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Matched-filterbank approaches to spectral analysis and parameter estimation

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Matched-filterbank approaches to spectral analysis and parameter estimation
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Li, Hongbin, 1970-
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xii, 122 leaves : ; 29 cm.

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Apes ( jstor )
Capons ( jstor )
Covariance ( jstor )
Estimation bias ( jstor )
Estimation methods ( jstor )
Estimators ( jstor )
Signals ( jstor )
Sine waves ( jstor )
Statistical discrepancies ( jstor )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 116-121).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Hongbin Li.

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MATCHED-FILTERBANK APPROACHES TO SPECTRAL
ANALYSIS AND PARAMETER ESTIMATION












By

HONGBIN LI












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


1999















This work is dedicated to my wife, Hong.














ACKNOWLEDGEMENTS




I would like to express my sincere gratitude to my advisor, Dr. Jian Li, for her constant support, encouragement, enthusiasm, and patience in guiding this research. My deepest appreciation goes to Dr. Petre Stoica for his numerous insightful remarks and suggestions which comprehensively influenced this work.

Special thanks are due to Drs. Jose C. Principe, Fred J. Taylor, William W. Edmonson, and David C. Wilson for serving on my supervisory committee and for their contribution to my graduate education at the University of Florida.

I also wish to thank Zhaoqiang Bi, Robert Stanfill, and other fellow graduate students with whom I had the great pleasure of interacting. Drs. Zheng-She Liu and Guoqing Liu have my gratitude for sharing many interesting discussions with me.

I would like to gratefully acknowledge all the people who helped me during my Ph.D. program.




















iii















TABLE OF CONTENTS




ACKNOWLEDGEMENTS ............................ iii

LIST OF TABLES ................................. vii

LIST OF FIGURES ................................ viii

A BSTRACT . . . . . . . . . . . . . . . . . . xi

CHAPTERS

1 INTRODUCTION .................. .......... 1

1.1 Background and Scope of the Work ................. 1
1.1.1 Capon, APES, and MAFI Spectral Estimators ...... 1 1.1.2 Efficient Implementation .................... 4
1.1.3 Amplitude Estimation ................... 5
1.1.4 Covariance Sequence Estimation ............... 5
1.2 Significance and Contributions .................. 6
1.3 Organization of the Dissertation ..................... 7

2 LITERATURE SURVEY ................. ....... 9

2.1 Filterbank Approaches and Capon Estimator .......... 9
2.2 Covariance Estimation ................. ..... 13

3 MAFI APPROACH TO SPECTRAL ESTIMATION .......... 16

3.1 Forward and Backward Data Vectors ............... 16
3.2 MAFI Filters....... ............. ....... 19
3.2.1 Capon Filter ....... .... ......... 20
3.2.2 APES Filter ........ .... .. ...... 21
3.2.3 Another Matched Filter .................. 22
3.3 Analyses of MAFI Approaches .. .................. 24
3.3.1 Computational Complexity ........ ......... 24
3.3.2 Statistical Performance .............. 25
3.4 2-D Extensions .......... .. ............. 27
3.5 Numerical Examples .............. ...... .. 31
3.5.1 1-D Complex Spectral Estimation .... .... .... 31
3.5.2 2-D Complex Spectral Estimation ......... . 33
3.6 Summary ................. ............ 36




iv









4 EFFICIENT IMPLEMENTATION OF CAPON AND APES ...... 39

4.1 Introduction .......... ... .. ....... 39
4.2 Efficient Implementation of APES .................. 39
4.3 Extension to Capon . ....... .................. 43
4.4 Numerical and Experimental Examples ............ 44
4.5 Summary ........... ...... ............. 45

5 AMPLITUDE ESTIMATION ................... ... 47

5.1 Introduction .......... .. ........... 47
5.2 LS Amplitude Estimators ......... ...... ... . 50
5.2.1 LSE(1, 0, K ) .. .... .. .. . .. .. . . 50
5.2.2 LSE (1,0,1) .......... ... .. ....... 52
5.3 WLS Amplitude Estimators ................... .. 54
5.3.1 W LSE(L, 0, K) ........... .. ... ...... .. 54
5.3.2 WLSE(L, 0,1) ........................57
5.4 MAFI Amplitude Estimators ................... 58
5.5 Numerical Examples ......................... 62
5.5.1 Estimation Performance versus SNR ........... 63
5.5.2 The Effect of M ......... ............ .. 68
5.6 Application to System Identification ............... 68
5.6.1 System Identification Using Amplitude Estimation . . 70 5.6.2 Numerical Examples . . . .... ....... 73
5.7 Summary ................... . ............ 78

6 CAPON ESTIMATION OF COVARIANCE SEQUENCES ...... 80

6.1 Introduction . . . . . . . . . . . . . . 80
6.2 Standard Covariance Estimator and Outlook .......... 82
6.3 Capon PSD Estimator ........................ 84
6.4 Capon Covariance Estimator ................... 87
6.4.1 Exact Method ................. ...... 87
6.4.2 Approximate Method ................... 89
6.4.3 Computational Aspects .................. 90
6.5 Numerical Results ................... ....... 91
6.5.1 ARMA Covariance Estimation .............. 92
6.5.2 AR Coefficient Estimation for ARMA Signals ....... 97 6.5.3 MA Model Order Determination ............. 98
6.6 Sum m ary . . . . . . . . . . . . . . .. 100

7 CONCLUSIONS ............................ 102

7.1 Summary Remarks ..... ................... .. 102
7.2 Future Work ................. .......... 105

APPENDIXES

A PROOF OF (3.42) AND (3.43) ................ ...... 106


B PROOF OF THEOREM 3.3.1 ...... ................ 108




V









C PROOF OF THEOREM 3.3.2 ..................... 110


D PROOF OF THEOREM 6.4.1 ......... .............. 114


REFERENCES ............................. 116

BIOGRAPHICAL SKETCH ................... ........ 122















































vi















LIST OF TABLES




5.1 Choice of M for the WLS and MAFI amplitude estimators ...... 70


6.1 Comparison of the computational burdens of the standard and Capon

methods with N = 256 and N = 512. .................. 91

6.2 The ARMA processes used in the numerical simulations. ....... 92




































vii














LIST OF FIGURES




1.1 Synthetic Aperture Radar (SAR) images of a simulated MIG-25 airplane obtained by using (a) 2-D FFT, (b) 2-D windowed FFT, (c) 2-D

Capon, and (d) 2-D APES ......................... 3

3.1 The I-D complex amplitude of the sum of 15 sinusoids used in the

simulations. (a) Real part; (b) Imaginary part. ........ . 33

3.2 Empirical bias and variance of the 1-D MAFI estimators as SNR1

varies when N = 64 and M = 15. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of the imaginary part of the estimated

am plitude . . . . . . . . . . . . . . . . . 34

3.3 Empirical bias and variance of the 1-D MAFI estimators as the filter

length, M, varies when N = 64 and SNR1 = 20 dB. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of the imaginary part of the

estimated amplitude. ........... ......... ....... 35

3.4 Empirical bias and variance of the 2-D MAFI estimators as the SNR1

varies when N1 = N2 = 32 and M1 = M2 = 8. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of the imaginary part of the

estimated amplitude. .............. ........... ..37




viii








3.5 Empirical bias and variance of the 2-D MAFI estimators as the filter

length, M = M = M2, varies when N1 = N2 = 32, and SNR1 = 20 dB. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of

the imaginary part of the estimated amplitude. ......... . 38


4.1 SAR images obtained from the ERIM data by using (a) 2-D FFT, (b)

2-D windowed FFT, (c) 2-D Capon, and (d) 2-D APES. ....... 46

5.1 PSD of the test data that consist of three sinusoids and an AR(1) noise

process .................................. 63

5.2 Empirical MSEs and the CRB versus local SNR when N = 32, M = 8,

and the observation noise is colored (an AR(1) process). (a) a3. (b) a,. 65

5.3 Empirical MSEs and the CRB versus local SNR when N = 32, M = 8,

and the observation noise is white. (a) a3. (b) al. ........ 67

5.4 Empirical MSEs and the CRB versus M when N = 32 and the observation noise is colored (an AR(1) process with a2 = 0.001). (a) a3.

(b) a ................. ................ 69

5.5 Averaged RMSEs and the number of flops versus N for the first system

when the observation noise is white (a2 = 0.01) and M = 20 for APES1 and MAFI1. (a) RMSE of a-parameters. (b) RMSE of b-parameters.

(c) Number of flops. ................... ........ 76

5.6 PSD estimate of the output of the first system corrupted by white

noise with a2 = 0.01 and N = 200. ................... 77

5.7 Averaged RMSEs and the number of flops versus N for the second

system when the observation noise is colored (an AR(1) process with a2 = 0.01) and M = 20 for APES1 and MAFI1. (a) RMSE of aparameters. (b) RMSE of b-parameters. (c) Number of flops ... 78


ix









6.1 Power spectral density estimates by using Capon-i and Capon-2. The

plots are the averages of 100 independent realizations. (a) N = 256, M =

50; (b) N = 32,M = 10. ....... ..... ... ........ .. 86

6.2 Pole-zero diagrams for ARMA test cases. (a) ARMA1; (b) ARMA2;

(c) ARMA3; (d) ARMA4..... ........ ............ ..93

6.3 True power spectral densities. (a) ARMAl; (b) ARMA2; (c) ARMA3;

(d) ARMA4 ............... .. ........... 94

6.4 True covariance sequences. (a) ARMA1; (b) ARMA2; (c) ARMA3;

(d) ARM A4...................... ............ 95

6.5 Covariance sequence estimation with N = 256 and M = 50. The

mean-squared errors (MSEs) of the covariance estimates, normalized with respect to r(0), are based on 100 independent realizations. (a)

ARMA1; (b) ARMA2; (c) ARMA3; (d) ARMA4 ............ 96

6.6 The AR coefficient estimation of the ARMA signals via the overdetermined modified Yule-Walker method with N = 256 and M = 32.

The curves are the summations of the mean-squared errors (MSE) of all the AR coefficient estimates versus the numbers of included equations, which have been set as 4, 8, 16, 32, 64 and 128, respectively. The MSE curves are based on 100 independent realizations. (a) ARMA1;

(b) ARMA2; (c) ARMA3; (d) ARMA4 .................. 99

6.7 10 superimposed realizations of the MA covariance sequence estimates

with N = 64 and M = 6. (a) The standard method; (b) The Capon

method. ...... ...... .... ............... 100










x















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.




MATCHED-FILTERBANK APPROACHES TO SPECTRAL
ANALYSIS AND PARAMETER ESTIMATION By

Hongbin Li

May 1999


Chairman: Jian Li
Major Department: Electrical and Computer Engineering

MAtched-FIlterbank (MAFI) estimators represent a general class of methods that make use of a set of matched filters for various estimation purposes. This dissertation investigates using MAFI approaches for complex spectral analysis, amplitude estimation for sinusoidal signals, and covariance sequence estimation.

For complex spectral analysis, we show that the widely used Capon and the recently introduced APES estimators are both members of the MAFI approach, though neither was originally derived in the MAFI framework. We prove that, to within a second-order approximation, Capon is biased downward whereas APES is unbiased, and that the bias of the forward-backward Capon is one half that of the forward-only Capon. We also show that Capon and APES are of similar computational complexities and both are more involved than most Fourier-based methods, especially for 2-Dimensional (2-D) data. Efficient implementation schemes which substantially




xi









reduce the computational requirement are presented for the Capon and APES estimators.

For amplitude estimation, we describe a large number of estimators which can be categorized as the Least Squares (LS), Weighted Least Squares (WLS), and MAFI methods. While all these methods are asymptotically statistically efficient, their performances in finite length of data samples are quite different. Specifically, we show that the WLS and MAFI methods outperform the LS methods whenever the observation noise is colored; we also show that the MAFI approach is very general and includes the WLS approach as a special case.

For covariance sequence estimation, we present a Capon covariance estimator by Fourier inverting the Capon Power Spectral Density (PSD) estimates. We describe the Fourier inversion in both an exact and approximate ways, of which the latter is computationally more attractive but with some minor accuracy loss. We show that the Capon covariance sequence estimates are in general better than the widely used sample covariance sequences.

























xii














CHAPTER 1
INTRODUCTION

This dissertation is concerned with spectral analysis and parameter estimation using MAtched-FIlterbank (MAFI) approaches. This chapter serves as a general introduction to the background and scope of the work. Significance and contributions are also highlighted in this chapter.


1.1 Background and Scope of the Work

1.1.1 Capon, APES, and MAFI Spectral Estimators

Spectral estimation is an important data analysis tool which has found applications in many diverse fields including speech analysis, telecommunications, radar and sonar systems, biomedical and seismic signal processing, and economics [75]. There are two broad classes of approaches to spectral analysis: non-parametric approaches and parametric approaches. While non-parametric methods typically postulate no model for the observed data, parametric approaches do assume some model so that the spectrum is represented by a set of parameters, thereby reducing the spectral estimation problem to that of estimating the parameters of the model. Parametric methods may offer more accurate spectral estimates than non-parametric methods, provided that the data indeed observe the model assumed by the former. In the more likely cases where an accurate data model is not available, parametric methods will be sensitive to model mismatch; hence using the robust non-parametric approaches in such cases may be a better choice. Recent studies in the literature show that there has been renewed interest in non-parametric approaches to spectral estimation.




1






2

Filterbank approaches to spectral estimation belong to the class of nonparametric spectral estimators. An important member of filterbank approaches is the Capon spectral estimator [16] [18]. Unlike the classical Fourier-based methods which are data-independent, the Capon spectral estimator adapts to the processed data in a manner so that the noise components of the data are rejected substantially. During the past few decades, the Capon spectral estimator has been widely used because its higher resolution and lower sidelobes give it an advantage over the Fourier-based methods. Additionally, its robustness and less variability make it preferable to the parametric methods [75] [41].

In a recent study, it was empirically observed that using the Capon estimator for complex spectral estimation gives biased spectral estimates whereas the newly introduced APES (Amplitude and Phase EStimation) method appears to be unbiased [52] (see Figures 1.1(a) to 1.1(d) for performance differences of using the Fourier-based methods, Capon, and APES for spectral estimation). The fact that both Capon and APES make use of a set of finite impulse response (FIR) matched filters was observed in [70]. A number of results on the performance differences of the Capon and APES estimators were also reported therein. However, the study in [70] was somewhat limited since it only considered the forward-only (FO) Capon and APES estimators. Owing to the general belief that forward-backward (FB) approaches to spectral estimation usually provide more accurate results and are used more often than their FO counterparts, it is of interest to conduct a study that investigates how the FB Capon and APES spectral estimators perform when compared with one another as well as with their FO counterparts.

In some sense, filterbank approaches reduce the problem of spectral estimation to a filter design problem subject to some constraints [46]. In Section 3.2, we discuss a general rule for choosing the impulse responses of the filterbank so that the signal-tonoise ratios (SNRs) at the outputs of the filterbank are maximized. It follows that the






3






















(a) (b)













(c) (d) Figure 1.1: Synthetic Aperture Radar (SAR) images of a simulated MIG-25 airplane obtained by using (a) 2-D FFT, (b) 2-D windowed FFT, (c) 2-D Capon, and (d) 2-D APES.






4

resulting filters are matched filters and we refer to this class of approaches to spectral estimation as MAtched FIlterbank (MAFI) approaches. We show that both Capon and APES are members of MAFI approaches, although none of them was originally derived in the MAFI framework (see [16] [52] for their original derivations). MAFI approaches to spectral estimation may also be used to devise new spectral estimators. Even though we show in Section 3.2.3 that a reasonable implementation of a seemingly novel MAFI spectral estimator reduces back to APES as well, it remains an open issue whether other interesting MAFI spectral estimators exist.

The MAFI interpretation also provides insights into the Capon and APES estimators and the relationship between them. Specifically, in the framework of MAFI approaches, we show by means of a higher-order expansion technique in Section 3.3 that the 1-D (one-dimensional) Capon estimator indeed underestimates the true complex spectrum while the 1-D APES is unbiased (to within a second-order approximation); we also show that the bias of the FB Capon is one half that of the FO Capon. Furthermore, we show that these results can be extended to 2-D (two-dimensional) Capon and APES estimators.

1.1.2 Efficient Implementation

For 2-D applications, a major concern of using Capon or APES is their computational loads which in general are much heavier than those of the traditional Fourier-based approaches. For a SAR image of size 128 x 128 formed from a 32 x 32 data matrix, as those in Figure 1.1, the number of flops required by Capon and APES implemented in the conventional way are 2.3 x 104 and, respectively, 3.0 x 104 times that of those by the Fourier-based methods. It should be mentioned that a SAR image of 128 x 128 is relatively small. As the size of the image increases, the amount of computation by Capon or APES increases drastically. In Chapter 4, we address the issue of how to efficiently implement Capon and APES for spectral estimation.






5

1.1.3 Amplitude Estimation

Another problem related to complex spectral estimation is amplitude estimation for sinusoidal signals where it is assumed that the number and frequencies of the sinusoids are known a priori. The observation noise is stationary and maybe colored. In the case that the noise can be modeled exactly, the Maximum Likelihood (ML) methodology can be used and the ML amplitude estimates are statistically efficient. However, an exact model of the observation noise is usually not available and, moreover, ML methods are in general very sensitive to inaccurate model information. As such it may be a better choice to use methods that do not model the noise exactly.

In Chapter 5, we describe a relatively large number of amplitude estimators which assume no model except for stationarity for the observation noise. The amplitude estimators can be categorized as three general classes, namely Least Squares

(LS), Weighted Least Squares (WLS), and MAFI approaches to amplitude estimation, which are all asymptotically statistically efficient. For finite length of data samples, however, their behaviors are quite different. We show that under certain circumstances, the MAFI approach to amplitude estimation is equivalent to the WLS approach, and yet the former is more general and includes the latter as a special case. We also show that the MAFI and WLS methods in general give more accurate amplitude estimates than the LS methods.

1.1.4 Covariance Sequence Estimation

The fact that covariance (the terms covariance and autocorrelation are used interchangeably with one another in this study) function and Power Spectral Density (PSD) are a Fourier transform pair makes the problem of covariance sequence estimation a research topic that is closely related to spectral estimation. Sample covariance sequences have been widely used in signal processing because of its computational simplicity and its consistency supposing that the given signals are ergodic






6

to the second order. When only a finite number of samples are available, using sample covariance sequences implies that the data beyond the observed duration either is zero or repeats itself periodically, which is certainly not a realistic assumption. There have been several attempts in the literature to derive other covariance estimators (see Chapter 2 for some details). It is known that the sample covariance sequence and the data periodogram constitute a Fourier transform pair. It is also known that the periodogram is a statistically inefficient (in fact inconsistent) PSD estimator [75]. This observation suggests that better covariance estimators may be obtained by Fourier inverting better PSD estimators. Since the MAFI PSD estimates are in general more accurate than periodograms, we propose a new covariance sequence estimator by Fourier inverting the MAFI PSD estimates. Specifically, we make use of the Capon PSD and such an approach is referred to as the Capon covariance sequence estimator. The same methodology can be similarly applied to the APES PSD estimates, though the APES covariance estimates are usually similar to the Capon covariance estimates. The reason is that, despite their different performance for complex spectral estimation, the Capon and APES PSD estimators usually behave quite similarly, especially for continuous spectra (also see the discussions in Section 3.3.2). It is observed that Capon covariance estimates are in general better than sample covariance estimates in terms of mean-squared errors (MSEs).


1.2 Significance and Contributions

The main results of this dissertation are as follows.

1. We introduce a new general class of MAFI approaches to spectral estimation.

We show that the Capon and APES estimators, though originally not derived

in the MAFI framework, are both members of the MAFI class.

2. To within a second-order approximation, we prove that the Capon estimator

gives biased (downward) complex spectral estimates whereas the APES method






7

is unbiased; we also prove that the bias for the FB Capon is one half that of the FO Capon. These theoretical results, supplemented with the empirical observation that Capon usually underestimates the spectrum in samples of practical length while APES is nearly unbiased, are believed to provide compelling reasons for preferring APES to Capon in most practical applications.

3. We present efficient implementation techniques for the MAFI spectral estimators. We show that by using such techniques the amount of computation

required by the Capon or APES estimator is significantly reduced.

4. The MAFI idea is also extended for amplitude estimation for sinusoidal signals

in colored noise. Specifically, we make extensions to the Capon and APES algorithms to multiple sinusoids with known frequencies. Furthermore, we describe

a generalized MAFI approach to amplitude estimation for multiple sinusoids.

5. A new covariance sequence estimator is presented by Fourier inverting the

Capon spectral estimates. The Capon covariance sequence estimates are shown to be more accurate than the conventional sample covariance sequence estimates.


1.3 Organization of the Dissertation

The dissertation is organized as follows. Chapter 2 gives a literature survey of such topics as filterbank approaches to spectral analysis, the Capon spectral estimator and its applications, and covariance estimation. In Chapter 3, we first introduce the MAFI approach to complex spectral estimation. We next show that the Capon and APES estimators are both members of the MAFI approach, followed by computational and statistical analyses of the MAFI spectral estimators. Extensions to the 2-D case are also included. Chapter 4 addresses the implementation issue of the MAFI spectral estimators. Chapter 5 discusses amplitude estimation for sinusoidal






8


signals from observation corrupted by colored noise. The MAFI concept introduced in Chapter 3 is extended for amplitude estimation. In Chapter 6, we investigate covariance sequence estimation using the Capon spectral estimates. Finally, we summarize this work and outline future work in Chapter 7.














CHAPTER 2
LITERATURE SURVEY

Historical and modern perspectives on the general topics of spectral analysis and parameter estimation have been well documented in the literature [42] [62]. Many classical articles, both theoretical and application-oriented, have been reprinted [20] [43]. Excellent texts are also available [75] [41] [55]. In this chapter, we give a brief review of a number of subjects that are related to our work, namely filterbank approaches to spectral estimation, the Capon method and applications, and covariance estimation.


2.1 Filterbank Approaches and Capon Estimator

Unless the observed signal can be modeled with a finite number of parameters, estimating the spectrum of a signal based on a finite length of observations is an ill-posed problem from a statistical standpoint, since we are required to estimate an infinite number of independent spectral values based on a finite number of samples. An assumption made by filterbank approaches and most other non-parametric methods is to assume that the PSD of the observed signal is (nearly) constant over a narrowband around any given frequency. Naturally one can proceed by passing the observed signal through a bandpass filter, which is swept through the frequency band of interest, and estimating the complex amplitude if complex spectral estimation is of interest (or measuring the filter output power and dividing it by the filter bandwidth if PSD estimation is the desired goal), a procedure adopted by all filterbank approaches. Obviously, how to choose the narrowband filters is a critical issue of filterbank approaches. Even though some of the classical Fourier-based methods,



9






10


such as periodogram by Schuster [67] and its various variations including the famous Blackman-Tuckey's method [5], can also be cast in the framework of filterbank approaches, originally they were not designed in such a manner. In other words, those methods made no attempt to purposely design a good bandpass filter to achieve some desired characteristics.

A notable example of filterbank approaches to spectral estimation is the REFIL (REfined FILter) method which was first introduced in [77] and was further developed in [56] (also discussed in [8] [59] [61]). The REFIL method is close in spirit to the Daniell approach [22] to reducing the variance of the periodogram. That is, REFIL does not split the available samples in shorter stretches. The REFIL idea is to design a bank of filters which pass the signal components within the passbands as much as possible relative the total power and, in the mean time, attenuate the frequencies outside the corresponding passbands. It turns out that REFIL in general gives better spectral estimates than the traditional Fourier-based methods. Nevertheless, the REFIL filters share a common characteristic with the Fourier-based ones: they are all data-independent in the sense that they do not adapt to the received data in any way. Presumably, it would be beneficial to take the data properties into account when designing filterbanks.

The Capon method is one (and perhaps the most famous one) of such datadependent filterbank spectral estimators. The Capon estimator was first named as the Maximum Likelihood Method (MLM) due to the the ML and Gaussian process context used in Capon's original work [16]. It turns out that MLM is a misnomer since it is not an ML spectral estimator and it does not possess any of the properties of an ML estimator. In [41] and [55], Capon's method was referred to as the Minimum Variance Spectral Estimator (MVSE) because it is derived by minimizing the variance of the output of a narrowband filter. Even the name MVSE is inaccurate in the sense






11

that Capon spectral estimates do not possess the minimum variance property. In this study it is simply named the Capon spectral estimator.

The Capon spectral estimator was originally introduced by Capon [16] [18] for applications in multi-dimensional seismic array frequency-wavenumber analysis. It is reformulated by Lacoss [44] for applications to 1-D time-series problems. The constraints adopted by the Capon method are such that the signal at the current frequency is passed undistorted (with unit gain) while the output power of the overall frequency domain is minimized. A number of researches have been conducted to study the Capon spectral estimator. Specifically, Capon and Goodman demonstrated [19] [17] that the Capon spectrum has a mean and a variance that behaves like the averaged periodogram; that is, Capon spectra are usually less variable than periodograms. Additionally, Lacoss's empirical study [44] suggested that the resolution of the Capon method is between that of Burg's AutoRegressive (AR) spectral estimator [9] [12] and that of the periodogram. Lacoss's study also suggested that the statistical variability of Capon is less than that of the AR estimator. Burg later proved that the reciprocal of the Capon spectrum of order M, the length of the Capon filter, is equal to the average of the reciprocals of the AR spectra from order 0 to M [11]. Such an averaging effect explains the empirical observations made by Lacoss. Further researches in quantizing the resolution properties of Capon spectra were reported in [21] [58].

Since Capon's method is a basic spectral estimator, it may be used in any application where the spectrum of the studied signal plays an important role. Indeed, ever since its first appearance, the Capon spectral estimator has been widely used in many areas including radar, sonar, communications, imaging, geophysical exploration, astrophysical exploration, and biomedical engineering (see [35] [79] and the references therein). Rather than making a futile attempt at making a full document






12

in great length, we describe in the following a few typical applications of the Capon estimator.

An interesting application of Capon's method is beamforming. Beamforming is used in conjunction with an array of sensors to provide a versatile form of spatial filtering. The objective of beamforming is to estimate the signal arriving from a desired direction in the presence of noise and interfering signals. If the desired and interfering signals occupy the same (temporal) frequency band, then temporal filtering cannot be applied to distinguish signal from interference. However, since the desired signal and interfering signals typically originate from different locations, such spatial diversity can be exploited to separate signal from interference using a spatial filter. In 1972, Frost made use of a linear constrained optimization technique and introduced an adaptive beamformer [27], referred to as the LCMV (Linearly Constrained Minimum Variance) beamformer in the array signal processing community. The basic idea of the LCMV beamforming is to constrain the response of the beamformer so that signals from the direction of interest are passed with specified gain while minimizing the output power due to interfering signals and noise arriving from other directions. One would immediately notice the similarity to the constraints adopted by Capon. Indeed, the LCMV beamformer is a direct extension of the temporal Capon filter to the spatial domain. Among the so-called statistically optimum beamformers, LCMV is perhaps the most popular one since it needs no auxiliary channels as required by the Multiple Sidelobe Canceller (MSC) [2], and, unlike the class of optimum beamformers proposed by Widrow et al. [83] which require reference signals, it is blind. A useful structure for LCMV implementation is the Generalized Sidelobe Canceller (GSC) [32]. GSC represents an alternative formulation of the LCML beamformer which changes the constrained optimization problem of LCML to an unconstrained one. The unconstrained nature lends GSC to adaptive implementation more readily than the original LCMV beamformer and hence GSC is the one used more often






13


in practise. GSC also found applications in direct-sequence Code-Division MultipleAccess (CDMA) communication systems for blind multiuser detection [36] [66] [65]. An important extension of the LCML beamformer was made in [7] [6] for applications in airborne radar systems. A typical signal environment faced by such systems consists of strong clutter/interference of complicated angle-Doppler spectrum which is unknown and may be varying in both time and space. The extension made by Brennan el al is to simultaneously combine the signals received on multiple elements of an antenna array (the spatial domain) and from multiple pulse repetition periods (the temporal domain). Such a technique is referred to as space-time adaptive processing that has received much research interest recently [80] [82].

Another interesting application of the Capon spectral estimator is SAR imaging [57] [3]. Conventional SAR imaging techniques are the FFT (Fast Fourier Transform) or windowed-FFT methods. A number of parametric spectral estimation methods have also been used for SAR imaging [26] [38] [33], though their interest in SAR imaging is limited because of their sensitivity to model errors. A comparative study in [24] showed that adaptive filterbank approaches such as Capon offers good SAR images and enjoys the advantage of robustness as compared to parametric methods. A number of modified Capon methods have also been suggested for SAR imaging [25] [4].


2.2 Covariance Estimation

One type of the covariance estimation problems is to estimate the covariance sequence from a finite number of data samples. A standard technique for estimating covariance sequences uses the biased or unbiased sample covariance estimator. The biased covariance estimator is more commonly used because it provides smaller MSEs than the unbiased one and guarantees the covariance estimates to be non-negative [55]. The problem of the sample covariance estimators is the unrealistic windowing






14

they assume on the observed data. By exploiting the AR spectral estimator, Burg proposed a technique that can offer covariance sequence estimates for any desired lag and hence avoids the windowing problem suffered by the sample covariance estimators [10]. However, it was shown in [78] that the Burg covariance estimates are less accurate and more variable than the sample covariance estimates. In Chapter 6, we describe a new method for covariance sequence estimation based on Capon spectra.

Estimating structured covariance matrix of the observed data vectors represents another type of the covariance estimation problem. For example, the covariance matrix of a stationary complex signal is Hermitian and Toeplitz. However, the sample covariance matrix obtained from a finite number of data samples seldom has this structure. Structured covariance matrix estimation is of importance in a variety of applications including array signal processing and time series analysis [28]. An intuitive way to obtain structured covariance estimates is to force the desired structures on the sample covariance matrix, a methodology adopted by the Iterated Toeplitz Approximation Method (ITAM) [81] [15] [84]. Specifically, ITAM alternatively makes use of rank approximation (via singular value decomposition) and Toeplitzation along the diagonals until convergence is reached. Obviously, such a method is by nature heuristic and no optimality can be associated with it, though the ITAM covariance matrix estimate is in general closer to the true covariance matrix than the sample covariance matrix in the Frobenius norm sense. Optimum structured covariance matrix estimate may be obtained by maximizing the corresponding likelihood function as considered in [1] [13] [23] [29] [85]. However, since there exists no closed-form solution to the complicated nonlinear ML estimation problem for Hermitian Toeplitz matrices, the ML methods proposed in these studies are iterative and computationally involved, and, moreover, they are not guaranteed to yield the global optimal solution, which to some degree limits the interest in using the ML structured covariance matrix estimate in practical applications. An approximate ML method that makes use of the






15

Extended Invariance Principle (EXIP) [76] was recently presented in [49] [50]. This method provides asymptotic (for large samples) ML estimation for structured covariance matrices. A closed-form solution for the estimation Hermitian Toeplitz matrices is obtained which makes the proposed method computationally much simpler than most existing Hermitian Toeplitz matrix estimation algorithms. Additionally, it was also shown that using the technique in such array processing algorithms as MUSIC [64] and ESPRIT [63] makes them achieve the Crambr-Rao Bound (CRB) for angle estimation, i.e., the best performance for any unbiased methods.














CHAPTER 3
MAFI APPROACH TO SPECTRAL ESTIMATION

Filterbank approaches decompose the observations {y(n)} 1 of a stationary signal y(t) as [51] [47] [73] [72]

y(n) = a(w)ewn +e,(n), n = 0,1,...,N 1; w [0,2r), (3.1) where a(w) denotes the complex amplitude of the sinusoidal signal with frequency w and e,(n) denotes the noise (or residual) term at frequency w, assumed to be zero-mean. The problem of interest is to estimate a(w) for any given w.

Briefly stated, most filterbank spectral approaches address the aforementioned problem by following two main steps: (a) pass the data {y(n)} through a bandpass filter with varying center frequency w; and (b) obtain the estimates, &(w), for w E [0, 27r), of the complex amplitude from the filtered data. The bandpass filter used is usually an M-tap FIR filter with its coefficient vector given by h = hi h2 ... (3.2) where (.)T denotes the transpose. (The choice of M is discussed in Section 3.5.) Observe that the notation emphasizes the dependence of the vector in (3.2) on the center frequency w. Although rules for choosing he vary, a rather general one for the choice of a matched filter is discussed in Section 3.2.


3.1 Forward and Backward Data Vectors

Let

y(1) y) ( ) y( M-) 1, (3.3)


16






17

be the overlapping vectors constructed from the data {y(n)}, where L = N-M+1. In what follows y(1) is referred to as the forward data vector. Let e,(1), 1 = 0, 1,..., L-1, be formed from {e,(n)} in the same manner as y(l) are from {y(n)}. Then the forward vectors can be written as y(1) = [a(w)aM(w)]ewl + (1), (3.4) where aM(w) is the steering vector and is given by aM(w) = 1 e ... ej(M-1) (3.5) Likewise, the backward data vectors are constructed as


(1) y*(N-1-1) y*(N-1-2) ... y*(N - M)

S= 01,. L 1, (3.6) where (-)* denotes the complex conjugate. Let e,(1), 1 = 0, 1,..., L 1, be formed from {e,(n)} the same way as (1l) from {y(n)}. Then the backward vector can be written as

?r(1) = [d(w)aM(w)]ejwil e,(1), (3.7) where

d(w) = a*(w)e-(1)W. (3.8) It is straightforward to verify that the forward and backward vectors are related by the following complex conjugate symmetry property: k(1) = Jy*(L 1 1), (3.9)


where J denotes the exchange matrix whose anti-diagonal elements are ones and all the others are zero.

Suppose that the initial phase of the sinusoidal signal in (3.1) is a random variable uniformly distributed over the interval [0, 21r) and independent of the noise






18

term. By making use of this assumption as well as (3.9), the covariance matrix of y(1) or, equivalently, of f(1), is given by

R ~ {y(l)yH(1)} = E {(l)H(1)} = 2a()a)HM(w) + Q(w), (3.10)


where (.)H denotes the conjugate transpose and Q(w) is the noise covariance matrix and is given by

Q(w) E H()} = E {()H()} (3.11) Note that both R and Q are Hermitian Toeplitz matrices.

The Forward-Only (FO) filterbank approaches use only the forward data vector to estimate R, which is the forward sample covariance matrix:
1 L-1
RFO = Y'(1)yH(1). (3.12) 1=0

The Forward-Backward (FB) filterbank approaches use the average of the forward and backward sample covariance matrices to obtain the estimate of R:


RFB = 1(RFO + iBO), (3.13) where BO denotes the backward sample covariance matrix:
1 L-1
RBO = Jl(H(1)H = JftoJ. (3.14) 1=0

The RFB in (3.13) is Hermitian but no longer Toeplitz. By using (3.9), one can show that RFB is a persymmetric matrix [30], i.e.,


IFB = JFBJ. (3.15) Since R is persymmetric, one would expect that RFB is a better estimate of R than the non-persymmetric RFO.






19

3.2 MAFI Filters

By definition, the matched filter is designed such that the corresponding signal-to-noise (SNR) ratio in the filter output is maximized; that is, hw = arg max h (w) (3.16) he hHQ(w)h,
The solution is obtained by making use of the Cauchy-Schwartz inequality (see, e.g., [75]):
Q-' (w)a(w)
he = (3.17) where Q(w) is assumed to be invertible. It is readily checked that the solution in (3.17) satisfies

h HaM(w) = 1, (3.18) which implies that the filter given in (3.17) passes the frequency w undistorted. By making use of this observation and of (3.4) and (3.7), we have

hH( (w)' + hH (), I = 0, 1, ..., L 1, (3.19) and

hH~) =e-j(N-1)W (w) +hHe,(1), 1= 0,1,..., L 1. (3.20) The least squares (LS) estimate of a(w) obtained by using only (3.19), i.e., the forward data vectors, is given by

FO(W) = hHg(w), (3.21) whereas the least squares (LS) estimate of a(w) obtained by using (3.19) and (3.20), that is, both the forward and backward data vectors, is given by <^FB(W) = 2 [hHg(w) + e-j(N-)wgH(w)h] (3.22) where g(w) and g(w) are, respectively, the normalized Fourier transforms of the forward and backward data vectors:
1
g(w) = Ey~(1)e, (3.23) 1=0






20

1 L-1
g(w) = y(1)e-j"W. (3.24) 1=0
Since Q(w) is Toeplitz and, therefore, persymmetric, one can show that the h, in (3.17) satisfies [52]

Jh* = hee-j(M-1)w. (3.25) Consequently, it follows after some calculation that (3.22) is equivalent to


&FB(W) = hH (w). (3.26) Hence, due to the persymmetry of Q(w), the FB estimate of a(w) has the same form as the FO estimate of a(w) (see (3.21) (3.26)) However, the filter vector h, obtained with the FO approach is in general different from that corresponding to the FB approach [52].

Although neither Capon nor APES was derived in the MAFI framework (for original derivations of these methods, we refer to [16] [75] [52]), in what follows we show that two natural estimators of Q(w) in (3.17) lead to the Capon and APES filters, respectively. More interestingly, we also show that even though a third natural estimator of Q(w) gives a new filter which is different from the former two, the spectral estimator corresponding to the new filter turns out to be equivalent to APES as well.

3.2.1 Capon Filter

By (3.10), one natural choice is to estimate Q(w) as


QCapon(W) I a(w)12 aM(w)aH(w), (3.27) where &(w) denotes an estimate of a(w) and R denotes either RFO or RFB, which, in turn, leads to FOC or FBC, respectively. (For notational simplicity, in what follows the forward-only Capon, APES, and MAFI spectral estimators will be referred to as FOC, FOA, and FOM, respectively. Likewise, FBC, FBA, and FBM represent the corresponding forward-backward counterpart estimators.) By making use of the






21

matrix inversion lemma [30], one can see that the second term in (3.27) has no influence on the he in (3.17). Hence, when (3.27) is substituted into (3.17), the matched filter reduces to the Capon filter [16] [44]: hCapon -'aM (w)(3.28) a (w) aM ) (3.28)

Observe that Q(w) is persymmetric for either FOC or FBC. By substituting (3.28) into (3.21) or (3.26), we obtain the Capon estimate of ca(w): am (w) 1(w)
&Capon(w) = aH(w)RlaM(w ) (3.29)

3.2.2 APES Filter

Ignoring the fact that aM(w) is known, we obtain the LS estimate of the vector a(w)aM(w) in (3.4) as
[a(w)aM(w)] = g(w). (3.30) Inserting (3.30) into (3.27) along with RFO substituted yields the FOA estimate of Q(w):

QFOA(W) = IFO g(w)gH(w). (3.31) A persymmetric estimate of Q(w) can be obtained by using both the forward and backward data vectors:


QFBA(w) = + JQFOA(w)J = RFB G(w)GH(w), (3.32) where

G(w) = g(w) g() (3.33) and we have used the fact that Jg*(w) = ej(L-1)g(w). Hence by (3.17), we obtain the the APES filter [52]:

hAPES = Q-(w)aM(w)
h' (w)Q (w)a(w)(3.34)
aM ()-1 )RM






22

where Q(w) denotes either QFOA or QFBA which corresponds to FOA or FBA. Consequently, the APES estimate of a(w) is given by (see (3.22) and (3.26)) &APES(w) = (3.35)

3.2.3 Another Matched Filter

Equations (3.4) and (3.7) suggest another way to estimate the FO and FB estimate of Q(w) (in what follows we sometimes omit the dependence on w for notational convenience):

QFOM( )
1 L-1
L= [y(l) &(w)aM(w)e"w] [Y(l) &(w)aM(w)ejw ]H
1=0
= RFO a -aaMgH + &aMaHM, (3.36) nFO

QFBM (w)
L-1
2L E (Y() &(w)a(w)e ) (() &(w)a(w)ew)H
1=0

+ (() a(w)aM(w)e3w) ((1) (W)aM(w)eiw)H]

= IFB [& + g a H HM1 M[& ~1H g+12 aMaHM, (3.37) QFB
where &(w) and &(w) = &*(w)e-j(N-1) denote some estimates of a(w) and &(w), respectively. By using the matrix inversion lemma (twice), one can see that the last and the third terms of (3.36) or (3.37) can be dropped without affecting the matched filter vector. Then, by using the matrix inversion lemma once again, we have

F aM = (RIFO &*aH)-1

= a

R-1am &*(aH O1 am + &*(aH Oa)R O
= FO M F0- 41 H- I UF-I (3.38)
1 &*aHm FO






23

and

g + a a ] aM = -F am= + FB


[g + ~RR (&*g + HaM FB





S1FOM ,FOa
1 a- a HM M






Fa + &*ag aM &*aHR







aH Fa
lft-am Hlf1 a A-- am + 1 H ,-1 amft-1 *
hFBM- 2a FB + 2aMFBa FB

S+ aFB [&*g + (3.39)

which gives the following expressions for the matched filter vectors:


hFFOM OMFOA() (3.42)aM
Hm(l) = Aa (w). (3.43)1 f-'am + a aM FoaMRfo-lg *a aM1FogRFOaM(34
aHROFOaM (3.40)


and

hFBM FBaM
aM 'FBaM
Fa 2 M FaMFB [&*g +1 __ 2* ] l+FhaM
amRFB aM

(3.41)

The previous MAFI filters are in general different from both the Capon and APES filters, since neither the Capon nor the APES filters depend on an estimate of a(w) which the new MAFI filters need to know. In spite of this fact, in Appendix A, we prove that, for a certain natural choice of &(w) in (3.36) and (3.37), the following equalities hold true: &FOM(W) = 6FOA(W), (3.42) &FBM(W) = &FBA(w). (3.43)





24

3.3 Analyses of MAFI Approaches

3.3.1 Computational Complexity

Let fF1/2 and ft /2 denote the Hermitian square roots of the positive definite matrices R f) and i1F, respectively. Define
FO FB--D

i (W) = FO/2 aM(w), (3.44) 2() = tF2g(), (3.45) Then FOC and FOA can be expressed as relatively simple functions of ,l (w) and pz2(w) (see (3.29) and (3.35)): Foc() =C1 \( ) 2( )
11 1( ) 112 (3.46) and

HFOA)(w) = (3.47) FOA(W) 1(w)~2 (l1 (W)12 112 H2 (w)2()12 Applying the matrix inversion lemma to (3.32) yields

BA(W) R G(w) GH ) G(w) I-GH (w)t (3.48)

where I is the 2 x 2 identity matrix. Next define Vl(W) = FB/2aM(w), (3.49) V2(W) B= F/2g(w), (3.50) v3(W) = /2g(w). (3.51) Then the FBC and FBA spectral estimators can be expressed as (see (3.29), (3.35), and (3.48)):

&FBC(W) =2 IvHw)V2W) (3.52)






25

and

x i. [~. (u ~ L) l V 2(W ) 12 1'(W)V2(W) 1 H'W () H'(w)V(W)] ^FBA(W) =
Il(W)i12 I 2H2
1 2 I 1 W)2() H(()w3(-)] 1 (3.53)
where
1 II2() 112 2 (W) 3(W)
(w) = 2 Il(W)V2(W) HV3()I I. (3.54) Hence computationally APES, especially FOA, is only slightly more involved than Capon. (Also see Section 3.5 for the simulation results.) More specifically, the amount of computations required by Capon or APES is dominated by calculating fl-1/2 and the matrix-vector products in (3.44)-(3.45) or (3.49)-(3.51). We mention that conventional Capon or APES implementation makes use of ((3.46), (3.47)), ((3.52), or (3.53)), which requires calculating the matrix-vector products in (3.44)(3.45) or (3.49)-(3.51) for each w of interest, thereby becoming computationally more and more intensive as the number of frequency samples increases. This is especially so in 2-D applications such as when forming SAR images. It is thus of great interest if other efficient implementation schemes for the MAFI approaches can be found.

3.3.2 Statistical Performance

All estimators under study, i.e., FOC, FOA, FBC, and FBA, can be shown to have the same asymptotic variance under the following condition:

C: The signal y(n) can be written as in (3.1), where e,(n) is a zero-mean stationary

random process with finite spectral density at w:

e(W) < 00. (3.55) In more exact terms, the following result holds true.






26


Theorem 3.3.1 Under Condition C and the additional assumption that e,(n) is circularly symmetrically distributed, the estimation errors in the Capon and APES spectral estimators are asymptotically circularly symmetrically distributed with zeromean and the following common variance:

lim LE {I&(w) a(w)|2} = e(w). (3.56) L--oo



Proof: See Appendix B.

The need to enforce Condition C limits, to some extent, the importance of the previous result. Indeed the assumption made in C is satisfied if (and essentially only if) the signal y(n) has a mixed spectrum and w is the location of a spectral line. The result of Theorem 1 is relevant to the spectral analysis of a target with dominant point scatterers in the presence of distributed clutter (see [52] and the references therein). In some other applications, however, the main interest is in the continuous component of the spectrum. For example, Condition C does not hold exactly for a target with distributed scatterers since the signature spectrum is continuous at w.

That the previous result is of a somewhat limited interest is also due to its asymptotic character. Indeed, in applications with medium or small-sized data samples, the spectral estimators under study have been found to behave quite differently in contradiction with what is predicted by the (asymptotic) result of Theorem 1 (see the numerical examples in Section 5). The finite-sample analysis of the spectral estimators under discussion would consequently be of considerable interest. However, a complete analysis, if possible, appears to be rather difficult at best. A partial one, by making use of a higher-order Taylor expansion technique, is nevertheless feasible. The result is as follows.

Theorem 3.3.2 To within a second-order approximation and under the mild assumption that the third-order moments of e,(n) and e,(n) are zero, the Capon and






27

APES estimators are related by
LE {&FBC(W) a(w)} LE {&FOC(W) aM(w)}
(< 0w (3.57) and

LE {&FBA(W) a(w)} = LE {&FOA(W) a(w)} = 0, (3.58) for sufficiently large values of L.

Proof: See Appendix C.

We believe that (3.57) and (3.58) provide a theoretical motivation for preferring APES to Capon in most spectral estimation exercises. Moreover, Theorem 2 also suggests that FBC should be preferred over FOC. While both FOA and FBA are similarly unbiased (within a second-order approximation), the latter is usually observed with slightly better resolution and sidelobe properties [52] at the cost of slightly more computations.


3.4 2-D Extensions

We briefly describe the 2-D extensions of the MAFI spectral estimators. We first decompose the observations {y(ni, n2)} as

y(ni, n2) = 1(W1, W2)ej(wlnl+w2n2) + e,(n, 12), (359)
(3.59)
n =, 1,...,N-1; n2 =0, 1,...,N2 1; W1, W2 E [0,27r),
where a(wl, w2) denotes the complex amplitude of a 2-D sinusoidal signal with frequency (wl, w2) and e,,,, (nl, n2) denotes the noise (or residual) term at frequency (w, w2), assumed to be zero-mean. Next, in a manner similar to the 1-D case, we form the M1 x M2 forward and backward data matrices:

Y(I1, 12) = y(nl,n2), n= 1, 1 + M -+M- 1; n2 = 12,...,12 + M2

Y(I1, 12) = y*(nl,n2), n1 = 1 ,..., N 11 M1;

n2 = N2 12 1,..., N2- 12- M2} ,

11 =0,1,..., L 1; 12=0,1,...,L2-1, (3.60)






28

where L =N1 M1 +1 and L2 =N2 -M2 + 1. Let Y(11, 12) = vec[Y (1,12)], (3.61) :(l, 12) = vec[rY(1,/2)], (3.62) where vec[-] denotes the operation of stacking the columns of a matrix on top of each other. Let

aM1,M2 (W1, W2) = aM2 (W2) aM 1(1), (3.63) where 0 denotes the Kronecker matrix product, and

aMk k) = 1 eW ... ej(Mk-1)k k = 1,2. (3.64) Then Y(11,12) and S(l1, 12) can be written as


Y(ll, 12) = [0(wl,,W2)aM1,M2 (w 2)]ej(w1li+212) + ek ,2 (l, 12), (3.65) r(11, 12) = [&(Wl W2)aM,M2(w1, W2)l]e(wl l1+212) + 1,W2(11, 12), (3.66) where

&(w1, w2) = *(w,2)ei-j(N-1)we-j(N2-l) 2, (3.67) and 4,, (11, 12) and e~,,,2 (i, 12) are, respectively, formed from {e,,,, (ni, n2) } in the same ways as Y(11, 12) and y(l1, 12) are made from {y(nj, n2)}.

Suppose that the initial phase of the sinusoidal signal of (3.59) is a random variable uniformly distributed over the interval [0, 27r) and independent of the noise term. Then the covariance matrix of y(1~, 12) or, equivalently, of k(l1, 12) is given by


R = la(wl, W2)2aM1,M2(W1,W2)aM1,M2 (W1, W2) + Q(W1,W2), (3.68) where Q(wl,W2) is the covariance matrix of e,1 (11, 12) or e61,, (11, /2). By making use of the fact that

r(l1,/2) = Jy*(L1 11 1, L2 12 1), (3.69)






29

one can see that R is persymmetric. Similarly, Q is also persymmetric.

The forward-backward sample covariance matrix takes the form:


iFB = I(RFO + IRBO), (3.70) where FO and RBO denote the sample covariance matrices of {y(l1, 12)} and {((11, 12)}, respectively, given by
SL1-1 L2-1
RFO = (11,12)H1, 2), (3.71) 11=0 12=0
L1-1 L2-1
RBO = (1112)yH(1 12). (3.72) 11=0 12=0

By making use of (3.69), one can see that RFB is also persymmetric.

Let H,,,,2 denote the impulse response of an M1 x M2 2-D FIR filter, and let hl,.2 = vec[H,,,,,2]. (3.73) Like in the I-D case, the impulse response of the matched filter is given by Q-1 (w1, W2)aM1,M2 (w, w2)
= a H,M2 (1, W2 Q-1(W, W2)aM1,M2 (W1, W2) 7 Note that

hH,2 aM1,M2(W1,W2) = 1. (3.75) The LS estimates of a(w1, w2) obtained by using only the forward data vectors and by using both the forward and backward data vectors are given by (similarly to (3.21) and (3.22) in the 1-D case)


FO(W1, W2) = h,'g(w), (3.76) and


FB 1, W2) [h1,W2g(wl, 2) + e-j(N-l1)w ej(N2-l)w2gH(1, W2)hw,,w2] (3.77)





30

where
SL-1 L2-1
E(w1,w2) LL2 Z Y(, 12)e-j(w'' w2'), (3.78)
11=0 12=0
L1-1 L2-1
(w,0w2) LjL2 1 2 Y(ll, 12)e-j(ll+w1 212). (3.79) 1i=0 12=0
Since Q is persymmetric, (3.77) can be written as &FB (1, W2) = h~ 1, 2). (3.80)


The Capon method estimates the noise covariance matrix as

(Capon (W1, W2) = i- IL(1,2) I2aM,M2 (w1,2)aH,M2( W1, 2), (3.81) where &(w1, w2) denotes some estimate of a(W1, w2), and R denotes either RiFO or IRFB, which correspond to 2-D FOC or FBC. Thus, the Capon estimate of a(w1, w2) is obtained as

&Capon(w1, 2) = aMI'M2 (W1, 2) 1- 2 ((3.82) aMl ,M2 (W1, w2)i-laM1,M2 (W1, W2) The FO and FB APES estimates of Q(w1, w2) take the form:

QFOA (1, W2) rFO (w1,W2)H(W1, W1), (3.83) and

QFBA (w1,2) = FB G(w1, 2)GH (W, W1), (3.84) where

G(w,W2) = 2g(W,L02) 9(l, 12) (3.85)

Hence we obtain &APES (W1, w2) as aHM ( 1, l) -1J( 1, L)l( 1 2) &APES (W1, W2) =- M21, 2(3.86) a .1 (W, 12)Q -1(W1, 2)aM1, ((,r 1 2) where ((i,W2) denotes either QFOA(Wi, 2) Or Q(FBA 2) -






31


Based on the 2-D extensions described above, it is not difficult to see that all the results of the previous section also hold true for the 2-D Capon and APES estimators. Indeed, the proofs for the 2-D estimators follow a similar pattern to those for the 1-D case shown in Appendixes B and C.


3.5 Numerical Examples

In the following, we study the Capon and APES complex amplitude estimates in a number of cases of interest. For both the 1-D and 2-D examples given below, we compare the performance of the forward-only Capon and APES as well as the forward-backward Capon and APES, which are, for simplicity, referred to as FCapon, FAPES, FBCapon and FBAPES, respectively.

3.5.1 1-D Complex Spectral Estimation

The 1-D data used in the examples consists of a sum of 15 complex sinusoids, with the real and imaginary parts shown in Figures 3.1(a) and 3.1(b), respectively, corrupted by a zero-mean complex white Gaussian noise. The data length is chosen as N = 64. In what follows we are interested in the bias and variance properties of the estimators under study. The bias and variance results shown below correspond to the frequency of the first sinusoid and they are obtained from 100 independent realizations.

We begin by studying the performance of the estimators as the signal-to-noise ratio (SNR) varies. The SNR for the kth sinusoid is defined as


SNR = 10log1 a (dB), (3.87)
0 e(Wk)

where COk is the complex amplitude of the kth sinusoid and Pe(wk) is the spectral density of the additive noise at frequency wk. The filter length is chosen as M = 15. The real and imaginary parts of the bias are shown in Figures 3.2(a) and 3.2(b), respectively, as a function of SNR1. As seen from these figures, FOA and FBA are






32

almost unbiased, while FOC and FBC are biased downward. In addition, we notice that the bias for FOC is approximately twice that of FBC. All these observations are consistent with the prediction of the theory. The variances of the real and imaginary parts of the amplitude estimates are shown in Figures 3.2(c) and 3.2(d), respectively. It appears that all of the estimators display similar variances. However, as shown in the next example, the variance of Capon becomes notably larger than that of APES as M increases.

Next we study the effect of the filter length, M, on the estimators. The SNR1 is fixed at 20 dB. As M varies, the real and imaginary parts of the bias are shown in Figures 3.3(a) and 3.3(b), respectively. From these figures, one can see that both FOA and FBA are unbiased for all practical filter lengths, whereas the bias of Capon grows significantly with increasing M. (A practical filter length means that M should not be too small [52]. In fact, all filterbank methods reduce to the Fourier transform approach when M = 1, and only when M is sufficiently large, the filterbank approach shows noticeable improvement over the Fourier method [52].) All estimators seem to perform similarly for M up to a fourth of the data length, with Capon being slightly biased downward. As the filter length increases further, the performance of Capon degrades rapidly, while that of APES remains unaffected. This observation is strengthened by the variance results shown for the real and imaginary parts of the amplitude estimates in Figures 3.3(c) and 3.3(d), respectively.

It is known that, as M increases, all of the estimators under study achieve better spectral resolution and that the best resolution is obtained at M = N/2 [52]. This fact, along with the statistical results shown in the previous examples, indicates that the choice of M for Capon should be made by a tradeoff between resolution and statistical stability. Usually we choose N/4 < M < N/2. While the choice of M for Capon is difficult to make, it is easy to see that APES achieves the best performance at M = N/2, since with this choice, APES achieves the highest possible






33


1-D Complex Sinusoids 1-D Complex Sinusoids
0.9 0.9 0.8 0.8
r V
0.7 0.7
0.6- 0.6
I .5I I 0.5







(a) (b) = 0.4 0.4


0.1 -D complex amplitude of the sum of 15 sinusoids 0.1used in the
0 0.2 0.4 0.6 0.8 1 0 0.2 OA 0.6 0.8 1
Frequency Hz Frequency Hz
(a) (b) Figure 3.1: The 1-D complex amplitude of the sum of 15 sinusoids used in the simulations. (a) Real part; (b) Imaginary part. resolution as well as the best statistical properties in terms of bias and variance. The previous examples also show that FOA and FBA perform similarly in terms of bias and variance properties for the frequency of interest.

To compare the computational complexities of the estimators under study, we count the flops required by each of them for the case where N = 64, M = 24, and the complex spectra are evaluated at 256 equally spaced points. The flops required by FOC and FBC are approximately the same, whereas the flops needed by FOA and FBA are, respectively, 1.08 and 1.41 times of that by the Capon estimators.

3.5.2 2-D Complex Spectral Estimation

As was mentioned in Section 3.4, the 2-D Capon and APES estimators behave rather similarly to their I-D counterparts. Since the problems encountered in applications such as synthetic aperture radar imaging are concerned with 2-D complex spectral estimation, we include a couple of 2-D numerical examples here. The data employed consists of three 2-D sinusoids corrupted by a 2-D zero-mean complex white Gaussian noise, with N1 = N2 = 32. The sinusoids are located in the frequency domain at (0.2, 0.2), (0.25, 0.25) and (0.4, 0.7) and their amplitudes are ej,/4, ej1/4









34



















x10-a 1-D Complex Amplitude Estimation 1-D Complex Amplitude Estimation
5 I 0.015
oo o. 9 I-- FOC 94". ': :' 0"0, o 0.01
0 a 0.01


- 0 .FBC
-0 00 '0' 0.0 0





S-0.01

-15- FBC


-0.01

5 10 15 20 25 30 5 10 15 20 25 30 SNR dB SNR dB

(a) (b)

x 10-3 1-D Complex Amplitude Estimation 10- 1-D Complex Amplitude Estimation

FOC FOC
--- FC 1. --- FC
+ FOA + FOA 2 FBA 1.4 o o FBA

0. 0. 1.2
1.5
E 1

a_0.8

0.6

0.5 0.4
0.2
( C
5 10 15 20 25 30 5 10 15 20 25 30 SNR dB SNR dB

(c) (d) Figure 3.2: Empirical bias and variance of the 1-D MAFI estimators as SNR1 varies when N = 64 and M = 15. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of the imaginary part of the estimated amplitude.









35


















1-D Complex Amplitude Estimation 1-D Complex Amplitude Estimation
0.1 0.3

0.2.---0.1
n0 ... o ... *o


-0.1




0
-0.4 FOC

- FBC -0.4-0.5 o FBA -0.5I I I I I I
-0. 5 10 15 20 25 30 35 0.60 5 10 15 20 25 30 35 Filter Length Filter Length

(a) (b)
1-D Complex Amplitude Estimation 1-D Complex Amplitude Estimation
0.012 0.012

FOC FOC
0.01 --- FBC 0.01 - FBC
* FOA + FOA o o FBA I o FBA

0.008 - 0.008


i 0.006 o0.006



>I >

0.002 0.002



0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Filter Length Filter Length

(c) (d)


Figure 3.3: Empirical bias and variance of the 1-D MAFI estimators as the filter length, M, varies when N = 64 and SNR1 = 20 dB. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude;

(d) Variance of the imaginary part of the estimated amplitude.






36

and 0.7ej,/4, respectively. The bias and variance for the amplitude estimate of the first 2-D sinusoid are obtained from 100 independent realizations. The SNR for the kth 2-D sinusoid is similarly defined as in (3.87). The bias and variance of the four estimators under study versus SNR1 are shown in Figures 3.4(a) to 3.4(d), respectively, where M1 = M2 = 8. Figures 3.5(a) to 3.5(d) show the statistical results as the 2-D FIR filter length varies, where SNR1 is fixed at 20 dB. We assume in Figures 3.5(a) to 3.5(d) that M, = M2. As seen from these plots, the performance of the 2-D MAFI estimators indeed resembles that of their 1-D counterparts and, therefore, we refer the readers to the 1-D examples for comments.


3.6 Summary

This chapter discusses using the MAFI approach for complex spectral estimation. The Capon and APES estimators are shown to be members of the MAFI class. By using a higher-order expansion technique, it is proved that to within a secondorder approximation Capon is biased (downward) while APES is unbiased, and that the bias of the forward-backward Capon is one half that of the forward-only Capon. It is also shown that the above conclusions carry over to the 2-D MAFI estimators as well. Since computationally APES is only slightly more involved than Capon, the preference of APES to Capon in practical applications follows logically because of the better statistical properties associated with the former.








37

















2-D Complex Amplitude Estimation 2-D Complex Amplitude Estimation
0.005 0.005

0.* .
0 : ........ O. "1 : ". o. .,.o ..


1 is *
SNR dB SNR dB
SComplex Amplitude Estimation Complex Amplitude Estimation






-- FBC - FBC







-0. FOA --- FOA 2 a o FBA a FBA
0.025 0.025




















0.5- -0.4
5 0 5 10 15 -5 0 5 10 15 SNR dB SNR dB














(a) (b)

2.Figure 3.4: E2-D Complex irical bias and variance of the 2-D MAFIeAmplitude Estimat honors as the SNR varies when NFC N2 = 32 and M 8. (a) Real part of the bias; (b)
(d) Variance of the imaginary part of the estimated amplitude.FBA














5 0 5 10 15 -5 0 5 10 15 SNR-dB SNR-dB
(c) (d)


Figure 3.4: Empirical bias and variance of the 2-D MAFI estimators as the SNR1 varies when NI = N2 = 32 and Mt = M2 = 8. (a) Real part of the bias; (b)








38


















2-D Complex Amplitude Estimation 2-D Complex Amplitude Estimation
0.1 iI I 0.1




S-0.1 -0.1




-0.3- -0.3
'-0.4 FOC -0.4 F-FOC A


-0.54 FBA -0.5-0.c I I I
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Filter Size Filter Size

(a) (b)

x10' 2-D Complex Amplitude Estimation 10- 2-D Complex Amplitude Estimation

FOC FOC - FBC - FBC FOA + FOA 6 OFBA 7 0 OFBA

U
4 I







53-
1 1


2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Filter Size Filter Size

(c) (d)


Figure 3.5: Empirical bias and variance of the 2-D MAFI estimators as the filter length, M = M1 = M2, varies when N1 = N2 = 32, and SNR1 = 20 dB. (a) Real part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of the imaginary part of the estimated amplitude.














CHAPTER 4
EFFICIENT IMPLEMENTATION OF CAPON AND APES

4.1 Introduction

In the previous chapter, we rederived the Capon and APES spectral estimators using the MAFI approach. In the MAFI framework, a number of statistical properties of Capon and APES were obtained. However, we did not address very carefully the implementation of the Capon and APES estimators. Equations (3.46), (3.47), (3.52), and (3.53) give the intuitive ways of implementing Capon and APES. However, such intuitive implementations are computationally expensive, especially for 2-D spectral estimation from 2-D data sequences. In this chapter, we study how to implement Capon and APES efficiently. For simplicity's sake, we only consider 2-D spectral estimation since 1-D spectral estimation is a special case of the former; we also only consider forward-backward Capon and APES since they are more often used than their forward-only counterparts.


4.2 Efficient Implementation of APES

First we rewrite the 2-D forward-backward APES estimator as aM,M2 (W1, (2)1-l(W1, W2) (U1, W2)
&APES(w, w) = a (4.1) L1L2HM1,M2 (W1, W2) -1 (W1, w2)aM1,M2 (W1, W2) where
(W, K [(w) ftw, )+ ( ) [W, C(WA] (4.2)


with fR denoting the forward-backward sample covariance matrix given by (3.70), Li-1 L2-1
(W1,0)2)= (ll1,2)e-(1I1+w212), (4.3) 11=0 12=0

39






40

and
L1-1 L2-1
I(wl,w2)= 7 1 2)e-j(wil+w212). (4.4) [1=0 12=0
Let

Z = [ y(0,0)... y(L- 1,0)... (0, L2 -1)- (Li-1, L2 1) (4.5)

and Z = JZ*J, where J denotes the exchange matrix. We can then rewrite (4.3) and (4.4) as

(W1, W2) = Za*L,L2 W2), (4.6) and

I(w1,w2) = ZaL1,L2 (1, W2) (4.7) By applying the matrix inversion lemma, we obtain

(1, 2) -1(w1, 2) + Q-(W1, 1, 2) (1, W2)Q(W1, W2) (4.8) L1L2 i (W1, W2)Q-1 (W, 12)I(WI, W2) where
R (, w2)IP H()Q-1(wl, 2) = I + H (4.9) LL w (W1, 2)R-1R(w1, W2) (For notational convenience, we sometimes drop the dependence on w1 and w2 below.) Hence,


aHM,,M2 (1, 1)Q-1 (W1, 02) (1, W2) H H-1 aMt,M2Q H+ I jL H --1SL MM2Q (4.10) L1L2 HQand


aM2 (w1, W2) -1aM,M2 (W1, W2) H --I + aM,M2Q aM,M2 a1,M2 + (4.11) L1L2 1 Q-'V)






41

It follows that (4.1) can be rewritten as



(L1L2 H- aM,M2Q- aM1,M2+ aM1,M2Q2

Since R is Hermitian and positive definite, we can obtain an upper triangular matrix C by Cholesky factorization such that 1-1 = ( -1)H [30]. Let
bT(w1, w2) a,M2 2)C-, (4.13)


d(wi,w2) = DaL,L2(Wl, 2), (4.14) and
e(wl, w2) = Ea*L,L2 (wl, w2), (4.15) where

D = (-)H, (4.16) and
E = (C)HZ. (4.17) We have [53]



aMH,M2 aM1,M2 I P
HH M -1 H -1 =1 M1,2 ~ +Rt-'L

bbTw2) b(ww2)e(, W2)H (W1, w2)d(w, w2) = bb(Wl, W2)d(W),2)2)w2), (4.18) L1L2 le(wl,w2) 112 H Q-1p

= H f-1 +
L1L2 1Hi-l'

= d(, 2) 112 + W 2)e(w 2) 2 (4.19) L1L2 jle(wl,W2) 112






42

and

aM1,M2Q-1 aM,M2

H -1 H, -1M2 H -1MM2
aM1,M2 m, M1 ,M2 J ,2 LIL2 R a
= Ilb(w,,w2)12 + bT(wj,W2)e(wj,W2)(0
= 2b(WWe(ww2)22 (4.20) L1L2 Ie(w,w2)I12

Next we observe that b(wl,w2), d(wi,w2), and e(wi,w2) can be calculated via 2-D FFT. Specifically, we partition C-1, D and E as follows:

-1= vec{B1} ... vec{BMM2} (4.21)


D= vec{Di} ... vec{DM 2} (4.22) and

E= vec{Ei} ... vec{EM1M2} (4.23) where Bk C CM1XM2, Dk E CLlxL2 and Ek E CLjxL2. Then (again, we drop the dependence on wl and w2)

b= aM1 ,M2 (a2 H ai )vec{B} ... (a12 aH)vec{B jm2}


= aHMBMa2 ... aH1BM Ma 2 a* (4.24) Likewise, we have

d Da1 = D aH 1Djia ... aH DMM2a* (4.25) and

e = Ea,,,2 = aH Ea2 ... aEMM2a2 .T (4.26) Note that aH Bka* represents the 2-D discrete Fourier transform of Bk at (w, w2). Hence APES can be efficiently implemented by first calculating b(w1,w2), d(wl, w2)






43

and e(wi, w2) via 2-D FFT, then using Equations (4.18), (4.19), and (4.20) to determine aH,,M2Q-1, ~HQ-1 and aH,M2Q- aM1,M2 respectively, and finally using (4.12) to obtain &APES(Wl,W2).

The structure of (3.53) is similar to that of (4.12). However, the amount of computation required by the former is much larger than that by the latter. The reason is that, even though 4(w1, w2) and P(l,W 2) can be obtained by 2-D FFT, for each (wl, 2) pair, we have to compute the additional matrix-vector products t-l/21(wl, w2) and W-1/2 M1,MM (W1, W2) (recall that E-i/2 E CM1M2 M1M2, 2(W1, 2) and 'I(w1,W2) E CM1M2x1) to obtain A2(w,2 W2) and pa (w1,w2). On the other hand, by computing D and E first (which are computed only once), we bypass calculating such matrix-vector products and save a large amount of computation. The larger the number of samples in the 2-D frequency domain, the more the amount of computation we will save. These discussions also apply to the implementation of Capon.


4.3 Extension to Capon

Note that APES becomes Capon when Q(wi, w2) is replaced by fR. Hence the efficient implementation of Capon can readily be achieved by modifying (4.12) as follows:
bT (w, w2)d(wl, 0w2)
Capon(w1, 2) (w, w2) (4.27) LL2 IIb(wl, w2)112
More specifically, the efficient implementation of Capon is by using (4.13) and (4.14) to calculate b(wl, 02) and d(wl, w2), respectively, and then using them in (4.27). Since the amount of computation required to calculate b(w, C) in (4.13), d(wi, w2) in (4.14), or e(wi, w2) in (4.15) is approximately the same and calculating &APES(W1, 2) by (4.12) and &Capon(W1, W2) by (4.27) are much less involved than obtaining b(w1, w2), d(wl, w2), and e(wi, 02), the total amount of computation required by APES is about 1.5 times of that required by Capon, as verified by the numerical and experimental examples in Section 4.4.






44

For similar reasons as for APES, the intuitive implementation of Capon given in (3.52) is computationally more involved than the efficient implementation of Capon proposed above.


4.4 Numerical and Experimental Examples

We present numerical and experimental examples comparing the performances of APES and Capon with the FFT methods [52] for SAR imaging. In the following examples, we choose M = N/2 and M = N/2 for both Capon and APES. For the windowed FFT method, we use the Kaiser window with parameter 4.

We first consider SAR imaging of a simulated MIG-25 airplane. The 32 x 32 data matrix was provided by the Naval Research Laboratory. The 128 x 128 SAR image obtained by using 2-D FFT, 2-D windowed FFT, 2-D Capon, and 2-D APES are shown in Figures 1.1(a) to 1.1(d), respectively. We note that Capon and APES outperform the FFT methods. The number of flops required by our efficient ways of implementing Capon and APES are about 950 and 1500 times of those required by the FFT methods, while those required by the intuitive ways of implementing Capon and APES are about 22800 and 30000 times, respectively, of those required by the FFT methods. That is, the number of flops required by the intuitive ways of implementing Capon and APES are, respectively, about 24 and 20 times of those required by our efficient ways of implementing them. If we increase the size of the image to 256 x 256 and 512 x 512, respectively, the ratios of the needed flops between the intuitive ways and our new ways for implementations are 36 & 40 for Capon, and 31 & 34 for APES, respectively.

We now consider an example of SAR imaging with experimental data. The data matrix is 64 x 64 and is obtained from the experimental data collected by one of the two apertures of the ERIM's (Environmental Research Institute of Michigan's) DCS IFSAR (interferometric SAR). The 256 x 256 SAR image obtained by using 2-D






45


FFT, 2-D windowed FFT, 2-D Capon, and 2-D APES are shown in Figures 4.1(a) to 4.1(d), respectively. Again, Capon and APES outperform the FFT methods. The number of flops required by the intuitive ways of implementing Capon and APES are, respectively, about 38 and 32 times of those required by our efficient ways of implementing them. If we increase the size of the image to 512 x 512, the ratios of the needed MATLAB flops between the intuitive ways and our new ways of implementing Capon and APES are 86 and 73, respectively.


4.5 Summary

This chapter addresses the implementation of the Capon and APES spectral estimators. The amount of computation required by APES is shown to be about 1.5 times that required by Capon. By using a technique proposed in this chapter, the computational complexities of Capon and APES are significantly reduced.






46























(a) (b)













(c) (d) Figure 4.1: SAR images obtained from the ERIM data by using (a) 2-D FFT, (b) 2-D windowed FFT, (c) 2-D Capon, and (d) 2-D APES.














CHAPTER 5
AMPLITUDE ESTIMATION

5.1 Introduction

Consider the noise-corrupted observations of K complex-valued sinusoids [71] [74]
K
x(n) = Eak k' + v(n), n = 0,1,...N 1, (5.1) k=1
where ak denotes the complex amplitude of the kth sinusoid having frequency wk, N is the number of available data samples, and v(n) is the observation noise which is complex-valued and assumed to be stationary (and possibly colored) with zero-mean and finite unknown Power Spectral Density (PSD) q(w). We assume that {Wk =1 are known, with wk : w1, for k : 1. The problem of interest is to estimate {kkK=1 from the observations {x(n)}I-01. In this chapter, we describe a relatively large number of methods for solving this problem.

Section 5.2 discusses least squares (LS) methods. LS methods are widely used for amplitude estimation because they are simple and easy to implement. If we restrict ourselves to estimating only one amplitude at a time, then the LS method reduces to the Discrete Fourier Transform (DFT) of the data at the frequency of the desired sinusoid and is computationally more efficient than the LS method that estimates K amplitudes simultaneously. Moreover, estimating one amplitude at a time does not necessarily require exact knowledge of the number of sinusoids in the data and of the frequency location of each sinusoid, which is a desired property in some applications. The disadvantage, however, is that using this one-at-a-time technique in general gives rather poor amplitude estimates when some sinusoids (that



47






48

are of interest to us) are close to one another. Statistical analyses that compare the merits of the two LS methods are also provided in Section 5.2.

Since the LS methods completely ignore the correlation of the observation noise, they are in general suboptimal. By splitting the data vector into a number of subvectors, the covariance matrix of the noise-only part of the data subvectors can be estimated, which makes it possible to use a Markov-like estimator that is optimal in the class of Weighted Least Squares (WLS) techniques [68]. We describe in Section 5.3 several ways for estimating the aforementioned covariance matrix, which lead to different WLS amplitude estimators. Additionally, we show that, if the restriction of estimating one amplitude at a time is again imposed, we obtain two WLS amplitude estimators that are equivalent to the Capon [16] [44] and APES [52] methods extensively used for spectral analysis.

The observation that some general spectral estimators, such as Capon and APES, can be used to solve the problem posed in (5.1) motivated us to seek other relatively sophisticated spectral analysis techniques for amplitude estimation. Both Capon and APES belong to the general class of filterbank approaches to spectral estimation [75], which involve splitting the data into subvectors, passing them through a set of narrowband filters (filterbank) whose center frequencies correspond to those that are of interest to us, and, finally, estimating the spectral density function at those frequencies from the filtered and, hopefully, signal-enhanced data. As one can see, the key issue of filterbank approaches is the design of the filters. A recent study has suggested the choice of matched filters, which gave rise to the MAtched-FIlterbank (MAFI) approach to spectral estimation [70]. Even though neither Capon nor APES was derived in the MAFI framework (see [16] and [52] for their original derivations), it was found that both are members of the MAFI approach [70]. In the light of the work of [70], we derive in Section 5.4 a generalized MAFI approach to amplitude estimation. Interestingly enough, we show that, under certain circumstances, MAFI






49

amplitude estimators have equivalent forms to the WLS methods. However, the MAFI approach is more general than the WLS technique in that the latter is a special case of the former. To show this, a new MAFI amplitude estimator that does not fall into the WLS category is described in Section 5.4. Other interesting MAFI amplitude estimators may exist and are yet to be discovered.

A common feature of the amplitude estimators considered in this chapter is that none of them models the observation noise exactly. Even so, all methods are asymptotically statistically efficient, that is, they all achieve the Crambr-Rao Bound (CRB) in large samples. However, their finite-sample properties, which are of primary interest to this work, are quite different. Since the finite-sample analysis is intractable in most cases, we use Monte-Carlo simulations in Section 5.5 to compare these methods with one another.

The amplitude estimation problem in (5.1) occurs in a variety of signal processing applications (see, e.g., [41] [55], and the references therein). In Section 5.6, we discuss its application to system identification. We show that, by using appropriate amplitude estimators, we can avoid the iterative search required by the standard system identification routines, such as the Output Error Method (OEM) [68], and achieve very good performance at a usually reduced computational load.

In concluding this section, we introduce the following notation to distinguish among the various amplitude estimators. For instance, LSE(1, 0, 1) denotes the LS estimator that does not split the data (and hence it uses one data "snapshot"), uses no prefiltering, and estimates one amplitude at a time. Likewise, MAFI(L, K, K) denotes the MAFI estimator that splits the data into L subvectors, utilizes a bank of K prefilters, and estimates K amplitudes simultaneously. The remaining amplitude estimators are similarly designated.






50

5.2 LS Amplitude Estimators

We consider two LS methods in this section, namely LSE(1, 0, K) and LSE(1, 0, 1).

5.2.1 LSE(1, 0, K)

This is perhaps the most direct approach. Let us write the available data sequence in the following form

x(0) 1 ... 1 cal v(0) x(1) ewl ... ei K a2 v(1)
= + (5.2) x(N 1) ej(N-l1)w ... ei(N-1)K K v(N 1) or, with obvious definitions, x = Aa + v, (5.3)

which is a linear regression equation. The LS estimate of a is & = (-AHA)-lAHx, (5.4) where (.)H denotes the conjugate transpose. Note that the noise is not modeled, even though it may be correlated. Despite this fact, LSE(1, 0, K) is asymptotically efficient [31]. A relatively simple manner to see this is as follows. First, note that E{&} = a, (5.5) where E{.} denotes the statistical expectation. The Mean Squared Error (MSE) of & is

MSE{&} = cov{&}J E{( & )(& a)H

= (AHA)- lAHWA(AHA)-, (5.6) where

W E{vvH. (5.7)





51

Next, since (see, e.g., [34])
1
lim I(AHA)= IN, (5.8) N-oo N
where IN denotes the N x N identity matrix, and

[(w1) 0 lim 1 lim N (AHWA)= [ .. (5.9) N--so N
0 O(WK)

the asymptotic MSE is given by

(W1) 0 lim NMSE{&} = ".. (5.10)
N-+oo
0 O(WK) Under the mild assumption that v(n) is circularly symmetric Gaussian, the CRB for a is given by (see, e.g., [68])

CRB{a} = (AHW-1 )-1. (5.11)

Using the following result (see [34] once again)



lim -(AHW-1A) I (5.12)
N->oo N
0 O-1(WK) we obtain
(WP) 0 lim NCRB{a} = ] (5.13) N--+oo
0 O(WK) which coincides with (5.10).
Remark: It can be readily checked from (5.6) and (5.11) that if v(n) is white, i.e. W IN, then LSE(1, 0, K) is statistically efficient for all N > K.






52

5.2.2 LSE(1, 0, 1)

Since the observation noise v(n) is not modeled, an idea that reduces the computational burden quite a bit is to include K 1 sinusoids in the noise term, and hence estimate only one amplitude at a time. In some signal processing applications, the frequencies {Wk}kK=1 may be unknown. A typical way to estimate both {k} i=1 and {Wk K= would consist of estimating just one amplitude for varying frequency w and, then, detecting the peaks in the so-obtained spectrum [75] [41] [55]. As such, the assumption made in Section 5.1 that {Wk K=1 are known a priori may be relaxed when using the one-at-a-time technique.

There is a somewhat subtle problem with the above technique: the sum of v(n) and K 1 sinusoids no longer has a finite PSD, and hence one of the previously made assumptions fails. Nevertheless, the idea still works as long as no two sinusoids (that are of interest) are spaced too close to one another, as shown below and later in Section 5.5.

The LSE(1, 0, 1) is easily derived as N-1
k k= x(n)e-'"n k = 1,2,...,K, (5.14) n=o
which is recognized as the DFT of {x(n)} Io' at Wk. The two estimates in (5.4) and (5.14) will be close to one another if Iwk W1 > 1/N (V k, 1; k $ 1) [75].

An analysis of LSE(1, 0, 1) runs as follows. Without loss of generality, let us consider (5.14) for k = 1. The LSE(1, 0, 1) estimate of al is given by & = ( ) -l Hx, (5.15) where

= 1 ej ... eJ(N-1)wl (5.16) and where (.)T denotes the transpose. Taking the expectation of (5.15) yields

1
E{&I} = al + AHA, (5.17)
N






53

where a = [ C2 ... aK ]T, and A is defined through


[a A]A. (5.18) Hence, LSE(1, 0, 1) is biased. However, it is asymptotically unbiased (that is, its bias goes to zero as N -+ oo). We next calculate the MSE of &1:

MSE{&,} = (.H ,)-lH (AaHAH + W) (dHi)-1. (5.19) Making use of (5.9) once again, along with the fact that AH&/v/N -- 0 as N -4 oc, we have

lim NMSE{&1} = (wl). (5.20) N-oo

Hence, LSE(1, 0, 1) is also asymptotically efficient. On the other hand, in finite samples (5.14) may be better or worse than (5.4), depending on the characteristics of the scenario under study.

The fact that (5.4) may be better than (5.14) comes as no surprise. As an example, let us assume that the Signal-to-Noise Ratio (SNR) is high. Then, the bias of (5.14) dominates the variance part. On the other hand, (5.4) has no bias and its variance will be smaller than the bias of (5.14) if the SNR is large enough. Consequently, the MSE of (5.4) will be smaller than that of (5.14).

The fact that (5.14) may be better than (5.4) is however a surprise. For an example of such a case, assume SNR < 1 and W = IN. Then, for (5.14),


MSE {&I} (dH)-1, (5.21) whereas for (5.4),

MSE({&} =[(AHA)-] (5.22) which can be much larger than (5.21) (e.g., if Iwk 1/N for some k > 2). In (5.22), []i,j denotes the ij-th element of the matrix argument.






54

Note that, for most cases of interest, LSE(1, 0, K) will give more accurate amplitude estimates than LSE(1, 0, 1), and that the difference between these two estimators is small for large N. On the other hand, LSE(1, 0, 1) is computationally more efficient than LSE(1, 0, K) since the matrix multiplication and inversion in (5.4) are avoided. Hence LSE(1, 0, 1) may still be worth considering.


5.3 WLS Amplitude Estimators

If we split the data vector x into subvectors, then the covariance matrix of the noise part of the subvectors may be estimated and can hence be used to derive an optimal WLS estimator (i.e., a Markov-like estimator) [68]. In this section, we describe a number of such WLS estimators that split the data into vectors of shorter length, utilize no prefiltering, and estimate either one or K amplitudes at a time.

5.3.1 WLSE(L, 0,K)

We define the following subvectors

y(1)= x(1) x(l+1) ... x(l+M-1) l=0,1,...,L-1, (5.23) where

L = N- M + 1. (5.24) The choice of M (M can be chosen smaller than K. See Figure 5.4. Moreover, when M = 1, all WLSE(L, 0, K) reduce to LSE(1, 0, K)) or, equivalently, of L is discussed in Section 5.5. We have

1 ... 1 ajlejwl v(l1)

ejW ... ejdK 2ej21 v(l + 1)
y(1) = + (5.25)


ej(M-1)w1 ... e(M-)wK CKejwKl v(l + M 1) or, with obvious notation,

y(l) = As(l) + e(l). (5.26)






55

Alternatively, we can rewrite (5.26) as y(1) = Ala + (1l), (5.27) where
ejwll 0

Al A ADz. (5.28)

0 ewKl

We will use (5.26) mostly for analysis and (5.27) for estimation.

The WLS (Markov-like) estimate of a in (5.27) is given by L-1-1 L-1
a = [ A -lA A[H y(1)] (5.29)
1=0 l=0

where Q is an estimate of Q = E{e(1)e(1)}. (5.30) To estimate Q, we may proceed as follows. Let L-1

R = y ()yH (1). (5.31) l=0

One can verify that as L -4 oo, R goes to R = APAH + Q, (5.32) where


P = ". (5.33)


Hence, one way to estimate Q is as Q = A APAH, (5.34) where P is made from some initial estimates of {ak }K= obtained for instance via one of the LS amplitude estimators. The need for initial amplitude estimates is a






56

drawback of Q in (5.34). In the following we try to circumvent this need in two different ways.
First, we show a way to simplify the WLSE(L, 0, K) that uses (5.29) with (5.34). From (5.34), we have that

](Q-A = APAH-1A + A = AF, (5.35)

where

Sr PAHQ-1A + IK. (5.36) For sufficiently large N and M, F is approximately diagonal since AHQ-1A is so (see, e.g., (5.12)). Consequently,

(Q-Al = i-'AFD, Ri-1ADjr = R-tAr. (5.37)

Inserting (5.37) into (5.29) yields (observe that TH cancels out)


& [ f 1AH -I1A [ AH -ly(l) (5.38) 1=0 L=0

which, unlike using (5.29) with (5.34), does not require any initial estimate of {ak}k=1The amplitude estimator in (5.38) can be interpreted as an extension of the Capon algorithm in [16] [44] to multiple sinusoids.

A different estimate of Q can be obtained as described next. Observe that K K APAH = Z[aka(wk)][aka(wk)]H k Z kH(, (5.39) k=1 k=1
where

a(w) = 1 ejw ... ei(M-1)w (5.40) We can use the vectors {ikkK=1 introduced above to rewrite (5.27) as
K
y(l) = Z3ke k'+ E(l). (5.41) k=1






57

From (5.41), we can estimate 3k one at a time via LS as L-1
Sk = EY(e-jw," g(wk), k = 1,2,...,K. (5.42) l=0
(Note that we could estimate all {13k}k=1 simultaneously via LS, which however appears to perform even worse than using (5.42), especially for small N.) The use of (5.42) in (5.32) and (5.39) leads to the following estimate of Q
K
S= R g(wk)gH(wk). (5.43) k=1
The WLSE(L, 0, K) that uses (5.29) with (5.43) does not require any initial estimate of {ak }=1. It is an extension of the APES algorithm in [52] to multiple sinusoids with known frequencies.

Remark: We note that E(k) and E(1) in (5.27) are correlated (for k 5 1), which implies that (5.29) is suboptimal (as it takes into account only the correlation between the elements of E(1), but ignores the correlation between E(1) and E(k), for k = 1). Yet, the WLS methods are likely to outperform the LS methods because the latter completely ignore the correlation in v(n).

5.3.2 WLSE(L, 0, 1)

The particularization of WLSE(L, 0, K) to WLSE(L, 0, 1) is straightforward. Specifically, the WLSE(L, 0, 1) that corresponds to using (5.29) with (5.34) can be readily verified (by using the matrix inversion lemma) to be aH (Wk) f-1 (Wk)
&k = k k = 1, 2,..., K, (5.44) aH(wk)if-a(Wk)
whereas the WLSE(L, 0, 1) that corresponds to using (5.29) with (5.43) is given by

aH(wk) g(wk)gH(wk) g(wk)
&k = k = 1, 2,..., K. (5.45)
aH(wk) [i g(wk)gH(k) a(wk)

Note that (5.44), like (5.38), does not depend on P. However, unlike (5.38), the equation (5.44) is exactly equivalent to using (5.29) with (5.34). Equations (5.44) and






58

(5.45) are recognized to have the same form as the Capon [16] [44] and, respectively, the APES [52] spectral estimators. The two estimators were derived in [70] [47] by a different approach, namely the MAFI approach, which we will consider in a generalized form in the next section.

It is interesting that the above two amplitude estimators, while both asymptotically efficient (and hence equivalent), have quite different finite-sample properties. Specifically, it was shown in [70] [47] that (5.44) is biased downward, whereas (5.45) is unbiased (within a second-order approximation) and in general has a better performance than the former.


5.4 MAFI Amplitude Estimators

In this section, we derive a generalized MAFI approach to amplitude estimation. Let HH E CKxM be a matrix each row of which is a Finite Impulse Response (FIR) filter (for some 1 < K < M yet to be specified). The MAFI idea can be explained as follows:

a) Design HH so that, when applied to {y(l)}, it maximizes the SNR at the K

filter outputs.

b) Estimate the amplitudes from the filtered data (whose SNR should be higher

than that in the raw data) by, e.g., the LS or WLS technique. Mathematically, H can be obtained as follows: H = arg max tr [(HHQH)- HH(APAH)H (5.46) "Generalized SNR"

where H is constrained in a way that is specified later (in particular, to guarantee that H is finite), and tr(-) denotes the trace of a matrix. Let XH = (HHQH)-1/2HHQ1/2, (5.47)






59

where (.)1/2 denote the Hermitian square root of the positive definite matrix argument. Observe that X is semi-unitary, i.e., XHX = IR. (5.48) The cost function in (5.46) can now be rewritten as f = tr [XH-1/2iAPAH -1/2X]. (5.49) It follows from the Poincar6 separation theorem (or the generalized Rayleigh quotient theorem) [37] that


max f = Ak (Q-1/2APAH -1/2) (5.50) k=1

where {Ak(')}k=1 denote the eigenvalues of the matrix between the parentheses, ordered such that A1 > A2 > -... > AR; furthermore, the columns of the maximizing X are equal to the eigenvectors corresponding to {Ak k=1.

Next, note that post-multiplying X by any unitary matrix of appropriate dimensions yields another valid solution for X. One such solution having a simple form can be obtained as follows. Observe that rank (^-1/2APAH -/2) = K, (5.51) which implies that we cannot improve the generalized SNR by choosing K > K since AK+1 = ... = Ag = 0. On the other hand, the larger the K the more filtered data will be available for amplitude estimation. Hence, we choose K = K. (5.52) In such a case, the maximizing X is given by X = Q-1/2AT, (5.53)






60

where T denotes some nonsingular matrix that makes X semi-unitary. One such T is

T = (AHQ-A)-1/2. (5.54) Hence,

X = Q-1/2A(AH -1A)-1/2 (5.55) We next observe that

H = (-1/2X (5.56) satisfies (5.47). Consequently, we have H = Q-IA(AH -A)-1/2. (5.57) The final step is to observe that post-multiplying H by a nonsingular matrix does not change the generalized SNR criterion. Then, it follows immediately that H = Q(-'A(AHQ-'A)-' (5.58) maximizes the generalized SNR and it also satisfies the constraint HHA = IK. (5.59) The constraint (5.59) says that each (row) filter in HH passes one sinusoid undistorted, and completely annihilates the others.

From (5.27) and (5.28), the filtered data corresponding to (5.58) is given by

z(1) HHy(l) = Da + HHe(1) Dla + v(1), 1= 0,1,..., L 1. (5.60) The covariance matrix of v(1) can be estimated as HH QH = (AHQ-1A)-1. (5.61) It follows that the WLS (Markov-like) estimate of ca in (5.60) is given by
L-1 -1 L-1
a = i DI(AH -1A)D] DH(AH -1A)(AHQ -A)-1AHQ-1 1=0 1=0
L-1 1 1
= E AH Q-1A, AHQ-ly(1) (5.62)
l=0 1=0






61


which shows that

MAFI(L, K, K) = WLSE(L, 0, K). (5.63) The MAFI interpretation of the WLS method, afforded by the above analysis, is interesting. In particular, it makes a clear connection between using the MAFI and the WLS techniques for amplitude estimation. The MAFI approach is however more general than the WLS technique. As an example, we derive a new MAFI amplitude estimator that does not belong to the WLS class as follows. Let zk(1) and vk(l) denote the k-th element of z(l) and, respectively, v(l) in (5.60). Then Zk(l) = ake' + vk(1), k = 1,2,...,K. (5.64) The above equations are related to one another only via the correlation between vk (1) and vp(l) (for k p). If we ignore the correlation, then the MAFI(L, K, 1) estimate of the ak via LS is given by
1L-1
k= L zk (l1)e (5.65) l=0
Unlike the Capon (5.44) and APES (5.45) estimators (which can also be shown to be members of the MAFI(L, K, 1) class [70]), the above MAFI(L, K, 1) estimator does require the knowledge of the number and frequencies of the sinusoids, which makes it behave more like a MAFI(L, K, K) estimator. In particular, it performs quite well for cases where some sinusoids are closely spaced, as will be seen in Section 5.5.

Other interesting MAFI amplitude estimators may be devised by using some other choices of H in lieu of the one given in (5.58), as the solution to (5.46) is not unique. Specifically, one may introduce certain unitary transform on the H in (5.58), or choose K < K, or replace the IK in (5.59) by another nonsingular matrix, which all lead to solutions that are different from (5.58). Furthermore, one could even change the criterion in (5.46) to another reasonable definition of the "generalized SNR". However, such variations on the theme of MAFI are beyond the scope of the present work.






62

5.5 Numerical Examples

In what follows, we investigate the performances of the various amplitude estimators described in the previous sections. For notational simplicity, we will refer to these methods as follows:

LSE1: LSE(1, 0, 1) using (5.14); LSEK: LSE(1, 0, K) using (5.4);

Capon1: WLSE(L, 0, 1) using (5.44); APES1: WLSE(L, 0, 1) using (5.45);

CaponK: WLSE(L, 0, K) using (5.38);

APESK: WLSE(L, 0, K) using (5.29) along with (5.43); MAFII: MAFI(L, K, 1) using (5.65) along with (5.43).

We will compare these methods with one another as well as the CRB given in (5.11). Since all these methods are asymptotically efficient, we only consider the case when N is relatively small. Specifically, we choose N = 32. The data consist of three complex sinusoids corrupted by a complex Gaussian noise v(n) (to be specified): x(n) = s(n) + v(n), n= 0,1,...,N- 1, (5.66) where
3
s(n) = Z kej2xfk. (5.67) k=1
The frequencies of the sinusoids are fl = 0.1, f2 = 0.11, and f3 = 0.3. Also, cl = ejr/4, o2 ej/3, and a3 = ej/4.

All examples are based on 200 Monte-Carlo simulations. The MSE figures shown in what follows are obtained as
1 200
MSE{jk} 200 &Ik(i) k 12, (5.68)
i= 1






63



30 ........
I 20







-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency

Figure 5.1: PSD of the test data that consist of three sinusoids and an AR(1) noise process.

where ^kk(i) is the estimate of ak derived in the ith simulation run.

5.5.1 Estimation Performance versus SNR

First, we consider the case where v(n) is colored. More exactly, v(n) is described by the following AutoRegressive (AR) process v(n) = 0.99v(n 1) + e(n), (5.69) with e(n) being a complex white Gaussian noise with zero-mean and variance O. The PSD of the test data is shown in Figure 5.1, where a2 = 0.01. The local SNR of the k-th sinusoid is defined as [41] lak 2 1 k 12 1SNR=1010g k+1/(2N) 10 10 (570)

k-1(2N)
Note the occurrence of N in the above SNR formula. For those methods that depend on M, we choose M = N/4 = 8, giving L = 25 (see Section 5.5.2 for a study of the effect of M on the performance).

Figure 5.2(a) shows the MSEs of the seven amplitude estimators for 3, aloun.ng with the corresponding CRB, as the SNR varies. As one can see, APESc, APESK, and MAFI1 are very close to the CRB, while LSEK, which ignores the noise correlationh is evidently away from the CRB. CaponK also deviates from the CRB for relation, is evidently away from the CRB. CaponK also deviates from the CRB for






64

most SNRs. The reason is that the approximation made in (5.37) is valid only for large N and M, which is not the case in this example. Figure 5.2(a) also shows that both LSE1 and Capon1 are inconsistent (in SNR). Their inconsistency is not surprising because both are biased estimators. Recall that the bias of LSE1, as given in (5.17), does not vanish unless N goes to infinity. Similarly, Capon1 is always biased (downward) for finite N [70] [47].

Figure 5.2(b) shows the counterpart curves for al. (The results for a2 are omitted because they resemble those for al.) Note that f2 f1 = 0.01, which is smaller than 1/N c 0.03, the Fourier resolution limit. The performance degrades for all estimators under study, especially for LSE1, Capon1, and APES1, which estimate only one amplitude at a time. As shown in Figure 5.2(b), LSE1 and Capon1 essentially fail for all SNRs considered due to their large MSEs. APES1 is no longer close to the CRB but, unlike the previous two estimators, it still appears to be consistent (in SNR). As in Figure 5.2(a), CaponK again deviates away from the CRB at high SNRs. It appears that the approximation made in (5.37) introduces a bias (at small N and M) that may be negligible at low SNRs but dominates the variance at high SNRs. The bias does not disappear as the SNR increases, which causes the divergence of CaponK from the CRB. APESK performs quite well for high SNRs; however, it is not very stable at low SNRs (due to large variance). The best estimator in this example is MAFI1. The knowledge of the number and locations of the sinusoids, which the other one-at-a-time estimators may spare but is indispensable to MAFI1, appears to play an important role in its good performance in the current case that fails the other one-at-a-time estimators.

As stated in Section 5.2, LSEK is statistically efficient, i.e. it achieves the CRB for any N > K, when the observation noise is white. To see how the other suboptimal (in finite samples) methods perform in such a case, we consider an example which is similar to the previous one except that v(n) is replaced by a zero-mean complex white







65






N=32,M=8,f3=0.3,colored Noise 10II

100




10



- LSE1
c -4 LSEK
10 ...... a Capon S x ....... x APES1
10 0o ..... CaponK
S..... o APESK
10-6 +...... + MAFI1 CRB
10-7
20 25 30 35 40 45 50 Local SNR dB

(a)
N=32,M=8,f1=0.1 ,colored Noise 101 I 1
10


10 : v ...
V..... ..... 13 ...
10-1 ......... .
0 ..... .........
-2 ..... ... ......
10-2 .......



a -- LSE1 ..
S10

C v v..... LSEK S......a Capon1
10 x....... APES1
S...... CaponK
10-5 ..... o APESK +...... + MAFI1
10-6 CRB

10-7 I I I I
15 20 25 30 35 40
Local SNR dB

(b)

Figure 5.2: Empirical MSEs and the CRB versus local SNR when N = 32, M = 8, and the observation noise is colored (an AR(1) process). (a) a3. (b) Cai.






66


Gaussian noise. The SNR is defined in the same manner as in (5.70). Figures 5.3(a) and 5.2(b) show the MSEs of the amplitude estimates of a3 and, respectively, al, and the corresponding CRB as the SNR increases. As one can see, the APES1, APESK, and MAFI1 estimates of a3 are again very close to the CRB; whereas for al, all suboptimal methods suffer from some performance loss as compared to the optimal LSEK, and yet the differences between LSEK and MAFI1 for all SNRs considered here are fairly small.

A brief summary based on the previous study is as follows. APES1 is recommended in applications where it is known a priori that no two sinusoids are closely spaced (see, e.g., the application discussed in the next section), or when the closelyspaced sinusoids are of no interest. The reason to prefer APES1 to APESK or MAFI1 in such cases is that the former is more flexible than the latter two since APES1 does not necessarily require the knowledge of the sinusoidal frequencies. In terms of computational cost, APES1 and MAFI1 are similar to one another and both are simpler than APESK. When it is desired to estimate closely spaced sinusoids in colored noise, however, MAFI1 may be preferred. In general, we do not recommend the use of Caponl since it has a computational complexity similar to that of APES1 but is biased. Although we did notice that CaponK gives close-to-CRB performance at very low SNRs, in most cases of interest, other methods like APES1 or MAFI1 may be preferred. LSEK is statistically efficient and may be preferred when the observation noise is white; in cases where the white noise assumption is invalid, it is preferable to use APES1 or MAFI1. LSE1 gives comparatively rather poor estimation accuracy but is computationally quite simple. The performance differences stated so far occur only when N is relatively small. As N increases, all methods tend to the CRB, independent of the noise correlation. Hence, when N is sufficiently large, LSE1 should be preferred because of its computational simplicity.







67






N=32,M=8,f3=0.3,white Noise 10 I I


100


10-1

W10
-7-3
=10
- LSE1 S ..... v LSEK
10 .... a Capon1 S ....... x APES1
10- ..... CaponK
e... o APESK 10-6 +...... + MAFI1
CRB 10-7 I I I I
20 25 30 35 40 45 Local SNR dB

(a)
N=32,M=8,f1=0.1,white Noise 101 I

10" ..... ........... . .. .
.. ...........
100
.. 3 ..



.. K ........
10 - LSE1 ......
-4 .... ..


10 CapO


w x ....... x APES1

10-s o ..... CaponK e .... o APESK + ...... MAFI1
10 CRB

10-7 I I
20 25 30 35 40 45 Local SNR dB

(b)

Figure 5.3: Empirical MSEs and the CRB versus local SNR when N = 32, M = 8, and the observation noise is white. (a) a3. (b) ao.






68

5.5.2 The Effect of M

All WLS and MAFI amplitude estimators studied in this chapter depend on the choice of M, the subvector length. It is known that as M increases, all of them can better deal with the case of closely spaced sinusoids, but their statistical stability in general decreases [75]. Hence, there is a tradeoff to to kept in mind when choosing M. Note that M should also be smaller than N/2; otherwise, the estimated covariance matrix will be rank deficient. The following example examines the effect of M on the performances of these estimators. LSE1 and LSEK do not depend on M and are thus not considered in this example.

The scenario is similar to the first example (AR noise) except that we fix a2 = 10-2, which corresponds to a local SNR of 30.8 dB for the first sinusoid (at fi = 0.1) and 39.2 dB for the third sinusoid (at f3 = 0.3). M is varied from 1 to 16 for all estimators except MAFI1, which requires that M > K (see (5.43) and (5.58)). The MSEs of the amplitude estimates of a3 and, respectively, al, and the corresponding CRBs are shown in Figures 5.4(a) and 5.4(b). As can be seen from these figures, all estimators are sensitive to the choice of M, to a smaller or larger extent. When no sinusoids are close to the one being estimated, such as the third sinusoid in this example, APES1, APESK, and MAFI1 perform quite well for a wide range of M. For the more difficult case as shown in Figure 5.4(b), the choice of M becomes very critical. Based on our empirical experience, a rule of thumb for the choice of M is given in Table 5.1.


5.6 Application to System Identification

Consider the linear discrete-time system described by the following equation [68]

x(n) = H(z-)u(n) + v(n), n = 0,1,..., N 1, (5.71)







69






N=32,a2=10-2,f 3=0.3,Colored Noise 101 I I I I S...... Capon1
10 x ....... x APES1 S...... CaponK
o ...... o APESK
210- +.. + MAFI1
LL CRB

10-2 ..:
as -2 "
o 10


C103 -10-s.
0 2 4 6 8 10 12 14 16 Subvector Length M

(a)
N=32, 2=10-2 f=0.1,Colored Noise 104 I I I I


10 3 ......a Caponi x ....... APES1
102 ...... CaponK S...... o APESK
.......+ + MAFI1
I 101 CRB




= 10 ..x :
10 ..... ...

10

10-4 I I I I I I

0 2 4 6 8 10 12 14 16 Subvector Length M

(b)

Figure 5.4: Empirical MSEs and the CRB versus M when N = 32 and the observation noise is colored (an AR(1) process with a2 = 0.001). (a) C3. (b) c1.






70


Estimator M APES1 N/4 < M < N/2 APESK N/4 < M < N/3 MAFI1 N/8 < M < 2N/5 Capon1 or CaponK N/8 < M < N/4


Table 5.1: Choice of M for the WLS and MAFI amplitude estimators where the input u(n) is a sinusoidal (probing) signal
K
u(n)= ykeiwk", n= 0,1,...,N-1, (5.72) k=1
and the transfer function is rational: B(z-') blz-' +... + bz-q (5.73) A(z-1) 1 + alz-1 +... + apz-P' We assume that

K > p + q. (5.74) Even if p and q were unknown, K could still be chosen sufficiently large to satisfy (5.74). The problem of interest in this section is to estimate {ai}lP and {bj},=l from {x(n) }yo'5.6.1 System Identification Using Amplitude Estimation

The commonly-used Output Error Method (OEM) does not model v(n) and obtains estimates of {ai} ,1 and {bj}j=l by minimizing the criterion N-I
COEM(a, b) = E Ix(n) H(z-')u(n)l2, (5.75) n=o

wherea=[ al ... ap ]T andb= [ bi ... bq ]T. Let


ak(a, b) = ykH(e'"k). (5.76)






71

For sufficiently large N (so that the transient response in the output can be neglected), the cost function COEM(a, b) is approximately equivalent to N-1 K 2
C, (a, b) = E x(n) E Ok(a, b) e4kn (5.77) n=0 k=1
The method that we propose for estimating a and b is based on (5.77) and consists of two steps:

First estimate {ak }=i in an unstructured/non-parametric form.

Then fit {ak(a, b)}K=1 to the amplitude estimates obtained in the previous step

by taking into account the statistical variance of the latter. In what follows, we detail the above two steps.

Step 1: Use an appropriate amplitude estimator to obtain estimates {&k k=1 of {ak kK=1 from the measurements {x(n)}N=01.. APES1 may be recommended in this case because we have control over the probing signal and we have no reason to choose any of the sinusoids too close to one another. The large-sample variance of the estimated amplitudes {&k JK=1 is proportional to {k(wk)}K=1 (see Section 5.2). To obtain estimates of {(Wk)}K=1, we can first calculate
K
i(n) = x(n) ,&k kn, n= O,,...,N-1, (5.78) k=1
and then utilize either a parametric or a non-parametric PSD estimator [75] [41] on (5.78) to obtain {(Wk)kK=1. Specifically, in the examples given in Section 5.6.2 we use the Capon PSD estimator [16] [44] [75] (also see Section 6.3), which determines { (Wk) K=1 as
M
(Wk) = k = 1, 2,..., K, (5.79) aH(wk)ii a(Wk)'
where a(wk) is defined in (5.40) and R, is the sample covariance matrix of the estimated noise vectors

v )= I(l) (+1)... (+M-1) = 0,1...L 1, (5.80)






72


that is,
L-1

L1 (5.81)

Step 2: Obtain estimates of {ai, bj} by minimizing
K
C2 (a, b) = E &k ak(a,b)12. (5.82) k=1 O(Wk)
To do so we can use a host of methods, provided that we have good initial estimates of a and b. To obtain such estimates and then minimize (5.82), we assume that p and q are known. (Standard techniques for system order determination can be found in, e.g., [68] [54].) We pick up the p+q largest {&k} (if the SNR is low, an alternative is to choose those {&k} that have the largest ratio &k 12 (Wk), assuming that (wk) was estimated) and define a criterion made from the corresponding terms of (5.82) P+q
C3(a, b) = E I&k ak(a, b)12 (5.83) k=1 O(Wk)

where we have assumed, for notational simplicity, that {&k} }(i are the p + q chosen amplitudes. Now, the minimization of (5.83) is simple. Indeed, almost always one can choose a and b to satisfy


&k = Ok(a,b), k = 1, 2,..., p + q. (5.84) Equation (5.84) is equivalent to GkA(eJWk) = B(ejwk), k = 1,2,... ,p + q, (5.85) 'Yk

which can be rewritten as a linear system of p + q equations with p + q unknowns {ai, bj}. That system will generally have a unique solution that makes (5.83) equal to zero, and which therefore gives our initial estimates of {ai, bj}.

As shown in the following numerical examples, the initial estimates are usually quite good. Hence, one can even skip the step of minimizing (5.82) to save computations.






73


Remark: According to the Extended Invariance Principle (EXIP) [76], the estimates of {ai, bj} obtained by minimizing (5.82) achieve the CRB asymptotically, and hence they have a better asymptotic accuracy than the OEM estimates whenever v(n) is colored. It also follows from this observation that in the case of K = p + q, the estimates obtained from (5.85) are asymptotically efficient. This latter result (of a somewhat limited interest, due to the requirement that K = p + q) was first proved in [40] in a much more complicated way.

5.6.2 Numerical Examples

The following examples assume that p and q are known to facilitate performance comparison. It is reasonable to do so since both OEM and the proposed method use similar techniques to determine the model orders. Also, we adopt the strategy to choose the p + q largest {rk} in Step 2 of the proposed method.


Example 5.6.2.1 The system considered in this example is given by (5.73) with

A(z-1) = 1 1.6019z-1 + 0.9801z-2, (5.86) and

B(z-1) = z-1 + 0.2472z-2 + 0.1600z -3. (5.87) The probing signal is given by

u(n) =2 cos(2r0.05n) + 2 cos(2w0.15n) + 2 cos(270.25n)

+2cos(27r0.35n) + 2cos(2rO0.45n), n = 0,1,...,N 1. (5.88) We consider using a real-valued probing signal because this is the usual case in practice. (A subtle question arises as the amplitude estimation techniques discussed in the previous sections all assume that the sinusoids are complex-valued. One might impose certain conjugate symmetry constraint and derive similar techniques that are specifically tailored for real-valued sinusoidal amplitude estimation so that, if






74


w1 = -w2, the estimators will give &i = &]. Yet, our experience shows that the gain would most often be minor and hence the effort is not worthwhile. See [39], for example.) Note that K = 10 for this case. The noise v(n) is a real-valued white Gaussian noise with zero-mean and variance o2 = 0.01. We estimate the system parameters using the proposed technique and OEM (OEM is provided in the System Identification Toolbox of MATLAB). For the proposed technique, we compute both the initial estimates given by solving (5.85) and the minimizer of (5.82). The minimizer of (5.82) is found by using the solution of (5.85), obtained by APES1, as the initial condition and then evoking a standard gradient-type nonlinear optimization routine provided by MATLAB. To reduce the number of graphs, we only show the averaged Root Mean Squared Error (RMSE) for the a-parameters


RMSE{i} = E RMSE{&i} (5.89)
i=1

and similarly for the b-parameters. All results are based on 200 Monte-Carlo simulations. Figures 5.5(a) and 5.5(b) show the averaged RMSEs of the a-parameters and, respectively, the b-parameters obtained by using OEM and the proposed technique, as N increases. Figure 5.5(c) shows the required number of flops as N increases. (APES1 and MAFI1 uses M = 20 in this and the following example, which does not fall in the range given in Table 5.1. The reason is that APES1 or MAFI1 with M = 20 is quite acceptable for the probing signal in (5.88) that contains well-separated sinusoids and, moreover, choosing a larger M would result in additional computations.) Finding the minimizer of (5.82) or the OEM estimates involve iterative searches which give variable flop counts from trial to trial. The number of flops needed by each of these two methods, as shown in Figure 5.5(c), is the average over 200 trials. As one can see, the initial estimates of {ai, bj} given by solving (5.85) with APES1 or MAFI1 have similar RMSEs to those obtained by OEM. The estimates obtained by minimizing (5.82) are slightly better than the initial estimates obtained by APES1






75

or MAFI1, but at a significantly increased computational cost. Due to this observation, we do not recommend using this approach, i.e., minimizing (5.82), for refined estimation accuracy. Other more sophisticated techniques for system identification (see, e.g., [68] [54]) may be preferred in that event. Figure 5.5(c) also shows that, as compared to OEM, there is little computational advantage of using the initial estimates obtained by APES1 and MAFI1. The reason may be that the system in this example is quite simple (it has white output errors, etc.) and, apparently, OEM reaches convergence in a relatively small number of iterations. For a more complex system, such as the one used in the next example, OEM may need more iterations to converge. It should be mentioned that we did not program our method very carefully and hence our code is unlikely to be as efficient as the OEM code in MATLAB. Regarding the estimation accuracy, we shall stress that in the current case where the noise v(n) is white, OEM coincides with the optimal Maximum Likelihood Method (MLM) [68] [54]. When v(n) is colored, OEM is no longer MLM. In that case, the initial system parameter estimates obtained by APES1 or MAFI1 may outperform those by OEM, as in fact shown in the next example.

Recall that LSEK is statistically efficient when the observation noise is white. Then, one might wonder why the initial estimates given by LSEK may be notably worse in such a case than those given by APES1 or MAFI1, as happened in the previous example (especially when N is small). The reason is that the transient response of this system cannot be neglected for small N. To show this, the PSD of x(n) is estimated by using the Capon PSD estimator, with N = 200 and M = 20, and is plotted in Figure 5.6. It shows two extra peaks (which behave like two sinusoids) at 0.1. The extra peaks are attributed to the response of the system (which has poles at 0.99ej2r0.1) to the initial conditions. Since it is essential for LSEK to have the accurate knowledge of the number and frequencies of the sinusoids frequencies to give reliable amplitude estimates, its performance in the previous example is considerably









76




















x 10- a-Parameters b-Parameters

(85) with LSEK
7 v (85) with LSEK 0.07- (85) with APES1 S. (85) with APES1 0 (85) wth MAFI 6 V o a (85) with MAFi 0.06 (82) with APES1
(82) with APES1 OEM U0 5- OEM 0.05

4 0.04



0.02

1 0 .0 1 -*
,... V ...


10 10 10 10 Data Length N Data Length N

(a) (b)

Computational Complexity


S v (85) with LSEK
(85) with APES1
ac (85) with MAFI1
4 4 (82) with APES1





2 ......o.
1 0 s







10 10 Data Length N

(c)


Figure 5.5: Averaged RMSEs and the number of flops versus N for the first system when the observation noise is white (a.2 = 0.01) and M = 20 for APES1 and MAFI1.

(a) RMSE of a-parameters. (b) RMSE of b-parameters. (c) Number of flops.






77


I I i I I
15
Ca 10






-20
I I I I I I I I
-.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Frequency

Figure 5.6: PSD estimate of the output of the first system corrupted by white noise with a2 = 0.01 and N = 200.

deteriorated. On the other hand, the above knowledge is not necessarily needed by APES1 and hence its performance is not affected. Unlike APES1, MAFIl does require this knowledge. Yet, the initial condition response is substantially weakened through the frequency selective filtering employed by MAFI1 and hence has little, if any, effect on the amplitude estimates and the system parameter estimates. As N increases, the transient effect becomes less severe, and, consequently, the initial system parameter estimates obtained by using LSEK approach those obtained by APES1 or MAFI1.


Example 5.6.2.2 We now consider a second system with

A(z-1) = 1 1.9109z-1 + 1.7251z-2 0.7033z-3 + 0.2450z-4, (5.90)

and

B(z-1) = z-1 + 1.0562z-2 + 0.6100z-3 + 0.1912z-4 + 0.0400z-5. (5.91)

The noise v(n) is an AR(1) signal as in (5.69) except that e(n) is now replaced by a real-valued white Gaussian noise with zero-mean and variance a2 = 0.01. The probing signal is the same as in the previous example. Figures 5.7(a) to 5.7(c) show the averaged RMSEs of the a-parameters and the b-parameters, as well as the number









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a-Parameters b-Parameters
0.7 0.9
v (85) with LSEK v (85) with LSEK 0.6 ........ (85) with APES1 I0.8 (85) with APES1 o o (85) with MAFIl I ao o (85) with MAFI1 (82) with APES1 0.7 o. (82) with APES1
0.5- OEM - OEM L PEM 0.6 PEM
0.4 0.


<> 0.3

0.2
0.2
0 .1 ".... .



10 103 10 10, Data Length N Data Length N

(a) (b)

Computational Complexity







10

............. ..

E V .... .
S v v... (85) with LSEK ........ (85) with APES1
10s- o (85) with MAFI1 0 (82) with APES1
OEM
o a PEM

104
10 10a
Data Length N

(c)


Figure 5.7: Averaged RMSEs and the number of flops versus N for the second system when the observation noise is colored (an AR(1) process with .2 = 0.01) and M = 20 for APES1 and MAFI1. (a) RMSE of a-parameters. (b) RMSE of b-parameters.

(c) Number of flops.




of flops, as N increases. As one can see, the initial system parameter estimates given


by APES1 or MAFI1 are significantly better than those given by OEM, and yet the


former two are computationally more efficient that the latter.




5.7 Summary



This chapter examines the problem of amplitude estimation of sinusoidal signals in colored noise. Three general classes of estimators, namely the LS, WLS, and


MAFI approaches to amplitude estimation, are discussed. It is shown that, under






79

certain circumstances, the MAFI approach to amplitude estimation is equivalent to the WLS approach, and yet the former is more general and includes the latter as a special case. The amplitude estimators under discussion can be further categorized depending on whether they estimate one amplitude at a time or all amplitudes simultaneously. MAFI or WLS methods, such as APES1, APESK, and MAFII, in general give more accurate amplitude estimates for sinusoids in colored noise. Methods that estimate only one amplitude at a time, such as APES1, mostly do not require the exact knowledge of the number and locations of the sinusoids and hence are more robust than those that estimate all amplitudes simultaneously.

As an application example, a system identification application using sinusoidal probing signals is discussed. A new technique for system identification is presented that can avoid iterative searches through fitting the system parameters to the output amplitude estimates. It is shown that, by using this technique with appropriate amplitude estimators, such as APES1 or MAFI1, we can obtain results that are generally better than those corresponding to the widely-used iterative OEM, yet usually at a reduced computational cost.















CHAPTER 6
CAPON ESTIMATION OF COVARIANCE SEQUENCES

6.1 Introduction

Covariance sequence estimation is a ubiquitous task in digital signal processing. A standard technique for estimating the covariance sequences is the so-called the standard sample covariance estimator. The standard covariance estimates are consistent provided that the given signals are ergodic to the second order. However, there is a major concern of using the standard estimator due to the unrealistic windowing of the observed data it assumes; that is, it assumes that the data beyond the observed duration either is zero or repeats itself periodically. Partly for this reason, there have been several attempts in the literature to derive more accurate covariance estimates than the standard ones. A notable example is the approach based on the Burg autoregressive (AR) spectral estimator. However, the so-obtained covariance estimator was found even less accurate than the standard sample covariance estimator [78]. To be more exact, the Burg approach was shown to have larger variances than the standard method. Another approach, which has generated a whole new research direction, relies on the maximum likelihood (ML) principle [13]. However, the ML estimation of covariance sequences is a computationally involved problem which does not have a closed-form solution. The solution given in [13] is iterative and not guaranteed to be globally optimum. Apparently, there exist no compelling alternatives to the standard method that can be recommended for general use.

In this chapter we present a new method, namely the Capon method [48], for covariance sequence estimation. The Capon method obtains the covariance sequence estimates by Fourier inverting the Capon power spectral density (PSD) estimates.


80






81


There are basically two Capon PSD estimators, referred to as Capon-1 [16] [44] and Capon-2 [45] herein. We find that, while Capon-2 is capable of finer spectral resolution around the peaks of a spectrum, it is generally a globally poorer spectral estimator than Capon-1. We hence concentrate our interest on Capon-1 for covariance sequence estimation in this chapter. Since the Capon spectra, i.e., the PSD estimates, are shown to be equivalent to AR or autoregressive moving-average (ARMA) spectra, the inversion procedure for computing the exact covariance sequences corresponding to the Capon spectra can be implemented in a rather convenient way. (Note that the calculation of the covariance sequences corresponding to the Capon spectra is an interesting problem by itself.) We also present an FFT-based approximate method to compute the covariance sequences from the Capon spectra. It has been found that the approximate method provides covariance estimates that are almost identical with those obtained by the exact method, while the computational complexity is greatly reduced.

Our primary interest is to apply the Capon method as well as the standard approach to ARMA signals. To that end, a few ARMA signals with typical pole and zero locations are studied in our numerical examples. The studies show that considerable improvements are attained by the new Capon method.

The Capon covariance estimation method can be readily used in many applications. One important class is the ARMA spectral estimation. Since most ARMA spectral estimators rely on the Yule-Walker equations to determine the AR coefficients, it may be expected that the better the covariance estimates used, the more accurate the AR coefficient estimates yielded. We examine how the Capon covariance estimates can be used with the overdetermined modified Yule-Walker (OMYW) method [14] [69] to compute more accurate AR coefficients. We find that the performances of the usage are influenced by the pole and zero locations and, still, generally






82

better AR coefficient estimates are obtained by using the Capon covariance estimates than by the standard ones. Another application discussed in this chapter is the moving-average (MA) model order determination by making use of the Capon and the sample covariance estimates, where we find that better performance is achieved by the former.

It should be mentioned that we can obtain APES covariance sequence estimates similarly by Fourier inverting the APES PSD estimates. However, in spite of the evident difference of the Capon and APES complex spectral estimates for discrete spectral components, their performances in continuous PSD estimation are similar to one another, which implies that the APES covariance estimates are similar to the Capon covariance estimates. As such the APES covariance estimation is not discussed herein.


6.2 Standard Covariance Estimator and Outlook

With no other assumptions made on the signal under study, except for assuming the second order ergodicity, there are two ways to obtain the standard sample covariances of the signal, namely, the biased and the unbiased covariance estimators. However, the biased covariance estimator is more commonly used since it provides smaller mean-squared errors (MSE) than the unbiased one and guarantees the covariance estimates to be positive semidefinite [75].

The biased sample covariance estimator of a wide-sense stationary signal with zero-mean has the form
N-k
f(k)= y=Y*(n)y(n+k), k=0,1,...,K. (6.1) n=l
where {y(n)}N1 are the observed data samples, N is the number of samples, f(k) denotes the estimate of the covariance function r(k), K is the largest lag desired (0 < K < N 1), and (.)* denotes the complex conjugate. Note that (6.1) is asymptotically unbiased.






83

The estimator given in (6.1) is consistent if the ergodicity assumption is satisfied. A study on whether the standard sample covariance estimator is also asymptotically statistically efficient, i.e., whether it asymptotically achieves the Crambr-Rao Bound (CRB), has been undertaken in [60]. Let y(t) be an ARMA(p, q) signal. If p q, then the sample covariance estimate i(k) is asymptotically statistically efficient if and only if 0 < k < p q; in particular, for AR processes of order p, f(k) is asymptotically efficient for 0 < k < p, but inefficient for all other k. If p < q, none of ?(k) is asymptotically efficient; in particular, none of ?(k) is asymptotically efficient for an MA process.

It is known that the sample covariance sequence {f(0), ..., (N 1), 0, 0,...) and the data periodogram constitute a Fourier transform pair. It is also known that the periodogram is a statistically inefficient (in fact inconsistent) estimator of the PSD [75]. This observation suggests that better covariance estimators might be obtained by Fourier inverting better PSD estimators. However, this is not necessarily so. Briefly stated, the reason is that the Fourier transform and the inverse Fourier transform are integral transforms and hence small errors in one domain may be associated with large errors in the other domain or vice versa. The fact that the covariance estimates in (6.1) are consistent whereas the periodogram is not illustrates this observation. Also note that, while the Burg estimate of the PSD is typically more accurate than the periodogram, the corresponding Burg estimate of the covariance sequence is generally poorer than (6.1) [78].

In spite of the fact briefly discussed above, in the following we consider estimating the covariance sequences by inverting a PSD estimate that is often much more accurate than the periodogram, namely the Capon PSD estimate. Like the periodogram, no model is assumed in the Capon PSD estimator, which makes it more robust than the parametric estimators in many situations. Although it has lower spectral resolution than the AR spectral estimator, it generally exhibits less variance






84


than the latter [44] [75]. Another reason that we consider the Capon PSD estimator is that it does not exhibit the so-called correlation matching property [75] [41]; that is, the inverse Fourier transform of the Capon PSD estimates does not yield the same covariance sequences used to obtain the Capon PSD estimates. This fact allows us to obtain a new covariance estimator from the Capon spectra.


6.3 Capon PSD Estimator

We have derived in Section 3.2.1 the Capon filters and Capon amplitude and phase spectra. In this section, we derive expression for the Capon PSD estimates. The Capon filter is rewritten below for easy reference (see (3.28)): hCapon R- (w) (62) S aH (w)RLaM am.

The filter output power is given by

E IhH(l) = hHRh 1 (6.3) SaH(w)RaM( (6.3)) Let 3 denote the bandwidth of the filter given by (6.2). Then the Capon PSD estimate has the form
E { hHf(l)2} 1
(w) =. (6.4)

Since the (equivalent) time-bandwidth product is equal to unity, one way is to choose 3 as the reciprocal of the temporal length of the Capon filter; that is

1
0 = 1. (6.5)
M

By choosing the filter bandwidth as given by (6.5), we obtain the so-called Capon-1 PSD estimator [44] [75]:

Capon-1: ((w aM(W (6.6) where we have replaced R by the sample covariance matrix fR. We may use the forward-only sample covariance matrix, but we prefer using the forward-backward






85

sample covariance matrix, given in (3.13), for better statistical properties associates with the latter [41].

Another more elaborate choice of / is obtained as the equivalent bandwidth of IH(w)12, where H(w) is the filter's frequency response: H(w) [hCapon HaM(W). (6.7) This specific bandwidth choice leads to the Capon-2 PSD estimator [45] [75]: Capon-2: O(w) = a I-M (6.8) aH(w)R-2aM (w)

Burg showed that the inverse of Capon-1 spectrum is equal to the average of the inverses of the estimated AR spectra of orders from 0 to M [11]. This observation reveals the fact that Capon-1 has less statistical variation as well as lower spectral resolution than the AR estimator. A similar but more involved relationship between Capon-2 and the AR estimators was derived in [75]. Theoretically, the performance of Capon-2 is hard to quantify. However, it is generally believed that Capon-2 possesses finer resolution and hence is a better spectral estimator than Capon-1 [45]. We will show here, with a typical example, that even though Capon-2 has better resolution locally around the power peaks, it is globally a more biased estimator than Capon1. Our experience also shows that Capon-2 generally gives much poorer covariance estimates than Capon-1. Therefore, Capon-2 is not recommended for covariance sequence estimation.

To illustrate the above claim, consider an ARMA(4,2) signal

y(n) = 2.76y(n 1) 3.809y(n 2) + 2.654y(n 3) 0.924y(n 4)

+e(n) 0.9e(n 1) + 0.81e(n 2), (6.9) where e(n) is a real white Gaussian random process with zero-mean and unit variance. The Capon-i and Capon-2 spectral estimates with N = 256 and M = 50 are shown in Figure 6.1(a), where the dashed curve stands for the true PSD of the ARMA






86

ARMA(4,2)(N=256, M=51, 100 realizations.) ARMA(4,2)(N=32, M=1 1, 100 realizations.)


0 40








plots are the averages of 100 independent realizations. (a) N = 256, M = 50; (b) N 32, M = 10.












and Capon-2 estimates. It is obvious that Capon-2 is highly biased and suffers from
0 0.1 0.2 0.3 0. 0.5 0 0.1 0.2 0.3 0.4 0.5
Frequency Frequency
(a) (b) Figure 6.1: Power spectral density estimates by using Capon-i and Capon-2. The plots are the averages of 100 independent rer of the overal) N = 256,M = 50; (b) N = 32,M = 10.


signal, while the solid and the dashdotted lines, respectively, indicate the Caponand Capon-2 estimates. It is obvious that Capon-2 is highly biased and suffers from a significant power loss. This observation has also been made with other signals, especially if the signal is a narrowband signal. An explanation of this behavior follows.

The calculation of the filter bandwidth in Capon-2 is applicable only if the Capon filter is a narrowband filter. Recall that the Capon method aims to find the Capon filter that minimizes the total output power of the overall frequency band while passes the current frequency w undistorted. No effort has been taken to make sure that the Capon filter is narrowband. Let the lobe of the Capon filter frequency response where the current frequency of interest w is located be called as the mainlobe, while all the others are called the sidelobes. It has been found that the steering frequency w is not necessarily at the maximum or the center of the mainlobe [75] [52]. Furthermore, if the input signal is a narrowband signal, there may exist "sidelobes", located at frequency bands where the power level of the input signal is low, that are even larger than the "mainlobe" of the Capon filter frequency response. Note






87


that the large "sidelobes" do not make any significant contributions to the filter output power so that the filter design criterion is still satisfied; that is, the output power is minimized, while the frequency response at w is one. In all such cases, the Capon filter is not a narrowband filter and hence it calculates an overestimated filter bandwidth. Hence the Capon-2 PSD estimates become highly biased. However, it is interesting to note that Capon-2 does possess higher resolution capability, around the power peaks, than Capon-1. This is illustrated in Figure 6.1(b), which shows the PSD estimates of the same ARMA signal as used in Figure 6.1(a) but with N = 32 and M = 10. The Capon-1 estimator cannot resolve the two power peaks this time, while Capon-2, albeit biased, still can.

For the preceding reasons we do not consider using Capon-2 for covariance sequence estimation in the sequel.


6.4 Capon Covariance Estimator

We describe below how the Capon PSD estimates can be Fourier inverted in a rather convenient manner yielding the Capon covariance sequence estimates. The study of the covariance sequences corresponding to the Capon spectra is an interesting endeavor by itself, which apparently has not been undertaken in the literature before. We also present an approximate but computationally more efficient method to calculate the Capon covariance estimates from the Capon PSD estimates.

6.4.1 Exact Method

Theorem 6.4.1 Let F = {fi,j} C(M+1)x(M+1) and let aM(w) 1 e ... ej(M-1) (6.10) then
M-1
a (w)raM(w) seJ, (6.11) s=-(M-1)






88

where
min(M+s-1,M-1)
Is = rk,k-s. (6.12) k=max(0,s)
If r is Hermitian, then pL = p*,.


Proof: See Appendix D.

It is obvious that whenever P is non-negative definite we have aH (w)FaM(w) > 0 for any w. Thus aHM(w)aM(w) is a valid power spectrum. Furthermore, Theorem 6.4.1 indicates that in such a case 1/aH(w)raM(w) is in fact the power spectrum of an (M 1)th-order AR process. Consequently, Capon-1 yields an equivalent AR(M 1) process (whereas Capon-2 yields an equivalent ARMA(M 1, M 1) process). By making use of (6.12), we can find the coefficients of the equivalent AR process. The calculation of the exact covariance sequences from the AR coefficients is a standard problem and can be solved, for example, via the inverse Levinson-Durbin algorithm (See [75] [41] and the references therein for more details). Hence the implementation of the Capon method for covariance estimation runs as outlined below:

Step 1: Pick up a value for M(M < N/2) and compute I by (3.13).

Step 2: Compute p, associated with P = -1 by (6.12). Factorize M-1

m=-(M-1)

(say, by using the Newton-Raphson algorithm) and obtain the (minimum-phase) spectral factor.

Step 3: Compute the corresponding covariance sequence f{temp(k) } from the spectral factor (or, equivalently, the AR model) by, for example, the inverse Levinson-Durbin algorithm.

While using the Fourier inverting method for covariance estimation, it is necessary that the integral of the PSD estimate over all frequencies gives a good estimate of the signal power; otherwise there may be scaling errors in the covariance estimates.




Full Text
74
l\ CO2, the estimators will give on = Yet, our experience shows that the
gain would most often be minor and hence the effort is not worthwhile. See [39],
for example.) Note that K 10 for this case. The noise v(n) is a real-valued
white Gaussian noise with zero-mean and variance er2 = 0.01. We estimate the
system parameters using the proposed technique and OEM (OEM is provided in the
System Identification Toolbox of MATLAB). For the proposed technique, we compute
both the initial estimates given by solving (5.85) and the minimizer of (5.82). The
minimizer of (5.82) is found by using the solution of (5.85), obtained by APESl, as the
initial condition and then evoking a standard gradient-type nonlinear optimization
routine provided by MATLAB. To reduce the number of graphs, we only show the
averaged Root Mean Squared Error (RMSE) for the a-parameters
(5.89)
i= 1
and similarly for the -parameters. All results are based on 200 Monte-Carlo simula
tions. Figures 5.5(a) and 5.5(b) show the averaged RMSEs of the a-parameters and,
respectively, the -parameters obtained by using OEM and the proposed technique,
as N increases. Figure 5.5(c) shows the required number of flops as N increases.
(APESl and MAFI1 uses M = 20 in this and the following example, which does not
fall in the range given in Table 5.1. The reason is that APESl or MAFI1 with M = 20
is quite acceptable for the probing signal in (5.88) that contains well-separated sinu
soids and, moreover, choosing a larger M would result in additional computations.)
Finding the minimizer of (5.82) or the OEM estimates involve iterative searches which
give variable flop counts from trial to trial. The number of flops needed by each of
these two methods, as shown in Figure 5.5(c), is the average over 200 trials. As
one can see, the initial estimates of {a,j} given by solving (5.85) with APESl or
MAFI1 have similar RMSEs to those obtained by OEM. The estimates obtained by
minimizing (5.82) are slightly better than the initial estimates obtained by APESl


42
and
Q 'aw,,
\H
.H T} 1
m2
Mi ,M2
aMi,M2R aMuM2 4
LxL2 V^R-1^
= iibK,^)f+lbT("1-1J2)e^'^)f.
11 ^ 2j" L1L2-||e(a;1,a,2)||2
(4.20)
Next we observe that b(wi,u;2), d(u>i, ci,u;2) can be calculated via 2-D
FFT. Specifically, we partition C_1, D and E as follows:
C_1 =
vec{Bi} ... vec{BMlM2}
and
D
E
~\ T
vec{Di} ... vec{DMlM2}
iT
(4.21)
(4.22)
(4.23)
vec{E!} ... vec{EMlM2}
where B*, G CMlxM2, CLlXl2 and E*, G CilX2. Then (again, we drop the
dependence on ux and o>2)
(aM2 aM JvecjB!} ... (a^r <8> )vec{BMlA2}
aMi-^laM2 aM1BM1M2a*M2
(4.24)
Likewise, we have
and
d = Dai1|La
e = Ea *LlM
a^Dia l2
a1DMiM2al2
afEia. ... a? Ej - -
(4.25)
(4.26)
1Li-Eji1l2 aLih;M1M2aL2
Note that a^B^a^ represents the 2-D discrete Fourier transform of Bfc at (a>i,a;2).
Hence APES can be efficiently implemented by first calculating b(u>i,u;2), d(cji,o>2)


79
certain circumstances, the MAFI approach to amplitude estimation is equivalent to
the WLS approach, and yet the former is more general and includes the latter as a
special case. The amplitude estimators under discussion can be further categorized
depending on whether they estimate one amplitude at a time or all amplitudes simul
taneously. MAFI or WLS methods, such as APES1, APESK, and MAFI1, in general
give more accurate amplitude estimates for sinusoids in colored noise. Methods that
estimate only one amplitude at a time, such as APESl, mostly do not require the
exact knowledge of the number and locations of the sinusoids and hence are more
robust than those that estimate all amplitudes simultaneously.
As an application example, a system identification application using sinusoidal
probing signals is discussed. A new technique for system identification is presented
that can avoid iterative searches through fitting the system parameters to the output
amplitude estimates. It is shown that, by using this technique with appropriate
amplitude estimators, such as APESl or MAFI1, we can obtain results that are
generally better than those corresponding to the widely-used iterative OEM, yet
usually at a reduced computational cost.


109
where {Re(z)} is the covariance sequence of ew(/) or ew(/). Note that (B.7) and (B.8)
are due to the circularly symmetric distribution assumption. It then follows from
(B.9), (B.10), and Condition C that:
OO
lim Le\sSH\ = lim Le{~S~SH\ = V Re{i)e-juji = e[ui)aMa^, (B.ll)
Ltoo L J Lt oo l J z'
ioo
where the last equality follows from the standard results on the transfer of spectral
densities through linear systems. Among others, the previous calculations imply that,
as L oo,g and g tend to aa^ and oOlm (in the mean square sense), respectively,
and, therefore, GGfi goes to \oi\2aLM&M- Hence hBBC and hBBA have the same limit
as L oo.
Let h denote a generic FIR vector and let denote the deterministic vector
that is the limit of (the possibly random) h when L goes to infinity. Observe that
for all methods under study, the associated h and hoo vectors satisfy
hHaM = 1, and h^,aM = 1. (B.12)
By using this observation with (3.26) and (B.l), we obtain
= hHg = a + hHS. (B.13)
Since 5 tends to zero as L goes to infinity, it follows from (B.13) that the estimation
error can be asymptotically written as (to a first-order approximation)
a. a ~ h^d. (B-14)
Then it readily follows that
LE {(a a)2} 0, as L oo, (B.15)
and
lim LE {\a- o:|2} = h^, lim LE {<5<5/7) = <£e(cj) |h^aM|2 = 0e(w), (B.16)
L-* oo L>oo L J 11
and the proof is concluded.


97
6.5.2 AR Coefficient Estimation for ARMA Signals
As an application of the Capon method for covariance estimation, we include
here an example on how to use the Capon covariance estimates to find the AR
coefficients of ARMA signals via the overdetermined modified Yule-Walker (OMYW)
method [14] [69] [75]. We first briefly explain the OMYW method.
For an ARMA(p, q) process, the covariance sequence and the AR coefficients
are related by
r(q) r(q-l)
r(q-p + 1)
a i
r(q + l)
r{q + 1) r(q)
r(q-p + 2)
2
=
r{q + 2)
r(q + L 1) r(q + L 2)
r(q + L-p) _
dp
_ r(q + L) _
(6.21)
If L = p in (6.21), then we have a system of q equations with q unknowns. These
equations are referred to as the modified Yule-Walker equations [14] since they con
stitute a generalization of the Yule-Walker equations for the AR signals. Replacing
the theoretical covariances (r(A:)} by their sample estimates {r(k)} in (6.21) yields
f(q) r(q-l)
1
+
h1
1
1
r(q + 1)
r(q + 1) r(q)
f(q-p + 2)
2

r{q + 2)
r(q + L 1) f(q + L- 2)
r(q + L p) _
dp
_ r(q + L) _
(6.22)
The overdetermined case of L > p in (6.22) is motivated by the fact that
additional information in the higher-lag covariances can be exploited to improve
the accuracy of the AR coefficient estimates; that is, we can make use of the additional
information by choosing L > p and solving the so-obtained overdetermined system of
equations, either in a least-squares (LS) or in a total-least-squares (TLS) sense [30].


Cl
which shows that
MAFI(L, K, K) = WLSE(L, 0, K). (5.63)
The MAFI interpretation of the WLS method, afforded by the above analysis,
is interesting. In particular, it makes a clear connection between using the MAFI and
the WLS techniques for amplitude estimation. The MAFI approach is however more
general than the WLS technique. As an example, we derive a new MAFI amplitude
estimator that does not belong to the WLS class as follows. Let zk{l) and vk{l) denote
the fc-th element of z(V) and, respectively, u(l) in (5.60). Then
zk{l) = ake^1 + uk(l), k = 1,2,..., K. (5.64)
The above equations are related to one another only via the correlation between vk{l)
and up(l) (for k ^ p). If we ignore the correlation, then the MAFI(L, K, 1) estimate
of the ak via LS is given by
1 L~l
&k = ~J^zk(l)e-^1. (5.65)
^ 1=0
Unlike the Capon (5.44) and APES (5.45) estimators (which can also be shown to be
members of the MAFI(L,iF, 1) class [70]), the above MAFI(L, K, 1) estimator does
require the knowledge of the number and frequencies of the sinusoids, which makes
it behave more like a MAFI(L, K, K) estimator. In particular, it performs quite well
for cases where some sinusoids are closely spaced, as will be seen in Section 5.5.
Other interesting MAFI amplitude estimators may be devised by using some
other choices of H in lieu of the one given in (5.58), as the solution to (5.46) is not
unique. Specifically, one may introduce certain unitary transform on the H in (5.58),
or choose K < K, or replace the Ik in (5.59) by another nonsingular matrix, which
all lead to solutions that are different from (5.58). Furthermore, one could even
change the criterion in (5.46) to another reasonable definition of the generalized
SNR. However, such variations on the theme of MAFI are beyond the scope of the
present work.


TABLE OF CONTENTS
ACKNOWLEDGEMENTS iii
LIST OF TABLES vii
LIST OF FIGURES viii
ABSTRACT xi
CHAPTERS
1 INTRODUCTION 1
1.1 Background and Scope of the Work 1
1.1.1 Capon, APES, and MAFI Spectral Estimators 1
1.1.2 Efficient Implementation 4
1.1.3 Amplitude Estimation 5
1.1.4 Covariance Sequence Estimation 5
1.2 Significance and Contributions 6
1.3 Organization of the Dissertation 7
2 LITERATURE SURVEY 9
2.1 Filterbank Approaches and Capon Estimator 9
2.2 Covariance Estimation 13
3 MAFI APPROACH TO SPECTRAL ESTIMATION 16
3.1 Forward and Backward Data Vectors 16
3.2 MAFI Filters 19
3.2.1 Capon Filter 20
3.2.2 APES Filter 21
3.2.3 Another Matched Filter 22
3.3 Analyses of MAFI Approaches 24
3.3.1 Computational Complexity 24
3.3.2 Statistical Performance 25
3.4 2-D Extensions 27
3.5 Numerical Examples 31
3.5.1 1-D Complex Spectral Estimation 31
3.5.2 2-D Complex Spectral Estimation 33
3.6 Summary 36
IV


6.1 Power spectral density estimates by using Capon-1 and Capon-2. The
plots are the averages of 100 independent realizations, (a) N = 256, M =
50; (b) N = 32, M = 10 86
6.2 Pole-zero diagrams for ARMA test cases, (a) ARMAl; (b) ARMA2;
(c) ARM A3; (d) ARMA4 93
6.3 True power spectral densities, (a) ARMAl; (b) ARMA2; (c) ARMA3;
(d) ARMA4 94
6.4 True covariance sequences, (a) ARMAl; (b) ARMA2; (c) ARMA3;
(d) ARMA4 95
6.5 Covariance sequence estimation with N 256 and M = 50. The
mean-squared errors (MSEs) of the covariance estimates, normalized
with respect to r(0), are based on 100 independent realizations, (a)
ARMAl; (b) ARMA2; (c) ARMA3; (d) ARMA4 96
6.6 The AR coefficient estimation of the ARMA signals via the overde
termined modified Yule-Walker method with N = 256 and M = 32.
The curves are the summations of the mean-squared errors (MSE) of
all the AR coefficient estimates versus the numbers of included equa
tions, which have been set as 4, 8, 16, 32, 64 and 128, respectively. The
MSE curves are based on 100 independent realizations, (a) ARMAl;
(b) ARMA2; (c) ARMA3; (d) ARMA4 99
6.7 10 superimposed realizations of the MA covariance sequence estimates
with N = 64 and M = 6. (a) The standard method; (b) The Capon
method 100
x


17
By borrowing the notations defined in (3.49)(3.51), the solution fbm to (A.4) is
straightforward to obtain:
1 1
FBM
12 1
/[ikiir
ln "2"- 1,2 1|-H- '2 M2IM2 + jIM2IM2M3
ii^h2+^ i^f^i
1.. ..91 u 12 1 1 ff |2 1 I, 11 a I H 12
inr k ^ - ihi i*! h
2 \ H I2
K ^3
(A.5)
Next we evaluate the FBA estimator as given by (3.53). We first compute the E :
-i = i
A
IMf
w**y imi -2
(A.6)
where
A = IMt IKI|2 2 ||i/2||2 2 H^H2 pf 1/3I + 4.
(A.7)
Substituting (A.7) into (3.53) and performing some simple manipulations, we obtain
FBA = [4*^2 2 ||i/3||2 {^1^2) + 2(i/fi/3)(i/fi/3)*] /[Ikill2 ll^ll2 ll^sll2
-2 llalli2 lk2||2 2 ||izi||2 ||i/3||2 + 4 ||i/i||2 H^ill2 \v2t/3\2
- ||l23||2 ||/f I/2|2 + 2 |l/f l/2|2 + (l/f l/2)*(i/f I/3)(l2^i23)*
+ (i/f */2)(vf V3)*{v%v3) \\v2\\2 Il/f 1/3|2 + 2 |i/f I/3|2] (A.8)
Comparing (A.5) and (A.8) confirms that FBA and FBM coincide. Hence (3.43)
follows.


65
(a)
(b)
Figure 5.2: Empirical MSEs and the CRB versus local SNR when N = 32, M = 8,
and the observation noise is colored (an AR(1) process), (a) 0:3. (b) 1.


34
(a) (b)
(c) (d)
Figure 3.2: Empirical bias and variance of the 1-D MAFI estimators as SNRj varies
when N = 64 and M = 15. (a) Real part of the bias; (b) Imaginary part of the
bias; (c) Variance of the real part of the estimated amplitude; (d) Variance of the
imaginary part of the estimated amplitude.


119
[49] H. Li, P. Stoica, and J. Li. Computationally efficient maximum likelihood esti
mation of structured covariance matrices. Proceedings of the IEEE International
Conference on Acoustics, Speech, and Signal Processing, pages 2325-2328, Seat
tle, May 1998.
[50] H. Li, P. Stoica, and J. Li. Computationally efficient maximum likelihood estima
tion of structured covariance matrices. IEEE Transactions on Signal Processing,
47(5), May 1999.
[51] H. Li, P. Stoica, J. Li, and A. Jakobsson. On the performance analysis of forward-
only and forward-backward matched-filterbank spectral estimators. Proceedings
of 31st Annual Asilomar Conference on Signals, Systems, and Computers, pages
1210-1214, Pacific Grove, California, November 1997.
[52] J. Li and P. Stoica. An adaptive filtering approach to spectral estimation and
SAR imaging. IEEE Transactions on Signal Processing, 44(6): 1469-1484, June
1996.
[53] Z.-S. Liu, H. Li, and J. Li. Efficient implementation of Capon and APES for
spectral estimation. IEEE Transactions on Aerospace and Electronic Systems,
34(4):1314-1319, October 1998.
[54] L. Ljung. System Identification: Theory for the User. Prentice-Hall, Englewood
Cliffs, NJ, 1987.
[55] S. L. Marple, Jr. Digital Spectral Analysis with Applications. Prentice-Hall,
Englewood Cliffs, NJ, 1987.
[56] C. T. Mullis and L. L. Scharf. Quadratic estimators of the power spectrum, in S.
Haykin, editor, Advances in Spectrum Analysis and Array Processing, Englewood
Cliffs, NJ: Prentice-Hall, 1991.
[57] D. C. Munson, Jr., J. D. OBrian, and W. K. Jenkins. A tomographic formulation
of spot-light mode synthetic aperture radar. Proceedings of the IEEE, 71(8):917-
925, August 1983.
[58] B. Musicus. Fast MLM power spectrum estimation from uniformly spaced cor
relations. IEEE Transactions on Acoustics, Speech Signal Processing, 33:1333-
1334, October 1985.
[59] R. Onn and A. O. Steinhardt. Multi-windows spectrum estimation a linear
algebraic approach. International Journal on Adaptive Control and Signal Pro
cessing, 7:103-116, 1993.
[60] B. Porat. Some asymptotic properties of the sample covariance of Gaussian
autoregressive moving-average processes. Journal of Time Series Analysis,
8(2):205-220, 1987.
[61] K. Riedel and A. Sidorenko. Minimum bias multiple taper spectral estimation.
IEEE Transactions on Signal Processing, 43(1): 188195, January 1995.
[62] E. A. Robinson. A historical perspective of spectrum estimation. Proceedings of
the IEEE, 70(9):885-907, September 1982.
[63] R. Roy and Thomas Kailath. ESPRIT-estimation of signal parameters via ro
tational invariance techniques. IEEE Transactions on Acoustics, Speech, and
Signal Processing, 37(7):984-995, July 1989.


13
in practise. GSC also found applications in direct-sequence Code-Division Multiple-
Access (CDMA) communication systems for blind multiuser detection [36] [66] [65].
An important extension of the LCML beamformer was made in [7] [6] for applica
tions in airborne radar systems. A typical signal environment faced by such systems
consists of strong clutter/interference of complicated angle-Doppler spectrum which
is unknown and may be varying in both time and space. The extension made by
Brennan el al is to simultaneously combine the signals received on multiple elements
of an antenna array (the spatial domain) and from multiple pulse repetition peri
ods (the temporal domain). Such a technique is referred to as space-time adaptive
processing that has received much research interest recently [80] [82].
Another interesting application of the Capon spectral estimator is SAR imag
ing [57] [3]. Conventional SAR imaging techniques are the FFT (Fast Fourier Trans
form) or windowed-FFT methods. A number of parametric spectral estimation meth
ods have also been used for SAR imaging [26] [38] [33], though their interest in SAR
imaging is limited because of their sensitivity to model errors. A comparative study
in [24] showed that adaptive filterbank approaches such as Capon offers good SAR
images and enjoys the advantage of robustness as compared to parametric methods.
A number of modified Capon methods have also been suggested for SAR imaging
[25] [4],
2.2 Covariance Estimation
One type of the covariance estimation problems is to estimate the covariance
sequence from a finite number of data samples. A standard technique for estimating
covariance sequences uses the biased or unbiased sample covariance estimator. The
biased covariance estimator is more commonly used because it provides smaller MSEs
than the unbiased one and guarantees the covariance estimates to be non-negative
[55]. The problem of the sample covariance estimators is the unrealistic windowing


28
where L\ = Ni Mi 4- 1 and L2 N2 M2 + 1. Let
y(/i,/2) = vec[Y (h,l2)\, (3.61)
y(h,l2) = vec[Y(/i, 2)], (3.62)
where vec[-] denotes the operation of stacking the columns of a matrix on top of each
other. Let
aM!,M2(wi>^2) 3-m2(w2) aMl(u}i),
where denotes the Kronecker matrix product, and
aMjfe(^fc)
1 ejk
T
ei{Mk-\)uk
k = 1,2.
(3.63)
(3.64)
Then y(Zi, l2) and y(h, h) can be written as
y(b, h) = [tt(wi,w2)aMllM2(w1,w2)]d(ll+22) + ea,li(J2(Zi, l2), (3.65)
y(h,l 2) = [a{uJi,u2)aMuM2{vi,u2)]eullll+J2l2) + eUuul2(h,l2), (3.66)
where
a(u)i,u2) = a*(£Ji,a;2)e-J^7Vl-1^Wle-J^JV2-1^W2, (3.67)
and WliW2(Zi, Z2) and eWliW2(Zi, Z2) are, respectively, formed from {eWli,2(ni, n2)} in the
same ways as y(Zi,Z2) and y{li,l2) are made from {y(ni,n2)}.
Suppose that the initial phase of the sinusoidal signal of (3.59) is a random
variable uniformly distributed over the interval [0, 27t) and independent of the noise
term. Then the covariance matrix of y(l\,l2) or, equivalently, of y(Zi,Z2) is given by
R= \a(u}Uuj2)\2aMuM2{^\,^2)a^lM2{u)i,u>2) + Q(uq,u;2), (3.68)
where Q(cji,cu2) is the covariance matrix of eu)li,2(Z1,Z2) or eWliW2(Zi, Z2). By making
use of the fact that
y(Zi, h) Jy*(Ti Zi 1, L2 l2 1),
(3.69)


60
where T denotes some nonsingular matrix that makes X semi-unitary. One such T
IS
T = (AH CT1 A)~1/2.
(5.54)
Hence,
X = Q-1/2A(AhQ-1A)-1/2.
(5.55)
We next observe that
H = Q-1/2X
(5.56)
satisfies (5.47). Consequently, we have
H = Q~1A(AhQ~1A)~1/2.
(5.57)
The final step is to observe that post-multiplying H by a nonsingular matrix does
not change the generalized SNR criterion. Then, it follows immediately that
H = Q^AA^Q-^A)-1 (5.58)
maximizes the generalized SNR and it also satisfies the constraint
H^A = I*. (5.59)
The constraint (5.59) says that each (row) filter in HH passes one sinusoid undis
torted, and completely annihilates the others.
From (5.27) and (5.28), the filtered data corresponding to (5.58) is given by
z(0 = HHy(/) = D*a + HHe(Z) = Da + i/(Z), / = 0,1,..., L 1. (5.60)
The covariance matrix of i/(/) can be estimated as
H"QH = (AhQ-1A)_1. (5.61)
It follows that the WLS (Markov-like) estimate of a in (5.60) is given by
a =
L1
D//i(AifQ1 A)D;
11=0
L-1
-1 r
L1
X]Df(A/Q-1A)(A/Q-1A)-1AiQ-1y(0
1=0
XAfQ-'A,
(=0
L-1
^AfQ-'y (i)
11=0
(5.62)


50
5.2 LS Amplitude Estimators
We consider two LS methods in this section, namely LSE(1,0, K) and
LSE(1, 0,1).
5.2.1 LSEfl, 0. K)
This is perhaps the most direct approach. Let us write the available data
sequence in the following form
x(0)
1
1
1
1
o'
p
*(1)

ejwi
ejUK
2
+
n(l)
I
tH
1
ej(N- 1)cli
g j(N-l)uiK
aK
1
CS
1
h-1
1
or, with obvious definitions,
x = a + v, (5.3)
which is a linear regression equation. The LS estimate of a is
= (^^^x, (5.4)
where (-)H denotes the conjugate transpose. Note that the noise is not modeled,
even though it may be correlated. Despite this fact, LSE(1,0, K) is asymptotically
efficient [31]. A relatively simple manner to see this is as follows. First, note that
E{ct} a, (5.5)
where E{-} denotes the statistical expectation. The Mean Squared Error (MSE) of
L S
MSE{q:} covjo:} = E {(o; a) (a oi)H}
= (")-1/W()-1, (5.6)
where
w =
(5.7)


69
10
1
10
o
£10'
LJJ
~o
CD
lio'2
cr
C/D
I
c
C0 -3
0>10
10"
10"
N=32,o2=1 O-2,f3=0.3,Colored Noise
T
a
Capon 1
:
X-
X
APES1
_
0
0
CaponK
:
O-
o
APESK
:i
+
+
MAFI1
;
CRB
.....0" :
.9
J I I I I L
0 2 4 6 8 10 12 14 16
Subvector Length M
(a)
N=32,a2=10-2,f1=0.1 .Colored Noise
= I
i
1
i
1
I
i
it run

a
Caponl
-=
:
X-
X
APES1
:
1.
0.
4
CaponK
j-
!
G
o
APESK
:
-
O
+
+
MAFI1
/ -
_
CRB
.'
=
:
r
-x..,
D
J
-
+.
x-

: o.
4
,9
0
o'
:
e
o...
+''
:
9
+
:
r
1
i
i
i
L
I
i
i
1
0 2 4 6 8 10 12 14 16
Subvector Length M
(b)
Figure 5.4: Empirical MSEs and the CRB versus M when N = 32 and the observation
noise is colored (an AR,(1) process with a2 = 0.001). (a) a3. (b) a\.


62
5.5 Numerical Examples
In what follows, we investigate the performances of the various amplitude
estimators described in the previous sections. For notational simplicity, we will refer
to these methods as follows:
LSE1: LSE(1,0,1) using (5.14);
LSEK: LSE(1,0,K) using (5.4);
Caponl: WLSE(L,0,1) using (5.44);
APES1: WLSE(L, 0,1) using (5.45);
CaponK: WLSE(L,0,/F) using (5.38);
APESK: WLSE(L,0, K) using (5.29) along with (5.43);
MAFI1: MAFI(L, K, 1) using (5.65) along with (5.43).
We will compare these methods with one another as well as the CRB given in (5.11).
Since all these methods are asymptotically efficient, we only consider the case when
N is relatively small. Specifically, we choose N = 32. The data consist of three
complex sinusoids corrupted by a complex Gaussian noise v(n) (to be specified):
x(n) = s(n) + v(n), n 0,1,..., N 1, (5.66)
where
3
s(n) = '}Takej2*h. (5.67)
fc=i
The frequencies of the sinusoids are /i = 0.1, /2 = 0.11, and = 0.3. Also, ai =
e-771'/4, o2 = eJir/3, and 0:3 = eJ7r/4.
All examples are based on 200 Monte-Carlo simulations. The MSE figures
shown in what follows are obtained as
1 200
MSE{cbJ = goo ~ ak
u =1
(5.68)


48
are of interest to us) are close to one another. Statistical analyses that compare the
merits of the two LS methods are also provided in Section 5.2.
Since the LS methods completely ignore the correlation of the observation
noise, they are in general suboptimal. By splitting the data vector into a number
of subvectors, the covariance matrix of the noise-only part of the data subvectors
can be estimated, which makes it possible to use a Markov-like estimator that is
optimal in the class of Weighted Least Squares (WLS) techniques [68]. We describe
in Section 5.3 several ways for estimating the aforementioned covariance matrix,
which lead to different WLS amplitude estimators. Additionally, we show that, if the
restriction of estimating one amplitude at a time is again imposed, we obtain two
WLS amplitude estimators that are equivalent to the Capon [16] [44] and APES [52]
methods extensively used for spectral analysis.
The observation that some general spectral estimators, such as Capon and
APES, can be used to solve the problem posed in (5.1) motivated us to seek other
relatively sophisticated spectral analysis techniques for amplitude estimation. Both
Capon and APES belong to the general class of interbank approaches to spectral
estimation [75], which involve splitting the data into subvectors, passing them through
a set of narrowband filters (filterbank) whose center frequencies correspond to those
that are of interest to us, and, finally, estimating the spectral density function at those
frequencies from the filtered and, hopefully, signal-enhanced data. As one can see,
the key issue of filterbank approaches is the design of the filters. A recent study has
suggested the choice of matched filters, which gave rise to the MAtched-FIlterbank
(MAFI) approach to spectral estimation [70]. Even though neither Capon nor APES
was derived in the MAFI framework (see [16] and [52] for their original derivations),
it was found that both are members of the MAFI approach [70]. In the light of the
work of [70], we derive in Section 5.4 a generalized MAFI approach to amplitude
estimation. Interestingly enough, we show that, under certain circumstances, MAFI


120
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101
the approximate Capon method, which avoids the spectral factorization and makes
use of FFT, may be preferred in practical application.


20
1 L_1
g(w) = ZY1 y(l)e~JUl- (3-24)
^ =o
Since Q(a>) is Toeplitz and, therefore, persymmetric, one can show that the
hw in (3.17) satisfies [52]
- hwe-j(M-1)w. (3.25)
Consequently, it follows after some calculation that (3.22) is equivalent to
&FB(u) = h*g(u). (3.26)
Hence, due to the persymmetry of Q(u>), the FB estimate of a{u) has the same
form as the FO estimate of a(u>) (see (3.21) (3.26)) However, the filter vector hw
obtained, with the FO approach is in general different from that corresponding to the
FB approach [52].
Although neither Capon nor APES was derived in the MAFI framework (for
original derivations of these methods, we refer to [16] [75] [52]), in what follows
we show that two natural estimators of Q(w) in (3.17) lead to the Capon and APES
filters, respectively. More interestingly, we also show that even though a third natural
estimator of Q(cj) gives a new filter which is different from the former two, the spectral
estimator corresponding to the new filter turns out to be equivalent to APES as well.
3.2.1 Capon Filter
By (3.10), one natural choice is to estimate Q(u;) as
QCaponM = R ~ |(w)|2 aM(w)a£( where (w) denotes an estimate of q¡(oj) and R denotes either Rpo or R-FBi which, in
turn, leads to FOC or FBC, respectively. (For notational simplicity, in what follows
the forward-only Capon, APES, and MAFI spectral estimators will be referred to
as FOC, FOA, and FOM, respectively. Likewise, FBC, FBA, and FBM represent
the corresponding forward-backward counterpart estimators.) By making use of the


APPENDIX D
PROOF OF THEOREM 6.4.1
Lemma D.0.1
MlM-1
M-1M-1 M-lM-j-1
(D.l)
Proof:
M-1W-1 Ml Ml
f(k>k j)= k
k=0 j=0 j=0 fc=0
Ml Ml Ml j1
= + (D-2)
j0 kj j= 1 fc=0
Let s = k j. Note that the first term of the right side of (D.2) corresponds to s > 0
and the second term corresponds to s < 0. Since 0 < k < M 1 and 0 < j < M 1,
it is obvious that
M 1 > k = s + j > s, for s > 0;
0 < k = s + j < s + M 1 = M |s| 1, for s < 0.
Therefore, we can rewrite (D.2) as
Ml Ml -1 M-\s\-l
'¡>2J2f{k,s)+ £ /(m)
s=0 k=s s=(M1) k=0
Ml Ml Ml Msl
= ^2^2f(k^s) + f(k~s)>
s=0 k=s s=l fc=0
which concludes the proof of Lemma D.0.1.
(D.3)
(D.4)
114


MATCHED-FILTERBANK APPROACHES TO SPECTRAL
ANALYSIS AND PARAMETER ESTIMATION
By
HONGBIN LI
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
1999
J


35
1-D Complex Amplitude Estimation 1-D Complex Amplitude Estimation
1-D Complex Amplitude Estimation
1-D Complex Amplitude Estimation
(C) (d)
Figure 3.3: Empirical bias and variance of the 1-D MAFI estimators as the filter
length, M, varies when N = 64 and SNRi = 20 dB. (a) Real part of the bias; (b)
Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude;
(d) Variance of the imaginary part of the estimated amplitude.


67
(b)
Figure 5.3: Empirical MSEs and the CRB versus local SNR when N = 32, M = 8,
and the observation noise is white, (a) 0:3. (b) a\.


83
The estimator given in (6.1) is consistent if the ergodicity assumption is satis
fied. A study on whether the standard sample covariance estimator is also asymptot
ically statistically efficient, i.e., whether it asymptotically achieves the Cramr-Rao
Bound (CRB), has been undertaken in [60]. Let y(t) be an ARMA(p,q) signal. If
p > q, then the sample covariance estimate r(k) is asymptotically statistically effi
cient if and only if 0 < k < p q\ in particular, for AR processes of order p, r(k) is
asymptotically efficient for 0 < k < p, but inefficient for all other k. If p < q, none of
r(k) is asymptotically efficient; in particular, none of r(k) is asymptotically efficient
for an MA process.
It is known that the sample covariance sequence (r(0),..., r(N 1), 0, 0,...}
and the data periodogram constitute a Fourier transform pair. It is also known
that the periodogram is a statistically inefficient (in fact inconsistent) estimator of
the PSD [75]. This observation suggests that better covariance estimators might be
obtained by Fourier inverting better PSD estimators. However, this is not necessarily
so. Briefly stated, the reason is that the Fourier transform and the inverse Fourier
transform are integral transforms and hence small errors in one domain may be
associated with large errors in the other domain or vice versa. The fact that the
covariance estimates in (6.1) are consistent whereas the periodogram is not illustrates
this observation. Also note that, while the Burg estimate of the PSD is typically more
accurate than the periodogram, the corresponding Burg estimate of the covariance
sequence is generally poorer than (6.1) [78].
In spite of the fact briefly discussed above, in the following we consider es
timating the covariance sequences by inverting a PSD estimate that is often much
more accurate than the periodogram, namely the Capon PSD estimate. Like the pe
riodogram, no model is assumed in the Capon PSD estimator, which makes it more
robust than the parametric estimators in many situations. Although it has lower
spectral resolution than the AR spectral estimator, it generally exhibits less variance


78
a-Parameters b-Parameters
Computational Complexity
v...
V
(85) with LSEK
...
...
(85) with APES1
o
(85) with MAFI1
0-
o
(82) with APES1
OEM
Q...
a
PEM
Data Length N
(c)
Figure 5.7: Averaged RMSEs and the number of flops versus N for the second system
when the observation noise is colored (an AR(1) process with a2 = 0.01) and M = 20
for APES1 and MAFI1. (a) RMSE of a-parameters. (b) RMSE of 6-parameters,
(c) Number of flops.
of flops, as N increases. As one can see, the initial system parameter estimates given
by APES1 or MAFI1 are significantly better than those given by OEM, and yet the
former two are computationally more efficient that the latter.
5.7 Summary
This chapter examines the problem of amplitude estimation of sinusoidal sig
nals in colored noise. Three general classes of estimators, namely the LS, WLS, and
MAFI approaches to amplitude estimation, are discussed. It is shown that, under


15
Extended Invariance Principle (EXIP) [76] was recently presented in [49] [50]. This
method provides asymptotic (for large samples) ML estimation for structured covari
ance matrices. A closed-form solution for the estimation Hermitian Toeplitz matrices
is obtained which makes the proposed method computationally much simpler than
most existing Hermitian Toeplitz matrix estimation algorithms. Additionally, it was
also shown that using the technique in such array processing algorithms as MUSIC
[64] and ESPRIT [63] makes them achieve the Cramr-Rao Bound (CRB) for angle
estimation, i.e., the best performance for any unbiased methods.


117
[15] J. A. Cadzow. Signal enhancement a composite property mapping algorithm.
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of the IEEE, 57(8): 14081418, August 1969.
[17] J. Capon. Correction to probability distributions for estimators of the frequency-
wavenumber spectrum. Proceedings of the IEEE, 59(1): 112, January 1971.
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[20] D. G. Childers, editor. Modern Spectrum Analysis. IEEE Press, New York, 1978.
[21] H. Cox. Resolving power and sensitivity to mismatch of optimum array proces
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ceedings on Optical Engineering in Aerospace Sensing, 2230, Orlando, FL, April
1994.
[25] S. R. DeGraaf. Sidelobe reduction via adaptive FIR filtering in SAR imagery.
IEEE Transactions on Image Processing, 3(3):292-301, May 1994.
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Annual ASSP Workshop on Spectrum Estimation and Modeling, pages 391-396,
August, 1988.
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[28] D. R. Fuhrmann. Application of Toeplitz covariance estimation to adap
tive beamforming and detection. IEE Transactions on Signal Processing,
39(10):2194-2198, October 1991.
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likelihood estimates of structured covariance matrices. IEEE Transactions on
Information Theory, 34(4):722-729, July 1988.
[30] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins Uni
versity Press, Baltimore, MD, 1996.
[31] U. Grenander and M. Rosenblatt. Statistical Analysis of Stationary Time Series.
Almqvist och Wiksell, Stockholm, 1956.


64
most SNRs. The reason is that the approximation made in (5.37) is valid only for
large N and M, which is not the case in this example. Figure 5.2(a) also shows that
both LSE1 and Caponl are inconsistent (in SNR). Their inconsistency is not sur
prising because both are biased estimators. Recall that the bias of LSE1, as given in
(5.17), does not vanish unless N goes to infinity. Similarly, Caponl is always biased
(downward) for finite N [70] [47],
Figure 5.2(b) shows the counterpart curves for aq. (The results for a2 are
omitted because they resemble those for oq.) Note that /2 /i = 0.01, which is
smaller than l/N 0.03, the Fourier resolution limit. The performance degrades
for all estimators under study, especially for LSE1, Caponl, and APES1, which
estimate only one amplitude at a time. As shown in Figure 5.2(b), LSE1 and Caponl
essentially fail for all SNRs considered due to their large MSEs. APES1 is no longer
close to the CRB but, unlike the previous two estimators, it still appears to be
consistent (in SNR). As in Figure 5.2(a), CaponK again deviates away from the
CRB at high SNRs. It appears that the approximation made in (5.37) introduces
a bias (at small N and M) that may be negligible at low SNRs but dominates the
variance at high SNRs. The bias does not disappear as the SNR increases, which
causes the divergence of CaponK from the CRB. APESK performs quite well for high
SNRs; however, it is not very stable at low SNRs (due to large variance). The best
estimator in this example is MAFI1. The knowledge of the number and locations of
the sinusoids, which the other one-at-a-time estimators may spare but is indispensable
to MAFI1, appears to play an important role in its good performance in the current
case that fails the other one-at-a-time estimators.
As stated in Section 5.2, LSEK is statistically efficient, i.e. it achieves the CRB
for any N > K, when the observation noise is white. To see how the other suboptimal
(in finite samples) methods perform in such a case, we consider an example which is
similar to the previous one except that v(n) is replaced by a zero-mean complex white


58
(5.45) are recognized to have the same form as the Capon [16] [44] and, respectively,
the APES [52] spectral estimators. The two estimators were derived in [70] [47]
by a different approach, namely the MAFI approach, which we will consider in a
generalized form in the next section.
It is interesting that the above two amplitude estimators, while both asymp
totically efficient (and hence equivalent), have quite different finite-sample properties.
Specifically, it was shown in [70] [47] that (5.44) is biased downward, whereas (5.45)
is unbiased (within a second-order approximation) and in general has a better per
formance than the former.
5.4 MAFI Amplitude Estimators
In this section, we derive a generalized MAFI approach to amplitude estima
tion. Let 6 CKxM be a matrix each row of which is a Finite Impulse Response
(FIR) filter (for some 1 < K < M yet to be specified). The MAFI idea can be
explained as follows:
a) Design so that, when applied to {y(Z)}, it maximizes the SNR at the K
filter outputs.
b) Estimate the amplitudes from the filtered data (whose SNR should be higher
than that in the raw data) by, e.g., the LS or WLS technique.
Mathematically, H can be obtained as follows:
H = argmaxtr (HhQH)_1H/(APA/)H
H
(5.46)
Generalized SNR
where H is constrained in a way that is specified later (in particular, to guarantee
that H is finite), and tr(-) denotes the trace of a matrix. Let
X" = (H"QH)-1/2H''Q1/2,
(5.47)


49
amplitude estimators have equivalent forms to the WLS methods. However, the
MAFI approach is more general than the WLS technique in that the latter is a
special case of the former. To show this, a new MAFI amplitude estimator that does
not fall into the WLS category is described in Section 5.4. Other interesting MAFI
amplitude estimators may exist and are yet to be discovered.
A common feature of the amplitude estimators considered in this chapter
is that none of them models the observation noise exactly. Even so, all methods
are asymptotically statistically efficient, that is, they all achieve the Cramr-Rao
Bound (CRB) in large samples. However, their finite-sample properties, which are of
primary interest to this work, are quite different. Since the finite-sample analysis is
intractable in most cases, we use Monte-Carlo simulations in Section 5.5 to compare
these methods with one another.
The amplitude estimation problem in (5.1) occurs in a variety of signal pro
cessing applications (see, e.g., [41] [55], and the references therein). In Section 5.6,
we discuss its application to system identification. We show that, by using appropri
ate amplitude estimators, we can avoid the iterative search required by the standard
system identification routines, such as the Output Error Method (OEM) [68], and
achieve very good performance at a usually reduced computational load.
In concluding this section, we introduce the following notation to distinguish
among the various amplitude estimators. For instance, LSE(1,0,1) denotes the LS
estimator that does not split the data (and hence it uses one data snapshot), uses
no prefiltering, and estimates one amplitude at a time. Likewise, MAFI(L, K, K)
denotes the MAFI estimator that splits the data into L subvectors, utilizes a bank of
K prefilters, and estimates K amplitudes simultaneously. The remaining amplitude
estimators are similarly designated.



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PAGE 135

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PAGE 136

7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0D\ :LQIUHG 0 3KLOOLSV 'HDQ &ROOHJH RI (QJLQHHULQJ 0LKUDQ 2KDQLDQ 'HDQ *UDGXDWH 6FKRRO


46
Figure 4.1: SAR images obtained from the ERIM data by using (a)
2-D windowed FFT, (c) 2-D Capon, and (d) 2-D APES.
2-D FFT, (b)


93
(c) (d)
Figure 6.2: Pole-zero diagrams for ARMA test cases, (a) ARMA1; (b) ARMA2; (c)
ARM A3; (d) ARMA4.
In the following numerical examples, we set N 256 (shorter data lengths
have also been considered, and the results are similar) and M 50 (see Sections 3.5.1
and 5.5.2 for how to choose M). The performances of the standard and the Capon
methods for each of the four ARMA models are shown in Figures 6.5(a) to 6.5(d),
where the curves show the mean-squared errors (MSE) of the covariance estimates,
normalized with respect to r(0), versus the time-lag of the covariance sequences.
(Only the results from the exact Capon method are demonstrated owning to space
limitation.) The MSE values are based on 100 independent realizations. It has
been found that the Capon method generally gives better results than the standard
method, especially for large time-lags. We also note that, when the poles are close
to the unit circle, as in Figure 6.5(b), the performance differences between the two
methods are not so large as in the other cases. (Though, the Capon method is still
better than the standard method.)


ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor, Dr. Jian Li, for her
constant support, encouragement, enthusiasm, and patience in guiding this research.
My deepest appreciation goes to Dr. Petre Stoica for his numerous insightful remarks
and suggestions which comprehensively influenced this work.
Special thanks are due to Drs. Jose C. Principe, Fred J. Taylor, William W.
Edmonson, and David C. Wilson for serving on my supervisory committee and for
their contribution to my graduate education at the University of Florida.
I also wish to thank Zhaoqiang Bi, Robert Stanfill, and other fellow graduate
students with whom I had the great pleasure of interacting. Drs. Zheng-She Liu and
Guoqing Liu have my gratitude for sharing many interesting discussions with me.
I would like to gratefully acknowledge all the people who helped me during
my Ph.D. program.
iii


86
ARMA(4,2)(N=256, M=51, 100 realizations.) ARMA(4,2)(N=32, M=11,100 realizations.)
(a) (b)
Figure 6.1: Power spectral density estimates by using Capon-1 and Capon-2. The
plots are the averages of 100 independent realizations, (a) N = 256, M = 50; (b)
iV = 32,M = 10.
signal, while the solid and the dashdotted lines, respectively, indicate the Capon-1
and Capon-2 estimates. It is obvious that Capon-2 is highly biased and suffers from
a significant power loss. This observation has also been made with other signals,
especially if the signal is a narrowband signal. An explanation of this behavior
follows.
The calculation of the filter bandwidth in Capon-2 is applicable only if the
Capon filter is a narrowband filter. Recall that the Capon method aims to find the
Capon filter that minimizes the total output power of the overall frequency band
while passes the current frequency u> undistorted. No effort has been taken to make
sure that the Capon filter is narrowband. Let the lobe of the Capon filter frequency
response where the current frequency of interest uj is located be called as the mainlobe,
while all the others are called the sidelobes. It has been found that the steering
frequency uj is not necessarily at the maximum or the center of the mainlobe [75] [52],
Furthermore, if the input signal is a narrowband signal, there may exist sidelobes,
located at frequency bands where the power level of the input signal is low, that
are even larger than the mainlobe of the Capon filter frequency response. Note


24
3.3 Analyses of MAFI Approaches
3.3.1 Computational Complexity
Let RpQ2 and Rpg 2 denote the Hermitian square roots of the positive definite
matrices RpQ and Rpg, respectively. Define
//.i(iu) = RpQ awf(o;), (3.44)
M2(w) = Rpo (3.45)
Then FOC and FOA can be expressed as relatively simple functions of /^(w) and
/z2(u;) (see (3.29) and (3.35)):
FOcM=
IIalHII2
and
foaM =
Ml hm2m
IImiMII2 (IImiHII2 IIm2M||2 Im?(w)m2MI2)
Applying the matrix inversion lemma to (3.32) yields
i -i
QFBAM ^FB %BG(w) G (w)RpBG(kO ^ Gff((j)Rpg,
(3.46)
(3.47)
(3.48)
where I is the 2x2 identity matrix. Next define
'lM = Rpg/2aM(w),
v2(u) = RpjfgH,
u3(u) = Rpg/2g(w).
Then the FBC and FBA spectral estimators can be expressed as (see (3.29),
and (3.48)):
fbcM
Ui(u})u2{uj)
IkiHII2
(3.49)
(3.50)
(3.51)
(3.35),
(3.52)


22
where Q(w) denotes either Qfqa or QfbA which corresponds to FOA or FBA.
Consequently, the APES estimate of a(u) is given by (see (3.22) and (3.26))
aM(w)Q-1(w)g(w)
apesM =
(3.35)
3.2.3 Another Matched Filter
Equations (3.4) and (3.7) suggest another way to estimate the FO and FB esti
mate of Q(w) (in what follows we sometimes omit the dependence on u> for notational
convenience):
QfomM
\l~i
= jYl [^(0 (^)aM(w)eJ,i] [y(l) (u)aM(w)eJji]H
=o
Rpo *ga^ -aMgH + ||2aMa^,
H
~|2.
(3.36)
a
FO
QfbmM
L-1
= [(y(0 (w)aAf(w)e,wi) (y(0 a(u)aM(u)ejul)
^ 1=0
H
+
(y(0 a(w)aM(w)eJU,j (y(0 ~ (w)aM(w)eJiJ^
H
= RpB
1
g + ag
H 1
aM 2&M
l H
A*g + & g
+ \\aMa%f, (3.37)
2
FB
where a(u) and a(u>) a*(o;)e_^Ar_1^ denote some estimates of a(u>) and a(co),
respectively. By using the matrix inversion lemma (twice), one can see that the last
and the third terms of (3.36) or (3.37) can be dropped without affecting the matched
filter vector. Then, by using the matrix inversion lemma once again, we have
OpoaM = (RfO *ga^) 1
-1 1
R-fo +
O
aM
1 dda^RpQg _
_ RpQaM ~ &*{aM Rfo S) Rpo ajw + a* (a^rRpQaAf )RpQg
1 A*a" RpJjg
(3.38)


70
Estimator
M
APES1
N/A < M < N/2
APESK
N/A < M < N/3
MAFI1
N/8 < M < 27V/5
Caponl or CaponK
N/8 Table 5.1: Choice of M for the WLS and MAFI amplitude estimators
where the input u(n) is a sinusoidal (probing) signal
K
u(n) = ^2,lke]Ukn, n = 0,
k=l
and the transfer function is rational:
H(-u = B(z ') = biz l + ... + bqz l
A{z~l) 1 + aiz-1 + ... + apz~P
We assume that
K > p + q.
(5.72)
(5.73)
(5.74)
Even if p and q were unknown, K could still be chosen sufficiently large to satisfy
(5.74). The problem of interest in this section is to estimate {a}^=:1 and {6^}y=1 from
5.6.1 System Identification Using Amplitude Estimation
The commonly-used Output Error Method (OEM) does not model v(n) and
obtains estimates of {aj}=1 and {bj}9j=l by minimizing the criterion
Nl
<^OEM(a, b) = Hn) ~ H{z~l)u{n) |2 (5.75)
710
where a = [ fll ... ap ]T and b = [ bx ... bq ]T Let
a*(a,b) = jkH(eJUJk).
(5.76)


113
Next note that
Qfba = R-fb GG// = R-fb it [ggH + gg7/]
= |a|2aMa^ + ^a*Sa^ + |*5a^ + |aaM |2"Maf
L1
2L
+ V7 !C MZ)ew(0 + w(0^(0] H2aMa^ ^aaM"
1=0
1 raaH +
1 ~SH 1 _* % //
--aaM5 --adaM--
E M)e?W + ¡W0¡£(0] **" + m"
1=0
2 L
h
Ai
(C.ll)
where we have made use of (C.4) and (B.1)-(B.2). Using again the assumption of
zero third-order moments of ew(n) and eu(n), and combining (C.10) and (C.ll) yields
(to a second-order approximation)
E {fbA a} ~ 0. (C.12)
Likewise, by replacing Qfba by QfOA in (C.9)-(C.10) followed by some straightfor
ward manipulations, we have (to a second-order approximation)
^{FOA ~ 0,
(C.13)
and the proof of (3.58) is complete.
To motivate the normalizing factor L used in both (3.57) and (3.58), we men
tion the fact that both 6 and S are 0(l/VL) and this implies that the second-order
approximation previously used is 0(1/L).


Power Spectral Density-dB Power Spectral Density-dB
94
ARMA1 ARMA2
(a)
ARMA3
(C)
(b)
ARMA4
(d)
Figure 6.3: True power spectral densities, (a) ARMA1; (b) ARMA2; (c) ARMA3;
(d) ARMA4.


100
(a) (b)
Figure 6.7: 10 superimposed realizations of the MA covariance sequence estimates
with N = 64 and M = 6. (a) The standard method; (b) The Capon method.
We use both the standard and the Capon methods to determine its covariance esti
mates. The results are shown in Figures 6.7(a) and 6.7(b) where N 64 and M = 6,
and where 10 superimposed realizations for both methods are displayed. As expected,
the Capon covariance estimates decay to almost zero after k = 5, while the standard
covariance estimates are much more erratic. Consequently the inference that the
process under study is an MA(4) is easier to make by using the Capon covariance
estimator.
6.6 Summary
The Capon method for covariance sequence estimation makes use of Fourier
inversion of the Capon spectral estimates. It is shown that the Capon covariance
estimates are usually more accurate than the standard biased sample covariance
estimates, especially for large lags. The Capon spectrum is equivalent to an AR
spectrum and hence the corresponding covariance sequence can be exactly calculated
via the inverse Levinson-Durbin algorithm. Since the spectral factorization needed
for the exact Capon covariance estimation is in general computationally expensive,


CHAPTER 7
CONCLUSIONS
7.1 Summary Remarks
In this work, we have discussed using MAtched-FIlterbank (MAFI) approaches
for spectral analysis and parameter estimation. Specifically, the problems of complex
spectral estimation, sinusoidal amplitude estimation, and covariance sequence esti
mation have been addressed.
In Chapter 3, we first introduced the MAFI concept, which is of primary
importance to this study, by choosing the filterbank as matched-filterbank, i.e., each
filter in the filterbank is a matched filter. Since we do not model the observation noise
exactly, there are various ways of estimating the covariance matrix of the observation
noise, which lead to different MAFI estimators for specific estimation purposes.
For complex spectral estimation, it was shown in Chapter 3 that the Capon
and APES estimators, while originally not derived in the MAFI framework, are both
members of the MAFI complex spectral estimators. Moreover, the MAFI concept
may be used to devise other interesting spectral estimators. Although Capon and
APES are both asymptotically efficient and are of similar computational complexity,
their performances are quite different for finite length of data samples. Specifically,
we proved that, to within a second-order approximation, Capon underestimates the
true spectrum whereas APES is unbiased, and that the bias of the forward-backward
Capon is one half that of the forward-only Capon. These results are believed to pro
vide compelling reasons for preferring APES to Capon in most practical applications.
It should be stressed that the better estimation accuracy obtained by Capon
or APES over the Fourier-based methods is achieved at the cost of computational
102


APPENDIX C
PROOF OF THEOREM 3.3.2
Proof of (3.57): We first consider the bias for the FBC. By using (3.28) and
(B.13), we obtain
P _1
aAfrLFB rn i\
FBC ~ a = -TTWTi (C-1)
RpBaAi
In what follows, we use the symbol ~ to denote an asymptotic equality that holds
to within a second-order approximation. A straightforward manipulation of (C.l)
yields
FBC a
aM(RfB ^ ^ aM^
FB
>H-d~ 1
+
aM^FBaM aM-^FBaM
a^R^(R-RFB)R-^
aM-RFBaM
1 1
X
Ta^R-^
+
a^RpaM a^R-!aM a"R-1aM_
a^R-HRFB-R)^-1^
a^R-xaw "r a^R-xaM
^R-^Rfb R)R-'5
a&R"1* a^R-^R-1 R^)aM
I IT i I
(a^RFBaM)(aMR 1&Ai)
a^R-xa m
+
a£R-^
a^R~xa M
(a^R-13)a^R~1(RFB ~ R)R~laM
(a^R-xaM)2
(C.2)
which, in turn, implies that
E {FBC }
E
BC 1
(a^R-1RFBR~1)(a^R-1aM) + (a^R-1RFBR~1aM)(a^R-1)
(a^R-xaM)2
(C.3)
110


77
Figure 5.6: PSD estimate of the output of the first system corrupted by white noise
with o2 = 0.01 and N = 200.
deteriorated. On the other hand, the above knowledge is not necessarily needed
by APES1 and hence its performance is not affected. Unlike APES1, MAFI1 does
require this knowledge. Yet, the initial condition response is substantially weakened
through the frequency selective filtering employed by MAFI1 and hence has little,
if any, effect on the amplitude estimates and the system parameter estimates. As
N increases, the transient effect becomes less severe, and, consequently, the initial
system parameter estimates obtained by using LSEK approach those obtained by
APES1 or MAFI1.
Example 5.6.2.2 We now consider a second system with
A(z-1) = 1 1.9109z-1 + 1.7251z~2 0.7033z-3 + 0.2450z~4, (5.90)
and
B(z~l) = z~l + 1.0562z~2 + 0.6100z-3 + 0.1912z~4 + 0.0400z5. (5.91)
The noise v(n) is an AR(1) signal as in (5.69) except that e(n) is now replaced by
a real-valued white Gaussian noise with zero-mean and variance a2 0.01. The
probing signal is the same as in the previous example. Figures 5.7(a) to 5.7(c) show
the averaged R.MSEs of the a-parameters and the 6-parameters, as well as the number


98
This is especially the case if there are poles in the ARMA model close to the unit
circle, since the covariance sequence decays to zero very slowly in such a case.
The performances of the AR coefficient estimation for the ARMA signals via
the OMYW method by using both the standard and Capon covariance estimates are
shown in Figures 6.6(a) to 6.6(d), where N = 256 and M = 32, and where the plots
are, again, based on 100 independent trials. The OMYW equations are solved by
using the LS method. To reduce the number of figures, the curves are the sum of
the MSEs of all AR coefficient estimates versus the number of equations used in the
OMYW method. When using the OMYW method, we assume that the AR orders
are known. The numbers of equations used in the curves are L = 4,8,16,32,64, and
128. It has been found that the AR coefficient estimates obtained by using the Capon
covariance estimates are usually better than those obtained by the sample covariance
estimates. Note that for the OMYW estimator based on the sample covariance
estimates, the estimation performance may significantly deteriorate with increasing
L (a large value of L may be used for lack of a priori information on the ARMA
signal in question), whereas the Capon-OMYW estimators performance is much less
affected by the increase of L.
6.5.3 MA Model Order Determination
For an MA(g) process, the covariances with lags larger than q are all zeros [75]
[41]. However, we can expect that the sample covariance estimates for an MA signal
will not decay to zero fast enough, while the Capon covariance estimates will usually
give f(k) m 0, for k > q, since the Capon method gives better covariance estimates
for higher lags. Hence the inference about the type of signal we are dealing with and
its order will be easier to make with the Capon method.
We consider an MA(4) signal
y{n) = e(n) 2.76e(n 1) + 3.809e(n 2) 2.654e(n 3) + 0.924e(n 4). (6.23)


68
5.5.2 The Effect of M
All WLS and MAFI amplitude estimators studied in this chapter depend on
the choice of M, the subvector length. It is known that as M increases, all of them can
better deal with the case of closely spaced sinusoids, but their statistical stability in
general decreases [75]. Hence, there is a tradeoff to to kept in mind when choosing M.
Note that M should also be smaller than N/2; otherwise, the estimated covariance
matrix will be rank deficient. The following example examines the effect of M on
the performances of these estimators. LSE1 and LSEK do not depend on M and are
thus not considered in this example.
The scenario is similar to the first example (AR noise) except that we fix
a2 = 10~2, which corresponds to a local SNR of 30.8 dB for the first sinusoid (at
fi = 0.1) and 39.2 dB for the third sinusoid (at /3 = 0.3). M is varied from 1 to
16 for all estimators except MAFI1, which requires that M > K (see (5.43) and
(5.58)). The MSEs of the amplitude estimates of a3 and, respectively, op, and the
corresponding CRBs are shown in Figures 5.4(a) and 5.4(b). As can be seen from
these figures, all estimators are sensitive to the choice of M, to a smaller or larger
extent. When no sinusoids are close to the one being estimated, such as the third
sinusoid in this example, APESl, APESK, and MAFI1 perform quite well for a wide
range of M. For the more difficult case as shown in Figure 5.4(b), the choice of M
becomes very critical. Based on our empirical experience, a rule of thumb for the
choice of M is given in Table 5.1.
5.6 Application to System Identification
Consider the linear discrete-time system described by the following equation
[68]
x(ri) = H(z l)u(n)+v(n),
n = 0,1,..., N 1,
(5.71)


81
There are basically two Capon PSD estimators, referred to as Capon-1 [16] [44]
and Capon-2 [45] herein. We find that, while Capon-2 is capable of finer spectral
resolution around the peaks of a spectrum, it is generally a globally poorer spectral
estimator than Capon-1. We hence concentrate our interest on Capon-1 for covariance
sequence estimation in this chapter. Since the Capon spectra, i.e., the PSD estimates,
are shown to be equivalent to AR or autoregressive moving-average (ARMA) spectra,
the inversion procedure for computing the exact covariance sequences corresponding
to the Capon spectra can be implemented in a rather convenient way. (Note that
the calculation of the covariance sequences corresponding to the Capon spectra is an
interesting problem by itself.) We also present an FFT-based approximate method
to compute the covariance sequences from the Capon spectra. It has been found that
the approximate method provides covariance estimates that are almost identical with
those obtained by the exact method, while the computational complexity is greatly
reduced.
Our primary interest is to apply the Capon method as well as the standard
approach to ARMA signals. To that end, a few ARMA signals with typical pole
and zero locations are studied in our numerical examples. The studies show that
considerable improvements are attained by the new Capon method.
The Capon covariance estimation method can be readily used in many appli
cations. One important class is the ARMA spectral estimation. Since most ARMA
spectral estimators rely on the Yule-Walker equations to determine the AR coeffi
cients, it may be expected that the better the covariance estimates used, the more
accurate the AR coefficient estimates yielded. We examine how the Capon covari
ance estimates can be used with the overdetermined modified Yule-Walker (OMYW)
method [14] [69] to compute more accurate AR coefficients. We find that the perfor
mances of the usage are influenced by the pole and zero locations and, still, generally


26
Theorem 3.3.1 Under Condition C and the additional assumption that ew(n) is
circularly symmetrically distributed, the estimation errors in the Capon and APES
spectral estimators are asymptotically circularly symmetrically distributed with zero-
mean and the following common variance:
lim LE (|(cj) a(o;)|2} = (3.56)
Proof: See Appendix B.
The need to enforce Condition C limits, to some extent, the importance of the
previous result. Indeed the assumption made in C is satisfied if (and essentially only
if) the signal y(n) has a mixed spectrum and uj is the location of a spectral line. The
result of Theorem 1 is relevant to the spectral analysis of a target with dominant
point scatterers in the presence of distributed clutter (see [52] and the references
therein). In some other applications, however, the main interest is in the continuous
component of the spectrum. For example, Condition C does not hold exactly for a
target with distributed scatterers since the signature spectrum is continuous at u>.
That the previous result is of a somewhat limited interest is also due to its
asymptotic character. Indeed, in applications with medium or small-sized data sam
ples, the spectral estimators under study have been found to behave quite differently
in contradiction with what is predicted by the (asymptotic) result of Theorem 1 (see
the numerical examples in Section 5). The finite-sample analysis of the spectral es
timators under discussion would consequently be of considerable interest. However,
a complete analysis, if possible, appears to be rather difficult at best. A partial one,
by making use of a higher-order Taylor expansion technique, is nevertheless feasible.
The result is as follows.
Theorem 3.3.2 To within a second-order approximation and under the mild as
sumption that the third-order moments of ew(n) and ew(n) are zero, the Capon and


6
to the second order. When only a finite number of samples are available, using sam
ple covariance sequences implies that the data beyond the observed duration either is
zero or repeats itself periodically, which is certainly not a realistic assumption. There
have been several attempts in the literature to derive other covariance estimators (see
Chapter 2 for some details). It is known that the sample covariance sequence and the
data periodogram constitute a Fourier transform pair. It is also known that the pe-
riodogram is a statistically inefficient (in fact inconsistent) PSD estimator [75]. This
observation suggests that better covariance estimators may be obtained by Fourier
inverting better PSD estimators. Since the MAFI PSD estimates are in general more
accurate than periodograms, we propose a new covariance sequence estimator by
Fourier inverting the MAFI PSD estimates. Specifically, we make use of the Capon
PSD and such an approach is referred to as the Capon covariance sequence estimator.
The same methodology can be similarly applied to the APES PSD estimates, though
the APES covariance estimates are usually similar to the Capon covariance estimates.
The reason is that, despite their different performance for complex spectral estima
tion, the Capon and APES PSD estimators usually behave quite similarly, especially
for continuous spectra (also see the discussions in Section 3.3.2). It is observed that
Capon covariance estimates are in general better than sample covariance estimates
in terms of mean-squared errors (MSEs).
1.2 Significance and Contributions
The main results of this dissertation are as follows.
1. We introduce a new general class of MAFI approaches to spectral estimation.
We show that the Capon and APES estimators, though originally not derived
in the MAFI framework, are both members of the MAFI class.
2. To within a second-order approximation, we prove that the Capon estimator
gives biased (downward) complex spectral estimates whereas the APES method


37
2-D Complex Amplitude Estimation 2-D Complex Amplitude Estimation
(a) (b)
(c)
(d)
Figure 3.4: Empirical bias and variance of the 2-D MAFI estimators as the SNRi
varies when Ni = N2 = 32 and Mi = M2 = 8. (a) Real part of the bias; (b)
Imaginary part of the bias; (c) Variance of the real part of the estimated amplitude;
(d) Variance of the imaginary part of the estimated amplitude.


14
they assume on the observed data. By exploiting the AR spectral estimator, Burg
proposed a technique that can offer covariance sequence estimates for any desired lag
and hence avoids the windowing problem suffered by the sample covariance estimators
[10]. However, it was shown in [78] that the Burg covariance estimates are less
accurate and more variable than the sample covariance estimates. In Chapter 6, we
describe a new method for covariance sequence estimation based on Capon spectra.
Estimating structured covariance matrix of the observed data vectors repre
sents another type of the covariance estimation problem. For example, the covariance
matrix of a stationary complex signal is Hermitian and Toeplitz. However, the sam
ple covariance matrix obtained from a finite number of data samples seldom has this
structure. Structured covariance matrix estimation is of importance in a variety of ap
plications including array signal processing and time series analysis [28]. An intuitive
way to obtain structured covariance estimates is to force the desired structures on the
sample covariance matrix, a methodology adopted by the Iterated Toeplitz Approxi
mation Method (ITAM) [81] [15] [84]. Specifically, ITAM alternatively makes use of
rank approximation (via singular value decomposition) and Toeplitzation along the
diagonals until convergence is reached. Obviously, such a method is by nature heuris
tic and no optimality can be associated with it, though the ITAM covariance matrix
estimate is in general closer to the true covariance matrix than the sample covariance
matrix in the Frobenius norm sense. Optimum structured covariance matrix estimate
may be obtained by maximizing the corresponding likelihood function as considered
in [1] [13] [23] [29] [85]. However, since there exists no closed-form solution to the
complicated nonlinear ML estimation problem for Hermitian Toeplitz matrices, the
ML methods proposed in these studies are iterative and computationally involved,
and, moreover, they are not guaranteed to yield the global optimal solution, which
to some degree limits the interest in using the ML structured covariance matrix es
timate in practical applications. An approximate ML method that makes use of the


23
and
4 PBaM
p 1
^FB
1 / Vg + agj a^Rpg
-Rpg (a*g
1 ~~ oaM
H-o-l
rrtFB
(d*g + agj
aM
R-pg aM 2 aM ^FB
a*g + ag
R-pBaM + 2a'Ai'^'FBaM'^'FB
a*g + ag
1 aH R-1
1 2 dAirLFB
a*g + ag
(3.39)
which gives the following expressions for the matched filter vectors:
^pQaM
and
FOM
aM^ FOaw
R-pOaAi + *aM^'FOaM'^'FOS a;*aM-^FC)S'^'FOaM
a^R-pQaM
(3.40)
,FBM
^PBaAi
aW^FBaM
R-pBaM + 9aM-^'FBaM'^'FB
*g + g
- -jCR
H D-l
M FB
&*g + a g
R'FBaAi
aM-^FBaM
(3.41)
The previous MAFI filters are in general different from both the Capon and APES
filters, since neither the Capon nor the APES filters depend on an estimate of a(oj)
which the new MAFI filters need to know. In spite of this fact, in Appendix A, we
prove that, for a certain natural choice of a(u>) in (3.36) and (3.37), the following
equalities hold true:
FOmM = TOaM. (3-42)
fbmM = fbaM-
(3.43)


115
Proof of Theorem 6.4.1: By making use of Lemma D.0.1 we obtain:
M-1M-1
aff(w)ra(w) = alr^ai
k=0 1=0
M-1M-1
= EE^-h
k=0 =0
M-1 M-1 M1 M-s-1
= E(Er-)e,'" + E( E rM+.)e-<
s=0 k=s s=l fc=0
M-1
= E
s=-{M-l)
with Us defined in (6.12).
If = r, then we have
Ms1
H-S = r*>fc+*> for s > 0- (D.5)
k=0
Taking complex conjugation of both sides yields
Ms1 Ms1
(/-.) = E rP+ = E r+*.*
fc=0 fc=0
Ml
= r*>*-s = (D-6)
l=s
and the proof is complete.


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Jian Li Chairman
Associate Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Jose C. Principe
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
William W. Edmonson
Assistant Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Frederick J. Taylor
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
David C. Wilson
Professor of Mathematics


5
1.1.3 Amplitude Estimation
Another problem related to complex spectral estimation is amplitude estima
tion for sinusoidal signals where it is assumed that the number and frequencies of the
sinusoids are known a priori. The observation noise is stationary and maybe colored.
In the case that the noise can be modeled exactly, the Maximum Likelihood (ML)
methodology can be used and the ML amplitude estimates are statistically efficient.
However, an exact model of the observation noise is usually not available and, more
over, ML methods are in general very sensitive to inaccurate model information. As
such it may be a better choice to use methods that do not model the noise exactly.
In Chapter 5, we describe a relatively large number of amplitude estimators
which assume no model except for stationarity for the observation noise. The am
plitude estimators can be categorized as three general classes, namely Least Squares
(LS), Weighted Least Squares (WLS), and MAFI approaches to amplitude estima
tion, which are all asymptotically statistically efficient. For finite length of data
samples, however, their behaviors are quite different. We show that under certain
circumstances, the MAFI approach to amplitude estimation is equivalent to the WLS
approach, and yet the former is more general and includes the latter as a special case.
We also show that the MAFI and WLS methods in general give more accurate am
plitude estimates than the LS methods.
1.1.4 Covariance Sequence Estimation
The fact that covariance (the terms covariance and autocorrelation are used
interchangeably with one another in this study) function and Power Spectral Den
sity (PSD) are a Fourier transform pair makes the problem of covariance sequence
estimation a research topic that is closely related to spectral estimation. Sample
covariance sequences have been widely used in signal processing because of its com
putational simplicity and its consistency supposing that the given signals are ergodic


52
5.2.2 LSEl.O.ll
Since the observation noise v(n) is not modeled, an idea that reduces the
computational burden quite a bit is to include K 1 sinusoids in the noise term, and
hence estimate only one amplitude at a time. In some signal processing applications,
the frequencies {u)k}k=1 may be unknown. A typical way to estimate both {ak}%=l
and {uk}£=1 would consist of estimating just one amplitude for varying frequency u>
and, then, detecting the peaks in the so-obtained spectrum [75] [41] [55]. As such,
the assumption made in Section 5.1 that {Luk}k=1 are known a priori may be relaxed
when using the one-at-a-time technique.
There is a somewhat subtle problem with the above technique: the sum of
v(n) and K 1 sinusoids no longer has a finite PSD, and hence one of the previously
made assumptions fails. Nevertheless, the idea still works as long as no two sinusoids
(that are of interest) are spaced too close to one another, as shown below and later
in Section 5.5.
The LSE(1,0,1) is easily derived as
n=0
which is recognized as the DFT of {^(n)}^1 at ujk. The two estimates in (5.4) and
(5.14) will be close to one another if \cok | 7§> 1 /N (V k,i,k ^ l) [75].
An analysis of LSE(1,0,1) runs as follows. Without loss of generality, let us
consider (5.14) for k = 1. The LSE(1,0,1) estimate of a.\ is given by
ai = (7) 1 ^x,
(5.15)
where
T
= l eiWi _ ej(N-i)wi
(5.16)
and where (-)1 denotes the transpose. Taking the expectation of (5.15) yields
(5.17)


4 EFFICIENT IMPLEMENTATION OF CAPON AND APES 39
4.1 Introduction 39
4.2 Efficient Implementation of APES 39
4.3 Extension to Capon 43
4.4 Numerical and Experimental Examples 44
4.5 Summary 45
5 AMPLITUDE ESTIMATION 47
5.1 Introduction 47
5.2 LS Amplitude Estimators 50
5.2.1 LSE(1,0, K) 50
5.2.2 LSE(1,0,1) 52
5.3 WLS Amplitude Estimators 54
5.3.1 WLSEL, 0, K) 54
5.3.2 WLSE(L, 0,1) 57
5.4 MAFI Amplitude Estimators 58
5.5 Numerical Examples 62
5.5.1 Estimation Performance versus SNR 63
5.5.2 The Effect of M 68
5.6 Application to System Identification 68
5.6.1 System Identification Using Amplitude Estimation .... 70
5.6.2 Numerical Examples 73
5.7 Summary 78
6 CAPON ESTIMATION OF COVARIANCE SEQUENCES 80
6.1 Introduction 80
6.2 Standard Covariance Estimator and Outlook 82
6.3 Capon PSD Estimator 84
6.4 Capon Covariance Estimator 87
6.4.1 Exact Method 87
6.4.2 Approximate Method 89
6.4.3 Computational Aspects 90
6.5 Numerical Results 91
6.5.1 ARMA Covariance Estimation 92
6.5.2 AR Coefficient Estimation for ARMA Signals 97
6.5.3 MA Model Order Determination 98
6.6 Summary 100
7 CONCLUSIONS 102
7.1 Summary Remarks 102
7.2 Future Work 105
APPENDIXES
A PROOF OF (3.42) AND (3.43) 106
B PROOF OF THEOREM 3.3.1 108
v


87
that the large sidelobes do not make any significant contributions to the filter
output power so that the filter design criterion is still satisfied; that is, the output
power is minimized, while the frequency response at to is one. In all such cases, the
Capon filter is not a narrowband filter and hence it calculates an overestimated filter
bandwidth. Hence the Capon-2 PSD estimates become highly biased. However, it
is interesting to note that Capon-2 does possess higher resolution capability, around
the power peaks, than Capon-1. This is illustrated in Figure 6.1(b), which shows the
PSD estimates of the same ARMA signal as used in Figure 6.1(a) but with N 32
and M = 10. The Capon-1 estimator cannot resolve the two power peaks this time,
while Capon-2, albeit biased, still can.
For the preceding reasons we do not consider using Capon-2 for covariance
sequence estimation in the sequel.
6.4 Capon Covariance Estimator
We describe below how the Capon PSD estimates can be Fourier inverted in
a rather convenient manner yielding the Capon covariance sequence estimates. The
study of the covariance sequences corresponding to the Capon spectra is an interest
ing endeavor by itself, which apparently has not been undertaken in the literature
before. We also present an approximate but computationally more efficient method
to calculate the Capon covariance estimates from the Capon PSD estimates.
6.4.1 Exact Method
Theorem 6.4.1 Let T = {rij} E cW+PxW+i) and let
d-M (w)
1 ejuJ
then
M l
aj^(w)raM(w) = Y hseJSUJi
s=-(M-1)
(6.10)
(6.11)


103
complexity. This becomes more evident in 2-D spectral estimation where the amount
of computation required by either Capon or APES is much more than by the Fourier-
based methods. Efficient implementation schemes were presented in Chapter 4 for
both the Capon and APES estimators. It was shown that the efficient implementa
tion scheme of Capon or APES reduces the computational complexity substantially,
especially when the number of frequency samples are large.
In complex spectral estimation, the locations of the spectral components that
are of interest to us are in general unknown. Hence, the center frequency of the
prefilter must sweep through all frequency bands to obtain the corresponding spec
tral estimates. One problem that is closely related to complex spectral estimation
is amplitude estimation for sinusoidal signals with known frequencies. Conventional
methods for amplitude estimation are based on the technique of Least Squares (LS)
fitting. However, LS methods typically perform quite poorly whenever the observa
tion noise is colored. By splitting the data vectors into subvectors of shorter length,
Weighted Least Squares (WLS) technique can be used for amplitude estimation. We
presented in Chapter 5 several WLS amplitude estimators and, interestingly enough,
they are closely related to the Capon and APES estimators for complex spectral es
timation. We also proposed a generalized MAFI approach to amplitude estimation,
where all prefilters corresponding to the different frequencies are simultaneously de
termined by defining and maximizing a generalized SNR. It was found that the
generalized MAFI approach to amplitude estimation is very general and includes the
WLS approach as a special case. We derived an interesting MAFI amplitude estima
tors that does not fall into the WLS class. But other MAFI amplitude estimators
may exist and are yet to be discovered.
Another problem we addressed is covariance sequence estimation. Specifically,
we presented in Chapter 6 a new covariance sequence estimator by Fourier inverting
the Capon Power Spectral Density (PSD) estimator. We described how to Fourier


Ill
Next we note that
L-l
R-FB = [{aaMe3ul1 + e(l)) (aaMe?ul + e(l))H
n 1=0
+ (aaMej! + e(l)) (a&Mejul + e(l))H
= \a\2a.M&M + + *)4 + M(o6H + H)
+E[gWe"(') + M0"(¡)].
^ 1=0
(C.4)
We also remind the reader of (B.5) and the assumption that eu(n) and ew(n) have
zero third-order moments. Since
e(l) = JUL-1-l),
(C.5)
which is like (3.9), we have
= = ^L l -
^ 1=0 ^ 1=0
= e~ML-1]J
k=0
e-ML-
By using these facts, along with (C.3), we can write
(C.6)
(aMR aM) E {pBC ~ }
1
£j(a" R-'aM)
1
(aj^R 1aM)(aSH + a<5)R *<5
+(a£R-15)
1
a^R 1(a*d + ci*d)(a^R xaM)
+ |(a^R 1aM)(Q!5/ + <5H)R 13lm
= \E {a*(a^R-15)2(a^R-1aM) + *(a^R-1)(a^R1)(a^R1aM)
+(aR xaw)
a^R-^
+ (a£fR 1aM)(a^R 1)(R xaM)
.-i.


C PROOF OF THEOREM 3.3.2 110
D PROOF OF THEOREM 6.4.1 114
REFERENCES 116
BIOGRAPHICAL SKETCH 122
vi


7
is unbiased; we also prove that the bias for the FB Capon is one half that of the
FO Capon. These theoretical results, supplemented with the empirical obser
vation that Capon usually underestimates the spectrum in samples of practical
length while APES is nearly unbiased, are believed to provide compelling rea
sons for preferring APES to Capon in most practical applications.
3. We present efficient implementation techniques for the MAFI spectral esti
mators. We show that by using such techniques the amount of computation
required by the Capon or APES estimator is significantly reduced.
4. The MAFI idea is also extended for amplitude estimation for sinusoidal signals
in colored noise. Specifically, we make extensions to the Capon and APES algo
rithms to multiple sinusoids with known frequencies. Furthermore, we describe
a generalized MAFI approach to amplitude estimation for multiple sinusoids.
5. A new covariance sequence estimator is presented by Fourier inverting the
Capon spectral estimates. The Capon covariance sequence estimates are shown
to be more accurate than the conventional sample covariance sequence esti
mates.
1.3 Organization of the Dissertation
The dissertation is organized as follows. Chapter 2 gives a literature survey of
such topics as filterbank approaches to spectral analysis, the Capon spectral estima
tor and its applications, and covariance estimation. In Chapter 3, we first introduce
the MAFI approach to complex spectral estimation. We next show that the Capon
and APES estimators are both members of the MAFI approach, followed by com
putational and statistical analyses of the MAFI spectral estimators. Extensions to
the 2-D case are also included. Chapter 4 addresses the implementation issue of the
MAFI spectral estimators. Chapter 5 discusses amplitude estimation for sinusoidal


76
(a)
Computational Complexity
Data Length N
(c)
(b)
Figure 5.5: Averaged RMSEs and the number of flops versus N for the first system
when the observation noise is white (a2 = 0.01) and M = 20 for APESl and MAFI1.
(a) RMSE of a-parameters. (b) RMSE of 5-parameters, (c) Number of flops.


3
(c)
(d)
Figure 1.1: Synthetic Aperture Radar (SAR) images of a simulated MIG-25 airplane
obtained by using (a) 2-D FFT, (b) 2-D windowed FFT, (c) 2-D Capon, and (d) 2-D
APES.


4
resulting filters are matched filters and we refer to this class of approaches to spectral
estimation as MAtched Fllterbank (MAFI) approaches. We show that both Capon
and APES are members of MAFI approaches, although none of them was originally
derived in the MAFI framework (see [16] [52] for their original derivations). MAFI
approaches to spectral estimation may also be used to devise new spectral estimators.
Even though we show in Section 3.2.3 that a reasonable implementation of a seemingly
novel MAFI spectral estimator reduces back to APES as well, it remains an open issue
whether other interesting MAFI spectral estimators exist.
The MAFI interpretation also provides insights into the Capon and APES
estimators and the relationship between them. Specifically, in the framework of
MAFI approaches, we show by means of a higher-order expansion technique in Sec
tion 3.3 that the 1-D (one-dimensional) Capon estimator indeed underestimates the
true complex spectrum while the 1-D APES is unbiased (to within a second-order
approximation); we also show that the bias of the FB Capon is one half that of
the FO Capon. Furthermore, we show that these results can be extended to 2-D
(two-dimensional) Capon and APES estimators.
1.1.2 Efficient Implementation
For 2-D applications, a major concern of using Capon or APES is their com
putational loads which in general are much heavier than those of the traditional
Fourier-based approaches. For a SAR image of size 128 x 128 formed from a 32 x 32
data matrix, as those in Figure 1.1, the number of flops required by Capon and APES
implemented in the conventional way are 2.3 x 104 and, respectively, 3.0 x 104 times
that of those by the Fourier-based methods. It should be mentioned that a SAR
image of 128 x 128 is relatively small. As the size of the image increases, the amount
of computation by Capon or APES increases drastically. In Chapter 4, we address
the issue of how to efficiently implement Capon and APES for spectral estimation.


73
Remark-. According to the Extended Invariance Principle (EXIP) [76], the
estimates of {a,i,bj} obtained by minimizing (5.82) achieve the CRB asymptotically,
and hence they have a better asymptotic accuracy than the OEM estimates whenever
v(n) is colored. It also follows from this observation that in the case of AT = p + q,
the estimates obtained from (5.85) are asymptotically efficient. This latter result (of
a somewhat limited interest, due to the requirement that K p + q) was first proved
in [40] in a much more complicated way.
5.6.2 Numerical Examples
The following examples assume that p and q are known to facilitate perfor
mance comparison. It is reasonable to do so since both OEM and the proposed
method use similar techniques to determine the model orders. Also, we adopt the
strategy to choose the p + q largest {or*} in Step 2 of the proposed method.
Example 5.6.2.1 The system considered in this example is given by (5.73)
with
A^z-1) = 1 1.6019Z-1 + 0.9801z"2, (5.86)
and
B(z~1) = z-1 + 0.24722-2 + 0.1600z3. (5.87)
The probing signal is given by
u(n) = 2 cos(27r0.05n) + 2 cos(27r0.15n) 4-2 cos(27r0.25n)
+2 cos(27r0.35n) + 2 cos(27r0.45n), n = 0,1,..., N 1. (5.88)
We consider using a real-valued probing signal because this is the usual case in
practice. (A subtle question arises as the amplitude estimation techniques discussed
in the previous sections all assume that the sinusoids are complex-valued. One might
impose certain conjugate symmetry constraint and derive similar techniques that
are specifically tailored for real-valued sinusoidal amplitude estimation so that, if


o6
drawback of Q in (5.34). In the following we try to circumvent this need in two
different ways.
First, we show a way to simplify the WLSE(L, 0, K) that uses (5.29) with
(5.34). From (5.34), we have that
RQ_1A = APA^Q^A + A = AI\ (5.35)
where
r = PA^Q^A + I*. (5.36)
For sufficiently large N and M, T is approximately diagonal since AHQ~1A is so
(see, e.g., (5.12)). Consequently,
Q_1A; = R_1ArD; R-^Djr = RxA;r.
Inserting (5.37) into (5.29) yields (observe that VH cancels out)
n -1 rr
(5.37)
ex
EifR-'A,
L/=o
L-l
U=o
(5.38)
which, unlike using (5.29) with (5.34), does not require any initial estimate of {ex*,}^!.
The amplitude estimator in (5.38) can be interpreted as an extension of the Capon
algorithm in [16] [44] to multiple sinusoids.
A different estimate of Q can be obtained as described next. Observe that
K
APAh = ^2[aka{ujk)}[aka{uk)]H =
K
k=1
k=1
where
a(cj) =
1 e
(5.39)
(5.40)
We can use the vectors {/3k}£=l introduced above to rewrite (5.27) as
K
y(0 =
k=1
(5.41)


38
2-D Complex Amplitude Estimation 2-D Complex Amplitude Estimation
(a) (b)
(c)
(d)
Figure 3.5: Empirical bias and variance of the 2-D MAFI estimators as the filter
length, M = M\ = M2, varies when N\ = N2 = 32, and SNRi = 20 dB. (a) Real
part of the bias; (b) Imaginary part of the bias; (c) Variance of the real part of the
estimated amplitude; (d) Variance of the imaginary part of the estimated amplitude.


one can see that R is persymmetric. Similarly, Q is also persymmetric.
The forward-backward sample covariance matrix takes the form:
29
rFB = 2(RFO + rB0)>
(3.70)
where Rpo and Rbo denote the sample covariance matrices of {y(1, 2)} and {y(h, Z2)},
respectively, given by
Rpo =
R-BO =
1
L\L2
L\ \ L/2~ 1
Y Y y(h,h)yH{h,i2),
h= 012=0
1
LiL2
EEWuWduk).
ll= 0 ¡2=0
(3.71)
(3.72)
By making use of (3.69), one can see that Rfb is also persymmetric.
Let HWlA,2 denote the impulse response of an x M2 2-D FIR filter, and let
h-o;1,i2 vecfH^j^].
(3.73)
Like in the 1-D case, the impulse response of the matched filter is given by
, Q-1(aJi,a¡2)aMi,M2(wi> w2)
XWl,J2 h
aMi ,m2 (wi ) w2)Q 1 (a¡i, o¡2)aMl ,m2 (wi ^2)
(3.74)
Note that
hwi, U2aMi,M2(wi ,V2) I-
(3.75)
The LS estimates of u>2) obtained by using only the forward data vectors and by
using both the forward and backward data vectors are given by (similarly to (3.21)
and (3.22) in the 1-D case)
Fo(wi,a¡2) = h"g(w),
(3.76)
and
-j(Ni-l)u¡ie-j(N2-
(3-77)


92
Processes
ARMA1
ARMA2
ARMA3
ARMA4
ai
-1.1824
-1.7351
-0.2000
-0.2000
0-2
0.6651
1.7829
0.0400
0.0400
a3
-0.0895
-0.9616
0.0000
0.0000
<24
0.0049
0.3969
0.0000
0.0000
b\
-0.2000
-0.2000
-1.1824
-1.7351
b2
0.0400
0.0400
0.6651
1.7829
^3
0.0000
0.0000
-0.0895
-0.9616
b4
0.0000
0.0000
0.0049
0.3969
a2
1.0000
1.0000
1.0000
1.0000
Table 6.2: The ARMA processes used in the numerical simulations.
6.5.1 ARMA Covariance Estimation
First we comment on the generation of the ARMA signals. To eliminate the
initial transient caused by improper initialization of the ARMA system we proceed
as follows. Given the coefficients of an ARMA(p,q), we can determine the covariance
matrix (see e.g., [75])
r(0) r(l) r(p 1)
r*(l) r(0) :
I r(l)
r*(p 1) r*(l) r(0)
(6.20)
We use C1/2e as the initial condition for the ARMA system, where e G Cpxl is a
zero-mean Gaussian random vector with identity covariance matrix.
Four ARMA signals are chosen for our study. In the selection of these ARMA
signals, efforts have been made to make them representative of a large class of ARMA
signals. The coefficients of the ARMA signals are listed in Tables 6.2. The pole and
zero diagrams are shown in Figures 6.2(a) to 6.2(d). Figures 6.3(a) to 6.3(d) give the
PSDs of the ARMA signals, while Figures 6.4(a) to 6.4(d) show the corresponding
covariance plots for these signals.


31
Based on the 2-D extensions described above, it is not difficult to see that
all the results of the previous section also hold true for the 2-D Capon and APES
estimators. Indeed, the proofs for the 2-D estimators follow a similar pattern to those
for the 1-D case shown in Appendixes B and C.
3.5 Numerical Examples
In the following, we study the Capon and APES complex amplitude estimates
in a number of cases of interest. For both the 1-D and 2-D examples given below,
we compare the performance of the forward-only Capon and APES as well as the
forward-backward Capon and APES, which are, for simplicity, referred to as FCapon,
FAPES, FBCapon and FBAPES, respectively.
3.5.1 1-D Complex Spectral Estimation
The 1-D data used in the examples consists of a sum of 15 complex sinusoids,
with the real and imaginary parts shown in Figures 3.1(a) and 3.1(b), respectively,
corrupted by a zero-mean complex white Gaussian noise. The data length is chosen
as N = 64. In what follows we are interested in the bias and variance properties of
the estimators under study. The bias and variance results shown below correspond
to the frequency of the first sinusoid and they are obtained from 100 independent
realizations.
We begin by studying the performance of the estimators as the signal-to-noise
ratio (SNR) varies. The SNR for the kth sinusoid is defined as
SNRfe = 101og10-^h- (dB), (3.87)
0e(^/c)
where is the complex amplitude of the kth sinusoid and Pe{u)k) is the spectral
density of the additive noise at frequency 04. The filter length is chosen as M = 15.
The real and imaginary parts of the bias are shown in Figures 3.2(a) and 3.2(b),
respectively, as a function of SNRi. As seen from these figures, FOA and FBA are


33
1-D Complex Sinusoids 1-D Complex Sinusoids
(a) (b)
Figure 3.1: The 1-D complex amplitude of the sum of 15 sinusoids used in the
simulations, (a) Real part; (b) Imaginary part.
resolution as well as the best statistical properties in terms of bias and variance. The
previous examples also show that FOA and FBA perform similarly in terms of bias
and variance properties for the frequency of interest.
To compare the computational complexities of the estimators under study, we
count the flops required by each of them for the case where N = 64, M = 24, and the
complex spectra are evaluated at 256 equally spaced points. The flops required by
FOC and FBC are approximately the same, whereas the flops needed by FOA and
FBA are, respectively, 1.08 and 1.41 times of that by the Capon estimators.
3.5.2 2-D Complex Spectral Estimation
As was mentioned in Section 3.4, the 2-D Capon and APES estimators be
have rather similarly to their 1-D counterparts. Since the problems encountered in
applications such as synthetic aperture radar imaging are concerned with 2-D com
plex spectral estimation, we include a couple of 2-D numerical examples here. The
data employed consists of three 2-D sinusoids corrupted by a 2-D zero-mean complex
white Gaussian noise, with N\ = N2 = 32. The sinusoids are located in the frequency
domain at (0.2, 0.2), (0.25, 0.25) and (0.4, 0.7) and their amplitudes are e-J7r'/4, eJ7r,/4


45
FFT, 2-D windowed FFT, 2-D Capon, and 2-D APES are shown in Figures 4.1(a)
to 4.1(d), respectively. Again, Capon and APES outperform the FFT methods. The
number of flops required by the intuitive ways of implementing Capon and APES
are, respectively, about 38 and 32 times of those required by our efficient ways of
implementing them. If we increase the size of the image to 512 x 512, the ratios of the
needed MATLAB flops between the intuitive ways and our new ways of implementing
Capon and APES are 86 and 73, respectively.
4.5 Summary
This chapter addresses the implementation of the Capon and APES spectral
estimators. The amount of computation required by APES is shown to be about 1.5
times that required by Capon. By using a technique proposed in this chapter, the
computational complexities of Capon and APES are significantly reduced.


This work is dedicated to my wife, Hong.


118
[32] L. J. Griffiths and C. W. Jim. An alternative approach to linearly constrained
adaptive beamforming. IEEE Transactions on Antennas and Propagation, AP-
30:27-34, January 1982.
[33] I. J. Gupta. High-resolution radar imaging using 2-D linear prediction. IEEE
Transactions on Antennas and Propagation, 42(l):31-37, January 1994.
[34] E. J. Hannan and B. Wahlberg. Convergence rates for inverse Toeplitz matrix
forms. Journal of Multivariate Analysis, 31:127-135, October 1989.
[35] S. Haykin, editor. Array Signal Processing. Prentice-Hall, Englewood Cliffs, NJ,
1985.
[36] M. Honig and U. Madhow. Blind adaptive multiuser detection. IEEE Transac
tions on Information Theory, 41(4):944-960, July 1995.
[37] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press,
Cambridge, 1985.
[38] Y. Hua. High resolution imaging of continuously moving object using stepped
frequency radar. Signal Processing, 35(l):33-40, January 1994.
[39] A. Jakobsson, T. Ekman, and P. Stoica. Capon and APES spectrum estima
tion for real-valued signals. Proceedings of the 8th IEEE DSP Workshop, Bryce
Canyon, August 1988.
[40] P. V. Kabaila. On output-error methods for system identification. IEEE Trans
actions on Automatic Control, 28(l):12-23, January 1983.
[41] S. M. Kay. Modern Spectral Estimation: Theory and Application. Prentice-Hall,
Englewood Cliffs, NJ, 1988.
[42] S. M. Kay and S L. Marple, Jr. Spectrum analysis a modern perspective.
Proceedings of the IEEE, 69(11): 13801419, November 1981.
[43] S. B. Kesler, editor. Modern Spectrum Analysis II. IEEE Press, New York, 1986.
[44] R. T. Lacoss. Data adaptive spectral analysis methods. Geophysics, 36(4):661-
675, August 1971.
[45] M. A. Lagunas and A. Gasull. An improved maximum likelihood method for
power spectral density estimation. IEEE Transactions on Acoustics, Speech
Signal Processing, ASSP-32(1):170-173, February 1984.
[46] M. A. Lagunas, M. E. Santamaria, A. Gasull, and A. Moreno. Maximum like
lihood filters in spectral estimation problems. Signal Processing, 10(1): 19-34,
January 1986.
[47] H. Li, J. Li, and P. Stoica. Performance analysis of forward-backward
matched-filterbank spectral estimators. IEEE Transactions on Signal Process
ing, 46(7):1954-1966, July 1998.
[48] H. Li, P. Stoica, and J. Li. Capon estimation of covariance sequences. Circuits,
Systems, and Signal Processing, 17:29-49, January 1998.


43
and e(ujuuj2) via 2-D FFT, then using Equations (4.18), (4.19), and (4.20) to deter
mine a^i M2Q-1t/>, 'tp11 Q~lip, and a^1>M2Q~1&mum2, respectively, and finally using
(4.12) to obtain APES^i)^2)-
The structure of (3.53) is similar to that of (4.12). However, the amount of
computation required by the former is much larger than that by the latter. The
reason is that, even though ?/>(u>i,u>2) and ^(uj \,u2) can be obtained by 2-D FFT,
for each (coi,tu2) pair, we have to compute the additional matrix-vector products
R-1/2t/>(o>i,w2) and R_1/2'0MllM2(a;i,^2) (recall that R_1/2 e CMlM2XMlM2, %j){ui,u2)
and ip(u>i,u2) CMlMiXl) to obtain /x2(w 1^2) and u2). On the other hand,
by computing D and E first (which are computed only once), we bypass calculating
such matrix-vector products and save a large amount of computation. The larger the
number of samples in the 2-D frequency domain, the more the amount of computation
we will save. These discussions also apply to the implementation of Capon.
4.3 Extension to Capon
Note that APES becomes Capon when Q^,^) is replaced by R. Hence
the efficient implementation of Capon can readily be achieved by modifying (4.12) as
follows:
bT(wi,W2)d(£Ji,W2) .
CaponKwa) Ll£2 ||b(Wl>W2)||2
More specifically, the efficient implementation of Capon is by using (4.13) and (4.14)
to calculate b(aq, o>2) and d(uq, u>2), respectively, and then using them in (4.27). Since
the amount of computation required to calculate b(a>,cD) in (4.13), d(ui,u)2) in (4.14),
or e(a)i,cu2) in (4.15) is approximately the same and calculating APES O-A >^2) by
(4.12) and dQap0n(wi, lo2) by (4.27) are much less involved than obtaining b(ui,u2),
d(uq,u;2), and e(ui,u)2), the total amount of computation required by APES is about
1.5 times of that required by Capon, as verified by the numerical and experimental
examples in Section 4.4.


LIST OF TABLES
5.1 Choice of M for the WLS and MAFI amplitude estimators 70
6.1 Comparison of the computational burdens of the standard and Capon
methods with N = 256 and 512 91
6.2 The ARMA processes used in the numerical simulations 92
vii


84
than the latter [44] [75]. Another reason that we consider the Capon PSD estimator
is that it does not exhibit the so-called correlation matching property [75] [41]; that
is, the inverse Fourier transform of the Capon PSD estimates does not yield the same
covariance sequences used to obtain the Capon PSD estimates. This fact allows us
to obtain a new covariance estimator from the Capon spectra.
6.3 Capon PSD Estimator
We have derived in Section 3.2.1 the Capon filters and Capon amplitude and
phase spectra. In this section, we derive expression for the Capon PSD estimates.
The Capon filter is rewritten below for easy reference (see (3.28)):
R-1aM(w)
^ Capon
1AcJ
The filter output power is given by
£{|hfy(¡)|2} =h"Rh,
(6.2)
1
(6.3)
a^(w)R-1aM(u>)
Let ¡5 denote the bandwidth of the filter given by (6.2). Then the Capon PSD estimate
has the form
ir f\uH,-.nW2\ i
0(u>)
{ihJyWl2}
(6.4)
P /3a^(cu)R-1aM(u;)'
Since the (equivalent) time-bandwidth product is equal to unity, one way is
to choose P as the reciprocal of the temporal length of the Capon filter; that is
1
0 =
M'
(6.5)
By choosing the filter bandwidth as given by (6.5), we obtain the so-called Capon-1
PSD estimator [44] [75]:
M
Capon-1: <¡>(u) = , (6-6)
a" (cu)R-1aM(a;)
where we have replaced R by the sample covariance matrix R. We may use the
forward-only sample covariance matrix, but we prefer using the forward-backward


96
ARMA1 (N=256, M=51,100 realizations.) ARMA2 (N=256, M=51, 100 realizations.)
(a)
(b)
ARMA3 (N=256, M=51,100 realizations.) ARMA4 (N=256, M=51,100 realizations.)
(c)
(d)
Figure 6.5: Covariance sequence estimation with N = 256 and M = 50. The mean-
squared errors (MSEs) of the covariance estimates, normalized with respect to r(0),
are based on 100 independent realizations, (a) ARMA1; (b) ARMA2; (c) ARM A3;
(d) ARMA4.


41
It follows that (4.1) can be rewritten as
APES(W1.W2)
lmum2
Q
[LiLi if) Q &muM2Q laMi,M2 + aM!,M2Q
? (4-12)
Since R is Hermitian and positive definite, we can obtain an upper triangular matrix
C by Cholesky factorization such that R_1 = Cri^C-1)^ [30]. Let
br(wi,w2) = a£ (wi.waJC \
(4.13)
d(o>i,w2) Dai2(wi,w2),
(4.14)
and
e(wi,w2) Ea1|£j(wi,W2),
(4.15)
where
D = (C-1)HZ,
(4.16)
and
E = (C-1)HZ.
(4.17)
We have [53]
aMi ,M2 Q
= aMi,M2R~ i^/) "b
b^1 (iJi, cu2)d(iu2, cj2) d-
bT(cJi, tu2)e(q>i, cu2)e//(n;1, a>2)d(a>i, oj2)
LiL2 ||e(cji,iu2)||2
, (4.18)
ipH Q 1tp
= R V> + TTfX
L\L2 V R-1^
||d(wi,a;2)||2+
|d7i(a;i, a;2)e(a;i, w2) |
L\L2 ||e(£Ji,u;2)||
2
2_ >
(4.19)


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.
MATCHED-FILTERBANK APPROACHES TO SPECTRAL
ANALYSIS AND PARAMETER ESTIMATION
By
Hongbin Li
May 1999
Chairman: Jian Li
Major Department: Electrical and Computer Engineering
MAtched-FIlterbank (MAFI) estimators represent a general class of methods
that make use of a set of matched filters for various estimation purposes. This disser
tation investigates using MAFI approaches for complex spectral analysis, amplitude
estimation for sinusoidal signals, and covariance sequence estimation.
For complex spectral analysis, we show that the widely used Capon and the re
cently introduced APES estimators are both members of the MAFI approach, though
neither was originally derived in the MAFI framework. We prove that, to within a
second-order approximation, Capon is biased downward whereas APES is unbiased,
and that the bias of the forward-backward Capon is one half that of the forward-only
Capon. We also show that Capon and APES are of similar computational com
plexities and both are more involved than most Fourier-based methods, especially
for 2-Dimensional (2-D) data. Efficient implementation schemes which substantially
xi


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
May 1999
Winfred M. Phillips
Dean, College of Engineering
Mihran J. Ohanian
Dean, Graduate School


66
Gaussian noise. The SNR is defined in the same manner as in (5.70). Figures 5.3(a)
and 5.2(b) show the MSEs of the amplitude estimates of 0:3 and, respectively, aq, and
the corresponding CRB as the SNR increases. As one can see, the APES1, APESK,
and MAFI1 estimates of a3 are again very close to the CRB; whereas for an, all
suboptimal methods suffer from some performance loss as compared to the optimal
LSEK, and yet the differences between LSEK and MAFI1 for all SNRs considered
here are fairly small.
A brief summary based on the previous study is as follows. APES1 is recom
mended in applications where it is known a priori that no two sinusoids are closely
spaced (see, e.g., the application discussed in the next section), or when the closely-
spaced sinusoids are of no interest. The reason to prefer APESl to APESK or MAFI1
in such cases is that the former is more flexible than the latter two since APESl does
not necessarily require the knowledge of the sinusoidal frequencies. In terms of com
putational cost, APESl and MAFI1 are similar to one another and both are simpler
than APESK. When it is desired to estimate closely spaced sinusoids in colored
noise, however, MAFI1 may be preferred. In general, we do not recommend the use
of Caponl since it has a computational complexity similar to that of APESl but is
biased. Although we did notice that CaponK gives close-to-CRB performance at very
low SNRs, in most cases of interest, other methods like APESl or MAFI1 may be
preferred. LSEK is statistically efficient and may be preferred when the observation
noise is white; in cases where the white noise assumption is invalid, it is preferable
to use APESl or MAFI1. LSE1 gives comparatively rather poor estimation accuracy
but is computationally quite simple. The performance differences stated so far occur
only when N is relatively small. As N increases, all methods tend to the CRB, in
dependent of the noise correlation. Hence, when N is sufficiently large, LSE1 should
be preferred because of its computational simplicity.


12
in great length, vve describe in the following a few typical applications of the Capon
estimator.
An interesting application of Capons method is beamforming. Beamforming
is used in conjunction with an array of sensors to provide a versatile form of spa
tial filtering. The objective of beamforming is to estimate the signal arriving from a
desired direction in the presence of noise and interfering signals. If the desired and in
terfering signals occupy the same (temporal) frequency band, then temporal filtering
cannot be applied to distinguish signal from interference. However, since the desired
signal and interfering signals typically originate from different locations, such spatial
diversity can be exploited to separate signal from interference using a spatial filter. In
1972, Frost made use of a linear constrained optimization technique and introduced
an adaptive beamformer [27], referred to as the LCMV (Linearly Constrained Mini
mum Variance) beamformer in the array signal processing community. The basic idea
of the LCMV beamforming is to constrain the response of the beamformer so that
signals from the direction of interest are passed with specified gain while minimizing
the output power due to interfering signals and noise arriving from other directions.
One would immediately notice the similarity to the constraints adopted by Capon.
Indeed, the LCMV beamformer is a direct extension of the temporal Capon filter to
the spatial domain. Among the so-called statistically optimum beamformers, LCMV
is perhaps the most popular one since it needs no auxiliary channels as required by
the Multiple Sidelobe Canceller (MSC) [2], and, unlike the class of optimum beam-
formers proposed by Widrow et al. [83] which require reference signals, it is blind.
A useful structure for LCMV implementation is the Generalized Sidelobe Canceller
(GSC) [32]. GSC represents an alternative formulation of the LCML beamformer
which changes the constrained optimization problem of LCML to an unconstrained
one. The unconstrained nature lends GSC to adaptive implementation more readily
than the original LCMV beamformer and hence GSC is the one used more often


36
and 0.7eJ7r/4, respectively. The bias and variance for the amplitude estimate of the
first 2-D sinusoid are obtained from 100 independent realizations. The SNR for the
kth 2-D sinusoid is similarly defined as in (3.87). The bias and variance of the four
estimators under study versus SNRi are shown in Figures 3.4(a) to 3.4(d), respec
tively, where Mi = M2 = 8. Figures 3.5(a) to 3.5(d) show the statistical results as
the 2-D FIR filter length varies, where SNRi is fixed at 20 dB. We assume in Figures
3.5(a) to 3.5(d) that Mi = M2. As seen from these plots, the performance of the 2-D
MAFI estimators indeed resembles that of their 1-D counterparts and, therefore, we
refer the readers to the 1-D examples for comments.
3.6 Summary
This chapter discusses using the MAFI approach for complex spectral estima
tion. The Capon and APES estimators are shown to be members of the MAFI class.
By using a higher-order expansion technique, it is proved that to within a second-
order approximation Capon is biased (downward) while APES is unbiased, and that
the bias of the forward-backward Capon is one half that of the forward-only Capon.
It is also shown that the above conclusions carry over to the 2-D MAFI estimators
as well. Since computationally APES is only slightly more involved than Capon, the
preference of APES to Capon in practical applications follows logically because of
the better statistical properties associated with the former.


21
matrix inversion lemma [30], one can see that the second term in (3.27) has no
influence on the hw in (3.17). Hence, when (3.27) is substituted into (3.17), the
matched filter reduces to the Capon filter [16] [44]:
R
aj^(u;)R-1aM(a;)
(3.28)
Observe that Q(tu) is persymmetric for either FOC or FBC. By substituting (3.28)
into (3.21) or (3.26), we obtain the Capon estimate of
3.2.2 APES Filter
^CaponC^)
aj^(o;)R-^(w)
(3.29)
Ignoring the fact that am(lo) is known, we obtain the LS estimate of the vector
a(w)aM(w) in (3.4) as
[a(u)a.M(uj)] = g(u>). (3.30)
Inserting (3.30) into (3.27) along with Rpo substituted yields the FOA estimate of
QM:
QfOaM = R-FO g(w)gH(w). (3-31)
A persymmetric estimate of Q(u) can be obtained by using both the forward and
backward data vectors:
QfbaM = ^ [QfOaM + JQfOa(w)J = R-FB G(w)G/i(u;), (3.32)
where
G(w) =
V2
g(cu) g(u) (3.33)
and we have used the fact that Jg*(w) = e^L~^g(oj). Hence by (3.17), we obtain
the the APES filter [52]:
,APES Q^l(^)aM(w)
h
aw(w)Q -1MaM(w)
(3.34)


30
where
g(wi,w2) = y{h,h)e J'(wi,1+W2ia),
1 2 i1=0 /2=o
g(wi,w2)
1
L1 1 Z/2 1
L\L-
y(/i>/2)e"
j(wii+W2/2)
(3.78)
(3.79)
;1=o 2=o
Since Q is persymmetric, (3.77) can be written as
Fb(wi> ^2) h5liW2g(wi, w2).
The Capon method estimates the noise covariance matrix as
(3.80)
Qcapon^i)^2) R |a(wi,W2)|2aAi1)M2(^1,^2)a^-liM2(a;i,w2), (3.81)
where ¡¡(cjx,^) denotes some estimate of g:(cji,j2), and R denotes either Rpo or
Rpg, which correspond to 2-D FOC or FBC. Thus, the Capon estimate of a(u>i,cj2)
is obtained as
^Capon (wl) ^2)
aMuM2(ui,V2)R Xg(wi,t^)
aMi,M2(Wli(J2)R W2)
The FO and FB APES estimates of Q(u>i,a;2) take the form:
QfOA(W1)W2) = RfO g(^l;^2)g(^l)^l),
(3.82)
and
where
Qfba(w1i ^2) Rfb G(wi, cj2)Gh(cji, uji),
G(cji,cj2)
y/2
g(uJi,W2) g(idi,U>2)
(3.83)
(3.84)
(3.85)
Hence we obtain APES (^l) ^2) as
aM1,M2(Wl>a;2)Q 1(Wl^2)g(^l,W2)
APEs(wi)w2) =
aMi ,M2 (w1 > w2)Q 1 (^1 > ^2)aMi ,M2 (t^l, ^2)
where Q(u,i,cj2) denotes either QfOa(wi>cj2) or Qfba(wi> ^2)-
(3.86)


CHAPTER 4
EFFICIENT IMPLEMENTATION OF CAPON AND APES
4.1 Introduction
In the previous chapter, we rederived the Capon and APES spectral estimators
using the MAFI approach. In the MAFI framework, a number of statistical properties
of Capon and APES were obtained. However, we did not address very carefully the
implementation of the Capon and APES estimators. Equations (3.46), (3.47), (3.52),
and (3.53) give the intuitive ways of implementing Capon and APES. However, such
intuitive implementations are computationally expensive, especially for 2-D spectral
estimation from 2-D data sequences. In this chapter, we study how to implement
Capon and APES efficiently. For simplicitys sake, we only consider 2-D spectral
estimation since 1-D spectral estimation is a special case of the former; we also only
consider forward-backward Capon and APES since they are more often used than
their forward-only counterparts.
4.2 Efficient Implementation of APES
First we rewrite the 2-D forward-backward APES estimator as
aMuM2(vi,u>2)Q~l(ui,u)2)ip(uJi,U2)
APEs(w>w) =
L\L2a.Ml iM2 (cji a;2) Q 1 (uq, ui2)aMl )A2 o>2)
where
Q(cu,u;) = R -
L\L2
^ H ii
r/>(ai,))tj> (w,Co) + '0(w,w)t/> (w,il>)
H ,
(4.1)
(4.2)
with R denoting the forward-backward sample covariance matrix given by (3.70),
#(!,<*) = (4.3)
1=0 2=0
39


APPENDIX A
PROOF OF (3.42) AND (3.43)
Using the MAFI filters given in (3.40) and (3.41) for spectral estimation re
quires an initial estimate of a{to). However, this turns out unnecessary.
Proof of (3.42): Inserting (3.40) into (3.21) yields
poaM) = a^RioS + (a^RFO^Xg^Rpog) la^Rpogl2 > (A-1)
from which we obtain (by assuming that the os in the right-hand and left-hand sides
are identical, i.e., both are o:fom):
FOM
iH
lM
Rpog
aMRFOaM (a^Rpo ) (g W Rpog) la^RpoSl2
Next we rewrite the FOA spectrum (see (3.35)) as follows:
a£(Rp0 ggH)1g
(A.2)
FOA
a£(RF0 ggH)_laM
\H
R
FO
+
i g^R-iog )
g
R-gg^R-\
F0 l g"Ri,g ) M
Rp +
,-1 s
RFQg
(A.3)
aMRFOaM (aMR'FOaM)(s//Rpog) ~ la^Rpogl
which is identical to (A.2). Hence the equality in (3.42) is proved.
Proof of (3.43): Substituting (3.41) into (3.26), and after simple manipula
tions, we obtain
aM%BaM ~ ^(g/7RFBg)(aMRFBaM) + q la^rRpgg|2 ApBM
+
J(aMRFBg)(g//Rpj3aM) ~ F(g/7RFBg)(aMRFBaM)
^FBM a^Rpgg,
(A.4)
106


8
signals from observation corrupted by colored noise. The MAFI concept introduced in
Chapter 3 is extended for amplitude estimation. In Chapter 6, we investigate covari
ance sequence estimation using the Capon spectral estimates. Finally, we summarize
this work and outline future work in Chapter 7.


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Point of View of Communications Engineering. Dover, New York, 1958.
[6] L. E. Brennan, J. D. Mallett, and I. S. Reed. Adaptive arrays in airborne
MTI radar. IEEE Transactions on Antennas and Propagation, 24(5):607-615,
September 1976.
[7] L. E. Brennan and I. S. Reed. Theory of adaptive radar. IEEE Transactions on
Aerospace and Electronic Systems, 9(2):237-252, 1973.
[8] T. B. Bronez. On the performance advantage of multitaper spectral analysis.
IEEE Transactions on Signal Processing, 40(12):2941-2946, December 1992.
[9] J. P. Burg. Maximum entropy spectral analysis. Proceedings of the 37th Meeting
Society of Exploration Geophysicists, Oklahoma City, OK, October 1967.
[10] J. P. Burg. A new analysis technique for time series data. NATO Advanced
Study Institute on Signal Processing with Emphasis on Underwater Acoustics,
Enschede, the Netherlands, August 1968.
[11] J. P. Burg. The relationship between maximum entropy and maximum likelihood
spectra. Geophysics, 37(2):375-376, April 1972.
[12] J. P. Burg. Maximum Entropy Spectral Analysis. Ph.D. Dissertation, Stanford
University, 1975.
[13] J. P. Burg, D. G. Luenberger, and D. L. Wenger. Estimation of structured
covariance matrices. Proceedings of the IEEE, 70(9):963-974, September 1982.
[14] J. A. Cadzow. Spectral estimation: An overdetermined rational model equation
approach. Proceedings of the IEEE, 70(9):907-939, September 1982.
116


CHAPTER 5
AMPLITUDE ESTIMATION
5.1 Introduction
Consider the noise-corrupted observations of K complex-valued sinusoids [71]
[74]
K
x(n) = '^2akejWkn-\-v(n), n = 0,1,... N 1, (5.1)
k=1
where a¡t denotes the complex amplitude of the kth. sinusoid having frequency Uk, N
is the number of available data samples, and v(n) is the observation noise which is
complex-valued and assumed to be stationary (and possibly colored) with zero-mean
and finite unknown Power Spectral Density (PSD) We assume that {<>k}k=i are
known, with u>k ^ toi, for k ^ l. The problem of interest is to estimate {ak}^=1 from
the observations {^(n)}^^1. In this chapter, we describe a relatively large number
of methods for solving this problem.
Section 5.2 discusses least squares (LS) methods. LS methods are widely used
for amplitude estimation because they are simple and easy to implement. If we
restrict ourselves to estimating only one amplitude at a time, then the LS method
reduces to the Discrete Fourier Transform (DFT) of the data at the frequency of
the desired sinusoid and is computationally more efficient than the LS method that
estimates K amplitudes simultaneously. Moreover, estimating one amplitude at a
time does not necessarily require exact knowledge of the number of sinusoids in the
data and of the frequency location of each sinusoid, which is a desired property
in some applications. The disadvantage, however, is that using this one-at-a-time
technique in general gives rather poor amplitude estimates when some sinusoids (that
47


88
where
min(M+s1,M1)
/A ^ r k,k-s- (6-12)
fe=max(0,s)
IfT is Hermitian, then is = ji*_s.
Proof: See Appendix D.
It is obvious that whenever T is non-negative definite we have aj^(o;)raM(w) >
0 for any to. Thus a^(w)raM(ij) is a valid power spectrum. Furthermore, Theorem
6.4.1 indicates that in such a case l/aj^(u;)raA/(cj) is in fact the power spectrum of an
(M l)th-order AR process. Consequently, Capon-1 yields an equivalent AR(M 1)
process (whereas Capon-2 yields an equivalent ARMA(M 1, M 1) process). By
making use of (6.12), we can find the coefficients of the equivalent AR process. The
calculation of the exact covariance sequences from the AR coefficients is a standard
problem and can be solved, for example, via the inverse Levinson-Durbin algorithm
(See [75] [41] and the references therein for more details). Hence the implementation
of the Capon method for covariance estimation runs as outlined below:
Step 1: Pick up a value for M{M < N/2) and compute R by (3.13).
Step 2: Compute ns associated with T = Rr1 by (6.12). Factorize
Ml
m=(M1)
(say, by using the Newton-Raphson algorithm) and obtain the (minimum-phase)
spectral factor.
Step 3: Compute the corresponding covariance sequence (ftemp(^)} from
the spectral factor (or, equivalently, the AR model) by, for example, the inverse
Levinson-Durbin algorithm.
While using the Fourier inverting method for covariance estimation, it is nec
essary that the integral of the PSD estimate over all frequencies gives a good estimate
of the signal power; otherwise there may be scaling errors in the covariance estimates.


75
or MAFI1, but at a significantly increased computational cost. Due to this observa
tion, we do not recommend using this approach, i.e., minimizing (5.82), for refined
estimation accuracy. Other more sophisticated techniques for system identification
(see, e.g., [68] [54]) may be preferred in that event. Figure 5.5(c) also shows that,
as compared to OEM, there is little computational advantage of using the initial
estimates obtained by APES1 and MAFI1. The reason may be that the system in
this example is quite simple (it has white output errors, etc.) and, apparently, OEM
reaches convergence in a relatively small number of iterations. For a more complex
system, such as the one used in the next example, OEM may need more iterations
to converge. It should be mentioned that we did not program our method very care
fully and hence our code is unlikely to be as efficient as the OEM code in MATLAB.
Regarding the estimation accuracy, we shall stress that in the current case where the
noise v{n) is white, OEM coincides with the optimal Maximum Likelihood Method
(MLM) [68] [54]. When v(n) is colored, OEM is no longer MLM. In that case, the
initial system parameter estimates obtained by APES1 or MAFI1 may outperform
those by OEM, as in fact shown in the next example.
Recall that LSEK is statistically efficient when the observation noise is white.
Then, one might wonder why the initial estimates given by LSEK may be notably
worse in such a case than those given by APES1 or MAFI1, as happened in the
previous example (especially when N is small). The reason is that the transient
response of this system cannot be neglected for small N. To show this, the PSD of
x(n) is estimated by using the Capon PSD estimator, with N = 200 and M = 20, and
is plotted in Figure 5.6. It shows two extra peaks (which behave like two sinusoids) at
0.1. The extra peaks are attributed to the response of the system (which has poles
at 0.99e:D27r0'1) to the initial conditions. Since it is essential for LSEK to have the
accurate knowledge of the number and frequencies of the sinusoids frequencies to give
reliable amplitude estimates, its performance in the previous example is considerably


104
inverting the Capon PSD estimates in both the exact and approximate manners.
The approximate Capon covariance estimates are computationally more convenient
to obtain than the exact Capon covariance estimates, and the accuracy loss is in
general very small provided that sufficient large frequency samples are used. It was
found that the Capon covariance sequence estimator gives more accurate covariance
estimates than the widely-used sample covariance sequences, especially for the higher
lags.
In summary, we have
introduced the MAFI idea for spectral analysis and parameter estimation.
shown that the Capon and APES complex estimators are both members of the
MAFI approach to complex spectral estimation.
proved that the estimation errors of the Capon and APES complex spectral
estimators have similar asymptotic distribution.
proved that for finite length of data samples, the Capon complex spectral esti
mator is biased (downward) whereas the APES complex spectral estimator is
unbiased, and that the bias of the forward-backward Capon is one half that of
the forward-only Capon (to within a second-order approximation).
devised efficient implementation schemes for the Capon and APES complex
spectral estimators.
presented a generalized MAFI approach, along with several other interesting
methods, for amplitude estimation.
demonstrated that the MAFI approach to amplitude estimation includes the
WLS approach.


CHAPTER 2
LITERATURE SURVEY
Historical and modern perspectives on the general topics of spectral analysis
and parameter estimation have been well documented in the literature [42] [62], Many
classical articles, both theoretical and application-oriented, have been reprinted [20]
[43]. Excellent texts are also available [75] [41] [55]. In this chapter, we give a brief
review of a number of subjects that are related to our work, namely filterbank ap
proaches to spectral estimation, the Capon method and applications, and covariance
estimation.
2.1 Filterbank Approaches and Capon Estimator
Unless the observed signal can be modeled with a finite number of parame
ters, estimating the spectrum of a signal based on a finite length of observations is
an ill-posed problem from a statistical standpoint, since we are required to estimate
an infinite number of independent spectral values based on a finite number of sam
ples. An assumption made by filterbank approaches and most other non-parametric
methods is to assume that the PSD of the observed signal is (nearly) constant over
a narrowband around any given frequency. Naturally one can proceed by passing
the observed signal through a bandpass filter, which is swept through the frequency
band of interest, and estimating the complex amplitude if complex spectral estima
tion is of interest (or measuring the filter output power and dividing it by the filter
bandwidth if PSD estimation is the desired goal), a procedure adopted by all filter-
bank approaches. Obviously, how to choose the narrowband filters is a critical issue
of filterbank approaches. Even though some of the classical Fourier-based methods,
9


72
that is,
1 i_1
Rvv = 7^ v(/)vH(/). (5.81)
^ 1=0
Step 2: Obtain estimates of {a,5,} by minimizing
K 1
C2(a,b) = ^^-- \ak afc(a,b)|2. (5.82)
*=i nuk)
To do so we can use a host of methods, provided that we have good initial estimates
of a and b. To obtain such estimates and then minimize (5.82), we assume that p
and q are known. (Standard techniques for system order determination can be found
in, e.g., [68] [54].) We pick up the p + q largest {d*} (if the SNR is low, an alternative
is to choose those {a*,} that have the largest ratio |djt|2/^(wfe), assuming that (u)k)
was estimated) and define a criterion made from the corresponding terms of (5.82)
P+Q ,
^3(a,b) = ^^-- \ak afc(a,b)|2, (5.83)
jfc=i where we have assumed, for notational simplicity, that {d*,}^ are the p + q chosen
amplitudes. Now, the minimization of (5.83) is simple. Indeed, almost always one
can choose a and b to satisfy
d* = afc(a,b), A: = 1, 2,... ,p + q. (5.84)
Equation (5.84) is equivalent to
T(e^) = R(e^fc), k = 1,2,... ,p + q, (5.85)
7 k
which can be rewritten as a linear system of p + q equations with p + q unknowns
{cii,bj}. That system will generally have a unique solution that makes (5.83) equal
to zero, and which therefore gives our initial estimates of {a,6j}.
As shown in the following numerical examples, the initial estimates are usu
ally quite good. Hence, one can even skip the step of minimizing (5.82) to save
computations.


55
Alternatively, we can rewrite (5.26) as
y (0 = A iOl + e(/),
where
A, = A
pjwii
0
pJUKl
= AD,.
We will use (5.26) mostly for analysis and (5.27) for estimation.
The WLS (Markov-like) estimate of a in (5.27) is given by
a =
L-1
i -1 r
L/=o
L-1
£AfQ-'y(0
where Q is an estimate of
Q = £{e(0e"(/)}.
To estimate Q, we may proceed as follows. Let
R=}Ey(0yffW.
1=0
One can verify that as L > oo, R goes to
R = APA" + Q,
where
Hence, one way to estimate Q is as
Q = R APA",
(5.27)
(5.28)
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
(5.34)
where P is made from some initial estimates of {ak}k=\ obtained for instance via
one of the LS amplitude estimators. The need for initial amplitude estimates is a


BIOGRAPHICAL SKETCH
Hongbin Li was born on March 30, 1970. He received the B.S. and M.S. degrees
in electrical engineering from the University of Electronic Science and Technology of
China (USETC) in July 1991 and April 1994, respectively. He was affiliated as an
Assistant Lecturer at the UESTC from 1994 to 1995. Since 1996, he has been working
toward his Ph.D. degree in electrical engineering at the University of Florida.
122


reduce the computational requirement are presented for the Capon and APES esti
mators.
For amplitude estimation, we describe a large number of estimators which
can be categorized as the Least Squares (LS), Weighted Least Squares (WLS), and
MAFI methods. While all these methods are asymptotically statistically efficient,
their performances in finite length of data samples are quite different. Specifically,
we show that the WLS and MAFI methods outperform the LS methods whenever the
observation noise is colored; we also show that the MAFI approach is very general
and includes the WLS approach as a special case.
For covariance sequence estimation, we present a Capon covariance estimator
by Fourier inverting the Capon Power Spectral Density (PSD) estimates. We describe
the Fourier inversion in both an exact and approximate ways, of which the latter is
computationally more attractive but with some minor accuracy loss. We show that
the Capon covariance sequence estimates are in general better than the widely used
sample covariance sequences.
xii


CHAPTER 1
INTRODUCTION
This dissertation is concerned with spectral analysis and parameter estimation
using MAtched-FIlterbank (MAFI) approaches. This chapter serves as a general
introduction to the background and scope of the work. Significance and contributions
are also highlighted in this chapter.
1.1 Background and Scope of the Work
1.1.1 Capon. APES, and MAFI Spectral Estimators
Spectral estimation is an important data analysis tool which has found applica
tions in many diverse fields including speech analysis, telecommunications, radar and
sonar systems, biomedical and seismic signal processing, and economics [75]. There
are two broad classes of approaches to spectral analysis: non-parametric approaches
and parametric approaches. While non-parametric methods typically postulate no
model for the observed data, parametric approaches do assume some model so that
the spectrum is represented by a set of parameters, thereby reducing the spectral
estimation problem to that of estimating the parameters of the model. Parametric
methods may offer more accurate spectral estimates than non-parametric methods,
provided that the data indeed observe the model assumed by the former. In the more
likely cases where an accurate data model is not available, parametric methods will
be sensitive to model mismatch; hence using the robust non-parametric approaches
in such cases may be a better choice. Recent studies in the literature show that there
has been renewed interest in non-parametric approaches to spectral estimation.
1


CHAPTER 6
CAPON ESTIMATION OF COVARIANCE SEQUENCES
6.1 Introduction
Covariance sequence estimation is a ubiquitous task in digital signal process
ing. A standard technique for estimating the covariance sequences is the so-called
the standard sample covariance estimator. The standard covariance estimates are
consistent provided that the given signals are ergodic to the second order. However,
there is a major concern of using the standard estimator due to the unrealistic win
dowing of the observed data it assumes; that is, it assumes that the data beyond the
observed duration either is zero or repeats itself periodically. Partly for this reason,
there have been several attempts in the literature to derive more accurate covariance
estimates than the standard ones. A notable example is the approach based on the
Burg autoregressive (AR) spectral estimator. However, the so-obtained covariance
estimator was found even less accurate than the standard sample covariance estima
tor [78]. To be more exact, the Burg approach was shown to have larger variances
than the standard method. Another approach, which has generated a whole new
research direction, relies on the maximum likelihood (ML) principle [13]. However,
the ML estimation of covariance sequences is a computationally involved problem
which does not have a closed-form solution. The solution given in [13] is iterative
and not guaranteed to be globally optimum. Apparently, there exist no compelling
alternatives to the standard method that can be recommended for general use.
In this chapter we present a new method, namely the Capon method [48], for
covariance sequence estimation. The Capon method obtains the covariance sequence
estimates by Fourier inverting the Capon power spectral density (PSD) estimates.
80


APPENDIX B
PROOF OF THEOREM 3.3.1
(We give below a proof of Theorem 3.3.1 only for the FB case. The proof for
the FO case follows a similar (and simpler) pattern.) From (3.4) and (3.7), we have
g = aaM + <5,
g = &&m + ,
where and S, respectively, are defined as
S =
^ 1=0
i = 4xs(0e_i"'-
^ 1=0
First we calculate the first and second-order moments of <5 and :
E{6}
E{~6}
£<[<5 e{6h}
= 0,
= 0,
= 0,
= 0,
;=o fc=o
L-i
= J X ( ||)R.(<)e-^,
i=-(L-l)
1=0 k=0
L-i
= 5 X ( |i|)R.(i)e->*.
fe-(L-l)
(B.l)
(B.2)
(B.3)
(B-4)
(B.5)
(B.6)
(B.7)
(B-3)
(B.9)
(B. 10)
108


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54
Note that, for most cases of interest, LSE(1,0,A') will give more accurate
amplitude estimates than LSE(1,0,1), and that the difference between these two
estimators is small for large N. On the other hand, LSE(1,0,1) is computationally
more efficient than LSE(1,0, K) since the matrix multiplication and inversion in (5.4)
are avoided. Hence LSE(1,0,1) may still be worth considering.
5.3 WLS Amplitude Estimators
If we split the data vector x into subvectors, then the covariance matrix of
the noise part of the subvectors may be estimated and can hence be used to derive
an optimal WLS estimator (i.e., a Markov-like estimator) [68]. In this section, we
describe a number of such WLS estimators that split the data into vectors of shorter
length, utilize no prefiltering, and estimate either one or K amplitudes at a time.
5.3.1 WLSEfL, 0. K)
We define the following subvectors
y (0 =
where
1 T
x(l) x (l + 1) ... x{l + M 1)
L = N -M + 1.
I = 0,1,..., L- 1, (5.23)
(5.24)
The choice of M (M can be chosen smaller than K. See Figure 5.4. Moreover, when
M 1, all WLSE(L, 0, K) reduce to LSE(1, 0, K)) or, equivalently, of L is discussed
in Section 5.5. We have
, (5.25)
i
1
axe^1
v(l)
y(0 =
ejwi
eJ0JK
+
v(l + 1)
ej(M-l)un
ej{M-l )ujk
apce^1*1
v(l + M 1)
or, with obvious notation,
y(0 = As(0 + (0-
(5.26)


Figure 6.4: True covariance sequences, (a) ARMA1; (b) ARMA2; (c) ARMA3;
Covariance Sequences
p f
J l l l
Covariance Sequences
Q- 3
?
01
(Q A _
O
1
ARMA3 ARMA4
Covariance Sequences
CD
Cn


44
For similar reasons as for APES, the intuitive implementation of Capon given
in (3.52) is computationally more involved than the efficient implementation of Capon
proposed above.
4.4 Numerical and Experimental Examples
We present numerical and experimental examples comparing the performances
of APES and Capon with the FFT methods [52] for SAR imaging. In the following
examples, we choose M = N/2 and M = /2 for both Capon and APES. For the
windowed FFT method, we use the Kaiser window with parameter 4.
We first consider SAR imaging of a simulated MIG-25 airplane. The 32 x 32
data matrix was provided by the Naval Research Laboratory. The 128 x 128 SAR
image obtained by using 2-D FFT, 2-D windowed FFT, 2-D Capon, and 2-D APES
are shown in Figures 1.1(a) to 1.1(d), respectively. We note that Capon and APES
outperform the FFT methods. The number of flops required by our efficient ways
of implementing Capon and APES are about 950 and 1500 times of those required
by the FFT methods, while those required by the intuitive ways of implementing
Capon and APES are about 22800 and 30000 times, respectively, of those required
by the FFT methods. That is, the number of flops required by the intuitive ways
of implementing Capon and APES are, respectively, about 24 and 20 times of those
required by our efficient ways of implementing them. If we increase the size of the
image to 256 x 256 and 512x512, respectively, the ratios of the needed flops between
the intuitive ways and our new ways for implementations are 36 & 40 for Capon, and
31 & 34 for APES, respectively.
We now consider an example of SAR imaging with experimental data. The
data matrix is 64 x 64 and is obtained from the experimental data collected by one of
the two apertures of the ERIMs (Environmental Research Institute of Michigans)
DCS IFSAR (interferometric SAR). The 256 x 256 SAR image obtained by using 2-D


19
3.2 MAFI Filters
By definition, the matched filter is designed such that the corresponding
signal-to-noise (SNR) ratio in the filter output is maximized; that is,
\h%aM(u)\2
arg max
tr h"QMhw (316)
The solution is obtained by making use of the Cauchy-Schwartz inequality (see, e.g.,
[75]):
_ Q-a mM ,
" a£(u>)Q-i(u)a(w) 10
where Q(u>) is assumed to be invertible. It is readily checked that the solution in
(3.17) satisfies
h"aMM = l, (3.18)
which implies that the filter given in (3.17) passes the frequency uo undistorted. By
making use of this observation and of (3.4) and (3.7), we have
h5y(Z)=a(w)e'wi + h^iw(0, / = 0,1,..., L 1, (3.19)
and
h?y(0 = e-*lf-1*a'{u>)eul + l = 0,1,..., L 1. (3.20)
The least squares (LS) estimate of (a;) obtained by using only (3.19), i.e., the forward
data vectors, is given by
FoM = (3-21)
whereas the least squares (LS) estimate of a(u) obtained by using (3.19) and (3.20),
that is, both the forward and backward data vectors, is given by
fbM = ^ [h^fg(w) + e~N-1)uJgH(u)K] (3.22)
where g(w) and g(w) are, respectively, the normalized Fourier transforms of the
forward and backward data vectors:
iM
1
L
1=0
(3.23)


40
and
Let
L\ 1 Z/2 1
V>(wi,w2) = ^2y{h,l2)e~UJlll+U2l2).
h= 0 2=0
(4.4)
Z =
y(0,0) y{L\ 1,0) y(0,L2 1) y(Li 1,Z,2 1)
(4.5)
and Z = JZ*J, where J denotes the exchange matrix. We can then rewrite (4.3) and
(4.4) as
^(wi,W2) = ZaliL2(o>i,u;2), (4.6)
and
^(wi,w2) = Za12 (wi,o;2). (4.7)
By applying the matrix inversion lemma, we obtain
Q-(a)l,^) = Q"K,^)+^1("1-a,!)y"1U,2)j,"(u,1'U,2)^1(a1'^), (4.8)
LiL2-V> (wi,w2)Q-1(a;i,a;2)'0(cj1,a;2)
where
Q-1(a,^)=R-1+ (49)
LL ip (a;i, a;2)R_1V> (a>i, cj2)
(For notational convenience, we sometimes drop the dependence on cji and cj2 below.)
Hence,
and
aMx,M2(w iwi)Q 1(c^i, o;2)t/j(c^i, a^2)
k-i
*mumA lWHQ >
,Q_1^ +
aMuM2
LiL2&mum2Q 1^
LiL2 tpHQ-1xp
L\L2 ipnQ-'-'ip
(4.10)
aMi,M2(Wliu;2)Q 'aMi.Mj^l,^)
lMi,M2
Q 'aMi,M2 +
,M2
Q-1^
H,
LiLi-ip Q_1i/>
(4.11)


27
APES estimators are related by
LE {FBC(u) a(uj)} LE{aF0C(u) a(w)} /0
7 \ = 0 7 T < 0, (3.57)
a[io) 2 Oi(ij)
and
LE {AFbaM MI = EE {foa(w) ~ a(w)} = 0, (3.58)
for sufficiently large values of L.
Proof: See Appendix C.
We believe that (3.57) and (3.58) provide a theoretical motivation for prefer
ring APES to Capon in most spectral estimation exercises. Moreover, Theorem 2
also suggests that FBC should be preferred over FOC. While both FOA and FBA
are similarly unbiased (within a second-order approximation), the latter is usually
observed with slightly better resolution and sidelobe properties [52] at the cost of
slightly more computations.
3.4 2-D Extensions
We briefly describe the 2-D extensions of the MAFI spectral estimators. We
first decompose the observations {y(ni,n2)} as
t/(ni,n2) = o:(a;i,W2)e-j(Jini+a;2n2)+eWliW2(n1,n2),
(3.59)
nx = 0,1,..., W 1; n2 = 0,1,...,N2 1; uq, cj2 where a(u>i,u2) denotes the complex amplitude of a 2-D sinusoidal signal with fre
quency (u>i,u>2) and eUliU2(ni, n2) denotes the noise (or residual) term at frequency
(uq,u;2), assumed to be zero-mean. Next, in a manner similar to the 1-D case, we
form the Mi x M2 forward and backward data matrices:
Y(u2) = {y(n\,n2), n\ = Zi,..., l\ + M\ 1; n2 Z2,..., l2 + M2 1} ,
Y(Zi,Z2) = {y*(ni,n2), nx = Ni-li-l,...,Ni-li-Mi]
n2 = N2 l2 1,..., N2 l2 M2} ,
h = 0,1,..., Li 1; l2 = 0,1,..., L2 1,
(3.60)


LIST OF FIGURES
1.1 Synthetic Aperture Radar (SAR) images of a simulated MIG-25 air
plane obtained by using (a) 2-D FFT, (b) 2-D windowed FFT, (c) 2-D
Capon, and (d) 2-D APES 3
3.1 The 1-D complex amplitude of the sum of 15 sinusoids used in the
simulations, (a) Real part; (b) Imaginary part 33
3.2 Empirical bias and variance of the 1-D MAFI estimators as SNRi
varies when IV = 64 and M = 15. (a) Real part of the bias; (b)
Imaginary part of the bias; (c) Variance of the real part of the esti
mated amplitude; (d) Variance of the imaginary part of the estimated
amplitude 34
3.3 Empirical bias and variance of the 1-D MAFI estimators as the filter
length, M, varies when N = 64 and SNRi = 20 dB. (a) Real part of
the bias; (b) Imaginary part of the bias; (c) Variance of the real part
of the estimated amplitude; (d) Variance of the imaginary part of the
estimated amplitude 35
3.4 Empirical bias and variance of the 2-D MAFI estimators as the SNRi
varies when Nx = N2 = 32 and Mx = M2 = 8. (a) Real part of the
bias; (b) Imaginary part of the bias; (c) Variance of the real part of
the estimated amplitude; (d) Variance of the imaginary part of the
estimated amplitude 37


59
where (-)1/2 denote the Hermitian square root of the positive definite matrix argu
ment. Observe that X is semi-unitary, i.e.,
XHX = lR. (5.48)
The cost function in (5.46) can now be rewritten as
/ = tr
x"q-1/2apahq-1/2x] .
(5.49)
It follows from the Poincar separation theorem (or the generalized Rayleigh quotient
theorem) [37] that
R
max / = Xk (CT1/2APA^Q-1/2) (5.50)
k=1
where {Afc(-)}j^.1 denote the eigenvalues of the matrix between the parentheses, or
dered such that Ai > A2 > ... > furthermore, the columns of the maximizing X
are equal to the eigenvectors corresponding to {Afc}^=1.
Next, note that post-multiplying X by any unitary matrix of appropriate
dimensions yields another valid solution for X. One such solution having a simple
form can be obtained as follows. Observe that
rank (q1/2APAhQ-1/2) = K, (5.51)
which implies that we cannot improve the generalized SNR by choosing K > K since
= ... = XR = 0. On the other hand, the larger the K the more filtered data
will be available for amplitude estimation. Hence, we choose
K = K.
In such a case, the maximizing X is given by
(5.52)
X = Q"1/2AT
(5.53)


51
Next, since (see, e.g., [34])
JSo^(") = I"' <5-8>
where 1^ denotes the N x N identity matrix, and
lim 4("W) =
7V->oo N
0 <>{wk)
the asymptotic MSE is given by
(5.9)
M 0
limIVMSE{a} =
NÂ¥00
0 <¡>{uk)
(5.10)
Under the mild assumption that v(n) is circularly symmetric Gaussian, the CRB for
a is given by (see, e.g., [68])
CRB{a} = (iW_1)_1.
(5.11)
Using the following result (see [34] once again)
lim 4(HW_1) =
N-+oo N
(/) L(W!)
(j) 1 (to k)
(5.12)
we obtain
M 0
lim 7VCRB{tx} =
N>oo
0 0(wk)
(5.13)
which coincides with (5.10).
Remark: It can be readily checked from (5.6) and (5.11) that if v(n) is white,
i.e. W ~ 1N, then LSE(1,0,R) is statistically efficient for all N > K.


53
where a = U ... CtK ]T and A is defined through
A
(5.18)
Hence, LSE(1, 0,1) is biased. However, it is asymptotically unbiased (that is, its bias
goes to zero as N oo). We next calculate the MSE of on:
MSE{i} = (H) 1aH [Acta11 AH + W) (H) 1. (5.19)
Making use of (5.9) once again, along with the fact that AHa/\f 0 as N > oo,
we have
lim AMSE{d;i} = (¡>{l\). (5.20)
N-+oo
Hence, LSE(1,0,1) is also asymptotically efficient. On the other hand, in finite
samples (5.14) may be better or worse than (5.4), depending on the characteristics
of the scenario under study.
The fact that (5.4) may be better than (5.14) comes as no surprise. As an
example, let us assume that the Signal-to-Noise Ratio (SNR) is high. Then, the
bias of (5.14) dominates the variance part. On the other hand, (5.4) has no bias
and its variance will be smaller than the bias of (5.14) if the SNR is large enough.
Consequently, the MSE of (5.4) will be smaller than that of (5.14).
The fact that (5.14) may be better than (5.4) is however a surprise. For an
example of such a case, assume SNR MSElon} (^)-1,
(5.21)
whereas for (5.4),
MSE{d!}
(A^A)-1
i,i
(5.22)
which can be much larger than (5.21) (e.g., if \u>k Wi| ~ l/N for some k > 2). In
(5.22), [-jij- denotes the ij-th. element of the matrix argument.


105
presented a new covariance sequence estimator by Fourier inverting the Capon
PSD estimates.
described how to Fourier inverting the Capon spectra exactly, which is an in
teresting endeavor by itself.
7.2 Future Work
One interesting area of future work is amplitude estimation for 2-D sinusoidal
signals (with application to 2-D system identification). However, unlike the extension
of the MAFI complex spectral estimators from the 1-D case to the 2-D case, the
extension for amplitude estimation is not so straightforward. A first question would
be the analysis of LSE(1,0, K) and LSE(1, 0,1) in the colored noise case. There may
be no compact expression for the asymptotic accuracy that is similar to the one in
the 1-D case. The reason is that there exist in the 2-D case no such expressions
as those shown in (5.8) and (5.9). Another important question would be how to
apply WLS to the 2-D data. This involves how to create the 2-D subvectors from
the data matrix, and how to estimate the covariance matrix of the noise subvectors,
etc. One can expect that the 2-D WLS amplitude estimators may be considerably
more complex than their 1-D counterparts. Yet, as in the 1-D case, the 2-D WLS (or
MAFI) amplitude estimates should in general be more accurate than the simple LS
amplitude estimates.


121
[80] H. Wang and L. Cai. On adaptive spatial-temporal processing for airborne
surveillance radar systems. IEEE Transactions on Aerospace and Electronic
Systems, 30(3):660-670, July 1994.
[81] H. Wang and G. H. Wakefield. Signal-subspace approximation for line spectrum
estimation. Proceedings of the International Conference on Acoustics, Speech,
and Signal Processing, pages 2054-2057, Dallas, TX, April 1987.
[82] J. Ward. Space-time adaptive processing for airborne radar. Technical Report
1015, Lincoln Laboratory, MIT, December 1994.
[83] B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode. Adaptive antenna
systems. Proceedings of the IEEE, 55(12) :21432159, December 1967.
[84] D. M. Wilkes and M. H. Hayes. Iterated Toeplitz approximation of covariance
matrices. Proceedings of the International Conference on Acoustics, Speech, and
Signal Processing, pages 1663-1666, New York, April 1988.
[85] D. B. Williams and D. H. Johnson. Robust estimation of structured covariance
matrices. IEEE Transactions on Signal Processing, 41(9):2891-2906, September
1993.


63
Figure 5.1: PSD of the test data that consist of three sinusoids and an AR(1) noise
process.
where fc(z) is the estimate of a.k derived in the zth simulation run.
5.5.1 Estimation Performance versus SNR
First, we consider the case where v(n) is colored. More exactly, v(n) is de
scribed by the following AutoRegressive (AR) process
v(n) = 0.99u(n 1) + e(n), (5.69)
with e(n) being a complex white Gaussian noise with zero-mean and variance a2.
The PSD of the test data is shown in Figure 5.1, where a2 0.01. The local SNR of
the k-th. sinusoid is defined as [41]
Iry, N\cu P
SNRfc = 10 log10 10 logio -rpr. (5.70)
/ Hf)df nJk>
Note the occurrence of N in the above SNR formula. For those methods that depend
on M, we choose M = N/4 = 8, giving L = 25 (see Section 5.5.2 for a study of the
effect of M on the performance).
Figure 5.2(a) shows the MSEs of the seven amplitude estimators for 0:3, along
with the corresponding CRB, as the SNR varies. As one can see, APES1, APESK,
and MAFI1 are very close to the CRB, while LSEK, which ignores the noise cor
relation, is evidently away from the CRB. CaponK also deviates from the CRB for


82
better AR coefficient estimates are obtained by using the Capon covariance esti
mates than by the standard ones. Another application discussed in this chapter is
the moving-average (MA) model order determination by making use of the Capon and
the sample covariance estimates, where we find that better performance is achieved
by the former.
It should be mentioned that we can obtain APES covariance sequence esti
mates similarly by Fourier inverting the APES PSD estimates. However, in spite
of the evident difference of the Capon and APES complex spectral estimates for
discrete spectral components, their performances in continuous PSD estimation are
similar to one another, which implies that the APES covariance estimates are similar
to the Capon covariance estimates. As such the APES covariance estimation is not
discussed herein.
6.2 Standard Covariance Estimator and Outlook
With no other assumptions made on the signal under study, except for assum
ing the second order ergodicity, there are two ways to obtain the standard sample
covariances of the signal, namely, the biased and the unbiased covariance estimators.
However, the biased covariance estimator is more commonly used since it provides
smaller mean-squared errors (MSE) than the unbiased one and guarantees the co-
variance estimates to be positive semidefinite [75].
The biased sample covariance estimator of a wide-sense stationary signal with
zero-mean has the form
JV-fc
(6.1)
k = 0,1,..., K.
72=1
where {y(n)}%=1 are the observed data samples, N is the number of samples, f(k)
denotes the estimate of the covariance function r(k), K is the largest lag desired
(0 < K < N 1), and ()* denotes the complex conjugate. Note that (6.1) is
asymptotically unbiased.


89
So after Step 3, we use a power compensation approach to obtain our ultimate Capon
covariance estimates. Our final estimates {^Caponi^)} have the form
n \ f) ^temp(^)
rCPnWPV,e1p(0)
where P0 is the estimated power of the signal

n=0
(6.13)
(6.14)
6.4.2 Approximate Method
Since the Capon-1 spectrum is equivalent to an AR spectrum, the covariance
sequence can be computed exactly as described above. With some accuracy loss,
the computational demand of the Capon method can be reduced. According to the
Wiener-Khintchine theorem, we have
r(k)
(6.15)
We can rewrite (6.15) as
1 r2n
r(k) / (t){u)elkudu. (6.16)
Jo
Let 3> N, and let the Capon spectrum be evaluated at u> = 2ir/\ that
is, we calculate (o>), = 0,..., 1. Then we can approximate the computation
of the covariance sequence corresponding to 4>(uj) by the following equation
f(k)
1 2tt
27t IV
Nl
J2
1
N
N-1
J2 j>(un)eikU,
=0
(6.17)
which can be evaluated by using FFT. Since the error in approximating the integra
tion by the summation above is 0(l/iV), the errors introduced are quite small for
large enough . Our computer simulations also confirm this observation.
The evaluation of (f)(uj) in (6.17) by directly using (6.6) is computationally
burdensome. We can instead make use of (6.11) as follows. Since R_1 is Hermitian,


85
sample covariance matrix, given in (3.13), for better statistical properties associates
with the latter [41].
Another more elaborate choice of (3 is obtained as the equivalent bandwidth
of \H(u>)\2, where H(u>) is the filters frequency response:
l h
H(w) = [h^"] a M(w).
(6.7)
This specific bandwidth choice leads to the Capon-2 PSD estimator [45] [75]:
a&(w)R-1aM(w)
Capon-2:
=
(6.8)
a^(o>)R-2a M{u)
Burg showed that the inverse of Capon-1 spectrum is equal to the average of
the inverses of the estimated AR spectra of orders from 0 to M [11]. This observation
reveals the fact that Capon-1 has less statistical variation as well as lower spectral
resolution than the AR estimator. A similar but more involved relationship between
Capon-2 and the AR estimators was derived in [75]. Theoretically, the performance of
Capon-2 is hard to quantify. However, it is generally believed that Capon-2 possesses
finer resolution and hence is a better spectral estimator than Capon-1 [45]. We will
show here, with a typical example, that even though Capon-2 has better resolution
locally around the power peaks, it is globally a more biased estimator than Capon-
1. Our experience also shows that Capon-2 generally gives much poorer covariance
estimates than Capon-1. Therefore, Capon-2 is not recommended for covariance
sequence estimation.
To illustrate the above claim, consider an ARMA(4,2) signal
y(n) = 2.76y(n 1) 3.809y(n 2) + 2.654y(n 3) 0.924y(n 4)
+e(n) 0.9e(n 1) + 0.81e(n 2), (6.9)
where e(n) is a real white Gaussian random process with zero-mean and unit variance.
The Capon-1 and Capon-2 spectral estimates with N = 256 and M = 50 are shown
in Figure 6.1(a), where the dashed curve stands for the true PSD of the ARMA


32
almost unbiased, while FOC and FBC are biased downward. In addition, we notice
that the bias for FOC is approximately twice that of FBC. All these observations are
consistent with the prediction of the theory. The variances of the real and imaginary
parts of the amplitude estimates are shown in Figures 3.2(c) and 3.2(d), respectively.
It appears that all of the estimators display similar variances. However, as shown in
the next example, the variance of Capon becomes notably larger than that of APES
as M increases.
Next we study the effect of the filter length, M, on the estimators. The SNRi
is fixed at 20 dB. As M varies, the real and imaginary parts of the bias are shown
in Figures 3.3(a) and 3.3(b), respectively. From these figures, one can see that both
FOA and FBA are unbiased for all practical filter lengths, whereas the bias of Capon
grows significantly with increasing M. (A practical filter length means that M should
not be too small [52]. In fact, all filterbank methods reduce to the Fourier transform
approach when M 1, and only when M is sufficiently large, the filterbank approach
shows noticeable improvement over the Fourier method [52].) All estimators seem
to perform similarly for M up to a fourth of the data length, with Capon being
slightly biased downward. As the filter length increases further, the performance of
Capon degrades rapidly, while that of APES remains unaffected. This observation is
strengthened by the variance results shown for the real and imaginary parts of the
amplitude estimates in Figures 3.3(c) and 3.3(d), respectively.
It is known that, as M increases, all of the estimators under study achieve
better spectral resolution and that the best resolution is obtained at M = N/2 [52].
This fact, along with the statistical results shown in the previous examples, indicates
that the choice of M for Capon should be made by a tradeoff between resolution
and statistical stability. Usually we choose N/4 < M < N/2. While the choice
of M for Capon is difficult to make, it is easy to see that APES achieves the best
performance at M = N/2, since with this choice, APES achieves the highest possible


CHAPTER 3
MAFI APPROACH TO SPECTRAL ESTIMATION
Filterbank approaches decompose the observations {y(n)}^=01 of a stationary
signal y(t) as [51] [47] [73] [72]
y{n) o(u)ejun + ew(n), n = 0,1,..., N 1; wG[0,27r), (3.1)
where a(co) denotes the complex amplitude of the sinusoidal signal with frequency
u> and ew(n) denotes the noise (or residual) term at frequency u>, assumed to be
zero-mean. The problem of interest is to estimate a(u) for any given u>.
Briefly stated, most filterbank spectral approaches address the aforementioned
problem by following two main steps: (a) pass the data {y(n)} through a bandpass
filter with varying center frequency u; and (b) obtain the estimates, (w), for u G
[0, 27t), of the complex amplitude from the filtered data. The bandpass filter used is
usually an M-tap FIR filter with its coefficient vector given by
r -\T
hw =
h\ h 2
h
M
(3.2)
where (-)T denotes the transpose. (The choice of M is discussed in Section 3.5.)
Observe that the notation emphasizes the dependence of the vector in (3.2) on the
center frequency lo. Although rules for choosing h^, vary, a rather general one for the
choice of a matched filter is discussed in Section 3.2.
3.1 Forward and Backward Data Vectors
Let
y (0 =
y(0 y( + !)
y(l + M -1)
Z = 0,1,..., L 1,
(3.3)
16


25
and
ug(uj)v2(u)
fbaM
II^iHII2 \ K(w)^H 1/f (w)i/3(w)] S ^w)
(3.53)
where
(3.54)
Hence computationally APES, especially FOA, is only slightly more involved than
Capon. (Also see Section 3.5 for the simulation results.) More specifically, the
amount of computations required by Capon or APES is dominated by calculating
R 1//2 and the matrix-vector products in (3.44)-(3.45) or (3.49)-(3.51). We mention
that conventional Capon or APES implementation makes use of ((3.46), (3.47)),
((3.52), or (3.53)), which requires calculating the matrix-vector products in (3.44)-
(3.45) or (3.49)-(3.51) for each w of interest, thereby becoming computationally more
and more intensive as the number of frequency samples increases. This is especially
so in 2-D applications such as when forming SAR images. It is thus of great interest
if other efficient implementation schemes for the MAFI approaches can be found.
3.3.2 Statistical Performance
All estimators under study, i.e., FOC, FOA, FBC, and FBA, can be shown to
have the same asymptotic variance under the following condition:
C: The signal y(n) can be written as in (3.1), where ew(n) is a zero-mean stationary
random process with finite spectral density at u:
4>e{u) < OO.
(3.55)
In more exact terms, the following result holds true.


99
ARMA1 (N=256, M=33,100 realizations.) ARMA2 (N=256, M=33,100 realizations.)
(a) (b)
(c)
(d)
Figure 6.6: The AR coefficient estimation of the ARMA signals via the overdeter
mined modified Yule-Walker method with N = 256 and M = 32. The curves are
the summations of the mean-squared errors (MSE) of all the AR coefficient estimates
versus the numbers of included equations, which have been set as 4, 8, 16, 32, 64 and
128, respectively. The MSE curves are based on 100 independent realizations, (a)
ARMA1; (b) ARMA2; (c) ARM A3; (d) ARMA4.


90
by Theorem 6.4.1 we have ¡is = ¡i*_s. Then
M1
Mf1^) = Y, ^eJSUJ
s=(M1)
M-1
0
E + E ft)
M1
s=0 s=(M1)
M-l Af-1
E ^ /_se-^* /io
,-JSiOfi
s=0
^(W) + 01 (k>fi) A0,
(6.18)
where
1V-1
(6.19)
s=0
which, again, can be evaluated by using FFT with zero-padding, i.e., = 0, for s =
M + 1,..., 1.
6.4.3 Computational Aspects
We briefly discuss the computational aspects of the standard method and the
Capon method for covariance sequence estimation. Recall that K, M, and N denote
the largest lag of the covariance estimates, the length of the Capon filter, and the
number of data samples, respectively. Assume that the data is real, N K, N M,
and K M. Then the standard method involves approximately 2KN flops.
The number of flops needed by the exact Capon method is difficult to deter
mine. For a moderate M, say N/6 < M < N/3, we find that the computationally
most demanding part of the exact Capon method is the spectral factorization. Since
the algorithm (Newton-Raphson) used to find the minimum-phase spectral factor is
iterative, the number of flops needed is hard to quantify. Another significant compu
tational demand is the calculation of the sample covariance matrix R, which involves
approximately 2NM2 + 4M3 flops. However, we note that the amount of computa
tions needed by the factorization is much larger than that needed for computing the
sample covariance matrix for a moderate or large M.


71
For sufficiently large N (so that the transient response in the output can be neglected),
the cost function CoEM(a)b) is approximately equivalent to
N-l
Cd a,b) = £
n=0
K
x(n) a,b)eJI
fc=i
(5.77)
The method that we propose for estimating a and b is based on (5.77) and consists
of two steps:
First estimate {ak}kz=1 in an unstructured/non-parametric form.
Then fit {(a, b)}^ to the amplitude estimates obtained in the previous step
by taking into account the statistical variance of the latter.
In what follows, we detail the above two steps.
Step 1: Use an appropriate amplitude estimator to obtain estimates {k}k=1
of {oik}k=i from the measurements {^(n)}^1. APESl may be recommended in
this case because we have control over the probing signal and we have no reason to
choose any of the sinusoids too close to one another. The large-sample variance of
the estimated amplitudes {&k}%=l is proportional to {(f)[oJk)}k=i (see Section 5.2). To
obtain estimates of {(¡>{^k)}k=v we can first calculate
K
v(n) x(n) y] ke3Lkn, n = 0,1,..., N 1, (5.78)
k=1
and then utilize either a parametric or a non-parametric PSD estimator [75] [41] on
(5.78) to obtain {(uk)}k=1. Specifically, in the examples given in Section 5.6.2 we
use the Capon PSD estimator [16] [44] [75] (also see Section 6.3), which determines
{ 4>{uk) =
M
k = l,2,...,K,
(5.79)
where a(uk) is defined in (5.40) and Rvv is the sample covariance matrix of the
estimated noise vectors
v(0 =
v(l) v(l + 1)
v(l + M 1)
1 = 0,1,...,L-1, (5.80)


57
From (5.41), we can estimate f3k one at a time via LS as
1 L_1
ft = 7Ey^e_Wi^g^ k = 1,2,..., K. (5.42)
L 1=0
(Note that we could estimate all {/3k}k=i simultaneously via LS, which however
appears to perform even worse than using (5.42), especially for small N.) The use of
(5.42) in (5.32) and (5.39) leads to the following estimate of Q
K
Q = R- (5-43)
k=1
The WLSE(L, 0, K) that uses (5.29) with (5.43) does not require any initial estimate
of {ak}£=v It is an extension of the APES algorithm in [52] to multiple sinusoids
with known frequencies.
Remark: We note that e(k) and e(Z) in (5.27) are correlated (for k ^ l),
which implies that (5.29) is suboptimal (as it takes into account only the correlation
between the elements of e(Z), but ignores the correlation between e(l) and e(fc), for
k ^ l). Yet, the WLS methods are likely to outperform the LS methods because the
latter completely ignore the correlation in v(n).
5.3.2 WLSEL, 0.11
The particularization of WLSE(L, 0,K) to WLSE(L,0,1) is straightforward.
Specifically, the WLSE(L, 0,1) that corresponds to using (5.29) with (5.34) can be
readily verified (by using the matrix inversion lemma) to be
a7i(wfc)R_1g(u;fc)
£*Jfc
A; = 1,2,..., AT,
(5.44)
af7(u;fc)R-1a(a;fc)
whereas the WLSE(L, 0,1) that corresponds to using (5.29) with (5.43) is given by
a H{u>k)
R g(wfc)g/i(wA;)
-l
g(wjt)
a"K)
R g{iok)gH(cok)
-l
a(wfc)
k = 1,2,..., K.
(5.45)
Note that (5.44), like (5.38), does not depend on P. However, unlike (5.38), the
equation (5.44) is exactly equivalent to using (5.29) with (5.34). Equations (5.44) and


17
be the overlapping vectors constructed from the data {y(n)}, where L = N M+l. In
what follows y(Z) is referred to as the forward data vector. Let ew(Z), l = 0,1,..., L1,
be formed from {ew(n)} in the same manner as y(Z) are from {y(n)}. Then the
forward vectors can be written as
y(0 = WuNMje1 +e(0,
(3.4)
where is the steering vector and is given by
a m(^)
1 T
1 eJU
(3.5)
Likewise, the backward data vectors are constructed as
m
y*(N-l- 1) y*(N-l- 2) ... y*{N-l-M)
Z = 0,1,..., L 1,
(3.6)
where ()* denotes the complex conjugate. Let aJ(Z), Z = 0,1,..., L 1, be formed
from {ew(n)} the same way as y(Z) from {y(n)}. Then the backward vector can be
written as
y(Z) = [tt(w)aM(w)]e,'wi + ea,(Z), (3.7)
where
(a;) = a*(a;)e_^JV1^. (3.8)
It is straightforward to verify that the forward and backward vectors are related by
the following complex conjugate symmetry property:
y(z) = Jy*(T z l),
(3.9)
where J denotes the exchange matrix whose anti-diagonal elements are ones and all
the others are zero.
Suppose that the initial phase of the sinusoidal signal in (3.1) is a random
variable uniformly distributed over the interval [0, 2tv) and independent of the noise


10
such as periodogram by Schuster [67] and its various variations including the famous
Blackman-Tuckeys method [5], can also be cast in the framework of filterbank ap
proaches, originally they were not designed in such a manner. In other words, those
methods made no attempt to purposely design a good bandpass filter to achieve some
desired characteristics.
A notable example of filterbank approaches to spectral estimation is the RE-
FIL (REfined FILter) method which was first introduced in [77] and was further
developed in [56] (also discussed in [8] [59] [61]). The REFIL method is close in spirit
to the Daniell approach [22] to reducing the variance of the periodogram. That is,
REFIL does not split the available samples in shorter stretches. The REFIL idea is
to design a bank of filters which pass the signal components within the passbands
as much as possible relative the total power and, in the mean time, attenuate the
frequencies outside the corresponding passbands. It turns out that REFIL in general
gives better spectral estimates than the traditional Fourier-based methods. Neverthe
less, the REFIL filters share a common characteristic with the Fourier-based ones:
they are all data-independent in the sense that they do not adapt to the received
data in any way. Presumably, it would be beneficial to take the data properties into
account when designing filterbanks.
The Capon method is one (and perhaps the most famous one) of such data-
dependent filterbank spectral estimators. The Capon estimator was first named as
the Maximum Likelihood Method (MLM) due to the the ML and Gaussian process
context used in Capons original work [16]. It turns out that MLM is a misnomer
since it is not an ML spectral estimator and it does not possess any of the properties
of an ML estimator. In [41] and [55], Capons method was referred to as the Minimum
Variance Spectral Estimator (MVSE) because it is derived by minimizing the variance
of the output of a narrowband filter. Even the name MVSE is inaccurate in the sense


3.5Empirical bias and variance of the 2-D MAFI estimators as the filter
length, M = Mi = M2, varies when Ni = TV2 = 32, and SNRx = 20
dB. (a) Real part of the bias; (b) Imaginary part of the bias; (c)
Variance of the real part of the estimated amplitude; (d) Variance of
the imaginary part of the estimated amplitude 38
4.1 SAR images obtained from the ERIM data by using (a) 2-D FFT, (b)
2-D windowed FFT, (c) 2-D Capon, and (d) 2-D APES 46
5.1 PSD of the test data that consist of three sinusoids and an AR(1) noise
process 63
5.2 Empirical MSEs and the CRB versus local SNR when TV = 32, M = 8,
and the observation noise is colored (an AR(1) process), (a) a3. (b) ax. 65
5.3 Empirical MSEs and the CRB versus local SNR when TV = 32, M = 8,
and the observation noise is white, (a) a3. (b) ax 67
5.4 Empirical MSEs and the CRB versus M when TV = 32 and the obser
vation noise is colored (an AR(1) process with o2 = 0.001). (a) o;3.
(b) ax 69
5.5 Averaged RMSEs and the number of flops versus TV for the first system
when the observation noise is white (o2 = 0.01) and M 20 for APES1
and MAFI1. (a) RMSE of a-parameters. (b) RMSE of 6-parameters.
(c) Number of flops 76
5.6 PSD estimate of the output of the first system corrupted by white
noise with a2 = 0.01 and TV = 200 77
5.7 Averaged RMSEs and the number of flops versus TV for the second
system when the observation noise is colored (an AR(1) process with
a2 = 0.01) and M = 20 for APESl and MAFI1. (a) RMSE of a-
parameters. (b) RMSE of 6-parameters, (c) Number of flops 78
ix


18
term. By making use of this assumption as well as (3.9), the covariance matrix of
y(l) or, equivalently, of y(l), is given by
R = E {y(l)yH(l)} = E {y(Z)yH(0} = lM|2 aM(w)a^(w) + Q(w), (3.10)
where (-)H denotes the conjugate transpose and Q(cu) is the noise covariance matrix
and is given by
QM = B{M0"(0} = B{§(0§5(0} (3.11)
Note that both R and Q are Hermitian Toeplitz matrices.
The Forward-Only (FO) interbank approaches use only the forward data vec
tor to estimate R, which is the forward sample covariance matrix:
1 L-1
R-fo = y y(0yH(0- (3-12)
^ 1=0
The Forward-Backward (FB) filterbank approaches use the average of the forward
and backward sample covariance matrices to obtain the estimate of R:
R-fb = ^(Rfo + Rbo)i (3-13)
where R-bo denotes the backward sample covariance matrix:
1 L_1
RBO = 7E3'Wy"(,)=jftFOJ- (3-!4)
^ 1=0
The Rfb in (3.13) is Hermitian but no longer Toeplitz. By using (3.9), one can show
that Rfb is a persymmetric matrix [30], i.e.,
RFb = JRfbj- (3.15)
Since R is persymmetric, one would expect that R-fb is a better estimate of R than
the non-persymmetric Rpo-


2
Filterbank approaches to spectral estimation belong to the class of nonpara-
metric spectral estimators. An important member of filterbank approaches is the
Capon spectral estimator [16] [18]. Unlike the classical Fourier-based methods which
are data-independent, the Capon spectral estimator adapts to the processed data in
a manner so that the noise components of the data are rejected substantially. During
the past few decades, the Capon spectral estimator has been widely used because
its higher resolution and lower sidelobes give it an advantage over the Fourier-based
methods. Additionally, its robustness and less variability make it preferable to the
parametric methods [75] [41].
In a recent study, it was empirically observed that using the Capon estimator
for complex spectral estimation gives biased spectral estimates whereas the newly
introduced APES (Amplitude and Phase Estimation) method appears to be unbiased
[52] (see Figures 1.1(a) to 1.1(d) for performance differences of using the Fourier-based
methods, Capon, and APES for spectral estimation). The fact that both Capon
and APES make use of a set of finite impulse response (FIR) matched filters was
observed in [70]. A number of results on the performance differences of the Capon
and APES estimators were also reported therein. However, the study in [70] was
somewhat limited since it only considered the forward-only (FO) Capon and APES
estimators. Owing to the general belief that forward-backward (FB) approaches to
spectral estimation usually provide more accurate results and are used more often
than their FO counterparts, it is of interest to conduct a study that investigates
how the FB Capon and APES spectral estimators perform when compared with one
another as well as with their FO counterparts.
In some sense, filterbank approaches reduce the problem of spectral estimation
to a filter design problem subject to some constraints [46]. In Section 3.2, we discuss a
general rule for choosing the impulse responses of the filterbank so that the signal-to-
noise ratios (SNRs) at the outputs of the filterbank are maximized. It follows that the


91
Flop Ratio £
Exact Capon
Approximate Capon
M = K = 32
281
47
M = K = 50
655
82
II
II
a*
1020
115
Table 6.1: Comparison of the computational burdens of the standard and Capon
methods with N = 256 and = 512.
The approximate Capon method no longer needs to perform the spectral fac
torization, and the computational demand is greatly reduced. The approximate
Capon method requires approximately 2NM2 + 6M3 flops, a large part of which
comes from the computation of the sample covariance matrix R.
To illustrate quantitatively the computational burdens of the above three
methods, we show a few simulation results in Table 6.1, where we define £ as the
ratio of the flops needed by the exact or approximate Capon method to that corre
sponding to the standard method. We remark that we did not pay special attention
to the coding of the algorithms, hence the numbers provided here should be only
taken as indicative of the computational complexities of the methods.
6.5 Numerical Results
In this section, we present numerical examples showing the performance of the
Capon method for covariance estimation. The first problem addressed is the ARMA
covariance estimation. Several ARMA signals with different pole and zero locations
are generated to compare the performances of the standard and the Capon methods.
We also consider AR coefficient estimation of the ARMA signals by using the Capon
and standard covariance estimates. Finally, we give an example illustrating the MA
model determination with the standard and Capon methods.


11
that Capon spectral estimates do not possess the minimum variance property. In
this study it is simply named the Capon spectral estimator.
The Capon spectral estimator was originally introduced by Capon [16] [18]
for applications in multidimensional seismic array frequency-wavenumber analysis.
It is reformulated by Lacoss [44] for applications to 1-D time-series problems. The
constraints adopted by the Capon method are such that the signal at the current
frequency is passed undistorted (with unit gain) while the output power of the over
all frequency domain is minimized. A number of researches have been conducted
to study the Capon spectral estimator. Specifically, Capon and Goodman demon
strated [19] [17] that the Capon spectrum has a mean and a variance that behaves
like the averaged periodogram; that is, Capon spectra are usually less variable than
periodograms. Additionally, Lacosss empirical study [44] suggested that the resolu
tion of the Capon method is between that of Burgs AutoRegressive (AR) spectral
estimator [9] [12] and that of the periodogram. Lacosss study also suggested that
the statistical variability of Capon is less than that of the AR estimator. Burg later
proved that the reciprocal of the Capon spectrum of order M, the length of the
Capon filter, is equal to the average of the reciprocals of the AR spectra from order
0 to M [11]. Such an averaging effect explains the empirical observations made by
Lacoss. Further researches in quantizing the resolution properties of Capon spectra
were reported in [21] [58].
Since Capons method is a basic spectral estimator, it may be used in any ap
plication where the spectrum of the studied signal plays an important role. Indeed,
ever since its first appearance, the Capon spectral estimator has been widely used
in many areas including radar, sonar, communications, imaging, geophysical explo
ration, astrophysical exploration, and biomedical engineering (see [35] [79] and the
references therein). Rather than making a futile attempt at making a full document


112
-a*(a"R-1aM)(a"R-ia)2 *(a^R-1aM)(a"R-1)(a"R"^)
-a(a^R_1aM)2(//R-15) (a^R1aM)2(WR15)|
= l-a{a.HM-R-1 *m)E {|a* R1^2 (a^R-1aM)(R~15)} (C.7)
By the Cauchy-Schwartz inequality, the quantity between the curly parentheses in
(C.7) is negative and so is its expectation.
The bias for the FOC spectral estimate can be obtained by replacing the FB
sample covariance Rpp in (C.1)-(C.3) by the FO sample covariance matrix Rpo> as
defined in (3.12), and by following a similar treatment we did in (C.7). The result is
as follows (to a second-order approximation):
(a^R-1aM)2£ {FOC a}
~ {la^R-^f (a&R-VK^R"1*)} (C.8)
Hence to within a second-order approximation, the bias of the FBC is one half that
of the FOC, and (3.57) follows.
Proof of (3.58): Again, we consider FBA first. Observe that QfbA Q
0(1/VZ). Then, by (3.34) and (B.13), we have (see (C.2))
FBA =
smQfba^
aMQFBAaM
a^Q-1(QFBA-Q)Q^+ a^Q-M
amQ a^Q JaM
(afcQ-^K CTHQfba Q)Q~laM
(a^Q'^Ai)2
(C.9)
Therefore, we obtain
(awQ 1*m?E {AFba a}
E {-(a^Q_1QFBAQ_15)(a^Q_1aM) + (a^Q1QFBAQ^1aM)(a^Q15)| .
(C.10)