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Knowledge-Aided Signal Processing

Permanent Link: http://ufdc.ufl.edu/UFE0022491/00001

Material Information

Title: Knowledge-Aided Signal Processing
Physical Description: 1 online resource (137 p.)
Language: english
Creator: Zhu, Xumin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: beamforming, knowledge, mimo, radar, stap, waveform
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

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. 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 is a poor estimate of the true covariance matrix. In KA adaptive beamforming, the a priori covariance matrix 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, the prior covariance matrix is often rank deficient. We consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix, the sample covariance matrix, and an identity matrix to get an enhanced estimate of the true one. Both approaches are fully automatic and they can be formulated into convex optimization problems that can be e?ciently 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xumin Zhu.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022491:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022491/00001

Material Information

Title: Knowledge-Aided Signal Processing
Physical Description: 1 online resource (137 p.)
Language: english
Creator: Zhu, Xumin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: beamforming, knowledge, mimo, radar, stap, waveform
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

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. 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 is a poor estimate of the true covariance matrix. In KA adaptive beamforming, the a priori covariance matrix 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, the prior covariance matrix is often rank deficient. We consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix, the sample covariance matrix, and an identity matrix to get an enhanced estimate of the true one. Both approaches are fully automatic and they can be formulated into convex optimization problems that can be e?ciently 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xumin Zhu.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022491:00001


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102bf6105d86445be16a40fd1775495272fdbb4f







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

2008

































2008 Xumin Zhu























To my parents









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 alvb--7

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 C'!i i:- 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.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF FIGURES .................................... 7

A BSTRA CT . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION ...................... .......... 12

1.1 Knowledge-Aided Space-Time Adaptive Processing .... ........ 12
1.2 Knowledge-Aided Adaptive Beamforming ............... 15
1.3 Knowledge-Aided Waveform Synthesis ........ ............ 16

2 LITERATURE REVIEW FOR KA-STAP ............ .......... 18

2.1 D irect U se . . . . . . . .. . 18
2.2 Indirect U se . .. . . . . . . .. 20
2.3 Contributions of Our Work ......... .......... ...... 21

3 ON USING A PRIORI KNOWLEDGE IN STAP ............... ..23

3.1 Introduction and Preliminaries .................. ..... .. 23
3.2 Minimum MSE Covariance Matrix Estimation .. . . ..... 25
3.2.1 Convex Combination (CC) .................. .. 25
3.2.2 General Linear Combination (GLC) ................. .. 28
3.3 Application to KASSPER Data .................. ... .. 30
3.4 Conclusions .................. ................ .. 34

4 MAXIMUM LIKELIHOOD-BASED KA-STAP ................. .. 38

4.1 Introduction and Preliminaries .................. ..... .. 38
4.2 Problem Formulation .................. ........... .. 40
4.3 Weight Determination of Prior Knowledge ............. .. 41
4.3.1 Convex Combination ............... ..... .. 42
4.3.2 Maximum Likelihood ............... ..... .. 44
4.4 Numerical Results ............... ............ .. 47
4.5 Conclusions ............... .............. .. 51

5 KNOWLEDGE-AIDED ADAPTIVE BEAMFORMING . . 58

5.1 Introduction .................. ................ .. 58
5.2 Problem Formulation ............ . . . ...... 60
5.3 Knowledge-Aided Covariance Matrix Estimation . . 61










5.3.1 MGLC and MCC .. ...........
5.3.2 Extensions .. .. ... .. .. .. .. ...
5.4 Using R for Adaptive Beamforming ........
5.5 Numerical Examples .. .............
5.5.1 Relatively Weak Interferences .. .....
5.5.2 Relatively Strong Interferences .......
5.6 Conclusions . . . . . .

6 WAVEFORM SYNTHESIS FOR DIVERSITY-BASED
PATTERN DESIGN .. ................


TRANSMIT BEAM-


6.1 Introduction . . . . . . . .
6.2 Formulation of the Signal Synthesis Problem .. ...........
6.3 Cyclic Algorithm for Signal Synthesis .. ..............
6.4 Numerical Case Studies .. .....................
6.4.1 Beampattern Matching Design .. ..............
6.4.2 Minimum Sidelobe Beampattern Design .. ..........
6.4.3 Waveform Diversity-Based Ultrasound Hyperthermia .
6.5 Concluding Rem arks .. .....................

7 MIMO RADAR WAVEFORM SYNTHESIS .. .............


Introduction ...... .............
Formulation of the Signal Synthesis Problem .
7.2.1 Without Time Correlation Considerations .
7.2.2 With Time Correlation Considerations .
Cyclic Algorithm for Signal Synthesis . .
Numerical Examples . ...........
Concluding Remarks . ...........


. . . . 0 8
.. . 09
. . . 110
. . 110
. .. . 112
. .. . 114
. .. . 118


8 CONCLUSIONS AND FUTURE WORK. ............


8.1 Conclusions . . . . . . .
8.2 Future Work ....... ... .......
8.2.1 KA Waveform Optimization for MIMO STAP .


. . 127
. . . 128
. . 128


8.2.2 Asymptotic ML Method for Optimal Weight Determination .

REFEREN CES . . . . . . . . .

BIOGRAPHICAL SKETCH . ........................


. 137









LIST OF FIGURES


Figure page

1-1 Angle-Doppler interference image for a single range bin . ...... 13

1-2 A general block diagram for a space-time processor. .............. 13

1-3 MIMO radar versus phased array .................. ....... .. 16

1-4 Reflected pulses from different range bins can overlap significantly. . ... 17

3-1 GLC estimates of ao and /0 as functions of range bin index. . 35

3-2 Estimation error ratio. ...................... . .... 36

3-3 Comparison of the ROC curves corresponding to the "ideal" detector, the "ini-
tial" detector, and the GLC detector. ............... .. .. 37

4-1 Log-likelihood function as a function of a. ............... .. .. 55

4-2 CC and ML estimates of a as functions of the range bin index. . ... 56

4-3 Comparison of the ROC curves corresponding to the "ideal" detector, the "prior"
detector, the "CC" detector, the \!I," detector, the S; l" detector, and the
"equal-. i;,ii detector. . . . . .. .......... .. 57

5-1 Averaged MGLC1 estimates of A, B and C versus the snapshot number N. 74

5-2 Averaged MGLC2 estimates of A, B and C versus the snapshot number N. 75

5-3 SINR versus the snapshot number N when weak interference present and for
cases i iv .................. ................... .. 76

5-4 Averaged MGLC2 estimates of A(l/l2a, A(2)/oJ, B and C versus the snapshot
number N .................. ................... .. 77

5-5 SINR versus the snapshot number N when weak interference present and for
cases v vi .................. ................... .. 77

5-6 SINR versus the snapshot number N when strong interference present and for
cases i iv .................. ................... .. 78

5-7 SINR versus the snapshot number N when strong interference present and for
cases v vi .................. ................... .. 79

6-1 Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. ................. 98

6-2 PAR values for CA synthesized waveforms with optimal R and for colorized
Hadamard code. .................. ................ .. 98









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.. 100

6-4 MSE of the difference between R and R as a function of sample number L. 100

6-5 Beampattern matching design with each desired beam width of 200 and under
the uniform elemental power constraint. ................ ..... 101

6-6 Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and under the relaxed elemental power constraint. ........... ..102

6-7 Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and a -40 dB null at -300, under the relaxed elemental power constraint. 104

6-8 Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and a null from -550 to -450, under the relaxed elemental power con-
straint .. . . . . . . . . .. 105

6-9 Breast model .................. .................. .. 106

6-10 PAR values for CA synthesized waveforms with optimal R. . . .... 106

6-11 Temperature distribution for N = 50 and L = 128. .............. ..107

7-1 Beampattern matching design. The probing signals are synthesized for N = 10,
L = 256 and P = 1 by using CA. ............... ...... 119

7-2 Correlation levels versus p for the CA synthesized waveforms with N = 10, L
256 and P = 1 (R / I) ................ ........... .. 120

7-3 Beampattern matching design. The probing signals are synthesized for N = 10,
L = 256 and P = 10 by using CA ................ .... .. .. 121

7-4 Correlation levels versus p for the CA synthesized waveforms with N = 10, L
256 and P = 10 (R / I) ................ .......... 122

7-5 Beampattern matching design. The probing signals are synthesized for N = 10,
L = 512 and P = 1 by using CA. ............... ...... 123

7-6 Correlation levels versus p for the CA synthesized waveforms with N = 10, L
512 and P = 1 (R / I) ................ .......... 124

7-7 Beampattern matching design. The probing signals are synthesized for N = 10,
L = 512 and P = 10 by using CA ............... ...... 125

7-8 Correlation levels versus p for the CA synthesized waveforms with N = 10, L =
512 and P = 10 (R / I) ................ .......... 126

7-9 Correlation levels versus p for the CA synthesized orthogonal waveforms with
N = 10, L = 256 and P = 10 (R I). .................. .... 126









7-10 Correlation levels versus p for the CA synthesized orthogonal waveforms with
N = 10, L = 512 and P 10 (R = I). .................. .. 126









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

By

Xumin Zhu

May 2008

C'!i ri: Jian Li
Major: Electrical and Computer Engineering

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 (\ MI\ 0O) 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 1.i, i-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 prior 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 prior 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.









Next, we consider applying KA signal processing algorithms for adaptive beamform-

ing. In array processing, when the available snapshot number is comparable with or even

smaller than the sensor number, the sample covariance matrix R is a poor estimate of the

true covariance matrix R. In KA adaptive b, ,ivnr. i in-ir,- the a priori covariance matrix

Ro usually represents prior knowledge on dominant sources or interference. Since the

noise power level is unknown, and thus cannot be included into the a priori covariance

matrix, Ro is often rank deficient. We consider both modified general linear combinations

(ILGLC) and modified convex combinations (\!CC) of the a priori covariance matrix

Ro, the sample covariance matrix R, and an identity matrix I to get an enhanced esti-

mate 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 prior

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.









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 interference 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 ri~, (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 E CMX1, with

M = SP being the degrees of freedom (DOFs) of STAP, used to maximize the output




















0
sine(azimuth)


Figure 1-1. Angle-Doppler interference image for a single range bin.


S Antennas
P pulses

Th-rl


P pulses

lfi


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-a, (1-1)

where R E CMXM is the true clutter-and-noise covariance matrix for that particular

range bin of current interest, and a E CMxI 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)}j 1, associated with the range bins close to the range









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 R can then be obtained by the well-known formula (see,

e.g., [5, 6, 7, 8]):

R 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 pre-

whitening) to enhance the performance of STAP (see, e.g., [8, 10, 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 Ro.









This can be obtained either by previous radar probings or by a map-based study. In this

dissertation, assuming that Ro 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-and-

noise covariance matrix R based on the a priori clutter covariance matrix Ro and the

sample covariance matrix R. 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 pro'. --iir 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-interference-

plus-noise ratio (SINR) adaptively. However, when the available snapshot number is

comparable with or even smaller than the sensor number, the sample covariance matrix

R 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 prior covariance matrix Ro. In practice, Ro usually represents prior knowledge on

dominant sources or interference. Unlike in some literature, for example, see [20, 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 Ro, hence Ro is often rank deficient. In the dissertation, we consider

both modified general linear combinations (I GLC) and modified convex combinations

(\!CC) of the a priori covariance matrix Ro, the sample covariance matrix R, 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










Targets
* /


C o m b i n a t i o n o f f{ n ( t ) } 1 2 P h
N(T) MIMO Receive Array N ( 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 R 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 prior 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










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 Return pulse
from near range from scene center


Return pulse
from far range


Fast time


Figure 1-4. Reflected pulses from different range bins can overlap significantly.









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 de-

tection 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-,I' .:1/,/ a prior

knowledge directly in space-time adaptive beamformer. The linearly-constrained minimum

variance (LC'\ V) space-time beamformer proposed in this paper includes an additional

a prior knowledge-based constraint to force nulls at the clutter locations. The authors

showed that the knowledge-aided constraint resulted in "(o .1. i. loading of the adaptive

covariance matrix estimate, which can be expressed as


RCL = R + Ro + /3ld, (2-1)

where RCL is the colored loading covariance matrix and f3 and 3d, 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, applica-

tion of the colored loading approach to both full DOF and reduced DOF beamformer is

considered and the performance is demonstrated by using I,.:il,-I/7, 1. :/; 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









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 Ro. 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 [10]. 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 Ro. The method proposed in the cited paper combines the following

techniques: covariance tall, iiir- which is used to compensate for internal clutter motion

(IC'\ ); adaptive correction of channel mismatches; colored loading as described in [23],

and eigenvalue scaling (this is based on the assumption that RCL (see (2-1)) can provide

better space-time response, and hence RCL 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 Ro 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









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 Ro obtained by KACE is combined with the sample covariance matrix R as

Ro + R 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

prior 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 IC'\

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 data-

adaptive 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 [30], 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.









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 benefits 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 ('!, lpter 4), and the look-ahead radar scheduling approach for real-time

KA-STAP are discussed.

In [18], and also in [17], the authors first discuss 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 (D3) 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 Ro from the available environmental knowledge [13, 11, 10, 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 Ro. 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 pli, vI!i, iii, approach proposed in [8] -,--, --1; to optimize the weight on Ro

so as to maximally whiten the observed interference data. However, as we will show in

C'!i lpter 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 v--,v to enhance the









detection performance of STAP. The performance of the algorithms are evaluated using

the KASSPER data.









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 J- {y(n)} 1, by

means of the well-known formula (see, e.g., [5, 6, 7, 8, 16]):
1 N
R =y(n)y*(n), (3-1)

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 R 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 ,- Ro, is

obtained in this way (see, e.g., [8, 16]). The problem is then to "(I 1iidoii, R and Ro into

an estimate of R, preferably much more accurate than both R and Ro.

In this note, we will consider a "linear combination" of R and Ro (see, e.g., [34, 11,

13]):

R = Ro + 3R; a> 0 and 3 > 0. (3-2)

Because a and 3 are constrained to be positive, and as typically Ro > 0 (positive definite),

we have that R > 0, which is a desirable feature. We will also consider the following

"convex combination":

R cRo + (1 a)R; a (0,1). (3-3)

The constraint in (3-3) on a is imposed, once again, to guarantee that R > 0. In general,

there is no obvious reason why 3 in (3-2) should be constrained to equal 1 a, like in

(3-3); however, the convex combination, (3-3), is a more parsimonious description of R









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 a and f that minimize the mean-squared

error (\! S I) of R:

MSE= E{|R- RI2}, (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 ao and /0 denote the optimal values of a and 3 that minimize (3-4). The second

goal of this note is to discuss how to obtain estimates, do and /o, of ao and /0 from the

available data (as we will see shortly, and as expected, both ao and /o 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 a = do and / = )o, 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 pro-

posed 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 applica-

tions 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 Ro matrix ([34] considers the case of Ro = I only). Also our proofs are more

explicit than those in [34], despite the more general, complex-valued data case we consider









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 pa-

per [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)}j 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)}j 1, such as the usual Gaussian assumption that would not be warranted in STAP.
Also, we assume that Ro is a fixed (non-random) matrix. However, if we have reasons to

assume that Ro 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 I|Ro RI| or the MSE E{I|Ro R1|2}; because in KA-

STAP such a knowledge is usually unavailable, we will focus on the case of non-random Ro

here.

3.2.1 Convex Combination (CC)

For (3-3), a simple calculation yields:

E{||R- R|2} = E{Ia(Ro-R)+(R-R)|2}

const + 2E{||Ro R||2} 2Re {tr LE{(R R)(R Ro)* }

const + a2E{||Ro RI|2} 2aE{|R R1|2}, (3-5)

where tr denotes the trace, Re stands for the real part, and where we have used the fact

that R is an unbiased estimate:

E{R} = R. (3-6)

The unconstrained minimization of (3-5) with respect to a gives:

E{I|R- R|2} E{|| R- |2}
CQ0 = 2 E{ R2 (3-7)
E{||lR-R R } E{||R-R R }+||2 IR-Rol2









Because co above belongs evidently to the interval (0, 1), it is also the solution to the

constrained minimization of the MSE. Observe the intuitive character of the expression for

co in (3-7): when R is much closer to R than to Ro (in a MSE sense), then co is close to

0, and viceversa.

To estimate co from the available data, we need an estimate of E{IR R |2}. Let ,r

and r, denote the mth columns of R and R, respectively. Consequently, we have


rm = Ny ()i(),


(3-8)


and


rm= E{y(n)y*(n)},


(3-9)


where ym(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 rm and r, above (we will reinstate this

index later on, when needed). Let


(3-10)


It follows from the assumptions made on {y(n)}j that {x(n)} ,1 are i.i.d. random

vectors with mean t = r. Because:


E{||R R||2}


our generic problem is to estimate E{IIr

variance of the sample mean of {x(n)}



S N 2


2 N 2}
- r|2} = E Yx(n) (i.e., the
n=
1). A simple calculation gives:

t N 2
n=1
1NN
NE E {[x(n) i]*[x() p}
n=l n=l 1
1
NE{ x(n) A112}. (3-12)
N


M
ZE{m
mr 1


r, II- ,


(3-11)


x(n) = y(n)y*{(n).









The variance Ef{|x(n) p, 2} in (3-12) can be estimated as:


1 N 1 N
N N (313)
n~l n~l

It follows that we can estimate E{||rm r., I } as

N
-N2 y(n)yr (n) r|-. (3-14)
n=l

Let

p E{|R- RI2}. (3-15)

Using (3-14) as an estimate of E{||rm r, I||- leads to the following estimate for p:

N M

N

N2 y(n)y*(yn)- R12 (3 16)
n=l

The above expression for p has a certain theoretical appeal, but its direct calculation may

be somewhat slow. A more attractive expression for p, from a computational standpoint,

can be obtained by observing that

1 M 1 N
1P N N ly(n)Y)( n) .1
rnm= 1 n= 1
M N
N Y N I [ i y(n) (l ) In 12 l, K1 (3 17)


and therefore that

1 Rly12 (3-18)
N2 y

Using p as an estimate of p and I|R Ro||2 as an estimate for I|R Ro 2 yields the

following estimate of co in (3-7):


o' = (319)
p+ R -Ro1 2








Alternatively, we can use the unbiased estimate I|R Ro012 for E{IIR Rol2} in (3 7),
and the resulting estimate of ao:

ao" (3-20)
|R- Ro0l2
Unlike (3-19), the estimate of ao in (3-20) is not guaranteed to be less than 1, as required;
thus we recommend using min(1, do") in lieu of (3-20).
3.2.2 General Linear Combination (GLC)
For (3-2), we have that:

E{ ||R-R|2}I E{|jRo (1 -3)R+3 (R R)I|2}

lRo (1 )RI2 + f 2E{lR- R112}

2 11Ro112 2a(1 3)Re [tr(R*R)] + (1 3)211R 2

+/2E{i|R R||2}, (3-21)

where Re (the real part) can be omitted because tr(R*R) > 0. The unconstrained
minimization of (3-21) with respect to a, for fixed 3, gives:

(1 /3o) tr(R R)
ao (322)

where so is the minimizer of the function that is obtained by inserting (3-22) (with 9o
replaced by 3) in (3-21), viz.:

(3 1)2 [IIRII211Roll2 tr2(R;R)]
( t)2 [R o 2 2E{IR R1122}. (3 23)
IIR oll2

The unconstrained minimization of (3-23) with respect to f3 yields:

A0o (3-24)
P+7

where p has been defined previously in (3-15) and

IIRII2R112- t2(RR)
S= (3-25)
JIRo 112









By the C ,i h!-Schwartz inequality, we have that 7 > 0. This observation implies that:


0o e (0,1). (3-26)

Furthermore, (3-26) along with the fact that tr(R*R) > 0 means that ao > 0. Therefore,

ao and /0 above are also the constrained minimizers of the MSE.

It is interesting that /o above belongs to the interval (0, 1), like in the CC case. How-

ever, ao above does not necessarily belong to (0, 1), unlike in the CC case. Additionally,

observe that as /o approaches 1, ao approaches 0 (see (3-22)), again like in the CC case!

However, here ao + so may well be different from 1 the constraint a + 3 = 1, which leads

to the CC case, is a bit hard to motivate in general, as already mentioned in Section 3.1.

Let
tr(R R)
V= (3-27)
IIRo 11
A simple calculation shows that:

R RI2 tr2(R.R) 2tr2(R)
IIv1Ro R112 lr 2 2 7. (3-28)
IIRo 1 12 IIRo12

Using this fact, we can estimate ao and 3o as follows:


o' = (3-29)

and

a0o' v(1 -o'), (3-30)

where p is given by (3-18),

IIRo- R2, (331)

and
tr(RfR)
S- tr (3-32)
IIR o 2ll









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:


7 +p IRo R112+ E{iR- R1|2} E{|Ri- VRo12}, (333)

an estimate of which is given by I|R Ro l2. Consequently, we can also estimate co as:


ao" = (3-34)
IIR- yRo012'

and 3o as

o" 1 (3-35)

However, the estimate of jo in (3-35) is not guaranteed to be positive, unlike (3-29). To

guarantee that 3o" > 0, we can either replace (3-34) by min(&o", v) or use max(3o", 0) in

lieu of (3-35). Following a sIt---- -1 i. o 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 (3-20)), GLC' (see (3-29), (3-30)), 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 -I ... --time steering; vector

corresponding to a target with spatial frequency w, and (normalized) Doppler frequency









wD was a slightly perturbed version of the following M x 1 "nominal" steering vector:


a(w), D) = a(w) d-(Un), (3-36)


where M = PS = 352, 0 denotes the Kronecker matrix product,
F iT
a( ) 1 ejw ... j(s-1)O (3-37)


and
T
a(wD) C JWD ... j(P ) -) (3-38)

with (.)T denoting the transpose. The spatial frequency w, has a fixed value corre-

sponding to steering towards 750 relative to the array normal, but the Doppler fre-

quency wD is only known to belong to the following set of 64 possible Doppler values

{0, 27/64,. 1267/64}. The number of range bins considered is equal to 1,000; of these

784 range bins contain no target, whereas the rest of the 216 bins contain at least one

target (at certain values of wD). 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 1,000 range bins, the nominal steering vectors, and a perturbed

version of R (that mimics the uncertainty in the a priori knowledge that alv--,i exists in

practice). The said perturbed clutter-and-noise matrix, Ro, is generated in the following

way:

Ro = R (tt*, (3-39)

where ( denotes the Hadamard matrix product, and t is a vector of i.i.d. Gaussian ran-

dom variables with mean 1 and variance of. Note that the perturbed matrices generated

in this way remain positive definite (we have also tried two other forms of perturbation of









R to obtain Ro, 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* (s, oD)Ru -ly H1
(340)
a* (o, WD)R -la( ,W) Ho

where Ho is the null hypothesis (i.e., no target), H1 is the alternative hypothesis (i.e.,

Ho is false), is a threshold, and R is an estimate of the clutter-and-noise covariance

matrix, R, for the range bin in question. The threshold 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 64,000 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 R 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" (3-40) with R = Ro 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 = 50 range bins from B whose covariance matrices Ro's are the closest (in

the Frobenius norm sense) to the covariance matrix Ro associated with the range bin of

current interest; let {y(n)}j 1 denote the secondary space-time snapshots corresponding to

the selected range bins from B; then, we obtain R using the standard formula in (3-1). (It

is interesting to observe that the above procedure for computing R makes also use of the a

prior information embedded in Ro's).









Figure 3-1 shows the GLC estimates of ao and 0o obtained for different range bins

and for four values of ao (see (3-39)). As expected, )o increases and do decreases as ao

increases. The smaller variability of ao and )0 observed in the figure for the larger values

of ao can be explained by the fact that these estimates approach 0 and, respectively, 1 as

f increases. Figure 3-2 dip-~'i the estimation error ratio

min1 Ilt R12 (3
(3 41)
min.2 IH2R- R112

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 (3-40) 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 R when ao takes

on small values (i.e., Ro 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 R even for a relatively

large perturbation level -see Figure 3-2(d), corresponding to af = 0.5, where (3-41)

fluctuates around 0.9.

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., (3-40) with R = R), the

"initial detector" (i.e., (3-40) with R = Ro), the sample matrix inversion (SMI) detector

(i.e., (3-40) 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 ao. Observe

also that, as at 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 Ro.









Finally, note that even when the initial detector has a performance close to that of "coin

-.1-- ;_, (see Figure 3-3(d)), the GLC detector has a reasonable performance -a fact

which -i--.- -r that the GLC detector is robust to the selection of the set B of I i -' .-

free" range bins.

3.4 Conclusions

In this ('!i Ipter, which extends the approach of [34] to complex-valued data and to

arbitrary "initial guess" matrices Ro, 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 R and Ro.

While the said method is ,-..i. l /. 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 R by carefully tuning the

weights of Ro and R in R = cRo + /3R according to the i I ive distances" of Ro and

R 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 Ro-based detector in

the practically interesting case of a relatively unreliable Ro.




































200 400 600 800 1000


200 400 600 800 1000


Figure 3-1. The GLC estimates of ao and Po as functions of range bin index, for four
different values of the perturbation level applied to R to obtain Ro: (a)
a2 0, (b) a = 0.01, (c) 2 = 0.1, (d) 2 = 0.5.


-0





















0.4 0.4


o.2I ll N it 1 0.2l
200 400 600 800 1000 200 400 600 800 1000
(a) (b)



0.8 1 0.8
0.6 0.6
0.4- 0.4-
0.2- 0.2-

200 400 600 800 1000 200 400 600 800 1000
(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 Ro: (a) of 0, (b) o = 0.01,
(c) 1a 0.1, (d) a? =0.5.





























(a) (b)


PFA PFA
(c) (d)
Figure 3-3. Comparison of the ROC curves corresponding to the "ideal" detector (with
R R), the "initial" detector (with R Ro), the SMI detector, and the GLC
detector, for four different values of the perturbation level applied to R to
obtain Ro: (a) a 0, (b) a = 0.01, (c) a = 0.1, (d) a = 0.5.









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 x P data matrix for a

certain range bin is then stacked column-wise to form an SP x 1 vector, which is called

a space-time snapshot. It is well-known that the optimal weight vector, w E CMX1, with

M = SP being the degrees of freedom (DOFs) of STAP, used to maximize the signal-

to-clutter-and-noise ratio (SCNR) of the beamformer output of this snapshot is given by

[4]:

w = R-la(w, ,D), (4 1)

where R E CMXM is the true clutter-and-noise covariance matrix for that particular

range bin of current interest, and a(w ~D) E CMxl denotes the steering vector, which

is a function of the spatial frequency ua and the (normalized) Doppler frequency wD 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 R is estimated from the training

or secondary data, {y(n)}j i, 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 yi below to denote the snapshot for the ith range bin, and y(n) to denote the

nth seco,:J.li snapshot for a particular ROI. The sample covariance matrix can then be









obtained by the well-known formula (see, e.g., [5, 6, 7, 8]):


1 N
R ^ i y()y*(n), (4-2)
n=l

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 [40, 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 prior 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, 10, 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









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 prior

knowledge and how much emphasis one should place on it. To address this problem, in

this paper, we propose a maximum likelihood (\l I) 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 -i Ro, is available. Ro can be obtained either by a previous scanning of the same

region or by a map-based study (see, e.g., [8, 10, 11, 12, 23]). This a priori Ro can then

be incorporated into STAP to improve its clutter suppression capability. As in [8], we

consider a "convex combination" of R and Ro, which corresponds to the direct use of prior

knowledge, as follows:

R caRo + (1 a)fR; a E(0, 1), (4-3)

where a is the weighting factor on the a priori knowledge, and the constraint on a is

imposed to guarantee that R > 0. This is also known as the "colored loadiin approach

[23]. However, in [23], the "(o !. ,I [ and "diagonal" loading levels were chosen manually.

Similarly, the authors of [15] considered equal weighting of R and R as a direct use of

prior knowledge: R = 0.5Ro + 0.5R (we refer to this as the "equal-,., i,!i1 approach









herein). Our goal is to devise an algorithm that can adaptively adjust the weighting factor

a, 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 a in (4-3) can be devised, depending on the

degree of accuracy of the a prior covariance matrix Ro.

4.3 Weight Determination of Prior Knowledge

A i.--l!tenini; approach for solving the problem of optimal weight determination

was proposed in [8], where the authors determined the weighting factor a of the a prior

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:

min Z,(a), (4-4)
a

where

Z,(a) = ys- I (45)

with 1- II denoting the Frobenius norm, and


yi = -1/2(i)y; R(i) QRo(i) + (1- a)R(i). (4-6)

In (4-6), yi denotes the space-time snapshot for the ith ROI; R1-1/2() is the whitening

matrix for that particular range bin with (.)-1/2 denoting the inverse square root of a

matrix, and yi is the whitened space-time snapshot; in (4-5), the summation is done over

a set of (target-free) snapshots of length I, and IM E CMXM 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

I !Litenin; 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)









where C(i) E C MX is a square root of R(i), and Q(i) E CMXM is an arbitrary unitary
matrix satisfying Q(i)Q*(i) Q*(i)Q(i) IM. Consequently the whitening matrix can be
expressed as:
R-1/2() Q*()C-1(i). (4 8)

Since Q(i) is an arbitrary unitary matrix, R-1/2(i) in (4-8) is not unique. The non-
uniqueness 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),

Zi(a) Q*(1)C- (1)yly* (C*())- Q(1) -Q*(1)Q(1)

C- (1)yly(C*(1))-1 iM (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 v! 1i. iii::, and R(i) is dependent on
i, the cost function in (4-5) becomes


Z,(a) = Q(i)*C- (i)yy*(C*(i)) Q(i) IM (4-10)

Note that the value of the cost function ZI(a) depends on the choice of {Q(i)}. Conse-
quently, different choices of the square root matrix may yield rather different values for a,
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].
4.3.1 Convex Combination

In the CC approach [48], a in (4-3) is allowed to vary with the range bin index. Be-
low, we omit the dependence of R, Ro, R, R and a on the range bin index for notational
simplicity. The CC approach considers a convex combination of R and Ro (see (4-3)) to









obtain an optimal estimate of R so as to minimize the MSE of R:


MSE= E{|R- RI2}.


(4-11)


We denote the optimal value of a obtained by the CC approach as acc.

Combining (4-3) and (4-11) and using the fact that R is an unbiased estimate of R

yields:


E{ |a(Ro- R) + (R- R) |2}

const + 2E{ Ro RI|2} 2aRe {tr [E{(R-

const + a2E{lRo RI|2} 2aE{|IR R||2},


R)(R -Ro)*}]

(4-12)


where tr denotes the trace, and Re stands for the real part. The minimization of (4-12)

with respect to a gives:


E {IIR- R1|2} E{IIR R112
a'" ceE = {-- R R2} .-
cc E{||IR-Ro||2} E{||IR-R||2}+ IIR-R 2



p E{i|R-R R2}.


(4-13)



(4 14)


To estimate co from the available data, we need an estimate of p. Note that


M
P Ef,


r, II ,


(4-15)


where


and


(4-16)


SN
rm= N y y() (n),
nrl


rnm = E{y(n)y*(n)},


(4-17)


E{I|R- RI|2}









denote the mth columns of R and R, respectively, with ym(n) being the mth element of

y(n). It has been shown in [48] that E{ Ir, r, ||-} can be estimated as

SN
N2 Iy(n)y(n) r. | (4 18)
n=l

It follows from (4-15) and (4-18) that

1M N

m= 1 n= 1
M N- t
m=l n=l

N N y(n) L', (n) 2 I 1 (4 19)


and therefore that
1 1
NN
S N2 ||y(u)||4 [R[2. (4-20)
n=l
Using (4-20) and the unbiased estimate ||R Ro 2 for E{ R Ro 12} in (4-13) yields the

following estimate of acc:

ace = (4-21)
||R R0o 2
To ensure that the estimate of acc in (4-21) is less than 1, as required, we recommend

using min(1, &cc) in lieu of (4-21).

We refer the reader to [48] for the detailed derivation of CC. Note that the CC ap-

proach is fully automatic and has a simple closed form solution which is computationally

efficient.

4.3.2 Maximum Likelihood

In this subsection, we introduce an ML approach to determine an optimal value

of a by assuming that a in (4-3) is constant for all range bins (for the sake of reduced

computational complexity). We denote the optimal value of a obtained by the ML

approach as &ML.

Since Ro, R, 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) = aRo(i) + (1 a)R(i) be the estimate of









the true clutter-and-noise covariance matrix R(i) for the ith range bin of interest (ROI).
As already mentioned, a 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 {yj}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 prior covariance matrices Ro's are the closest (in the Frobenius norm sense) to the

covariance matrix Ro(i); let Bi denote the range bin set containing the selected bins for

the ith ROI; and let E7 index the set Bi; then, the secondary training snapshots {y(n)}N 1

for the ith ROI in (4-2) can be chosen as {yn}ne,; consequently, R(i) obtained with SDS

can be computed as:

R(i) Z yny* (4-22)
nETi
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 a. Assume

that {y}$ 1, are independent identically distributed (i.i.d.) with the following distribution:


yi ~ Cn'(0, R(i)); R(i) aRo(i) + (1 a)R(i), (4-23)

where CA'(0, R(i)) denotes the circularly symmetric complex Gaussian distribution with

mean zero and covariance matrix R(i). The log-likelihood function of {yi}i is then

proportional to:

C = {ln 1 (i)(i )yi}
i=

Iln aRo(i) + (1 )R(i) + y aRo(i) +(1 )R(i)] yi (4-24)
i= 1( -









where I I denotes the determinant of a matrix. Our problem is that of finding the optimal

&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 &ML can be found by a search method (see,

e.g., [49]). With the optimal &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 I | and

(*)-1 operations for R(i) E CMxM in (4-24), which are required for each i and each

candidate a. We present in the Appendix a fast algorithm that can be used to reduce the

computational complexity significantly for KA-STAP.

In ir ini, 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 trans-

form matrix, T E CMXD, 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
after the transformation. In the numerical examples, we consider a multi-bin element-

space 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, v the

pth Doppler bin, we can write the transform matrix T as:


T = Fp Is, (4-25)


where Fp is a P x Q matrix whose columns are discrete Fourier transform filters at Q

.ilIi i.:ent Doppler frequencies with the pth Doppler bin as the central frequency bin, and

0 denotes the Kronecker matrix product. The resulting snapshot for the ith range bin,









is given by T*yi, which we still refer to as yi, for notational simplicity. Then all of our

previous discussions apply to the RD case.

Note that the optimal &ML in the ML approach for KA-STAP is constant for all

range bins. We have also tried to find optimal &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 &ML, whereas the computational

complexity was greatly increased. Consequently, the case in which &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 '02 dataset [35]. The simulated airborne radar system has P = 32

pulses and S = 11 spatial channels, yielding M = PS = 352 DOFs. The mainbeam of the

radar is steered to an azimuth of 1950 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 = 1000 range bins are sampled covering a range swath of interest from 35

km to 50 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:


Yi R1/2(i)Vi, i 1,-. ,I, (4-26)


where (.)1/2 denotes the Hermitian square root of a matrix, and {vl} E CMxl are i.i.d.

circularly symmetric complex Gaussian random vectors with mean 0 and covariance

matrix I.









The a priori covariance matrix, Ro(i), is constructed as a perturbed version of

the true R(i) (to mimic the uncertainty in the a priori knowledge that alv--, exists in

practice):

Ro(i) R(i) O tit, (4-27)

where 0 denotes the Hadamard matrix product, and ti is a vector of i.i.d. Gaussian ran-

dom variables with mean 1 and variance af. 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 uw and (normalized) Doppler frequency CD can be expressed as follows:


a(,, D) a(ws) a(wD), (4-28)

where
F iT
a(a;,) 1 w ... te(s-t1>) (4-29)

and
T
a(wD) WD ... j(P-)WD (4-30)

with (.)T denoting the transpose. The spatial frequency us has a known value for

a given azimuth angle, but since the target velocity is unknown, the Doppler fre-

quency UD is only known to belong to the following set of 32 possible Doppler values

{-327/32, -307/32, ... ,307/32}. Thus, there are totally L x K = 1000 x 32 = 32000

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 50 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 a~(i) denote the power

of the test target at the ith range bin. Define the signal-to-clutter-and-noise ratio (SCNR)









for the ith range bin as
tr [o(i)a(u, D (4 31)
tr [R(i)]
where wD0 is the Doppler frequency bin in which the test targets are inserted. Hence the

average SCNR over I = 1000 range bins can be calculated as

1 tr [1o(i)a(, wD0)a* ( ) (4 32)
I 1 tr [R(i)]

In our numerical illustrations, we choose the training sample number as N = 50,

which is much less than the full DOFs, M = 352, of the simulated radar system. Hence

R(i) is severely rank deficient. We consider a multi-bin post-Doppler STAP using three

.,.i ,i:ent 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 di- p-, the log-likelihood function in (4-24) as a function of a for four

values of of. The optimal weighting factors obtained by the ML approach for the four

different perturbation levels are listed in Table 4-1. As expected, &ML decreases as of

increases. Observe that aML = 0.8 when of = 0, and &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 a for different values of of.
of 0 0.01 0.1 0.5
&ML 0.80 0.35 0.08 0.014


Figure 4-2 shows the CC estimate &cc as a function of the range bin index, and the

ML estimate &ML, for the four values of of It is clear that &cc decreases as af increases.

The average values of &cc's over the 1000 range bins, denoted as acc, for the different

perturbation levels are given in Table 4-2.

Table 4-2. Average values of the CC estimates of a for different values of of.
2
of 0 0.01 0.1 0.5
acc 0.343 0.077 0.004 0.0002









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 at increases, as it should. Hence the

weighting factor of the prior knowledge determined by either the ML approach (aML) or

by the CC approach (ac,) 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:
2
a*(wo, uD)R -(i)y H1
aD --, ^(4-33)
a*(w, WD)R-1(i)a(w D) Ho

where Ho is the null hypothesis (i.e., no target), H1 is the alternative hypothesis (i.e., Ho is

false), ( 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 In our simulations, 1000 test targets are inserted in the 21st

Doppler bin and the average SCNR of the targets is -8.34 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 "e1,1 ,!- '.v, i!i detector, and to the "prior" detector, where only

prior knowledge is used for target detection. The "ideal" detector refers to using the true









clutter-and-noise covariance matrix. The sample covariance matrix inversion ( S\!l")

detector is obtained by using the sample covariance matrix, where no a prior 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 a and they give better performance than the SMI detector for the four values of

of. As compared to the ML and equal-weight detectors, CC results in poorer performance,

which is due to some of the occ(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 a 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 ao 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 a. 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 prior 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









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

prior 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 a 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 R and Ro on the range bin index is omitted in this Appendix for

notational simplicity. Let the columns of the matrix Y E CMXN contain the secondary

training snapshots for the ROI. The sample covariance matrix in (4-22) can be re-written

as

R YY*. (4-34)
N
Note that


aRo + (1 a)R aRo [IM + laRo1YY*

S aRo0 IN+(Y*Ro Y

oM Ro IN +(GI, (4-35)


where (assuming a > 0)

= (436)
aN








and


G = Y*RoY. (4-37)

Let the eigen-decomposition of G be given by

G UAU*, (4-38)

where the columns of U are the eigenvectors of G, and A is a diagonal matrix formed by
the eigenvalues of G. Then, (4-35) becomes:

aRo + (1 a)R aM |Ro| IN + (A|. (4-39)

Note that for each range bin index, as a varies, we need only to update IIN + (A|, which
is computationally convenient.
Next, we consider computing R-1. It follows from (4-34) and using the matrix
inversion lemma that:

[aRo + 1 a)Rl aRo + YY*j1

1 {Rol (R lY [I + Y*RY] Y*Ro1}

S{Rol -RoY [IN + (G]- Y*Ro}

S{Ro Ro1YU [IN + A]- U*Y*R-1 }, (4-40)

Let
P = RO1YU. (4-41)

Then (4-40) becomes


[ -( 1
aR + (1 a)R 1 -{Ro P [IN + A] P*} (4-42)

As we can see, |Ro R-1 and P in (4-39) and (4-42) are not related to a, and hence they
only need to be calculated once for each range bin index. Furthermore, IN + (A is an
N x N diagonal matrix (usually N < M), whose I I and (.)-1 can be calculated easily.









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 R(i) and Ro(i), which corresponds to the
direct use of prior knowledge, as follows:

R(i) = (i)Ro(i) + (1- a(i))R(i); (i) e (0,), (4-43)

where a(i) is the weighting factor on the a priori knowledge for the ith range bin, and the
constraint on a(i) is imposed to guarantee that R(i) > 0.
Note that the covariance matrix of yj can be expressed as {aiRo(i) + (1 a)R(i)}.
Assume that N secondary snapshots, {yj}nc,i for the ith range bin are i.i.d. and share
the same covariance property as yi. Next, we apply LOOCV: for the nth range bin (for
any n E Bi), we test the "likelihood" of

Y ~ CV {o, aQRo(i) + (1 a) R(i) (4-44)

where the sample covariance matrix [R(i)j is estimated from the remaining N 1
L Jn
secondary range bins except for the nth one. Let


R(i) = oiRo(i) + (1 a,) [R(i) (4-45)
L J n










The log-likelihood function of {y,},,s is hence proportional to:


C(i = {n |R(i) + yA '(i)y,}. (4-46)
nEiB

Consequently, the optimal ai 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.

x 105 1 x105


-1.2 -1.2-

-1.4 -1.4

-1.6- -1.6

-1.8 -1.8


0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
(a) (b)


Figure 4-1.


(c) (d)

The log-likelihood function as a function of a, for four different values of the
perturbation level of the a prior knowledge: (a) of 0, (b) f = 0.01, (c)
tf 0.1, and (d) of 0.5.












































200 400 600 800 1000


.8


.6


.4-


.2



200 400 600 800 1000


Figure 4-2.


0.6-


0.2-


200 400 600 800 1000


The CC and ML estimates of a as functions of the range bin index, for four
different values of the perturbation level of the a prior knowledge: (a) 2 = 0,
(b) af2 0.01, (c) 2 = 0.1, and (d) 2 = 0.5.


I I II





























































Figure 4-3.


-- Ideal (Prior)
M- ML
---CC
Equal-weight
-. SMI
. . I . . .


PFA

(b)


PFA PFA


Comparison of the ROC curves corresponding to the "ideal" detector, the
"prior" detector, the "CC" detector, the \!I/" detector, the S\!I" detector,
and the "e,1iu !-v. i!,ii detector, for four different values of the perturbation
level of the a prior knowledge: (a) a 0, (b) f = 0.01, (c) f = 0.1, and (d)
a2 0.5.









CHAPTER 5
KNOWLEDGE-AIDED ADAPTIVE BEAMFORMING

5.1 Introduction

Given the true array covariance matrix R and the steering vector ao 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 [50] 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 R, where
1 N
R 8=y()y*(n), (5-2)

with N denoting the snapshot number. However, when N is comparable with or even

smaller than M, R 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 Ro [10, 11, 8]. When Ro 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 R and

Ro. In this paper, we consider the case of Ro being rank deficient in a general adaptive









beamforming application. This case occurs frequently in practice when we only have

prior knowledge on dominant sources or interference. Unlike in some literature, for

example, see [20, 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 Ro to make it full rank. We consider both modified

general linear combinations (\!GLC) and modified convex combinations (\ CC) of the

a prior covariance matrix Ro, the sample covariance matrix R, 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 xn. The inverse of a

matrix R is denoted as R- We use (.)T to denote the transpose, and (.)* the conjugate

transpose. The Frobenius norm is denoted as I|| |. 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 > 0 means that R is positive semidefinite.









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 (\ GLC)

of the a priori covariance matrix Ro, the sample covariance matrix R, and the identity

matrix I to get an enhanced estimate, let us call it R, of the true array covariance matrix

R:

R = ARo + BR + CI. (5-3)

The combination weights A, B and C in (5-3) should be chosen to guarantee that R > 0.

We also consider the following modified convex combination (\ICC) of the three terms

(i.e., Ro, R and I):

R = ARo + BR + CI; A + B + C = 1. (5-4)

Again, constraints should be enforced to ensure that R obtained by (5-4) satisfies R > 0.

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 loadiin 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 (\ Sl' ) of R:

MSE= E{|R- RI|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 R, in SCB to improve the array output

SINR.








5.3 Knowledge-Aided Covariance Matrix Estimation
5.3.1 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) El{|R- R1|2}

SE{IIAR+CI-(1- B)R+ B(R- R)12}. (5-6)

Using the fact that R is an unbiased estimate of R, we get

MSE(R) E{IB(R-R)2}1 +I ARo+ CI (1- B)R 2

SB2E{R R|2} + (1- B)2RI2 2(1 B) tr [R*(ARo + CI)]

+||ARo + CI|2,

(5-7)

where we have used the fact that for any two positive semidefinite matrices E and F,

tr(EF) > 0, for E > 0, F > 0. (58)

Let

p E{I|R- RI|2}. (5-9)

Consequently, (5-7) can be rewritten as

MSE(R) pB2+ IIRIB2 + I|ARo + CI|2 + 2B tr [R*(ARo + CI)]

-2B IIR2 2 tr [R*(ARo + CI)] + const. (5-10)

Define 0 as
0 =[A BC]T. (5-11)









Then the term of (5-10) that is a linear function of 0 can be written more compactly as

-2bT0, where

bT [tr(R*Ro) |R|2 tr(R*)] (5-12)

The quadratic term in (5-10) is:

pB2+ ARo +CI + BR 12 OTAO, (5-13)

where
IIRo 2 tr(R-R) tr(R;)
A tr(R*R) R112 +p tr(R*) (5-14)

tr(Rn) tr(R*) ||I||2
Hence,

MSE(R) 0TAO 2bT0 + const. (5-15)

Note that the matrix A > 0. The simplest way to prove this follows from (5-13):

OTAO > 0, VO / 0 if and only if there are no A and C, not both equal to 0, such that

ARo + C( = 0. This amounts to assuming that Ro is not a scaled version of I, which is a

very natural assumption. In view of A > 0, (5-15) has a unique minimum solution given

by:
0o = [Ao Bo Co] = A-b. (5-16)

Interestingly, we show in the Appendix that 0 < Bo < 1. However, 0o obtained by (5-16)

may not guarantee that R > 0. Consequently, we rewrite (5-15) as:

[0 0o]TA[0 Oo] + const, (5-17)









and consider the following MSE minimization problem for MGLC with the constraint

R > 0 enforced:

min 6
6,0


[0 o] TA-8

R(0) > 0. (5-18)


The above formulation is equivalent to


min MSE R(0)] s.t. R(0) > 0,
0


(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


(5-20)


to the MGLC formulation in (5-18). We use ul to denote a vector of l's of length 1. The

resulting problem is still a SDP.

In practice, 0o must be replaced by its estimate 0o, where


Oo A-1b,


with A and b being the estimates of A and b, respectively. Then (5-18) becomes:


min
s,0


6

0R >0o
R(o) > 0.


[ 0I
0-0o >0
A-1


(5-22)


(5-21)


u o = 1, u3 = [1 1 1]T,









One way to obtain A and b is to replace p and R in A and b by p and R, where p is

an estimate of p, which can be obtained as [51]:

1 1
p 2 jE y()4 R2. (5-23)
n=l
We refer to [51] for a detailed derivation of p.

As -i-.-i- -i. I in [34] (and also in [54, 36]), to estimate 0 and R consistently, we prefer

to use an alternative estimation scheme to obtain A and b as follows. Because


p + ||R||2 E{R RR 2} + RI2 E {|RI2},

we can estimate p + I||R|2 by I||R|2, which is an unbiased estimate that is smaller than

the previously s-l_-_. -1. 1 p + I|R||2; we can estimate l|R||2 by I|R||2 p, again, a smaller

estimate. We also replace R by R in tr(R) and tr(R*Ro). We refer to the resulting

SDP problem in (5-22) for MGLC as MGLC1, and similarly, (5-22) with the additional

constraint (5-20) for MCC as MCC1.

Alternatively, we can enforce in (5-22) A > 0, B > 0, and C > 0. Then the constraint

R(0) > 0 is trivially satisfied and (5-22) becomes a quadratic program (QP):

min ( o A( o

s.t. i > 0, i 1,2,3. (5-24)

We denote this formulation for MGLC as MGLC2. Similarly, adding (5-20) as an addi-

tional constraint to (5-24) yields a QP problem for MCC, which we denote as MCC2.
5.3.2 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 }')I denote S a priori covariance matrices representing knowledge about S

dominant sources or strong interference. We consider the following linear combination of









{R }) 1, Rand I:
s
R = A(8)R ) + BR + CI, (5-25)
S= 1
where {A(")}s are the weights applied to the a priori covariance matrices {R1 }s1.
Constraints again need to be imposed to ensure that R > 0. 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 = tr(R I s) ... il tr(S) tr(R)*) (5-26)

tr(R)*R) .. tr(Rf)) IRl+p tr(R*)
tr(R *) ... tr(R 5s)*) tr(R*) ||I||2

Similarly, b in (5-12) now has the form:

b [tr(R*R')) tr(R*Rs) |R||2 tr(R*) (5-27)

We redefine 0 as
0 [A(') .. A(s) B C]T. (5 28)

With A and b given in (5-26) and (5-27), the corresponding A and b can be

obtained by using the same scheme as before. Consequently, an estimate of 0 in (5-28)

can be obtained by solving (5-22) for extended MGLC1, and (5-24) for extended MGLC2
(with the constraints being replaced by 0i > 0, i = 1, 2, ... S, S + 1,S + 2). By solving

(5-22) and (5-24) with the following additional constraint:

u+20 (5-29)

we get extended MCC1 and MCC2, respectively.
Let

0MGLCi AMGLCi SLCi MLCCi CMGLCi Oi 1t, 2, (5-30)










and

[ATci = s,-Mcc = 1, 2, (5-31)
0McC, MCCi MCCi C

be the solutions to the extended MGLCi and MCCi problems, i 1, 2, respectively. Given

(5-30) and (5-31), the resulting R can be expressed as:

S
RMGLC i ( LCi + MGLCR + MLCi, 1,2, (5-32)
s 1

and
S
s
ItMCC, = () MCCRO + ^MCCR + CMccI, i = 1,2. (5-33)
S= 1
Remark: We note from the unconstrained solution of 0 (see (5-16) with A and

b given in (5-26) and (5-27), respectively) that the value of CMGLC increases as tr(R)

increases. Specifically, tr(R) will be large in the presence of strong interference, resulting

in a very high weighting value on I in RMGLC. Consequently, when such a RMGLC is used

in adaptive beamformers, their adaptive capability is greatly reduced because CMGLC 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):


0S+2 < 7Amin, (5-34)


where Amin is the smallest non-zero eigenvalue of R, and 7 is a scaling factor. Noting

the fact that 0s+2 should decrease as N/M increases, we may choose 7 to be inversely

proportional to N/M. In our simulations, we choose 7 = 103M/N. The constraint in

(5-34) is usually inactive when tr(R) 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 oaoao + kaka* + Q, (5-35)
k= 1









where o 2 and a 2, respectively, are the powers of SOI and of the kth interference imping-

ing on the array, ao and ak are the corresponding steering vectors, and Q is the noise

covariance matrix.

The array weight vector w can be determined by the SCB approach:


minw*Rw s.t. w*ao = 1. (5-36)
w

The solution to (5-36) is:
R-lao
wo0 R lac (5-37)
a*R-lao
The beamformer output signal-to-interference-plus-noise ratio (SINR) can be expressed as

2 1w*ao 2
SINR a l0 (5-38)
Wo ( i aka + Q) w

Note that the SINR in (5-38) is independent of the scaling factor of the weight vector wo.

By inserting (5-37) in (5-38) and using the matrix inversion lemma, we get the optimal

array output SINR:
K -1
SINR ao oa0 a(akak + Q a0. (539)


In SCB, the sample covariance matrix R is used in lieu of R. As a result, the weight

vector of SCB is given by
R-lao
WSCB (5-40)
a*R-lao
Then SINRsCB follows from (5-38) by replacing the wo in (5-38) with wSCB *

As discussed in the previous subsections, we can use MGLC and MCC to get en-

hanced KA-estimates of R (see (5-32) and (5 33)). If we use {RMGLCi}i 1,2 in lieu of R in

the SCB formulation, the corresponding array weight vector for MGLC can be obtained as
n-1

WMGLCi R LCia i= 1, 2. (5-41)
a0RMGLCi ao

Similarly, the weight vector for using {RMcc}i= 1,2 in lieu of R can be obtained by

replacing the subscript \IGLC with \!CC", i = 1,2, in (5-41). The corresponding









SINR values can be obtained by using (5-38) with the wo 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 = 10 sensors and half-wavelength spacing between

.,.i i:ent elements. Assume a spatially white Gaussian noise whose covariance matrix Q

is given by 10I. We assume that the direction-of-arrival (DOA) of the SOI relative to

the array normal is 0o = 0 and that there are K = 2 interference whose DOAs are

01 -40, 02 = 200. The power of the SOI is 20 dB, i.e., a2 20 dB. The powers of the

two interference, i.e., a, and aoj, will be specified later on. Also, we assume knowledge of

the steering vector ao of the SOI. In our simulations, averaged SINR values obtained from

200 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

prior knowledge. To investigate the effects of the accuracy of Ro on the performance of

MGLC, we consider the following six cases (we use Roi to denote the a priori covariance

matrix for the ith case, i = 1, 6):

(i). Accurate a priori knowledge, i.e.,

2
Rol R -Q aaoa + aka( (5-42)
k= 1

(ii). Accurate a priori knowledge of the interference, i.e.,

2
R02 Jkakak. (5-43)
k-l









(iii). Only the DOA of the first interference is accurately known:


R03 ala. (5-44)

(iv). Inaccurate a priori knowledge. We consider the case where the a prior knowl-

edge on the DOAs of the interference is totally vA. ni:. i.e.,
R a4 72 + a* a (5-45)
R04 3a3a3 + 4a4a4, (545)

where a = a a4 a and a3 and a4 are the steering vectors for two uncorrelated

signals impinging on the array from -55 and 60.

(v). The DOAs of the interference are accurately known:

R1 alat, (5-46)

and

R =- a2al. (5-47)

(vi). The DOA of the first interference is accurately known, but we assume wrongly

that the DOA of the second interference is 60:

RI = alat, (5-48)

and

RS = a4a. (5-49)

5.5.1 Relatively Weak Interferences

We first consider a scenario in which the power of the SOI and the powers of the

two interference are all equal to 20 dB, i.e., a = 20 dB, k = 0, 1,2. In the presence of

relatively weak interference, tr (R) is small and hence the constraint in (5-34) is inactive

for this case.

Figures 5-1 5-2 show the averaged estimates of A, B and C from 200 Monte-Carlo

realizations when the a priori covariance matrices {Roi} 1 (see (5-42)-(5-45)) are used









as Ro for MGLC1 and MGLC2, respectively. (Note that the averaged estimate of A/ at is

shown in Figures 5-1(c) and 5-2(c).) We note that by enforcing R > 0, the so-obtained

AMGLC1) BMGLCI and CMGLCI 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 -> oc. Recall that R converges to R N -- o. For MGLC1, A = 1 B

and = 1 B for any B will result in R =R as N -> oc. MGLC2 is, however, more

restrictive. For MGLC2, A = 1 B and C = 1 B for any 0 < B < 1 will result in

R = R as N -- o. For all other cases of Figures 5-1 and 5-2, there is no ambiguity and

we observe that A and C approach 0 and B approaches 1 as N -- o, as desired. We

also observe from Figures 5-1(d) and 5-2(d) that A is close to 0 when inaccurate a priori

covariance matrix R04 is used as Ro. That is to ~-v, little or no emphasis is placed on the

a prior 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 Ro = Roi, 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 Ro = Rol, MGLC2 significantly

outperforms MGLC2 with A = 0. MGLC1, however, is inferior to MGLC2 and will not be

considered further. Observe that with accurate knowledge of the interference, i.e., with

Ro = R02, both MGLCi and MGLC2 outperform MGLC2 with A = 0, as shown in Figure

5-3(b). Observe also that with only the DOA information on the first interference, i.e.,

with Ro = R03, slight improvements over MGLC2 with A = 0 can still be achieved by

using MGLC1 or MGLC2. It is interesting to note that when the a priori knowledge is

inaccurate, i.e., when R04 is used as Ro, the performance of MGLC2 is similar to that of

MGLC2 with A = 0, as shown in Figure 5-3(d). This is an appealing feature, indicating

that MGLC2 is robust to the error in the a priori knowledge. For comparison purposes,

we also consider equal weighting of Ro, R and I for the four cases. For example, one can









set A = B = C = 1, and hence R = Ro + R + I (we refer to this as "equal v i!il ).

Observe that by adaptively choosing the combination weights, the performance of MGLC2

is alv--i- 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)/J~, A(2)/a, B and C versus the snapshot number N

when {R } 2 (see (5-46) and (5-47)) and {R0} 1 (see (5-48) and (5-49)) are used as

the a priori covariance matrices for MGLC2. Observe that when a2 = a2 = 20 dB and

both DOAs of the two interference are accurately known, A(1)/a7 and A(2)/a2 obtained

by MGLC2 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 } are given by {R?} } MGLC2 can give a proper weight on

the accurate component, i.e., RW(, and at the same time suppress the wrong component,

i.e., R) 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 1} MGLC2 outperforms MGLC2 with A) = A) = 0, especially when N is small

(see Figure 5-5(a)). When {R6 } 1 are used as the a prior covariance matrices, the
performance of MGLC2 is still slightly better than that of MGLC2 with A1 = A(2) = 0, as

shown in Figure 5-5(b).

5.5.2 Relatively Strong Interferences

We now consider a scenario where the powers of the interference are much stronger
than that of the SOI. Assume that a2 20 dB, a2 70 dB, and a = 60 dB. The

constraint in (5-34) is now active due to the presence of strong interference.

We first consider the Cases (i) (iv) when Roi, i = 1, 4, is used as the a priori co-

variance matrix. A's, B's and C's obtained by MGLC2 for this case of strong interference

have similar properties as those shown in Figures 5-1 and 5-2, except that C's now take

on larger values, which are mainly determined by the constraint (5-34). Figure 5-6 shows









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 prior

knowledge gives greater SINR improvement in the presence of strong interference.

We next consider the Cases (v) and (vi) for this case of strong interference. Fig-

ure 5-7 shows the averaged SINR versus the snapshot number N. Again, an obvious

improvement of MGLC2 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 interference, as N oo, all SINR curves (including

those of MGLC1) approach the optimal SINR curve, as expected. Moreover, the SINR

curves of MGLC2 ahv-- stay above those of MGLC with A = 0 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 Ro, 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 R with the a

prior covariance matrix Ro 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 knowledge-

aided adaptive b, ,il-r. .ii nii.r 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









accurate a priori knowledge can significantly improve the performance of the adaptive

beamformers, especially in the presence of strong interference.

Appendix: A Property of Bo in (5 16)

In this appendix, we show that 0 < Bo < 1. Note that 0o in (5-16) satisfies:


A0o = b,


(5-50)


where A and b are given in (5-14) and (5-12), respectively.

associated with (5-50) is [Alb], where

I|Ro 12 tr(RR) tr(R-)
[Alb] tr(R-R) |R112+p tr(R*)

tr(R*) tr(R*) ||I||2

Let
tr(RoR)
D = 2 R o0 R,
||IRo 1


The augmented matrix


tr(R*Ro)


tr(R*)


and
tr (Ro)
D2 = Ro I.
I I Ro 12
Then making use of the Gaussian elimination, it follows that (5-51) is equivalent to

I|Ro 02 tr(RR) tr(R*) tr(R*Ro)
0 ||Dl|2 +p tr(DDD2) I||D |2
0 tr(D*D2) I|D2 2 tr (DD2)

and consequently it is also equivalent to


IIRo012
0
0


tr(R|R)

rDtr2 (DD2)
IIDll1 +p DII
tr(DTD2)


tr(RO)
0

|ID2 12


tr(R*Ro)
IDl 112 tr2(DD2)
tr (D ID2
tr (DTD2)


(5-51)


(5-52)


(5-53)





(5-54)


(5-55)




























-0.5- 1 i--i---- I ---I- -I
20 40 60 80 1
Snapshot Number

(a)



22



1.5 -

1 ...-- --------------------

0.5

n-


20 40 60
Snapshot Number


80 100


-A

--C


20-.
1.51- ""-... .-


-- -- -- -- -- -- -- -- -----
0.5-
fi._


20 40 60
Snapshot Number


20 40 60
Snapshot Number


80 1(


80 100


Figure 5-1.


The averaged MGLC1 estimates of A, B and C versus the snapshot number N
when ao = = o- = 20 dB, and the a prior covariance matrix is given by (a)

Ro = Rol (see (5-42)), (b) Ro = Ro2 (see (5-43)), (c) Ro = R03 (see (5-44))

and (d) Ro Ro4 (see (5-45)).


Hence Bo can be obtained as



BD I211D112 2 tr2(D*D2)
Bo = (5-56)
pllD2 2 + IDi121D2112 tr2(DD2)


Given (5-56) and by using the Cauchy-Schwartz inequality, we have that 0 < Bo < 1.


--A
---B

("


I I ~I


--A

L--B.
'''. --C






























3 --A

2.5- E-i-B
-C

2-

1.5-




0.5-


20 40 60
Snapshot Number


80 11


-2-
2 -.


1.5-

1 ---------------------

0.5 "

0-

-0.5 .


20 40 60
Snapshot Number


80 100


--A
2 .5 -,

2-
--



1.5 -



0-

-0.5 -


20 40 60
Snapshot Number


80 1(


--B
2.5-

2-

1.5 -. ...... -

1 -__-__-..-- ------- -----=

0.5




-0.5-


20 40 60
Snapshot Number


80 100


Figure 5-2.


The averaged MGLC2 estimates of A, B and C versus the snapshot number N

when ao2 = a = o = 20 dB, and the a priori covariance matrix is given by (a)

Ro = Rol (see (5-42)), (b) Ro = Ro2 (see (5-43)), (c) Ro = Ro3 (see (5-44))

and (d) Ro = Ro4 (see (5-45)).


...,

















































20 40 60
Snapshot Number


80 100


20



10S. o o ----....
15 -


0 -----'"
5- o -.-..
5
o ,'
0 oo ---SCB
-.MGLC2 w/A=0
MGLC
-10 MGLC2

-15 o Equal weight
-SINRopt
-20 I I I o


20 40 60
Snapshot Number


80 100


20 40 60
Snapshot Number


80 100


20



10- -' o a
1 o o o
5 ,- 0 -
0o
0 0o, ---SCB
-5 .- -.MGLC w/A=0
MGLC1
-10 MGLC2

-15- o Equal weight
-SINRopt
-20 I I opt


20 40 60
Snapshot Number


80 100


Figure 5-3.


SINR versus the snapshot number N when ao = a = oa = 20 dB, and the a

prior covariance matrix is given by (a) Ro = Rol (see (5-42)), (b) Ro = R02

(see (5-43)), (c) Ro = R03 (see (5-44)) and (d) Ro = R4 (see (5-45)).


>- -. --- -0 ----0
o 0o O
So ---- -


oo00 -
---SCB
i' -..MGLC2 w/A=
SM GLC


SMGLC2
o Equal weight
-SINRopt
opt


> I > -I I : .__I.-.. .1- -.__,--'-1

.-' o"" o o _o__0
0 o o -
0 *---
o000

-- -SCB
.--.MGLC2 w/A=0
/MGLC


1'--
SMGLC2
o Equal weight
-SINRopt
opt


I I I


I I I





























20 40 60 80 100
Snapshot Number


20 40 60 80 100
Snapshot Number


Figure 5-4. The averaged MGLC2 estimates of A(1)/o~, A(2)/o, B and C versus the
snapshot number N when o-2 = = o 20 dB, and the a priori covariance
matrices are given by (a) {R }2 {RR) l (see (5 46) and (5-47)), and

(b) {R }8) {R }2 (see (5-48) and (5-49)).









20 20


5- 5-
~. 5 ,-- 5 -

0 C w 0 .
---SCB -5' ---SCB

-10- --MGLC2 wI A =A(2)0- -10- -.MGLC2WIA =A(2=0
-15 MGLC2 -15 MGLC2
opt opt
-20 -20
20 40 60 80 100 20 40 60 80 100
Snapshot Number Snapshot Number
(a) (b)


prior covariance matrices are given by (a) {R } { } 1 (see (5-46)
and (5 47)), and (b) M{R }GLC (see (5 48) and (5 49)).





























20 -

15- ---

10- -----



0- -
-5- -
-- ---SCB
-10- .-..MGLC2 w/A=0
MGLC2
-1o
-SINRopt
-2 I I I I


20 40 60
Snapshot Number


80 100


20








0-


/ -SCB
-10- ..-.MGLC2 w/A=0

-15- MGLC2
-SINRo
nfl Iopt


20 40 60
Snapshot Number


80 100


20 -- --------

15 -- -- -- -- -- ------" "
15

10- "'' ... --..
.- -


0

-5- ,
---SCB
-10- .-.-.MGLC2 w/A=0
> MGLC2
-SINRopt
r_ __ I I I I


20 40 60
Snapshot Number


80 100


20 40 60 80 100
Snapshot Number


Figure 5-6.


SINR versus the snapshot number N when a~ = 20 dB, Ia = 70 dB and

o = 60 dB, and the a prior covariance matrix is given by (a) Ro = Ro (see

(5-42)), (b) Ro = R02 (see (5-43)), (c) Ro = R03 (see (5-44)) and (d)

Ro = R04 (see (5-45)).





































-5-
-10-
-15
-20




Figure 5-7.


100


100


(a) (b)


SINR versus the snapshot number N when ao 20 dB, 2 = 70 dB and
a 60 dB, and the a prior covariance matrices are given by (a)

{R } {R 1 (see (5-46) and (5-47)), and (b) {RIO} 1- J {R 1
(see (5-48) and (5-49)).









CHAPTER 6
WAVEFORM SYNTHESIS FOR DIVERSITY-BASED TRANSMIT BEAMPATTERN
DESIGN

6.1 Introduction

Waveform diversity has been utilized both in multiple-input multiple-output (MI\ !O)

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

[60], better parameter identifiability [67, 78], and direct applicability of adaptive array

techniques [67, 69, 80]; 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.









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 .I..il 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 diversity-

based 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









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,,,. A Hermitian square root of a matrix

R is denoted as R1/2. 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 e CLxN be the transmitted waveforms, where N is the number

of the transmitters, and L denotes the number of samples in each waveform. Let


R A X*X (6-1)
L

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. N2

The class of (unconstrained) signal waveform matrices X that realize a given covari-

ance matrix R is given by

X* R1/2U*, (62)
VIL
where U* is an arbitrary semi-unitary N x 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 .;,,.l,,. .. ..,
matrix X is as follows:
2
min X- LUR12 (63)
XEC;U









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 X to (6-3) as the signal waveform matrix (in which case it will satisfy

the constraints, but it will only approximately realize R) or vALUR1/2 (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 E 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 i.-.i-. I.1 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 0: Set U to an initial value (e.g., the elements of U can be independently
drawn from a complex Gaussian distribution with mean 0 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: Obtain the matrix X E C that minimizes (6-3) for U fixed at its most recent value.

Step 2: Determine the matrix U (U*U = I) that minimizes (6-3) for X fixed at its most
recent value.

Iteration: 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








difference between the U matrices at two consecutive iterations is less than or equal
to 10-4.
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 v--,v (see, e.g., [88, 89]). A simple derivation
of it runs as follows. For given X, we have that

X VUR/ 2 const 2Re tr [ R1/2X*U] (6-4)

Let
LR1/2X* UEU* (6-5)

denote the singular value decomposition (SVD) of VR1/2X*, where U is N x N, E is
N x N, and U is L x N. Then

Re {tr [LR1/2X*U] }

Re {tr [Lu*UUE1 (6-6)
N
= Re { [U*UU] } ,,n. (6-7)
n=l
Because

(u*uu) (u*u*u) *uu*

< U*U

I, (6-8)

it follows that

Re2 uu _}< L[u*UUI

< [(u*uu) (u*u*u)1]
< 1, (6-9)









and therefore that
2 N
X UR12 > const 2 (6-10)
n~l
The lower bound in (6-10) is achieved at


U = UU*, (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,(l)}f

denote the elements in the nth column x, of the signal waveform matrix X. Then the

constant-modulus requirement means that:


x,(l)| = c, for some given constant c

and for / 1,... L. (6-12)


(For example, we can choose c = R~ 2; 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:


mmin cej z 2 ,(6-13)


where c > 0 and z E C are given numbers. Because


cej z 2 const 2clz cos [ arg(z)] (6-14)

the minimizing i is evidently given by


= arg(Q). (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









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 X*X
L
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,) max Ix 2 < p, for a given p [1, L], (6-16)
S 1=1 |Xn(1|2l)

(where, once again, we omit the dependence of p on n for notational simplicity). If we add

to (616) a power constraint, viz.

L
S x(l)|2 1 (e.g., 7 = Rn), (6-17)

then the set C is described by the equations:


(6-18)
Ix 2(1)2 l=1 ,--- ,L.

While the above constraint set is not convex, an efficient ili'', .:hm for solving the corre-

sponding 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. is not deemed necessary, we can relax









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.

6.4.1 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 0 can be

shown to be (see, e.g., [61, 69, 72]):


P() a*(O)Ra(O), (6-19)


where R is as defined before,

a(0) --'.I -L- () c -"' "- ) -N (e ) (6 20)


and where fo is the carrier frequency of the transmitted signal, and 7-,(0) is the time

needed by the signal emitted via the nth transmit antenna to arrive at the focal point;

unless otherwise stated, 0 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,
C
R N, = n =l, N, (6-21)

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 approximat-

ing) a (scaled version of a) desired transmit beampattern.









(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 0 and 0 is given by a*(0)Ra(0).

Assume that we are given a desired transmit beampattern 0(0) defined over a region

of interest Q. Let {pg}2 1 be a fine grid of points that cover Q2. As indicated above, our

goal is to choose R such that the transmit beampattern, a*(0)Ra(0), matches or rather

approximates (in a least squares (LS) sense) the desired transmit beampattern, ((0),

over the region of interest f, and also such that the cross-correlation (beam)pattern,

a*(0)Ra(0) (for 0 / 0), is minimized (once again, in a LS sense) over a given set {Ok}kI

Mathematically, we therefore want to solve the following problem:


m in "'K' [o(pg) a*(pg)Ra(pg)]2
a,R G
K-1 K
K a*(0k)Ra(P)12}
K2 K |( >2
k l1 p k+l

s.t. R,, N' n 1,- ,N

R > 0, (622)


where a is a scaling factor, i,., > 0, g = 1, ,G, is the weight for the gth grid point and

t > 0 is the weight for the cross-correlation term. Note that by choosing nm-:, i'., > i,

we can give more weight to the first term in the design criterion above, and vice versa for

nri::., i,,, < w,. 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 /L-iR1/2,

where U 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 -;,'Il, -..,. waveforms with optimal R (abbreviated as optimal R). When we use the









solution X to (6-3) obtained via CA as the transmitted signal waveform matrix, then X

will satisfy the PAR constraints, but will realize R only approximately. We refer to the

so-synthesized waveforms as the CA -;,',.'l,. -.., waveforms with PAR < p (abbreviated as

PAR < p).

In the following examples, the transmit array is assumed to be a uniform linear array

(ULA) comprising N = 10 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 Q, we choose a mesh grid size of 0.1. Finally, the CA algorithm

is initialized using the U described in Step 0.

In the first example, the desired beampattern has one wide main-beam centered

at 0 with a width of 600. The weighting factor i,-, in (6-22) is set to 1 and ", is set to

0. 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

0(0) scaled by the optimal value of a. 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 -;;,ill,.-

sized waveforms with optimal R are slightly different from those obtained using the CA

-,.Ii, ., .: I, ,,:, fi.,n' with PAR < p. Let

R X*X (6-23)
L

be the sample covariance matrix corresponding to the CA -;;,',/ /, -..,/ n, forms with PAR

< p. Let

6 R- R (6-24)

denote the norm of the difference between R and R. Then we have 6 = -29.7891 dB,

-41.7237 dB, and -119.5251 dB for Figures 6-1(a), 6-1(b), and 6-1(c), respectively. As









expected, the difference decreases as the PAR value increases. For the case of PAR = 2,

the difference is essentially zero. The mean-squared error (\!Sl) of R (i.e., the average

value of 62), obtained under PAR = 1 and estimated via 100 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 0, as an ensemble of realizations obtained from the 100 Monte-

Carlo trials. In each Monte-Carlo trial, the initial value for U in Step 0 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 p increases.

Figure 6-2 shows the actual PAR values of the CA -; ,,Ih .:.../, a..r. I., ,,m- 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 R1/2 with a 256 x 10 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 -;;u,.1 f,../ ;. .., f., -, with optimal R have much lower PAR values than

the colored Hadamard code sequences. Note also that the actual PAR values of the CA

..;.,, ..,/ ., [. -i, 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 01 = -400, 02 = 0, and 03 = 400, each with a width of 200. The same mesh grid is used

as before, and we choose the weighting factors as "'., = 1 and w, = 1. Figure 6-5 shows the

corresponding beampatterns. Remarks similar to those on Figure 6-1 can be made for this

example as well.

6.4.2 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:









(a) 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 00 and the prescribed 3-dB angles are

01 and 02 (i.e., the 3-dB mainbeam width is 02 01, with 01 < 00 < 02). Let ts denote the

sidelobe region of interest and {pg} a grid covering it. Then the design problem of interest

in this section can be mathematically formulated as follows:

min -t
t,R
s.t. a*(0o)Ra(0o) a*(p,)Ra(p,) > t, Vp, E Q,

a*(01)Ra(01) 0.5a*(0o)Ra(0o)

a*(02)Ra(02) 0.5a*(0o)Ra(0o)

R>0

0.8 ( 5) R & 1.2 ) ,
N
n= ,N, "R" = C. (6-25)
n-l

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 semi-

definite 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 00 = 0 with a

3-dB width equal to 200 (01 = -100 and 02 = 100). The sidelobe region is chosen to

be [-900, -200] U [200, 90], which allows for some transition between the main-

beam and sidelobe region. The same mesh grid size of 0.10 is used here. Figure 6-6 shows









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 -;;,ql,. -: ..,/ i,. f. .I I,, with optimal R are similar to those obtained

from the CA -;;, ,.1,. -. waveforms with PAR < p 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 -40 dB or deeper null at pT = -30. To do this,

we add the following constraint to the minimum sidelobe beampattern design problem in

(6-25):

a*(p,n)Ra(pn) < -40 dB, p, = -300, (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 -.;i.I-,' :.,,/ waveforms with PAR

< p, the null depths at -30 for the three different PAR values shown in Figure 6-7 are

-22.7769 dB, -33.9913 dB and -39.9657 dB, respectively. Hence a stringent PAR con-

straint can have a significant impact on the null depth. For PAR < 1.2, the beampatterns

obtained with the CA -;.,'/ //. ..,/ "i. f.-,i with PAR < p and, respectively, with the CA

,h-ii, .I, .i f. -.i 1. with optimal R are almost identical.

Finally, consider an example where we wish to form a broad null over the region Q =

[-550, -450], where the power gain must be at least 30 dB lower than the power gain at

00. To do this, we add the following constraint to the minimum sidelobe beampattern

design in (6-25):

a*(p.)Ra(p.) < 10 -a*(Oo)Ra(Oo), Vp, c Q,. (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.









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 hyperther-

mia. Of all women diagnosed with breast cancer, 21 have locally advanced disease and

even with ., i-iressive 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 [90, 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 43C 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 hun-

dreds 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 L. i11 !: breast tissue

region, under a uniform elemental power constraint. The beampattern design problem is









therefore to choose the covariance matrix R of the transmitted waveforms to achieve the

following goals:

(a) Realize a predetermined main-beam width that is matched to the entire tumor region;

in the said region the power should be within 10' of the power deposited at the

tumor center;

(b) Minimize the peak sidelobe level in a prescribed region (the surrounding healthy

breast tissue region).

This problem can be mathematically formulated as:


min -t
t,R
s.t. a*(Oo)Ra(Oo) a*(tp)Ra(/p) > t, V e a

a*(v)Ra(v) > 0.9a*(0o)Ra(0o), V v e QT

a*(v)Ra(v) < 1.1a*(0o)Ra(0o), V V E QT

R>0

R N n =, 2, ,N, (6-28)

where 0o is the tumor center location (0o is now a coordinate vector), and QT and QB

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 10 cm diameter, which includes breast tissues, skin, chest wall, and a

16 mm diameter tumor whose center is located at x = 0 mm, y = 50 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









densities and temperature distributions inside the breast model when the synthesized

waveforms are transmitted via the acoustic transducers.

Figure 6-10 shows the actual PAR values of the CA -;;,,i/,. -.. ./ waveforms with

optimal R. Figure 6-11 shows the temperature distributions within the breast model, with

Figure 6-11(a) corresponding to the CA -;;,'/,1, -..,1 ; ,.ri. [.., ,, with PAR = 1 and Figure

6-11(b) to the CA -;u,,'/l,. ..: ,1 i.forms 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 43C, while the

temperature of the surrounding normal tissues is below 40 C. In contrast with this, when

a phased-array is used for transmission and the d. 1 i- ,il-sum technique is employ, .1 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 dir './/l; 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









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 Is z112 s.t. PAR(s) < p, (6-29)
s

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 = ex; l||x|2 = 1; where c > 0 is a variable. (6-30)


Using (6-30) in (6-29) yields:


||s z||2 = cx z 12 C2 2cRe(x*z) + const. (6-31)

If Re(x*z) < 0, then the minimum value of (6-31) with respect to c > 0 occurs at c = 0. If

Re(x*z) > 0, then the minimization of (6-31) with respect to c > 0 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 = 0. Because PAR(s) = PAR(x) does not depend on the phases of the elements of x,

we can ah--,v- choose x such that Re(x*z) > 0 -so that we achieve a smaller value of

(6-31). Consequently, the minimizing value c > 0 of (6-31) is alv--,v- given by (6-32). The

remaining problem is:


maxRe(x*z) s.t. l||x|2 = 1 and PAR(x) < p, (6-33)
x

or equivalently

min |x z|2 s.t. IlX| 2 = 1 and PAR(x) < p, (6-34)
x
























-50 0 50
Angle (degree)
(a)


-50 0 50
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 I||s|2 = I|z|2 (which is the case

when we choose = R,, in (6-17)). The use of this constraint is logical if we want

to match R,, exactly (for strict transmission power control, for example). However, if

matching R, exactly is not a necessary condition, then a smaller matching error between

s and z is obtained using (6-32) and (6-34).




























-50 0 50
Angle (degree)

(c)


Figure 6-1.


Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. The probing signals are
synthesized for N = 10 and L = 256 by using CA under (a) PAR = 1
(resulting in 6 = -29.7891 dB), (b) PAR < 1.1 (resulting in 6 = -41.7237 dB),
and (c) PAR < 2 (resulting in 6 = -119.5251 dB).


7

6

5

a0 4
a 3


4 6
Index of Transmit Antenna


Figure 6-2. PAR values for CA synthesized waveforms with optimal R and for colorized
Hadamard code.


-e-CA (PAR = 1): Optimal R
- -CA (PAR< 1.1): Optimal R
CA (PAR< 2): Optimal R
-Colored Hadamard


tit I






















0.3












CD
0.2


2 0.1


E 0



E -0.1


-0.2 -



-50 0 50
Angle (degree)

(a)
0.3


0.2


0 0.1


0
C
a
Q.
E -0.1
m

-0.2-



-50 0 50
Angle (degree)

(b)


*

















0.2F


a)

- 0.1
a

0)
Q.
EC


E -0.1
a0

-0.2


I I I
-50 0 50
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.


-YI r -11 --1


-251


-26

v -27
LU
n) -28

-29


-30

-31


-_I-
101


102
Sample Number L


Figure 6-4. MSE of the difference between R and R (CA synthesized constant modulus
waveforms) as a function of sample number L obtained with 100 Monte-Carlo
trials. R is obtained from the CA synthesized constant modulus waveforms.

























-











Angle (degree)

(a)
3.5 i
S: ---Desired
I . . I










S .......... .. .......... .......... ..... ......... ...................- C A O p tim a l
0 CA PAR11.1



-50 0 50
Angle (degree)

(b)
Figure 6-5. Beampattern matching design with each desired beam width of 20 and under
the uniform elemental power constraint. The probing signals are synthesized
for and L -C 256 by using C A under (a) PAR 1 and (b) PAR
2.5


2 -r- -

2- I I i
0u1.5 -.-
I I
I I "I

1- -

































- 2 0 ... . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .


0
Angle (degree)


-5

1.-10
E
-15
nn


-20 -



-30
-50 0 50
Angle (degree)


Figure 6-6.


Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and under the relaxed elemental power constraint. The probing signals
are synthesized for N = 10 and L = 256 by using CA under (a) PAR = 1 and
(b) PAR < 1.1.


.- . . . . . .

.- . . . . . .

.- . . . . .


































-50 0 50
Angle (degree)
(a)


-50 0 50
Angle (degree)
(b)






































-50 0 50
Angle (degree)
(c)

Figure 6-7. Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and a -40 dB null at -300, under the relaxed elemental power
constraint. The probing signals are synthesized for N = 10 and L = 256 by
using CA under (a) PAR 1, (b) PAR < 1.1, and (c) PAR < 1.2.






























Angle (degree)
(a)


-50 0 50
Angle (degree)
(b)


Figure 6-8. Minimum sidelobe beampattern design with the 3-dB main-beam width equal
to 200 and a null from -550 to -450, under the relaxed elemental power
constraint. The power gain difference between 0 and the null is constrained to
be less than or equal to 30 dB. The probing signals are synthesized for N = 10
and L = 256 by using CA under (a) PAR = 1 and (b) PAR < 1.1.




















70

60

50-
E
E 40

30

20

10


-60







Figure 6-9. Breast model.


Acoustic transducer array
..... .. ...............


-4U -2U


U
x (mm)


2U 4U bu


5",


Figure 6-10. PAR


-r.
10 20 30 40
Index of Acoustic Transducer


values for CA synthesized waveforms with optimal R.


r w^''^


























aO"""""'I


20 40 60 80
x (mm)


100 120 140


60

-50
E
E 40

30

20

10


20 40 60 80 100 120 140
x (mm)

(b)


Figure 6-11. Temperature distribution for N = 50 and L = 128. (a): CA synthesized
constant modulus signals, and (b): CA synthesized signals with optimal R).


70-

60

-50
E
E 40

30

20-


anr


...~~~~..,
.
.
.
.
.
. -- .









CHAPTER 7
MIMO RADAR WAVEFORM SYNTHESIS

7.1 Introduction

MIMO radar is an emerging technology that has significant potential for advanc-

ing the state-of-the-art of modern radar. Unlike a standard phased-array radar, which

transmits scaled versions of a single waveform, a MIMO radar system can transmit via

its antennas multiple probing signals that may be chosen at will. An interesting current

research topic in MIMO radar is the optimal synthesis of the transmitted waveforms.

For MIMO radar with widely separated antennas, waveform designs without 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 tech-

niques 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].

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 per-

formance 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 com-

pute 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 I..'l 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).









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) too. We present a cyclic algorithm (CA)

for the synthesis of such an X. We also consider the case where the synthesized waveforms

are required to have good auto- and cross-correlation properties in time. Several numerical

examples are provided to demonstrate the effectiveness of the proposed methodology.

7.2 Formulation of the Signal Synthesis Problem

Consider a MIMO radar equipped with N transmit antennas. Let the columns of

X e CLxN be the transmitted waveforms, where L denotes the number of samples in each

waveform.

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,(1) be the

(1, n)th-element of X. The constant-modulus requirement means that:


x,(l) = c, for some given constant c and for = 1, L. (7-1)


(For example, we can choose c = R~1, where R, is the nth diagonal element of R; we

omit the dependence of c on n for notational simplicity.) However, (7-1) may be too hard

a requirement on the signal matrix. 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,) max 2 < p, for a given p [1, L], (7-2)


(where, once again, we omit the dependence of p on n for notational simplicity). If we add

to (7-2) a power constraint, viz.

L
S (e.g., 7- R ), (7-3)









then the low PAR constraint is described by the equations:


L IX(12 (7 4)

1X. (1)|~2 = 1, < L.

In what follows, we use C to denote the set of signal matrices X that satisfy these con-

straints.

7.2.1 Without Time Correlation Considerations

Let

R A X*X (7-5)
L

be the (sample) covariance matrix of the transmitted waveforms, where (.)* denotes the

conjugate transpose. The class of (unconstrained) signal waveform matrices X that realize

a given covariance matrix R is given by

X* = R1/2U, (7-6)


where U* is an arbitrary semi-unitary N x L matrix (U*U = I), and R1/2 is a Hermitian

square root of the matrix R. Then a possible mathematical formulation of the problem

of .;1,'i,//. -:.;:,1 the probing
follows [48]:
2
min X- LUR1/2 (7 7)
XEC;U

where || || denotes the Frobenius norm.

7.2.2 With Time Correlation Considerations

In i] ,ni real applications such as least-squares (LS) (i.e., matched filtering) based

MIMO radar in i-_- an important requirement on the synthesized waveforms is that

they have good correlation properties. The synthesis of constant-modulus transmit signals

with good auto- and cross-correlation properties and R = I is considered in [93] for

MIMO synthetic aperture radar (SAR) imaging. We now extend the problem in [93] to

synthesize a waveform matrix that realizes an arbitrary covariance matrix R under both









the constant-modulus and low PAR constraints, and which also has good correlation

properties in time.

Let
L
rnn(p) = x,(1)xlx(1 -p) r *(-p), p = 0, 2, (7-8)
I=p+l
denote the (cross)-correlation of x,(1) and x,(l) at lag p. To ensure the good auto- and

cross-correlation properties of X in time, we must have that:
N N P-1
EE E -(p) 2 (7-9)
n= n=1lp=-P+1 '

where P is an integer selected on an application-dependent basis. (7-9) means that we

want the sidelobes of the auto-correlation functions {rn,(p)} (i.e., p / 0) and also the

cross-correlation functions {rnn(p)}nj n (i.e., p / 0) to be -i1i II" at all lags p (note that

rn(0) = LRn, Vn and n, where Rn denotes the (n, fn)th element of R).

We now reformulate the problem slightly. Let X* be the following block-Toeplitz

matrix:
P+L-1

xi(1) ... xi(L) 0



0 x1(1) ... xi(L)
X*= (7-10)

XN(1) ... XN(L) 0


0 XN(1) ... XN(L)

Note that X* is NPx (P+L-1) and in what follows, we assume that NP < P+L-1.

The auto- and cross-correlations that appear in (7-9) are the elements of the positive

semi-definite matrix X*X. Consequently, we can write (7-9) more compactly as follows:


X*X- LR -II i!! (7-11)









where R = R I with & denoting the Kronecker matrix product.

Consequently, a mathematical formulation of the problem of -;, ./I,,. ..:.', the probing

'.:.,il matrix X with time correlation considerations can be written as follows:


min X VUR1/2 2 (7-12)
XEC;U

where U* is an arbitrary semi-unitary NP x (P + L 1) matrix (U*U = I).

We observe that (7-7) is a special case of (7-12) with P = 1 and hence we only

consider the minimization problem in (7-12) hereafter. Note that (7-12) is non-convex

due to the non-convexity of the constraint U*U = I and possibly of the set C, too. we

present in the next section a computationally attractive cyclic (or alternating) optimiza-

tion algorithm for solving (7-12), as sI--.-, -1. 1 in a related context in [86, 87]. We refer to

the cited papers for more details on this type of algorithm and its properties.

7.3 Cyclic Algorithm for Signal Synthesis

We first summarize the steps of the cyclic minimization algorithm for solving (7-12)

and then describe each step in detail.

StepO: Set U to an initial value; alternatively we can start with an initial value for X

(e.g., the elements of U or X can be independently drawn from a complex Gaussian
distribution with mean 0 and standard deviation 1), in which case the sequence of

the next steps should be inverted.

Stepl: Obtain the matrix X E C that minimizes (7-12) for U fixed at its most recent

value.

Step2: Determine the matrix U (U*U = I) that minimizes (7-12) for X fixed at its

most recent value.

Iter.: 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

difference between the U matrices at two consecutive iterations is less than or equal

to 10-4.









An important advantage of the above algorithm is that Step 2 has a closed-form

solution. A simple derivation of it runs as follows. For given X (X can be constructed
from X using (7-10)), we have that

X L- vUR12 2 const- 2Re tr [LR1/2X*U} (7-13)



Y = -LR1/2 (7 14)

where Re(.) denotes the real part of a complex-valued vector or matrix. Let

v/LR1/2X* UUJ* (7-15)

denote the singular value decomposition (SVD) of R1/2X*, where U is NP x NP, E is
NP x NP, and U is (P + L 1) x NP. Then the solution to the minimization problem in
Step 2 is given by [88, 89]:

U = UU*. (716)

The solution to the problem in Step 1 naturally depends on the constraint set C.

Note that the constraints in (7-1) and (7-4) 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. Under the constant-modulus constraint in (7-1), Step 1 of the algorithm

also has a closed-form solution. The generic problem in such a case is:
P
min m e P 2
p=1

Smin const 2Re -zi ej }
ma ( P (717)
= max cos arg YZ, -p (7-17)
\(p=I








where {zp} are given numbers. The solution to (7-17) is evidently given by

= arg Yz, (7-18)
(P-i 1
Therefore, under the constant-modulus constraint, both steps of the cyclic algorithm have
closed-form solutions.
Under the low PAR constraint in (7-4), the generic problem in this case is:
P
mm |x zp2
p=1

= min const 2Re(Z )z, x + Px2
S(Pp=i ) X

m in 1- z 2 (7 19)
x P

where x is an element of the waveform matrix X. Let x, denote the nth column of X.
Then given (7-19), the column-wise minimization problem in Step 1 of the CA algorithm
under the low PAR constraint is as follows:

min I|x, z||2
xn
L
s.t. ftx(1) 2
l=1
Ix,(1)2 < y, = 1,.. ,L, (7-20)

where the elements of z can be calculated from {zp} in (7-19). While the above mini-
mization problem is not convex, an efficient ild.,rithm for solving it has been proposed in
[86, 87].
7.4 Numerical Examples
We present several numerical examples to demonstrate the effectiveness of CA for
signal synthesis.









For the transmitted waveforms to have good auto- and cross- correlation properties in

time, we need that:


IIX*JpX112 = -in Il" (ideally, zero),

for p= -P+1,... ,-1,1,.- ,P- (7-21)


where
p+1
1 0


p= i (Lx L) (7-22)



0

is a "shift" matrix. The above definition of J, is valid for p > 0. For p < 0, J, should

be defined as the transpose of the matrix in (7-22). Hence, we use the normalized values

of (7-21), denoted as "correlation 1., v !- to evaluate the time correlation quality of X

hereafter.

We consider an example of MIMO radar beampattern matching design. We first

review briefly the beampattern matching design (more details can be found in [72, 76]).

The power of the probing signal at a generic focal point with coordinates 0 can be

shown to be (see, e.g., [61, 69, 72]):


P() a*(0)Ra(0), (7-23)


where R is as defined before, and a(0) is the steering vector. The beampattern matching

design problem consists of choosing R, under a uniform elemental power constraint,

C
R N'n= n= 1,-. ,N, (7-24)

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 approximat-

ing) a (scaled version of a) desired transmit beampattern.

(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 0 and 0 is given by a*(0)Ra(0).

Assume that we are given a desired transmit beampattern 0(0) defined over a region of

interest Q. Let 1{/} 1, be a fine grid of points that cover 2. Let {0k}j = be a given set of

signal locations. Mathematically, we can formulate the design problem as follows:


min ,,, [ao1(,ig) a*(,ig)Ra(,ig)]2
aR G
g K- K

+ 22, K a*(k)Ra( 2}
k1 p= k+l
C
s.t. R"= n= 1,...,N
N
R > 0, (725)


where a is a scaling factor,, > 0, g = 1, G, is the weight for the gth grid point and

it' > 0 is the weight for the cross-correlation term. Note that by choosing rin:, i,'., > i,

we can give more weight to the first term in the design criterion above, and vice versa for

r-n::., i,', < i,' 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.

In this example, the transmit array is assumed to be a uniform linear array (ULA)

comprising N = 10 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 f, we choose a mesh grid size of 0.1. The desired beampattern

has one wide main-beam centered at 0 with a width of 600. The weighting factor i,',

in (7-25) is set to 1 and it, is set to 0. Figures 7-1(a), 7-1(b), and 7-1(c) show the

beampatterns using the CA synthesized waveforms under the constraints of PAR = 1









(constant-modulus), PAR < 1.1, and PAR < 2 with P = 1, respectively. For comparison

purposes, we also show the desired beampattern 0(0) scaled by the optimal value of a

as well as the beampattern obtained from the optimal R. Note that the beampattern

obtained using the CA synthesized waveforms is close to the desired one even under the

constant-modulus constraint, and it approaches the one obtained from the optimal R as

the PAR value increases.

Figure 7-2 shows the correlation levels (normalized values of (7-21)) versus p for

the CA synthesized waveforms with P = 1. Recall that for P = 1, the time correlation

property of the waveforms is not considered, and hence the resulting waveforms under

different constraints (i.e. PAR = 1, PAR < 1.1 and PAR < 1.2) all have relatively high

correlation levels.

We next consider the same waveform synthesis problem but with P = 10. By using

the same initial values in Step 0 of the CA algorithm (the initial value for X when P = 10

is constructed from the initial value for X when P = 1 by using (7-10)), the resulting

beampatterns are similar to those shown in Figure 7-1, as shown in Figure 7-3. Figure

7-4 shows the corresponding correlation levels (normalized values of (7-21)) versus p. The

correlation levels decrease as the PAR value increases, as expected. As we can see, the

correlation levels of the waveforms shown in Figure 7-4 are (much) lower than those shown

in Figure 7-2 due to optimizing the auto- and cross-correlations of the waveforms.

Figures 7-5 7-8 are similar to Figures 7-1 7-4, except that the sample number L

is set to be 512. As compared to the case where L = 256, improvements on the resulting

beampatterns as well as on the time correlations of the waveforms can be observed when L

is large.

Finally we consider an example with R = I and P = 10. The array is the same as

before. The corresponding correlation levels versus p with the sample number L = 256

and L = 512 are shown in Figures 7-9 and 7-10, respectively. We note from Figures 7-9

and 7-10 that even under the constant-modulus constraint, the synthesized orthogonal









waveforms have very low correlation levels. Note also that the correlation improvements

associated with relaxed practical constraints (larger PAR values) are not so obvious in this

case. Again, due to the increased sample number L, lower correlation levels are achieved in

Figure 7-10 as compared to those shown in Figure 7-9.

7.5 Concluding Remarks

Transmit waveform synthesis pl '. an important role in many MIMO radar ap-

plications including MIMO SAR imaging and transmit beampattern design. We have

presented a computationally attractive 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. We have also considered the case

where the synthesized waveforms are required to have good auto- and cross-correlation

properties in time. A number of numerical examples have been provided to demonstrate

that the proposed algorithm is quite effective.































0 50 -50 0
Angle (degree) Angle (degree)
(a) (b)


-50 0 50
Angle (degree)


Figure 7-1.


Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. The probing signals are
synthesized for N = 10, L = 256 and P = 1 by using CA under (a) PAR = 1,
(b) PAR < 1.1, and (c) PAR < 2 (the two curves corresponding to "Optimal
R" and "CA: PAR < 2" coincide with each other).







































Figure 7-2. Correlation levels versus p for the CA synthesized waveforms with N = 10,
L 256 and P 1 (R / I).































0 50 -50 0
Angle (degree) Angle (degree)
(a) (b)


-50 0 50
Angle (degree)


Figure 7-3.


Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. The probing signals are
synthesized for N = 10, L = 256 and P = 10 by using CA under (a) PAR = 1,
(b) PAR < 1.1, and (c) PAR < 2 (the two curves corresponding to "Optimal
R" and "CA: PAR < 2" coincide with each other).













































c 60

S-80
0
-100 -

-120-

-140 I I I
-10 -5 0 5 10
P


Figure 7-4. Correlation levels versus p for the CA synthesized waveforms with N = 10,
L 256 and P 10 (R / I).















































-50 0 50
Angle (degree)


-50 0 50
Angle (degree)
(b)


Figure 7-5.


Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. The probing signals are
synthesized for N = 10, L = 512 and P = 1 by using CA under (a) PAR = 1,
(b) PAR < 1.1, and (c) PAR < 2 (the two curves corresponding to "Optimal
R" and "CA: PAR < 2" coincide with each other).







































Figure 7-6. Correlation levels versus p for the CA synthesized waveforms with N = 10,
L 512 and P 1 (R / I).















































-50 0 50
Angle (degree)


-50 0 50
Angle (degree)
(b)


Figure 7-7.


Beampattern matching design with the desired main-beam width of 600 and
under the uniform elemental power constraint. The probing signals are
synthesized for N = 10, L = 512 and P = 10 by using CA under (a) PAR = 1,
(b) PAR < 1.1, and (c) PAR < 2 (the two curves corresponding to "Optimal
R" and "CA: PAR < 2" coincide with each other).

















c -60 -
0

2; -80 -

-100-

-120-

-140
-10 -5 0 5 10
P


Figure 7-8. Correlation levels versus p for the CA synthesized waveforms with N = 10,
L = 512 and P = 10 (R / I).



-CA: PAR=1
-20 --CA, PAR< 1.1
--e-CA: PARE2
-40 -

c -60-

S-80
u >~Sii ^<


Figure 7-9.


Correlation levels versus p for the
N = 10, L = 256 and P = 10 (R -


CA synthesized orthogonal waveforms with
= I).


-10 -5 0 5 10
P


Figure 7-10. Correlation levels versus p for the
N = 10, L = 512 and P = 10 (R


CA synthesized orthogonal waveforms with
= I).









CHAPTER 8
CONCLUSIONS AND FUTURE WORK

8.1 Conclusions

In this dissertation, we have studied several knowledge-aided (KA) signal processing

applications. The contributions of our work are listed as follows:

First, we have investigated KA space-time adaptive processing (STAP) algorithms in

ground moving target indication radar for wide area surveillance. We have proposed two

shrinkage methods and an ML approach to determine the optimal weight that should be

applied on the a priori knowledge. All of the proposed approaches are fully automatic,

and we have shown that they can adaptively adjust the weighting factor according to the

degree of accuracy of the a priori knowledge based on collected data. We have also shown

that the a priori knowledge can be used both directly or indirectly to improve target

detection performance of the radar.

Second, we have considered KA signal processing in a general adaptive beamforming

application. We have extended the shrinkage methods to their modified versions, in which

the array covariance matrix is considered to be a linear combination of multiple positive

semidefinite matrices. These modified approaches are still fully automatic, and they can be

formulated as convex optimization problems which can be efficiently solved with globally

optimal solution guaranteed. We have also shown that providing (partially) accurate a

prior knowledge can improve the performance of the adaptive beamformers, sometimes

significantly.

Third, we have proposed a simple and efficient cyclic algorithm for KA MIMO

radar waveform synthesis. Given the covariance matrix of the waveforms as the a priori

knowledge, which can be obtained in a previous (optimization) stage or simply prescribed,

we can synthesize a set of waveforms that, under the constant modulus or low PAR

constraints, realize (at least approximately) the given covariance matrix, and also have

good auto- and cross-correlation properties.









8.2 Future Work

In what follows, we discuss some further possible directions that are closely related to

our work.

8.2.1 KA Waveform Optimization for MIMO STAP

We can consider how the emerging MIMO radar techniques can be used with STAP

to advance the stat-of-the-art of STAP. The concept of MIMO radar STAP is first

introduced in [55]. MIMO radar STAP for multipath clutter mitigation is considered

in [70]. Design issues of the transmit waveforms for MIMO STAP based on the MIMO

ambiguity function are discussed in [94].

One interesting research topic in MIMO STAP would be the optimal transmit

waveform design. Given the a priori interference information, we can study waveform

optimization in MIMO radar STAP for improved target parameter estimation based on

various performance metrics (for example, the Cramer-Rao bound (CRB) matrix for target

parameters) by using the prior knowledge.

8.2.2 Asymptotic ML Method for Optimal Weight Determination

In C'! lpter 4, we have considered a linear combination of two covariance matrices

(the a priori covariance matrix Ro and the sample covariance matrix R) into an enhanced

estimate of the true clutter-and-noise covariance matrix in KA-STAP. And we have

presented an ML approach to determine the combination coefficients for the two terms.

We can also consider a linear combination of multiple semi-definite matrices instead of just

two terms:
s
R= A(s)R ) + BR + CI. (8-1)
s=l 1
We can again consider ML estimation of these unknown combination weights in the above

equation. The problem of estimating covariance matrices with a linear structure has been

studied in [95], and an .i-vmptotically efficient ML approach is proposed to determine the

combination coefficients in an iterative manner. One needs to investigate if it is possible to

apply the method proposed in [95] to optimal weight determination in (8-1).









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BIOGRAPHICAL SKETCH

Xumin Zhu received the B.Eng. degree in electrical engineering and information

science from University of Science and Technology of C('!i Hefei, C'ii i, in 2004. She

is expected to receive her Ph.D degree with the Department of Electrical and Computer

Engineering, University of Florida, Gainesville in 2008. Her research interests include

statistical and array processing, space-time adaptive processing, and multiple-input-

multiple-output radar.