w) log( R I-RI(w)yH(w ) = log(1 - yH(pw)f-1y(w)) + loglRI. (3.48) Hence, 0, = arg maxH ()-ly(w), (3.49) and Q. = R- y H(Cj u = R- && . (3.50) where subscript u corresponds to the estimates of the unstructured data model. Because, according to [72], On in (3.49) is asymptotically equivalent to the c-, in (3.39), we can replace js, by ~, in (3.38) and (3.39). The resulting estimates for b and 0, be = ,(, U)lI (3.51) 2 a = argmax (3.52) 0 gaH(O, U )I-I(0, D)' 28 should have the same asymptotic variance as the structured ML estimates in (3.38) and (3.39). From (3.50) and the matrix inversion lemma [18], we have (see [72] for detail) -1- - 1 H H 1 __-I Qu = R- I Htf + R- 1 + &HR-16& and 1 + a )R-& Thus 2 O H 1, Ou = arg max 0 H 2/, C(0, 01/ 2 1 + R aCH )R-l.(O, Wu) - I H . 1+(, R-')= arg max ,H(O, .)-a(O, )' O gH )u)R) ( 10 C ) = O,. (3.53) Since (3.50) implies that Q 2 = R,, immediately, we have aH( e, a KQn1l be = (3.54) a&(e, ),'a(, a) Note that the simplicity of the final results of be, Ge, and O, are quite remarkable. A 2-D search (see (3.39)) is simplified to two 1-D searches (see (3.49) and (3.52), respectively). Since neither the search for 0 nor w is a search on a concave, we first find an approximate location with coarse grids and then use it as the initial condition to find a more accurate estimate via a 1-D search method, such as the FMIN function in MATLAB. Since the estimates {be, Ge, JO} have the same asymptotic variance as the ML estimates in (3.38) and (3.39) (see [72] for more details), they are also statistically efficient estimates of {b, 0, w} for the target model in (3.14). 29 3.5 Numerical Results We present several numerical examples to demonstrate the effectiveness of the VAR filtering approach for clutter suppression. The following parameters are common to all of the simulations: * The target complex amplitude b = 1, the normalized Doppler frequency w = 0.27r, the DOA 0 = 600, the wavelength A = 0.03 m and the interelement distance ( = A/2. * The VAR filter order P = 2 and the number of secondary range bins S = 5. * The number of antenna elements M = 8 and the number of pulses in a CPI N = 16. * The noise is the zero-mean white Gaussian random process with variance xa. The clutter is the zero-mean colored Gaussian random process with the covariance matrix Qc. The jamming is assumed to be zero-mean and temporally white Gaussian with variance a and its DOA is Oj. In our simulations, the SNR, jamming-to-noise ratio (JNR), and CNR, respectively, are defined as: SNR = 10log10 lb12 JNR= 10logo0 2 ' and CNR = 10loglo Qc(l ' where Qc(i, j) is the ijth element of Qc and the JNR and CNR are set to be 25 dB and 40 dB in the simulations, respectively. If the clutter ridge is along the diagonal, the clutter ridge slope ' = (/(2vT) = 1. For comparison purposes, we have also included the associated CRBs for the moving target parameters. First, consider the case of non-fluctuating clutter, no system mismatches, and no array calibration errors. We assume that there is also no jamming. For this case, 30 since DPCA uses the a priori correct knowledge of ( = 2vT and ULA, DPCA can achieve complete clutter rejection. VAR filtering cannot achieve complete clutter rejection since it does not rely on this knowledge, but instead, it relies on a limited amount of the secondary data adaptively. In Figure 3.2, however, we note that when SNR increases from -5 dB to 20 dB, the performances of DPCA and VAR filtering are quite similar. The MSEs of the estimates of the target parameters {b, 0, w} can be quite close to the corresponding CRBs at high SNR. We now assume that there is jamming from Oj = 1200. Figure 3.3 shows that the performances of both VAR filtering and DPCA are similar to the corresponding ones in Figure 3.2. Note that VAR filtering and DPCA cannot suppress the temporally white jamming. However, the unstructured parameter estimation method presented in [72] can deal with the jamming type of interferences that is temporally white but a point source in space via estimating the spatial covariance matrix of the interference along with the target parameters. Next, consider the case of system mismatches. Let 7 = 1.2 and the other parameters are the same as those in Figure 3.2. Note that DPCA is very sensitive to the system mismatches, as shown in Figure 3.4, because DPCA still assumes = 1 and only when 7 = (/(2vT) is an integer can DPCA cancel the clutter effectively. Since VAR filtering is an adaptive method, it can handle this error without any problem. We remark that by using some interpolation methods, the mismatches can be compensated out to some degree of success, but at a price of more complicated signal processing. As discussed in Section 3.2, fluctuating clutter causes a broadening of clutter ridge and makes the clutter rejection more difficult. In Figure 3.5, we set the velocity standard derivation av = 0.05. Note that VAR filtering is much more robust than DPCA against clutter fluctuations, although VAR filtering also suffers from some performance degradations. 31 Finally, we take into account of array calibration errors. Since the VAR filter 7-(z-1) does not rely on the knowledge of the array manifold a(O) and is estimated adaptively, VAR filtering is robust against array calibration errors. Without loss of generality, we only consider the mutual coupling effects among antenna elements, which occurs often in practice. Mutual coupling among antenna elements [13] is present when the signal output of an individual antenna element has not only one dominant component due to the direct incident plane wave, but also several lesser components due to scattering of the incident wave at the neighboring elements. In Figure 3.6, we set the mutual coupling coefficient to be p = 0.1, i.e., the a(O) in (3.15) is now pre-multiplied by a Hermitian Toeplitz matrix formed from [1, p, p2, ... pM-1]. Note that in the presence of mutual coupling errors, VAR filtering suffers little performance degradations as can be seen by comparing the Doppler frequency estimates in Figures 3.5(c) and 3.2(c). Note that the poor estimates of b and 0 in Figures 3.5(a) and 3.5(b) are due to the mutual coupling errors since these estimates are based on the ULA assumption of a(0) in (3.15) (see [72] for details). Due to mutual coupling, the MSEs of the estimates of {b, 0} do not decrease as fast as the increase of SNR since the errors due to the presence of mutual coupling start to dominate at high SNR. Since the estimation of the Doppler frequency w via the unstructured method in [72] is not based on the ULA assumption, w can still be estimated accurately. 3.6 Summary In this chapter, we have compared using VAR filtering and DPCA for airborne radar clutter suppression and moving target parameter estimation. We have shown that since the VAR filtering technique is an adaptive method, it is much more robust than DPCA against system uncertainties including system mismatches, array calibration errors, and clutter fluctuations. 32 -1 a0 -0.8 -5 -0.6 -10 -0.4 -15 -0.2 -20 -25 0.2 -30 0.4 S-35 08 -40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 08 1 (a) -1 0 -0.8 -0 6 -10 -0.4 -15 -0.2 --20 0 0 .2 -25 0.4 -30 0.6 -35 0.8 ~-.-40 -1 -08 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 2 fd/,fr (b) -1. 0 -0.8 -5 -0.6 -0.2 - -20 0 -25 0.2 0.4 0.6 -35 0.8 -40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (c) Figure 3.1: Clutter ridge for (a) = 1, (b) 7 = 1.2, and (c) a = 0.2 and 5 = 33 20 -10 SNR (dB) (a) CRB - -0- OPCA -+ - VAR 110 -10 -20 --5 5--SNR (dB) o (b) Figure 3.2: Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB and there is no system mismatches, no array calibration errors, and no jamming. 34 CRB -- DPCA SNR (dB) (a) --- DPCA -- VAR a10 -0 t 10 1 SNR (d) (b) SNR (M0) (c) Figure 3.3: Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB, the jammer has JNR=25 dB, and there is no system mismatches and no array calibration errors. 35 - CRB DPCA - VAR - 0 15 SNR (dB (a) -30 ----- -- - - - -010 -10 - ---ACRB VAR 0 5 10 15 SNR (dB) (b) 10 -5C-F- CRB SNR (d) (c) Figure 3.4: Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency in the presence of system mismatches (' = 1.2) and when the clutter is non-fluctuating with CNR = 40 dB and there is no array calibration errors and no jamming. 36 oRB 0 -- VAR 40 S 0 5 10 15 -10 -20 -0 15 0 15 SNR (lB) (b) - -*-- VAR 40 -50 005 10 I5 0 SNR (dB) (c) Figure 3.5: Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is fluctuating with o, = 0.05 and CNR = 40 dB and there is no system mismatches, no array calibration errors, and no jamming. 37 0- VAR -.-.-.- - - - - - -A SNR (dB) (a) -- 1CRB -CA - - - - - - - - -------- ---- a 5 10 SNR (dB) (b) - 4- DPCA S -* VAR D-0 40 0-~ SNR (dB) (c) Figure 3.6: Comparison of the MSEs of the target parameter estimates after VAR filtering ("*") and DPCA processing ("+") with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB in the presence of antenna mutual coupling with p = 0.1 and there is no system mismatches and no jamming. 38 CHAPTER 4 MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR 4.1 Introduction Airborne radars are used to detect the presence of moving targets and estimate their parameters in the presence of noise, ground clutter, and jammers. In applications of conventional Low Range Resolution (LRR) airborne wide-area surveillance phased array radars, the moving target parameters of interest include the target range, Radar Cross Section (RCS), Direction-Of-Arrival (DOA), and Doppler frequency due to the relative motion between the target and the radar [13]. With the development of modern radar technology, especially with the dramatic improvement of the radar range resolution capability, the attention has been shifted from simple target detection and tracking to target recognition, mapping, and imaging capabilities [82, 42]. Future airborne radars will be required to provide increasingly high range resolution features of ground targets, which makes the signal processing needed by airborne high range resolution (HRR) phased array radars more important. Clutter suppression is critical for airborne radar signal processing. The ground clutter observed by an airborne radar is spread over two dimensions of both the range and spatial angle and the clutter spectrum also covers a certain Doppler region due to the platform motion even though the clutter itself may be stable [80]. Many signal processing methods have been proposed to suppress ground clutter for airborne LRR phased array radars. With the non-adaptive Displaced-Phase-Center-Antenna (DPCA) processing [10], the clutter is canceled by simply subtracting outputs from different apertures with the same effective phase center on a pulse-to-pulse basis. DPCA is sensitive to both antenna element mismatch and velocity controlling errors. 39 Space-Time Adaptive Processing (STAP) based techniques [80, 79] simultaneously process the signals received from multiple elements of an antenna array and from multiple pulses. STAP is data-adaptive and can outperform DPCA in the presence of sensor mismatch or velocity controlling errors. However STAP can require a significant amount of computations due to the need of using a bank of filters and the inversions of matrices of large dimensions. A Vector Auto-Regressive (VAR) filtering technique was recently proposed in [72] to suppress the clutter adaptively. The VAR filtering technique whitens the correlated clutter only temporally, and can be computationally simpler than STAP. Although the technique can be easily used for spatial whitening as well, it is not needed since the VAR-filtered interference is assumed to be spatially colored with an unknown covariance matrix, which automatically achieves jamming suppression when the VAR filter output is used with the Maximum Likelihood (ML) methods presented in [72] to estimate the target parameters for LRR radar. Compared to a conventional airborne LRR radar [58], an airborne HRR radar can not only enhance the radar's capability of detecting, locating and tracking moving targets, but can also provide more features for applications including Automatic Target Recognition (ATR) [23, 25]. Two important technical issues associated with the signal processing of an airborne HRR radar are clutter suppression and feature extraction. It appears that few techniques have been reported for HRR clutter suppression [24, 17]. Moreover, the range migration resulted from the radial motion between the radar and the moving target will accumulate from pulse to pulse and destroy the range alignment. The range migration makes it impossible to directly use the LRR target parameter estimation approaches proposed in [72]. In this paper, we present moving target feature extraction algorithms in the presence of temporally and spatially correlated ground clutter for airborne HRR phased array radars. In addition to the common DOA and Doppler frequency of all 40 scatterers of a rigid-body target, the parameters of interest also include those related to the RCS and range of each scatterer. To avoid the range migration problems that occur in HRR radar data, we first divide the HRR range profiles into LRR segments. Since each LRR segment contains a sequence of HRR range bins, no information is lost due to the division and hence no loss of resolution occurs. We show how to use the VAR filtering technique to suppress the ground clutter. Then a parameter estimation algorithm is proposed for target feature extraction. From the VAR-filtered data, the target Doppler frequency and the spatial signature vectors are first estimated by using a ML method. The target phase history and DOA (or the array steering vector for unknown array manifold) are then estimated from the spatial signature vectors by minimizing a Weighted Least Squares (WLS) cost function. The RCS related complex amplitude and range related frequency of each target scatterer are then extracted from the estimated target phase history by using RELAX [38], a relaxation-based high resolution feature extraction algorithm. The remainder of this Chapter is organized as follows. In Section 4.2, we introduce the data model and formulate the problem of interest. The VAR filtering technique for clutter suppression is also described in that section. Section 4.3 presents the moving target feature extraction algorithm for airborne HRR phased array radars. In Section 4.4, numerical examples are presented to illustrate the performance of the proposed algorithm. Finally, Section 4.5 contains our conclusions. 4.2 Data Model and VAR Filtering The range resolution of a radar is determined by the transmitted signal bandwidth. To achieve high range resolution, a radar must transmit wide-band pulses, which are often linear frequency modulated (LFM) chirp signals [82]. The range resolution of an LRR radar is much larger than the length of a target so that the target occupies only one LRR range bin, while the range resolution of an HRR radar 41 is so small that the target occupies several HRR range bins. The data collected by an LRR radar is only a small subset of the data collected by an HRR radar. Consider an airborne HRR radar having a one-dimensional (1-D) antenna array with M elements uniformly spaced along the flight path of an airborne platform. A cluster of N chirp pulses is transmitted during the coherent processing interval (CPI). After dechirping, sampling, and Fourier transformation of the signals in each element of the array, we obtain the HRR range profiles. These profiles may be blurred by the ground clutter so severely that without clutter suppression, they are not useful for any applications. To avoid range migration problems, we divide each HRR profile into non-overlapping LRR segments so that each LRR segment contains L HRR range bins, as shown in Figure 1. We choose L to be much larger than the maximum number of range bins over which any target can possibly expand and migrate during the CPI. We then apply inverse Fourier transform to each segment. For the segment of interest, where a target may be present, the inverse Fourier transform (IFFT) yields the primary data, which can be written as (see Appendix A for the model derivation): x/(n) = a(O) akej27rfkl ejvnlejwOn + el(n), 1k=1 l= 0,---,L - 1, n= 0,--.,N - 1, (4.1) where K is the number of scatterers of the target, N is the number of pulses transmitted during the CPI, and L is the number of HRR bins per LRR cell as well as the number of target phase history samples. The complex amplitude ok and the frequency fk are, respectively, proportional to the RCS and range of the kth scatterer of the target. The vectors x1(n), a(O), and ee(n) are M x 1; x1(n) is the array output vector of the lth phase history sample due to the nth pulse; a(O) is the array manifold and is a function of the target DOA 0 relative to the flight path; el(n) is the interference including ground clutter, possibly a jammer that is temporally white 42 but a point source in space, and spatially and temporally white Gaussian noise. We assume that the clutter and noise in different HRR range bins are independent and identically distributed. The real-valued scalars vo and wo are, respectively, the scaled radial velocity and the normalized Doppler frequency of the target. The presence of vo due to the radial motion between the radar and target results in range migration. For notational convenience, let wz = wo(l + rl), I = 0, , L - 1, (4.2) where r = vo/lwo is a constant independent of the target motion (see (7.9) in Appendix A) and is usually very small (<0.01). Then (4.1) can be expressed as xl(n) =a [ akej27rfkl] ewIn + el(n), 1= 0, * * *, L - 1, n = 0, , N - 1, (4.3) where we sometimes drop the dependence of a on 0 for notational brevity. Let K bl = Z Okej27rfk, l = 0,..., L - 1, (4.4) k=1 be the target phase history sequence. Note that L > K is required to allow the estimation of {ak, fk}K=1 from {bl}l0 1. Then inserting (4.4) into (4.3) gives, x,(n)=blae jwn +er(n), l=0,---,L-1, n=0,---,N-1. (4.5) When L = 1, the model in (4.5) reduces to the data model used in [72] for the LRR radar. For L > 1, we have a phase history sequence per LRR range bin and no loss of range resolution occurs since no information is lost. The secondary data are obtained from segments adjacent to the segment of interest in the same way as the primary data are obtained from the segment of interest (see Figure 1). The secondary data are assumed to be target free and are modeled as a VAR random process (see [72]). The VAR filter has the following structure, P N(z-) = I + E H,z-', (4.6) p=1 43 where z-1 denotes the unit delay operator, I is the identity matrix, and P is the filter order. We assume that the statistics of the ground clutter are the same for the primary and secondary data. The goal of the VAR filtering technique is to estimate the VAR filter coefficients, H = [H1, - - , Hp], from the secondary data and then filter the primary data so that the clutter component of the filter output is temporally white. Let I denote the number of adjacent segments we use to obtain the secondary data and let eil(n), i = 1, ---, I, 1 = 0, ***, L-l1, n = 0, ***, N-l, denote the secondary data. The VAR filter coefficients are estimated based on the following least squares criterion, which is an extension of the criterion in [72] to the case of L> 1, L-1 I N-1 P 2 HI1,... Hp = arg min ei(n) + Hpeiz(n - p) , (4.7) H1,...,Hp l=0 i=1 n=P p=1 where |1. denotes the Euclidean norm. The solution to (4.7) is given by HI = EWH ( H)- , (4.8) where IF = [%F10 ... %Pit ... I(L-1)], (4.9) Tit = [ it (P) .. Vi,(N - 1)], (4.10) il (n) = - [e(n - 1) ... eT(n - P)]T , (4.11) E = [El0 ... Et ". EI(L-1)], (4.12) and Eil = [ei (P) ... eil(N - 1)], (4.13) with (.)T and (.)H denoting the transpose and the conjugate transpose, respectively. Once the VAR filter coefficients are determined, we use the filter to suppress the clutter in the primary data. The VAR filter output for the primary data has the 44 form yz(n) = ?i(z-1)xl(n) = bliaej'n + Ei(n), I= 0,-...,L-1, n= P,..,N-1, (4.14) where i(z-1) has the same form as 'H(z-1) in (4.6) except that the {HP}_l in (4.6) are replaced by {^Ip},P=, P li = I+ E Ipe-'P, = 0,...,L- 1, (4.15) p=1 and El(n) = (z-1)[el(n)], = 0, , L - 1, n = P, - --, N -1. (4.16) Note that, after VAR filtering the number of temporal data samples is N = N - P. Since the VAR filter whitens the clutter only in the temporal domain, the VAR filtered clutter is still spatially colored. Like in [72], the parameter estimation algorithm developed herein can deal with the spatially colored VAR filtered clutter present in the filtered primary data. Let al = bl tia, 1 = 0, , L - 1, (4.17) be referred to as the spatial signature vector of the target for the lth phase history sample. We can rewrite (4.14) as yl (n) = ale""'"jn + E (n), I = 0, - - -, L - 1, n = P, - --, N - 1. (4.18) 4.3 Target Feature Extraction The target feature extraction algorithm we present below will first obtain estimates {a}t0 of the spatial signature vectors and an estimate co of the Doppler frequency wo. Then the target phase history sequence {b}L1 and the target DOA O will be estimated from {a }[o=. Finally, the estimates {bil}o are used to extract the target range features {(k, fk}K=1. Figure 2 outlines the steps of this algorithm, with each step described in detail in the following subsections. 45 4.3.1 Doppler Frequency and Spatial Signature Estimation We assume that the VAR filtered interference is a zero-mean temporally white Gaussian random process with an unknown covariance matrix Q. Then the negative log-likelihood function is proportional to C1 = In |QI + Tr (Q-1C), (4.19) where | - and Tr(.) denote, respectively, the determinant and the trace of a matrix, and L-1 N-1 C E y [y((n) - a ei" "]H 1=0 n=P L-1 NL [Y - at3H] [Y - alH]H , (4.20) 1=0 with Yj = [y,(P) .. yl(N-1)], l=0,...,L-1, (4.21) and 31 = [ew'P ... ew(N-1)]H, l=,.,L-1. (4.22) Minimizing C1 using the structure of al in (4.17) is a highly nonlinear optimization problem that would require a search over a multidimensional parameter space. To simplify the problem, we assume that {a}Lo1 are all unknown. In particular, the array manifold information a(O) is not used at this stage, and hence it can be arbitrary and unknown, which increases the robustness of the algorithm derived in this section against array calibration errors. Since the number of unknowns is increased by assuming that {a}foI1 are unknown, the performance of our algorithm will be worse than if the structure of {al}t-1 was used, according to the parsimony principle [69]. Yet by doing so, we obtain a much simpler and faster, and hence more practical, algorithm. 46 Minimizing C1 in (4.19) with respect to Q results in Q = C. The cost function in (4.19) with Q replaced by Q becomes: C2 = In IC = In L- N1 at - Y1) at - Y1)H 1=0 +YY - . (4.23) The minimization of C2 in (4.23) with respect to a, gives at - NY1,, I= 0, - - -, L - 1. (4.24) Then the estimate of the Doppler frequency wo is obtained by minimizing the concentrated cost function, 1 Y,0, gH H C3=ln 1 = k YH y1 1 H1 (4.25) L =0 N which requires a 1-D search. Note that the spatial signature estimate in (4.24) can be interpreted as the temporal average of the Doppler shift and range migration compensated VAR filtered spatial measurements. 4.3.2 Phase History Sequence and Array Steering Vector Estimation The estimate Q of the covariance matrix Q of the VAR filtered clutter-plusnoise sequence can be expressed as L-1 Q NL [YYH _ NiiH] . (4.26) l=0 Given {at}[=o and Q, the estimates of {b1}[Lo1 and a (or the target DOA 0 if the array manifold is known) can be obtained by minimizing the following Weighted Least Squares (WLS) cost function, L-1 C4 = (d - bilt-a) H (A - blitia) . (4.27) l=0 Note that this cost function is similar to the one used for the unstructured method in [721 since the Fisher information matrix (FIM) for at in (4.18) is also proportional to Q-1 (see [72] for more details). 47 Method 1 To estimate 0, we must know the array manifold a(0) as a function of 0. The method for estimating {b1}Lo1 and 0 by using the array manifold is referred to as Method 1. Without loss of generality, we consider a uniform linear array. Then a(0) has the form a(0) = [1 ej cosO'' eJA-(M-1)cose , (4.28) where d denotes the spacing between two adjacent antenna sensors, A is the radar wavelength, and 0 is the azimuth angle of the target relative to the flight path. Minimizing C4 in (4.27) gives H-H aH ^ -1I a b =, 1 = 0,...,L- 1, (4.29) a aHi and L-1 gH 1 -1 2 S= argmax H - (4.30) S1=0 aHli Q-1r1n Once 0 is determined, {bt}) o is obtained with (4.29) by replacing a with a(0). In the simulations, we use fmin(), a built-in function in MATLAB, to search for 0. Method 2 To achieve robustness against array calibration errors, we can assume that a is completely unknown. The corresponding method is used to estimate both {bt}l=0L and a and is referred to as Method 2. Note that since replacing {b1t}L- by {Pbl}L[-1 and a by a/0 in (4.27), where 3 is any non-zero complex scalar, does not change C4, {b}=I1 and a can only be determined up to an unknown multiplicative complex constant. This unknown complex scalar, similar to the unknown gain and initial phase of a radar system, does not affect most practical applications including ATR. Moreover, the presence of this ambiguity does not affect the minimization of C4 (see below). 48 Given bl, minimizing C4 in (4.27) with respect to a yields L-1 -1 L-1 a= t 1H -1 1 b*AH a-1, (4.31) L=0 l=0 where (.)* denotes the complex conjugate. Given a, bl can be estimated by using (4.29). Hence we can cyclically iterate (4.29) and (4.31) to obtain the estimates of {b}zL=1 and a. To start the iteration, we must have an initial estimate of either a or {bd L1 which in our initialization approach is obtained by using the Singular Value Decomposition (SVD) [63] as described next. Rewriting C4 in (4.27), we have L-1 C5 = d, - ba)0 ( -ba) H t ab), (4.32) 1=0 where Qg = Al Q-', I = 0, .. , L - 1. (4.33) To retain the weight of the term that may give the largest error in (4.32), we -1 choose W = Qio, where 10 is selected such that itol ato is the largest among { l }/I. Let a = W It, (4.34) and g = WEa. (4.35) We obtain the initial estimates by minimizing the following approximate cost function L-1 C6 = E (& - blB)H (l - bra), (4.36) 1=0 which is equivalent to C7 = A - b T112, (4.37) where I IF denotes the Frobenius norm, b = [bo ... bL_1]T, (4.38) 49 and A = [ao .. aL-1] . (4.39) The C7 in (4.37) is minimized if abT = U101VH, (4.40) where ul and vl are, respectively, the left and right singular vectors associated with the largest singular value al of A. Then either the initial estimate of b, i.e., g(o) = vx, (4.41) or the initial estimate of a, i.e., i(0) = W- U1, (4.42) can be used to initialize the alternating optimization approach. We use the former in our numerical examples. The steps of Method 2 are as follows: Step 0: Obtain the initial estimate b(o) of b with (4.41). Step 1: Update {b1}L[i with (4.29) by replacing a in (4.29) with the most recently determined a. Step 2: Update a with (4.31) by replacing {b1}[ o1 in (4.31) with the most recently determined { }o LStep 3: Iterate Steps 1 and 2 until practical convergence occurs which is determined by checking the relative change (1 of the cost function C4 in (4.27) between two consecutive iterations. We remark that if the range migration is negligible, i.e., r = 0 in (4.2), the 7iz's in (4.15) do not depend on 1. Then Step 0 alone gives the solution that minimizes the C4 in (4.27). 50 4.3.3 Target Range Feature Estimation Once the sequence {bl}L=0 is available, the range feature estimates, {k, fk} kl, can be obtained by minimizing the following cost function, C8 = $b - Fall2, (4.43) where a = [L1 a2 '' OK]T, (4.44) and F = [f f2 .. fK], (4.45) with fk [1 e j2xfk ... ej27rfk(L-1)]T, k 1, ..., K. (4.46) This is a sinusoidal parameter estimation problem in which a large number of solutions are available in the literature (see, e.g., [66]). Here we will use RELAX [38] that appears to be among the best available algorithms that can be used to solve the minimization problem in (4.43). To make the paper self-contained, we briefly outline the steps involved in RELAX. Assume that {&I, fi },lik are given, where K denotes the assumed number of scatterers. Let bk =- fi, k=l, ---, , (4.47) i=1,ij4k Then minimizing C8 in (4.43) with respect to ak yields the estimate fkHbk k = L ifk k' (4.48) where the estimate fk of fk is determined from fk = arg max fHk . (4.49) 51 Note that fk corresponds to the dominant peak location of the squared magnitude of the Fourier spectrum fkl~bk, and that dk is just the complex height (scaled by 1/L) of the same Fourier spectrum at its peak location. With the above preparations, the RELAX algorithm used to estimate the target range features {ak, fk K=- can be outlined as follows: Step (1): Assume K = 1. Obtain {&k, fk}k=1 from b by using (4.48) and (4.49). Step (2): Assume K = 2. Compute b2 with (4.47) using {&k, fk}k=l obtained in Step (1). Obtain {&k, fk k=2 from b2. Next, compute b1 with (4.47) using {&k, fk k=2 and then redetermine {&k, fk}k=1 from b1. Iterate the previous two substeps until "practical convergence" is achieved (see below). Step (3): Assume K = 3. Compute b3 with (4.47) using {&k, fk 1=1 obtained in Step (2). Obtain {&k, fk}k=3 from b3. Next, compute b1 with (4.47) using {&k, fk}k=2 and then redetermine {&k, fk}k=1 from b1. Then, compute b2 with (4.47) using {&k, fk}k=1,3 and redetermine {&k, fk}k=2 from b2. Iterate the above three substeps until "practical convergence" is reached. Remaining Steps: Continue similarly until K is equal to the desired or estimated number of sinusoids, which can be determined by using the Generalized Akaike Information Criterion (GAIC) [38, 61, 64]. The "practical convergence" in the iterations of the above algorithm can be determined by checking the relative change 2 of the cost function C8 in (4.43) between two consecutive iterations. We remark that for very closely spaced target scatterers, RELAX can converge slowly. To avoid this problem, we may use another method to provide initial estimates to RELAX [83]. 4.4 Numerical Examples We present several numerical examples to illustrate the performance of the proposed clutter suppression and moving target feature extraction algorithms. In 52 the following examples, we assume that the array is a uniform linear array (ULA) with M = 8. The interelement distance is d = A/2. The number of pulses in a CPI is N = 16. The phase history sample number is L = 16, i.e. an LRR range segment contains 16 HRR range bins. We assume that the target consists of two closely spaced scatterers (K = 2) with parameters al = 1, a2 = 1, fl = 0.1, and f2 = fl + -. The motion parameters of the target are wo = 0.27r and vo = rwo = 0.001wo. The target DOA is 0 = 600. We use (1 = (2 = 10- to test the "practical convergence" for both Method 2 and the range feature extraction algorithm using RELAX. The mean-squared errors (MSEs) of the various estimates are obtained from 200 Monte Carlo trials. We simulate the ground clutter as a temporally and spatially correlated Gaussian random process (see Appendix B for the generation of the clutter). We use the clutter ridge slope y = d/(2VT) = 1 in the simulations for the case when the system mismatch is absent (see Appendix B for details), and y = 1.1 for a case when the system mismatch exists. The Clutter-to-Noise Ratio (CNR) is defined as the ratio of the clutter variance to the noise variance. Herein we set CNR = 40 dB in all examples although our algorithms can also deal with much stronger CNRs. The noise is a zero-mean spatially and temporally white Gaussian random process. The VAR filter order is chosen as P = 2, and only I = 3 secondary LRR range bins are used to estimate the VAR filter coefficients. When array calibration errors exist, the errors for different elements are assumed to be independent and identically distributed complex Gaussian variables. More specifically, a complex Gaussian random vector with zero-mean and covariance matrix 0.04I is added to the array manifold a(O) to simulate array calibration errors, which implies that the variance of the calibration error for each element is 0.04. Further, in the simulations when a point source jammer is introduced, the jamming signal is assumed to be a zero-mean temporally white 53 Gaussian random process. Figure 4.3 shows the mesh plot of the power spectral density of the clutter, jamming, noise, and the target observed by the airborne radar, when the clutter ridge is along the diagonal (-y = 1), the jamming impinges from 01 = 450 with JNR = 30 dB, and the target is located at DOA = 600 with wo = 0.27r. We first present an example in the absence of array calibration errors, jamming, and system mismatch. Note that the complex amplitude estimates {6kkK=1 Of {tk kK=1 obtained via Method 2 are all scaled by a common unknown complex scalar. To calculate their best possible MSEs, we scale them to minimize C9 = |ao - &112, (4.50) where a0o is the true value of a. Minimizing (4.50) with respect to # yields &Ho 3 = 2 . (4.51) Note that this scaling scheme is only used to illustrate the complex amplitude estimation performance; it is not a necessary step in a practical application including ATR since only the relative amplitudes are important. For comparison purpose, the MSEs of the estimates {&k }K=1 obtained via Method 1 are presented in two ways: with and without a scaling scheme similar to (4.50). Figures 4.4(a)-(d) show the MSEs of the estimates of the Doppler frequency, target DOA, and the complex amplitude and range frequency for the first scatterer when compared with the Cram6r-Rao Bounds (CRBs), which are derived in Appendix C. The horizontal axis in the figures shows the values of the Signal-to-Noise Ratio (SNR), which is defined as the ratio of lai12 to the noise variance. Due to using the scaling scheme in (4.50), the MSEs for {&k }K= may be better than the CRBs (which do not account for such a scaling). Figure 4.4 shows that the MSEs of the parameter estimates are very close to the CRBs as the SNR increases, which indicates that the clutter suppression scheme works well and the parameter estimation algorithm is highly accurate. 54 We next consider the case of a point source at 450 as a jammer, where all other parameters are the same as for Figure 4.4. The Jammer-to-Noise Ratio (JNR), which is the ratio of the jammer's temporal variance to the noise variance, is chosen as JNR=30 dB. Figures 4.5(a) and (b) show the MSEs of the complex amplitude and range frequency estimates as a function of SNR. Note that the VAR filtering technique cannot suppress the jammer since the jamming signal is already temporally white. The jamming is suppressed in the parameter estimation algorithm due to the assumption that the interference is spatially colored with arbitrary and unknown covariance matrix. Next, we present an example in which array calibration errors are present in the array steering vector. All other parameters are kept the same as for Figure 4.5. Figures 4.6(a)-(d) show the MSEs of the estimates of the Doppler frequency, target DOA, and complex amplitude and range frequency of the first scatterer, as functions of SNR. Note that the MSEs of the complex amplitude and range frequency estimates obtained via Method 2 are close to the CRBs as the SNR increases, and that the MSEs of the complex amplitude estimates for Method 1 fail to follow the CRBs if the scaling scheme is not used, though the range frequency is still well estimated via Method 1. Hence both Methods 1 and 2 are robust against array calibration errors as far as the relative complex amplitude and range frequency of the scatterer are concerned. Our final example concerns the case where three degradation factors, i.e. array calibration error, jamming, and system mismatch (y = 1.1) are present. Figures 4.7(a) and (b) exhibit the curves of the MSEs and the CRBs of the complex amplitude and range frequency estimates versus SNR. Comparing Figures 4.7 with 4.6, we note that our parameter estimation algorithms are robust against system mismatch and can obtain accurate range feature estimates. These results also verify the robustness of using the VAR filtering technique for clutter suppression in the presence of system mismatch. 55 We remark that the above simulation results show that Methods 1 and 2 provide similar performances for target relative complex amplitude and range frequency estimation. Method 2 avoids the 1-D search over the DOA space and usually requires only a few (3 - 6) iterations when SNR < 0 dB and no iteration when SNR > 0 dB. To estimate the target phase history sequence {bito'1 from the spatial signature estimates t} 0, Method 2 needs only about 6% (for the no iteration case) or 30% (for 4 iterations) of the amount of computations measured in MATLAB flops required by Method 1 (this difference is mainly due to the fact that the latter requires a 1-D search over the DOA space). Hence if the array calibration errors are known to be significant enough to result in a useless target DOA estimate or if the target DOA is not of interest, Method 2 is to be preferred over Method 1. 4.5 Summary We have presented a moving target feature extraction algorithm for the airborne HRR phased array radar in the presence of temporally and spatially correlated ground clutter, jamming interference that is a temporally white, spatially point source, and spatially and temporally white Gaussian noise. To avoid the range migration problems that occur in HRR radar data, we divided the HRR range profiles into LRR segments. We have shown in detail how to use a VAR filtering technique to suppress the ground clutter for the HRR radar. To gain robustness against the array calibration errors, we assumed in a first stage that the array manifold is completely unknown and arbitrary. From the VAR-filtered data, the target Doppler frequency and the spatial signature vectors were first estimated by using a ML method. The target phase history and DOA (or the array steering vector for unknown array manifold) were then estimated from the spatial signature vectors by minimizing a WLS cost function. The RCS related complex amplitude and range related frequency of each target scatterer were finally extracted from the estimated target phase history by using RELAX, a relaxation-based high resolution feature extraction algorithm. 56 Numerical simulations have shown that our moving target feature extraction algorithm performs well in the presence of jamming and is robust against array calibration errors and system mismatch. 57 L HRR Range Bins Ill III III- III III III I I HRR Range Secondary Segment Segment of Interest Profile IFFT IFFT Secondary Data Primary data {w(,,)} Figure 4.1: HRR range profiles are divided into LRR segments with each segment containing L HRR range bins. 58 HRR Range Profile Divide into LRR Segments and Inverse Fourier Transform Each Segment VAR Filtering for Clutter Suppression Doppler Frequency and Spatial Signature Estimation Target Phase History Estimation High Resolution Target Feature Extraction _To ATR Applications Figure 4.2: Flow chart of the proposed clutter suppression and target parameter estimation algorithm for HRR phased array radars. 59 06 0.6 -10, 04 -230 -40 -50 -02 -04 -0.5 -1 -1 -0 -6 -0 4 -02 0 02 04 06 08 cos(W) 2 fdr 2 fd/fr (a) (b) Figure 4.3: (a) Mesh plot and (b) projection of the power spectral density of the interference (including clutter, jamming, and noise) and the target when the clutter ridge is along the diagonal (y = 1), the jamming comes from Gj = 450, and the target is located at DOA = 600 with w0o = 0.27r. The axes V0 and w are for the spatial frequency and the normalized Doppler frequency, respectively, (see Equations (3.1) and (3.2)). 60 S40 I CRB F CRB1 \ -o- MSE 3 -o MSE -10 20 -20 ' 10 S-30 b -50 a \-20 -50 -70 -40 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 SNR (dB) SNR (dB) (a) (b) CRB -- CRB -0- Method 1 (no scalln -o- Method 1 A Method 1 (scaling) -20- - Method 2 - -- Method2 \-30 \ -10 . -40 ' 0 -40 S-20 -30 -60 -40 -70 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 SNR (dB) SNR (dB) (c) (d) Figure 4.4: Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, no jammer, no array calibration errors, no system mismatch.) 61 21, _,o, S CRB CRB - Method 1 no sca-- - Meth (no ca - Method 1 S Method 1 (scaling) -20 - 4- Method2 10 - - Method 2 -70 -40 -30 N -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 SNR (dB) SNR (dB) (a) (b) Figure 4.5: Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 450 with JNR=30 dB, no array calibration errors , no system mismatch.) 62 4O - CRB I- CRB -o- - MSE " - E- MSE -10 20 \ S -20 \ 10 0 -30 0 -50 a -l -60 -20 -70 -30 -80 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 SNR (dB) SNR (dB) (a) (b) 20CRB '-1 CRB S- -- Method 1 (no scalin - o - Method 1 A Method 1 (scaling) 20 -- Method 2 - - Method 2 -30 -70 -40 -so -40-o -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 SNR (dB) SNR (dB) (c) (d) Figure 4.6: Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, a jammer at 450 with JNR=30 dB, array calibration error covariance matrix 0.04I, no system mismatch.) 63 2-10 2 _ CRB - - CRB - o Method 1 (no scalin -0- - Method 1 A Method 1 (scaling) - -*- Method 2 - -- Method 2 -2 10 -30 N 0 -so 'As -20 -so -30 -70 -40 -20 -15 -10 - 0 5 10 is 20 o 20 - -10 -5 5 10 15 20 SNR (dB) SNR (dB) (a) (b) Figure 4.7: Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 450 with JNR=30 dB, array calibration error covariance matrix 0.041, system mismatch with clutter ridge slope -y = 1.1.) 64 CHAPTER 5 MULTIPLE MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR 5.1 Introduction In this Chapter, we extend the single target approaches to the case of multiple targets. The multiple moving target scenario occurs frequently in radar applications. Yet to the best of our knowledge, little research on the topic has been reported in the literature. We present a relaxation-based algorithm for multiple moving target feature extraction. Each of the targets is assumed to have a rigid-body and the scatterers of the same target have the same DOA. The relaxation-based algorithm is used to minimize a nonlinear least squares fitting function by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently determined values. Thus the problem of multiple target feature extraction is reduced to the feature extraction of a single target in a relaxation-based iteration step. We use numerical examples to demonstrate the performance of this algorithm for clutter suppression and multiple moving target feature extraction. The remainder of this Chapter is organized as follows. In Section 5.2, we establish the multiple moving target data model for airborne HRR phased array radar, which is followed by a brief discussion of the VAR filtering technique. In Section 5.3, we present the relaxation-based multiple moving target feature extraction method. Simulation results and their analysis are presented in Section 5.4. Finally, we give the conclusions in Section 5.5. 5.2 Data Model and VAR Filtering The range resolution of a radar is determined by the transmitted signal bandwidth. To achieve high range resolution, a radar must transmit wideband pulses, 65 which are often linear frequency modulated (LFM) chirp signals [821. The range resolution of a LRR radar is much larger than the length of a target so that the target occupies only one LRR range bin, while the range resolution of a HRR radar is so small that each target occupies several HRR range bins. Consider an airborne HRR radar having a one-dimensional (1-D) antenna array with M elements uniformly spaced along the flight path of an airborne platform. A cluster of N chirp pulses is transmitted during a coherent processing interval (CPI). After dechirping, sampling, and Fourier transforming the signals in each element of the array, we obtain the HRR range profiles. Without clutter and jamming suppression, these clutter and jamming dominated profiles are not useful for any applications. The HRR radar also has the range migration problems, which occur when the target scatterers migrate from one range bin to another during the CPI. To avoid the range migration problems, we divide each HRR range profile into non-overlapping LRR segments so that each LRR segment contains L HRR range bins, as shown in Figure ??. We choose L to be much larger than the maximum number of range bins over which all targets can possibly expand and migrate during the CPI. We then apply the inverse Fourier transform to each segment. For the segment of interest, where the targets may be present, the inverse Fourier transform yields the primary data. We assume that D targets are present in the primary data with the dth target consisting of Kd scatterers. We assume that the scatterers of each target have the same Doppler frequency and the same DOA, but different complex amplitudes and range frequencies. (If the scatterers of a target have different Doppler frequencies due to, for example, turning, they are treated as belonging to multiple targets.) Then the primary data model can be written as (see Appendix A for the model derivation): x,(n) = dI adkeJ2 fdk ejd"nl je na(Od) + el(n), (5.1) d=1 k=1 l=0,.--,L-1, n= 0,---,N-1, 66 where xl(n) is the array output vector of the lth phase history sample due to the nth pulse; a(Od) is the array manifold and is a function of the dth target DOA 9d relative to the flight path; el(n) is the interference including the temporally and spatially correlated Gaussian ground clutter, both temporally and spatially white Gaussian noise, and possibly a jammer that is temporally white but a point source in space. We assume that the clutter, noise, and jamming in different HRR range bins are independent and identically distributed. The complex amplitude adk and the frequency fdk are, respectively, proportional to the RCS and range of the kth scatterer of the dth target. The vd and wd are, respectively, the scaled radial velocity and the normalized Doppler frequency of the dth target. Range migration occurs due to the radial motion between the radar and target and the high range resolution of the HRR radar. For notational convenience, let Wdl = Wd(1 + rl), l = 0,---, L - 1, d = 1,., D, (5.2) where r = vd/wd is a known constant independent of the target motion (see Appendix A) and is usually very small (< 0.01). Then (5.2) can be written as D xt(n) = bdlewd'nad + e(n), 1=0,---,L-1, n= 0, , N - 1, (5.3) d=1 where we have dropped the dependence of ad on Od for notational brevity and Kd bdl =) adk eej27rfdkl, = 0, ,L - 1, d = 1..., D. (5.4) k=1 Note that, when L = 1, the model in (5.3) reduces to the data model for the LRR case. For L > 1, we have a phase history sequence for each LRR segment and no loss of range resolution occurs because of no information loss. Similar to Section ??, once the VAR filter coefficients are determined, we use the filter to suppress the clutter in the primary data. The VAR filter output for the primary data has the form yi(n) = li(z-1)xl(n) 67 D E Z bd(?'tdiadeJidIn + Ej(n), d= 1 l= 0,---,L-1, n= P,---,N-1, (5.5) where Wl(z-') has the same form as W(z-') in (??) except that the {Hp}pp in (??) are replaced by {HP}P,= P Akdl = I + Hpe-jdIP, l = 0, ,L-1, d = 1,...,D, (5.6) p=l and E(n) = (z)e(n), l=0, - - -, L - 1, n = P, - --, N -1. (5.7) Let adl = bdtdlad, l = 0,..,L-1, d= 1, , D, (5.8) be referred to as the spatial signature vector of the dth target for the lth phase history sample. Then (5.5) can be rewritten as D Y1(n) = E adleJdrn + El(n), = 0, ,L 1, n= P,..., N - 1. (5.9) d= 1 Our problem of interest is to estimate {wd, Od, {adk, fdkd kK1 D= if the array manifold {a(0d)}D=l is known or {Wd, ad, {jdk, fdk kKd I1 if the array manifold {a(0d)}D1 is unknown from the VAR filter output y1(n), = 0,.-, L-1, n = P, , N-1, by minimizing a Nonlinear Least Squares (NLS) fitting criterion using a relaxation-based optimization algorithm. 5.3 Feature Extraction of Multiple Moving Targets Our feature extraction algorithm consists of the following two separate steps: Step I: Estimate the target space-time parameters {wd, Od, {bdl}fr-}lD=1 if the array manifold {a(0d)}d=1 is known or {Wd, ad, {bdl}= }dD=l1 if the array manifold is unknown from the VAR filter output {yi(n)}, 1 = 0,.. , L - 1, n = P,.. ,N- 1. Step II: Estimate the target range parameters {Qdk, fdk jkK1, d = , ., D, from the estimate {bdl}L-1 of {bdl} L-I obtained in Step I. 68 5.3.1 Space-Time Parameter Estimation The NLS estimates of the space-time parameters {Wd, Od, {bdl}= }fr0dD=1 or {Wd, ad, {bd }[dD=1 can be obtained by minimizing the following NLS criterion: L-1 D 2 C1 = y1n) - Z adleiwdin . (5.10) 1=0 d=1 The minimization of the cost function C1 in (5.10) is a highly nonlinear complicated optimization problem. Here we present a cyclic optimization approach, which is a conceptually and computationally simple method for multiple moving target feature extraction [84]. The relaxation-based algorithm is used to minimize C1 by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently determined values. Therefore, the feature extraction of multiple moving targets is reduced to the feature extraction of a single moving target in a relaxation-based iteration step. We first consider the space-time parameter estimation of the dth target and then give detailed steps of our approach for multiple targets. Space-Time Parameter Estimation of the dth Target Let D Yd(n) = yI(n) - E bit 6 iiegln, 1= 0,*** ,L- 1, n = P,*. ,N- 1, (5.11) i=1,i d where {Oi, a 1, i{bi}l=o }-d, ifd is assumed available. Note that if the array manifold is known, ai is replaced by a(6i), with {0}D,1 i~ d assumed available. Hence YdI(n) can be written as Ydl (n) = adlejwdIn + -dl(n), = 0,... , L - 1, n = P,... , N - 1, (5.12) where Edl (n) denotes the interference due to clutter, noise, and contributions from other targets. We assume that {Edl(n)} is a zero-mean temporally white Gaussian random process with an unknown arbitrary spatial covariance matrix Qd. Then the 69 negative log-likelihood function of ydl(n) in (5.12), is proportional to C2 = in Qd + Tr (Qd'Cd) , (5.13) where I " I and Tr(.) denote, respectively, the determinant and the trace of a matrix and L-1 N-1 Cd = 1 [ydl(n) - adleijdln] [ydl(n) - adlej'dln] H NL 1=0 n=P 1 L-1 - E [Ydl - adldHI [Ydl - adl H]H, (5.14) 1=0 with Ydl = [Ydt(P) '.. Ydl(N - 1)], 1 0, - - L 1, (5.15) and dl = [ejwdIP ejdl(N-1)]H , 1 = 0, , L - 1. (5.16) Minimizing C2 in (5.13) with respect to Qd results in Qda= Cd. The cost function in (5.13) with Qd replaced by Qd becomes: C3 = In ICd 1 L-1 Ydldl Ydl)3dl H = In 1 T N ad - adt N N N +Y HY Ydl N)dl(dH +YdY - I (5.17) The minimization of C3 in (5.17) with respect to adl gives 1 1 wdl=wd(l+rl) adl = N-Ydl dld 1 ) = 0,..., L - 1. (5.18) The estimate of the Doppler frequency Wd of the dth target is obtained by minimizing the concentrated cost function, cYln - - [YdY dldl dtlYdj l (5.19) C4 = In Ydl ,0 (5.19) L=0 70 which requires a 1-D search. Note that the spatial signature estimate in (5.18) can be interpreted as the temporal average of the Doppler shift and range migration compensated VAR filtered spatial measurements. The estimate Qd of the covariance matrix Qd of the VAR filtered interference sequence can be written as L-1 Q N = [YdIYdHI- _ dlH . (5.20) 1=0 Given dl{a 1}= and Qd, the estimates of {bdl},I1 and DOA Od (if the array manifold is known) or ad (if the array manifold is unknown) can be obtained by minimizing the following Weighted Least Squares (WLS) cost function, L-1 C5 = (dl - bdl'tsdlad) d1 d- bdl'adlad) . (5.21) 1=0 Note that this cost function is similar to the one used for the unstructured method in [72] since the Fisher information matrix (FIM) for adl in (5.12) is also proportional to Qd1 (see [72] for more details). Method 1: To estimate 0d, we must know the array manifold a(0d) as a function of Od. The method for estimating {bd}lf1 and Od by using the array manifold is referred to as Method 1. Without loss of generality, we consider a uniform linear array (ULA), where a(Od) has the form, a(Od) = [1 eJ"cosOd ... ej(M-1) COSOd ]T, (5.22) with A being the radar wavelength and ( being the spacing between two adjacent sensors. Minimizing C5 in (5.21) gives ^ aHtH I b a1 = d dQd , I = 0, ' , L - 1, (5.23) adH 'd dl-dig and L-1 ad idlQdH 1 dl 2 d = arg max dI d (5.24) d l= ad-I Od 1 dlad 71 Once ld is determined, {bde}L=O is obtained with (5.23) by replacing ad with a(0d). Method 2: To achieve robustness against array calibration errors, we can assume that ad is completely unknown. The corresponding method used to estimate both {bdI}L- and ad is referred to as Method 2. Note that since replacing {bd,}lL- by {bd IL 01 and ad by ad/3 in (5.21), where 3 is any non-zero complex scalar, does not change C5, {bdl}tLo1 and ad can only be determined up to an unknown multiplicative complex constant. This unknown complex scalar, similar to the unknown gain and initial phase of a radar system, does not affect most practical applications such as ATR. Given {bd}[ fo1, minimizing C5 in (5.21) with respect to ad yields L-1 -1 L-1 ad [ Ibdl Qd dlQd d (5.25) l=0 l=0 where (.)* denotes the complex conjugate. Given ad, {bdl}lo can be estimated by using (5.23). Hence we can cyclically iterate (5.23) and (5.25) to obtain the estimates of {bdl},L 1 and ad. To start the iteration, we must have an initial estimate of either ad or {bdl}lo1. Our initialization approach is obtained by using the Singular Value Decomposition (SVD) [63]. Rewriting C5 in (5.21), we have L-1 ^ -1 H ^ -1 C6 =E (addl - bdlad) Qdl d ldl - bdlad) , (5.26) 1=0 where Qdt = HQd-1dl,, 1 = 0,..., L - 1. (5.27) To place the most weight on the term that is associated with the largest signal energy, we choose Wd = Qdlo, where 10 is selected such that /dloadlo is the largest among S d dl } = . Let Adl t W dV l adl, (5.28) 72 and ad = W ad. (5.29) We minimize the following approximate cost function L-1 C7 = (d, - bdad)H (dl - bdLd) , (5.30) 1=0 which is equivalent to CO = Ad - adb F, (5.31) where II1 - IF denotes the Frobenius norm, bd = [bdo ... bd(L-1]T, (5.32) and Ad= [ado ... * d(L-1)] (5.33) The Ca in (5.31) is minimized if adbT = dlUdlVH, (5.34) where Udl and Vdl are, respectively, the left and right singular vectors associated with the largest singular value adl of Ad. Then either the initial estimate of bd, i.e., d) vl, (5.35) or the initial estimate of ad, i.e., a(ï¿½) = Wd2Ud, (5.36) can be used to initialize the alternating optimization approach. We use the one in (5.35) in our numerical examples. The steps of Method 2 are summarized as follows: Step (0): Obtain the initial estimate 0) of bd with (5.35). 73 Step (1): Update {bdl}o=0 with (5.23) by replacing ad in (5.23) with the most recently determined ad. Step (2): Update ad with (5.25) by replacing {bdl}1f0 in (5.25) with the most recently determined {bde}{ 0 L. Step (3): Iterate Steps (1) and (2) until practical convergence, which is determined by checking the relative change ( of the cost function C5 in (5.21) between two consecutive iterations. We remark that if the range migration is negligible, i.e., r = 0 in (5.2), then {fidl}L-1 in (5.6) do not depend on 1. Then Step (0) alone gives the solution that minimizes the C5 in (5.21). Summary of the Steps of Space-Time Parameter Estimation The space-time parameter estimates {Wd, ^d (or ad), bd}D=1 of the multiple moving targets can be obtained as follows: Step I.1: Assume D = 1. Obtain {Wd, 0d (or ad), bd}d=l from y1(n). Step 1.2: Assume D = 2. Compute y21(n) with (5.11) using {2d, Od (or ad), bd}d=1 obtained in Step 1.1. Obtain {Wd, Od (or id), bd}d=2 from y21(n). Next, compute ylz(n) with (5.11) using {wd, 0d (or ad), bd}d=2 and then re-determine {Wd, Od (or ad), bd}d=l from yll(n). Iterate the previous two substeps until "practical convergence" is achieved (to be discussed later on). Step 1.3: Assume D = 3. Compute y31(n) with (5.11) using {wd, 0d (or ad), bd}d=1,2 obtained in Step 1.2. Obtain {Wd, 0d (or dd), b dd=3 from y31(n). Next, compute yl(n) with (5.11) using {Wd, 0d (or ad), bd}d=2,3 and then re-determine {Wd, 0d (or ad), bdd=l from yll(n). Then, compute y21(n) with (5.11) using {Wd, 0d (or id), bd}d=1,3 and re-determine {Wd, 0d (or ad), bd}d=2 from y21(n). Iterate the above three substeps until "practical convergence" is reached. Remain Steps: Continue similarly until D is equal to the desired or estimated 74 number of targets, which is assumed to be known or can be determined by using the Generalized Akaike Information Criterion (GAIC) [38, 61]. The "practical convergence" in the iterations of the above algorithm can be determined by checking the relative change e of the cost function C1 in (5.10) between two consecutive iterations. The steps leading to the last step are needed to provide good initial conditions for the last step of the algorithm. The cyclic optimization approach [5] converges under mild conditions although the global minimum is not a gaurantee. If the multiple targets are closely spaced, the convergence speed could be slow. To increase the convergence rate, we can either provide better initial estimates or use certain speed acceleration methods [5]. Estimating the target range features {&dk, ^dkk1 from b, d = 1 , D, is our next concern. 5.3.2 Target Range Feature Estimation Once the sequences {bdl z} o, d = 1, ..., D, for all targets are available, the range feature estimates, {&dk fdk } k 1, d = 1, ... , D, can be obtained by minimizing the following cost function, C9(Odk, fdkk=) = ld - Fdad l2, d = 1,.. , D, (5.37) where bdl is the lth element of vector bd, ad = [Ldl *** OdKd]T, d= 1,-..., D, and Fd= [fdl ... fdKd], d=1,...,D, with fdk = [1 ej2fdk ... ej2fdk(L-1)] T, d = 1,'",D, k = 1,-.,Kd. The estimates {&dk, fdk}.F= of {adk, fdk} Kd1 can be obtained by using the RELAX algorithm (see [38] for more details), which has a similar structure as the approach used for the space-time parameter estimation. 75 We remark that our multiple moving target feature extraction algorithm above may have used more unknowns than necessary at certain steps. We choose to do so to simplify and speed up the algorithm. For example, to use the cyclic optimization algorithm, we could estimate both the space-time and the range parameters of the dth target and subtract out the dth target based on the parameter estimates {wd, Od (or ad), {&dk, fdk} kK1 }D=I for the iteration steps. However, since estimating the range parameters {&dk, fdk k1 can be computationally demanding, we choose to separate the range parameter estimation from the space-time parameter estimation. Our numerical results have shown little accuracy degradation but reduced computations, especially for large Kd, as a result of the separate space-time and range parameter estimation. 5.4 Numerical Examples We present several numerical examples to illustrate the performance of our proposed algorithm. In the following examples, we assume that the array is a ULA with M = 8; the interelement distance between two antennas is ( = A/2; the number of pulses in a CPI is N = 16; the phase history sample number is L = 16, i.e., an LRR range segment contains 16 HRR range bins. Consider two targets (D = 2) with DOAs 01 = 30' and 02 = 1500, and Doppler frequencies w1 = 0.27 and w2 = 0.47 with r = 0.01. Each of the two targets consists of two closely spaced scatterers (K1 = K2 = 2) with parameters all = 1, a12 = 1, a21 = 1, ae22 = 1, f =l = 0.1, f12 = 0.1+1/2L, f21 = 0.3, and f22 = 0.3+1/2L, where the subscript ij means the jth scatterer of the ith target. The VAR filter order is P = 2. (No obvious performance improvement is obtained by using higher orders.) The number of secondary range bins is S = 4. We set C = ( = 10-3 to determine the "practical convergence" in the simulations. The mean-squared errors (MSEs) of the various estimates are obtained from 100 Monte Carlo trials. 76 We simulate the ground clutter as a temporally and spatially correlated Gaussian random process [80]. The Clutter-to-Noise Ratio (CNR), defined as the ratio of the clutter variance to the noise variance, is set to be CNR= 40 dB. A jamming signal, which is a zero-mean temporally white Gaussian random process, also exists. The Jammer-to-Noise Ratio (JNR), which is the ratio of the jammer's temporal variance to the noise variance, is chosen as JNR= 25 dB and the jamming signal impinges from 0j = 1200. When array calibration errors exist, the errors for different elements are assumed to be independent and identically distributed complex Gaussian random variables. In our simulations, a complex Gaussian random vector with zero-mean and covariance matrix 0.041 is added to the array manifold to simulate array calibration errors, which implies that the variance of the calibration error for each element is 0.04. Another degradation factor, the system mismatch error, which is due to the antenna spacing, platform velocity, and/or pulse repetition frequency (PRF) mismatch errors, is also considered. We use the clutter ridge slope [36] - = 1 in the simulations for the case when the system mismatch error is absent and -y = 1.1 for the case when the system mismatch error exists. We first present an example of neither array calibration errors nor system mismatch errors. Note that the complex amplitude estimates {<(dk}k1d of {Odk}kKd obtained via Method 2 are all scaled by a common unknown complex scalar. To calculate their best possible MSEs, we scale them to minimize C10 = 1ad0 - a d112, (5.38) where adO is the true value of ad. Minimizing (5.38) with respect to 3 yields: = d 1dO. (5.39) Note that this scaling scheme is only used to illustrate the complex amplitude estimation performance; it is not a necessary step in a practical application including ATR since only the relative amplitudes are of interest. For comparison purposes, the 77 MSEs of the estimates {&dk}k=l obtained via Method 1 are presented both with and without a scaling scheme similar to (5.38). Figures 5.2(a)-5.2(d) show the MSEs of the estimates of the Doppler frequency, target DOA, complex amplitude, and range frequency of the first scatterer of the first target as a function of the Signal-to-Noise Ratio (SNR), which is defined as the ratio of la11 2 to the noise variance, and compare them with the corresponding CramerRao bounds (CRBs) (see Appendix B for the CRB derivation). (The results for the other scatterer and the other target are similar.) Due to using the scaling scheme in (5.38), the MSEs of {&dk}k=1 may be better than the CRBs (which do not account for such a scaling). We note that as the SNR increases, the MSEs can approach the corresponding CRBs, which indicates that the clutter suppression scheme works well and the parameter estimation algorithm is highly accurate. To further illustrate this point, we show the modulus of a corrupted HRR range profile due to clutter and jamming in Figure 5.1(a), and the modulus of the true HRR range profile compared with the modulus of the estimated HRR profile generated from the estimates of the amplitudes and the range frequencies obtained via Method 1 without the scaling scheme in Figure 5.1(b). We set SNR= 0 dB and keep all the other parameters the same as those in Figure 5.2. Note from Figure 5.1 that our algorithm performs well for clutter suppression and feature extraction. Next, we present an example when both the array calibration and the system mismatch errors exist. All other parameters are kept the same as for Figure 5.2. Figures 5.3(a)-5.3(d) show the MSEs of the estimates of the Doppler frequency, target DOA, complex amplitude, and range frequency of the first scatterer of the first target, as a function of SNR. Note that the MSEs of the complex amplitude and range frequency estimates obtained via Method 2 are close to the CRBs as the SNR increases. The MSEs of the complex amplitude estimates for Method 1 fail to follow the CRBs if the scaling scheme is not used. The range frequency is still well 78 estimated via Method 1. Hence from Figure 5.3, we note that both Methods 1 and 2 are robust against both array calibration and system mismatch errors as far as the Doppler frequency, relative complex amplitude, and range frequency of the scatterer are concerned. We remark that the above simulation results show that Methods 1 and 2 provide similar performances for target relative complex amplitude and range frequency estimation. Method 2 avoids the 1-D search over the DOA space and usually requires only a few (3 - 6) iterations. To estimate the target phase history sequence {bll}L from the spatial signature estimates (1}[ , Method 2 needs only about 10% - 30% of the amount of computations measured in MATLAB flops required by Method 1. (This difference is mainly due to the fact that the latter requires a 1-D search over the DOA space.) Hence if the array calibration errors are known to be significant enough to result in a useless target DOA estimate or if the target DOA is not of interest, Method 2 is preferred over Method 1. Consider next an example using the experimental data generated by using the Moving and Stationary Target Acquisition and Recognition (MSTAR) Slicy data collected by imaging a slicy object shown in Figure 5.4(a). The data were collected by the Sandia National Laboratory using the STARLOS sensor. The field data were collected by a spotlight-mode SAR with a carrier frequency of 9.559 GHz and bandwidth of 0.591 GHz. The radar was about 5 km away from the ground object. The data were collected when the object was illuminated by the radar from approximately the azimuth angle 0' and elevation angle 300. The pulse width is 3 ps; the PRF is 3000 Hz; the sampling frequency is 130 MHz. We assume that the target is moving toward the radar with a radial speed of 14 meters/sec. According to [26], approximately, we have the normalized Doppler frequency wd = 0.19 and the scaled radial velocity Vd = 0.0005Wd. 79 The 32 x 32 SAR data matrix we used to generate the HRR data has a resolution of 0.51 m in range and 0.54 m in cross-range. The 32 x 32 SAR data can be treated as the samples from one array element of a phased-array radar. Since the PRF of the SAR system is quite different from that of a GMTI/HRR radar system, we set the PRF ratio of the GMTI/HRR radar system and the SAR system to be 5. We assume that the range samples L = 16, temporal samples N = 16, and the number of antenna elements M = 8. Since the PRF ratio between the two radar systems is 5, four times zero-padding is used to interpolate the cross-range samples of the 32 x 32 SAR data matrix and an N x L data matrix is saved as the data from one array element of the phased-array radar during the CPI. We then can generate the samples for the other array elements similarly and build up the entire data observed by the phased-array radar. Finally strong clutter and jamming are added to the data. Figure 5.5(a) shows the average of the modulo of the N = 16 range profiles in the presence of the strong clutter, jamming, and noise. Figure 5.5(b) shows the average of the modulo of the N = 16 original normalized range profiles in the absence of the clutter, jamming, and noise, which is compared with the modulus of the estimated range profile generated from the estimates of the amplitudes and range frequencies obtained via Method 1 without the scaling scheme. The effectiveness of our algorithm is again evident. 5.5 Summary We have presented a robust and accurate method for the clutter suppression and feature extraction of multiple moving targets for airborne HRR phased array radar. To avoid the range migration problems that occur in HRR radar data, we divide the HRR range profiles into LRR segments. We have shown how to use the VAR filtering technique to suppress the ground clutter and use the cyclic or relaxationbased algorithm to extract the features of multiple moving targets. The multiple moving target feature extraction problem is reduced to the feature extractions of 80 a single moving target in a relaxation-based iteration step. For each target and in each iteration, the target phase history sequence and DOA (or the unknown array manifold) are estimated from the spatial signature vectors by minimizing a Weighted Least Squares (WLS) cost function. Numerical results have demonstrated that our multiple moving target feature extraction algorithm performs well in the presence of strong interference including clutter, noise, and jammer and is robust against array calibration and system mismatch errors. 81 1.2 - - estimated profil original profile 0.8 0.6 -i 0.4 0.2 -u t" -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 5 normalized range frequency (a) 800 700600 500 Z 400 300 200 100 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Normalized Range Frequency (b) Figure 5.1: (a) The real HRR range profile without interference (clutter and jammer) and the estimated HRR range profile after clutter suppression and feature extraction, and (b) the cluttered HRR range profile with CNR= 40 dB, SNR= 0 dB. 82 -- CRB f - CRB l- - * MSEt *-- * MSE a10 1 -0 -15 -0 -5 0 5 0 5 -15 -10o -5 0 10 5 SR (i) SNR (dB) (a) (b) CRB - CRB - Method 1 (no scaling) 0- - Method 1 - Method 1 (scaling) Method2 - v Method 2 -30 -0 0 - -10 -5 0 5 1 1 S -0 -15 -1 -1 1 SNR(dB) NR (dB) (c) (d) Figure 5.2: Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and no array calibration error. 83 S CRB CR o 10 -20 -15 -10 -5 0 5 1o 5 3- -15 -10 -5 0 5 10 15 SNR (,) SNR (dB) (a) (b) - CRB - CR - - e Method 1 (no scaling) e- - 0 Method 1 2o - - Method 1 (scaling) -0 -- Method2 --v Method 2 10 -30 , 2 -70 -20 -15 -10 -5 0 5 10 15 -20 -15 -10 -5 0 5 10 15 SNR (d) SNR (dB) (c) (d) Figure 5.3: Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and array calibration error (covariance matrix = 0.041) . 84 (a) Figure 5.4: (a) Target photo taken at 450 azimuth angle. 85 45 -0.4 -0.3 -2 -01 0 01 02 03 04 05 Nonrozd Rabg F.quenoy (a) -- Normalized FFT Etimaon R 03 02 0.15 0.1 II (b) Figure 5.5: (a) the HRR range profile corrupted by clutter and jamming and (b) the average of the normalized FFT range profiles without interference (clutter and jammer) and the estimated HRR range profile by using the algorithms presented. 86 CHAPTER 6 MULTIPLE MOVING TARGET DETECTION FOR AIRBORNE HRR RADAR 6.1 Introduction In this chapter, we first introduce a single moving target detector in the presence of spatially and temporally correlated ground clutter for airborne HRR phased array radar [41]. Due to the high range resolution of the HRR radar, the scatterers of a moving target may move out of their range bins during the coherent processing interval (CPI) and hence results in the range migration problem. We divide the HRR range profiles into non-overlapping large range segments to avoid these range migration problems that occur in HRR radar data. In each range segment, we apply inverse Fourier transform (IFT) to obtain a set of HRR phase history data. Since the estimated HRR range profile in each range segment can be obtained by applying the Fourier transform (FT) to the estimated HRR phase history data after clutter and jamming suppression, no information is lost due to the division and hence no loss of resolution occurs. We show how to use the VAR filtering technique to suppress the ground clutter. Then a moving target detector based on a Generalized Likelihood Ratio Test (GLRT) detection strategy is derived. The detection threshold is determined according to the desired false alarm rate, which is made possible via an asymptotic statistical analysis. After the target Doppler frequency and spatial signature are estimated from the VAR-filtered data as if a target were present, a simple detection variable is computed and compared to the detection threshold to render a decision on the presence of a target. We next extend this single moving target detection approach to the case of multiple moving targets. Each of the targets is assumed to have a rigid-body and the 87 scatterers of the same target have the same DOA. By combining the multiple moving target feature extraction algorithms presented in the previous chapter and the single moving target detection strategy introduced herein, we present a multiple moving target detection approach. We consider the detection of one target at a time based on a relaxation-based algorithm and hence the problem of multiple target detection is reduced to a sequence of single target detection problems. In the numerical examples, we consider the effects of the degradation factors including jamming, calibration error, and system mismatch on our detector. We also consider the impact of contaminated secondary data on the detector performance. The remainder of this chapter is organized as follows. In Section 6.2, we establish the multiple moving target data model for airborne HRR phased array radar, which is followed by a brief discussion of the VAR filtering technique. In Section 6.3, we introduce the GLRT detector for single moving target detection for airborne HRR phased array radar [41]. We present the multiple moving target detection algorithm in Section 6.4. Simulation results and their analyses are presented in Section 6.5. Finally, we give the conclusions in Section 6.6. 6.2 Data Model and VAR Filtering Consider the same data model as in (5.2): D ( Kd x,(n) = E adkej d) enledna(Od) + e(n), d=1 k=1 D b di e Uld ad+ el(n), d=1 = 0,---,L-1, n= 0,---,N-1, (6.1) where x, (n) is the array output vector of the lth sample due to the nth pulse; bdl is the target phase history sequence due to the dth target; ad (dropping the dependence of ad on Od for notational brevity) is the array manifold and is a function of the dth target DOA, Od, relative to the flight path; el(n) is the interference including 88 clutter, noise, and possibly jamming; the complex amplitude adk and the frequency fdk are, respectively, proportional to the RCS and range of the kth scatterer of the dth target; WdI = Wd(1 + rl) with r = Vd/Wd being a known constant independent of the target motion and Vd and wd being, respectively, the scaled radial velocity and the normalized Doppler frequency of the dth target. The secondary data are obtained from the LRR segments adjacent to the LRR segment of interest in the same way as the primary data are obtained from the segment of interest. The secondary data are assumed to be target free and are modeled as a VAR random process [72, 49]. The VAR filter has the form: P lI(z-1) = I + E Hz-P, (6.2) p=1 where P is the VAR filter order, z-1 denotes the unit delay operator, and I is the identity matrix. The goal of the VAR filtering technique is to estimate the VAR filter coefficients, H = [H1, ... , Hp], from the secondary data and then filter the primary data so that the clutter component of the filter output is temporally white. Based on the following least squares criterion, we have L-1 S N-1 P 2 H = arg Hmin es,(n) + Hea(n - p) , (6.3) 1=0 s=1 n=P p=1 where -1 I| denotes the Euclidean norm; S denotes the number of secondary data segments; and esl(n) denotes the secondary data. Once the VAR filter coefficients are determined (see Section 4.2 for more details), we obtain the VAR filter output for the primary data yj(n) = l(z-1)x/(n) D Sbdlid'dadejdl + E(n), d=1 D SadedlejwdIn + el(n), d=1 89 l= 0,--,L-1, n=P,---,N-1, (6.4) where ldt = bdl'itdad is referred to as the spatial signature vector of the dth target for the lth phase history sample; i(z-1) has the same form as W(z-1) in (6.2) except that the {Hp}P1 in (6.2) are replaced by the estimates {1^[,} ; and El(n) denotes the filtered output due to the interference. In the next two sections, we start with the single moving target detection and then extend it to the multiple moving target detection. 6.3 Detection of a Single Moving Target Let the dth target be the target of interest. Let D ydI(n) = y1(n) - E bililiei'n, I = 0, , L - 1, n = P, , N - 1, (6.5) i=l,i d where {wJ, ai, {bi i=o }=Dd, i d is assumed available. Note that if the array manifold is known, a is replaced by a(0i), with {}1, assumed available. Hence yd(n)D can be written as ydl(n) adlewdn + EdI(n), 1= 0,. , L - 1, n= P,... , N - 1, (6.6) where Edi(n) denotes the interference due to clutter, noise, and contributions from other targets. We assume that {EdL(n)} is a zero-mean temporally white Gaussian random process with an unknown arbitrary spatial covariance matrix QdThe problem of determining whether the radar receptions contain signals from the dth target is usually posed as a binary hypothesis test, Ho: ydL(n) = Edl(n), 1 = 0, , L - 1,(6.7) SH1: Ydl(n) adlejWdlnï¿½Edl(n), l= 0,---, L - 1. Consider the joint probability density function (PDF) of {yd(n)}. The joint PDF of the VAR filtered primary data vectors under H1 is fHi ( Yd L--; Qd, {ad}L=0I ,d) rMNL Qd L exp -NLTr [QdC ( 1{ade} _01,d) ] (6.8) 90 |

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CHAPTER 2 LITERATURE SURVEY Historical and modern perspectives on the general topics of clutter suppres sion, moving target detection, and parameter estimation for airborne radar have been well documented in the literature. Among them, there are many classical articles, both theoretical and application-oriented, [74, 51, 1, 31, 32, 71, 47, 12], excellent texts are also available [45, 62, 52, 76, 4, 85, 78, 43, 3, 80, 11, 27, 72]. In this Chapter, we give a brief review of a number of subjects that are related to our work, namely clut ter suppression and moving target detection and parameter estimation for airborne phased array radar. 2.1 Clutter Suppression For airborne radar, due to the strong ground clutter reflection, clutter sup pression is critical for airborne radar signal processing. The ground clutter seen by an airborne radar is extended in both range and angle. It also is spread over a region in Doppler due to the platform motion. A potential target may be obscured not only by mainlobe clutter that originates from the same angle as the target but also by sidelobe clutter that comes from different angles but has the same Doppler frequen cy. In recent years, many signal processing schemes have been proposed for ground clutter suppression for airborne low range resolution (LRR) phased array wide-area surveillance radar. 2.1.1 Displace-phase-center-antenna (DPCAl DPCA [10, 62] processing is one of the simplest methods for clutter suppression currently used in certain airborne surveillance radar systems. DPCA is designed to 10 4.3.1 Doppler Frequency and Spatial Signature Estimation We assume that the VAR filtered interference is a zero-mean temporally white Gaussian random process with an unknown covariance matrix Q. Then the negative log-likelihood function is proportional to C\ = In |Q| + Tr (Q_1C), (4.19) where | | and Tr(-) denote, respectively, the determinant and the trace of a matrix, and ^ ^ 1=0 n=P = E tY aA"l IY' uSftH > (42) 7VjL 1=0 with Y, = [yi{P) yi(N 1)], I = 0, , L 1, (4.21) and /3( = [eip ... / = 0, ,L 1. (4.22) Minimizing C\ using the structure of a in (4.17) is a highly nonlinear optimization problem that would require a search over a multidimensional parameter space. To simplify the problem, we assume that {a;}^,1 are all unknown. In particular, the array manifold information a(9) is not used at this stage, and hence it can be arbi trary and unknown, which increases the robustness of the algorithm derived in this section against array calibration errors. Since the number of unknowns is increased by assuming that {a/}^1 are unknown, the performance of our algorithm will be worse than if the structure of {a;}^1 was used, according to the parsimony principle [69]. Yet by doing so, we obtain a much simpler and faster, and hence more practical, algorithm. 46 We remark that the above simulation results show that Methods 1 and 2 pro vide similar performances for target relative complex amplitude and range frequency estimation. Method 2 avoids the 1-D search over the DOA space and usually requires only a few (3 ~ 6) iterations when SNR < 0 dB and no iteration when SNR > 0 dB. To estimate the target phase history sequence {bi}Â¡CQl from the spatial signature estimates {/}^1, Method 2 needs only about 6% (for the no iteration case) or 30% (for 4 iterations) of the amount of computations measured in MATLAB flops required by Method 1 (this difference is mainly due to the fact that the latter requires a 1-D search over the DOA space). Hence if the array calibration errors are known to be significant enough to result in a useless target DOA estimate or if the target DOA is not of interest, Method 2 is to be preferred over Method 1. 4.5 Summary We have presented a moving target feature extraction algorithm for the air borne HRR phased array radar in the presence of temporally and spatially correlat ed ground clutter, jamming interference that is a temporally white, spatially point source, and spatially and temporally white Gaussian noise. To avoid the range mi gration problems that occur in HRR radar data, we divided the HRR range profiles into LRR segments. We have shown in detail how to use a VAR filtering technique to suppress the ground clutter for the HRR radar. To gain robustness against the array calibration errors, we assumed in a first stage that the array manifold is completely unknown and arbitrary. From the VAR-filtered data, the target Doppler frequency and the spatial signature vectors were first estimated by using a ML method. The target phase history and DOA (or the array steering vector for unknown array man ifold) were then estimated from the spatial signature vectors by minimizing a WLS cost function. The RCS related complex amplitude and range related frequency of each target scatterer were finally extracted from the estimated target phase history by using RELAX, a relaxation-based high resolution feature extraction algorithm. 56 xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E7I32GL6A_UVG78R INGEST_TIME 2015-02-05T20:04:16Z PACKAGE AA00024538_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES and ad = Wd2ad. We minimize the following approximate cost function L-1 Cf = (tdi bdia.d)H (Ldi bdi&d), 1=0 which is equivalent to C8 = IIrf db 2 F where || ||f denotes the Frobenius norm, b d [bdo bd(L-i)]J (5.29) (5.30) (5.31) (5.32) and < = [do *d(L-i)] (5.33) The Cs in (5.31) is minimized if db = UdxOdiV^, (5.34) where ud\ and vdl are, respectively, the left and right singular vectors associated with the largest singular value od\ of Ad- Then either the initial estimate of b<, i.e., bf = vj (5.35) or the initial estimate of a<, i.e., 4 = W^uj,, (5.36) can be used to initialize the alternating optimization approach. We use the one in (5.35) in our numerical examples. The steps of Method 2 are summarized as follows: Step (0): Obtain the initial estimate b0) of bd with (5.35). 73 This is a dedication to my loving and supportive parents. upon the array design, the number of array elements could be several or a few thou sand. Because phase shifters and associated equipment are relatively expensive, the technique is expensive to implement. Its advantage is that the technique has signif icantly better performance than a parabolic reflector type of single antenna. Thus, the use of this technique requires a performance versus implementation cost trade-off. Unlike the traditional mechanically scanned radars, the phased arrays need not scan mechanically. Scanning is accomplished electronically by adjusting the phase within each transmit/receive (T/R) module (i.e., radiating element) to focus the wavefront, which is a line of equal phase radiation, in the desired direction. Key performance advantages of the phased arrays are the wide (octave) bandwidth, spectral purity, and the agile beam pointing capabilities. Fig. 1.2 shows a line array used as a re ceiving antenna. Each element is multiplied by a gain that phase shifts and amplifies the received signal from each array element. The array output is the sum of these weighted signals. In any usual radar environment, there are many reflectors. Some of these reflectors are of interest to the radar system and are called targets. Some of these reflectors are of no interest to the radar system and are called clutter. Reflectors that are of no interest to the radar system cause difficulties in two manners: (1) erroneously reporting clutter as target tends to overload the data processors capacity and (2) clutter can prevent the detection and parameter estimation of desired targets. For airborne phased array radar, due to the strong ground clutter reflection, clutter suppression is critical for airborne phased array radar signal processing. The ground clutter seen by an airborne radar is extended in both range and angle. It also is spread over a region in Doppler due to the platform motion. A mesh plot view of the interference environment seen by an airborne phased array radar is shown in Fig. 1.3(a). The Jammer in Fig. 1.3 is localized in an angle and distributed over all Doppler frequencies. The clutter echo from a single ground patch has a Doppler 2 Figure 6.3: Multiple moving target detection probability (for PÂ¡ = 10-6) under conditions of: (1) no target in the secondary data (only clutter and noise) (solid line), (2) the presence of two targets in the seondary data with different DOAs and different Doppler frequencies from the two targets in the primary data (dotted line), (3) the presence of two targets in the secondary data with different DOAs and the same Doppler frequencies from the two targets in the primary data (dashed line), and (4) the presence of two targets in the secondary data with both the same DOAs and Doppler frequencies as the two targets in the primary data (dashdotted line). 102 1.2 Scope of the Work Our work starts with the airborne LRR phased array radar. Due to the im portance of clutter suppression for airborne radar, in Section 3.2, we simulate the high fidelity ground clutter [80]. Then in Section 3.3, we introduce the Vector Auto- Regressive (VAR) filtering technique [72] which whitens the clutter temporally, as well as a robust unstructured maximum likelihood (ML) method for moving target parameter estimation. We demonstrate that the VAR filtering technique and the unstructured ML estimation method work effectively and robustly for clutter sup pression and parameter estimation. In Chapter 4, our discussion is extended to the airborne HRR phased array radar. To avoid the range migration problems that occur in HRR radar data, we first divide the HRR range profiles into LRR segments. Since each LRR segment contains a sequence of HRR range bins, no information is lost due to the division and hence no loss of resolution occurs. We show how to extend the VAR filtering technique to suppress the ground clutter for the HRR radar. Then a parameter estimation algorithm is proposed for target feature extraction. Numerical results are provided to demonstrate the effectiveness performances of the proposed algorithm. Since the multiple moving target scenario occurs frequently in radar appli cations yet little research work has been reported in the literature. In Chapter 5, our discussion is focused on multiple moving target feature extraction. The feature extraction is achieved by using a relaxation-based algorithm to minimize a nonlin ear least squares fitting function by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently de termined values. Thus the problem of multiple target feature extraction is reduced to the feature extraction of a single target, which is presented in Chapter 4, in a relaxation-based iteration step. Numerical examples are used to demonstrate the effectiveness of this multiple moving target feature extraction algorithm. 5 APPENDIX A THE DERIVATION OF MOVING TARGET DATA MODEL FOR HRR AIRBORNE RADAR We sketch below the derivation of the data model in (4.1). We first establish the data model for a single antenna and then extend it to the case of the phased array radar. Assume that the radar transmits a group of chirp pulses with the pulse width T0 and the pulse repetition interval (PRI) T. A normalized chirp signal has the form s(t) = e-j(2nfot+pt2), |i| < To/2, (7.1) where /0 denotes the carrier frequency and 2p denotes the frequency modulation rate. Let K and v denote the scatterer number of a ridig-body target of interest and the radial velocity between the radar and target, respectively. Then, at time t, the range of the kth scatterer is Rk(t) = Rk + vt, k = 1, , K, where Rk denotes the range location of the kth scatterer when the first pulse is transmitted. Let t t nT denote the fast time, where n is the pulse number. Then the received signal from the kth scatterer is: rk(t) = akexp |i| < To/2, n = 0,1, ,7V 1 2vt\ 2nf0 (t t0 Ark J + p ( t r0 Ark Â¥) (7.2) where ak is determined by the RCS of the scatterer, r0 = 2R0/c, Ark = 2 (Rk Ro)/c with R0 denoting a reference range (possibly corresponding to the center of the target), c is the speed of light, and N is the total number of pulses transmitted in a CPI. By using s(t r0), (|i r0| < T0/2), as the reference signal, the dechirped signal has the form: xk(t,n) = rk(t)s*(t r0) OLk exp . 2vt j2ivfo ( Ark + exp jp 2i 2t0 Ark 2 vt + 2 vt (7.3) 105 CHAPTER 5 MULTIPLE MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR 5.1 Introduction In this Chapter, we extend the single target approaches to the case of multiple targets. The multiple moving target scenario occurs frequently in radar applications. Yet to the best of our knowledge, little research on the topic has been reported in the literature. We present a relaxation-based algorithm for multiple moving target feature extraction. Each of the targets is assumed to have a rigid-body and the scatterers of the same target have the same DOA. The relaxation-based algorithm is used to minimize a nonlinear least squares fitting function by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently determined values. Thus the problem of multiple target feature extraction is reduced to the feature extraction of a single target in a relaxation-based iteration step. We use numerical examples to demonstrate the performance of this algorithm for clutter suppression and multiple moving target feature extraction. The remainder of this Chapter is organized as follows. In Section 5.2, we establish the multiple moving target data model for airborne HRR phased array radar, which is followed by a brief discussion of the VAR filtering technique. In Section 5.3, we present the relaxation-based multiple moving target feature extraction method. Simulation results and their analysis are presented in Section 5.4. Finally, we give the conclusions in Section 5.5. 5.2 Data Model and VAR Filtering The range resolution of a radar is determined by the transmitted signal band width. To achieve high range resolution, a radar must transmit wideband pulses, 65 REFERENCES [1] S. P. Applebaum. Adaptive arrays. IEEE Transactions on Antennas and Prop agation, AP-24(5):585-598, September 1976. [2] W.J. Bangs. Array processing with generalized beamformers. Ph.D. dissertation, Yale University, New Haven, CT, 1971. [3] Z. Bao, G. Liao, R. Wu, Y. Zhang, and Y. Wang. Adaptive spatial-temporal processing for airborne radars. Chinese Journal of Electronics, 2(1), April 1993. [4] E. C. Barde, R. L. Fante, and J. A. Torres. Some limitations on the effectiveness of airborne adaptive radar. IEEE Transactions on Aerospace and Electronic Systems, 28(4), October 1992. [5] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty. Nonlinear Programming Theory and Algorithms. John Wiley & Sons, Inc., New York, NY, 1992. [6] L. E. Brennan and I. S. Reed. Theory of adaptive radar. IEEE Transactions on Aerospace and Electronic Systems, AES-9:237-252, March 1973. [7] E. Conte, M. Lops, and G. Ricci. Radar detection in K-distributed clutter. IEE Proceedings F, 141 (2): 116-118, April 1994. [8] E. Conte, M. Lops, and G. Ricci. Asymptotically optimum radar detection in compound-Gaussian clutter. IEEE Transactions on Aerospace and Electronic Systems, 31 (2):617625, April 1995. [9] E. Conte and G. Ricci. Sensitivity study of GLRT detection in compound- Gaussian clutter. IEEE Transactions on Aerospace and Electronic Systems, 34, 1998. [10] F. R. Dickey, M. Labitt, and F. M. Staudaher. Development of airborne moving target radar for long range surveillance. IEEE Transactions on Aerospace and Electronic Systems, 27:959-971, November 1991. [11] A. Drosopoulos and S. Haykin. Adaptive Radar Parameter Estimation with Thomsons Multiple-Window Method. In daptive Radar Detection and Estima tion, S. Haykin and A. Steinhardt (eds.), Chapter 7. John Wiley and Sons, Inc., New York, NY, 1992. [12] J. Ender and R. Klemm. Airborne MTI via digital filtering. IEE Proceedings, 136(1):2229, February 1989. [13] A. Farina. Antenna-Based Signal Processing Techniques for Radar Systems. Artech House, Norwood, MA, 1992. 115 o (b) Figure 1.3: Mesh plot of (a) the power spectral density of the interference (including clutter, jamming, and noise) and (b) Fully adaptive response when the clutter ridge is along the diagonal (7 = 1) and the jamming is from 9j = 120. 9 (6.31); Step 2: Compute the sample covariance matrix R using (6.13); Step 3 : Obtain the estimate of the Doppler frequency Ud by minimizing (6.18) via a 1-D search at the frequencies 2nn/N for n 0, 1, , N 1, and then calculate Zdi(&d), 1 = 0, , L 1, according to (6.16); Step 4: Compute the detection variable Â£ and render a decision according to (6.23). Note that the above detector is derived with an asymptotic analysis, which suggests to determine the Doppler frequency via a 1-D search over the DFT grid 2irn/N, n = 0, 1, , N l. However, a targets Doppler frequency may not be very close to any of the DFT grid and searching over only N samples over the DFT grid may not yield an accurate Doppler frequency estimate, especially when N is small. The detection performance may degrade due to the poor parameter estimates. To alleviate this difficulty, we search over a finer grid 2ixn/(qN), n 0, 1, , qN 1, with q being an integer greater than 1, to obtain a more accurate Doppler frequen cy estimate. We still use the aforementioned approach to calculate the detection threshold, even though using a finer grid may cause the asymptotic distribution of the resulting test statistic to be more difficult to determine and the asymptotic prob ability of false alarm no longer guaranteed. Nevertheless, numerical examples show that using a finer grid yields a performance improvement of about 3 dB [41]. 6.4 Detection of Multiple Moving Targets The multiple moving target detection approach is based on a combination of the multiple moving target feature extraction in Chapter 5 and the single moving target detection in Section 6.3. The detection approach consists of the following steps: Step I: Assume that there is one target, i.e., D = 1. Estimate the target space-time parameters {ud, {adi}i=o)d=\ from VAR filter output {y(n)}, / = 0, , L - 1, n P, , N 1 by using the single moving target feature extraction algorithms 95 Figure 6.2: Multiple moving target detection probability (for PÂ¡ = 10-6) under conditions of: (1) no degradation factors except for the presence of clutter plus noise (solid line), (2) the presence of a point source jammer at 120 (dotted line), (3) the presence of both the array calibration errors and a point source jammer at 120 (dashed line), and (4) the presence of the system mismatch (7 = 1.1) in addition to the degradation factors in (3) (dashdotted line). 101 Figure 4.5: Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 45 with JNR=30 dB, no array calibration errors no system mismatch.) 62 where Rd is given in (6.13), and Zdl(Ud) -^pCdiPdli l 0, , L 1. (6.16) The minimization of C2 in (6.15) with respect to adi gives di = ZdiNLFi(i+rÂ¡)> Z = 0, ,Â£-1, (6.17) with the estimate Â¡>d of cod being determined by minimizing the following cost func tion, C3 In Rw 1 L Ll Zdi(ud)zdi{ud), 1=0 (6.18) which requires a 1-D search over the parameter space. Note that after the estimate ujd of the Doppler frequency is obtained, the spatial signature estimate in (6.17) is the temporal average of the Doppler shift and range migration compensated VAR filtered spatial measurements. 6.3.2 GLRT Detection Strategy We formulate the GLRT detection strategy as - flh {Yd/};=0 ;Qd,{d}I=0 H0 c H ({Y^Jto1;^ Ho) Ho > r0, Ri (6.19) where {^}^1 and u>d are, respectively, the maximum likelihood estimates of the spatial signature vectors and target Doppler frequency, r0 is a threshold, which is to be determined according to the desired false alarm rate, and C ({di}Â¡Lo, )d) under Fi, \ Rd, under H0, (6.20) with C ({rf/}^,1, u>d) having the same form as C ({adi}Â¡Ho, ^d) in (6.9) except that {adij^o1 and codi are replaced by {&di}jCo and >d( 1 + rl), respectively, and Rd is defined in (6.13). 92 [76] M. Viberg, B. Ottersten, and T. Kailath. Detection and estimation in sensor arrays using weighted subspace fitting. IEEE Transactions on Signal Processing, 39(ll):2436-2449, November 1991. [77] H. Wang and L. Cai. On adaptive spatial-temporal processing for airborne radar systems. IEEE Transactions on Aerospace and Electronic Systems, 30(3):660 670, July 1994. [78] H. Wang and C. J. Lee. Adaptive array processing for real-time airborne radar detection of critical mobile targets. Adaptive Antenna Systems Symposium, Melville, NY, November 1992. [79] H. Wang, H. R. Park, and M. C. Wicks. Recent results in space-time processing. Proceedings of the 1994 National Radar Conference, Atlanta, GA, pages 104-109, 1994. [80] J. Ward. Space-time adaptive processing for airborne radar. Technical Report 1015, MIT Lincoln Laboratory, December 1994. [81] S. Watts. Radar detection prediction in k-distribution sea clutter and thermal noise. IEEE Transactions on Aerospace and Electronic Systems, 23(1) :4045, January 1987. [82] D. R. Wehner. High Resolution Radar. Artech House, Norwood, MA, 1987. [83] R. Wu, J. Li, and Z.-S. Liu. Super resolution time delay estimation via MODE- WRELAX. IEEE Transactions on Aerospace and Electronic Systems, 35(1):294- 307, January 1999. [84] W. I. Zangwill. Nonlinear Programming: A Unified Approach. Prentice-Hall, Inc., Englewood Cliffs, NJ 07632, 1969. [85] M. D. Zoltowski. Beamspace ML Bearing Estimation for Adaptive Phased Array Radar. In Adaptive Radar Detection and Estimation, S. Haykin and A. Steinhardt (eds.), Chapter 5. John Wiley and Sons, Inc., New York, NY, 1992. 120 which is independent of the target velocity. For an HRR phased array radar with the array manifold a(0), (7.5) becomes K x,(n) = a(0) J2akej27rfkl ejvnle juj0n ,k=0 n 0, , N 1, l = 0, , L 1. (7.10) Finally, when the interference is included, we obtain the data model in (4.1). 107 Define the spatial and temporal steering vectors associated with the Ith clutter patch in the ith range bin to be, respectively, vs(, l) = [ 1 ej27n?ii (3.5) and vt{i, l) = [ 1 ei2irWil . ej2n(N-l)uru jT_ (3.6) Rewrite the clutter sequence Xnm, rn = 0, o' II e r-H 1 , N 1, in a 1-D vector Nr Nc = 1) Va(t, l) CMNx\ (3.7) =1 /=i Where Then the clutter is modeled as a Gaussian random process with the space-time covariance matrix expressed as Qc = E{xcXc} Nr Nc = 5^X)-{|o
1=1 1=1which is a Toeplitz-block-Toeplitz matrix with dimension MN x MN. Given Â£, v, and T, the clutter ridge slope 7 il C (3.9) zuu 2 vT is a constant. The 2-D clutter spectrum in the plane of normalized spatial frequency versus normalized Doppler frequency appears as a straight line with a slope of 7 (see Figure 3.1 (a) and (b) for the 2-D clutter spectrum with 7=1 and 7 = 1.2, respectively). So far, the clutter model in (3.8) is non-fluctuating, i.e., the returns from a clutter patch do not fluctuate from pulse to pulse. However, in practice, many factors, such as any pulse-to-pulse instabilities of the radar system components and the intrinsic clutter motion due to wind, may cause small pulse to pulse fluctuations 20 (a) Figure 5.5: (a) the HRR range profile corrupted by clutter and jamming and (b) the average of the normalized FFT range profiles without interference (clutter and jammer) and the estimated HRR range profile by using the algorithms presented. 86 Space-Time Adaptive Processing (STAP) based techniques [80, 79] simultaneously process the signals received from multiple elements of an antenna array and from multiple pulses. STAP is data-adaptive and can outperform DPCA in the presence of sensor mismatch or velocity controlling errors. However STAP can require a sig nificant amount of computations due to the need of using a bank of filters and the inversions of matrices of large dimensions. A Vector Auto-Regressive (VAR) filtering technique was recently proposed in [72] to suppress the clutter adaptively. The VAR filtering technique whitens the correlated clutter only temporally, and can be compu tationally simpler than STAP. Although the technique can be easily used for spatial whitening as well, it is not needed since the VAR-filtered interference is assumed to be spatially colored with an unknown covariance matrix, which automatically achieves jamming suppression when the VAR filter output is used with the Maximum Like lihood (ML) methods presented in [72] to estimate the target parameters for LRR radar. Compared to a conventional airborne LRR radar [58], an airborne HRR radar can not only enhance the radars capability of detecting, locating and tracking mov ing targets, but can also provide more features for applications including Automatic Target Recognition (ATR) [23, 25]. Two important technical issues associated with the signal processing of an airborne HRR radar are clutter suppression and feature extraction. It appears that few techniques have been reported for HRR clutter sup pression [24, 17]. Moreover, the range migration resulted from the radial motion between the radar and the moving target will accumulate from pulse to pulse and destroy the range alignment. The range migration makes it impossible to directly use the LRR target parameter estimation approaches proposed in [72], In this paper, we present moving target feature extraction algorithms in the presence of temporally and spatially correlated ground clutter for airborne HRR phased array radars. In addition to the common DOA and Doppler frequency of all 40 (c) Figure 3.4: Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a func tion of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency in the presence of system mismatches (7 = 1.2) and when the clutter is non-fluctuating with CNR = 40 dB and there is no array calibration errors and no jamming. 36 TABLE OF CONTENTS ACKNOWLEDGEMENTS iii LIST OF FIGURES vi ABSTRACT ix CHAPTERS 1 INTRODUCTION 1 1.1 Background 1 1.2 Scope of the Work 5 1.3 Contributions 6 1.4 Outline 7 2 LITERATURE SURVEY 10 2.1 Clutter Suppression 10 2.1.1 Displace-phase-center-antenna (DPCA) 10 2.1.2 Adaptive Filtering Based Space-time Adaptive Processing (STAP) 11 2.1.3 Vector Auto-regressive (VAR) Filtering 12 2.2 Moving Target Detection 13 2.3 Parameter Estimation 14 2.3.1 The RELAX algorithm 15 2.3.2 ML methods in Radar Array Signal Processing 16 3 MOVING TARGET PARAMETER ESTIMATION FOR AIRBORNE LRR RADAR 18 3.1 Introduction 18 3.2 Clutter Simulation 19 3.3 Clutter Suppression 21 3.3.1 Data model 21 3.3.2 VAR Filtering 22 3.3.3 DPCA 25 3.4 Parameter Estimation 26 3.5 Numerical Results 30 3.6 Summary 32 IV I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jian Li Chairman Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jose C. Principe Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Fredrick J. Taylor Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. John M. Anderson Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David C. Wilson Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Electrical and Computer Engineering in the College of Liberal Arts and Sciences and 4.3.3 Target Range Feature Estimation Once the sequence {6/I^q1 is available, the range feature estimates, {ak, fk}k-j, can be obtained by minimizing the following cost function, C8 = ||b-Fa||2, (4-43) where ex = [exi a2 aK]T, (4.44) and F = [f, f2 lK (4.45) with ffc [1 ej27rfk ej27Tfk{L~1)]T k = 1, , AL (4.46) This is a sinusoidal parameter estimation problem in which a large number of solutions are available in the literature (see, e.g., [66]). Here we will use RELAX [38] that appears to be among the best available algorithms that can be used to solve the minimization problem in (4.43). To make the paper self-contained, we briefly outline the steps involved in RELAX. Assume that {cq, are giverb where K denotes the assumed number of scatterers. Let R bfc = b k = (4.47) i=l, k Then minimizing C8 in (4.43) with respect to ak yields the estimate fkHbk Oik fkfk (4.48) where the estimate fk of fk is determined from fk = arg max fk bk (4.49) 51 form y i(n) = H(z 1)xi(n) = biiiiae^171 + ei(n), l 0, , L l, n = P, ,ZV- 1, (4.14) where H{z~l) has the same form as in (4.6) except that the {Hp}^ in (4.6) are replaced by {HP}Â£=1, p iii = I + ^Hpe-^iP, Z = 0, , L 1, (4.15) p=i and e/(n) = [e/(n)j, l = 0,---,L-l, n = P,---,N~ 1. (4.16) Note that, after VAR filtering the number of temporal data samples is N N P. Since the VAR filter whitens the clutter only in the temporal domain, the VAR filtered clutter is still spatially colored. Like in [72], the parameter estimation algorithm developed herein can deal with the spatially colored VAR filtered clutter present in the filtered primary data. Let a* = biHia, Z = 0, , L 1, (4.17) be referred to as the spatial signature vector of the target for the Zth phase history sample. We can rewrite (4.14) as yi(n) = aiejuJin + ei(n), Z = 0, , L 1, n = P, , N 1. (4.18) 4.3 Target Feature Extraction The target feature extraction algorithm we present below will first obtain estimates {/}^1 of the spatial signature vectors and an estimate luo of the Doppler frequency uj0. Then the target phase history sequence {z}^1 and the target DOA 9 will be estimated from {/}^1. Finally, the estimates are used to extract the target range features {ctfe, fk}k=i- Figure 2 outlines the steps of this algorithm, with each step described in detail in the following subsections. 45 (a) (b) (c) (d) Figure 5.2: Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and no array calibration error. 83 L HRR Range Bins Secondary Segment IFFT Segment of Interest IFFT HRR Range Profile Secondary Data Primary data i-i l-0 Figure 4.1: HRR range profiles are divided into LRR segments with each segment containing L HRR range bins. 58 (a) (b) (c) (d) Figure 5.3: Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and array calibration error (covariance matrix = 0.041) . 84 CHAPTER 7 SUMMARY AND CONCLUSIONS This dissertation has considered the clutter suppression and moving target detection and parameter estimation for airborne phased array radar. In Chapter 3, we have simulated the airborne LRR radar clutter which is spread in both range and angle due to the radar platform velocity. We have nu merically demonstrated the effectiveness of the VAR filtering technique for clutter suppression and an unstructured ML parameter estimation method [72] for airborne LRR phased array radar. The robustness of this algorithm is shown by comparing it to the well-known DPCA technique in the presence of various system mismatches. In Chapter 4, our discussion has extended to the airborne HRR phased array radar. To avoid the range migration problems that occur in HRR radar data, we first divide the HRR range profiles into Low Range Resolution (LRR) segments. Since each LRR segment contains a sequence of HRR range bins, no information is lost due to the division and hence no loss of resolution occurs. We show how to use the VAR filtering technique to suppress the ground clutter for HRR airborne radar. Then a parameter estimation algorithm is proposed for target feature extraction. From the VAR-filtered data, the target Doppler frequency and the spatial signature vectors are first estimated by using a Maximum Likelihood (ML) method. The target phase history and Direction-of-Arrival (DOA) (or the array steering vector for unknown array manifold) are then estimated from the spatial signature vectors by minimizing a Weighted Least Squares (WLS) cost function. The target Radar Cross Section 103 Given bi, minimizing C\ in (4.27) with respect to a yields a = L-1 n -1 ll=0 L-1 (4.31) 1=0 where ()* denotes the complex conjugate. Given a, bÂ¡ can be estimated by using (4.29). Hence we can cyclically iterate (4.29) and (4.31) to obtain the estimates of {bi}t~,o and a To start the iteration, we must have an initial estimate of either a or {bi}^ which in our initialization approach is obtained by using the Singular Value Decom position (SVD) [63] as described next. Rewriting C\ in (4.27), we have H L-1 c5 = ^ 6*a) Q (^/- bia) > 1=0 (4.32) where Qi = ^Q''Til, l = 0, ,L 1. (4.33) To retain the weight of the term that may give the largest error in (4.32), we choose W = Q0, where l0 is selected such that is the largest among r II i in > Let III lm=o (4.34) ; = W^, ln, and = Wa. (4.35) We obtain the initial estimates by minimizing the following approximate cost function L-1 Co = ^ (, btk)H (n bÂ¡), 1=0 (4.36) which is equivalent to C7=|jA-abT||2F, (4.37) where || ||jr denotes the Frobenius norm, b = [bo bL_ i]T, (4.38) 49 CHAPTER 6 MULTIPLE MOVING TARGET DETECTION FOR AIRBORNE HRR RADAR 6.1 Introduction In this chapter, we first introduce a single moving target detector in the pres ence of spatially and temporally correlated ground clutter for airborne HRR phased array radar [41]. Due to the high range resolution of the HRR radar, the scatterers of a moving target may move out of their range bins during the coherent process ing interval (CPI) and hence results in the range migration problem. We divide the HRR range profiles into non-overlapping large range segments to avoid these range migration problems that occur in HRR radar data. In each range segment, we apply inverse Fourier transform (IFT) to obtain a set of HRR phase history data. Since the estimated HRR range profile in each range segment can be obtained by applying the Fourier transform (FT) to the estimated HRR phase history data after clutter and jamming suppression, no information is lost due to the division and hence no loss of resolution occurs. We show how to use the VAR filtering technique to suppress the ground clutter. Then a moving target detector based on a Generalized Likeli hood Ratio Test (GLRT) detection strategy is derived. The detection threshold is determined according to the desired false alarm rate, which is made possible via an asymptotic statistical analysis. After the target Doppler frequency and spatial sig nature are estimated from the VAR-filtered data as if a target were present, a simple detection variable is computed and compared to the detection threshold to render a decision on the presence of a target. We next extend this single moving target detection approach to the case of multiple moving targets. Each of the targets is assumed to have a rigid-body and the 87 Let z(n) = [ Zo(n) zm-2(n) ]T- Rewriting (3.27) in a vector form, we have z(n) = ba(9)e^n + e(n), = aejujn + e(n) eC[M~l)x\ n = 0, , N 2, (3.28) where b = >(e?27^AC0S6, eJW), a(9) is a subvector of a(9) containing the first M 1 elements of a(9), and e(n) is the interference vector after the DPCA processing. For the ideal case of non-fluctuating clutter and no system mismatches, z(n) is free of clutter, but the noise level is doubled. By applying the unstructured approach presented in [72] to z(n), we can estimate the moving target parameters {6, 9, uj}. 3.4 Parameter Estimation We outline the robust unstructured ML parameter estimation method [72] here. The negative log-likelihood function for N samples of data from model (3.25) is shown as: VN(b,9,u) = log|Q| +Tr{CT1C (b,9,u)}, (3.29) where C(b,9,w) = ^(y(n) b(9, u>)eJJn)(y(n) b(9, cj)eJun)f. (3.30) n=0 Let |Q- = % with Qij being the (ij)th element of Q. By using some matrix computation rules, we have and So and <91n|Q| dQij tr(Q-1Qy) dQ dQ, dQi dc dQij = -tr(Q-1QL)+tr(Q-1QLQ-1C) tr(Q-1Qy=[Q-1]... (3.31) (3.32) (3.33) (3.34) 26 We introdue a moving target detector based on a Generalized Likelihood Ratio Test (GLRT) detection strategy and extend it to the case of multiple moving target detection. 1.4 Outline The dissertation is organized as follows. Chapter 2 gives a literature survey of topics including clutter suppression for airborne phased array radar, target parameter estimation, and target detection. In Chapter 3, we first introduce the ground clutter observed by an airborne radar, VAR filtering based clutter suppression technique, and unstructure ML target parameter estimation algorithm. We next demonstrate the effectiveness of the VAR filtering technique and ML parameter estimation method for airborne LRR phased array radar clutter suppression and parameter estimation. Chapter 4 addresses the VAR filtering technique for airborne HRR phased array radar and the target feature extraction algorithms. Chapter 5 discusses multiple moving target feature extraction for airborne HRR phased array radar. In Chapter 6, we investigate the multiple moving target detection problem. Finally, we summarize this work in Chapter 7. 7 should have the same asymptotic variance as the structured ML estimates in (3.38) and (3.39). From (3.50) and the matrix inversion lemma [18], we have (see [72] for detail) = I R-1 - R-1^ j R aa R a l + HR_1/ 1 + a^R-1 and a"(0, c5u)Q1a(6, Cou) = aH{0, DU)R 1a(9, Cju) - "(0, w^Rr1* 1 + a^R^a Thus 9e = arg max aH{6, u)Qu1o e *h aH{9, wu)Q-1a(0, Qju) = arg max 9 aH(9, c5u)R ^l + aWR_1ay faH(0, >u)R_1a(0, uju) aH(0, ^r-1 ~rr~-T- i+a R-a = arg max aH(9, o)u)R 1a e aH(9, u>u)R-la(9, Â£>)' = 9U. (3.53) Since (3.50) implies that Qu = Ru, immediately, we have k = a"( QQ^j. (3.54) aH{9e, uu)Q-1a(9e, Cju) Note that the simplicity of the final results of be, 9e, and lju are quite remarkable. A 2-D search (see (3.39)) is simplified to two 1-D searches (see (3.49) and (3.52), respectively). Since neither the search for 0 nor uj is a search on a concave, we first find an approximate location with coarse grids and then use it as the initial condition to find a more accurate estimate via a 1-D search method, such as the FMIN function in MATLAB. Since the estimates {e, 9e, u)u} have the same asymptotic variance as the ML estimates in (3.38) and (3.39) (see [72] for more details), they are also statistically efficient estimates of {b, 0, lu} for the target model in (3.14). 29 due to the nth look of the N looks during a coherent processing interval (CPI) can be written as: x(n) = 6a(0)eJam + e(n) CMxl, n = 0, N 1, (3.14) where b is the complex amplitude of the signal proportional to the target RCS, uj is the normalized Doppler frequency due to the relative motion between the array platform and the target, a(6) is the array response due to the target from the DOA 6 relative to the flight path, and e(n) represents the interference due to clutter, jamming, and noise. Assume that the data is collected with a uniform linear array (ULA) of M sensors, i.e., a(*) = 1 cose ej(M-l)2fCOS0 (3.15) The data sequence {x(n)}^T01 is referred to as the primary data. The data from adjacent range bins are assumed to be target free and are referred to as the secondary data. Note that due to the jamming free secondary data, the conventional STAP methods fail and hence are not considered hereafter. (For the suppression of high- duty-cycle jammers where jammers exist in both primary and secondary data, both our method and other STAP approaches can handle it without difficulty and hence are ignored.) Rewrite (3.14) as x(n) = ad + e(n) CMxl, n = 0, N 1, (3.16) with a = ba(6). (3.16) and (3.14) are the so called unstructured and structured data model, respectively. 3.3.2 VAR Filtering By assuming that the statistics of the interference in the primary and sec ondary data are the same, the secondary data are used to adaptively estimate the 22 Hence the ML estimate of Q is given by Q~T + [Q-1CQ-1] |q=q = 0. (3.35) Then Qs = C(b,9,u). (3.36) where the subscript s corresponds to the estimates of the structured data model. Substitute (3.36) into (3.29), we have: VN(b,0,uj) = log|C(6,0,w)| + M. (3.37) As shown in [37], we have where Â£ ag(0a, }a)Kjy(}a) aH(6s, hs)R-1(0s, us) slh(0, /)R_1y(w) nax x x , (1 yH(u;)R_1y(u;))ag(0, u;)R_1a(0, u) N-1 yM = y(n)e_Jwn> n=0 R=^T.y(n)y"(n). n=0 = R-y(w)yff(w), (3.38) (3.39) (3.40) (3.41) (3.42) bs, 9S, and us are the ML estimates of 6, 9, and u in (3.25), respectively. However, the estimates 9S and Cjs are coupled in a 2-D search. To simplify the 2-D search algorithm, we turned to the unstructured data model (3.26). Similarly, the negative log-likelihood function for N samples of data is shown to be: VN{a,u) = log|Q| + Tr{Q-1C(a, u>)}, (3.43) where N-1 C{a,u) = ^2(y(n) eJam)(y(n) aeJU,n)H. n=0 (3.44) 27 [6] J. Li, G. Liu, N. Jiang, and P. Stoica, Clutter Suppression and Moving Tar get Detection and Feature Extraction for Airborne High Range Resolution Phased Array Radar, Proceedings of the SPIEs 14th Annual International Symposium on Aerospace/Defense Sensing, Simulation and Controls, Orlando, FL, April 2000. [7] N. Jiang and J. Li, Multiple Moving Target Feature Extraction for Airborne HRR Radar, Proceedings of the 4th Work Multiconference on Systemics, Cybernetics and Informatics (SCI) Orlando, FL, July 2000. [8] J. Li, G. Liu, N. Jiang, and P. Stoica, Airborne Phased Array Radar: Clutter and Jamming Suppression, and Moving Target Detection and Feature Extraction, Proceedings of the 1st IEEE Sensor Array and Multichannel Signal Processing Work shop, Cambridge, MA, March 2000. [9] N. Jiang and J. Li, Space-Time Processing for Airborne High Range Resolution Phased Array Radar, accepted by 2001 IEEE Radar Conference, Atlanta, GA, May 2001. 122 Then (3.19) can be rewritten as H = argmin ||E H\F||^, (3.20) H where || \\f denotes the Frobenius norm. Provided that the matrix has full rank, then H has the form [72] H = E^h 1. (3.21) A necessary condition for the existence of the matrix inverse in (3.21) is that \& has at least as many columns as rows, i.e., S(N P) > MP, a condition that can be easily satisfied by choosing S large enough. Let us take a look at the effect from the possiblely existing jammer in the secondary data on the filter coefficients. Assume the interference in the secondary data includes only jammer and noise. From (3.18), straightforwardly, we have E{es(n)ef(n) + --- + HLes(n-L)ef(n)} = E{es(n)ef (n)}, E{es(n)ef (n 1) H b He(n L)e*(n 1)} = E{es(n)ef (n 1)}, (3.22) E{es(n)ef (n L) H b HLes(n L)ef (n L)} = E{es(n)ef (n L)}, where E{x} denotes the expected value of x. Let E{es(n)e^(n /)} = Q(Z) Â£ CMxM, then rewrite (3.23) by matrices, we have [I Hi HL ] Hence, Q(0) Q(-l) Q(l) Q(0) Q (L) Q(L-l) Q (-L) Q(-L + l) Q(0) [ Q(i) 0 o]. [Hx Hl] = [ Q(-l) Q(-L)] 24 APPENDIX C PROOF OF (6.26) We prove below the following approximation for large N: 1 4^ 1-R'd1T2zdi{d)zZ{d) 1=0 4 1 L-1 1 ~~ 1 T Z^d)Zdl (Wd)Rrf 1=0 1 L~l 1 (7-47) Z=0 with which (6.23) is approximated as (6.26). Note that we have used |I AB| = |I BA | in the first equality of (7.47). Let i i L_1 G = RÂ¡1j-^2zdi(i)z^{i)R1. (7.48) ^ 1=0 We perform the eigndecomposition on G and obtain, G = UAIJ-1, (7.49) where U is a unitary matrix containing the eigenvectors of G and Ai O' A = (7.50) 0 A m with Ai, , Am being the corresponding eigenvalues of G. Inserting (7.49) into the left side of (7.47), we have |I-G| = |U(I A)U-11 |I A| M = (7-51) m= 1 Since G is 0(1/N) under H0, we can approximately express (7.51) as M |I-G| l-J^Am = l-Tr(G) m= 1 1 4^ = 1 i2lzdi^d)'Rd1Zdi(Od), (7.52) ^ 1=0 which concludes the proof. 114 which requires a 1-D search. Note that the spatial signature estimate in (5.18) can be interpreted as the temporal average of the Doppler shift and range migration compensated VAR filtered spatial measurements. The estimate Qd of the covariance matrix of the VAR filtered interference sequence can be written as 1 iVL L1 V KY2 - 1=0 (5.20) Given and Qd, the estimates of {bdi}Â¡CQ and DOA 9d (if the array manifold is known) or ad (if the array manifold is unknown) can be obtained by minimizing the following Weighted Least Squares (WLS) cost function, L-1 H C5 = ^ (i bfu'Hdia.dj 1 (adi ~ bdiiidiaSj (5-21) 1=0 Note that this cost function is similar to the one used for the unstructured method in [72] since the Fisher information matrix (FIM) for adl in (5.12) is also proportional to Q^1 (see [72] for more details). Method 1: To estimate 9d, we must know the array manifold a(9d) as a function of 9d. The method for estimating {bdi}^ and 9d by using the array manifold is referred to as Method 1. Without loss of generality, we consider a uniform linear array (ULA), where a(9d) has the form, a(0d) = ! ei2-?cosdd ej(M-l)2fÂ£cOS0d (5.22) with A being the radar wavelength and Â£ being the spacing between two adjacent sensors. Minimizing C5 in (5.21) gives i ad ''dl Q d 1 ^dl dl ~ . a-d'HdiQd1'H and H ~ , \^ndl^adl 9d = arg max > nj i=o a^dlQ-]diad (5.23) (5.24) 71 to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 2001 M. Jack Ohanian Dean, College of Engineering Winfred M. Phillips Dean, Graduate School Doppler frequencies as the two targets in the primary data. Figure 6.3 shows that the targets in the secondary data could cause severe problems to the targets in the primary data during the VAR filtering process only if the targets in the secondary data have both the same DOAs and Doppler frequencies as the targets in the primary data. As we have discussed in the previous chapters, we use the estimated VAR filter based on the assumed target-free secondary data to pre-whiten (null) the clutter in the primary data. If the clutter in the secondary data is stronger than the clutter in the primary data, the VAR filter overnulls the clutter in the primary data, which is even desirable because it increases the signal to interference ratio at the filter output. For the contaminated secondary data, the moving targets in the secondary data are treated as clutter. Hence if the moving targets in the secondary data happen to have the same DOAs and Doppler frequencies as the targets of interest in the primary data, the targets in the primary data will be nulled out due to the targets in the secondary data, which results in degraded performance. Otherwise, the moving targets in the secondary data are just some interferences that are not related to the targets in the primary data and hence will not cause them to be nulled out. Therefore, the presence of moving targets with different DOAs or Doppler frequencies in the secondary data have little effect on the targets in the primary data. 6.6 Summary We have introduced the single moving target detection strategy for the air borne HRR phased array radar in the presence of clutter, jammer, and noise. We have shown how to extend the single target detector to detect multiple moving tar gets. Numerical examples have shown that the HRR multiple moving target detector is robust against jammer, calibration errors, system mismatch, and the secondary data contaminated by the presence of other targets. 99 number of targets, which is assumed to be known or can be determined by using the Generalized Akaike Information Criterion (GAIC) [38, 61]. The practical convergence in the iterations of the above algorithm can be determined by checking the relative change e of the cost function C\ in (5.10) between two consecutive iterations. The steps leading to the last step are needed to provide good initial conditions for the last step of the algorithm. The cyclic optimization approach [5] converges under mild conditions although the global minimum is not a gaurantee. If the multiple targets are closely spaced, the convergence speed could be slow. To increase the convergence rate, we can either provide better initial estimates or use certain speed acceleration methods [5]. Estimating the target range features {dk, fdk}ki from bd, d = 1, , D, is our next concern. 5.3.2 Target Range Feature Estimation Once the sequences {bdi}^Q, d = 1, , D, for all targets are available, the range feature estimates, {adk, fdk)k=v d = 1, , D, can be obtained by minimizing the following cost function, C9({adk, fdk}Â¡) = ||bd Fdc*d||2, d = 1, , D, (5.37) where bdi is the Ith element of vector bd, Oid [dl adKd]T d=l,---,D, and Fd = [fdi dKd], d=l, ,Â£>, with idk [1 ej2nfdk ej2nfi^L~1'>]T d = 1, , D, k = 1, , Kd. The estimates {dk, fdk)kti f {adk, fdk}k=i can be obtained by using the RELAX algorithm (see [38] for more details), which has a similar structure as the approach used for the space-time parameter estimation. 75 in the clutter returns. Clutter fluctuation will broaden the clutter ridge and make the clutter rejection more difficult. Define, OLil = oa au (3.10) where a-) is the random complex amplitude for the /th clutter patch in the th range bin from the nth pulse. Then (3.7) becomes Nr Nc xc = Y^ v*(> *)) Vs^>(3-n) =i i=i where denotes the Hadamard matrix product [18]. Usually, the fluctuation is modeled as a wide-sense stationary Gaussian ran dom process. Thus we have [80]: 7u(n) = E = <72iaexp (3.12) where av is referred to as the velocity standard derivation. Plugging (3.12) into (3.8), we have the covariance of the fluctuating clutter: Nr Nc Qc = Y^[ra v( 0vt(*. l)H) (v*(b 0V(*> l)H), (3.13) ii ;=i where r = E{ol(ol)h} = 7(0) 7(l) 7ii(l) 7(0) 7(W-1) lu{N-2) 7a(iV-l) ju{N-2) 7(0) is the N x N covariance matrix describing the fluctuations for the /th patch of the th range bin (see Figure 3.1 (c) for the 2-D clutter ridge with ov 1 and 7 = 1). 3.3.1 Data model 3.3 Clutter Suppression For an airborne surveillance radar, where at most one target is assumed to be present in any given range bin, the sampled output vector of an array of M sensors 21 We simulate the ground clutter as a temporally and spatially correlated Gaussian random process [80]. The Clutter-to-Noise Ratio (CNR), defined as the ratio of the clutter variance to the noise variance, is set to be CNR= 40 dB. A jamming signal, which is a zero-mean temporally white Gaussian random process, also exists. The Jammer-to-Noise Ratio (JNR), which is the ratio of the jammers temporal variance to the noise variance, is chosen as JNR= 25 dB and the jamming signal impinges from 9j = 120. When array calibration errors exist, the errors for different elements are assumed to be independent and identically distributed complex Gaussian random variables. In our simulations, a complex Gaussian random vector with zero-mean and covariance matrix 0.041 is added to the array manifold to simulate array calibration errors, which implies that the variance of the calibration error for each element is 0.04. Another degradation factor, the system mismatch error, which is due to the antenna spacing, platform velocity, and/or pulse repetition frequency (PRF) mismatch errors, is also considered. We use the clutter ridge slope [36] 7 = 1 in the simulations for the case when the system mismatch error is absent and 7 = 1.1 for the case when the system mismatch error exists. We first present an example of neither array calibration errors nor system mismatch errors. Note that the complex amplitude estimates {dk}k=i f {o-dk^kt 1 obtained via Method 2 are all scaled by a common unknown complex scalar. To calculate their best possible MSEs, we scale them to minimize Co = Hado PctdW2, (5.38) where ctd0 is the true value of ad. Minimizing (5.38) with respect to /3 yields: a __ &d ad0 oq\ P m hi|2 (5.39) a Note that this scaling scheme is only used to illustrate the complex amplitude esti mation performance; it is not a necessary step in a practical application including ATR since only the relative amplitudes are of interest. For comparison purposes, the 77 Ground Moving Target Detection and Feature Extraction with Airborne Phased Array Radar By Nanzhi Jiang A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2001 [29] M. Kaveh and A. J. Barabell. The statistical performance of the MUSIC and the minimum-norm algorithms in resolving plane waves in noise. IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-34(2):331-341, April 1986. [30] E. J. Kelly. An adaptive detection algorithm. IEEE Transactions on Aerospace and Electronics Systems, 22(1):115127, March 1986. [31] R. Klemm. Adaptive clutter suppression for airborne phased array radars. IEE Proceedings, 130(1):125-731, February 1983. [32] R. Klemm. Adaptive airborne MTI: an auxiliary channel approach. IEE Pro ceedings, 134(3):269-276, June 1987. [33] R. Klemm. Adaptive airborne MTI with two-dimensional motion compensation. IEEE Proceedings-F, 138(6) :551558, December 1991. [34] J. Li. Improving ESPRIT via beamforming. IEEE Transactions on Aerospace and Electronic Systems, 28(2):520-528, April 1992. [35] J. Li and R. T. Compton, Jr. Angle and polarization estimation using ESPRIT with a polarization sensitive array. IEEE Transactions on Antennas and Prop agation, 39(9):1376-1383, September 1991. [36] J. Li, G. Liu, N. Jiang, and P. Stoica. Moving target feature extraction for air borne high range resolution phased array radar, to appear in IEEE Transactions on Signal Processing. [37] J. Li and P. Stoica. An adaptive filtering approach to spectral estimation and SAR imaging. IEEE Transactions on Signal Processing, 44(6): 1469-1484, June 1996. [38] J. Li and P. Stoica. Efficient mixed-spectrum estimation with applications to target feature extraction. IEEE Transactions on Signal Processing, 44:281-295, February 1996. [39] J. Li, P. Stoica, and D. Zheng. Angle and waveform estimation via RELAX. IEEE Transactions on Aerospace and Electronic Systems, 33:1077-1087, July 1997. [40] J. Li and D. Zheng. Parameter estimation using RELAX with a COLD array. Circuits, Systems, and Signal Processing, 17(4):471-481, July 1998. [41] G. Liu and J Li. Moving target detection via airborne hrr phased array radar. Submitted to IEEE Transactions on Aerospace and Electronic Systems, to appear 2001. [42] D. L. Mensa. High Resolution Radar Cross-Section Imaging. Artech House, Inc., Norwood, MA, 1991. [43] D. P. Michal and D. R. Fuhrmann. Multiple target detection for an antenna array using outlier rejection methods. 0 of ICASS93, 4:45-48, 1993. [44] J. H. Michels, P. Varshney, and D. D. Weiner. 0 signal detection involving temporal and cross-channel correlation. IEEE Transactions on Aerospace and Electronic Systems, 31 (3):866879, July 1995. 117 2.2 Moving Target Detection Target detection is critical for radar applications since without target detec tion, target feature extraction and automatic target recognition (ATR) are not possi ble. The radar target detection issue has been widely investigated since the beginning of the radar era. Most previous investigations on the detection of a radar target in clutter and noise are based on some statistical models of the clutter plus noise, which include Gaussian, Log-Normal [20], Weibull [19, 16], K-family [81, 46, 7, 56] distrib utions, and more recently, the so called alpha-stable (SctS) distribution [28, 60] and compound-Gaussian distribution [9, 8]. These statistical model based detection al gorithms consider mainly the temporal statistics of the radar measurements of the clutter. The non-Gaussian detection schemes may suffer from complicated model parameter estimation problems. A classical radar problem is to detect targets of unknown amplitude embedded in clutter with unknown statistical properties. It is assumed that returns from other range cells are available to provide an estimate of the statistical properties of the clutter. Kelly derived a decision rule for detection of a signal of unknown amplitude masked by Gaussian interference with unknown covariance [30]. The decision rule is a generalized maximum likelihood ratio test (GLRT) in the sense that the condi tional densities that constitute the likelihood ratio are maximized over the unknown parameters. In his paper, Kelly does not make any optimality claims for his detec tor. However, recent work by Scharf and Friedlander has shown that equivalence of the GLRT and uniformly most powerful invariant detectors, thereby conferring a sense of optimality to the GLRT [57]. Additionally, the GLRT exhibits the property of constant false alarm rate (CFAR) implying that the probability of false alarm is independent of the noise covariance matrix. More recently, Swindlehurst and Sto- ica develop a GLRT for target detection under an unstructured array model [72]. 13 where | | and Tr(-) denote, respectively, the determinant and trace of a matrix, and 1 L-1 JV-l c {{adi}f^o ,^d) = Wn) ~ a dieJUJdin] [y di{n) Adle]Udin]H 1=0 n=P L-1 = *7 E IY< a'^] [Y, a*0%]" (6.9) ^ L 1=0 with Ydi = [ydi{P) y
andPdi = [ejuldlP ej^N^]H, l 0, , L 1. (6.11) The joint PDF of the VAR filtered data vectors under H0 is U ({Y4S; Q-) = TiisrJ^pi p (Qi1^)} <612) where i ^ *- = 15fEy-y5- <6-13) 6.3.1 Doppler Frequency and Spatial Signature Estimation The negative log-likelihood function under H\ is proportional to C: = In |Qd| + Tr [Q^C ({a4,^0] (6-14) Minimizing Ci in (6.14) with respect to Qd results in Qd = C ({ad/}^1,^) under Hx. Similarly, we can obtain Qd = Rd under H0. With the estimate Qd under Hi, the cost function in (6.14) is equivalent to C2 = In | C ({a^I^Zq1 W(i) | In L-i 1 ^2 (adi zdi{ujd)) (aa zdi{ud)) H 1=0 1L_1 TRd 7 ZdlMz-dl (Wd) 1=0 (6.15) 91 3.5 Numerical Results We present several numerical examples to demonstrate the effectiveness of the VAR filtering approach for clutter suppression. The following parameters are common to all of the simulations: The target complex amplitude 6 = 1, the normalized Doppler frequency uj 0.27T, the DOA 9 = 60, the wavelength A = 0.03 m and the interelement distance Â£ = A/2. The VAR filter order P 2 and the number of secondary range bins S = 5. The number of antenna elements M = 8 and the number of pulses in a CPI N = 16. The noise is the zero-mean white Gaussian random process with variance a2N. The clutter is the zero-mean colored Gaussian random process with the covariance matrix Qc. The jamming is assumed to be zero-mean and temporally white Gaussian with variance a2 and its DOA is 0j. In our simulations, the SNR, jamming-to-noise ratio (JNR), and CNR, respectively, are defined as: and where Qc{i,j) is the ijth element of Qc and the JNR and CNR are set to be 25 dB and 40 dB in the simulations, respectively. If the clutter ridge is along the diagonal, the clutter ridge slope 7 = Â£/(2vT) 1. For comparison purposes, we have also included the associated CRBs for the moving target parameters. First, consider the case of non-fluctuating clutter, no system mismatches, and no array calibration errors. We assume that there is also no jamming. For this case, 30 which is equivalent to Ho -jr^Zdii^Qd'zdiid) > 1 e~T/(2iVi). (6.29) 1=0 H\ Since 1 exp(t/2(L)) ~ t/2(L) for large L, (6.29) is asymptotically equiva lent tO TT L-1 2'^2zjl(d)Qdlzdi(d) % r. (6.30) i=o Hi Note that zdi{^d) is an M x 1 zero-mean Gaussian random vector with the covari ance matrix Qd/7V. Hence for every ojd, Y^io zdi(ud)Qd lzdi{^d) is a sum of the squares of 2ML real-valued independent Gaussian random variables with zero-mean and variance 1/2. Therefore ZY^fo zdi(ud)Qdlzdi{UJd) is a central x2-distributed random variable with 2ML degrees of freedom. Then the Â£ in (6.23) is (asymptotical ly) the maximum value of a collection of x2(2ML) random variables. The threshold r can be determined according to a desired probability of false alarm, Pj. As shown in [72], if we restrict the evaluation of the detection variable to the Discrete Fourier Transform (DFT) frequencies n/, n = 0, 1, , 1, then Â£ is asymptotically distributed as the maximum value of a collection of independent x2(2ML) random variables. Let F(u) = P {p < u\p ~ x2(2ML)} (6.31) be the cumulative density function of the central x2-distribution with 2ML degrees of freedom, where P{-} expresses the probability that an event occurs. Then given Pf, we can use F(t) = (1 Pf)'!*, (6.32) to determine r by also using (6.31) and the MATLAB function CHI2INV. 6.3.4 Moving Target Detection Steps The steps of the moving target detection algorithm include (see Figure 6.1): Step 1: For a given Pf, determine the detection threshold by solving (6.32) and then 94 -1 Q(0) Q(1) Q(-L +1) Q(l) Q(0) Q(-L + 2) Q(L-l) Q(L-l) Q(0) (3.23) where Q(/) denotes the estimated covariance matrix from the available secondary data. If there are only noise and jammer in the secondary data, due to the properties of noise and jammer, Q(/) = 0, for l ^ 0. So from eqn. (3.23), we will have [Hx HL ] ~ 0. (3.24) Therefore, the jammer in the secondary data contributes little to the VAR filter coefficients and the VAR filter has little effect on the jammer in the primary data. The filtered array output will have the form, y(n) = 'H(z~1)x(n) = bH(e-^)a(0)ejbjn + e(n) = ba(9, uj)ejun + e(n) (3.25) = ejun + e(n) eCMx\ n = P, , N 1, (3.26) where a(0, u) = T-L{e~^)a{9) and = bJi(e~^)a[9). The moving target parameters {b, 0, a;} can then be estimated from the VAR filter output by using the unstructured approach presented in [72]. 3.3.3 DPCA Let xm(n) be the mth element of x(n) in (3.14). Then the DPCA processing yields zm(n) = xm+i(n) xm(n + 1), n = 0, , N 2, m = 0, , M 2. (3.27) 25 (c) (d) Figure 4.4: Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, no jammer, no array calibration errors, no system mismatch.) 61 They compare the unstructured GLRT with the standard test via simulation for var ious levels of array perturbation. Even in the absence of such perturbations, the unstructured GLRT may be useful in providing a more rapid initial scan of the environment prior to application of the structured model. In [41], Liu and Li in vestigate the moving target detection in the presence of ground clutter for airborne HRR phased array radar. Due to the high range resolution of the HRR radar, the scatterers of a moving target may move out of their range bins during the coherent processing interval (CPI) and hence results in the range migration problem. They di vide the HRR range profiles into non-overlapping large range segments to avoid these range migration problems that occur in HRR radar data. A moving target detector based on a GLRT detection strategy is derived in [41]. The detection threshold is determined according to the desired false alarm rate, which is made possible via an asymptotic statistical analysis. After the target Doppler frequency and spatial signa ture vectors are estimated, a simple detection variable is computed and compared to the detection threshold to render a decision on the presence of a target. Numerical results are provided to demonstrate the performance of the proposed moving target detection algorithm. 2.3 Parameter Estimation The essence of the spectral estimation problem is [66] From a finite record of a stationary data sequence, estimate how the total power is distributed over frequen cy. Spectral analysis is widely used in many diverse fields, including economics, meteorology, astronomy, speech analysis, and radar and sonar systems. Generally speaking, there are two branches of analysis methods: non-parametric and paramet ric methods. Just opposite to the non-parametric method, the parametric or model based methods of spectral estimation assume that the signal satisfies a generating model with known functional form, and then proceed by estimating the parameters 14 model for the array is that it provides robustness to errors in the model for the array response. 17 Once 6d is determined, {bdi}i=o IS obtained with (5.23) by replacing ad with a (dd). Method 2: To achieve robustness against array calibration errors, we can assume that did is completely unknown. The corresponding method used to estimate both {bdi}Â¡Z0] and ad is referred to as Method 2. Note that since replacing {bdi}jCQ by {PbditfJo1 and ^d by ad/(3 in (5.21), where (3 is any non-zero complex scalar, does not change C5, {bdi}^ and ad can only be determined up to an unknown multiplicative complex constant. This unknown complex scalar, similar to the unknown gain and initial phase of a radar system, does not affect most practical applications such as ATR. Given {bdi}^=Q, minimizing C5 in (5.21) with respect to ad yields a d = L-1 1=0 -1 L1 (5.25) =o where ()* denotes the complex conjugate. Given ad, {bdi}^1 can be estimated by using (5.23). Hence we can cyclically iterate (5.23) and (5.25) to obtain the estimates of {bdi}^ and ad. To start the iteration, we must have an initial estimate of either ad or Our initialization approach is obtained by using the Singular Value Decomposition (SVD) [63]. Rewriting C5 in (5.21), we have where Ll C*6 ^ (-di &di bdiad^j Qdi (idi &di bdiad^j , 1=0 .h ~ Qdi^ndlQ^ndl, 1 = 0, , l 1. (5.26) (5.27) To place the most weight on the term that is associated with the largest signal energy, we choose Wd = Qdlo, where l0 is selected such that L~dÂ¡0di0 is the largest among 1 W'M'dl S^dl r Let til IIJ/=0 'Wd.'tdl &dl, (5.28) 72 and Qj = o2jInl (ajaj ), (7.19) with oj 0., and Oj being, respectively, the variances of the noise, clutter, and jamming; IL being the identity matrix of dimension L; Qc being as given in (3.8); and a j denoting the jammer steering vector which has the same form as a (9) in (4.28) except that 9 in (4.28) is replaced by the jammer DOA 9j. According to the extended Slepian-Bangs formula, the zjth element of the Fisher information matrix (FIM) has the form [2, 66] (7.20) where Â¡i = [(fs 0 f<) 0 fr] 0 a, x- denotes the derivative of x with respect to the zth unknown parameter, and Re(x) the real part of x. Note that the FIM is block diagonal since the parameters in Q are independent of those in Â¡j, and vice versa. Hence, the CRB matrix for the target features and the motion parameters can be calculated from the second term on the right side of (7.20). Let rj = [ Re7 (a) Imr(a:) fT lu0 9 ]T , (7.21) where f is a vector consisting of the range frequencies of the scatterers, and Im(x) denotes the imaginary part of x. Let G, = {[(FI/c) 0 fd] (u^ 0 fr)} 0 a, (7.22) and G2 jG\. (7.23) where uk = [1 l]r G TZKxl. Let G3 = j2tt { [((d 0 olt) F) 0 fd] (uÂ£ 0 fr)} 0 a, (7.24) where d = [0 1 L 1]T. Let G4 = j {(fs 0 fd) [(uL + rdL) 0 dN] fr} 0 a, (7.25) 109 where = [1 l]2 G CLxl and Cd = j(7r2/180) sin(0d7r/18O) with 9d being measured in degrees. Let G = [ Gar GQi Gf G, Ge], (7.45) where Gar = [ gif g^j Sdi ''' Â§dkd ] > GQi = j Gar, G/ = [ g(i g{Kl Sdi gdkd ] > Gw = [ g? go ] , and G Then the CRB matrix for the parameter vector rj is given by CRB (rj) = [2Re(GHQ-1G)]_1. (7.46) 113 4.3 (a) Mesh plot and (b) projection of the power spectral density of the interference (including clutter, jamming, and noise) and the target when the clutter ridge is along the diagonal (7 = 1), the jamming comes from 9j = 45, and the target is located at DOA = 60 with u)q = 0.27T. The axes d and zo are for the spatial frequency and the normalized Doppler frequency, respectively, (see Equations (3.1) and (3.2)) 60 4.4 Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, no jammer, no array calibration errors, no system mismatch.) 61 4.5 Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 45 with JNR=30 dB, no array calibration errors , no system mismatch.) 62 4.6 Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, a jammer at 45 with JNR=30 dB, array calibration error covariance matrix 0.041, no system mismatch.) 63 4.7 Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 45 with JNR=30 dB, array calibration error covariance matrix 0.041, system mismatch with clutter ridge slope 7 = 1.1.) ... 64 5.1 (a) The real HRR range profile without interference (clutter and jam mer) and the estimated HRR range profile after clutter suppression and feature extraction, and (b) the cluttered HRR range profile with CNR= 40 dB, SNR= 0 dB 82 5.2 Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and no array calibration error 83 5.3 Comparison of MSEs with CRBs as a function of SNR for (a) Doppler frequency, (b) DOA, (c) amplitude, and (d) range frequency of the first scatterer of the first target with CNR= 40 dB, JNR= 25 dB, and array calibration error (covariance matrix = 0.041) 84 5.4 (a) Target photo taken at 45 azimuth angle 85 5.5 (a) the HRR range profile corrupted by clutter and jamming and (b) the average of the normalized FFT range profiles without interference (clutter and jammer) and the estimated HRR range profile by using the algorithms presented 86 6.1 Flow chart of the proposed clutter suppression and single moving tar get detection algorithm for HRR phased array radars 100 6.2 Multiple moving target detection probability (for PÂ¡ = 10~6) under conditions of: (1) no degradation factors except for the presence of clutter plus noise (solid line), (2) the presence of a point source jam mer at 120 (dotted line), (3) the presence of both the array calibration errors and a point source jammer at 120 (dashed line), and (4) the presence of the system mismatch (7 = 1.1) in addition to the degra dation factors in (3) (dashdotted line) vii 101 Figure 1.1: Echoes from an aircraft and a moving vehicle on the ground. Figure 1.2: Conceptual block diagram of line array antenna. 8 4 MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR 39 4.1 Introduction 39 4.2 Data Model and VAR Filtering 41 4.3 Target Feature Extraction 45 4.3.1 Doppler Frequency and Spatial Signature Estimation ... 46 4.3.2 Phase History Sequence and Array Steering Vector Esti mation 47 4.3.3 Target Range Feature Estimation 51 4.4 Numerical Examples 52 4.5 Summary 56 5 MULTIPLE MOVING TARGET FEATURE EXTRACTION FOR AIR BORNE HRR RADAR 65 5.1 Introduction 65 5.2 Data Model and VAR Filtering 65 5.3 Feature Extraction of Multiple Moving Targets 68 5.3.1 Space-Time Parameter Estimation 69 5.3.2 Target Range Feature Estimation 75 5.4 Numerical Examples 76 5.5 Summary 80 6 MULTIPLE MOVING TARGET DETECTION FOR AIRBORNE HRR RADAR 87 6.1 Introduction 87 6.2 Data Model and VAR Filtering 88 6.3 Detection of a Single Moving Target 90 6.3.1 Doppler Frequency and Spatial Signature Estimation ... 91 6.3.2 GLRT Detection Strategy 92 6.3.3 Asymptotic Statistical Analysis 93 6.3.4 Moving Target Detection Steps 94 6.4 Detection of Multiple Moving Targets 95 6.5 Numerical Examples 96 6.6 Summary 99 7 SUMMARY AND CONCLUSIONS 103 APPENDIXES REFERENCES 115 BIOGRAPHICAL SKETCH 121 v CHAPTER 1 INTRODUCTION This dissertation is concerned with clutter suppression and moving target detection and feature extraction for airborne phased array radar. This chapter serves as a general introduction to the background and scope of the work. Contributions are also highlighted in this chapter. 1.1 Background Airborne radars are used to detect the presence of moving targets and estimate their parameters in the presence of noise, ground clutter, and jammers. A typical application (see Fig. 1.1) of these radars is surveillance to detect low-flying aircrafts or vehicles moving over terrain through possible weather disturbances. The function of the airborne radar in this situation is to reject the returns from terrain (usually called ground clutter) and weather while retaining the return from the targets of interest, such as the aircrafts or vehicles, thereby allowing target detection and followed by target parameter estimation for possible automatic target recognition. Meanwhile, the ability to locate targets in angle is a key characteristic of vir tually every airborne radar. The high angle (spatial) resolution can be offered by phased arrays. Phased arrays have been used in fields as diverse as radar, commu nications, electronic warfare and radio astronony; furthermore, researchers plan to use phased array-based systems to enhance airport safety and traffic efficiency. The phased array technique is a much newer technology than a single antenna system. The antenna array consists of many elements such as dipoles. Each element of the phased array is phase shifted to steer the beam to the desired direction. Depending 1 (RCS) related complex amplitude and range related frequency of each target scat- terer are then extracted from the estimated target phase history by using RELAX, a relaxation-based high resolution feature extraction algorithm. We have extended the single target approaches to the case of multiple targets in Chapter 5. We present a relaxation-based algorithm for multiple moving target feature extraction. Each of the targets is assumed to have a rigid-body and the scatterers of the same target have the same DOA. The relaxation-based algorithm is used to minimize a nonlinear least squares fitting function by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently determined values. Thus the problem of multiple target feature extraction is reduced to the feature extraction of a single target in a relaxation-based iteration step. The final discussion has been the multiple moving target detection for airborne HRR phased array radar in Chapter 6. We have shown how to combine the multiple moving target feature extraction methods with the single moving target detection strategy to solve the problem. In summary, we have investigated the problems of clutter suppression and moving target detection and feature extraction for airborne phased array radar. Ef fective algorithms have been proposed and good simulation results have been accom plished. 104 negative log-likelihood function of ydi(n) in (5.12), is proportional to C'2 = ln|Qd| + Tr(Qd-1Cd), (5.13) where | | and Tr(-) denote, respectively, the determinant and the trace of a matrix and Cd = 777 W") ~ adieJUdin] [ydi{n) adie^d,n]U ^ ^ 1=0 n=P Ll ^ 177 T, 1Y^ [V* *05] (5.14) ^ ^ 1=0 with Ydi = [ydi{P) ydi{N 1)], Z = 0, , L 1, (5.15) and Pdl = [ejUdlP ejd,(N- 1}]H, Z = 0, , L 1. (5.16) Minimizing C2 in (5.13) with respect to Qd results in Qd = Cd. The cost function in (5.13) with Qd replaced by Qd becomes: In Ll J- NL ^ =0 N ad; Y di(3. dl N |V yH Yd;/3d//3^Yd(1 +Yd'Yd' _^ a di Yd,A dl N H (5.17) The minimization of C3 in (5.17) with respect to adi gives , Z = 0, , L 1. (5.18) Udl=&d(.l+rl) The estimate of the Doppler frequency ud of the dth target is obtained by minimizing the concentrated cost function, adi = -^Y diAcu C4 = In NL L-1 E 1=0 L YdlYdi - YdiAdiAdiY, dl N (5.19) 70 We next consider the case of a point source at 45 as a jammer, where all other parameters are the same as for Figure 4.4. The Jammer-to-Noise Ratio (JNR), which is the ratio of the jammers temporal variance to the noise variance, is chosen as JNR=30 dB. Figures 4.5(a) and (b) show the MSEs of the complex amplitude and range frequency estimates as a function of SNR. Note that the VAR filtering technique cannot suppress the jammer since the jamming signal is already temporally white. The jamming is suppressed in the parameter estimation algorithm due to the assumption that the interference is spatially colored with arbitrary and unknown covariance matrix. Next, we present an example in which array calibration errors are present in the array steering vector. All other parameters are kept the same as for Figure 4.5. Figures 4.6(a)-(d) show the MSEs of the estimates of the Doppler frequency, target DOA, and complex amplitude and range frequency of the first scatterer, as functions of SNR. Note that the MSEs of the complex amplitude and range frequency estimates obtained via Method 2 are close to the CRBs as the SNR increases, and that the MSEs of the complex amplitude estimates for Method 1 fail to follow the CRBs if the scaling scheme is not used, though the range frequency is still well estimated via Method 1. Hence both Methods 1 and 2 are robust against array calibration errors as far as the relative complex amplitude and range frequency of the scatterer are concerned. Our final example concerns the case where three degradation factors, i.e. array calibration error, jamming, and system mismatch (7 = 1.1) are present. Figures 4.7(a) and (b) exhibit the curves of the MSEs and the CRBs of the complex amplitude and range frequency estimates versus SNR. Comparing Figures 4.7 with 4.6, we note that our parameter estimation algorithms are robust against system mismatch and can obtain accurate range feature estimates. These results also verify the robustness of using the VAR filtering technique for clutter suppression in the presence of system mismatch. 55 6.3 Multiple moving target detection probability (for Pf = 10 6) under conditions of: (1) no target in the secondary data (only clutter and noise) (solid line), (2) the presence of two targets in the seondary data with different DOAs and different Doppler frequencies from the two targets in the primary data (dotted line), (3) the presence of two tar gets in the secondary data with different DOAs and the same Doppler frequencies from the two targets in the primary data (dashed line), and (4) the presence of two targets in the secondary data with both the same DOAs and Doppler frequencies as the two targets in the primary data (dashdotted line) viii 102 frequency that depends on its aspect angle with respect to the radar platform; clutter from all angles lies on the clutter ridge shown in Figure 1.3(a). A potential target may be obscured not only by mainlobe clutter that originates from the same angle as the target but also by sidelobe clutter that comes from different angles but has the same Doppler frequency. Therefore, clutter suppression is very challenging and important for airborne phased array radar. Clutter suppression has drawn a lot of attention during these years and many signal processing schemes have been proposed for it, which will be addressed in the following chapters. Target detection is critical for radar applications since without target detec tion, target feature extraction and target recognition are not possible. The radar target detection issue has been widely investigated since the beginning of the radar era. As we know, the degree of detectability of targets in a noise environment is con trolled by the ratio of the reflected signal energy and the thermal noise power. Often, however, even when the targets signal-to-noise ratio (SNR) value is very large, it is difficult to detect it because the reflected energy is less than that reflected from the clutter. That is, if the radar cross section (RCS) of the clutter in the resolution vol ume containing the target is greater than the RCS of the target, it may be impossible to detect the presence of the target since it is obscured by the clutter. Thus, there are two related problems. The first is to prevent clutter from being reported as a target. The second is to detect targets despite the presence of clutter in the same resolution cell. Note that if a signal processor, based on a discriminant, attenuates the clutter return while not affecting the target return, both problems are alleviated. Again, this is another reason why clutter suppression is critically important for airborne phased array radar. For a radar system, resolution is defined as the ability of the radar to distin guish between two closely spaced targets in at least one of the state variables used 3 I = O, , L 1, n = P, , N 1, (6.4) where ctdi bdii-Ldi&d is referred to as the spatial signature vector of the dth target for the Ith phase history sample; has the same form as ^(z-1) in (6.2) except that the {Hp}p=1 in (6.2) are replaced by the estimates {Hp}p=1; and e/(n) denotes the filtered output due to the interference. In the next two sections, we start with the single moving target detection and then extend it to the multiple moving target detection. 6.3 Detection of a Single Moving Target Let the dth target be the target of interest. Let D ydi(n) = yi(n) ^ b'Hte:IOJiin, Z = 0, ,L 1, n = P, , N l, (6.5) 1=1, i^:d where {>, , {bii}^1}^, i^d assumed available. Note that if the array manifold is known, 2 is replaced by a(d), with d assumed available. Hence ydi{n) can be written as ydi{n) = adieJuld,n + edi{n), l = 0, , L 1, n = P, , N 1, (6.6) where edi{n) denotes the interference due to clutter, noise, and contributions from other targets. We assume that {edi{n)} is a zero-mean temporally white Gaussian random process with an unknown arbitrary spatial covariance matrix Q. The problem of determining whether the radar receptions contain signals from the dth target is usually posed as a binary hypothesis test, H0: ydi{n) = edi{n), Z = 0, ,L 1, Hi : ydi(n) = adleJlJdin + edi(n), 1 = 0, ,L 1. Consider the joint probability density function (PDF) of {yd/(n)}. The joint PDF of the VAR filtered primary data vectors under H\ is 1 (6.7) .flit ({Ydj}^1; Q,i, {adl}t=o>Ud) = ~ 7T MNL Qd \NL exp j-iVZ/Tr Q C } (6.8) 90 is so small that the target occupies several HRR range bins. The data collected by an LRR radar is only a small subset of the data collected by an HRR radar. Consider an airborne HRR radar having a one-dimensional (1-D) antenna array with M elements uniformly spaced along the flight path of an airborne platform. A cluster of N chirp pulses is transmitted during the coherent processing interval (CPI). After dechirping, sampling, and Fourier transformation of the signals in each element of the array, we obtain the HRR range profiles. These profiles may be blurred by the ground clutter so severely that without clutter suppression, they are not useful for any applications. To avoid range migration problems, we divide each HRR profile into non-overlapping LRR segments so that each LRR segment contains L HRR range bins, as shown in Figure 1. We choose L to be much larger than the maximum number of range bins over which any target can possibly expand and migrate during the CPI. We then apply inverse Fourier transform to each segment. For the segment of interest, where a target may be present, the inverse Fourier transform (IFFT) yields the primary data, which can be written as (see Appendix A for the model derivation): xj(n) a(0) K jlirfkl Ufc=l ejv0nlejUon + ei(n)j l = 0,- ,L 1, n = 0, , N 1, (4.1) where K is the number of scatterers of the target, N is the number of pulses trans mitted during the CPI, and L is the number of HRR bins per LRR cell as well as the number of target phase history samples. The complex amplitude a and the frequency fk are, respectively, proportional to the RCS and range of the kth scat- terer of the target. The vectors xÂ¡(n), a(0), and e(n) are M x 1; x;(n) is the array output vector of the Ith phase history sample due to the nth pulse; a(9) is the array manifold and is a function of the target DOA 6 relative to the flight path; e(n) is the interference including ground clutter, possibly a jammer that is temporally white 42 in Section 4.3. Determine the existence of the target by using the single moving target detection strategy introduced in Section 6.3. If a target is present, go to the next step, otherwise, stop the detection process. Step II: Assume that there are two targets, i.e., D = 2. Estimate the target space-time parameters {ujd, {adi}^Jo}d=i frm the VAR filter output {yz(n)}, l 0, , L 1, n = P, , N 1 by using the relaxation-based multiple moving target feature extraction algorithms in Section 5.3. Then based on the estimated features for each possible target, use the single moving target detection strategy (see Section 6.3) to determine the existence of each target seperately. If the detected target num ber D is equal to D, go to the next step, otherwise, stop the detection process. Remain Steps: Set D D + 1. Continue similarly until D ^ D. The D obtained in the final step of the above approach is the estimated number of targets. The detection threshold r is a fixed number determined by (6.32) in the mul tiple target detection process because each detection is based on one target at a time. Assume that the true moving target number is D. Each target is assumed to be inde pendent of each other and independent of the interference also. Before the true target number is reached, i.e., if the assumed target number D is less than the true target number D, the remaining (D D) targets can be appoximated as colored Gaussian with zero-mean and treated as part of the clutter. This approximation results in little performance degradation due to the robustness of the data model (6.6). 6.5 Numerical Examples We present several numerical examples to illustrate the performance of the proposed clutter suppression and moving target detection algorithms. In the following examples, we assume that the array is a uniform linear array (ULA) with M = 8; the interelement distance between two antennas is ( A/2; the number of pulses in a CPI is N 18; the phase history is L 32, i.e., an LRR range segment contains 32 96 Gaussian random process. Figure 4.3 shows the mesh plot of the power spectral den sity of the clutter, jamming, noise, and the target observed by the airborne radar, when the clutter ridge is along the diagonal (7 = 1), the jamming impinges from 9j 45 with JNR = 30 dB, and the target is located at DOA = 60 with ujq = 0.27r. We first present an example in the absence of array calibration errors, jam ming, and system mismatch. Note that the complex amplitude estimates {0^}^ of {ak)k=i obtained via Method 2 are all scaled by a common unknown complex scalar. To calculate their best possible MSEs, we scale them to minimize C9 = \\a0-p\\\ (4.50) where a0 is the true value of a. Minimizing (4.50) with respect to Â¡3 yields 0 = ~ H a a0 Hall2 ' (4.51) Note that this scaling scheme is only used to illustrate the complex amplitude esti mation performance; it is not a necessary step in a practical application including ATR since only the relative amplitudes are important. For comparison purpose, the MSEs of the estimates {&k}k-\ obtained via Method 1 are presented in two ways: with and without a scaling scheme similar to (4.50). Figures 4.4(a)-(d) show the MSEs of the estimates of the Doppler frequency, target DOA, and the complex amplitude and range frequency for the first scatterer when compared with the Cramr-Rao Bounds (CRBs), which are derived in Appendix C. The horizontal axis in the figures shows the values of the Signal-to-Noise Ratio (SNR), which is defined as the ratio of |aq|2 to the noise variance. Due to using the scaling scheme in (4.50), the MSEs for {dfc}f=1 may be better than the CRBs (which do not account for such a scaling). Figure 4.4 shows that the MSEs of the parameter estimates are very close to the CRBs as the SNR increases, which indicates that the clutter suppression scheme works well and the parameter estimation algorithm is highly accurate. 54 the statistical distribution of the interference in the primary data, which is usually with targets of interest, is the same as (or at least close to) the distribution of the clutter in the secondary data, which is assumed to be target free. Unfortunate ly, the large number of desired secondary data often prevents the secondary data from containing truly homogeneous data, which could result in a dramatic perfor mance degradation. Further, STAP cannot be used for jamming suppression if the secondary data bins used to obtain the second-order statistics of the ground clutter do not contain jamming interference (deceptive jammers). 2.1.3 Vector Auto-regressive (VAR) Filtering The VAR filtering technique [44, 72] is proposed to model the ground clutter observed by an airborne radar as a vector auto-regressive random process, whose coefficients are estimated adaptively from the target-free secondary data. The VAR filter is then used to suppress the clutter in the primary data temporally where a tar get may be present. Although the technique can be easily used for spatial whitening as well, it is not needed since the VAR-filtered interference is assumed to be spatially colored with an arbitrary unknown covariance matrix, which automatically achieves jamming suppression when the VAR filter output is used with the Maximum Like lihood (ML) methods presented in [72] to estimate the target parameters for LRR radar. Different from the adaptive filtering based STAP, in which usually clutter suppression is performed first and then the target direction of arrival (DOA) and Doppler frequency are determined by the pointing angle of the beam (or monopulse processing), the VAR-filtered primary data can be used with a robust unstructured Maximum Likelihood (ML) parameter estimation method proposed in [72] to achieve asymptotically statistically efficient results. In this dissertation, we not only demon strate the effectiveness of the VAR filtering technique for airborne LRR phased array radar clutter suppression but also extend it to be used for airborne HRR phased array radar. 12 if Detecting a Target Figure 6.1: Flow chart of the proposed clutter suppression and single moving target detection algorithm for HRR phased array radars. 100 MSEs of the estimates {dk}ki obtained via Method 1 are presented both with and without a scaling scheme similar to (5.38). Figures 5.2(a)-5.2(d) show the MSEs of the estimates of the Doppler frequency, target DOA, complex amplitude, and range frequency of the first scatterer of the first target as a function of the Signal-to-Noise Ratio (SNR), which is defined as the ratio of |c*ii|2 to the noise variance, and compare them with the corresponding Cramr- Rao bounds (CRBs) (see Appendix B for the CRB derivation). (The results for the other scatterer and the other target are similar.) Due to using the scaling scheme in (5.38), the MSEs of {-dk}k=i may be better than the CRBs (which do not account for such a scaling). We note that as the SNR increases, the MSEs can approach the corresponding CRBs, which indicates that the clutter suppression scheme works well and the parameter estimation algorithm is highly accurate. To further illustrate this point, we show the modulus of a corrupted HRR range profile due to clutter and jamming in Figure 5.1(a), and the modulus of the true HRR range profile compared with the modulus of the estimated HRR profile generated from the estimates of the amplitudes and the range frequencies obtained via Method 1 without the scaling scheme in Figure 5.1(b). We set SNR= 0 dB and keep all the other parameters the same as those in Figure 5.2. Note from Figure 5.1 that our algorithm performs well for clutter suppression and feature extraction. Next, we present an example when both the array calibration and the sys tem mismatch errors exist. All other parameters are kept the same as for Figure 5.2. Figures 5.3(a)-5.3(d) show the MSEs of the estimates of the Doppler frequency, target DOA, complex amplitude, and range frequency of the first scatterer of the first target, as a function of SNR. Note that the MSEs of the complex amplitude and range frequency estimates obtained via Method 2 are close to the CRBs as the SNR increases. The MSEs of the complex amplitude estimates for Method 1 fail to follow the CRBs if the scaling scheme is not used. The range frequency is still well 78 VAR filter coefficients [72], which are assumed to have the form: p U{z-1) = l + Y,HPz~P, (3-17) p=i where P is the VAR filter order, z_1 denotes the unit delay operator, and I is the identity matrix. Next we outline the estimation of the VAR filter 77(z_1) using the target-free secondary data. We denote the secondary data by {es(n)}^=T01, s = 1, , S, where S is the number of secondary range bins used to estimate the VAR filter coefficients. The VAR filter is obtained such that [72] H{z 1)es(n) = es(n) + Hies(n 1) H b HPes(n P) = es(n), n = P, , N 1, (3.18) is temporally white for each s. This approach assumes that the statistics of the clutter is the same in all of the primary and secondary data sets. The problem of estimating the VAR filter coefficients is based on the following least-squares error criterion [72]: Hi, , Hp = arg min Hi S Nl EE s=l n=P es(n) + ^Hpes(n p) p=i (3.19) Let H = [H1..-HP], = -[ef(n-l) ef(n-P)]r, *. = [*a{P)--1>.{N- 1)], ^ = [^!- vE^s], Es = [es(P) es(N 1)], and E = [Ei Es], 23 Minimizing C\ in (4.19) with respect to Q results in Q = C. The cost function in (4.19) with Q replaced by Q becomes: Co = In IC | In L-1 E NL ^ 1=0 H +Y,Y - N The minimization of C2 in (4.23) with respect to a; gives a i 1 N (4.23) (4.24) , Z = 0, , L 1. wi=o{l+rl) Then the estimate of the Doppler frequency uj0 is obtained by minimizing the con centrated cost function, C3 = In L-1 E NL ^ 1=0 h Y ,/WY N (4.25) which requires a 1-D search. Note that the spatial signature estimate in (4.24) can be interpreted as the temporal average of the Doppler shift and range migration compensated VAR filtered spatial measurements. 4.3.2 Phase History Sequence and Array Steering Vector Estimation The estimate Q of the covariance matrix Q of the VAR filtered clutter-plus- noise sequence can be expressed as 1 L-1 Q = Wl Â£ VY" **Â¡ 1=0 (4.26) Given {;}^,1 and Q, the estimates of {fy}^1 and a (or the target DOA 0 if the array manifold is known) can be obtained by minimizing the following Weighted Least Squares (WLS) cost function, L-1 H CA = E ( Wia) CT1 (, bflha) (4.27) =o Note that this cost function is similar to the one used for the unstructured method in [72] since the Fisher information matrix (FIM) for aj in (4.18) is also proportional to Q-1 (see [72] for more details). 47 PAGE 1 *URXQG 0RYLQJ 7DUJHW 'HWHFWLRQ DQG )HDWXUH ([WUDFWLRQ ZLWK $LUERUQH 3KDVHG $UUD\ 5DGDU %\ 1DQ]KL -LDQJ $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 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The signals spectral characteristics of interest are then de rived from the estimated model. In those cases where the assumed model is a close approximation to the reality, it is no wonder that the parametric methods provide more accurate spectral estimates than the nonparametric techniques. The nonpara- metric approach to Power Spectral Density (PSD) estimation remains useful, though, in applications where there is little or no information about the signal in question. The parametric approach postulates a model for the data, which provides a means of parameterizing the spectrum, and thereby reduces the spectral estimation problem to that of estimating the parameters in the assumed model. Besides the Maximum Likelihood (ML) estimation method [65, 67], the parametric methods of ten used include, for example, autoregressive (AR) model based methods [15, 14], eigendecomposition based methods including MUSIC [59, 29, 48, 22, 50, 68] and E- SPRIT [55, 54, 34, 35], and nonlinear least squares (NLS) fitting based methods as CLEAN [21, 53, 73] and RELAX [38, 39, 40]. RELAX is used in the feature extrac tion of our work due to its conceptual and computational simplicity and its excellent asymptotic statistical properties. 2.3.1 The RELAX algorithm The RELAX algorithm [38]. To minimize the Nonlinear Least Square (NLS) cost function, the RELAX algorithm performs a complete search for the global min imum of the cost function by letting only the parameters of one scatterer vary at a time while freezing the parameters of all other scatterers at their most recently de termined values. The RELAX algorithm can also be referred to as SUPER CLEAN. If the number of reiterations in each step of the RELAX algorithm is set to be ze ro, then RELAX becomes the CLEAN algorithm, which was first proposed in radio astronomy [21]. CLEAN is computationally more efficient than RELAX but its reso lution and estimation accuracy are inferior to RELAX. Hence RELAX is preferred in 15 Step (1): Update {bdi}^~Q with (5.23) by replacing ad in (5.23) with the most recently determined ad. Step (2): Update ad with (5.25) by replacing {bdi}[lQ in (5.25) with the most recently determined {bdi}jCQ . Step (3): Iterate Steps (1) and (2) until practical convergence, which is determined by checking the relative change Â£ of the cost function C5 in (5.21) between two consecutive iterations. We remark that if the range migration is negligible, i.e., r = 0 in (5.2), then {/Hdi}Â¡'=o (5-6) do not depend on l. Then Step (0) alone gives the solution that minimizes the C3 in (5.21). Summary of the Steps of Space-Time Parameter Estimation The space-time parameter estimates {ujd, 9d (or ad), hd}d=\ f the multiple moving targets can be obtained as follows: Step LI: Assume D = 1. Obtain {Cjd, 9d (or ), bd}d=\ from yi(n). Step 1.2: Assume D = 2. Compute y2i(n) with (5.11) using {ibd, 9d (or ad), bd}d=i obtained in Step 1.1. Obtain {od, 9d (or d), bd}d=2 from y2i(n). Next, compute yu(n) with (5.11) using {d, 9d (or d), bd}d=2 and then re-determine {)d, 9d (or d), b(}d=1 from yu(n). Iterate the previous two substeps until practical convergence is achieved (to be discussed later on). Step 1.3: Assume D = 3. Compute y3;(n) with (5.11) using {0>d, 9d (or d), bd}d=i,2 obtained in Step 1.2. Obtain {)d, 9d (or ad), bd}d=3 from y3/(n). Next, compute yu(n) with (5.11) using {ud, 9d (or d), bd}d=2,3 and then re-determine {iud, 9d (or d), bd}d=i from yu{n). Then, compute y2i(n) with (5.11) using {ud, 9d (ord), bd}d=i,3 and re-determine {d>d, 9d (ord), bd}d=2 from y2/(n). Iterate the above three substeps until practical convergence is reached. Remain Steps: Continue similarly until D is equal to the desired or estimated 74 The discussion of multiple moving target detection follows the discussion of multiple moving target feature extraction. In Chapter 6, our discussion is concerned with multiple moving target detection for airborne HRR phased array radar, which is simplified to a sequence of single moving target detection problems based on a relaxation-based algorithm. Numerical examples have demonstrated the effectiveness of this multiple moving target detection algorithm even for contaminated secondary data. 1.3 Contributions The main results of this dissertation are as follows: We introduce the VAR filtering technique and a ML parameter estimation method. We show the effectiveness of clutter suppression and parameter es timation by using these schemes. We compare the robustness of VAR filtering techniques to (Displaced-phase-center-antenna) DPCA in the presence of dif ferent system mismatches. We present our data model of a moving target observed by an airborne HRR phased array radar in the presence of range migration. We extend the VAR filtering technique so that it can be used for airborne HRR phased array radar. We devise an effective parameter estimation method for moving target feature extraction. We present the data model of multiple moving targets observed by an airborne HRR phased array radar in the presence of range migration. We derive a relaxation-based parameter estimation method for multiple moving target fea ture extraction by taking advantage of the technique we devised for a single moving target 6 Method 1 To estimate 9, we must know the array manifold a(9) as a function of 9. The method for estimating {fy}^1 and 9 by using the array manifold is referred to as Method 1. Without loss of generality, we consider a uniform linear array. Then a(9) has the form a (9) = cos 6 l)cos0 (4.28) where d denotes the spacing between two adjacent antenna sensors, A is the radar wavelength, and 9 is the azimuth angle of the target relative to the flight path. Minimizing C4 in (4.27) gives bi = aH'HIl'ci lat I = 0, 1, (4.29) and ~.H JL 9 |aHUX Q 'ai arg max / ~.h (4.30) i=o aHnt Q-'Uia Once 9 is determined, {fy}^1 is obtained with (4.29) by replacing a with a(9). In the simulations, we use fminQ, a built-in function in MATLAB, to search for 9. Method 2 To achieve robustness against array calibration errors, we can assume that a is completely unknown. The corresponding method is used to estimate both and a and is referred to as Method 2. Note that since replacing {bi}jc(by {/3bi}jc0l and a by a/Â¡3 in (4.27), where (3 is any non-zero complex scalar, does not change C4, {bi}Â¡Zq1 and a can only be determined up to an unknown multiplicative complex constant. This unknown complex scalar, similar to the unknown gain and initial phase of a radar system, does not affect most practical applications including ATR. Moreover, the presence of this ambiguity does not affect the minimization of C4 (see below). 48 [60] D. R. Sheen, S. C. Wei, T. B. Lewis, and S. R. DeGraaf. Ultrawide bandwidth polarimetric SAR imagery of foliage-obscured objects. SPIE Proceedings on OE/LASE, Vol. 1875, Los Angeles, CA, January 1993. [61] T. Sderstrm and P. Stoica. System Identification. Prentice-Hall International, London, U.K., 1989. [62] F. Staudaher. Airborne MTI. Radar Handbook, Chapter 16, Editor M. Skolnik, McGraw-Hill, 1990. [63] G. W. Stewart. Introduction to Matrix Computations. Academic Press, Inc., New York, NY, 1973. [64] P. Stoica, D. Eykhoff, P. Janssen, and T. Sderstrm. Model structure selection by cross-validation. International Journal of Control, 43(6): 1841 1878, 1986. [65] P. Stoica, B. Friedlander, and T. Sderstrm. Approximate maximum-likelihood approach to ARMA spectral estimation. International Journal of Control, 45(4):1281-1310, 1987. [66] P. Stoica and R. L. Moses. Introduction to Spectral Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1997. [67] P. Stoica, R. L. Moses, B. Friedlander, and T. Sderstrm. Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements. IEEE Transactions on Signal Processing, 37:378-392, March 1989. [68] P. Stoica and A. Nehorai. MUSIC, maximum likelihood, and Cramer-Rao bound. IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP- 37(5):720-741, May 1989. [69] P. Stoica and T. Sderstrm. On the parsimony principle. International Journal of Control, 36:409-418, 1982. [70] P. Stoica and T. Sderstrm. On reparametrization of loss functions used in es timation and the invariance principle. Signal Processing, 17(4):383-387, August 1989. [71] L. B. Stotts. Moving-Target Detection Techniques for Optical-Image Sequences. Ph.D. thesis, University of California, San Diego, California, 1988. [72] A. Lee Swindlehurst and Petre Stoica. Maximum likelihood methods in radar array signal processing. Proceedings of the IEEE, 86:421-441, February 1998. [73] J. Tsao and B. D. Steinberg. Reduction of sidelobe and speckle artifacts in microwave imaging: The CLEAN technique. IEEE Transactions on Antennas and Propagation, 36:543-556, April 1988. [74] H. L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons, Inc., New York, NY, 1968. [75] M. Viberg, B. Ottersten, and O. Erikmats. A Comparison of model-based de tection and adaptive sidelobe cancelling for radar array processing. Chalmers University of Technology, Gothenburg, Sweden, Tech. Rep. CTH-TE-13, April 1994. 119 [45] B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai. Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing. In Radar Array Processing, S. Haykin et. al. (eds.), Chapter 4. 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IEEE Transactions on Aerospace and Electronic Systems, 35(2):445-458, April 1999. [57] L. L. Scharf and B. Friedlander. Matched subspace detectors. IEEE Transactions on Signal Processing, 42:2146-2157, August 1994. [58] D. Curtis Schleher. MTI and Pulsed Doppler Radar. Artech House, Inc., Nor wood, MA, 1991. [59] R. O. Schmidt. Multiple emitter location and signal parameter estimation. IEEE Transactions on Antennas and Propagation, AP-34(3):276-280, March 1986. 118 (c) (d) Figure 4.6: Comparison of MSEs with CRBs as a function of SNR, for (a) target Doppler frequency, (b) target DOA, (c) complex amplitude and (d) range frequency of the first scatterer. (CNR=40 dB, a jammer at 45 with JNR=30 dB, array calibration error covariance matrix 0.041, no system mismatch.) 63 estimated via Method 1. Hence from Figure 5.3, we note that both Methods 1 and 2 are robust against both array calibration and system mismatch errors as far as the Doppler frequency, relative complex amplitude, and range frequency of the scatterer are concerned. We remark that the above simulation results show that Methods 1 and 2 pro vide similar performances for target relative complex amplitude and range frequency estimation. Method 2 avoids the 1-D search over the DOA space and usually re quires only a few (3 ~ 6) iterations. To estimate the target phase history sequence from the spatial signature estimates {q/}^1, Method 2 needs only about 10% ~ 30% of the amount of computations measured in MATLAB flops required by Method 1. (This difference is mainly due to the fact that the latter requires a 1-D search over the DOA space.) Hence if the array calibration errors are known to be significant enough to result in a useless target DOA estimate or if the target DOA is not of interest, Method 2 is preferred over Method 1. Consider next an example using the experimental data generated by using the Moving and Stationary Target Acquisition and Recognition (MSTAR) Slicy data collected by imaging a slicy object shown in Figure 5.4(a). The data were collect ed by the Sandia National Laboratory using the STARLOS sensor. The field data were collected by a spotlight-mode SAR with a carrier frequency of 9.559 GHz and bandwidth of 0.591 GHz. The radar was about 5 km away from the ground object. The data were collected when the object was illuminated by the radar from approx imately the azimuth angle 0 and elevation angle 30. The pulse width is 3 /rs; the PRF is 3000 Hz; the sampling frequency is 130 MHz. We assume that the target is moving toward the radar with a radial speed of 14 meters/sec. According to [26], approximately, we have the normalized Doppler frequency ujd = 0.1 97t and the scaled radial velocity Vd = 0.0005^. 79 plus noise (solid line), (2) the presence of a point source jammer from 120 (dotted line), (3) the presence of both the array calibration errors and a point source jammer from 120 (dashed line), and (4) the presence of the system mismatch (7 = 1.1) in addition to the degradation factors in (3) (dashdotted line). Figure 6.2 shows that only very small degradation in detection performance occurs due to the presence of the jamming interference, the array calibration errors, and the system mismatch. This means that the clutter suppression and the detection scheme works well in the presence of degradation factors. Our next example concerns multiple moving target detection for the contami nated secondary data, i.e., there are two other targets in the secondary data. Figure 6.3 shows the probability of detection obtained under conditions of (1) no targets in the secondary data, (2) two targets in the secondary data with DOAs 9\ = 100 and 62 = 120, and Doppler frequencies Di = 0.6-7T and D2 = 0.87T with r = 0.001. Each of the two targets consists of two closely spaced scatterers (K\ K2 = 2) with parameters an = 1, a12 = 1, *21 = 1, 22 = 1, fn = 0.2, /12 = 0.2 + 1/2L, f21 = 0.4, and /22 = 0.4+1/2L, (3) two targets in the secondary data with DOAs 9i = 100 and 92 = 120, A 0.27T and D2 = 0.4-7T with r 0.001. Each of the two targets consists of two closely spaced scatterers (K\ = K2 = 2) with parameters &n = 1, \2 = 1, 2i = 1, 22 = 1, /11 = 0.2, /12 = 0.2 + 1/2L, /21 = 0.4, and f22 = 0.4 + 1/2L, (4) two targets in the secondary 9\ 30 and 62 = 150, and Doppler frequencies Di = 0.27T and D2 = 0.47T with r = 0.001. Each of the two targets consists of two closely spaced scatterers (K\ K2 = 2) 6in = 1, 2 = 1, 22 = 1, fu = 0.2, /12 = 0.2 + 1/2L, /21 = 0.4, and f22 0.4 + 1/2L. Note that for case (2), the two targets in the secondary data have different DOAs and Doppler frequencies from the two targets in the primary data; for case (3), the two targets in the secondary data have different DOAs but the same Doppler frequencies from the two targets in the primary data; and for case (4), the two targets in the secondary data have both the same DOAs and 98 but a point source in space, and spatially and temporally white Gaussian noise. We assume that the clutter and noise in different HRR range bins are independent and identically distributed. The real-valued scalars v0 and u0 are, respectively, the scaled radial velocity and the normalized Doppler frequency of the target. The presence of Vo due to the radial motion between the radar and target results in range migration. For notational convenience, let ui =u0(l + rl), Z = 0, ,Â£-!, (4.2) where r = vo/luq is a constant independent of the target motion (see (7.9) in Appendix A) and is usually very small (<^0.01). Then (4.1) can be expressed as x/(n) = a K ^Oike jZnfkl ,k=1 e^n + ei(n), Z = 0, ,L 1, n 0, , N 1, (4.3) where we sometimes drop the dependence of a on 9 for notational brevity. Let K = Z = 0, , L 1, (4.4) k=1 be the target phase history sequence. Note that L > K is required to allow the estimation of {afc, fk}k=i fro {MiS/- Then inserting (4.4) into (4.3) gives, x/(n) = fyae-^'71 + e;(n), Z = 0, ,Â£ 1, n = 0, , N 1. (4.5) When L = 1, the model in (4.5) reduces to the data model used in [72] for the LRR radar. For L > 1, we have a phase history sequence per LRR range bin and no loss of range resolution occurs since no information is lost. The secondary data are obtained from segments adjacent to the segment of interest in the same way as the primary data are obtained from the segment of interest (see Figure 1). The secondary data are assumed to be target free and are modeled as a VAR random process (see [72]). The VAR filter has the following structure, p H(z-1) = I + ^Hp^, (4.6) p= i 43 APPENDIX D DERIVATION OF THE ORBS FOR MULTIPLE MOVING TARGET DATA MODEL The CRB matrix corresponding to the data model in (5.2) is derived for the general case in which the interference term e((n) includes clutter, jamming, and noise. Equation (5.2) can be re-written as ( d= 1 \ where Kd (7.29) and with 4 [! .. . eJ'27r/dfc(L-l)jT , (7.30) = [1 e)d '' . ejud(N-l)jTj (7.31) [fj(0)fj(l) fJ(i-i)]1', (7.32) (dll) = [1 eiUirl fJ""";V" l-H 1 o' II '-O h (7.33) Let Q/v, Qc, and Qj be, respectively, the covariance matrices of the noise, ground clutter, and jamming, which are independent of each other. Then Q E{een} Q^ + Qc + Qj, (7.34) where Q n o-2n1mnli (7.35) Qc = (JcIl Qc, (7.36) and Qj = o]lNL (aja^), (7.37) with a2N, jamming, IL being the identity matrix of dimension L, Qc being as given in [80], and 111 HRR range bins. Consider two targets (D = 2) with DOAs 9\ = 30 and 92 = 150, and Doppler frequencies uq 0.2n and u>2 = OAtt with r = 0.001. Each of the two targets consists of two closely spaced scatterers (K\ = K2 = 2) with parameters an = 1, a.\2 1, Oi2i 1, a22 1, /n = 0.1, /i2 = 0.1 + 1/2L, /21 = 0.3, and f22 = 0.3 + 1/2L, where the subscript ij means the jth scatterer of the ith target. The VAR filter order is P = 2. We set e = Â£ = 10-3 to determine the practical convergence in the simulations. We simulate the ground clutter as a temporally and spatially correlated Gaussian random process [80]. The Clutter-to-Noise Ratio (CNR), defined as the ratio of the clutter variance to the noise variance, is set to be CNR= 40 dB in the primary data. A jamming signal, which is a zero-mean temporally white Gaussian random process, possibly also exists. The Jammer-to-Noise Ratio (JNR), which is the ratio of the jammers temporal variance to the noise variance, is chosen as JNR= 25 dB and the jamming signal impinges from 9j = 120. When array calibration errors exist, the errors for different elements are assumed to be independent and identically distrib uted complex Gaussian random variables. In our simulations, a complex Gaussian random vector with zero-mean and covariance matrix 0.041 is added to the array manifold to simulate array calibration errors, which implies that the variance of the calibration error for each element is 0.04. Another degradation factor, the system mismatch error, which is due to the antenna spacing, platform velocity, and/or pulse repetition frequency (PRF) mismatch errors, is also considered. We use the clutter ridge slope [36] 7 = 1 in the simulations for the case when the system mismatch error is absent and 7 = 1.1 for the case when the system mismatch error exists. First, consider the multiple moving target detection for the HRR radar in the absence of any degradation factor and in the presence of one or more degradation factors. The curves in Figure 6.2 show the detection probability of our detector under conditions of (1) no degradation factors except for the presence of clutter 97 CHAPTER 4 MOVING TARGET FEATURE EXTRACTION FOR AIRBORNE HRR RADAR 4.1 Introduction Airborne radars are used to detect the presence of moving targets and esti mate their parameters in the presence of noise, ground clutter, and jammers. In applications of conventional Low Range Resolution (LRR) airborne wide-area sur veillance phased array radars, the moving target parameters of interest include the target range, Radar Cross Section (RCS), Direction-Of-Arrival (DOA), and Doppler frequency due to the relative motion between the target and the radar [13]. With the development of modern radar technology, especially with the dramatic improvement of the radar range resolution capability, the attention has been shifted from simple target detection and tracking to target recognition, mapping, and imaging capabilities [82, 42], Future airborne radars will be required to provide increasingly high range resolution features of ground targets, which makes the signal processing needed by airborne high range resolution (HRR) phased array radars more important. Clutter suppression is critical for airborne radar signal processing. The ground clutter observed by an airborne radar is spread over two dimensions of both the range and spatial angle and the clutter spectrum also covers a certain Doppler region due to the platform motion even though the clutter itself may be stable [80]. Many sig nal processing methods have been proposed to suppress ground clutter for airborne LRR phased array radars. With the non-adaptive Displaced-Phase-Center-Antenna (DPCA) processing [10], the clutter is canceled by simply subtracting outputs from different apertures with the same effective phase center on a pulse-to-pulse basis. DPCA is sensitive to both antenna element mismatch and velocity controlling errors. 39 aj denoting the jammer steering vector which has the same form as a(?<) in (5.22) except that 9d in (5.22) is replaced by the jammer DOA Oj. According to the extended Slepian-Bangs formula, the ijth element of the Fisher information matrix (FIM) has the form [2, 66] r] = [ Re7 (a) ImT(a) i1 u> 6 ]T, (7.39) with a, f, u>, and 0 being the vectors consisting of the complex amplitudes, range frequencies, Doppler frequencies, and DOAs of the targets, respectively; ^de- notes the derivative of x with respect to the Ah parameter of r/; Re(x) is the real part of x and Im(x) is the imaginary part of x. Note that the FIM is block diagonal since the parameters in Q are independent of those in /x and vice versa. Hence, the CRB matrix for the target features and the motion parameters can be calculated from the second term on the right side of (7.38). Let AY'q-.(A drii) \drÂ¡i (7.38) g dk ad, (7.40) and Sdlk=JSdk- (7.41) Let g = Kd Y adkd-L USdk ,fc=l urd S (7.42) where = [0 1 L \}r. Let g d=J Kd Y a^dk ufd ,fc=i [(uL + rdL) dN] Qujrd > ad, (7.43) and Sd = Cd {usdk (7.44) 112 Numerical simulations have shown that our moving target feature extraction algo rithm performs well in the presence of jamming and is robust against array calibration errors and system mismatch. 57 (a) (c) Figure 3.3: Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB, the jammer has JNR=25 dB, and there is no system mismatches and no array calibration errors. 35 where x/(n) is the array output vector of the Ith. phase history sample due to the nth pulse; a(9d) is the array manifold and is a function of the dth target DOA 9d relative to the flight path; ei(n) is the interference including the temporally and spatially correlated Gaussian ground clutter, both temporally and spatially white Gaussian noise, and possibly a jammer that is temporally white but a point source in space. We assume that the clutter, noise, and jamming in different HRR range bins are independent and identically distributed. The complex amplitude adk and the frequency fdk are, respectively, proportional to the RCS and range of the A;th scatterer of the dth target. The vd and ud are, respectively, the scaled radial velocity and the normalized Doppler frequency of the dth target. Range migration occurs due to the radial motion between the radar and target and the high range resolution of the HRR radar. For notational convenience, let LOdi = ujd(l + rl), 1 = 0, 1, d=l, ,Â£>, (5.2) where r = vd/ud is a known constant independent of the target motion (see Appendix A) and is usually very small ( x-i(n) = '^2bdleJJJdinad +ei(n), l = 0,---,L-l, n = 0, , N 1, (5.3) d=l where we have dropped the dependence of ad on 6d for notational brevity and Kd bdi = J2adkej2nfdkl, l = 0,- ,L 1, d=l, ,Â£>. (5.4) fc=i Note that, when L = 1, the model in (5.3) reduces to the data model for the LRR case. For L > 1, we have a phase history sequence for each LRR segment and no loss of range resolution occurs because of no information loss. Similar to Section ??, once the VAR filter coefficients are determined, we use the filter to suppress the clutter in the primary data. The VAR filter output for the primary data has the form y i(n) = 67 since DPCA uses the a priori correct knowledge of Â£ = 2vT and ULA, DPCA can achieve complete clutter rejection. VAR filtering cannot achieve complete clutter rejection since it does not rely on this knowledge, but instead, it relies on a limited amount of the secondary data adaptively. In Figure 3.2, however, we note that when SNR increases from 5 dB to 20 dB, the performances of DPCA and VAR filtering are quite similar. The MSEs of the estimates of the target parameters {b, 9, u>} can be quite close to the corresponding CRBs at high SNR. We now assume that there is jamming from 9j = 120. Figure 3.3 shows that the performances of both VAR filtering and DPCA are similar to the corresponding ones in Figure 3.2. Note that VAR filtering and DPCA cannot suppress the tem porally white jamming. However, the unstructured parameter estimation method presented in [72] can deal with the jamming type of interferences that is temporally white but a point source in space via estimating the spatial covariance matrix of the interference along with the target parameters. Next, consider the case of system mismatches. Let 7 = 1.2 and the other parameters are the same as those in Figure 3.2. Note that DPCA is very sensitive to the system mismatches, as shown in Figure 3.4, because DPCA still assumes 7=1 and only when 7 = Â£/(2vT) is an integer can DPCA cancel the clutter effectively. Since VAR filtering is an adaptive method, it can handle this error without any problem. We remark that by using some interpolation methods, the mismatches can be compensated out to some degree of success, but at a price of more complicated signal processing. As discussed in Section 3.2, fluctuating clutter causes a broadening of clutter ridge and makes the clutter rejection more difficult. In Figure 3.5, we set the velocity standard derivation ov = 0.05. Note that VAR filtering is much more robust than DPCA against clutter fluctuations, although VAR filtering also suffers from some performance degradations. 31 ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my advisor, Dr. Jian Li, for her constant support, enthusiasm, and patience in guiding this research. Special thanks are due to Drs. Jose C. Principe, Fred J. Taylor, John M. Anderson, and David C. Wilson for serving on my supervisory committee and for their contribution to my graduate education at the University of Florida. I also wish to thank all the fellow graduate students in the spectral analysis laboratory with whom I had the great pleasure of interacting. Drs. Guoqing Liu, Renbiao Wu, Jianhua Liu, and Xi Li have my gratitude for sharing many interesting discussions with me. I would like to gratefully acknowledge all the people who helped me during my Ph.D. program. iii 5.3.1 Space-Time Parameter Estimation The NLS estimates of the space-time parameters {d, 9d, {bdi}i=o\d=i or {u)d, a<, {bdi}^~o}d=i can be obtained by minimizing the following NLS criterion: d 2 (5.10) L-1 C.-E /=0 Yi{n) ^2adie ,]udin d= 1 The minimization of the cost function C\ in (5.10) is a highly nonlinear complicated optimization problem. Here we present a cyclic optimization approach, which is a conceptually and computationally simple method for multiple moving target feature extraction [84]. The relaxation-based algorithm is used to minimize C\ by letting only the parameters of one target vary at a time while fixing the parameters of all other targets at their most recently determined values. Therefore, the feature extraction of multiple moving targets is reduced to the feature extraction of a single moving target in a relaxation-based iteration step. We first consider the space-time parameter estimation of the dth target and then give detailed steps of our approach for multiple targets. Space-Time Parameter Estimation of the dth Target Let D Ydi(n) = yi(n) biiHa^e3,Jliin, Z = 0, , L 1, n = P, ,N 1, (5.11) i=l,i^d where , {bu}^}?^ is assumed available. Note that if the array manifold is known, is replaced by a(?), with {0i}^=1 assumed available. Hence ydi{n) can be written as Ydi{n) = 3Ldlejdin + edi(n), Z = 0, ,L 1, n = P, , N 1, (5.12) where edi(n) denotes the interference due to clutter, noise, and contributions from other targets. We assume that (e,/(n)} is a zero-mean temporally white Gaussian random process with an unknown arbitrary spatial covariance matrix Qd. Then the 69 -i r Figure 3.1: Clutter ridge for (a) 7 = 1, (b) 7 = 1.2, and (c) av 0.2 and 7 = 1. 33 (a) Figure 5.4: (a) Target photo taken at 45 azimuth angle. 85 Note that fa corresponds to the dominant peak location of the squared magnitude of the Fourier spectrum f^bfc, and that k is just the complex height (scaled by 1 /L) of the same Fourier spectrum at its peak location. With the above preparations, the RELAX algorithm used to estimate the tar get range features {cqfc, fk}k=i can be outlined as follows: Step (1): Assume K = 1. Obtain {k,fk}k=i from b by using (4.48) and (4.49). Step (2): Assume K 2. Compute b2 with (4.47) using {k, fa}k=i obtained in Step (1). Obtain {k, fk}k=2 from b2. Next, compute bi with (4.47) using {k, fk}k=2 and then redetermine {k, fk}k=i from bj. Iterate the previous two substeps until practical convergence is achieved (see below). Step (3): Assume K = 3. Compute b3 with (4.47) using {k, fk}l=\ obtained in Step (2). Obtain {k, fk}k=3 from b3. Next, compute hq with (4.47) using {k, fk)l=2 and then redetermine {k,fk]k=1 from bi. Then, compute b2 with (4.47) using {oi-k, fk}k=1,3 and redetermine {k,fk}k=2 from b2. Iterate the above three substeps until practical convergence is reached. Remaining Steps: Continue similarly until K is equal to the desired or estimat ed number of sinusoids, which can be determined by using the Generalized Akaike Information Criterion (GAIC) [38, 61, 64]. The practical convergence in the iterations of the above algorithm can be determined by checking the relative change Â£2 of the cost function Cs in (4.43) between two consecutive iterations. We remark that for very closely spaced target scatterers, RELAX can converge slowly. To avoid this problem, we may use another method to provide initial estimates to RELAX [83]. 4.4 Numerical Examples We present several numerical examples to illustrate the performance of the proposed clutter suppression and moving target feature extraction algorithms. In 52 method is an effective approach for carrying out clutter suppression and parameter estimation, and being more robust against system mismatches than conventional displaced-phase-center-antenna (DPCA) processing. Compared to a conventional airborne LRR radar, an airborne High Range Resolution (HRR) radar can not only enhance the radars capability of detecting, locating and tracking moving targets, but can also provide valuable features for ap plications including automatic target recognition (ATR). We study moving target fea ture extraction algorithms in the presence of ground clutter for airborne HRR phased array radar. The VAR filtering technique is extended for airborne HRR phased array radar for clutter suppression. We also devise effective and robust feature extraction algorithms for the radar system. Multiple moving target scenarios occur frequently in radar applications. Yet, to the best of our knowledge, little research on the topic has been reported in the literature. We present a relaxation-based algorithm for multiple moving target feature extraction, which reduces the multiple moving target feature extraction problem to a sequence of single moving target feature extraction problems. Target detection is critical for every radar system since without target de tection, target feature extraction and ATR are impossible. Our final discussion is focused on multiple moving target detection for airborne HRR phased array radar. We combine the multiple moving target feature extraction methods with a single moving target detection algorithm for multiple target detection. Finally, the VAR filtering technique is demonstrated to be effective for clutter suppression via numerical examples. The proposed moving target detection and feature extraction algorithms are also shown to be both robust and accurate. x and = [0 L_i]. (4.39) The C^ in (4.37) is minimized if b7 = UiCTxvf, (4.40) where Ui and Vi are, respectively, the left and right singular vectors associated with the largest singular value or the initial estimate of a, i.e., (0) = W^ui, (4.42) can be used to initialize the alternating optimization approach. We use the former in our numerical examples. The steps of Method 2 are as follows: Step 0: Obtain the initial estimate b^0^ of b with (4.41). Step 1: Update with (4.29) by replacing a in (4.29) with the most recently determined . Step 2: Update with (4.31) by replacing {bi}Â¡Z0l in (4.31) with the most recently determined {bi}^. Step 3: Iterate Steps 1 and 2 until practical convergence occurs which is determined by checking the relative change of the cost function C4 in (4.27) between two consecutive iterations. We remark that if the range migration is negligible, i.e., r = 0 in (4.2), the 4ii s in (4.15) do not depend on l. Then Step 0 alone gives the solution that minimizes the C4 in (4.27). 50 D = ^ bdiiidia.de]Udin + (n), d=l / = 0, ,L 1, n = P,- ,N 1, (5.5) where i-L(z~l) has the same form as in (??) except that the {Hp}p=1 in (??) are replaced by {Hp}p=1, p Hdi = I + ^2 Hpe-^!P, Z = 0, , L 1, d=l,---,D, (5.6) p=i and e;(n) = l(z~l)ei(n), Z = 0, ,L 1, n = P, , AT 1. (5.7) Let a-di = bdi'HdiB.d, Z = 0, ,L 1, (5.8) be referred to as the spatial signature vector of the eZth target for the Zth phase history sample. Then (5.5) can be rewritten as D yi(n) = ^adie" + e;(n), Z = 0,---,L-1, n = P, , N 1. (5.9) d=l Our problem of interest is to estimate {u>d, 9d, {a:^, /dfc}f=i }Â£=i if the array manifold {a(0d)}^=1 is known or {ud, ad, {adk, /dfc}f=i}d=i if the array manifold {a^)}^ is unknown from the VAR filter output y/(n), 1 = 0, 1, n = P, , N 1, by minimizing a Nonlinear Least Squares (NLS) fitting criterion using a relaxation-based optimization algorithm. 5.3 Feature Extraction of Multiple Moving Targets Our feature extraction algorithm consists of the following two separate steps: Step I: Estimate the target space-time parameters {cod, 6d, {bdi}Â¡Po}d=i if the ar ray manifold {a(^d)}f=1 is known or {u)d, ad, {bdi}Â¡P0l}^=l if the array manifold is unknown from the VAR filter output {y/(n)}, Z = 0, , L 1, n = P, , N 1. Step II: Estimate the target range parameters {adk, /d/fc}f=i, d = 1, , D, from the estimate of obtained in Step I. 68 to describe the target vector. Conceptually, resolution is applicable to phased ar ray radar in the domains of range or angle or Doppler frequency. In applications of conventional Low Range Resolution (LRR) airborne wide-area surveillance phased array radars, the moving target parameters (state variables) of interest include the target range (distance between the radar and target), Radar Cross Section (RCS), Direction-Of-Arrival (DOA), and Doppler frequency due to the relative motion be tween the target and the radar [13]. With the development of modern radar technolo gy, especially with the dramatic improvement of the radar range resolution capability, the attention has been shifted from simple target detection and tracking to target recognition, mapping, and imaging capabilities [82, 42]. Future airborne radars will be required to provide increasingly high range resolution features of ground targets, which makes the signal processing needed by airborne high range resolution (HRR) phased array radars more important. Compared to a conventional airborne LRR radar [58], an airborne HRR radar can not only enhance the radars capability of de tecting, locating and tracking moving targets, but can also provide more features for applications including Automatic Target Recognition (ATR) [23, 25]. Two important technical issues associated with the signal processing of an airborne HRR radar are clutter suppression and feature extraction. It appears that few techniques have been reported for HRR clutter suppression [24, 17]. Moreover, the range migration result ed from the radial motion between the radar and the moving target will accumulate from pulse to pulse and destroy the range alignment. The range migration makes it impossible to directly use the LRR target parameter estimation approaches proposed in [72]. Therefore, the major signal processing techniques for airborne phased array radar are clutter suppression and moving target detection and parameter estimation, which are the motivation of our work. 4 the following examples, we assume that the array is a uniform linear array (ULA) with M = 8. The interelement distance is d = A/2. The number of pulses in a CPI is N 16. The phase history sample number is L = 16, i.e. an LRR range segment contains 16 HRR range bins. We assume that the target consists of two closely spaced scatterers (K = 2) with parameters au = 1, aÂ¡2 = T /i = 0.1, and = /i + The motion parameters of the target are coq = 0.27T and vq = ru>o = O.OOltuo- The target DOA is 9 = 60. We use Â£i = Â£2 = 10~3 to test the practical convergence for both Method 2 and the range feature extraction algorithm using RELAX. The mean-squared errors (MSEs) of the various estimates are obtained from 200 Monte Carlo trials. We simulate the ground clutter as a temporally and spatially correlated Gaussian random process (see Appendix B for the generation of the clutter). We use the clut ter ridge slope 7 = d/(2VT) = 1 in the simulations for the case when the system mismatch is absent (see Appendix B for details), and 7 = 1.1 for a case when the system mismatch exists. The Clutter-to-Noise Ratio (CNR) is defined as the ratio of the clutter variance to the noise variance. Herein we set CNR = 40 dB in all ex amples although our algorithms can also deal with much stronger CNRs. The noise is a zero-mean spatially and temporally white Gaussian random process. The VAR filter order is chosen as P = 2, and only 1 3 secondary LRR range bins are used to estimate the VAR filter coefficients. When array calibration errors exist, the er rors for different elements are assumed to be independent and identically distributed complex Gaussian variables. More specifically, a complex Gaussian random vector with zero-mean and covariance matrix 0.041 is added to the array manifold a(0) to simulate array calibration errors, which implies that the variance of the calibration error for each element is 0.04. Further, in the simulations when a point source jam mer is introduced, the jamming signal is assumed to be a zero-mean temporally white 53 (a) (b) (c) Figure 3.5: Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is fluctuating with av = 0.05 and CNR = 40 dB and there is no system mismatches, no array calibration errors, and no jamming. 37 LIST OF FIGURES 1.1 Echoes from an aircraft and a moving vehicle on the ground 1.2 Conceptual block diagram of line array antenna 1.3 Mesh plot of (a) the power spectral density of the interference (includ ing clutter, jamming, and noise) and (b) Fully adaptive response when the clutter ridge is along the diagonal (7 = 1) and the jamming is from dj = 120 3.1 Clutter ridge for (a) 7 = 1, (b) 7 = 1.2, and (c) av = 0.2 and 7=1. . 3.2 Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB and there is no system mismatches no array calibration errors, and no jamming 3.3 Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB, the jammer has JNR=25 dB, and there is no system mismatches and no array calibration errors 3.4 Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency in the presence of system mismatches (7 = 1.2) and when the clutter is non-fluctuating with CNR = 40 dB and there is no array calibration errors and no jamming 3.5 Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is fluctuating with av = 0.05 and CNR = 40 dB and there is no system mismatches, no array calibration errors, and no jamming 3.6 Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB in the presence of antenna mutual coupling with p = 0.1 and there is no system mismatches and no jamming 4.1 HRR range profiles are divided into LRR segments with each segment containing L HRR range bins 4.2 Flow chart of the proposed clutter suppression and target parameter estimation algorithm for HRR phased array radars 8 8 9 33 34 35 36 37 38 58 59 vi CHAPTER 3 MOVING TARGET PARAMETER ESTIMATION FOR AIRBORNE LRR RADAR 3.1 Introduction The ground clutter observed by an airborne radar is Doppler spread due to the platform motion. Because of the importance of clutter suppression for airborne radar, many clutter suppression methods are presented in the literature in recent years, which include DPCA and adaptive filtering based STAP. Their advantages and disadvantages are introduced in Chapter 2. In [44, 72], a new idea was proposed to model the ground clutter observed by an airborne radar as a VAR random process, whose coefficients are estimated adaptively from the target-free secondary data. The VAR filter is then used to suppress the clutter in the primary data temporally where a target may be present. Different from adaptive filtering based STAP, in which usually clutter suppression is performed first and then the target DOA and Doppler frequency are determined by the pointing angle of the beam (or monopulse processing), the VAR-filtered primary data can be used with a robust unstructured ML parameter estimation method proposed in [72] to achieve asymptotically statistically efficient results. In this chapter, we first simulate the high fidelity ground clutter [80]. Then we quantitatively demonstrate the effectiveness of the VAR filtering technique proposed in [72] for clutter and jamming suppression, which is missing in [72]. We also compare the performances of VAR filtering and DPCA when used with the robust unstructured ML method in [72] for moving target parameter estimation in the presence of various system mismatches. 18 Figure 4.2: Flow chart of the proposed clutter suppression and target parameter estimation algorithm for HRR phased array radars. 59 scatterers of the same target have the same DOA. By combining the multiple moving target feature extraction algorithms presented in the previous chapter and the single moving target detection strategy introduced herein, we present a multiple moving target detection approach. We consider the detection of one target at a time based on a relaxation-based algorithm and hence the problem of multiple target detection is reduced to a sequence of single target detection problems. In the numerical examples, we consider the effects of the degradation factors including jamming, calibration error, and system mismatch on our detector. We also consider the impact of contaminated secondary data on the detector performance. The remainder of this chapter is organized as follows. In Section 6.2, we establish the multiple moving target data model for airborne HRR phased array radar, which is followed by a brief discussion of the VAR filtering technique. In Section 6.3, we introduce the GLRT detector for single moving target detection for airborne HRR phased array radar [41]. We present the multiple moving target detection algorithm in Section 6.4. Simulation results and their analyses are presented in Section 6.5. Finally, we give the conclusions in Section 6.6. 6.2 Data Model and VAR Filtering Consider the same data model as in (5.2): d=i \fc=i D = ^6d/eJa;dinad + e/(n), l = 0, , L 1, n = 0, , iV 1 (6.1) where x(n) is the array output vector of the Zth sample due to the nth pulse; bdi is the target phase history sequence due to the dth target; ad (dropping the dependence of ad on 0d for notational brevity) is the array manifold and is a function of the dth target DOA, 9d, relative to the flight path; e;(n) is the interference including 88 The 32 x 32 SAR data matrix we used to generate the HRR data has a resolution of 0.51 m in range and 0.54 m in cross-range. The 32 x 32 SAR data can be treated as the samples from one array element of a phased-array radar. Since the PRF of the SAR system is quite different from that of a GMTI/HRR radar system, we set the PRF ratio of the GMTI/HRR radar system and the SAR system to be 5. We assume that the range samples L = 16, temporal samples N = 16, and the number of antenna elements M = 8. Since the PRF ratio between the two radar systems is 5, four times zero-padding is used to interpolate the cross-range samples of the 32 x 32 SAR data matrix and an N x L data matrix is saved as the data from one array element of the phased-array radar during the CPI. We then can generate the samples for the other array elements similarly and build up the entire data observed by the phased-array radar. Finally strong clutter and jamming are added to the data. Figure 5.5(a) shows the average of the modulo of the N = 16 range profiles in the presence of the strong clutter, jamming, and noise. Figure 5.5(b) shows the average of the modulo of the N = 16 original normalized range profiles in the absence of the clutter, jamming, and noise, which is compared with the modulus of the estimated range profile generated from the estimates of the amplitudes and range frequencies obtained via Method 1 without the scaling scheme. The effectiveness of our algorithm is again evident. 5.5 Summary We have presented a robust and accurate method for the clutter suppression and feature extraction of multiple moving targets for airborne HRR phased array radar. To avoid the range migration problems that occur in HRR radar data, we di vide the HRR range profiles into LRR segments. We have shown how to use the VAR filtering technique to suppress the ground clutter and use the cyclic or relaxation- based algorithm to extract the features of multiple moving targets. The multiple moving target feature extraction problem is reduced to the feature extractions of 80 (3.45) 1 iV_1 Qu = T7 <*e]wn)(y(n) aeJujn)H. Hence, VN(a,u) N = log = log = log n=0 N-1 i (y(n) ae*")(y(n) ' 71=0 R y(u)aH ayH(uj) + aa11 (a y(Â£j))(a y(w))ff + R y(w)ya(w) From (3.46), clearly, the estimate of a is, Si = y(u). The remaining term in (3.46) is a function of ui only, and can be written as log Hence, and Cju = argmaxyff(u;)R ly(u), Qu = R-y(w)y (w) -H = R aa . (3.46) (3.47) R-yMy^M = log( R I R 1y(cj)yH(w) ) = log(l-yii(u;)R1y(a;))+ log|R|. (3.48) (3.49) (3.50) where subscript u corresponds to the estimates of the unstructured data model. Be cause, according to [72], uu in (3.49) is asymptotically equivalent to the ljs in (3.39), we can replace ljs by Cju in (3.38) and (3.39). The resulting estimates for b and 9, aH{9U, ORua bu ~ ^ , aH {9Ui 5u)Rua(0u, cuu) 9U = argmax aH{9, >U)R 1a e aH{0, t5u)R_1a(0, cDu) (3.51) (3.52) 28 BIOGRAPHICAL SKETCH Nanzhi Jiang received the B.Sc. and M.Sc. degrees in electrical engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1995 and 1998, respectively. Since 1998, she has been working toward her Ph.D. degree in electrical engineering at the University of Florida. Ms. Jiang has the following list of publications: [1] N. Jiang, E. G. Larsson, J. Liu, and J. Li, Differential Space-Time Modulation for Frequency Hopping Systems, submitted to IEEE Signal Processing Letters, January 2001. [2] N. Jiang and J. Li, Multiple Moving Target Feature Extraction for Airborne HRR Radar, submitted to IEEE Transaction on Aerospace and Electronic Systems, February 2000. [3] N. Jiang, R. Wu, G. Liu, and J. Li, Clutter Suppression and Moving Target Parameter Estimation for Airborne Phased Array Radar, Electronic Letters, Vol. 36, No. 5, pp. 456-457, 2nd March 2000. [4] J. Li, G. Liu, N. Jiang, and P Stoica, Moving Target Feature Extraction for Air borne High Range Resolution Phased Array Radar, to appear in IEEE Transactions on Signal Processing. [5] N. Jiang, R. Wu, J. Li, and R. Williams, Super Resolution Range Feature Ex traction of Moving Targets in the Presence of Range Migration, to appear in IEEE Transactions on Aerospace and Electronic Systems. 121 Since v/c and Ark tend to be very small in practice, when the dwell time, (N 1 )T, is short, we can approximately express (7.3) as: Xk(t,n) Oik exp Attv i 4pr0v + + 2pArfe i exp 47T 4/9To A c . 4 pvT ~ exp l j tn k = l,---,K, n = 0,1, , N 1, vTn (7.4) where A denotes the radar wavelength, and a*, is scaled by a constant phase. Note that, in (7.4) above, the first exponential term, a linear function of i, corresponds to the phase change of the signal due to the kth scatterer within a chirp pulse, which is caused by the relative range and target velocity; the second one accounts for the phase shift (Doppler shift) between pulses, which is due to the radial velocity of the A:th scatterer; finally, the last term represents the accumulated phase shift from profile to profile, i.e., range migration. We assume that 2 (i?max Rmin) /c Let Ts denote the sampling period, then we can express the sampled dechirped signal as K xi(n) = ej2nfkl ejvonlejon, n = 0, , A 1, 1 = 0, \k=0 / where L denotes the number of data samples due to each pulse, ft = _(2 + 2pw + e*n\T" A 7TC 7T / ujq = 2nvT and v0 = -- 7TC 7T / 2 pro \ 7TC aJ 4 pvTTs ,L- 1, (7.5) (7.6) (7.7) (7.8) Since both u0 and lu0 depend on the relative speed between the radar and the target, if is known, so is v0, and vice versa. Defining vq = ru>0, we have: pTs r = = wo n fo pro (7.9) 106 1.2 estimated protilo original profile (a) Figure 5.1: (a) The real HRR range profile without interference (clutter and jammer) and the estimated HRR range profile after clutter suppression and feature extraction, and (b) the cluttered HRR range profile with CNR= 40 dB, SNR= 0 dB. 82 (a) (b) (c) Figure 3.2: Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB and there is no system mismatches no array calibration errors, and no jamming. 34 We remark that our multiple moving target feature extraction algorithm above may have used more unknowns than necessary at certain steps. We choose to do so to simplify and speed up the algorithm. For example, to use the cyclic optimiza tion algorithm, we could estimate both the space-time and the range parameters of the dth. target and subtract out the dth target based on the parameter estimates {u;d, 8d (or d),{dfc, fdk}k=i)d=i fr the iteration steps. However, since estimating the range parameters {dk, fdk)k 1 can be computationally demanding, we choose to separate the range parameter estimation from the space-time parameter estimation. Our numerical results have shown little accuracy degradation but reduced compu tations, especially for large Kd, as a result of the separate space-time and range parameter estimation. 5.4 Numerical Examples We present several numerical examples to illustrate the performance of our proposed algorithm. In the following examples, we assume that the array is a ULA with M 8; the interelement distance between two antennas is Â£ = A/2; the number of pulses in a CPI is N = 16; the phase history sample number is L = 16, i.e., an LRR range segment contains 16 HRR range bins. Consider two targets (D = 2) with DOAs 91 = 30 and 92 = 150, and Doppler frequencies u)i = 0.2ir and u;2 = 0.47T with r = 0.01. Each of the two targets consists of two closely spaced scatterers (Afi = K2 2) with parameters an = 1, ai2 1, a2i = 1, c*22 = 1, /n = 0.1, fi2 0.1 + 1/2L, /2i = 0.3, and /22 = 0.3+1/2L, where the subscript ij means the jth scatterer of the zth target. The VAR filter order is P = 2. (No obvious performance improvement is obtained by using higher orders.) The number of secondary range bins is S = 4. We set e = Â£ = 10-3 to determine the practical convergence in the simulations. The mean-squared errors (MSEs) of the various estimates are obtained from 100 Monte Carlo trials. 76 a single moving target in a relaxation-based iteration step. For each target and in each iteration, the target phase history sequence and DOA (or the unknown array manifold) are estimated from the spatial signature vectors by minimizing a Weighted Least Squares (WLS) cost function. Numerical results have demonstrated that our multiple moving target feature extraction algorithm performs well in the presence of strong interference including clutter, noise, and jammer and is robust against array calibration and system mismatch errors. 81 3.2 Clutter Simulation The ground clutter used in our simulations is generated according to the clut ter model in [80]. A continuous field of clutter is modeled in [80] as a superposition of a large number of uncorrelated point scatterers which are evenly distributed in azimuth around the radar in all possible ambiguous range bins. Assume that the number of range bins is Nr, each of which consists of Nc patches in azimuth. The center location of the Ith clutter patch in the zth range bin is specified by the slant range Ri, the elevation angle Assume that the airborne radar platform is moving with a velocity v. The spatial frequency and the normalized Doppler frequency of the Ith clutter patch in the zth range bin are, respectively, d = jcosfacos 9i, (3.1) and 2 vT Wii = cosfacosdi, (3.2) where Â£ is the spacing between two adjacent sensors. The clutter return due to the mth spatial antenna for the nth pulse is Nr Nc Xnm EE aiiexp{j2n(m'dii + n/cciii)}, n = 0, ,N 1, m 0, 1, (3.3) =i i=i where an denotes the random complex amplitude of the Ith clutter patch in the zth range bin. Let E{\a\2} = a%, (3.4) where Â£'{} represents the expectation, a2 is the noise power per element, and denotes the clutter-to-noise ratio (CNR) of the Ith clutter patch in the zth range bin, which is related to the terrain type and can be determined by the radar equation [80]. 19 (a) (b) (c) Figure 3.6: Comparison of the MSEs of the target parameter estimates after VAR filtering (*) and DPCA processing (+) with the ideal CRBs (solid line) as a function of SNR for (a) amplitude, (b) DOA, and (c) Doppler frequency when the clutter is non-fluctuating with CNR = 40 dB in the presence of antenna mutual coupling with p = 0.1 and there is no system mismatches and no jamming. 38 and G5 = C[(fa fd) Ofr] (dM a), (7.26) where Â£ = j(7r2/180) sin(#7r/180) with 6 being measured in degrees. Let G = [ Gi G2 G3 G4 G5 ] (7.27) Then the CRB matrix for the parameter vector 77 is given by CRB(tj) = [2Re(GwQ_1G)]_1. (7.28) 110 many applications where high resolution and high estimation accuracy of individual scatterers are desired. 2.3.2 ML methods in Radar Array Signal Processing Standard solutions to the problem of radar array signal processing involve the use of classical space-time filters, either data independent (e.g., as with a delay-and- sum beam former) or adaptive (e.g., using linear constraints, maximum signal- to-interference-plus-noise ratio criteria, etc.) [80]. ML approaches have not been extensively considered for this problem, mainly because they are perceived to be too computationally complex. Recent work has shown that for a single target source in Gaussian interference with unknown spatial covariance, the ML solution can be obtained via a two-dimensional (2-D) search over target DOA and Doppler [75]. In [72], Swindlehurst and Stoica demonstrate that the 2-D search required for the ML solution can be replaced with two simpler one-dimensional (1-D) searches without affecting the asymptotic accuracy of the estimates. If the array is uniform and linear, then only a single 1-D search is required. The resulting computational savings have the potential for making the ML approach more feasible for radar applications. The key idea behind the simplification of the algorithm is the use of the so-called extended invariance principle (EXIP) in estimation first introduced in [70]. This technique involves reparameterizing the ML criterion in a way that admits a simple solution and then refining that solution by means of a weighted least squares (WLS) fit. In particular, they initially use an unstructured model for the array response instead of one parameterized by the DOA of the target signal. Under this model, the ML solution reduces to a search over only the target Doppler frequency. Refined estimates of the target parameters, including DOA, are then obtained via WLS, where the optimal weighting is simply the Fisher information matrix (FIM) corresponding to the unstructured ML criterion. An additional advantage of using an unstructured 16 APPENDIX B DERIVATION OF THE ORBS FOR SINGLE MOVING TARGET DATA MODEL The CRB matrix corresponding to the data model in (4.1) is derived for the general case in which the interference term e/(n) includes the ground clutter, jamming that is temporally white but a point source in space, and spatially and temporally white Gaussian noise. Express (4.1) in a vector form, x = [(fs fd) fr] a + e fs = Fa, (7.12) with F and a as defined in (4.45) and (4.44), respectively, id = [1 eju, ejo{N-1)]T, (7.13) and fr = [f0T f? fi-l]T, (7.14) with f) = [1 ejuJrl ejorl(N-1)]T l = 0, , L 1. (7.15) Let Qat, Qc, and Qj be, respectively, the covariance matrices of the noise, ground clutter and jamming, which are independent of each other. Then Q = E{eeH} = + Qc + Qji (7-16) where Qn = cr2NlMNL, (7.17) Qc Oc^-l Qc, (7.18) 108 Finally, we take into account of array calibration errors. Since the VAR fil ter K(z~1) does not rely on the knowledge of the array manifold a(9) and is esti mated adaptively, VAR filtering is robust against array calibration errors. Without loss of generality, we only consider the mutual coupling effects among antenna el ements, which occurs often in practice. Mutual coupling among antenna elements [13] is present when the signal output of an individual antenna element has not on ly one dominant component due to the direct incident plane wave, but also several lesser components due to scattering of the incident wave at the neighboring ele ments. In Figure 3.6, we set the mutual coupling coefficient to be p = 0.1, i.e., the a(9) in (3.15) is now pre-multiplied by a Hermitian Toeplitz matrix formed from [1, p, p2, , pM-x]. Note that in the presence of mutual coupling errors, VAR filtering suffers little performance degradations as can be seen by comparing the Doppler frequency estimates in Figures 3.5(c) and 3.2(c). Note that the poor estimates of b and 9 in Figures 3.5(a) and 3.5(b) are due to the mutual coupling errors since these estimates are based on the ULA assumption of a(9) in (3.15) (see [72] for details). Due to mutual coupling, the MSEs of the estimates of {5, 9} do not decrease as fast as the increase of SNR since the errors due to the presence of mutual coupling start to dominate at high SNR. Since the estimation of the Doppler frequency uj via the unstructured method in [72] is not based on the ULA assumption, u can still be estimated accurately. 3.6 Summary In this chapter, we have compared using VAR filtering and DPCA for air borne radar clutter suppression and moving target parameter estimation. We have shown that since the VAR filtering technique is an adaptive method, it is much more robust than DPCA against system uncertainties including system mismatches, array calibration errors, and clutter fluctuations. 32 Figure 4.3: (a) Mesh plot and (b) projection of the power spectral density of the interference (including clutter, jamming, and noise) and the target when the clutter ridge is along the diagonal (7 = 1), the jamming comes from Oj = 45, and the target is located at DOA = 60 with lo0 = 0.27T. The axes and w are for the spatial frequency and the normalized Doppler frequency, respectively, (see Equations (3.1) and (3.2)). 60 clutter, noise, and possibly jamming; the complex amplitude adk and the frequency fdk are, respectively, proportional to the RCS and range of the kth scatterer of the dth target; c= w(l + rl) with r = vd/uod being a known constant independent of the target motion and vd and ujd being, respectively, the scaled radial velocity and the normalized Doppler frequency of the dth target. The secondary data are obtained from the LRR segments adjacent to the LRR segment of interest in the same way as the primary data are obtained from the segment of interest. The secondary data are assumed to be target free and are modeled as a VAR random process [72, 49]. The VAR filter has the form: p U(z~x) = I + ^rHpz-p, (6.2) p=i where P is the VAR filter order, z~l denotes the unit delay operator, and I is the identity matrix. The goal of the VAR filtering technique is to estimate the VAR filter coeffi cients, H = [Hi, , Hp], from the secondary data and then filter the primary data so that the clutter component of the filter output is temporally white. Based on the following least squares criterion, we have p L-1 s N-l ^/S^EEE =0 s=l n=P e,i(n) + Eh pesi(n p) p= i (6.3) where ||-|| denotes the Euclidean norm; S denotes the number of secondary data segments; and es/(n) denotes the secondary data. Once the VAR filter coefficients are determined (see Section 4.2 for more details), we obtain the VAR filter output for the primary data y i{n) = ii(z 1)xl(n) D = X] bdi'Hdi8ide:Udin + cj(n), d= 1 D = ^die3Od,n + et(w), d= 1 89 (a) (b) Figure 4.7: Comparison of MSEs with CRBs as a function of SNR, for (a) complex amplitude and (b) range frequency of the first scatterer. (CNR=40 dB, a jammer at 45 with JNR=30 dB, array calibration error covariance matrix 0.041, system mismatch with clutter ridge slope 7 = 1.1.) 64 which are often linear frequency modulated (LFM) chirp signals [82]. The range res olution of a LRR radar is much larger than the length of a target so that the target occupies only one LRR range bin, while the range resolution of a HRR radar is so small that each target occupies several HRR range bins. Consider an airborne HRR radar having a one-dimensional (1-D) antenna ar ray with M elements uniformly spaced along the flight path of an airborne platform. A cluster of N chirp pulses is transmitted during a coherent processing interval (CPI). After dechirping, sampling, and Fourier transforming the signals in each element of the array, we obtain the HRR range profiles. Without clutter and jamming suppres sion, these clutter and jamming dominated profiles are not useful for any applications. The HRR radar also has the range migration problems, which occur when the target scatterers migrate from one range bin to another during the CPI. To avoid the range migration problems, we divide each HRR range profile into non-overlapping LRR segments so that each LRR segment contains L HRR range bins, as shown in Figure ??. We choose L to be much larger than the maximum number of range bins over which all targets can possibly expand and migrate during the CPI. We then apply the inverse Fourier transform to each segment. For the segment of interest, where the targets may be present, the inverse Fourier transform yields the primary data. We assume that D targets are present in the primary data with the dth target con sisting of Kd scatterers. We assume that the scatterers of each target have the same Doppler frequency and the same DOA, but different complex amplitudes and range frequencies. (If the scatterers of a target have different Doppler frequencies due to, for example, turning, they are treated as belonging to multiple targets.) Then the primary data model can be written as (see Appendix A for the model derivation): (5.1) Z = 0, ,Â£ !, n = 0, , JV 1, 66 [14] A. Farina, A. Forte, F. Prodi, and F. Vinelli. Superresolution capabilities in 2D direct and inverse SAR processing. International Symposium on Noise and Clutter Rejection in Radars and Imaging Sensors, pages 151-156, Kanagawa Science Park, Japan 1994. [15] A. Farina, F. Prodi, and F. Vinelli. Application of superresolution techniques to radar imaging. Chinese Journal of Systems Engineering and Electronics, 5(1):1 14, January 1994. [16] A. Farina, A. Russo, F. Scannapieco, and S. Barbarossa. Theory of radar de tection in coherent weibull clutter. IEE Proceedings F, 134(2): 174-190, April 1987. [17] K. Gerlach. Spatially distributed target detection in non-Gaussian clutter. IEEE Transactions on Aerospace and Electronic Systems, 35:926-934, July 1999. [18] G. H. Golub and C. F. Van Loan. Matrix Computations. The John Hopkins University Press, Baltimore, MD, 3rd edition, 1996. [19] M. Guida, M. Longo, and M. Lops. Biparametric linear estimation for CFAR against weibull clutter. IEEE Transactions on Aerospace and Electronic Systems, 28(1): 138152, January 1992. [20] M. Guida, M. Longo, and M. Lops. Biparametric CFAR procedures for lognormal clutter. IEEE Transactions on Aerospace and Electronic Systems, 29(3):798-809, July 1993. [21] J. A. Hogbom. Aperture synthesis with a non-regular distribution of interfer ometer baselines. Astronomy and Astrophysics Supplements, 15:417-426, 1974. [22] Y. Hua. A pencil-MUSIC algorithm for finding two-dimensional angles and polar izations using crossed dipoles. IEEE Transactions on Antennas and Propagation, 41(3), March 1993. [23] S. Hudson and D. Psaltis. Correlation filters for aircraft identification from radar range profiles. IEEE Transactions on Aerospace and Electronic Systems, 29(3):741-748, July 1993. [24] P. K. Hughes. A high-resolution radar detection strategy. IEEE Transactions on Aerospace and Electronic Systems, 19(5):663-667, 1983. [25] S. P. Jacobs and J. A. OSullivan. High resolution radar models for joint tracking and recognition. IEEE National Conference on Radar, Syracuse, New York, pages 99-104, May 1997. [26] Nanzhi Jiang, Renbiao Wu, Jian Li, and Rob Williams. Super resolution range feature extraction of moving targets in the presence of range migration, to appear in IEEE Transactions on Aerospace and Electronic Systems. [27] S. Z. Kalson. Adaptive array CFAR detection. IEEE Transactions on Aerospace and Electronic Systems, 31(2):534-542, April 1995. [28] R. Kapoor, A. Banerjee, G. A. Tsihrintzis, and N. Nandhakumar. UWB radar detection of targets in foliage using alpha-stable clutter models. IEEE Transac tions on Aerospace and Electronic Systems, 35(3):819-834, July 1999. 116 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Ground Moving Target Detection and Feature Extraction with Airborne Phased Array Radar By Nanzhi Jiang May 2001 Chairman: Dr. Jian Li Major Department: Electrical and Computer Engineering Battlefield awareness is critical to the success of future military operations. Enabling radar signal processing techniques for battlefield awareness includes the detection and recognition of ground moving targets. Since the Gulf War, DARPA (Defence Advanced Research Projects Agency) has initiated several programs to im prove the detection and recognition capability of existing systems, including MSTAR (Moving and Stationary Target Acquisition and Recognition), MTE (Moving Target Exploitation), and SHARP (System-oriented HRR Automatic Recognition Program). Our efforts herein are partially funded by one of the DARPA programs. In this disser tation, we investigate the detection and feature extraction of ground moving targets with airborne phased array radar. The ground clutter observed by an airborne radar is spread over both the range and spatial angle. The clutter spectrum also covers a certain Doppler region due to the platform motion. Without clutter suppression, moving target detection and parameter estimation are impossible. For airborne low range resolution (LRR) phased array radar, we demonstrate that the combination of a vector auto-regressive (VAR) filtering technique and a maximum likelihood (ML) parameter estimation IX scatterers of a rigid-body target, the parameters of interest also include those related to the RCS and range of each scatterer. To avoid the range migration problems that occur in HRR radar data, we first divide the HRR range profiles into LRR segments. Since each LRR segment contains a sequence of HRR range bins, no information is lost due to the division and hence no loss of resolution occurs. We show how to use the VAR filtering technique to suppress the ground clutter. Then a parameter estimation algorithm is proposed for target feature extraction. From the VAR-filtered data, the target Doppler frequency and the spatial signature vectors are first estimated by using a ML method. The target phase history and DOA (or the array steering vector for unknown array manifold) are then estimated from the spatial signature vectors by minimizing a Weighted Least Squares (WLS) cost function. The RCS related complex amplitude and range related frequency of each target scatterer are then extracted from the estimated target phase history by using RELAX [38], a relaxation-based high resolution feature extraction algorithm. The remainder of this Chapter is organized as follows. In Section 4.2, we introduce the data model and formulate the problem of interest. The VAR filtering technique for clutter suppression is also described in that section. Section 4.3 presents the moving target feature extraction algorithm for airborne HRR phased array radars. In Section 4.4, numerical examples are presented to illustrate the performance of the proposed algorithm. Finally, Section 4.5 contains our conclusions. 4.2 Data Model and VAR Filtering The range resolution of a radar is determined by the transmitted signal band width. To achieve high range resolution, a radar must transmit wide-band pulses, which are often linear frequency modulated (LFM) chirp signals [82], The range resolution of an LRR radar is much larger than the length of a target so that the target occupies only one LRR range bin, while the range resolution of an HRR radar 41 compensate for the phase shift of the outputs of antenna sensors due to the platform motion. For example, for the simple case of an array of two antennas with spacing Â£ in the direction of the flight path, DPCA compensation requires that Â£ = 2vT be satisfied, where v is the radar platform speed and T is the radar pulse repetition interval (PRI). Hence the airborne clutter can be cancelled by subtracting out the second antenna sensor output due to the (i + l)st pulse from the first antenna sensor output due to the ith pulse. Since it is difficult to change the spacing among antenna sensors, the non-adaptive DPCA technique requires that the platform velocity v and PRI T be precisely controlled, which is difficult to achieve in practice. If Â£ = 2vT cannot be perfectly satisfied, then we have a system mismatch, to which DPCA is very sensitive. DPCA is also very sensitive to other uncertainties including array calibration errors and clutter fluctuations. 2.1.2 Adaptive Filtering Based Space-time Adaptive Processing (STAP) The adaptive filtering based STAP techniques [6, 31, 32, 12, 33, 77, 79, 80] simultaneously process the signals received from multiple elements of an antenna array and from multiple pulses. A basic illustration of space-time adaptive processing is given in Figure 1.3(b). A space-time adaptive processor may be thought of as a two- dimensional (2-D) filter that represents combined receiver beamforming and target Doppler filtering. Note the high gain at the target angle and Doppler, and the deep nulls along both the jamming and clutter lines. Applying this filter to the data will suppress the interference and enable target detection. A bank of adaptive filters is then formed to cover all potential target angles and velocities. STAP is an adaptive approach and can outperform DPCA in the presence of many system mismatches and uncertainties. However, STAP may require a sig nificant amount of computations because it needs a bank of filters and may require the inversion of matrices of large dimensions, which makes it often too complex for real-time implementation. STAP requires that the clutter is homogeneous. That is, 11 where z_1 denotes the unit delay operator, I is the identity matrix, and P is the filter order. We assume that the statistics of the ground clutter are the same for the primary and secondary data. The goal of the VAR filtering technique is to estimate the VAR filter coefficients, H = [HÂ¡, , Hp], from the secondary data and then filter the primary data so that the clutter component of the filter output is temporally white. Let / denote the number of adjacent segments we use to obtain the secondary data and let (n), i = 1, , I, 1 = 0, , L 1, n = 0, , N 1, denote the secondary data. The VAR filter coefficients are estimated based on the following least squares criterion, which is an extension of the criterion in [72] to the case of L > 1, L-1 I N-1 H1,...,HP = arg mir Â£Â£Â£ 1=0 i=1 n=P en(n) + Eh Peu{n-p) p=i (4.7) where || || denotes the Euclidean norm. The solution to (4.7) is given by H = (T'T'") 1, (4.8) where (4.9) (4.10) (4.11) (4.12) and * = [Â¥10 Vtl ^/(L-l)], = hMP) MN-1)], ^u(n) = [eJi{n- 1) e^(n P)]T , E = [Eio Eu E/(_!)], Eu = [e(P) e{N 1)], (4.13) with (-)T and (-)H denoting the transpose and the conjugate transpose, respectively. Once the VAR filter coefficients are determined, we use the filter to suppress the clutter in the primary data. The VAR filter output for the primary data has the 44 By taking the logarithm of both sides of (6.19), the GLRT formulation can be simplified to C = 2 L In Rr, Ho - In | C ({d Ifjo1, d) | > 2 In r0 = r. Hx (6.21) Given and cfid, we obtain from (6.15), In |C ({d}/Iq1, ujd) | = In Substituting (6.22) into (6.21) yields R-d + In 1 ^ 1 \ Zdl'^d)Zdl(Ud) L-l 1=0 (6.22) C = 2NL\n 1 i_1 1 1T zdi(ud)z%(UJd) z=o Ro < > Hx T. (6.23) Note that, when L 1, the HRR detector in (6.23) reduces to the unstructured detector proposed in [72] for a wide area surveillance LRR phased array radar. 6.3.3 Asymptotic Statistical Analysis Since under H0 (see [72]), E [zdiMz%(ud)] = Qj, (6.24) where E[-] denotes the expectation, then 1 L_1 -^zdi(o;d)z^(u;d) 1=0 ~ jvQi' (6.25) which is 0(l/iV). Hence for large , Â£ in (6.23) can be approximated as (see Ap pendix C): Â£ 2L\n 1 L_1 1 jJ2Zdli,^d)^dlzdl(^d) 1=0 Ho > r Since under H0, J H x Jim Rd1 = Q d1. ZV-> oo (6.26) (6.27) holds with probability 1, we have C ~ 2L\n 1 L-l 1 jJ^^i^Qd'zdiid) 1=0 Ho < r > r Hx (6.28) 93 |