KNOWLEDGE-AIDED SIGNAL PROCESSING

Transcription

1 KNOWLEDGE-AIDED SIGNAL PROCESSING By XUMIN ZHU 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 28 1

2 c 28 Xumin Zhu 2

3 To my parents 3

4 ACKNOWLEDGMENTS First, I would like to express my most sincere gratitude to my advisor, Dr. Jian Li, for her constant support, encouragement, and guidance. I am specially grateful for the opportunity she has offered me to pursue my research under her supervision. She is always willing to share her knowledge and career experience as an advisor, collaborator, and friend. The experience working with her has proven to be invaluable and memorable. I would also like to thank Dr. Henry Zmuda, Dr. Clint Slatton, and Dr. Sergei Shavanov for serving on my supervisory committee and for their valuable advice. Their wonderful teaching has widened my horizon to their research fields, and has benefited my research work a lot. My special gratitude is due to Dr. Petre Stoica at Uppsala University, Sweden, for his guidance in many interesting topics. I felt so fortunate to have the opportunity to work with him and to benefit from his insightful ideas and constructive advice. I also want to thank my groupmates in Spectral Analysis Lab: Yubo Cheng, Lin Du, Dr. Bin Guo, Jun Ling, Arsen Ivanov, William Roberts, Xiang Su, Dr. Yijun Sun, Xing Tan, Dr. Luzhou Xu, Ming Xue, Tarik Yardibi, Dr. Xiayu Zheng, for their friendship and support. Last but not least, I wish to thank my family for their constant love and support. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Knowledge-Aided Space-Time Adaptive Processing Knowledge-Aided Adaptive Beamforming Knowledge-Aided Waveform Synthesis LITERATURE REVIEW FOR KA-STAP Direct Use Indirect Use Contributions of Our Work ON USING A PRIORI KNOWLEDGE IN STAP Introduction and Preliminaries Minimum MSE Covariance Matrix Estimation Convex Combination (CC) General Linear Combination (GLC) Application to KASSPER Data Conclusions MAXIMUM LIKELIHOOD-BASED KA-STAP Introduction and Preliminaries Problem Formulation Weight Determination of Prior Knowledge Convex Combination Maximum Likelihood Numerical Results Conclusions KNOWLEDGE-AIDED ADAPTIVE BEAMFORMING Introduction Problem Formulation Knowledge-Aided Covariance Matrix Estimation

6 5.3.1 MGLC and MCC Extensions Using R for Adaptive Beamforming Numerical Examples Relatively Weak Interferences Relatively Strong Interferences Conclusions WAVEFORM SYNTHESIS FOR DIVERSITY-BASED TRANSMIT BEAM- PATTERN DESIGN Introduction Formulation of the Signal Synthesis Problem Cyclic Algorithm for Signal Synthesis Numerical Case Studies Beampattern Matching Design Minimum Sidelobe Beampattern Design Waveform Diversity-Based Ultrasound Hyperthermia Concluding Remarks MIMO RADAR WAVEFORM SYNTHESIS Introduction Formulation of the Signal Synthesis Problem Without Time Correlation Considerations With Time Correlation Considerations Cyclic Algorithm for Signal Synthesis Numerical Examples Concluding Remarks CONCLUSIONS AND FUTURE WORK Conclusions Future Work KA Waveform Optimization for MIMO STAP Asymptotic ML Method for Optimal Weight Determination REFERENCES BIOGRAPHICAL SKETCH

7 Figure LIST OF FIGURES page 1-1 Angle-Doppler interference image for a single range bin A general block diagram for a space-time processor MIMO radar versus phased array Reflected pulses from different range bins can overlap significantly GLC estimates of α and β as functions of range bin index Estimation error ratio Comparison of the ROC curves corresponding to the ideal detector, the initial detector, and the GLC detector Log-likelihood function as a function of α CC and ML estimates of α as functions of the range bin index Comparison of the ROC curves corresponding to the ideal detector, the prior detector, the CC detector, the ML detector, the SMI detector, and the equal-weight detector Averaged MGLC 1 estimates of A, B and C versus the snapshot number N Averaged MGLC 2 estimates of A, B and C versus the snapshot number N SINR versus the snapshot number N when weak interferences present and for cases i - iv Averaged MGLC 2 estimates of A (1) /σ 2 1, A (2) /σ 2 2, B and C versus the snapshot number N SINR versus the snapshot number N when weak interferences present and for cases v - vi SINR versus the snapshot number N when strong interferences present and for cases i - iv SINR versus the snapshot number N when strong interferences present and for cases v - vi Beampattern matching design with the desired main-beam width of 6 and under the uniform elemental power constraint PAR values for CA synthesized waveforms with optimal R and for colorized Hadamard code

8 6-3 Differences between the beampatterns obtained from optimal R and the CA synthesized waveforms under (a) PAR = 1, (b) PAR 1.1, and (c) PAR MSE of the difference between R and ˆR as a function of sample number L Beampattern matching design with each desired beam width of 2 and under the uniform elemental power constraint Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 2 and under the relaxed elemental power constraint Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 2 and a 4 db null at 3, under the relaxed elemental power constraint Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 2 and a null from 55 to 45, under the relaxed elemental power constraint Breast model PAR values for CA synthesized waveforms with optimal R Temperature distribution for N = 5 and L = Beampattern matching design. The probing signals are synthesized for N = 1, L = 256 and P = 1 by using CA Correlation levels versus p for the CA synthesized waveforms with N = 1, L = 256 and P = 1 (R I) Beampattern matching design. The probing signals are synthesized for N = 1, L = 256 and P = 1 by using CA Correlation levels versus p for the CA synthesized waveforms with N = 1, L = 256 and P = 1 (R I) Beampattern matching design. The probing signals are synthesized for N = 1, L = 512 and P = 1 by using CA Correlation levels versus p for the CA synthesized waveforms with N = 1, L = 512 and P = 1 (R I) Beampattern matching design. The probing signals are synthesized for N = 1, L = 512 and P = 1 by using CA Correlation levels versus p for the CA synthesized waveforms with N = 1, L = 512 and P = 1 (R I) Correlation levels versus p for the CA synthesized orthogonal waveforms with N = 1, L = 256 and P = 1 (R = I)

9 7-1 Correlation levels versus p for the CA synthesized orthogonal waveforms with N = 1, L = 512 and P = 1 (R = I)

10 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 KNOWLEDGE-AIDED SIGNAL PROCESSING Chair: Jian Li Major: Electrical and Computer Engineering By Xumin Zhu May 28 Abstract The concept of Knowledge-aided (KA) radar was first proposed about twenty years ago, where the a priori knowledge can be used to enhance radar signal processing. Recently, KA signal processing has been attracting increasing attention of researchers and practitioners alike. In this dissertation, we focus on its applications to space-time adaptive processing (STAP) for surveillance, to array processing for adaptive beamforming, and also to multiple-input-multiple-output (MIMO) radar for waveform syntehsis. We first study KA-STAP algorithms used in airborne radar for wide area surveillance. In STAP, the clutter covariance matrix is routinely estimated from secondary target-free data. Because this type of data is, more often than not, rather scarce, the so-obtained estimates of the clutter covariance matrix are typically rather poor. In KA-STAP, an a priori guess of the clutter covariance matrix (e.g., derived from the knowledge of the terrain probed by the radar) is available. The problem we focus on is to combine this a priori clutter covariance matrix and the sample covariance matrix estimated from the secondary data into a preferably more accurate estimate of the true clutter covariance matrix. We consider two shrinkage methods, called the general linear combination (GLC) and the convex combination (CC), as well as a maximum likelihood-based approach to solve this problem. 1

11 Next, we consider applying KA signal processing algorithms for adaptive beamforming. In array processing, when the available snapshot number is comparable with or even smaller than the sensor number, the sample covariance matrix ˆR is a poor estimate of the true covariance matrix R. In KA adaptive beamforming, the a priori covariance matrix R usually represents prior knowledge on dominant sources or interferences. Since the noise power level is unknown, and thus cannot be included into the a priori covariance matrix, R is often rank deficient. We consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix R, the sample covariance matrix ˆR, and an identity matrix I to get an enhanced estimate of R. Both approaches are fully automatic and they can be formulated into convex optimization problems that can be efficiently solved. Finally, we introduce KA waveform synthesis for MIMO radar in which the a priori knowledge is manifested as knowing a covariance matrix of the waveforms. We propose a cyclic algorithm (CA) to synthesize a set of waveforms that realize the given covariance matrix under practically motivated constraints, and which also have good auto- and cross-correlation properties in time, if desired. 11

12 CHAPTER 1 INTRODUCTION The concept of KA radar was first proposed by Vannicola in [1, 2] and Haykin in [3]. In KA radar, some a priori knowledge such as radar parameters and/or information about the environment is available. This a priori knowledge can then be incorporated into signal processing algorithms to improve the performance of the radar. In this dissertation, we study several KA signal processing applications: we first consider KA space-time adpative processing (KA-STAP) algorithms in airborne radar for wide area surveillance; then we discuss KA adaptive beamforming for array applications where knowledge on dominant sources or interferences is given; finally, we introduce KA waveform synthesis for waveform diversity-based systems (such as MIMO radar) where the a priori knowledge is manifested as knowing a covariance matrix of the waveforms. 1.1 Knowledge-Aided Space-Time Adaptive Processing STAP is widely used in ground moving target indication (GMTI) radar to detect moving targets in the presence of severe interference, such as clutter, jamming and noise [4, 5, 6]. Figure 1-1 shows a typical interference environment seen by the airborne radar for a single range bin. The jamming is located at one angle and spread over all Doppler frequencies. And the clutter lies on the diagonal clutter ridge. (For a side-looking airborne radar and small crab angle, it is known that the clutter Doppler frequency depends linearly on the sinusoidal value of the azimuth angle.) In GMTI radar, the velocity difference between moving targets and ground clutter is exploited (by using the Doppler effect) for target detection. A general block diagram for a space-time adaptive processor is shown in Figure 1-2. Assume that the system has S receive antennas and P pulses. A space-time processor can be viewed as an adaptive linear combiner or filter that combines all the snapshots from range bin of interest. It is well-known that the optimal weight vector, w C M 1, with M = SP being the degrees of freedom (DOFs) of STAP, used to maximize the output 12

13 Normalized Doppler Frequency.5 Target Clutter Jammer sine(azimuth) Figure 1-1. Angle-Doppler interference image for a single range bin. S Antennas P pulses P pulses T T T T Data w1 w2 wps Adaptive Linear Combiner w * y Figure 1-2. A general block diagram for a space-time processor. signal-to-clutter-and-noise ratio (SCNR) of the beamformer is given by [4]: w = R 1 a, (1 1) where R C M M is the true clutter-and-noise covariance matrix for that particular range bin of current interest, and a C M 1 denotes the steering vector of the target. In standard STAP, in order to estimate the clutter-and-noise covariance matrix R, the training (or secondary) data, {y(n)} N n=1, associated with the range bins close to the range 13

14 bin of interest (ROI) are selected, under the assumption that the training data have the same clutter and noise statistics as the ROI, i.e., they are target free and homogeneous. The sample covariance matrix ˆR can then be obtained by the well-known formula (see, e.g., [5, 6, 7, 8]): ˆR = 1 N N y(n)y (n), (1 2) n=1 where ( ) denotes the conjugate transpose. To achieve a performance within 3 db of the optimum STAP when R is used, compared to the dimension of R, at least twice the number of target free and homogeneous training samples is required [9]. However, such a large number of homogeneous samples is generally not available in practice. The heterogeneity of the data due to heterogeneous terrain can lead to a significant performance degradation of the standard STAP. Recently, KA-STAP has been studied to enhance detection performance of STAP in severe heterogeneous clutter environments. There can be various knowledge sources, including radar parameters, land coverage data, terrain information and manmade features of the area. These a priori environmental information can be used either in an indirect way (e.g., KA training and filter selection) or in a direct way (e.g., KA data prewhitening) to enhance the performance of STAP (see, e.g., [8, 1, 11, 12, 13, 14, 15, 16, 17, 18] and the references therein). It should be noted that the a priori environmental knowledge can be a double-edged sword in practical applications. Accurate a priori knowledge can significantly improve the target detection performance of the radar in heterogeneous clutter environments; however, inaccurate environmental knowledge, which is possible in practice due to environmental changes or outdated information, can significantly degrade the radar s performance by wasting adaptive array s DOF on false environmental clutter. Therefor, a fundamental issue in KA-STAP is to determine the degree of accuracy of the a priori knowledge and the optimal emphasis that should be placed on it. In KA-STAP, the a priori knowledge consists usually of an initial guess of the clutter covariance matrix, let us call it R. 14

15 This can be obtained either by previous radar probings or by a map-based study. In this dissertation, assuming that R is full rank, we consider two shrinkage methods, called the general linear combination (GLC) and the convex combination (CC), as well as a maximum likelihood-based approach to obtain an enhanced estimate of true clutter-andnoise covariance matrix R based on the a priori clutter covariance matrix R and the sample covariance matrix ˆR. Given the new estimate, it then can be used in the STAP beamformer for better target detection. 1.2 Knowledge-Aided Adaptive Beamforming We also consider KA signal processing algorithms in a general adaptive beamforming application. It is well-known that in array processing, given the true array covariance matrix R and the steering vector for the signal of interest (SOI), the standard Capon beamformer (SCB) [19] can be used to maximize the array output signal-to-interferenceplus-noise ratio (SINR) adaptively. However, when the available snapshot number is comparable with or even smaller than the sensor number, the sample covariance matrix ˆR is a poor estimate of the true covariance matrix R. To estimate R more accurately, we can make use of prior environmental knowledge, which again is manifested as knowing an a priori covariance matrix R. In practice, R usually represents prior knowledge on dominant sources or interferences. Unlike in some literature, for example, see [2, 21, 22], where the noise power level is treated as prior knowledge, we now consider a more practical case where the noise power level is assumed to be unknown, and thus cannot be included into R, hence R is often rank deficient. In the dissertation, we consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix R, the sample covariance matrix ˆR, and an identity matrix I to get an enhanced estimate of R, denoted as R. MGLC and MCC, respectively, are the modifications of the GLC and CC methods. Both MGLC and MCC can choose the combination weights fully automatically. Moreover, both the MGLC and MCC methods can be extended to deal with linear combinations of an arbitrary number of 15

16 Targets Targets MIMO Receive Array Receive Phased-Array MIMO Transmit Array Transmit Phased-Array (a) (b) Figure 1-3. (a) MIMO radar. (b) Phased array. positive semidefinite matrices. Furthermore, both approaches can be formulated as convex optimization problems that can be solved efficiently to obtain globally optimal solutions. Given R obtained from MGLC or MCC, it can be used instead of ˆR in standard Capon beamformer for KA adaptive beamforming. 1.3 Knowledge-Aided Waveform Synthesis In this dissertation, we study KA waveform synthesis for MIMO radar as well. The a priori knowledge in this case is manifested as knowing a covariance matrix of the waveforms. This can be obtained in a previous optimization stage (for example, in the stage of transmit beampattern design as we will discuss later) or simply pre-specified. Transmit beampattern design is a critically important task in many fields including defense and homeland security as well as biomedical applications. Unlike a standard phased-array, which transmits scaled versions of a single waveform, a MIMO radar system transmits multiple different waveforms that can be chosen at will (see Figure 1-3). Flexible transmit beampattern designs for MIMO radar can then be achieved by properly choosing how the transmit waveforms are correlated with one another. Recently proposed techniques for waveform diversity-based transmit beampattern design have focused on the optimization of the covariance matrix of the waveforms, as optimizing a performance metric directly with respect to the waveform matrix is a more complicated 16

17 operation. Given a covariance matrix as the a priori knowledge, the problem becomes that of determining a signal waveform matrix X whose covariance matrix is equal or close to the given covariance matrix, and which also satisfies some practically motivated constraints (such as constant-modulus or low peak-to-average-power ratio constraints). We propose a computationally efficient cyclic optimization algorithm for the synthesis of such an X. Moreover, in many radar imaging applications, the length (or sample support) of the transmitted waveforms can be rather large, hence the reflected waveforms from near and far ranges can overlap significantly, as shown in Figure 1-4. This in turn requires good correlation property of the transmit waveforms. Therefore we also consider synthesizing an X that has good auto- and cross-correlation properties. Return pulse from near range Return pulse from scene center Return pulse from far range Fast time Figure 1-4. Reflected pulses from different range bins can overlap significantly. 17

18 CHAPTER 2 LITERATURE REVIEW FOR KA-STAP In this chapter, we briefly review the KA-STAP literature. DARPA s recent KASSPER program aims at exploiting environmental knowledge to enhance the detection performance of STAP. The a priori knowledge can be used either in a direct way or in an indirect way. 2.1 Direct Use In this section, we first consider the direct use of the a priori knowledge in KA- STAP. In [23], the authors present a framework for incorporating the low-fidelity a priori knowledge directly in space-time adaptive beamformer. The linearly-constrained minimum variance (LCMV) space-time beamformer proposed in this paper includes an additional a priori knowledge-based constraint to force nulls at the clutter locations. The authors showed that the knowledge-aided constraint resulted in colored loading of the adaptive covariance matrix estimate, which can be expressed as R CL = ˆR + β c R + β d I, (2 1) where R CL is the colored loading covariance matrix and β c and β d, respectively, are the colored and diagonal loading levels. However, in this colored loading approach, the colored and diagonal loading levels are chosen manually. A reduced degree-of-freedom (DOF) colored loading approach is considered in [24]. In [13], also by the same authors, application of the colored loading approach to both full DOF and reduced DOF beamformer is considered and the performance is demonstrated by using high-fidelity a priori knowledge radar data. In [25], the authors consider a combination of the synthetic aperture radar (SAR) processing and the ground moving target indication (GMTI) processing. SAR is used to detect stationary targets with long CPIs, whereas GMTI is used to detect moving-targets with a short CPI. In [25], discrete scatterers are identified by applying a (low) threshold 18

19 to the SAR image. Then the space-time response for each discrete scatterer is calculated and the response is used to build the a priori clutter covariance matrix R. The colored loading approach proposed in [23] is then used for KA-STAP processing. Similarly, [26] describes an approach using SAR data to estimate clutter characteristics for generating the a priori clutter covariance matrix. In [11], a KA parametric covariance estimation (KAPE) method is introduced to improve the performance of STAP in heterogeneous clutter environments. The basis for KAPE is the availability of fairly accurate a priori knowledge. After the covariance matrix is constructed from the a priori knowledge, it is used directly for data prewhitenning on a range bin basis. A KA-STAP approach of using data corresponding to several independent CPIs is introduced in [1]. The multiple CPI radar data is used to form earth-based clutter reflectivity maps. The so-obtained maps are then utilized to calculate the a priori clutter covariance matrix R. The method proposed in the cited paper combines the following techniques: covariance tapering, which is used to compensate for internal clutter motion (ICM); adaptive correction of channel mismatches; colored loading as described in [23], and eigenvalue scaling (this is based on the assumption that R CL (see (2 1)) can provide better space-time response, and hence R CL may provide more accurate eigenvectors, whereas the a priori knowledge obtained form multiple CPI data can predict the spatial variations of clutter amplitudes more accurately, and hence the eigenvalues of R are preferred). Techniques for suppressing large discrete clutter returns are also proposed in this paper. In [15], the authors propose a combination of the KA covariance estimation (KACE) and the enhanced FRACTA (FRACTA.E) algorithm. KACE provides an a priori clutter covariance matrix by exploiting the known operating parameters of the radar. And FRACTA is a combination of several techniques to eliminate strong heterogeneities from the data, for covariance matrix estimation and for target detection as well [27]. It includes 19

20 fast maximum likelihood (FML) method [22], reiterative censoring (RC), adaptive power residue (APR) metric, concurrent block processing (CBP), two-weight method (TWM) and adaptive coherence estimate (ACE) metric [28]. The a priori clutter covariance matrix R obtained by KACE is combined with the sample covariance matrix ˆR as R + ˆR to get a new estimate of the covariance matrix, which belongs to the direct use of the a priori knowledge. 2.2 Indirect Use Now, we consider the indirect use of the a priori knowledge in KA-STAP. A KA spectral domain (or range Doppler domain) approach to estimate the clutter-and-noise covariance matrix is discussed in [14]. In this paper, the a priori knowledge is used indirectly for training data selection. The knowledge sources are first used to model the observed clutter scene (i.e., to form RCS map versus range and Doppler bins). This a priori knowledge of the clutter scene is then used to identify the homogeneous scattering regions. The clutter power is then estimated from the collected radar data of these homogeneous segments and the clutter-and-noise covariance matrix is generated from the so-obtained clutter power file. The effects of inaccurate a priori knowledge, such as ICM, channel calibration error, etc., on the proposed approach are also considered in this paper. In [29], a KA radar detector is discussed, which is composed of three components: the map based selector to eliminate range bins containing strong stationary scatterers from the training data, which corresponds to the indirect use of the a priori knowledge; a dataadaptive selection algorithm for removing dynamic outliers or targets; and an adaptive detector for target detection. The performance of the proposed algorithm is evaluated based on both simulated and measured data. In [3], the authors give a detailed description on the numerical implementation of the digital terrain data in airborne KA-STAP. The registration and corrections of the data are discussed. Numerical examples show that the performance of STAP can be improved by properly exploiting the a priori knowledge of the terrain probed by the radar. 2

21 In the literature, there are also several review papers on KA-STAP [8, 17, 18]. [8] gives an overview of the KASSPER program at DARPA. The benifits of KA adpative radar, the direct and indirect use of the a priori knowledge, the prewhitenning approach for determining the degree of accuracy of the a priori knowledge (we will discuss this in detail in Chapter 4), and the look-ahead radar scheduling approach for real-time KA-STAP are discussed. In [18], and also in [17], the authors first discusse individually how to deal with heterogeneous clutter training data selection and how to mitigate antenna array effects such as mutual coupling and channel mismatch, with the aid of a priori knowledge. They also introduce the use of hybrid processing for heterogeneous range bins (the hybrid algorithm is a cascade of the direct data domain (D 3 ) algorithm and the joint domain localization (JDL) training [31, 32, 33].) Finally, these separate STAP issues are combined into a preliminary, but comprehensive and practical KA-STAP approach. 2.3 Contributions of Our Work The direct use of the a priori knowledge usually consists of constructing the a priori clutter covariance matrix R from the available environmental knowledge [13, 11, 1, 15]. However, one of the important problems in KA-STAP is to determine the degree of accuracy of the a priori knowledge and the optimal emphasis that should be placed on it. That is, to determine the optimal weight on R. To our knowledge, there are no efficient algorithms in the literature for solving this problem. As we mentioned before, in the colored loading approach [23], the colored and diagonal loading levels are chosen manually. And the prewhitening approach proposed in [8] suggests to optimize the weight on R so as to maximally whiten the observed interference data. However, as we will show in Chapter 4, this approach may not work properly. In this dissertation, we propose several efficient algorithms to determine the optimal weight on the a priori knowledge. In our approaches, the a priori knowledge is used in both direct and indirect ways to enhance the 21

22 detection performance of STAP. The performance of the algorithms are evaluated using the KASSPER data. 22

23 CHAPTER 3 ON USING A PRIORI KNOWLEDGE IN STAP 3.1 Introduction and Preliminaries In standard STAP, the clutter-and-noise covariance matrix, let us call it R, is estimated from secondary data (presumed to be target free), let us say {y(n)} N n=1, by means of the well-known formula (see, e.g., [5, 6, 7, 8, 16]): ˆR = 1 N N y(n)y (n), (3 1) n=1 where ( ) denotes the conjugate transpose. However, frequently the dimension of R (denoted by M in what follows) is larger than, or at best comparable with, N. The result is that ˆR is, more often than not, a poor estimate of R (particularly so when M N, see Section 3.3 of this note for such a case). To estimate R more accurately, we can try to make use of prior knowledge on the terrain probed by the radar, acquired either from a previous scanning or from a map-based study. In knowledge-aided STAP (KA-STAP), an initial guess of R, let us say R, is obtained in this way (see, e.g., [8, 16]). The problem is then to combine ˆR and ˆR into an estimate of R, preferably much more accurate than both ˆR and R. In this note, we will consider a linear combination of ˆR and R (see, e.g., [34, 11, 13]): R = αr + β ˆR; α > and β >. (3 2) Because α and β are constrained to be positive, and as typically R > (positive definite), we have that R >, which is a desirable feature. We will also consider the following convex combination : R = αr + (1 α) ˆR; α (, 1). (3 3) The constraint in (3 3) on α is imposed, once again, to guarantee that R >. In general, there is no obvious reason why β in (3 2) should be constrained to equal 1 α, like in (3 3); however, the convex combination, (3 3), is a more parsimonious description of R 23

24 than the general linear combination in (3 2), which is probably the reason why (3 3) is commonly used in the literature (see, e.g., [8] and the references there). The first goal of this note is to obtain the α and β that minimize the mean-squared error (MSE) of R: MSE = E{ R R 2 }, (3 4) for both (3 2) and (3 3); hereafter, denotes the Frobenius matrix norm or the Euclidean vector norm, depending on the context. We stress that this is a constrained estimation problem: the two classes of covariance matrix estimates in (3 3) and (3 2) have only 1 and, respectively, 2 free parameters. This means that the estimate R takes values in a restricted set that in general does not contain R; therefore, the trivial but problematic minimizer R = R of (3 4) is in general avoided. Let α and β denote the optimal values of α and β that minimize (3 4). The second goal of this note is to discuss how to obtain estimates, ˆα and ˆβ, of α and β from the available data (as we will see shortly, and as expected, both α and β depend on the unknown matrix R). Finally, we will explain how to use the proposed estimates of R, viz. R in (3 2) or (3 3) with α = ˆα and β = ˆβ, in a KA-STAP exercise based on the KASSPER data set [8, 35] (KASSPER stands for knowledge-aided sensor signal processing and expert reasoning). Estimation of a large-dimension covariance matrix from a limited number of samples in the manner outlined above (and detailed in the next section) has been originally proposed in [34] and its main references (with an emphasis on applications in economics) and later on considered in several other papers, for example in [36] (with a focus on applications in bioinformatics) and in [37] (with an emphasis on array processing applications). However, the cited papers considered only the case of real-valued data. Our approach in this note is an extension of that in [34] to the complex-valued data case, as well as to a general R matrix ([34] considers the case of R = I only). Also our proofs are more explicit than those in [34], despite the more general, complex-valued data case we consider 24

25 here. Finally, to conclude this brief review of the relevant literature, we refer the reader interested in KA-STAP and in the problem described in this note, to the more detailed paper [38], also by the present authors, which presents a maximum likelihood based approach to KA-STAP. 3.2 Minimum MSE Covariance Matrix Estimation We will consider the MSE minimization problem first for (3 3) and then for (3 2). We assume that the secondary data {y(n)} N n=1 are i.i.d. random vectors with mean zero and covariance matrix R. However, we do not make any distributional assumptions on {y(n)} N n=1, such as the usual Gaussian assumption that would not be warranted in STAP. Also, we assume that R is a fixed (non-random) matrix. However, if we have reasons to assume that R is random, then we can modify the following results in a straightforward manner to accommodate a random R doing so can be useful if, and likely only if, we have some knowledge on the error R R or the MSE E{ R R 2 }; because in KA- STAP such a knowledge is usually unavailable, we will focus on the case of non-random R here Convex Combination (CC) For (3 3), a simple calculation yields: E{ R R 2 } = E{ α(r ˆR) + ( ˆR R) 2 } = const + α 2 E{ R ˆR 2 } 2αRe { [ tr E{( ˆR R)( ˆR ]} R ) } = const + α 2 E{ R ˆR 2 } 2αE{ ˆR R 2 }, (3 5) where tr denotes the trace, Re stands for the real part, and where we have used the fact that ˆR is an unbiased estimate: The unconstrained minimization of (3 5) with respect to α gives: E{ ˆR} = R. (3 6) α = E{ ˆR R 2 } E{ ˆR R 2 } = E{ ˆR R 2 } E{ ˆR R 2 } + R R 2. (3 7) 25

26 Because α above belongs evidently to the interval (, 1), it is also the solution to the constrained minimization of the MSE. Observe the intuitive character of the expression for α in (3 7): when ˆR is much closer to R than to R (in a MSE sense), then α is close to, and viceversa. To estimate α from the available data, we need an estimate of E{ ˆR R 2 }. Let ˆr m and r m denote the mth columns of ˆR and R, respectively. Consequently, we have ˆr m = 1 N N y(n)ym(n), (3 8) n=1 and r m = E{y(n)ym(n)}, (3 9) where y m (n) denotes the mth element of y(n). To simplify the notation, in what follows we omit the index m of some variables, such as ˆr m and r m above (we will reinstate this index later on, when needed). Let x(n) = y(n)y m(n). (3 1) It follows from the assumptions made on {y(n)} N n=1 that {x(n)} N n=1 are i.i.d. random vectors with mean µ = r. Because: E{ ˆR R 2 } = M E{ ˆr m r m 2 }, (3 11) m=1 our generic problem is to estimate E{ ˆr r 2 1 } = E N variance of the sample mean of {x(n)} N n=1). A simple calculation gives: 1 N 2 E x(n) µ N = E 1 N 2 [x(n) µ] N n=1 = 1 N 2 N n=1 n=1 ñ=1 N 2 x(n) µ (i.e., the n=1 N E{[x(n) µ] [x(n) µ]} = 1 N E{ x(n) µ 2 }. (3 12) 26

27 The variance E{ x(n) µ 2 } in (3 12) can be estimated as: 1 N N x(n) ˆµ 2 ; n=1 ˆµ = 1 N N x(n) = ˆr. (3 13) n=1 It follows that we can estimate E{ ˆr m r m 2 } as Let 1 N 2 N y(n)ym(n) ˆr m 2. (3 14) n=1 ρ = E{ ˆR R 2 }. (3 15) Using (3 14) as an estimate of E{ ˆr m r m 2 } leads to the following estimate for ρ: ˆρ = 1 N 2 = 1 N 2 N n=1 m=1 M y(n)ym(n) ˆr m 2 N y(n)y (n) ˆR 2. (3 16) n=1 The above expression for ˆρ has a certain theoretical appeal, but its direct calculation may be somewhat slow. A more attractive expression for ˆρ, from a computational standpoint, can be obtained by observing that ˆρ = 1 N = 1 N M 1 N y(n)y N m(n) ˆr m 2 n=1 [ ] M 1 N y(n) 2 y m (n) 2 ˆr m 2, (3 17) N m=1 m=1 n=1 and therefore that ˆρ = 1 N 2 N y(n) 4 1 N ˆR 2. (3 18) n=1 Using ˆρ as an estimate of ρ and ˆR R 2 as an estimate for R R 2 yields the following estimate of α in (3 7): ˆα = ˆρ ˆρ + ˆR R 2. (3 19) 27

28 Alternatively, we can use the unbiased estimate ˆR R 2 for E{ ˆR R 2 } in (3 7), and the resulting estimate of α : ˆα = ˆρ ˆR R 2. (3 2) Unlike (3 19), the estimate of α in (3 2) is not guaranteed to be less than 1, as required; thus we recommend using min(1, ˆα ) in lieu of (3 2) General Linear Combination (GLC) For (3 2), we have that: E{ R R 2 } = E{ αr (1 β)r + β( ˆR R) 2 } = αr (1 β)r 2 + β 2 E{ ˆR R 2 } = α 2 R 2 2α(1 β)re [tr(r R)] + (1 β) 2 R 2 +β 2 E{ ˆR R 2 }, (3 21) where Re (the real part) can be omitted because tr(r R) >. The unconstrained minimization of (3 21) with respect to α, for fixed β, gives: α = (1 β ) tr(r R) R 2, (3 22) where β is the minimizer of the function that is obtained by inserting (3 22) (with β replaced by β) in (3 21), viz.: (β 1) 2 [ R 2 R 2 tr 2 (R R)] R 2 + β 2 E{ ˆR R 2 }. (3 23) The unconstrained minimization of (3 23) with respect to β yields: β = where ρ has been defined previously in (3 15) and γ ρ + γ, (3 24) γ = R 2 R 2 tr 2 (R R) R 2. (3 25) 28

29 By the Cauchy-Schwartz inequality, we have that γ >. This observation implies that: β (, 1). (3 26) Furthermore, (3 26) along with the fact that tr(r R) > means that α >. Therefore, α and β above are also the constrained minimizers of the MSE. It is interesting that β above belongs to the interval (, 1), like in the CC case. However, α above does not necessarily belong to (, 1), unlike in the CC case. Additionally, observe that as β approaches 1, α approaches (see (3 22)), again like in the CC case! However, here α + β may well be different from 1 the constraint α + β = 1, which leads to the CC case, is a bit hard to motivate in general, as already mentioned in Section 3.1. Let A simple calculation shows that: ν = tr(r R) R 2. (3 27) νr R 2 = tr2 (R R) R 2 + R 2 2 tr2 (R R) R 2 = γ. (3 28) Using this fact, we can estimate α and β as follows: and where ˆρ is given by (3 18), ˆβ = ˆγ ˆγ + ˆρ, (3 29) ˆα = ˆν(1 ˆβ ), (3 3) ˆγ = ˆνR ˆR 2, (3 31) and ˆν = tr(r ˆR) R 2. (3 32) 29

30 Exactly as in the CC case, we can also derive alternative estimates for the above quantities. To see how this can be done, note first that: γ + ρ = νr R 2 + E{ ˆR R 2 } = E{ ˆR νr 2 }, (3 33) an estimate of which is given by ˆR ˆνR 2. Consequently, we can also estimate α as: and β as ˆρ ˆα = ˆν ˆR ˆνR, (3 34) 2 ˆβ = 1 ˆα ˆν. (3 35) However, the estimate of β in (3 35) is not guaranteed to be positive, unlike (3 29). To guarantee that ˆβ, we can either replace (3 34) by min(ˆα, ˆν) or use max( ˆβ, ) in lieu of (3 35). Following a suggestion in [34], we will use the first of these two alternatives in what follows. Remark: We have studied the behavior of the above four methods for estimating R, viz. CC (see (3 19)), CC (see (3 2)), GLC (see (3 29), (3 3)), and GLC (see (3 34), (3 35), and the subsequent discussion), in a number of numerical examples based on the KASSPER data set see the next section for details on this data set. In these examples, GLC (after appropriate scaling) provided slightly more accurate estimates of R than the other methods, but the detection results (which were the ones of more interest, see the next section) obtained with the four methods were quite similar to one another. Consequently, for the sake of conciseness, in the following we present only the results obtained with GLC which we will call GLC for short. 3.3 Application to KASSPER Data The KASSPER data set has been generated, as described in [35], using a uniform linear array with half-wavelength inter-element spacing and S = 11 sensors, and a number of P = 32 pulses in each coherent processing interval. The space-time steering vector corresponding to a target with spatial frequency ω s and (normalized) Doppler frequency 3

31 ω D was a slightly perturbed version of the following M 1 nominal steering vector: a(ω s, ω D ) = ā(ω s ) ã(ω D ), (3 36) where M = P S = 352, denotes the Kronecker matrix product, [ ] T ā(ω s ) = 1 e jωs e j(s 1)ωs, (3 37) and [ ã(ω D ) = 1 e jω D e j(p 1)ω D ] T, (3 38) with ( ) T denoting the transpose. The spatial frequency ω s has a fixed value corresponding to steering towards 75 relative to the array normal, but the Doppler frequency ω D is only known to belong to the following set of 64 possible Doppler values {, 2π/64,, 126π/64}. The number of range bins considered is equal to 1,; of these 784 range bins contain no target, whereas the rest of the 216 bins contain at least one target (at certain values of ω D ). Furthermore, the clutter-and-noise covariance matrix, R, is specified for each range bin (R varies quite a bit with the range bin index, but we omit its dependence on the bin index for notational simplicity). The final goal of this exercise is to detect the range and Doppler bins comprising targets. This, of course, has to be done without using the detailed knowledge of the scenario that was mentioned above. We will only use the noisy space-time snapshots corresponding to the 1, range bins, the nominal steering vectors, and a perturbed version of R (that mimics the uncertainty in the a priori knowledge that always exists in practice). The said perturbed clutter-and-noise matrix, R, is generated in the following way: R = R tt, (3 39) where denotes the Hadamard matrix product, and t is a vector of i.i.d. Gaussian random variables with mean 1 and variance σt 2. Note that the perturbed matrices generated in this way remain positive definite (we have also tried two other forms of perturbation of 31

32 R to obtain R, and the results were similar to those corresponding to (3 39) and thus they will not be presented). Let y denote the space-time snapshot corresponding to the range bin of current interest. We will use the adaptive matched filter (AMF) detector [39] for target detection: a (ω s, ω D ) R 1 y a (ω s, ω D ) R 1 a(ω s, ω D ) H 1 H ξ, (3 4) where H is the null hypothesis (i.e., no target), H 1 is the alternative hypothesis (i.e., H is false), ξ is a threshold, and R is an estimate of the clutter-and-noise covariance matrix, R, for the range bin in question. The threshold ξ will be varied to generate ROC (receiver operating characteristic) curves in the usual way (this time, using the fact that we know which ones of the 64, range-doppler bins contain targets). We will use the GLC method, described in the previous section, to obtain R. Note that the GLC method is fully automatic and computationally efficient (in particular, the estimate R of R obtained with it has a closed-form expression), which are appealing features for practical applications. In order to make use of the GLC method, we need an (unbiased) estimate ˆR of R, for each range bin. We obtain this estimate from secondary data as follows: in a first step (which is done once forever), we use the initial detector (3 4) with R = R to derive a set, let us call it B, of range bins that are thought of being target free (some of these bins might contain targets, but since they passed the initial detection step, these targets must be weak compared with the clutter-and-noise level); then, for each given range bin, we pick up N = 5 range bins from B whose covariance matrices R s are the closest (in the Frobenius norm sense) to the covariance matrix R associated with the range bin of current interest; let {y(n)} N n=1 denote the secondary space-time snapshots corresponding to the selected range bins from B; then, we obtain ˆR using the standard formula in (3 1). (It is interesting to observe that the above procedure for computing ˆR makes also use of the a priori information embedded in R s). 32

33 Figure 3-1 shows the GLC estimates of α and β obtained for different range bins and for four values of σ 2 t (see (3 39)). As expected, ˆβ increases and ˆα decreases as σ 2 t increases. The smaller variability of ˆα and ˆβ observed in the figure for the larger values of σ 2 t can be explained by the fact that these estimates approach and, respectively, 1 as σ 2 t increases. Figure 3-2 displays the estimation error ratio min ϕ1 ϕ 1 R R 2 min ϕ2 ϕ 2 ˆR, (3 41) R 2 for the same cases as in Figure 3-1. Note that we chose to express the estimation error ratio as in (3 41) because the AMF detector in (3 4) is invariant to the scaling of the estimated covariance matrix. As can be seen from the figure, the optimally scaled R can be a significantly more accurate estimate of R than the optimally scaled ˆR when σ 2 t takes on small values (i.e., R is not too far from R) see Figures 3-2(a) and 3-2(b); however, interestingly enough, the scaled R is more accurate than the scaled ˆR even for a relatively large perturbation level see Figure 3-2(d), corresponding to σt 2 =.5, where (3 41) fluctuates around.9. Finally, Figure 3-3 shows the ROC curves (we remind the reader that target detection is our main goal here) corresponding to the ideal detector (i.e., (3 4) with R = R), the initial detector (i.e., (3 4) with R = R ), the sample matrix inversion (SMI) detector (i.e., (3 4) with R equal to the sample covariance matrix), and the GLC detector. The SMI detector (with N = 375) is first used as initial detector to eliminate range bins with strong targets from the secondary data, and then the sample covariance matrix is computed, for each range bin, from the so-obtained secondary data corresponding to the 375 range bins that are nearest to the one of interest. The GLC detector outperforms significantly the SMI detector and the initial detector for the larger values of σt 2. Observe also that, as σ 2 t decreases, the performance of the GLC detector approaches that of the ideal detector, as one would desire note that this is done in an entirely data-adaptive manner, with the GLC detector using no direct information about the accuracy of R. 33

34 Finally, note that even when the initial detector has a performance close to that of coin tossing (see Figure 3-3(d)), the GLC detector has a reasonable performance a fact which suggests that the GLC detector is robust to the selection of the set B of targetfree range bins. 3.4 Conclusions In this Chapter, which extends the approach of [34] to complex-valued data and to arbitrary initial guess matrices R, we have described a computationally simple and fully automatic method that obtains an enhanced estimate of a covariance matrix R by linearly combining in an optimal MSE manner the sample covariance matrix ˆR and R. While the said method is completely general, here the focus was on its use for KA-STAP. In an application of this method to the KASSPER data set, we showed that the method can provide (much) more accurate estimates R of R than ˆR by carefully tuning the weights of R and ˆR in R = αr + β ˆR according to the relative distances of R and ˆR to R. The use of the so-obtained enhanced estimate of R in an STAP detector was shown to outperform significantly the SMI detector, as well as the R -based detector in the practically interesting case of a relatively unreliable R. 34

35 α α β β (a) (b) α α β β (c) (d) Figure 3-1. The GLC estimates of α and β as functions of range bin index, for four different values of the perturbation level applied to R to obtain R : (a) σ 2 t =, (b) σ 2 t =.1, (c) σ 2 t =.1, (d) σ 2 t =.5. 35

36 (a) (b) (c) (d) Figure 3-2. The ratio in (3 41) as a function of range bin index, for four different values of the perturbation level applied to R to obtain R : (a) σ 2 t =, (b) σ 2 t =.1, (c) σ 2 t =.1, (d) σ 2 t =.5. 36

37 PD PD.4.2 GLC SMI SADL Ideal/Initial PFA 1 (a).4 Initial GLC.2 SMI SADL Ideal PFA 1 (b) PD PD.4 Initial GLC.2 SMI SADL Ideal PFA (c).4 Initial GLC.2 SMI SADL Ideal PFA (d) Figure 3-3. Comparison of the ROC curves corresponding to the ideal detector (with R = R), the initial detector (with R = R ), the SMI detector, and the GLC detector, for four different values of the perturbation level applied to R to obtain R : (a) σ 2 t =, (b) σ 2 t =.1, (c) σ 2 t =.1, (d) σ 2 t =.5. 37

38 CHAPTER 4 MAXIMUM LIKELIHOOD-BASED KA-STAP 4.1 Introduction and Preliminaries Space-time adpative processing (STAP) is widely used in airborne radars for ground moving target detection [4, 5, 6]. In STAP, I range bins are sampled during a coherent processing interval (CPI), and for each range bin, the responses to P pulses are collected from each of the S elements of the receive antenna array. This S P data matrix for a certain range bin is then stacked column-wise to form an SP 1 vector, which is called a space-time snapshot. It is well-known that the optimal weight vector, w C M 1, with M = SP being the degrees of freedom (DOFs) of STAP, used to maximize the signalto-clutter-and-noise ratio (SCNR) of the beamformer output of this snapshot is given by [4]: w = R 1 a(ω s, ω D ), (4 1) where R C M M is the true clutter-and-noise covariance matrix for that particular range bin of current interest, and a(ω s, ω D ) C M 1 denotes the steering vector, which is a function of the spatial frequency ω s and the (normalized) Doppler frequency ω D of the target. Since a scaled version of w above does not change the SCNR, we simplify our notation by considering the form in (4 1) only. In practice, R is not known, and therefore must be estimated from the available data. In standard STAP, the sample covariance matrix ˆR is estimated from the training or secondary data, {y(n)} N n=1, associated with the range bins close to the range bin of interest (ROI), under the assumption that the training data have the same clutter and noise statistics as the ROI, i.e., they are target free and homogeneous. Note that we will use y i below to denote the snapshot for the ith range bin, and y(n) to denote the nth secondary snapshot for a particular ROI. The sample covariance matrix can then be 38

39 obtained by the well-known formula (see, e.g., [5, 6, 7, 8]): ˆR = 1 N N y(n)y (n), (4 2) n=1 where ( ) denotes the conjugate transpose. To achieve a performance within 3 db of the optimum STAP when R is used, N 2M target free and homogeneous training samples are required [9]. However, such a large number of samples are generally not available in practice. Moreover, the target free and homogeneous assumption on the training data is usually far from valid in practical applications. The heterogeneity of the data can be caused by many real-world effects, including the presence of strong discrete scatterers and dense targets, variations in the reflectivity properties of the scanned area, as well as nonlinear array responses. This, in turn, can lead to a significant performance degradation of the standard STAP. To lessen the demanding requirement of the availability of many homogeneous samples in the standard STAP, many data editing algorithms [4, 41, 42, 43, 44, 29] and reduced-dimensional STAP techniques [5, 45, 46, 47] have been developed. The former try to remove the heterogeneity from the available data before training; whereas the latter attempt to reduce the dimensionality of the problem and hence reduce the number of secondary samples required. Recently, knowledge-aided (KA) STAP has been developed, which aims at exploiting environmental knowledge to enhance the detection performance of STAP. In KA-STAP, some a priori knowledge about the radar s operating environment, including radar parameters, land coverage data, terrain information and manmade features of the area, are available. The a priori knowledge can be used either in an indirect way (e.g., KA training and filter selection) or in a direct way (e.g., KA data pre-whitening) to improve the STAP performance (see, e.g., [8, 1, 11, 12, 13, 14, 15, 16, 17, 18, 29] and the references therein). It should be noted that the a priori environmental knowledge can be a double-edged sword in practical applications. Accurate a priori knowledge can significantly improve the 39

40 target detection performance of the radar in heterogeneous clutter environments; however, inaccurate environmental knowledge, which is possible in practice due to environmental changes or outdated information, can significantly degrade the radar s performance by wasting adaptive array s DOF on false environmental clutter. Therefore, one of the important problems in KA-STAP is to determine the degree of accuracy of the a priori knowledge and how much emphasis one should place on it. To address this problem, in this paper, we propose a maximum likelihood (ML) based KA-STAP. The ML approach is compared with the convex combination (CC) approach recently proposed in [48] using numerically simulated data. The remainder of the paper is organized as follows. Section 4.2 formulates the problem of interest. Section 4.3 discusses the methods for determination of the optimal weight on the prior knowledge for KA-STAP. Section 4.4 presents numerical results obtained by using simulated radar data and Section 4.5 provides the conclusions. 4.2 Problem Formulation In KA-STAP, the prior knowledge usually consists of some information about the clutter-and-noise covariance matrix R. Therefore, we assume that an initial guess of R, let us say R, is available. R can be obtained either by a previous scanning of the same region or by a map-based study (see, e.g., [8, 1, 11, 12, 23]). This a priori R can then be incorporated into STAP to improve its clutter suppression capability. As in [8], we consider a convex combination of ˆR and R, which corresponds to the direct use of prior knowledge, as follows: R = αr + (1 α) ˆR; α (, 1), (4 3) where α is the weighting factor on the a priori knowledge, and the constraint on α is imposed to guarantee that R >. This is also known as the colored loading approach [23]. However, in [23], the colored and diagonal loading levels were chosen manually. Similarly, the authors of [15] considered equal weighting of ˆR and ˆR as a direct use of prior knowledge: R =.5R +.5 ˆR (we refer to this as the equal-weight approach 4

41 herein). Our goal is to devise an algorithm that can adaptively adjust the weighting factor α, i.e., the weight of the prior knowledge, based on the available data. Hence the problem of interest is to see if an optimal choice of α in (4 3) can be devised, depending on the degree of accuracy of the a priori covariance matrix R. 4.3 Weight Determination of Prior Knowledge A whitening approach for solving the problem of optimal weight determination was proposed in [8], where the authors determined the weighting factor α of the a priori knowledge so as to maximally whiten the observed interference data. However, as we will show shortly, this approach may not work properly. In [8], the optimization problem is formulated as follows: where with denoting the Frobenius norm, and min α Z I (α), (4 4) 1 Z I (α) = ỹ i ỹi I M, (4 5) I i ỹ i = R 1/2 (i)y i ; R(i) = αr (i) + (1 α) ˆR(i). (4 6) In (4 6), y i denotes the space-time snapshot for the ith ROI; R 1/2 (i) is the whitening matrix for that particular range bin with ( ) 1/2 denoting the inverse square root of a matrix, and ỹ i is the whitened space-time snapshot; in (4 5), the summation is done over a set of (target-free) snapshots of length I, and I M C M M denotes the identity matrix of dimension M. The optimization problem in (4 4) is highly non-linear and solving it requires a large amount of computations. Moreover, we will show in the following that this whitening approach may not work due to the non-uniqueness of the whitening matrix R 1/2 (i), and the dependence of R(i) on the range bin index i. Note that R(i) is a positive definite matrix that can be written as: R(i) = C(i)C (i) = C(i)Q(i)Q (i)c (i), (4 7) 41

42 where C(i) C M M is a square root of R(i), and Q(i) C M M is an arbitrary unitary matrix satisfying Q(i)Q (i) = Q (i)q(i) = I M. Consequently the whitening matrix can be expressed as: R 1/2 (i) = Q (i)c 1 (i). (4 8) Since Q(i) is an arbitrary unitary matrix, R 1/2 (i) in (4 8) is not unique. The nonuniqueness of the whitening matrix will not affect the cost function in (4 5) when I = 1 or when R(i) is independent of i. For example, when I = 1 (without loss of generality, we consider the range bin index i = 1), Z 1 (α) = Q (1)C 1 (1)y 1 y 1(C (1)) 1 Q(1) Q (1)Q(1) = C 1 (1)y 1 y 1(C (1)) 1 I M, (4 9) and therefore the value of the cost function does not depend on the choice of Q(1). However, when I > 1, which is usually required for whitening, and R(i) is dependent on i, the cost function in (4 5) becomes 1 Z I (α) = Q(i) C 1 (i)y i yi (C (i)) 1 Q(i) I M. (4 1) I i Note that the value of the cost function Z I (α) depends on the choice of {Q(i)}. Consequently, different choices of the square root matrix may yield rather different values for α, and there is no guideline as to which one to choose for good performance. Next, we introduce an effective weight determination procedure based on the ML approach. We also review the related approach recently proposed in [48] Convex Combination In the CC approach [48], α in (4 3) is allowed to vary with the range bin index. Below, we omit the dependence of ˆR, R, R, R and α on the range bin index for notational simplicity. The CC approach considers a convex combination of ˆR and R (see (4 3)) to 42

43 obtain an optimal estimate of R so as to minimize the MSE of R: MSE = E{ R R 2 }. (4 11) We denote the optimal value of α obtained by the CC approach as α CC. yields: Combining (4 3) and (4 11) and using the fact that ˆR is an unbiased estimate of R E{ R R 2 } = E{ α(r ˆR) + ( ˆR R) 2 } = const + α 2 E{ R ˆR 2 } 2αRe { [ tr E{( ˆR R)( ˆR ]} R ) } = const + α 2 E{ R ˆR 2 } 2αE{ ˆR R 2 }, (4 12) where tr denotes the trace, and Re stands for the real part. The minimization of (4 12) with respect to α gives: Let α CC = E{ ˆR R 2 } E{ ˆR R 2 } = E{ ˆR R 2 } E{ ˆR R 2 } + R R 2. (4 13) ρ = E{ ˆR R 2 }. (4 14) To estimate α from the available data, we need an estimate of ρ. Note that ρ = M E{ ˆr m r m 2 }, (4 15) m=1 where and ˆr m = 1 N N y(n)ym(n), (4 16) n=1 r m = E{y(n)y m(n)}, (4 17) 43

44 denote the mth columns of ˆR and R, respectively, with y m (n) being the mth element of y(n). It has been shown in [48] that E{ ˆr m r m 2 } can be estimated as 1 N 2 N y(n)ym(n) ˆr m 2. (4 18) n=1 It follows from (4 15) and (4 18) that ˆρ = 1 N = 1 N M 1 N y(n)y N m(n) ˆr m 2 n=1 [ ] M 1 N y(n) 2 y m (n) 2 ˆr m 2, (4 19) N m=1 m=1 n=1 and therefore that ˆρ = 1 N 2 N y(n) 4 1 N ˆR 2. (4 2) n=1 Using (4 2) and the unbiased estimate ˆR R 2 for E{ ˆR R 2 } in (4 13) yields the following estimate of α CC : ˆα CC = ˆρ ˆR R 2. (4 21) To ensure that the estimate of α CC in (4 21) is less than 1, as required, we recommend using min(1, ˆα CC ) in lieu of (4 21). We refer the reader to [48] for the detailed derivation of CC. Note that the CC approach is fully automatic and has a simple closed form solution which is computationally efficient Maximum Likelihood In this subsection, we introduce an ML approach to determine an optimal value of α by assuming that α in (4 3) is constant for all range bins (for the sake of reduced computational complexity). We denote the optimal value of α obtained by the ML approach as ˆα ML. Since R, ˆR, and hence R in (4 3) as well as R vary with the range bin index i, in this subsection, we will reinstate the dependence on the range bin index of these variables for presentation s clarity. Specifically, let R(i) = αr (i) + (1 α) ˆR(i) be the estimate of 44

45 the true clutter-and-noise covariance matrix R(i) for the ith range bin of interest (ROI). As already mentioned, α is assumed to be independent of i here. Note that the a priori knowledge can be used for secondary data selection (SDS) for each ROI, which corresponds to an indirect exploitation of the a priori knowledge. Let I be the number of the range bins sampled by the radar during a CPI and let {y i } I i=1 denote the corresponding clutter data set. Then for the ith ROI, we pick up N range bins (excluding the ith range bin which is the primary data) from the range bin set whose a priori covariance matrices R s are the closest (in the Frobenius norm sense) to the covariance matrix R (i); let B i denote the range bin set containing the selected bins for the ith ROI; and let T i index the set B i ; then, the secondary training snapshots {y(n)} N n=1 for the ith ROI in (4 2) can be chosen as {y n } n Ti ; consequently, ˆR(i) obtained with SDS can be computed as: ˆR(i) = 1 y n y N n. (4 22) n T i Note that this SDS procedure is also used in the CC approach to obtain the sample covariance matrix. Next, we introduce the ML approach to determine the optimal value of α. Assume that {y i } I i=1 are independent identically distributed (i.i.d.) with the following distribution: y i CN (, R(i)); R(i) = αr (i) + (1 α) ˆR(i), (4 23) where CN (, R(i)) denotes the circularly symmetric complex Gaussian distribution with mean zero and covariance matrix R(i). The log-likelihood function of {y i } I i=1 is then proportional to: C = = I {ln R(i) + y i R } 1 (i)y i i=1 I i=1 { ln αr (i) + (1 α) ˆR(i) [ + yi αr (i) + (1 α) ˆR(i) ] } 1 yi, (4 24) 45

46 where denotes the determinant of a matrix. Our problem is that of finding the optimal ˆα ML that maximizes the log-likelihood function in (4 24). A closed-form solution to this problem is not likely to exist, but the optimal ˆα ML can be found by a search method (see, e.g., [49]). With the optimal ˆα ML determined by the ML approach, R(i) in (4 23) can be readily obtained and used for target detection. Note that the main computational part of ML for KA-STAP is due to the and ( ) 1 operations for R(i) C M M in (4 24), which are required for each i and each candidate α. We present in the Appendix a fast algorithm that can be used to reduce the computational complexity significantly for KA-STAP. In many practical applications, when the computational resources and the available secondary samples are limited, one often considers a more practical reduced-dimension (RD) KA-STAP. One common approach to reduce the dimensionality is to apply a transform matrix, T C M D, to the original snapshot of ROI [5]. Hence, after transformation, the RD space becomes a D-dimensional subspace. Since D is usually much smaller than the total number of DOFs, i.e., D M, the dimensionality can be reduced significantly after the transformation. In the numerical examples, we consider a multi-bin elementspace post-doppler STAP to reduce the dimension. This RD technique preserves the number of spatial DOFs (equals to S), while reduces the number of the temporal DOFs from P to Q (Q < P ). Consequently, the dimensionality (or the DOFs) of the RD space is given by D = QS. Focusing on the adaptive processing for a specific Doppler bin, say the pth Doppler bin, we can write the transform matrix T as: T = F p I S, (4 25) where F p is a P Q matrix whose columns are discrete Fourier transform filters at Q adjacent Doppler frequencies with the pth Doppler bin as the central frequency bin, and denotes the Kronecker matrix product. The resulting snapshot for the ith range bin, 46

47 is given by T y i, which we still refer to as y i, for notational simplicity. Then all of our previous discussions apply to the RD case. Note that the optimal ˆα ML in the ML approach for KA-STAP is constant for all range bins. We have also tried to find optimal ˆα ML (i) as a function of the range bin index i by using an ML approach based on a leave-one-out cross-validation method, but the performance was similar to that of using a constant ˆα ML, whereas the computational complexity was greatly increased. Consequently, the case in which ˆα ML (i) is a function of i will not be presented in this paper. 4.4 Numerical Results In this section, we demonstrate the performance of the ML approach, as compared to the CC approach, based on simulated clutter-only data for KA-STAP. Similar to [24], in this paper, we focus on solving the problem of small sample support number in a heterogeneous clutter environment. In our simulations, we generate the clutter data based on the KASSPER 2 dataset [35]. The simulated airborne radar system has P = 32 pulses and S = 11 spatial channels, yielding M = P S = 352 DOFs. The mainbeam of the radar is steered to an azimuth of 195 and an elevation of -5 (the azimuth and elevation are measured clockwise from the true north and the local horizon, respectively). In each CPI, a total of I = 1 range bins are sampled covering a range swath of interest from 35 km to 5 km. The KASSPER data simulates a severe heterogeneous clutter environment and the true clutter-and-noise covariance matrix, R(i), is specified for each range bin. We simulate the clutter data for the ith range bin as: y i = R 1/2 (i)v i, i = 1,, I, (4 26) where ( ) 1/2 denotes the Hermitian square root of a matrix, and {v i } C M 1 are i.i.d. circularly symmetric complex Gaussian random vectors with mean and covariance matrix I. 47

48 The a priori covariance matrix, R (i), is constructed as a perturbed version of the true R(i) (to mimic the uncertainty in the a priori knowledge that always exists in practice): R (i) = R(i) t i t i, (4 27) where denotes the Hadamard matrix product, and t i is a vector of i.i.d. Gaussian random variables with mean 1 and variance σt 2. Note that the perturbed matrices generated in this way remain positive definite. The nominal space-time steering vector corresponding to a target with spatial frequency ω s and (normalized) Doppler frequency ω D can be expressed as follows: a(ω s, ω D ) = ā(ω s ) ã(ω D ), (4 28) where [ ā(ω s ) = 1 e jωs e j(s 1)ωs ] T, (4 29) and [ ã(ω D ) = 1 e jω D e j(p 1)ω D ] T, (4 3) with ( ) T denoting the transpose. The spatial frequency ω s has a known value for a given azimuth angle, but since the target velocity is unknown, the Doppler frequency ω D is only known to belong to the following set of 32 possible Doppler values { 32π/32, 3π/32,, 3π/32}. Thus, there are totally L K = 1 32 = 32 range-doppler bins. We use the same scheme described in [24] for target detection. A series of test targets are inserted in all the range bins that span from 35 km to 5 km for a fixed Doppler bin. The target power is the highest at the closest range bin, and reduces with range, proportional to a fourth power of the reciprocal of the range. Let σ(i) 2 denote the power of the test target at the ith range bin. Define the signal-to-clutter-and-noise ratio (SCNR) 48

49 for the ith range bin as where ω D tr [σ(i)a(ω 2 s, ω D )a (ω s, ω D )], (4 31) tr [R(i)] is the Doppler frequency bin in which the test targets are inserted. Hence the average SCNR over I = 1 range bins can be calculated as 1 I I i=1 tr [σ(i)a(ω 2 s, ω D )a (ω s, ω D )]. (4 32) tr [R(i)] In our numerical illustrations, we choose the training sample number as N = 5, which is much less than the full DOFs, M = 352, of the simulated radar system. Hence ˆR(i) is severely rank deficient. We consider a multi-bin post-doppler STAP using three adjacent adaptive Doppler bins (i.e., Q = 3), which reduces the dimensionality of the system to D = 33, and the resulting sample covariance matrix now has full rank (N > D). Figure 4-1 displays the log-likelihood function in (4 24) as a function of α for four values of σ 2 t. The optimal weighting factors obtained by the ML approach for the four different perturbation levels are listed in Table 4-1. As expected, ˆα ML decreases as σ 2 t increases. Observe that ˆα ML =.8 when σ 2 t =, and ˆα ML takes on small values at high perturbation levels, as desired. Similarly to the CC approach, the ML approach is also fully automatic. Table 4-1. ML estimates of α for different values of σ 2 t. σt ˆα ML Figure 4-2 shows the CC estimate ˆα CC as a function of the range bin index, and the ML estimate ˆα ML, for the four values of σ 2 t. It is clear that ˆα CC decreases as σ 2 t increases. The average values of ˆα CC s over the 1 range bins, denoted as ˆα CC, for the different perturbation levels are given in Table 4-2. Table 4-2. Average values of the CC estimates of α for different values of σ 2 t. σ 2 t ˆα CC

50 From the results in Tables 4-1 and 4-2, we note that the ML approach tends to put more emphasis on the prior knowledge as compared to the CC approach. For both approaches, the optimal weighting factor decreases as σ 2 t increases, as it should. Hence the weighting factor of the prior knowledge determined by either the ML approach (ˆα ML ) or by the CC approach (ˆα CC ) can be used as an indicator for the degree of accuracy of the environmental knowledge. We use the adaptive matched filter (AMF) detector [39] for target detection.let R(i) be an estimate of the true clutter-and-noise covariance matrix, R(i), of the ROI. The AMF detector has the form: a (ω s, ω D ) R 1 (i)y i 2 a (ω s, ω D ) R 1 (i)a(ω s, ω D ) H 1 H ξ, (4 33) where H is the null hypothesis (i.e., no target), H 1 is the alternative hypothesis (i.e., H is false), ξ is a target detection threshold. The AMF detector for RD is similar. The AMF detector is first applied to the clutter-only data. The AMF outputs are thresholded and the number of false alarms (range-doppler bins with the AMF outputs exceeding the threshold over the entire range-doppler map) is counted. Then with the test targets inserted, the AMF detector with the same threshold is applied and the number of detections (range-doppler bins with the AMF outputs exceeding the threshold in the fixed Doppler bin) is also recorded. Given the two numbers, the probability of detection (PD) and the probability of false alarm (PFA) can be calculated. The corresponding PD vs. PFA curves, i.e., the receiver operating characteristic (ROC) curves, can then be generated by varying the threshold ξ. In our simulations, 1 test targets are inserted in the 21st Doppler bin and the average SCNR of the targets is db. In Figure 4-3, we show the corresponding ROC curves for the ML approach as well as for the CC approach. For comparison purposes, we also show the ROC curves corresponding to the equal-weight detector, and to the prior detector, where only prior knowledge is used for target detection. The ideal detector refers to using the true 5

51 clutter-and-noise covariance matrix. The sample covariance matrix inversion ( SMI ) detector is obtained by using the sample covariance matrix, where no a priori knowledge is incorporated in the detector. As we can see, both the ML and CC detectors, as well as the equal-weight detector, outperform the prior detector significantly for relatively large values of σt 2, and they give better performance than the SMI detector for the four values of σt 2. As compared to the ML and equal-weight detectors, CC results in poorer performance, which is due to some of the ˆα CC (i) being very close to zero. It is also interesting to note that, at high perturbation levels, using prior knowledge alone leads to very poor ROC performance. This is also consistent with the sharp drops of the log-likelihood function at α = 1, as shown in Figures 4-1(c) - 4-1(d). In general, the ML detector gives a good overall performance for different values of σ 2 t and outperforms the CC and equal-weight detectors. Note that the performance of the ML detector achieves that of the ideal detector when the prior knowledge is accurate. Next, we compare the computational complexities of the CC and ML approaches. The computation of the optimal weighting factor in CC is straightforward, while the ML approach is computationally more intensive as it requires a search over the parameter space for α. However, in our simulations, the dimensionality D of the RD space is small, and hence the computational time needed for the ML searching is not significant. Without any optimality claims for our MATLAB codes, we found that the computational time needed for CC is comparable to, or a little bit more than, the equal-weight approach, and also that ML requires approximately twice as much PC time as the CC approach to process the simulated data. 4.5 Conclusions In KA-STAP, the a priori environmental knowledge can be utilized to improve the target detection performance of the radar in a variety of heterogeneous environments. The a priori knowledge usually consists of information about the clutter-and-noise covariance matrix. In this paper, the a priori knowledge was used both directly and indirectly. The 51

52 main problem was to determine the degree of accuracy of the a priori knowledge and how much emphasis one should place on it. We have considered a convex combination of the a priori clutter covariance matrix and the sample covariance matrix, which corresponds to the direct use of prior knowledge. An ML approach has been proposed to determine the optimal weighting factor of the a priori knowledge. The performance of the ML approach has been compared to that of the CC detector proposed in [48], based on the simulated clutter-only data for KA-STAP. We have shown that both the ML and CC approaches are fully automatic, and that the weighting factor α determined by either approach can be used as an indicator for the degree of accuracy of the prior knowledge. Simulation results have also been presented to show that the ML detector can adaptively adjust the emphasis placed on the a priori knowledge based on the available data, and that it can provide better ROC performance than CC and equal weighting. Appendix A. Efficient ML Computation The dependence of ˆR and R on the range bin index is omitted in this Appendix for notational simplicity. Let the columns of the matrix Y C M N contain the secondary training snapshots for the ROI. The sample covariance matrix in (4 22) can be re-written as ˆR = 1 N YY. (4 34) Note that αr + (1 α) ˆR = αr [ I M + 1 α ] αn R 1 YY = αr IN + ζy R 1 Y = α M R I N + ζg, (4 35) where (assuming α > ) ζ = 1 α αn, (4 36) 52

53 and Let the eigen-decomposition of G be given by G = Y R 1 Y. (4 37) G = UΛU, (4 38) where the columns of U are the eigenvectors of G, and Λ is a diagonal matrix formed by the eigenvalues of G. Then, (4 35) becomes: αr + (1 α) ˆR = α M R I N + ζλ. (4 39) Note that for each range bin index, as α varies, we need only to update I N + ζλ, which is computationally convenient. Next, we consider computing R 1. It follows from (4 34) and using the matrix inversion lemma that: [ ˆR] [ 1 αr + (1 α) = αr + 1 α ] 1 N YY = 1 { R 1 ζr 1 Y [ I N + ζy R 1 Y ] } 1 Y R 1 α = 1 { R 1 ζr 1 Y [I N + ζg] 1 } Y R 1 α = 1 { R 1 ζr 1 YU [I N + ζλ] 1 } U Y R 1, (4 4) α Let Then (4 4) becomes P = R 1 YU. (4 41) [ αr + (1 α) ˆR ] 1 1 { = R 1 ζp [I N + ζλ] 1 P }. (4 42) α As we can see, R, R 1 and P in (4 39) and (4 42) are not related to α, and hence they only need to be calculated once for each range bin index. Furthermore, I N + ζλ is an N N diagonal matrix (usually N M), whose and ( ) 1 can be calculated easily. 53

54 Therefore, the computational complexity of the ML approach can be reduced significantly by using (4 39) and (4 42). Appendix B. LOOCV-Based ML KA-STAP Note that the weighting factor obtained by the two-step ML approach is constant for all range bins. However, in practice, the clutter-and-noise covariance matrix varies with the range bins, which results different optimal weighting factors for different range bins. In this appendix, we consider the problem of determining the weighting factor as a function of the range bin index. Based on the clutter-only data, we apply a new ML approach by using the leave-one-out-cross validation (LOOCV) method in Step II to determine the optimal weighting factor. We consider a convex combination of ˆR(i) and R (i), which corresponds to the direct use of prior knowledge, as follows: R(i) = α(i)r (i) + (1 α(i)) ˆR(i); α(i) (, 1), (4 43) where α(i) is the weighting factor on the a priori knowledge for the ith range bin, and the constraint on α(i) is imposed to guarantee that R(i) >. Note that the covariance matrix of y i can be expressed as { α i R (i) + (1 α i ) ˆR(i) }. Assume that N secondary snapshots, {y n } n Bi, for the ith range bin are i.i.d. and share the same covariance property as y i. Next, we apply LOOCV: for the nth range bin (for any n B i ), we test the likelihood of { [ ] } y n CN, α i R (i) + (1 α i ) ˆR(i), (4 44) n [ ] where the sample covariance matrix ˆR(i) is estimated from the remaining N 1 secondary range bins except for the nth one. Let n [ ] R n (i) = α i R (i) + (1 α i ) ˆR(i). (4 45) n 54

55 The log-likelihood function of {y n } n Si is hence proportional to: C(i) = { ln R } n (i) + y 1 n R n (i)y n. (4 46) n B i Consequently, the optimal α i for the ith range bin can be determined by maximizing the cost function C(i) in (4 46). Note that the same fast computation methods as in Appendix A can be applied here to improve computation efficiency. 1 x 15 1 x x 15 (a) x 15 (b) (c) (d) Figure 4-1. The log-likelihood function as a function of α, for four different values of the perturbation level of the a prior knowledge: (a) σ 2 t =, (b) σ 2 t =.1, (c) σ 2 t =.1, and (d) σ 2 t =.5. 55

56 (a) (b) (c) (d) Figure 4-2. The CC and ML estimates of α as functions of the range bin index, for four different values of the perturbation level of the a prior knowledge: (a) σ 2 t =, (b) σ 2 t =.1, (c) σ 2 t =.1, and (d) σ 2 t =.5. 56

57 PD PD.4 Ideal (Prior).2 ML CC Equal weight SMI PFA (a).4 RD: Ideal ML.2 CC Equal weight Prior SMI PFA (b) PD PD.4 RD: Ideal ML.2 CC Equal weight Prior SMI PFA (c).4 RD: Ideal ML.2 CC Equal weight Prior SMI PFA (d) Figure 4-3. Comparison of the ROC curves corresponding to the ideal detector, the prior detector, the CC detector, the ML detector, the SMI detector, and the equal-weight detector, for four different values of the perturbation level of the a prior knowledge: (a) σ 2 t =, (b) σ 2 t =.1, (c) σ 2 t =.1, and (d) σ 2 t =.5. 57

58 CHAPTER 5 KNOWLEDGE-AIDED ADAPTIVE BEAMFORMING 5.1 Introduction Given the true array covariance matrix R and the steering vector a for the signal of interest (SOI), the standard Capon beamformer (SCB) [19] can be used to maximize the array output signal-to-interference-plus-noise ratio (SINR) adaptively. Since its first introduction almost forty years ago, SCB and its robustified versions [5] have been extensively studied and widely used in many applications, such as radar, sonar, wireless communications, and biomedical imaging. Let y(n) denote the nth output snapshot of an array comprising of M sensors. In practice, the true array covariance matrix R, where R = E{y(n)y (n)} (5 1) is unknown, and so it is usually replaced by the sample covariance matrix ˆR, where ˆR = 1 N N y(n)y (n), (5 2) n=1 with N denoting the snapshot number. However, when N is comparable with or even smaller than M, ˆR usually is a poor estimate of R. To obtain an improved estimate of R when the snapshot number N is limited, we can make use of prior environmental knowledge. The concept of knowledge-aided (KA) signal processing was first proposed by Vannicola et al. in [1, 2] and by Haykin in [3]. In a KA system, the a priori knowledge is manifested as having an initial guess of the true array covariance matrix R, denoted as R [1, 11, 8]. When R has full rank, we have considered two shrinkage approaches in [51], called the general linear combination (GLC) and the convex combination (CC) methods, as well as a maximum likelihood based approach in [38] to obtain an improved estimate of R based on ˆR and R. In this paper, we consider the case of R being rank deficient in a general adaptive 58

59 beamforming application. This case occurs frequently in practice when we only have prior knowledge on dominant sources or interferences. Unlike in some literature, for example, see [2, 21, 22], where the noise power level is treated as prior knowledge, we now consider a more practical case where the noise power level is assumed to be unknown, and thus cannot be included into R to make it full rank. We consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix R, the sample covariance matrix ˆR, and the identity matrix I to get an enhanced estimate of R, denoted as R. MGLC and MCC, respectively, are the modifications of the GLC and CC methods proposed in [51]. Both MGLC and MCC can choose the combination weights fully automatically. Moreover, both the MGLC and MCC methods can be extended to deal with linear combinations of an arbitrary number of positive semidefinite matrices. Furthermore, both approaches can be formulated as convex optimization problems that can be solved efficiently to obtain globally optimal solutions. The remainder of the paper is organized as follows. Section 5.2 formulates the problem of interest. In Section 5.3, we discuss how to obtain enhanced covariance matrix estimates by using MGLC and MCC via convex optimization formulations. In Section 5.4, we use the enhanced covariance matrices instead of the sample covariance matrix in SCB to improve the array output SINR. In Section 5.5 numerical examples are presented to demonstrate the effectiveness of the proposed algorithms. Finally, conclusions are given in Section 5.6. Notation. Vectors are denoted by boldface lowercase letters and matrices by boldface uppercase letters. The nth component of a vector x is written as x n. The inverse of a matrix R is denoted as R 1. We use ( ) T to denote the transpose, and ( ) the conjugate transpose. The Frobenius norm is denoted as. The real part of an argument is denoted as Re( ). The expectation operator is denoted as E( ) and the trace operator is denoted as tr( ). The notation R means that R is positive semidefinite. 59

60 5.2 Problem Formulation To make use of a priori knowledge and at the same time to ensure that the estimate of R is positive semidefinite, we consider a modified general linear combination (MGLC) of the a priori covariance matrix R, the sample covariance matrix ˆR, and the identity matrix I to get an enhanced estimate, let us call it R, of the true array covariance matrix R: R = AR + B ˆR + CI. (5 3) The combination weights A, B and C in (5 3) should be chosen to guarantee that R. We also consider the following modified convex combination (MCC) of the three terms (i.e., R, ˆR and I): R = AR + B ˆR + CI; A + B + C = 1. (5 4) Again, constraints should be enforced to ensure that R obtained by (5 4) satisfies R. We refer to the use of (5 3) and (5 4) (with optimized A, B and C, see below) to obtain an estimate of R as the MGLC and MCC approaches, respectively. With B normalized to 1, (5 3) is also known as the colored loading approach proposed in [23], with the colored loading level given by A and the diagonal loading level by C. However, in [23], the colored and diagonal loading levels were chosen only manually. The first goal of this paper is to obtain optimal estimates of the weighting factors A, B and C that minimize the mean-squared error (MSE) of R: MSE = E{ R R 2 }, (5 5) for both (5 3) and (5 4); the second goal of this paper is then to use R (with optimized A, B and C) as a new estimate of R, in lieu of ˆR, in SCB to improve the array output SINR. 6

61 5.3 Knowledge-Aided Covariance Matrix Estimation MGLC and MCC We will consider the MSE minimization problem first for (5 3) and then for (5 4). For (5 3), a simple calculation yields MSE( R) = E{ R R 2 } = E{ AR + CI (1 B)R + B( ˆR R) 2 }. (5 6) Using the fact that ˆR is an unbiased estimate of R, we get MSE( R) = E{ B( ˆR R) 2 } + AR + CI (1 B)R 2 = B 2 E{ ˆR R 2 } + (1 B) 2 R 2 2(1 B) tr [R (AR + CI)] + AR + CI 2, (5 7) where we have used the fact that for any two positive semidefinite matrices E and F, tr(ef), for E, F. (5 8) Let Consequently, (5 7) can be rewritten as ρ = E{ ˆR R 2 }. (5 9) MSE( R) = ρb 2 + R 2 B 2 + AR + CI 2 + 2B tr [R (AR + CI)] 2B R 2 2 tr [R (AR + CI)] + const. (5 1) Define θ as θ = [A B C] T. (5 11) 61

62 Then the term of (5 1) that is a linear function of θ can be written more compactly as 2b T θ, where b T = [ tr(r R ) R 2 tr(r ) ]. (5 12) The quadratic term in (5 1) is: ρb 2 + AR + CI + BR 2 = θ T Aθ, (5 13) where A = R 2 tr(r R) tr(r ) tr(r R) R 2 + ρ tr(r ) tr(r ) tr(r ) I 2. (5 14) Hence, MSE( R) = θ T Aθ 2b T θ + const. (5 15) Note that the matrix A >. The simplest way to prove this follows from (5 13): θ T Aθ >, θ if and only if there are no A and C, not both equal to, such that AR + CI =. This amounts to assuming that R is not a scaled version of I, which is a very natural assumption. In view of A >, (5 15) has a unique minimum solution given by: θ = [A B C ] T = A 1 b. (5 16) Interestingly, we show in the Appendix that B 1. However, θ obtained by (5 16) may not guarantee that R. Consequently, we rewrite (5 15) as: [θ θ ] T A[θ θ ] + const, (5 17) 62

63 and consider the following MSE minimization problem for MGLC with the constraint R enforced: min δ,θ s.t. δ δ [θ θ ] T [θ θ ] A 1 R(θ). (5 18) The above formulation is equivalent to min θ MSE [ R(θ) ] s.t. R(θ), (5 19) and it is a Semidefinite Program (SDP) that can be efficiently solved in polynomial time using public domain software [52, 53]. For MCC, we only need to add the following additional constraint u T 3 θ = 1, u 3 = [1 1 1] T, (5 2) to the MGLC formulation in (5 18). We use u l to denote a vector of 1 s of length l. The resulting problem is still a SDP. In practice, θ must be replaced by its estimate ˆθ, where ˆθ =  1ˆb, (5 21) with  and ˆb being the estimates of A and b, respectively. Then (5 18) becomes: min δ,θ s.t. δ δ [θ ˆθ ] [θ ˆθ ] T  1 R(θ). (5 22) 63

64 One way to obtain  and ˆb is to replace ρ and R in A and b by ˆρ and ˆR, where ˆρ is an estimate of ρ, which can be obtained as [51]: ˆρ = 1 N 2 N y(n) 4 1 N ˆR 2. (5 23) n=1 We refer to [51] for a detailed derivation of ˆρ. As suggested in [34] (and also in [54, 36]), to estimate θ and R consistently, we prefer to use an alternative estimation scheme to obtain  and ˆb as follows. Because ρ + R 2 = E{ ˆR R 2 } + R 2 = E { ˆR 2 }, we can estimate ρ + R 2 by ˆR 2, which is an unbiased estimate that is smaller than the previously suggested ˆρ + ˆR 2 ; we can estimate R 2 by ˆR 2 ˆρ, again, a smaller estimate. We also replace R by ˆR in tr(r) and tr(r R ). We refer to the resulting SDP problem in (5 22) for MGLC as MGLC 1, and similarly, (5 22) with the additional constraint (5 2) for MCC as MCC 1. Alternatively, we can enforce in (5 22) A, B, and C. Then the constraint R(θ) is trivially satisfied and (5 22) becomes a quadratic program (QP): ( min θ ˆθ ) T  (θ ˆθ ) θ s.t. θ i, i = 1, 2, 3. (5 24) We denote this formulation for MGLC as MGLC 2. Similarly, adding (5 2) as an additional constraint to (5 24) yields a QP problem for MCC, which we denote as MCC Extensions In the previous subsection, we considered a linear combination of three terms to get an enhanced estimate of R. The previous approach can be easily extended to a linear combination of an arbitrary number of positive semidefinite matrices. Let {R (s) } S s=1 denote S a priori covariance matrices representing knowledge about S dominant sources or strong interferences. We consider the following linear combination of 64

65 {R (s) } S s=1, ˆR and I: R = S s=1 A (s) R (s) + B ˆR + CI, (5 25) where {A (s) } S s=1 are the weights applied to the a priori covariance matrices {R (s) } S s=1. Constraints again need to be imposed to ensure that R. We observe that (5 3) is a special case of (5 25) with S = 1. Matrix A in (5 14) for this extended case becomes: A = R (1) 2 tr(r (1) R (S) ) tr(r (1) R) tr(r (1) ) tr(r (1) R (S) ) R (S) 2 tr(r (S) R) tr(r (S) ) tr(r (1) R) tr(r (S) R) R 2 + ρ tr(r ) tr(r (1) ) tr(r (S) ) tr(r ) I 2 Similarly, b in (5 12) now has the form: b T =. (5 26) [ ] tr(r R (1) ) tr(r R (S) ) R 2 tr(r ). (5 27) We redefine θ as θ = [ A (1) A (S) B C ] T. (5 28) With A and b given in (5 26) and (5 27), the corresponding  and ˆb can be obtained by using the same scheme as before. Consequently, an estimate of θ in (5 28) can be obtained by solving (5 22) for extended MGLC 1, and (5 24) for extended MGLC 2 (with the constraints being replaced by θ i, i = 1, 2,, S, S + 1, S + 2). By solving (5 22) and (5 24) with the following additional constraint: u T S+2θ = 1, (5 29) we get extended MCC 1 and MCC 2, repectively. Let ˆθ [Â(1) = MGLC  (S) i MGLCi MGLCi ˆB MGLC i Ĉ MGLC i] T, i = 1, 2, (5 3) 65

66 and ˆθ [Â(1) = MCC Â (S) i MCCi MCCi ˆB MCC i Ĉ MCC i] T i = 1, 2, (5 31) be the solutions to the extended MGLC i and MCC i problems, i = 1, 2, respectively. Given (5 3) and (5 31), the resulting R can be expressed as: R MGLC i = S s=1 Â (s) MGLCi R(s) + ˆB MGLC ˆR + ĈMGLCi I, i = 1, 2, (5 32) i and R MCC i = S s=1 Â (s) MCCi R(s) + ˆB MCC ˆR + ĈMCCi I, i = 1, 2. (5 33) i Remark: We note from the unconstrained solution of θ (see (5 16) with A and b given in (5 26) and (5 27), respectively) that the value of ĈMGLC increases as tr( ˆR) increases. Specifically, tr( ˆR) will be large in the presence of strong interferences, resulting in a very high weighting value on I in R MGLC. Consequently, when such a R MGLC is used in adaptive beamformers, their adaptive capability is greatly reduced because ĈMGLC is large. To avoid the said problem of MGLC, an additional constraint can be enforced in the MGLC formulations in (5 22) and (5 24): θ S+2 γλ min, (5 34) where λ min is the smallest non-zero eigenvalue of ˆR, and γ is a scaling factor. Noting the fact that θ S+2 should decrease as N/M increases, we may choose γ to be inversely proportional to N/M. In our simulations, we choose γ = 1 3 M/N. The constraint in (5 34) is usually inactive when tr( ˆR) is small. 5.4 Using R for Adaptive Beamforming Assume that the true covariance matrix R of the array output has the following form: K R = σa 2 a + σka 2 k a k + Q, (5 35) k=1 66

67 where σ 2 and σk 2, respectively, are the powers of SOI and of the kth interference impinging on the array, a and a k are the corresponding steering vectors, and Q is the noise covariance matrix. The array weight vector w can be determined by the SCB approach: The solution to (5 36) is: min w w Rw s.t. w a = 1. (5 36) w = R 1 a a R 1 a. (5 37) The beamformer output signal-to-interference-plus-noise ratio (SINR) can be expressed as SINR = w σ 2 wa 2 ( K ). (5 38) k=1 σ2 k a ka k + Q w Note that the SINR in (5 38) is independent of the scaling factor of the weight vector w. By inserting (5 37) in (5 38) and using the matrix inversion lemma, we get the optimal array output SINR: ( K 1 SINR opt = σa 2 σka 2 k a k + Q) a. (5 39) k=1 In SCB, the sample covariance matrix ˆR is used in lieu of R. As a result, the weight vector of SCB is given by ŵ SCB = ˆR 1 a a ˆR 1 a. (5 4) Then SINR SCB follows from (5 38) by replacing the w in (5 38) with ŵ SCB. As discussed in the previous subsections, we can use MGLC and MCC to get enhanced KA-estimates of R (see (5 32) and (5 33)). If we use { R MGLC i } i=1,2 in lieu of ˆR in the SCB formulation, the corresponding array weight vector for MGLC can be obtained as w MGLC i = 1 R a MGLCi 1 R a, i = 1, 2. (5 41) MGLCi a Similarly, the weight vector for using { R MCC i } i=1,2 in lieu of ˆR can be obtained by replacing the subscript MGLC i with MCC i, i = 1, 2, in (5 41). The corresponding 67

68 SINR values can be obtained by using (5 38) with the w in (5 38) replaced by the MGLC and MCC weight vectors. 5.5 Numerical Examples In this section, we present several numerical examples comparing the performance of SCB and MGLC. The performance is also compared with that obtained by setting A s to zero in MGLC, resulting in a diagonal loading approach with the diagonal loading level computed fully automatically. The performance of MCC was inferior to that of MGLC in all of our examples and hence only the MGLC results are presented hereafter. We consider a uniform linear array (ULA) with M = 1 sensors and half-wavelength spacing between adjacent elements. Assume a spatially white Gaussian noise whose covariance matrix Q is given by 1I. We assume that the direction-of-arrival (DOA) of the SOI relative to the array normal is θ = and that there are K = 2 interferences whose DOAs are θ 1 = 4, θ 2 = 2. The power of the SOI is 2 db, i.e., σ 2 = 2 db. The powers of the two interferences, i.e., σ1 2 and σ2, 2 will be specified later on. Also, we assume knowledge of the steering vector a of the SOI. In our simulations, averaged SINR values obtained from 2 Monte-Carlo trials are given. Note that unlike SCB, MGLC allows N to be less than the number of sensors M. We use N = 4, 6, 8 for the N < M case. Obviously, the performance of MGLC depends on the degree of accuracy of the a priori knowledge. To investigate the effects of the accuracy of R on the performance of MGLC, we consider the following six cases (we use R i to denote the a priori covariance matrix for the ith case, i = 1,, 6): (i). Accurate a priori knowledge, i.e., 2 R 1 = R Q = σa 2 a + σka 2 k a k. (5 42) k=1 (ii). Accurate a priori knowledge of the interferences, i.e., R 2 = 2 σka 2 k a k. (5 43) k=1 68

69 (iii). Only the DOA of the first interference is accurately known: R 3 = a 1 a 1. (5 44) (iv). Inaccurate a priori knowledge. We consider the case where the a priori knowledge on the DOAs of the interferences is totally wrong, i.e., R 4 = σ 2 3a 3 a 3 + σ 2 4a 4 a 4, (5 45) where σ 2 3 = σ 2 1, σ 2 4 = σ 2 2 and a 3 and a 4 are the steering vectors for two uncorrelated signals impinging on the array from -55 and 6. (v). The DOAs of the interferences are accurately known: R (1) 5 = a 1 a 1, (5 46) and R (2) 5 = a 2 a 2. (5 47) (vi). The DOA of the first interference is accurately known, but we assume wrongly that the DOA of the second interference is 6 : R (1) 6 = a 1 a 1, (5 48) and Relatively Weak Interferences R (2) 6 = a 4 a 4. (5 49) We first consider a scenario in which the power of the SOI and the powers of the two interferences are all equal to 2 db, i.e., σ 2 k = 2 db, k =, 1, 2. In the presence of relatively weak interferences, tr ( ˆR) is small and hence the constraint in (5 34) is inactive for this case. Figures show the averaged estimates of A, B and C from 2 Monte-Carlo realizations when the a priori covariance matrices {R i } 4 i=1 (see (5 42)-(5 45)) are used 69

70 as R for MGLC 1 and MGLC 2, respectively. (Note that the averaged estimate of A/σ1 2 is shown in Figures 5-1(c) and 5-2(c).) We note that by enforcing R, the so-obtained  MGLC 1, ˆBMGLC 1 and ĈMGLC1 can be negative, as shown in Figure 5-1. For Figures 5-1(a) and 5-2(a), where the prior knowledge is accurately known, we see clearly that ambiguities exist as N. Recall that ˆR converges to R N. For MGLC 1,  = 1 ˆB and Ĉ = 1 ˆB for any ˆB will result in R = R as N. MGLC 2 is, however, more restrictive. For MGLC 2,  = 1 ˆB and Ĉ = 1 ˆB for any ˆB 1 will result in R = R as N. For all other cases of Figures 5-1 and 5-2, there is no ambiguity and we observe that  and Ĉ approach and ˆB approaches 1 as N, as desired. We also observe from Figures 5-1(d) and 5-2(d) that  is close to when inaccurate a priori covariance matrix R 4 is used as R. That is to say, little or no emphasis is placed on the a priori knowledge when it is unreliable. Figures 5-3(a) - 5-3(d) show the averaged array output SINR (in db) versus the snapshot number N when the a priori covariance matrix is given by R = R i, i = 1,, 4, respectively. Note that we consider SINR because it is an important criterion in many array processing applications, such as communications. As we can see from Figure 5-3(a), with accurate a priori knowledge, i.e., with R = R 1, MGLC 2 significantly outperforms MGLC 2 with A =. MGLC 1, however, is inferior to MGLC 2 and will not be considered further. Observe that with accurate knowledge of the interferences, i.e., with R = R 2, both MGLC 1 and MGLC 2 outperform MGLC 2 with A =, as shown in Figure 5-3(b). Observe also that with only the DOA information on the first interference, i.e., with R = R 3, slight improvements over MGLC 2 with A = can still be achieved by using MGLC 1 or MGLC 2. It is interesting to note that when the a priori knowledge is inaccurate, i.e., when R 4 is used as R, the performance of MGLC 2 is similar to that of MGLC 2 with A =, as shown in Figure 5-3(d). This is an appealing feature, indicating that MGLC 2 is robust to the error in the a priori knowledge. For comparison purposes, we also consider equal weighting of R, ˆR and I for the four cases. For example, one can 7

71 set A = B = C = 1, and hence R = R + ˆR + I (we refer to this as equal weight ). Observe that by adaptively choosing the combination weights, the performance of MGLC 2 is always better than that of equal weighting. We now consider the Cases (v) and (vi). Figures 5-4(a) and 5-4(b), respectively, show the averaged estimates of A (1) /σ 2 1, A (2) /σ 2 2, B and C versus the snapshot number N when {R (s) 5 } 2 s=1 (see (5 46) and (5 47)) and {R (s) 6 } 2 s=1 (see (5 48) and (5 49)) are used as the a priori covariance matrices for MGLC 2. Observe that when σ 2 1 = σ 2 2 = 2 db and both DOAs of the two interferences are accurately known, Â (1) /σ 2 1 and Â(2) /σ 2 2 obtained by MGLC 2 are almost identical. They have relatively large values when N is small and decrease as N increases, as desired. Observe also that for the case where the a priori covariance matrices {R (s) } 2 s=1 are given by {R (s) 6 } 2 s=1, MGLC 2 can give a proper weight on the accurate component, i.e., R (1) 6, and at the same time suppress the wrong component, i.e., R (2) 6, in the a priori knowledge. In Figure 5-5, we display the averaged array output SINR as a function of the snapshot number N for the two cases. By making use of the a priori covariance matrices {R (s) 5 } 2 s=1, MGLC 2 outperforms MGLC 2 with A (1) = A (2) =, especially when N is small (see Figure 5-5(a)). When {R (s) 6 } 2 s=1 are used as the a priori covariance matrices, the performance of MGLC 2 is still slightly better than that of MGLC 2 with A (1) = A (2) =, as shown in Figure 5-5(b) Relatively Strong Interferences We now consider a scenario where the powers of the interferences are much stronger than that of the SOI. Assume that σ 2 = 2 db, σ 2 1 = 7 db, and σ 2 2 = 6 db. The constraint in (5 34) is now active due to the presence of strong interferences. We first consider the Cases (i) - (iv) when R i, i = 1,, 4, is used as the a priori covariance matrix. Â s, ˆB s and Ĉ s obtained by MGLC 2 for this case of strong interferences have similar properties as those shown in Figures 5-1 and 5-2, except that Ĉ s now take on larger values, which are mainly determined by the constraint (5 34). Figure 5-6 shows 71

72 the averaged SINR versus the snapshot number N for the four cases. It is interesting to note that as compared to Figure 5-5, providing accurate or partially accurate a priori knowledge gives greater SINR improvement in the presence of strong interferences. We next consider the Cases (v) and (vi) for this case of strong interferences. Figure 5-7 shows the averaged SINR versus the snapshot number N. Again, an obvious improvement of MGLC 2 over the other methods can be observed. Finally, we comment that our numerical examples (not presented herein) show that for both cases of weak and strong interferences, as N, all SINR curves (including those of MGLC 1 ) approach the optimal SINR curve, as expected. Moreover, the SINR curves of MGLC 2 always stay above those of MGLC with A = as well as those of SCB. 5.6 Conclusions In knowledge-aided adaptive beamforming, a priori environmental knowledge can be utilized to improve the performance of adaptive arrays. In practice, the a priori knowledge usually consists of (partial) information about the array covariance matrix. Consequently, we can obtain an initial guess of the array covariance matrix, denoted as R, which is usually rank deficient. In this paper, we have presented two fully automatic methods, namely MGLC and MCC, for combining the sample covariance matrix ˆR with the a priori covariance matrix R and the identity matrix I to get an enhanced estimate of R in the optimal mean squared error sense. We have also extended the MGLC and MCC methods to deal with linear combinations of an arbitrary number of positive semidefinite matrices. It has been shown that both MGLC and MCC can be formulated as convex optimization problems, which can be solved efficiently in polynomial time with global optimality guaranteed. When the MGLC and MCC techniques are used for knowledgeaided adaptive beamforming, we have shown via numerical examples that MGLC can choose the combination weights adaptively (based on the data) according to the accuracy of the a priori knowledge. We have also shown that providing accurate or partially 72

73 accurate a priori knowledge can significantly improve the performance of the adaptive beamformers, especially in the presence of strong interferences. Appendix: A Property of B in (5 16) In this appendix, we show that B 1. Note that θ in (5 16) satisfies: Aθ = b, (5 5) where A and b are given in (5 14) and (5 12), respectively. The augmented matrix associated with (5 5) is [A b], where R 2 tr(r R) tr(r ) tr(r R ) [A b] = tr(r R) R 2 + ρ tr(r ) R 2 tr(r ) tr(r ) I 2 tr(r ). (5 51) and Let D 1 = tr (R R) R 2 R R, (5 52) D 2 = tr (R ) R 2 R I. (5 53) Then making use of the Gaussian elimination, it follows that (5 51) is equivalent to R 2 tr(r R) tr(r ) tr(r R ) D ρ tr(d 1D 2 ) D 1 2, (5 54) tr(d 1D 2 ) D 2 2 tr (D 1D 2 ) and consequently it is also equivalent to R 2 tr(r R) tr(r ) tr(r R ) D ρ tr2 (D 1 D 2) D 2 D tr2 (D 1 D 2) D 2 2 tr(d 1D 2 ) D 2 2 tr (D 1D 2 ). (5 55) 73

74 3 2.5 A B C A B C Snapshot Number (a) A/σ 1 2 B C Snapshot Number (b) A B C Snapshot Number (c) Snapshot Number (d) Figure 5-1. The averaged MGLC 1 estimates of A, B and C versus the snapshot number N when σ 2 = σ 2 1 = σ 2 2 = 2 db, and the a priori covariance matrix is given by (a) R = R 1 (see (5 42)), (b) R = R 2 (see (5 43)), (c) R = R 3 (see (5 44)) and (d) R = R 4 (see (5 45)). Hence B can be obtained as B = D 1 2 D 2 2 tr 2 (D 1D 2 ) ρ D D 1 2 D 2 2 tr 2 (D 1D 2 ). (5 56) Given (5 56) and by using the Cauchy-Schwartz inequality, we have that B 1. 74

75 3 2.5 A B C A B C Snapshot Number (a) A/σ 1 2 B C Snapshot Number (b) A B C Snapshot Number (c) Snapshot Number (d) Figure 5-2. The averaged MGLC 2 estimates of A, B and C versus the snapshot number N when σ 2 = σ 2 1 = σ 2 2 = 2 db, and the a priori covariance matrix is given by (a) R = R 1 (see (5 42)), (b) R = R 2 (see (5 43)), (c) R = R 3 (see (5 44)) and (d) R = R 4 (see (5 45)). 75

76 SINR (db) 5 5 SCB MGLC 2 w/ A= SINR (db) 5 5 SCB MGLC 2 w/ A= MGLC 1 MGLC 2 Equal weight SINR opt Snapshot Number (a) MGLC 1 MGLC 2 Equal weight SINR opt Snapshot Number (b) SINR (db) 5 5 SCB MGLC 2 w/ A= SINR (db) 5 5 SCB MGLC 2 w/ A= MGLC 1 MGLC 2 Equal weight SINR opt Snapshot Number (c) MGLC 1 MGLC 2 Equal weight SINR opt Snapshot Number (d) Figure 5-3. SINR versus the snapshot number N when σ 2 = σ 2 1 = σ 2 2 = 2 db, and the a priori covariance matrix is given by (a) R = R 1 (see (5 42)), (b) R = R 2 (see (5 43)), (c) R = R 3 (see (5 44)) and (d) R = R 4 (see (5 45)). 76

77 A (1) /σ 1 2 A (2) /σ 2 2 B C A (1) /σ 1 2 A (2) /σ 2 2 B C Snapshot Number (a) Snapshot Number (b) Figure 5-4. The averaged MGLC 2 estimates of A (1) /σ1, 2 A (2) /σ2, 2 B and C versus the snapshot number N when σ 2 = σ1 2 = σ2 2 = 2 db, and the a priori covariance 5 } 2 s=1 (see (5 46) and (5 47)), and matrices are given by (a) {R (s) } 2 s=1 = {R (s) (b) {R (s) } 2 s=1 = {R (s) 6 } 2 s=1 (see (5 48) and (5 49)) SINR (db) SCB MGLC 2 w/ A (1) =A (2) = SINR (db) SCB MGLC 2 w/ A (1) =A (2) = 15 2 MGLC 2 SINR opt Snapshot Number (a) 15 2 MGLC 2 SINR opt Snapshot Number (b) Figure 5-5. SINR versus the snapshot number N when σ 2 = σ1 2 = σ2 2 = 2 db, and the a priori covariance matrices are given by (a) {R (s) } 2 s=1 = {R (s) 5 } 2 s=1 (see (5 46) and (5 47)), and (b) {R (s) } 2 s=1 = {R (s) 6 } 2 s=1 (see (5 48) and (5 49)). 77

78 SINR (db) SCB MGLC 2 w/ A= SINR (db) SCB MGLC 2 w/ A= 15 2 MGLC 2 SINR opt Snapshot Number (a) 15 2 MGLC 2 SINR opt Snapshot Number (b) SINR (db) SCB MGLC 2 w/ A= SINR (db) SCB MGLC 2 w/ A= 15 2 MGLC 2 SINR opt Snapshot Number (c) 15 2 MGLC 2 SINR opt Snapshot Number (d) Figure 5-6. SINR versus the snapshot number N when σ 2 = 2 db, σ 2 1 = 7 db and σ 2 2 = 6 db, and the a priori covariance matrix is given by (a) R = R 1 (see (5 42)), (b) R = R 2 (see (5 43)), (c) R = R 3 (see (5 44)) and (d) R = R 4 (see (5 45)). 78

79 SINR (db) SCB MGLC w/ A (1) =A (2) = 2 SINR (db) SCB MGLC w/ A (1) =A (2) = MGLC 2 SINR opt Snapshot Number (a) 15 2 MGLC 2 SINR opt Snapshot Number (b) Figure 5-7. SINR versus the snapshot number N when σ 2 = 2 db, σ1 2 = 7 db and σ2 2 = 6 db, and the a priori covariance matrices are given by (a) {R (s) } 2 s=1 = {R (s) 5 } 2 s=1 (see (5 46) and (5 47)), and (b) {R (s) } 2 s=1 = {R (s) 6 } 2 s=1 (see (5 48) and (5 49)). 79

80 CHAPTER 6 WAVEFORM SYNTHESIS FOR DIVERSITY-BASED TRANSMIT BEAMPATTERN DESIGN 6.1 Introduction Waveform diversity has been utilized both in multiple-input multiple-output (MIMO) communications and in MIMO radar. In the past decade, communications systems using multiple transmit and receive antennas have attracted significant attention from government agencies, academic institutions and research laboratories, because of their potential for dramatically enhanced throughput and significantly reduced error rate without spectrum expansion. Similarly, MIMO radar systems have recently received the attention of researchers and practitioners alike due to their improved capabilities compared with a standard phased-array radar. A MIMO radar system, unlike a standard phased-array radar, can transmit multiple probing signals that may be chosen at will. This waveform diversity offered by MIMO radar is the main reason for its superiority over standard phased-array radar; see, e.g., [55] - [83]. For colocated transmit and receive antennas, for example, MIMO radar has been shown to have the following appealing features: higher resolution (see, e.g., [55, 57]), superior moving target detection capability [6], better parameter identifiability [67, 78], and direct applicability of adaptive array techniques [67, 69, 8]; in addition, the covariance matrix of the probing signal vector transmitted by a MIMO radar system can be designed to approximate a desired transmit beampattern an operation that, once again, would be hardly possible for conventional phased-array radar [67, 72, 76]. Transmit beampattern design is critically important not only in defense applications, but also in many other fields including homeland security and biomedical applications. In all these applications, flexible transmit beampattern designs can be achieved by exploiting the waveform diversity offered by the possibility of choosing how the different probing signals are correlated with one another. 8

81 An interesting current research topic is the optimal synthesis of the transmitted waveforms. For MIMO radar with widely separated antennas, waveform designs without any practical constraint (such as the constant-modulus constraint) have been considered in [73]. For MIMO systems with colocated antennas, on the other hand, the recently proposed techniques for transmit beampattern design or for enhanced target parameter estimation and imaging have focused on the optimization of the covariance matrix R of the waveforms [61, 65, 67, 72, 75, 76, 79, 81]. For example, in a waveform diversity-based ultrasound system, R can be designed to achieve a beampattern that is suitable for the hyperthermia treatment of breast cancer [84]. Now, instead of designing R, as in the cited references, we might think of designing directly the probing signals by optimizing a given performance measure with respect to the matrix X of the signal waveforms. However, compared with optimizing the same performance measure with respect to the covariance matrix R of the transmitted waveforms, optimizing directly with respect to X is a more complicated problem. This is so because X has more unknowns than R and the dependence of various performance measures on X is more intricate than the dependence on R (as R is a quadratic function of X). In effect, there are several recent methods, as mentioned above, that can be used to efficiently compute an optimal covariance matrix R, with respect to several performance metrics; yet the same cannot be said about determining an optimal signal waveform matrix X, which is the ultimate goal of the designing exercise. Furthermore, in some cases, the desired covariance matrix is given (e.g., a scaled identity matrix), and therefore there is no optimization with respect to R involved (directly or indirectly). In this paper, we consider the synthesis of the signal waveform matrix X for diversitybased flexible transmit beampattern design. With R obtained in a previous (optimization) stage, our problem is to determine a signal waveform matrix X whose covariance matrix is equal or close to R, and which also satisfies some practically motivated constraints (such as constant-modulus or low peak-to-average-power ratio (PAR) constraints). We present a 81

82 cyclic optimization algorithm for the synthesis of such an X. We also investigate how the synthesized waveforms and the corresponding transmit beampattern design depend on the enforced practical constraints. Several numerical examples are provided to demonstrate the effectiveness of the proposed methodology. Notation. Vectors are denoted by boldface lowercase letters and matrices by boldface uppercase letters. The nth component of a vector x is written as x(n). The nth diagonal element of a matrix R is written as R nn. A Hermitian square root of a matrix R is denoted as R 1/2. We use ( ) T to denote the transpose, and ( ) for the conjugate transpose. The Frobenius norm is denoted as. The real part of a complex-valued vector or matrix is denoted as Re( ). 6.2 Formulation of the Signal Synthesis Problem Let the columns of X C L N be the transmitted waveforms, where N is the number of the transmitters, and L denotes the number of samples in each waveform. Let R 1 L X X (6 1) be the (sample) covariance matrix of the transmitted waveforms. We assume that L > N (typically L N). Note that X has 2NL real-valued unknowns, which is usually a much larger number than the number of unknowns in R, viz. N 2. The class of (unconstrained) signal waveform matrices X that realize a given covariance matrix R is given by 1 L X = R 1/2 U, (6 2) where U is an arbitrary semi-unitary N L matrix (U U = I). Besides realizing (at least approximately) R, the signal waveform matrix must also satisfy a number of practical constraints. Let C denote the set of signal matrices X that satisfy these constraints. Then a possible mathematical formulation of the problem of synthesizing the probing signal matrix X is as follows: min X C;U X LUR 1/2 2. (6 3) 82

83 Depending on the constraint set C, the solution X to (6 3) may realize R exactly or only approximately. Evidently as C is expanded (i.e., the constraints are relaxed), the matching error in (6 3) decreases. Whenever the matching error is different from zero, we can use either the solution ˆX to (6 3) as the signal waveform matrix (in which case it will satisfy the constraints, but it will only approximately realize R) or LÛR1/2 (which realizes R exactly but satisfies only approximately the constraints) the choice between these two signal waveform matrices may be dictated by the application at hand. The minimization problem in (6 3) is non-convex due to the non-convexity of the constraint U U = I and possibly of the set C, too. The constraint U U = I generates the so-called Stiefel manifold, and there are algorithms that can be used to minimize a function over the said manifold (see, e.g., [85]). However, these algorithms are somewhat intricate both conceptually and computationally, and their convergence properties are not completely known; additionally, in (6 3) we also have the problem of minimizing with respect to X C, which may also be non-convex. With the above facts in mind, we prefer a cyclic (or alternating) minimization algorithm for solving (6 3), as suggested in a related context in [86, 87]. We refer to the cited papers for more details on this type of algorithm and its properties. 6.3 Cyclic Algorithm for Signal Synthesis We first summarize the steps of the cyclic minimization algorithm and then describe each step in detail. Step : Set U to an initial value (e.g., the elements of U can be independently drawn from a complex Gaussian distribution with mean and standard deviation 1); alternatively we can start with an initial value for X, in which case the sequence of the next steps should be inverted. Step 1: Step 2: Iteration: Obtain the matrix X C that minimizes (6 3) for U fixed at its most recent value. Determine the matrix U (U U = I) that minimizes (6 3) for X fixed at its most recent value. Iterate Steps 1 and 2 until a given stop criterion is satisfied. In the numerical examples presented later, we terminate the iteration when the Frobenius norm of the 83

84 difference between the U matrices at two consecutive iterations is less than or equal to 1 4. An important advantage of the above algorithm is that Step 2 has a closed-form solution. This solution can be derived in a number of ways (see, e.g., [88, 89]). A simple derivation of it runs as follows. For given X, we have that X LUR 1/2 2 [ ]} = const 2Re {tr LR 1/2 X U. (6 4) Let LR 1/2 X = ŪΣŨ (6 5) denote the singular value decomposition (SVD) of LR 1/2 X, where Ū is N N, Σ is N N, and Ũ is L N. Then { [ ]} Re tr LR 1/2 X U { [Ũ ]} = Re tr UŪΣ = N n=1 (6 6) {[Ũ ] } Re UŪ Σ nn. (6 7) nn Because (Ũ ) (Ū UŪ U Ũ) = Ũ UU Ũ Ũ Ũ = I, (6 8) it follows that {[Ũ ] } Re 2 UŪ nn ] [Ũ UŪ [(Ũ UŪ 2 nn ) (Ū U Ũ)] 1, (6 9) nn 84

85 and therefore that The lower bound in (6 1) is achieved at X LUR 1/2 2 N const 2 Σ nn. (6 1) n=1 Û = ŨŪ, (6 11) which is thus the solution to the minimization problem in Step 2 of the cyclic algorithm. The solution to the problem in Step 1 naturally depends on the constraint set C. For example, in radar systems the need to avoid expensive amplifiers and A/D converters has led to the requirement that the transmitted signals have constant modulus. Let {x n (l)} L l=1 denote the elements in the nth column x n of the signal waveform matrix X. Then the constant-modulus requirement means that: x n (l) = c, for some given constant c and for l = 1,, L. (6 12) (For example, we can choose c = R 1/2 nn ; we omit the dependence of c on n for notational simplicity.) Under the constraint in (6 12), Step 1 of the algorithm has also a closed-form solution. Indeed, the generic problem to be solved in such a case is: where c > and z C are given numbers. Because min ψ ce jψ z 2, (6 13) ce jψ z 2 = const 2c z cos [ψ arg(z)], (6 14) the minimizing ψ is evidently given by ψ = arg(z). (6 15) Therefore, under the constant-modulus constraint, both steps of the cyclic algorithm have solutions that can be readily computed. However, (6 12) may be too hard a requirement 85

86 on the signal matrix in the sense that the corresponding minimum value of the matching criterion in (6 3) may not be as small as desired. In particular, this means that 1 L ˆX ˆX may not be a good approximation of R (see, e.g., [79], where it was shown that signals that have constant modulus and take on values in a finite alphabet may fail to realize well a given covariance matrix). With the above facts in mind, we may be willing to compromise and therefore relax the requirement that the signals have constant modulus. In effect, in some modern radar systems this requirement can be replaced by the condition that the transmitted signals have a low peak-to-average-power ratio (PAR). Mathematically, the low PAR requirement can be formulated as follows: PAR(x n ) = max l x n (l) 2 L l=1 x ρ, for a given ρ [1, L], (6 16) 2 n(l) 1 L (where, once again, we omit the dependence of ρ on n for notational simplicity). If we add to (6 16) a power constraint, viz. 1 L L x n (l) 2 = γ, (e.g., γ = R nn ), (6 17) l=1 then the set C is described by the equations: 1 L L l=1 x n(l) 2 = γ, x n (l) 2 ργ, l = 1,, L. (6 18) While the above constraint set is not convex, an efficient algorithm for solving the corresponding problem in Step 1 of the cyclic algorithm has been proposed in [86, 87]. Note that the constraints in (6 18) are imposed on X in a column-wise manner. Consequently, the solution to Step 1 is obtained by dealing with the columns of X in a one by one fashion. With the power constraint in (6 17) enforced, the diagonal elements of R can be synthesized exactly. If the exact matching of R nn is not deemed necessary, we can relax 86

87 the optimization by omitting (6 17). In the Appendix we show how to modify the algorithm of [86, 87] in the case where only (6 16) is enforced. (In all numerical examples presented in the following section, (6 17) will be enforced.) 6.4 Numerical Case Studies We present several numerical examples to demonstrate the effectiveness of CA for signal synthesis in several diversity-based transmit beampattern design applications Beampattern Matching Design We first review briefly the beampattern matching design (more details can be found in [72, 76]). We then present a number of relevant numerical examples. The power of the probing signal at a generic focal point with coordinates θ can be shown to be (see, e.g., [61, 69, 72]): P (θ) = a (θ)ra(θ), (6 19) where R is as defined before, [ a(θ) = e j2πf τ 1 (θ) e j2πf τ 2 (θ) e j2πf τ N (θ) ] T, (6 2) and where f is the carrier frequency of the transmitted signal, and τ n (θ) is the time needed by the signal emitted via the nth transmit antenna to arrive at the focal point; unless otherwise stated, θ will be a one-dimensional angle variable (expressed in degrees). The design problem under discussion consists of choosing R, under a uniform elemental power constraint, R nn = C, n = 1,, N, (6 21) N where C is the given total transmitted power, to achieve the following goals: (a) Control the spatial power at a number of given locations by matching (or approximating) a (scaled version of a) desired transmit beampattern. 87

88 (b) Minimize the cross-correlation between the probing signals at a number of given locations (a reason for this requirement is explained in [72, 76]); the cross-correlation between the probing signals at locations θ and θ is given by a (θ)ra( θ). Assume that we are given a desired transmit beampattern φ(θ) defined over a region of interest Ω. Let {µ g } G g=1 be a fine grid of points that cover Ω. As indicated above, our goal is to choose R such that the transmit beampattern, a (θ)ra(θ), matches or rather approximates (in a least squares (LS) sense) the desired transmit beampattern, φ(θ), over the region of interest Ω, and also such that the cross-correlation (beam)pattern, a (θ)ra( θ) (for θ θ), is minimized (once again, in a LS sense) over a given set {θ k } K k=1. Mathematically, we therefore want to solve the following problem: { 1 G min w g [αφ(µ g ) a (µ g )Ra(µ g )] 2 α,r G g=1 K 1 2w c K 2 K K k=1 p=k+1 a (θ k )Ra(θ p ) 2 } s.t. R nn = C N, n = 1,, N R, (6 22) where α is a scaling factor, w g, g = 1,, G, is the weight for the gth grid point and w c is the weight for the cross-correlation term. Note that by choosing max g w g > w c we can give more weight to the first term in the design criterion above, and vice versa for max g w g < w c. We have shown in [67, 72, 76] that this design problem is a semi-definite quadratic program (SQP) that can be efficiently solved in polynomial time. Once the optimal R has been determined, we can use CA to synthesize the waveform matrix X. As mentioned in Section 6.2, the CA solution to (6 3) may be chosen to realize R exactly or only approximately. When the signal waveforms are synthesized as LÛR1/2, where Û is the solution to (6 3) obtained via CA, then they realize R exactly, but satisfy the PAR constraints only approximately. We refer to the so-synthesized waveforms as the CA synthesized waveforms with optimal R (abbreviated as optimal R). When we use the 88

89 solution ˆX to (6 3) obtained via CA as the transmitted signal waveform matrix, then ˆX will satisfy the PAR constraints, but will realize R only approximately. We refer to the so-synthesized waveforms as the CA synthesized waveforms with PAR ρ (abbreviated as PAR ρ). In the following examples, the transmit array is assumed to be a uniform linear array (ULA) comprising N = 1 sensors with half-wavelength inter-element spacing. The sample number L is set equal to 256. The uniform elemental power constraint with C = 1 is used for the design of R. For Ω, we choose a mesh grid size of.1. Finally, the CA algorithm is initialized using the U described in Step. In the first example, the desired beampattern has one wide main-beam centered at with a width of 6. The weighting factor w g in (6 22) is set to 1 and w c is set to. Figures 6-1(a), 6-1(b), and 6-1(c) show the beampatterns using the CA synthesized waveforms under the constraints of PAR = 1 (constant-modulus), PAR 1.1, and PAR 2, respectively. For comparison purposes, we also show the desired beampattern φ(θ) scaled by the optimal value of α. Note that the beampattern obtained using the CA synthesized waveforms is close to the desired one even under the constant-modulus constraint. We also note from Figure 6-1 that the beampatterns obtained using the CA synthesized waveforms with optimal R are slightly different from those obtained using the CA synthesized waveforms with PAR ρ. Let ˆR = 1 L ˆX ˆX (6 23) be the sample covariance matrix corresponding to the CA synthesized waveforms with PAR ρ. Let δ = ˆR R (6 24) denote the norm of the difference between ˆR and R. Then we have δ = db, db, and db for Figures 6-1(a), 6-1(b), and 6-1(c), respectively. As 89

90 expected, the difference decreases as the PAR value increases. For the case of PAR = 2, the difference is essentially zero. The mean-squared error (MSE) of ˆR (i.e., the average value of δ 2 ), obtained under PAR = 1 and estimated via 1 Monte-Carlo trials, is shown in Figure 6-4 as a function of the sample number L. Note that, as also expected, the MSE decreases as L increases. Figures 6-3(a) - 6-3(c) show the corresponding beampattern differences as a function of θ, as an ensemble of realizations obtained from the 1 Monte- Carlo trials. In each Monte-Carlo trial, the initial value for U in Step of CA was chosen independently. Among other things, Figure 6-3 shows that CA is not very sensitive to the initial value of U used, and that this sensitivity decreases as ρ increases. Figure 6-2 shows the actual PAR values of the CA synthesized waveforms with optimal R corresponding to Figure 6-1. These PAR values are also compared to those associated with the waveform matrix obtained by pre-multiplying R 1/2 with a matrix whose columns contain orthogonal Hadamard code sequences of length 256. The colored Hadamard sequences also have the optimal R as their sample covariance matrix. Note that the CA synthesized waveforms with optimal R have much lower PAR values than the colored Hadamard code sequences. Note also that the actual PAR values of the CA synthesized waveforms with optimal R obtained under PAR 1.1 are slightly lower than the PAR values obtained under PAR = 1. Next, we consider a scenario where the desired beampattern has three pulses centered at θ 1 = 4, θ 2 =, and θ 3 = 4, each with a width of 2. The same mesh grid is used as before, and we choose the weighting factors as w g = 1 and w c = 1. Figure 6-5 shows the corresponding beampatterns. Remarks similar to those on Figure 6-1 can be made for this example as well Minimum Sidelobe Beampattern Design The minimum sidelobe beampattern design problem we consider here (see [72, 76] for more details) is to choose R, under the uniform elemental power constraint in (6 21) or rather a relaxed version of it (see later on), to achieve the following goals: 9

91 (ã) Minimize the sidelobe level in a prescribed region. ( b) Achieve a predetermined 3 db main-beam width. Assume that the main-beam is directed toward θ and the prescribed 3-dB angles are θ 1 and θ 2 (i.e., the 3-dB mainbeam width is θ 2 θ 1, with θ 1 < θ < θ 2 ). Let Ω s denote the sidelobe region of interest and {µ g } a grid covering it. Then the design problem of interest in this section can be mathematically formulated as follows: min t,r t s.t. a (θ )Ra(θ ) a (µ g )Ra(µ g ) t, µ g Ω s a (θ 1 )Ra(θ 1 ) =.5a (θ )Ra(θ ) a (θ 2 )Ra(θ 2 ) =.5a (θ )Ra(θ ) R ( ) C.8 N n = 1,, N, ( C R nn 1.2 N ), N R nn = C. (6 25) n=1 Note that the relaxed elemental power constraint in (6 25), while still quite practical, offers more flexibility than the strict elemental power constraint in (6 21). Note also that the total transmit power is the same for both (6 25) and (6 21), viz. C. In the examples below, we set C = 1. As shown in [76], this minimum sidelobe beampattern design problem is a semidefinite program (SDP) that can be efficiently solved in polynomial time. Once the optimal R has been determined, we can again use CA to synthesize the waveform matrix X. Consider first an example where the main-beam is directed toward θ = with a 3-dB width equal to 2 (θ 1 = 1 and θ 2 = 1 ). The sidelobe region is chosen to be Ω s = [ 9, 2 ] [2, 9 ], which allows for some transition between the mainbeam and sidelobe region. The same mesh grid size of.1 is used here. Figure 6-6 shows 91

92 the synthesized beampatterns obtained using the CA synthesized waveforms under the constraints of PAR = 1 and PAR 1.1. Note that the minimum sidelobe beampatterns obtained from the CA synthesized waveforms with optimal R are similar to those obtained from the CA synthesized waveforms with PAR ρ even for PAR = 1. We next consider a case with the same design parameters as in the above example except that now we also wish to place a 4 db or deeper null at µ n = 3. To do this, we add the following constraint to the minimum sidelobe beampattern design problem in (6 25): a (µ n )Ra(µ n ) 4 db, µ n = 3, (6 26) (the so-obtained problem is still a SDP). Figures 6-7(a) - 6-7(c) show the beampatterns obtained by using the CA synthesized waveforms under the constraints of PAR = 1, PAR 1.1, and PAR 1.2, respectively. For the CA synthesized waveforms with PAR ρ, the null depths at 3 for the three different PAR values shown in Figure 6-7 are db, db and db, respectively. Hence a stringent PAR constraint can have a significant impact on the null depth. For PAR 1.2, the beampatterns obtained with the CA synthesized waveforms with PAR ρ and, respectively, with the CA synthesized waveforms with optimal R are almost identical. Finally, consider an example where we wish to form a broad null over the region Ω n = [ 55, 45 ], where the power gain must be at least 3 db lower than the power gain at θ. To do this, we add the following constraint to the minimum sidelobe beampattern design in (6 25): a (µ n )Ra(µ n ) 1 3 a (θ )Ra(θ ), µ n Ω n. (6 27) Figures 6-8(a) and 6-8(b) show the beampatterns corresponding to the CA synthesized waveforms obtained under PAR = 1 and PAR 1.1, respectively. Similar remarks to those on Figure 6-7 can be made in this case as well. 92

93 6.4.3 Waveform Diversity-Based Ultrasound Hyperthermia In this final example, we consider an application of the waveform diversity-based transmit beampattern design to the treatment of breast cancer via ultrasound hyperthermia. Of all women diagnosed with breast cancer, 2% have locally advanced disease and even with aggressive treatments, the risk of distant metastases remains high. Thermal therapy provides a good treatment option for this type of cancer: the breast tumor is heated, and the resulting heat distribution sensitizes tumor tissues to the anti-cancer effects of ionizing radiation or chemotherapy [9, 91, 92]. Thermal therapy can also help achieve targeted drug delivery. A challenge in the local hyperthermia treatment of breast cancer is heating the malignant tumors to a temperature above 43 C for about thirty to sixty minutes, while maintaining a normal temperature level in the surrounding healthy breast tissue region. Ultrasound arrays have been recently used for hyperthermia treatment because they can provide satisfactory penetration depths in the human tissue. Note that the elemental power of an ultrasound array must be limited to avoid burning healthy tissue. As a result, a large aperture array is needed to deliver sufficient energy for heating the tumor without harming the healthy tissue. However, due to the short wavelength of the ultrasound, the focal spots generated by a large ultrasound array are relatively small and therefore hundreds of focal spots are required for complete tumor coverage, which results in excessively long treatment times. We have shown recently that flexible transmit beampattern design schemes can provide a sufficiently large focal spot under a uniform elemental power constraint, which can lead to more effective breast cancer therapies [84]. In the cited reference, the goal of the transmit beampattern design was to focus the acoustic power onto the entire tumor region while minimizing the peak power level in the surrounding healthy breast tissue region, under a uniform elemental power constraint. The beampattern design problem is 93

94 therefore to choose the covariance matrix R of the transmitted waveforms to achieve the following goals: (ǎ) Realize a predetermined main-beam width that is matched to the entire tumor region; in the said region the power should be within 1% of the power deposited at the tumor center; (ˇb) Minimize the peak sidelobe level in a prescribed region (the surrounding healthy breast tissue region). This problem can be mathematically formulated as: min t,r t s.t. a (θ )Ra(θ ) a (µ)ra(µ) t, µ Ω B a (ν)ra(ν).9a (θ )Ra(θ ), a (ν)ra(ν) 1.1a (θ )Ra(θ ), ν Ω T ν Ω T R R nn = C, n = 1, 2,, N, (6 28) N where θ is the tumor center location (θ is now a coordinate vector), and Ω T and Ω B denote the tumor and the surrounding healthy breast tissue regions, respectively. Once the optimal R has been determined, we use CA to synthesize the waveform matrix X under the constant-modulus constraint (PAR =1). We simulated a 2D breast model, as shown in Figure 6-9. The breast model is a semicircle with a 1 cm diameter, which includes breast tissues, skin, chest wall, and a 16 mm diameter tumor whose center is located at x = mm, y = 5 mm. There are 51 acoustic transducers arranged in a uniform array, as shown in the figure, with half wavelength (relative to the carrier frequency) inter-element spacing. The dots in Figure 6-9 mark the locations of the acoustic transducers. The sample number L is chosen to be 128. The finite-difference time-domain (FDTD) method [84] is used to simulate the power 94

95 densities and temperature distributions inside the breast model when the synthesized waveforms are transmitted via the acoustic transducers. Figure 6-1 shows the actual PAR values of the CA synthesized waveforms with optimal R. Figure 6-11 shows the temperature distributions within the breast model, with Figure 6-11(a) corresponding to the CA synthesized waveforms with PAR = 1 and Figure 6-11(b) to the CA synthesized waveforms with optimal R. As shown in Figures 6-11(a) and 6-11(b), by transmitting either of the synthesized diversity-based waveforms, the entire tumor region is heated to a temperature equal to or greater than 43 C, while the temperature of the surrounding normal tissues is below 4 C. In contrast with this, when a phased-array is used for transmission and the delay-and-sum technique is employed to ensure that the energy is focused on the tumor center, the temperature distribution is far from satisfactory (see [84]). 6.5 Concluding Remarks We have considered the problem of waveform synthesis for diversity-based flexible transmit beampattern designs. Optimization of a performance metric directly with respect to the signal matrix can lead to an intractable problem even under a relatively simple low PAR constraint. For this reason, we proposed the following strategy: first optimize the performance metric of interest with respect to the signal covariance matrix R; and then synthesize a signal waveform matrix that, under the low PAR constraint, realizes (at least approximately) the optimal covariance matrix derived in the first step. We have presented a cyclic optimization algorithm for the synthesis of a signal waveform matrix to (approximately) realize a given covariance matrix R under the constant-modulus constraint or the low PAR constraint. The output of the cyclic algorithm can be used to obtain either a waveform matrix whose covariance matrix is exactly equal to R but whose PAR is slightly larger than the imposed value, or a waveform matrix with the imposed PAR but whose covariance matrix may differ slightly from R the type of application will dictate which one of these two kinds of waveforms will be more useful. A number of 95

96 numerical examples have been provided to demonstrate that the proposed algorithm for waveform synthesis is quite effective. Appendix: On Enforcing Solely the PAR Constraint Consider the following generic form of the problem: min s s z 2 s.t. PAR(s) ρ, (6 29) where z is given and PAR(s) is as defined in (6 16). Hence we have omitted the power constraint (6 17), which should lead to a smaller matching error. Because PAR(s) is insensitive to the scaling of s, let us parameterize s as s = cx; x 2 = 1; where c is a variable. (6 3) Using (6 3) in (6 29) yields: s z 2 = cx z 2 = c 2 2cRe(x z) + const. (6 31) If Re(x z), then the minimum value of (6 31) with respect to c occurs at c =. If Re(x z) >, then the minimization of (6 31) with respect to c gives: c = Re(x z), (6 32) and the value of (6 31) corresponding to (6 32) is smaller than the value associated with c =. Because PAR(s) = PAR(x) does not depend on the phases of the elements of x, we can always choose x such that Re(x z) > so that we achieve a smaller value of (6 31). Consequently, the minimizing value c of (6 31) is always given by (6 32). The remaining problem is: or equivalently max x Re(x z) s.t. x 2 = 1 and PAR(x) ρ, (6 33) min x x z 2 s.t. x 2 = 1 and PAR(x) ρ, (6 34) 96

97 3 2.5 Desired CA: Optimal R CA: PAR = 1 Beampattern Angle (degree) (a) Desired CA: Optimal R CA: PAR 1.1 Beampattern Angle (degree) (b) which has the form required by the algorithm of [86, 87]. Therefore, we can solve (6 34) using the said algorithm and then compute s = cx with c given by (6 32). The alternative discussed in Sec. 6.3 is to constrain s 2 = z 2 (which is the case when we choose γ = R nn in (6 17)). The use of this constraint is logical if we want to match R nn exactly (for strict transmission power control, for example). However, if matching R nn exactly is not a necessary condition, then a smaller matching error between s and z is obtained using (6 32) and (6 34). 97

98 3 2.5 Desired CA: Optimal R CA: PAR 2 Beampattern Angle (degree) (c) Figure 6-1. Beampattern matching design with the desired main-beam width of 6 and under the uniform elemental power constraint. The probing signals are synthesized for N = 1 and L = 256 by using CA under (a) PAR = 1 (resulting in δ = db), (b) PAR 1.1 (resulting in δ = db), and (c) PAR 2 (resulting in δ = db) CA (PAR = 1): Optimal R CA (PAR 1.1): Optimal R CA (PAR 2): Optimal R Colored Hadamard PAR Index of Transmit Antenna Figure 6-2. PAR values for CA synthesized waveforms with optimal R and for colorized Hadamard code. 98

99 .3 Beampattern Difference Angle (degree) 5 (a).3 Beampattern Difference Angle (degree) (b) 99 5

100 .3 Beampattern Difference Angle (degree) (c) Figure 6-3. Differences between the beampatterns obtained from optimal R and the CA synthesized waveforms under (a) PAR = 1, (b) PAR 1.1, and (c) PAR 2. MSE (db) Sample Number L Figure 6-4. MSE of the difference between R and ˆR (CA synthesized constant modulus waveforms) as a function of sample number L obtained with 1 Monte-Carlo trials. ˆR is obtained from the CA synthesized constant modulus waveforms. 1

101 3.5 3 Desired CA: Optimal R CA: PAR = 1 Beampattern Angle (degree) (a) Desired CA: Optimal R CA: PAR 1.1 Beampattern Angle (degree) (b) Figure 6-5. Beampattern matching design with each desired beam width of 2 and under the uniform elemental power constraint. The probing signals are synthesized for N = 1 and L = 256 by using CA under (a) PAR = 1 and (b) PAR

102 15 1 CA: Optimal R CA: PAR = 1 Beampattern (db) Angle (degree) (a) CA: Optimal R CA: PAR 1.1 Beampattern (db) Angle (degree) (b) Figure 6-6. Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 2 and under the relaxed elemental power constraint. The probing signals are synthesized for N = 1 and L = 256 by using CA under (a) PAR = 1 and (b) PAR

103 1 CA: Optimal R CA: PAR = 1 Beampattern (db) Angle (degree) (a) 1 CA: Optimal R CA: PAR 1.1 Beampattern (db) Angle (degree) (b) 13

104 1 CA: Optimal R CA: PAR 1.2 Beampattern (db) Angle (degree) (c) Figure 6-7. Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 2 and a 4 db null at 3, under the relaxed elemental power constraint. The probing signals are synthesized for N = 1 and L = 256 by using CA under (a) PAR = 1, (b) PAR 1.1, and (c) PAR

105 1 CA: Optimal R CA: PAR = 1 Beampattern (db) Angle (degree) (a) CA: Optimal R CA: PAR 1.1 Beampattern (db) Angle (degree) (b) Figure 6-8. Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 2 and a null from 55 to 45, under the relaxed elemental power constraint. The power gain difference between and the null is constrained to be less than or equal to 3 db. The probing signals are synthesized for N = 1 and L = 256 by using CA under (a) PAR = 1 and (b) PAR

106 8 7 Acoustic transducer array 6 y (mm) Tumor Chest wall Breast x (mm) Figure 6-9. Breast model PAR Index of Acoustic Transducer Figure 6-1. PAR values for CA synthesized waveforms with optimal R. 16

107 C 8 y (mm) x (mm) (a) C 8 7 y (mm) x (mm) (b) Figure Temperature distribution for N = 5 and L = 128. (a): CA synthesized constant modulus signals, and (b): CA synthesized signals with optimal R). 17