A Physically Constrained Maximum Likelihood Method for
Snapshot Deficient Adaptive Array Processing
by
Andrea L. Kraay
B.S., Electrical Engineering
George Mason University, 1999
Submitted to the Department of Electrical Engineering and Computer Science in
partial fulfillment of the requirements for the dual degrees of
Master of Science in Electrical Engineering and Electrical Engineer
at the
BARKER
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
and the
WOODS HOLE OCEANOGRAPHIC INSTITUTION
@Massachusetts
February 2003
Institute of Technology. All rights reserved.
LIBRARIES
Author .....................................................
ng
Joint Program in Electrical Engineering/Applied Ocean Physics and Engi
Massachusetts Institute of Technology/Woods Hole Oceanographic Irs 'ution
February 15, 2003
C ertified by .....................................
ArthurNl3aggeroer
Ford Professor of Engineering, Secretary of the Navy/Chief of Naval Operations
Chair for Ocean Sciences, Dept. of Ocean Engineering, Thesis Supervisor
Accepted by ...................
Arthur C. Smith
Chairman, Committee on Graduate Students
Department of Electrical Engineering and Computer Science
...............................
Mark A. Grosenbaugh
Chairman, Joint Committee for Applied Ocean Physics and Engineering
Massachusetts Institute of Technology/Woods Hole Oceanographic Institution
Accepted by ...............................
2
A Physically Constrained Maximum Likelihood Method for
Snapshot Deficient Adaptive Array Processing
by
Andrea L. Kraay
B.S., Electrical Engineering
George Mason University, 1999
Submitted to the Department of Electrical Engineering and Computer Science on
February 15, 2003, in partial fulfillment of the requirements for the degrees of
Master of Science in Electrical Engineering and Computer Science and Electrical
Engineer and Master of Engineering in Electrical and Ocean Engineering
Abstract
This thesis presents a Physically Constrained Maximum Likelihood (PCML) method
for spatial covariance matrix estimation as a reduced-rank adaptive array processing
algorithm. The physical constraints of propagating energy imposed by the wave equation and the statistical nature of the snapshots are exploited to estimate the "true"
maximum-likelihood (full-rank and physically realizable) covariance matrix. The resultant matrix may then be used in adaptive processing for interference cancellation
and improved power estimation in non-stationary environments.
Power estimates for a given environment are computed using a variety of reducedrank methods for different levels of snapshot support. The PCML method is shown
to have less bias and a lower standard deviation at a given number of snapshots than
any of the other reduced-rank adaptive processing methods used. This is of particular importance in the low snapshot regime where reduced rank adaptive processing
methods are used to cope with non-stationary environments of moving ships with
high bearing rates.
Thesis Supervisor: Arthur B. Baggeroer
Title: Ford Professor of Engineering
3
4
Acknowledgments
This thesis is dedicated to the memory of my grandmother Margaret Harris Berning,
and to my mother and my two sisters. Thank you for all your love and support.
I'd also like to give many thanks to my advisor, Prof. Art Baggeroer. His teaching
and guidance were invaluable to me in both the production of this thesis, and in my
academic growth.
All work was completed under SBCNAV/CNO Chair and ONR/MIT Scholar of
Oceanographic Sciences/N00014-99-1-0087 through the Office of Naval Research.
5
6
Contents
1
Introduction
19
2
Problem Description
21
2.1
Frequency Wavenumber Spectra .....................
21
2.2
Frequency Wavenumber Power Estimation using Antenna Arrays . . .
23
2.3
The Sample Covariance Matrix
. . . . . . . . . . . . . . . . . . . . .
27
2.4
The Snapshot Problem . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3
4
5
Common Reduced-Rank ABF Techniques
33
3.1
Diagonal Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.2
Dominant Mode Rejection . . . . . . . . . . . . . . . . . . . . . . . .
34
3.3
Eigenvalue Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.4
Beamspace Processing
35
. . . . . . . . . . . . . . . . . . . . . . . . . .
Estimation of Covariance Matrices with A Priori Constraints
37
4.1
Statistical Data M odel . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.2
Physical Constraints on R . . . . . . . . . . . . . . . . . . . . . . . .
38
The Physically Constrained Maximum Likelihood (PCML) Algorithm
41
5.1
41
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
5.2
5.3
5.4
6
Algorithm Initialization . . . . . . . . . . . . . . . . . .
43
5.2.1
Covariance Matrix Initialization . . . . . . . . .
43
5.2.2
Power Spectrum Initialization . . . . . . . . . .
43
5.2.3
White Noise Initialization
. . . . . . . . . . . .
45
Iterative Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5.3.1
Covariance Matrix Update . . . . . . . . . . . . . . . . . . . .
45
5.3.2
Power Spectrum Update . . . . . . . . . . . . . . . . . . . . .
47
5.3.3
White Noise Update . . . . . . . . . . . . . . . . . . . . . . .
48
O utputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.4.1
Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . .
49
5.4.2
PCML-MVDR Power Estimate
49
. . . . . . . . . . . . . . . . .
Results
51
6.1
Test Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
6.1.1
A rray
. . . . . . . . . . . . . . . . . . . . . . .
51
6.1.2
Signal Environment . . . . . . . . . . . . . . . .
51
6.1.3
Sample Covariance Matrix Generation
. . . . .
55
6.1.4
Discrete Wavenumber Spacing . . . . . . . . . .
56
6.1.5
Covariance Matrix Taper . . . . . . . . . . . . .
57
6.2
Power Estimates vs. Snapshots
. . . . . . . . . . . . .
200, MVDR Power Estimates and Standard Deviations
58
6.2.1
L
6.2.2
L = 200, PCML Algorithm Performance
6.2.3
L
6.2.4
L = 150, PCML Algorithm Performance
6.2.5
L
6.2.6
L = 100, PCML Algorithm Performance
. . . . . . . . . . .
75
6.2.7
L = 50, MVDR Power Estimates and Standard Deviations .
78
=
=
=
. . . .
150, MVDR Power Estimates and Standard Deviations
. . . .
100, MVDR Power Estimates and Standard Deviations
8
60
63
66
69
72
6.2.8
L = 50, PCML Algorithm Performance . . . . . . . . . . . . .
81
6.2.9
L = 30, MVDR Power Estimates and Standard Deviations . .
84
6.2.10 L = 30, PCML Algorithm Performance . . . . . . . . . . . . .
87
6.2.11 L = 10, MVDR Power Estimates and Standard Deviations . .
90
6.2.12 L
10, PCML Algorithm Performance . . . . . . . . . . . . .
93
6.2.13 L = 5, MVDR Power Estimates and Standard Deviations . . .
96
6.2.14 L
99
=
=
5, PCML Algorithm Performance
. . . . . . . . . . . . .
6.2.15 L = 2, MVDR Power Estimates and Standard Deviations . . . 102
6.2.16 L
7
=
2, PCML Algorithm Performance
. . . . . . . . . . . . . 105
6.3
Noise Estimate at k, = k
= 0 vs. Snapshots . . . . . . . . . . . . . . 108
6.4
Low-Level Source Detection . . . . . . . . . . . . . . . . . . . . . . . 109
6.5
Variants of PCML Implementation
. . . . . . . . . . . . . . . . . . .111
6.5.1
Wavenumber Grid Spacing . . . . . . . . . . . . . . . . . . . .111
6.5.2
Covariance Matrix Taper . . . . . . . . . . . . . . . . . . . . .
114
6.5.3
Spreading of Discrete Sources . . . . . . . . . . . . . . . . . .
117
6.5.4
Constant Estimate of White Noise
119
. . . . . . . . . . . . . . .
Conclusions
127
7.1
The PCML Method as a Reduced Rank Adaptive Processor
. . . . .
127
7.2
Effects of Various Implementation Choices . . . . . . . . . . . . . . .
128
A Gradient Derivations
A.1
131
Gradient of Likelihood Function With Respect to Directional Power
Spectrum
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2 Gradients of Likelihood Function With Respect to Sensor Noise . . . 133
A .2.1
First Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.2.2
Second Gradient
. . . . . . . . . . . . . . . . . . . . . . . . . 134
9
B 2-D Windows for Integral Approximation
135
B .1 U niform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B .2 H anning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
B.3 Triangular ........................................
10
137
List of Figures
2-1
Snapshot Generation Process
. . . . . . . . . . . . . . . . . .
24
2-2
Array Processing Environment . . . . . . . . . . . . . . . . . .
25
2-3
Eigenvalue as a Function of Source Motion: oc
5-1
General PCML Algorithm Schematic . . . . . . . . . . . . . . . . . .
42
5-2
Sample Discrete Wavenumber Grids . . . . . . . . . . . . . . . . . . .
44
5-3
Sample Multiplicative Update Scale . . . . . . . . . . . . . . . . . . .
48
6-1
50 Element Circular Array, R = 5A . . . . . . . . . . . . . . . . . . .
52
6-2
Frequency Wavenumber Response of Array . . . . . . . . . . . . . . .
52
6-3
Ensemble MVDR Power Estimate . . . . . . . . . . . . . . . . . . . .
59
6-4
CBF, L
=
(Ttl 1),
7 1
200: (a) Mean Output Power (b) Standard Deviation of
O utput Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-5
DL, L
=
61
200: (a) Mean MVDR Power (b) Standard Deviation of
M V D R Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-7
60
PCML, L = 200: (a) Mean MVDR Power (b) Standard Deviation of
M V D R Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-6
30
61
DMR, L = 200: (a) Mean MVDR Power (b) Standard Deviation of
M V D R Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
62
6-8
EVF, L = 200: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-9
L
=
62
200: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6-10 L = 200: (a) Mean PCML White Noise Estimate (b) Standard Deviation of PCML White Noise Estimate
6-11 L
=
. . . . . . . . . . . . . . . . . .
64
200: (a) Mean PCML Likelihood Convergence (b) Standard De-
viation of PCML Likelihood Convergence . . . . . . . . . . . . . . . .
64
6-12 L = 200: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence . . . . . . . . . . . . . . . .
65
6-13 CBF, L = 150: (a) Mean Output Power (b) Standard Deviation of
O utput Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-14 PCML, L
=
66
150: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6-15 DL, L = 150: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-16 DMR, L
=
150: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-17 EVF, L
=
67
68
150: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6-18 L = 150: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCM L PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6-19 L = 150: (a) Mean PCML White Noise Estimate (b) Standard Deviation of PCML White Noise Estimate
. . . . . . . . . . . . . . . . . .
70
6-20 L = 150: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence . . . . . . . . . . . . . . . .
12
70
6-21 L = 150: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence . . . . . . . . . . . . . . . .
71
6-22 CBF, L = 100: (a) Mean Output Power (b) Standard Deviation of
O utput Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6-23 PCML, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
6-24 DL, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
6-25 DMR, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6-26 EVF, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6-27 L = 100: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-28 L
=
75
100: (a) Mean PCML White Noise Estimate (b) Standard Devia-
tion of PCML White Noise Estimate . . . . . . . . . . . . . . . . . .
76
6-29 L = 100: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence . . . . . . . . . . . . . . . .
76
6-30 L = 100: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence . . . . . . . . . . . . . . . .
6-31 CBF, L
=
77
50: (a) Mean Output Power (b) Standard Deviation of
O utput Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6-32 PCML, L = 50: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6-33 DL, L = 50: (a) Mean MVDR Power (b) Standard Deviation of MVDR
P ow er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
79
6-34 DMR, L = 50: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power .......
...............................
80
6-35 EVF, L = 50: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6-36 L = 50: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCM L PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6-37 L = 50: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate . . . . . . . . . . . . . . . . . . . . .
6-38 L
=
82
50: (a) Mean PCML Likelihood Convergence (b) Standard Devi-
ation of PCML Likelihood Convergence . . . . . . . . . . . . . . . . .
82
6-39 L = 50: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence . . . . . . . . . . . . . . . . .
83
6-40 CBF, L = 30: (a) Mean Output Power (b) Standard Deviation of
O utput Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-41 PCML, L
=
30: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-42 DL, L
P ower
=
84
85
30: (a) Mean MVDR Power (b) Standard Deviation of MVDR
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
6-43 DMR, L = 30: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6-44 EVF, L = 30: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-45 L
=
86
30: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6-46 L = 30: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate . . . . . . . . . . . . . . . . . . . . .
14
88
6-47 L
=
30: (a) Mean PCML Likelihood Convergence (b) Standard Devi-
ation of PCML Likelihood Convergence . . . . . . . . . . . . . . . . .
6-48 L
=
88
30: (a) Mean PCML Eigenvalue Convergence (b) Standard Devi-
ation of PCML Eigenvalue Convergence . . . . . . . . . . . . . . . . .
89
6-49 CBF, L = 10: (a) Mean Output Power (b) Standard Deviation of
O utput Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
6-50 PCML, L = 10: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
6-51 DL, L = 10: (a) Mean MVDR Power (b) Standard Deviation of MVDR
P ower
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-52 DMR, L
=
91
10: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
6-53 EVF, L = 10: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-54 L
=
92
10: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCM L PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6-55 L = 10: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate . . . . . . . . . . . . . . . . . . . . .
94
6-56 L = 10: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence . . . . . . . . . . . . . . . . .
94
6-57 L = 10: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence . . . . . . . . . . . . . . . . .
95
6-58 CBF, L = 5: (a) Mean Output Power (b) Standard Deviation of Output
Pow er
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
6-59 PCML, L = 5: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
97
6-60 DL, L = 5: (a) Mean MVDR Power (b) Standard Deviation of MVDR
P ow er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6-61 DMR, L = 5: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power .......
6-62 EVF, L
=
...............................
98
5: (a) Mean MVDR Power (b) Standard Deviation of MVDR
P ow er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
6-63 L = 5: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate .................................
99
6-64 L = 5: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate . . . . . . . . . . . . . . . . . . . . .
100
6-65 L = 5: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence
. . . . . . . . . . . . . . . . . 100
6-66 L = 5: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
6-67 CBF, L
=
. . . . . . . . . . . . . . . . . 101
2: (a) Mean Output Power (b) Standard Deviation of Output
Pow er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6-68 PCML, L = 2: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6-69 DL, L = 2: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Pow er
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6-70 DMR, L = 2: (a) Mean MVDR Power (b) Standard Deviation of
M VDR Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6-71 EVF, L = 2: (a) Mean MVDR Power (b) Standard Deviation of MVDR
P ow er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6-72 L
=
2: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCM L PSD Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
16
6-73 L = 2: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate . . . . . . . . . . . . . . . . . . . . . 106
6-74 L = 2: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence
. . . . . . . . . . . . . . . . . 106
6-75 L = 2: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
6-76 (a) Mean Power Estimate at u, =
. . . . . . . . . . . . . . . . .
107
y = 0 (b) Standard Deviation of
. . . . . . . . . . . . .
. . . . . . 108
6-77 Power Near Low Level Source r.e. Local Mean . . . . .
109
Power Estimate at u, = uy = 0
6-78 L = 200, Au = 0.2: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-79 L
=
30, Au
=
111
. . . . . . . . . . .
0.2: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
. . . . . . . . . . .
112
6-80 L = 200, Au = 0.05: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-81 L = 30, Au
=
. . . . . . . . . . .
112
0.05: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
. . . . . . . . . . .
113
6-82 L = 200, Hanning Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
. . . . . . . . . . . . . . . . . . . 114
6-83 L = 30, Hanning Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-84 L
=
200, Triangle Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-85 L
=
. . . . . . . . . . . . . . . . . . . 115
. . . . . . . . . . . . . . . . . . . 115
30, Triangle Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
. . . . . . . . . . . . . . . . . . . 116
6-86 L = 200, Source Spread 0,, = Oy = (27r)
1
.g
1 bo: (a) Mean PCML PSD
Estimate (b) Standard Deviation of PCML PSD Estimate
17
. . . . . .
117
6-87 L
=
30, Source Spread 0_, = O
=
(27) V.1,: (a) Mean PCML PSD
Estimate (b) Standard Deviation of PCML PSD Estimate
6-88 L = 200, dPCML
=
-2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-89 L
=
200, &PCML
. . . . . . 118
=
. . . . . . . . . . . . . . . . . . . 119
-2 dB: (a) Mean PCML Likelihood Convergence
(b) Standard Deviation of PCML Likelihood Convergence . . . . . . . 120
6-90 L
=
30, &PCML = -2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
. . . . . . . . . . . . . . . . . . . 120
6-91 L = 30, &PCML = -2 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence . . . . . . . . . 121
6-92 L = 200, &PCML
=
0 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-93 L = 200,
PCML =
. . . . . . . . . . . . . . . . . . . 121
0 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence . . . . . . . . . 122
6-94 L
=
30,
=-CML
0 dB:
(a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-95 L
=
. . . . . . . . . . . . . . . . . . . 122
30, dPCML = 0 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence . . . . . . . . . 123
6-96 L = 200,
&PCML =
2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-97 L
=
200,
=-CML
2 dB:
. . . . . . . . . . . . . . . . . . .
123
(a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence . . . . . . . . . 124
6-98 L
=
30, dPCML = 2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
6-99 L
=
30,
=-CML
2 dB:
. . . . . . . . . . . . . . . . . . . 124
(a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence . . . . . . . . . 125
18
Chapter 1
Introduction
Adaptive array processing with large aperture arrays is used in many fields where
high spatial resolution and sidelobe control are needed. To cope with non-stationary
environments of moving ships with high bearing rates, adaptive processing methods
require modification.
In non-stationary environments, the number of data snapshots available is often
insufficient to produce a well-conditioned sample covariance matrix needed for interference cancellation. The number of data snapshots available for forming a sample
covariance matrix is limited by the amount of time a discrete source is in a resolution
cell of the array. Methods currently used to compensate for snapshot-deficiency (often called reduced-rank methods) can add a significant amount of bias to the system,
thus raising the minimum level at which a target may be detected [2].
A Physically Constrained Maximum Likelihood (PCML) method for spatial covariance matrix estimation will be explored as a potential reduced-rank adaptive array
processing algorithm. The physical constraints of propagating energy imposed by the
wave equation and the statistical nature of the snapshots are exploited to estimate
the "true" maximum-likelihood (full-rank and physically realizable) covariance ma19
trix. The resultant matrix may then be used in adaptive processing for interference
cancellation and improved power estimation in non-stationary environments.
The goal of this research is to implement the PCML technique and to compare
its performance with current methods used to compensate for snapshot deficiency.
Performance will be assessed in terms of power estimate bias and standard deviation,
and ability to detect weak, discrete sources.
Chapter 2 provides basic background information on frequency-wavenumber spectral estimation in non-stationary environments and on array processing. This chapter
also defines the notations used throughout this thesis.
Chapter 3 summarizes several current approaches to the problem of snapshotdeficient power estimation. The output of these algorithms will be used for comparison with that of the PCML method.
Chapter 4 describes the statistical data model and physical constraints which
provide the basis of the PCML algorithm.
Chapter 5 details the implementation of the PCML algorithm used in this thesis.
Chapter 6 presents the performance of the PCML method relative to other methods in a variety of test scenarios, and explores several implementation variants of the
PCML method.
Chapter 7 summarizes the main findings of this thesis.
20
Chapter 2
Problem Description
This chapter contains basic background information on space-time random processes
and array processing, and is intended to motivate the problem of frequency-wavenumber
spectral estimation in non-stationary environments.
The general notations used
throughout this thesis are established here.
2.1
Frequency Wavenumber Spectra
A zero-mean random process is represented in space and time by x(t,z), and its spacetime covariance function is denoted as:
Kx(ti, t2 ,Pi, P 2 ) = E[x(ti, pI)x*(t 2 , P 2 )]
(2.1)
The random process is assumed to be at least wide-sense stationary, and the
spatial medium is assumed to be homogeneous, making the space-time covariance a
function of separation in time and space.
Kx(ti, t2,Pi, P2) = E[x(ti - t 2 , Pi - P2 )x* (0, 0)] = Kx(T, A z)
21
(2.2)
The temporal frequency spatial covariance function is obtained by applying a
Fourier transform along the dimension of the time lag, r:
Rx(w, Ap)
=
Ftime{Kx(T, Ap)} =
f
K,(r, Ap)e-wr dT
(2.3)
For simplicity, Rx(w, Ap) will be referred to as the spatial covariance function
and the dimension of temporal frequency will be assumed.
The frequency-wavenumber power spectrum is obtained by applying a threedimensional Fourier transform to the spatial covariance function along the dimensions
of spatial separations Ax, Ay, and Az, and represents the amount of power arriving
from the direction of wavenumber k 1 and temporal frequency w.
P(w, k)
=
Fspace{Rx(w, Ap)} =
J
Rx(w, Ap)e+jk T AP dAp
(2.4)
where
Ap
=
[Ax Ay Az]T
and k
=
[kx ky k,]T
(2.5)
and k is restricted to a sphere of radius 27r/A where propagating waves exist 2 .
Depending on the spatial dimensions where the spatial covariance function is sampled,
the dimensions of the wavenumber vector (and corresponding Ap vector) and Fourier
Transform may decrease from 3-D to 2-D or 1-D. And, instead of being restricted to
'The wavenumber vector has a magnitude of 27r/A and points in the direction of signal propagation
sin
2r
sin 0 sin#
k = 7
cos
_
[cos
1
where 9 and 4 are the angles of incidence referenced from the origin.
2
The 3-D wave equation in a homogeneous medium dictates that
||k|
= Vk2 + k 2+
22
k2
=
lie on a sphere, the wavenumber vector would then be restricted to lie within a disc
of radius 27r/A, or along a line of length 2(27r/A).
2.2
Frequency Wavenumber Power Estimation using Antenna Arrays
Antenna arrays are used in a wide variety of applications to elicit information contained in a received signal, such as its temporal spectrum and its direction and speed
of propagation. The geometrical arrangement of individual sensors in space allows
filtering algorithms to extract this information simultaneously by exploiting the temporal and spatial characteristics of the data [18].
Each sensor samples the space-time random process x(t,z) at N discrete sensor
positions Po, Pi, P2, ...
, PN--1
These samples are compiled into data snapshot vectors
which may be generated in either the frequency domain or the time domain, depending
on the time bandwidth product of the input data [18]. Frequency domain snapshots
are used in this thesis, and their model for generation is shown in Figure 2-1.
The generation of frequency domain snapshot vectors has two major steps: time
windowing, and Fourier Transforming.
The time series data from each sensor is
assumed to be a zero-mean, bandlimited process centered at we and is first broken up
into disjoint time blocks of length A T indexed in time by I 3
xj(t, po)
XAT
-
xj(t, pi)
1SX (t,
3
(2.6)
pN-1)
Other more elaborate schemes may involve overlapping data and using a non-uniform window,
such as a Hanning taper.
23
Narrowband
Fourier
10=-
Trandorm
at M
Frequenes
eaui)rmbu
Beuutarmw
(46p)
4
x(
whei *Ve)=30(ak)=[AoD,F,)
Nwbmid
Beatormw
.
N(cNpsjf
Figure 2-1: Snapshot Generation Process
where (I - 1) AT < t < lAT.
Each block is then Fourier Transformed at M
frequency bins:
XI(Pm, PO)
X, (WM) =
X1(Wm, P1)
(2.7)
XI(m, PN-1)
where Wm = wc + m-Jj is the center frequency of a frequency bin whose width is
AT. AT should be chosen to achieve an appropriate frequency resolution, and should
also be much larger than the maximum propagation time across the array.
Each X, (wm) may be processed by a narrowband frequency-domain beamformer.
Throughout this thesis, a narrowband beamformer will be assumed and w will be
used to denote the particular frequency being processed, i.e. Wm.
24
-N-I
*
ear
*
0
1
2
0
Figure 2-2: Array Processing Environment
The spatial covariance function is represented at discrete separation in space by
the spatial covariance matrix, R(w).
[w)i
=
R2 (w, Api,5)
=
R2 (w, pi
-
p3 )
(2.8)
Directional power estimation with an antenna array is done by applying a set
of complex weights to the input at each sensor and calculating the mean-squared
beamformer output.
See Figure 2-2.
The complex weights are chosen to pass a
plane wave signal 4' propagating in the direction of the unit vector a8 with temporal
frequency w, and to suppress signals arriving from all other directions.
The output power density of the array processor steered in the direction of k8 is:
P(w, k8 )
=
E[lY(w)I2 ]
= WHRW
(2.9)
where w is the complex weight vector and R is the ensemble spatial covariance matrix.
4 Alternate models for how a signal arrives at the array may incorporate more propagation physics,
such as matched-field processing or near-field processing.
25
The weights, w, may be constant (as in conventional beamforming) or a function of
the incoming data (as in adaptive beamforming). See Van Trees [18] for a detailed
discussion of beamformer weights.
In an environment of loud, discrete interference, adaptive beamforming can offer far superior performance in terms of power estimation. The Minimum Variance
Distortionless Response (MVDR) Beamformer is the most extensively used adaptive
weighting scheme and, with the ensemble covariance matrix, yields the maximum
likelihood power estimate at wavenumber k, [6].
WMvDR
PMVDR(W,
ks)
R-vk(k)
WMvDRRwMVDR
(2.10)
s
-
k
(2.11)
where vk(k) is the (Nxl) array response vector that describes how the array
responds to a unit-amplitude signal input. For the plane-wave propagation model,
[vk(k)],
(2.12)
e-
There are two key points about the MVDR power estimate:
1. Even though PMVDR(W, k) is often referred to as the maximum likelihood power
estimate, it is a power through beam estimate and is not a true frequencywavenumber power density spectrum estimate. That is, when inverted by
[R
=
(2i)
(2
PMVDR(W,
k)ejkT (pi-pj ) k
(2.13)
it does not yield the covariance matrix that was used to generate it.
2. PMVDR(W, k) is the maximum likelihood power estimate only when ensemble
26
quantities are used. When a sample covariance matrix made of complex normal
snapshots is substituted into the MVDR power estimate Capon and Goodman
demonstrated that
PMVDR(W,
k) has a complex chi-squared distribution with
bias and variance:
E[PMVDR(, k)]
PMVDR(w, k)
_
OPMVDR(Wk)
L- N + 1
L
VL - N + 1
(2.15)
L
k)
PMVDR(,
(2.14)
where N is the number of sensors and L the number of data snapshots used to
form the sample covariance matrix.
In practice, one rarely has access to the ensemble covariance matrix and must
form a sample covariance matrix from a finite amount of data.
2.3
The Sample Covariance Matrix
The sample covariance matrix, R, is given by
1L
I
1: = XXf
L1=1
(2.16)
and is substituted into the MVDR formula for power estimation.
1
H
PMVDR(W,ks)
vk
(2.17)
(ks)R-lVk(ks)
Many adaptive algorithms either explicitly or implicitly involve forming the sample
covariance matrix and its inverse. The expressions for the singular value decomposition of the ensemble quantities R and R- 1 are useful in understanding how forming a
27
sample covariance matrix with a limited number of snapshots can affect its invertibility. When L<N snapshots are used to form R, N-L eigenvalue estimates are zero. In
this case, R rank deficient and not invertible. Even when L>N snapshots are used,
L must still be large enough to well estimate the low eigenvalues. When the low
eigenvalues are not well estimated, the sample covariance matrix is invertible, but
poorly conditioned and sensitive to inversion.
N
R
=
(2.18)
E
n=1
N
R-=
Alunun'
(2.19)
n=1
To obtain good conditioning of the lower eigenvalue estimates, Brennan, Reed
and Mallat suggest L > 3N as a criteria [4]. In rank-reduced adaptive processing
methods (discussed in Chapter 3), more emphasis tends to be placed on the larger,
more dominant eigenvalues estimates. Assuming a loud discrete source manifests
itself as a single large eigenvalue in the sample covariance matrix, Cox suggests that
using L ~ 3. (Number of significantly loud sources) is sufficient for estimating the large
eigenvalues needed for canceling loud interference with reduced-rank algorithms.
In a stationary environment, using more snapshots to form the sample covariance
matrix yields better power estimates. However, the non-stationarity of many practical
environments limits the amount of data available to form a sample covariance matrix,
especially when the arrays in use have a large number of sensors.
2.4
The Snapshot Problem
This section draws heavily from [2]. There are two limits upon the number of snapshots which are available for adaptive processing with any array: 1) The duration of
28
environmental short-term stationarity; and 2) The bandwidth over which frequency
averaging can be done without introducing distortions in the phase estimates of the
sample covariance matrix.
The cross range extent of the array's resolution cell is a function of the broadside
angular resolution, AO:
ArX
rA
Larray
sin A 0
z ~xrrin/~O
where r is the range to the source,
Larray
(2.20)
is the array aperture, and A the operating
wavelength. A moving source, traveling tangential to the array with a bearing rate
of 0, will be in this resolution cell for a duration of
rA
T AX
AT A ==
Vsource
-
1
(2.21)
(.1
Larray
seconds.
The available bandwidth for frequency averaging is determined for signals close to
endfire. The estimate of the phase in the cross spectra is smeared when one averages
over too large a bandwidth. The available bandwidth is constrained by:
1
BW < I- .
8
1
c
_
Ttransit
=
(2.22)
8Larray
where Transit is the signal transit time across the array at endfire. The product of
these two constraints gives the number of snapshots available for forming the sample
covariance matrix:
L <
1
_C
8f 0
Larray
-
2
8
-
A_
2
(2.23)
Larray
Continuing to average more data snapshots beyond the prescribed limits spreads
the eigenvalue spectrum of a moving, discrete source. For example, a single stationary
29
Eigenvalues as a Function of Source Motion
5
.. . ...
-10 -
cc)
- 20 - - - ----
010
.o.
.
--
- - -
-W
s
- - - -
- - -
Source Motion inBeam Widths
Figure 2-3: Eigenvalue as a Function of Source Motion: oc
,
,
source would manifest itself as one distinct eigenvalue in the ensemble covariance
matrix.
Figure 2-3 referenced from [2] depicts the eigenvalue spread caused by a single
discrete source as it traverses the array. The quantity yi = "Larray
is target motion
during sample covariance formation relative to a beamwidth. As soon as the source
motion occupies one full beamwidth during snapshot averaging, the second eigenvalue
becomes comparable. With increasing speed, the source effectively splits into multiple
sources of decreased power.
The inverse square dependence on array length can severely limit the number of
available snapshots. This is of particular concern in applications where large aperture
arrays are used for their high resolution. For example, a 200 Hz source moving at 20
knt and a 10 km distance transiting a 100 wavelength array would yield approximately
3 snapshots. This is far less than then the L > 3N = 600 snapshots recommended for
full-rank adaptive processing, assuming a standard sensor spacing of A/2.
In order to deal with snapshot deficiency, reduced-rank methods may be used.
30
Several reduced-rank methods commonly found in practice are diagonal loading, dominant mode rejection, eigenvalue filtering and beamspace processing. All are presented
in Chapter 3.
31
32
Chapter 3
Common Reduced-Rank ABF
Techniques
3.1
Diagonal Loading
A simple method of addressing both target self-nulling and the limited snapshot
problem in MVDR processing is to apply diagonal loading to the sample covariance
matrix used in the weight computation [13]. The diagonally loaded matrix has full
rank and is invertible.
RDL
RDATA
+ E
(3.1)
HDLVk(k)
v
(k)Rjt'jvk(k)
The load level c is chosen to satisfy a white noise gain constraint and reduces
the amount of adaptive nulling in all directions. The MVDR output power is then
computed as:
33
P(k)MVDR,DL =
W1VDRDLRDATAWMVDR,DL
(3.3)
The additional sensor noise introduced by diagonal loading reduces the expected
SINR power loss normally incurred by using the sample covariance matrix. However,
it can also add a significant amount of bias, raising the minimum detectable level of
a target.
3.2
Dominant Mode Rejection
The Dominant Mode Rejection (DMR) algorithm tends to preserve adaptive nulling in
the "dominant" interference directions and increases it in the "noise" directions. An
approximation of the sample covariance matrix, denoted as R, is formed by retaining
the P largest eigenvalues of R and averaging the rest. For example, if the sample
covariance matrix has the eigenvalue decomposition
rnk
ZAuiuAZU
RDATA
where rnk
=
(3.4)
min(L,N), them the DMR approximation to the matrix inverse is
P
R1R
-
Z
rnk
l
+
i=1
Z
a-1 uiuO
(3.5)
i=P+1
and
1
a =rrk -
nk
E Ai
die+
The DMR weights and output power estimate are computed as:
34
(3.6)
WMVDR,DMR =
P(k) MVDR,DMR
3.3
RDMRvk(k
v(RVkk)
VH (RDMRVk~k
= WMVDR,DMR
(3.7)
DATAWMVDR,DMR
(3.8)
Eigenvalue Filtering
The Eigenvalue Filtering (EF) algorithm approximates the sample covariance matrix
A in a similar way to the DMR algorithm. Instead of averaging the lower, poorly
estimated eigenvalues, they are set to zero (filtered), and the matrix inverse is approximated as:
P
AT-uiuf
- =
3.4
(3.9)
Beamspace Processing
Beamspace processing allows for more adaptivity across a specified sector of space
(usually around the steering direction) where high levels of sidelobe leakage tend to
occur. the N-dimensional element space is reduced to a B-dimensional beam space
via the (NxB) transformation matrix:
To=
VO_
B2
...
v0AO
vO
VA
-- V-+AO
(3.10)
the columns of the matrix are tapered array response vectors steered symmetrically
about the look direction, 6. The columns need not be orthogonal to one another, but
should be normalized such that
35
v
(3.11)
H
The beamspace steering vector and covariance matrix are:
(3.12)
TO = TH
RBS = To RDATATo
(3.13)
And, the beamspace MVDR weights and output power estimate are:
ftii-i T 0
BS
WMVDR,BS
P(k)MVDR,BS
WMVDR,BsRBSWMVDR,BS
36
(3.14)
(3-15)
Chapter 4
Estimation of Covariance Matrices
with A Priori Constraints
4.1
Statistical Data Model
Multivariate statistics provide the framework for estimating a covariance matrix with
a specified structure from vector samples of a random process [16].
The complex
vector samples in our case are array snapshots, X 1 , X 2 , - - -, XL, and are modeled
as i.i.d. zero-mean complex normal random vectors. Their joint probability density
function is:
p(X 1 , X 2 ,
XL)
I=I
(27)N
IRI
(4.1)
XH1
Given the L data snapshots, the maximum likelihood estimate of the covariance
matrix R is given by:
RtML=
argmax
RCR
P
-XlX21
XL
argmax
=R
37
RCR
(2Lr)1R
(~NR~
~RX
(4.2)
This can be manipulated to
RML
a RR
AML
-log IRI - Tr - E R-XXI
(L 1_1
ar Rx
NML =
IRI
-log
argmax
RCR
-
Tr (R-IDATA)
L R, fDATA)
(4.3)
(4.4)
(4-5)
L (R, fDATA) is referred to as the likelihood function, and we wish to find the R,
subject to the appropriate constraints, that maximizes it. While there is no closed
form solution to this equation, it is possible to derive the first order derivative (See
Appendix A for derivation and interpretation) which suggests an iterative update
algorithm may be used to maximize the likelihood function. Note that the sample
covariance matrix,
RDATA,
is not inverted, so the likelihood function may still be
evaluated in snapshot deficient cases.
4.2
Physical Constraints on R
Physical constraints may be imposed on R and should reduce the number of snapshots required for adaptive array processing since they provide apriori restrictions [1].
Snyder and Miller [15] and Barton [3] used the above statistical snapshot structure in
conjunction with a Toeplitz constraint when the data is from a stationary time series
or an antenna array with some amount of linear, equally-spaced sensor geometry.
For the case of the PCML algorithm, an arbitrary array geometry is assumed which
prevents the Toeplitz constraint from being imposed, but will keep the algorithm
applicable to a broader range of cases where alternate array geometries are used or
38
when linearly arranged sensors deviate from their nominal positions.
The PCML algorithm will exploit the statistical data model above, and impose
the physical constraints of the wave equation on the structure of the sensor covariance
matrix. For problems in a homogeneous medium, this can be done by requiring that
the covariance matrix have a Fourier relationship with the power spectrum:
[R
, + (21)3
= o
f
P(w, k) [vk(k)]i [v'(k)]j d
(4.6)
where Q(k) is the support of the wavenumber field at frequency f imposed by the
wave equation.
The radially symmetric 3-D wave equation is:
1 0 2 P(r t)
V 2 P(r, t) =2
c2
(47)
(O't4.7
(at)2
where P(r,t) is the acoustic pressure field at range r and time t, and c is the speed
of sound in the medium of propagation.
The wave equation is derived under the
assumptions that a pulse maintains its shape as it propagates, and that the distance
propagated is r = ct. It is a linear differential equation, and therefore obeys the
superposition principle. Modeling an environment as a superposition of uncorrelated
plane waves and noise, a single propagating wave of frequency w and wavenumber
k = [kx ky k,]T is denoted as:
(4.8)
P(x, y, z, t) = Ae-j(wt-(kxx+kyy+kzz))
and can be substituted into the 3-D wave equation:
( 2
(aX) 2
92
2
+
x
(ay)2
+
(aZ) 2 )c
1
P(x, y, z, t)
39
=
2
-
2
(at)2
P(x, y, z, t)
(4.9)
(kX + ky + k) P(z,y,z,t) = (?
(k2+ k +k)
X
Z
(27r;
(Af )
2=
)P(x, y, z, t)
A
(4.10)
(4.11)
which defines the wavenumber support of propagating acoustic energy '. Separation of the sensor noise power from the power of the propagating field in the Inverse
Fourier Transform is necessary when implementing the PCML algorithm since the
sensor noise does not have the same physical constraints as the the propagating field,
and it ensures the positive definite structure of the covariance matrix.
This Chapter presents the fundamental components of the Physically Constrained
Maximum Likelihood (PCML) method. The algorithm computes the maximum likelihood estimate of the covariance matrix R within the class of covariances R whose
Fourier Transform consists of a white noise term and propagating energy only.
The next chapter outlines the iterative PCML method for power spectrum estimation, and presents the implementation used to generate the results of this thesis.
'If a 2-D array geometry is used, then the wavenumber would be restricted to lie within a disc
of radius (27r)/A. If a 1-D array geometry is used, then the wavenumber would be restricted to lie
within a line of length 2(27r)/A. Figure 5-2 illustrates these allowable regions.
40
Chapter 5
The Physically Constrained
Maximum Likelihood (PCML)
Algorithm
5.1
Introduction
This chapter is intended to outline the implementation of the PCML Algorithm used
in this thesis. A general schematic of the algorithm is shown in Figure 5-1. The
major components are the algorithm initialization and the iterative loop. The output
will be the PCML covariance matrix estimate, which will later be used in an MVDR
power estimate to compare with that of other reduced-rank ABF methods.
41
INITIAUZATION
INITIALIZE POWER
ESTMATE
INITIALIZE WHITE
NOISE ESTIMATE
RE(RPU)=M
UKELIHOOD
*T
FUNCTION
POWER SPECTRUM
____
R, (N)= 5- (PAA1k+0'1)
MVDR POWER
ESTMATE
(v"
COMPUTE GRADIENT
COMPUTE GRADIENTS
UPDATE POWER
EST1MATE
UPDATE WHITE NOISE
ESTPIATE
R.,1
P.0%k)=$ P.s,L
=o
{.3L~
Figure 5-1: General PC4L Algorithm Schematic
ve(')Y
5.2
5.2.1
Algorithm Initialization
Covariance Matrix Initialization
The initial PCML covariance matrix estimate will be initialized with the sample
covariance matrix obtained directly from the array data:
Ro = RDATA
There are many ways
RDATA
(5.1)
is estimated in practice, including exponentially
weighting the snapshots as a function of time, or applying a sliding window. In this
thesis, the unconstrained maximum likelihood estimate of the covariance matrix is
used1 .
Ro
5.2.2
=
EXIXH
1
L1=1
(5.2)
Power Spectrum Initialization
The PCML power spectrum estimate 2 will be initialized as the MVDR power estimate
in this thesis. Computations will take place at discrete wavenumber points, ka, to
keep the number of unknown parameters finite.
Po (w, kn)
= PMVDR(w,kn)
=
(vk (kn)R--vk(kn))
(5.3)
The points k, should be spaced closely enough to sample the power spectrum
over a sufficient amount of space, and sparsely enough to avoid estimating a large
'This is equivalent to using a uniformly weighted L-point sliding window.
2
The PCML power spectrum estimate is associated with propagating energy only. The white
noise power is estimated separately.
43
2-0 Wavenuwber Grid Points
1 -D Wonenurnber Grid Points
1
1
05
0
-0.5
-1
-1
-1
-0.5
1
0
0.5
u. - k/4kII
-1
-05
0
u. - klI 141
0.5
1
3-0 Wmrp.wnber Grd Points
1
-1
0
Y
-k/1kII
0
-1 -1
u -ylI
k|
Figure 5-2: Sample Discrete Wavenumber Grids
number of correlated variables3 . The spacing of the spatial sample points could
follow any number of schemes, such as a radial or linear pattern. The scheme used in
this thesis is the 2-D linearly spaced grid shown in Figure 5-2, with the Au spacing
approximately equal to the resolution of the array. Mann provides discussion and
analysis of wavenumber grid spacing in [14]. The effects of various grid spacings on
the power spectral density estimates are shown in the results section of this thesis.
3
The resolution of the array determines the spatial separation needed to avoid correlation between
adjacent sample points.
44
5.2.3
White Noise Initialization
The initial white noise power is restricted to lie somewhere between zero and the
average diagonal term of the sample covariance matrix. So the initial PCML white
noise power estimate (and all subsequent white noise power estimates) should be
confined to the range:
1K
<
0< a2
Ti E
[ADATA] nln
(5.4)
n=1
A general expression for or is
022 =a -
N
E
-
[fDATA I
n=1,
~
, 0< a < 1
(5.5)
[DATA
[f_ AA n,n
(5.6)
In this thesis, the white noise is initialized as:
a-2 = mino
(re 2
0
+
tue+
tt,
U 7N
where uO is a random number, uniformly distributed between 0 and 10.
5.3
Iterative Loop
5.3.1
Covariance Matrix Update
The first step in the iterative loop is to obtain the new covariance matrix estimate.
This is done by inverse Fourier Transforming the power spectrum estimate from the
previous iteration
".
4
The Fourier Transform should be done only in the appropriate dimensions. For example,
if k = [kx kY]T, then the equation for the Inverse Fourier Transform would become [Rm]jj =
fO(k) Pm-i (w, k) e-jkT
(PiPj)
dk +
O_1 6,j
45
[Rm]
I~mjj
=
(2_)
(27r)'
2(k)
1(k)
[Vs-pace
P
a2_1 1
i'm-i
(w, k) ++ Umi
(Pm-1(Li), k) +
1)
Pm-1 (w, k) e-jk T
(5.8)
Vk(k)vk(k) dk
Pm-1 (w, k) [vk(k)]i [Vk(k)]j dk +
(27r)3 I2(k)
(5.7)
(pi-p)
a2
dk +
2
(5.9)
_16,,j
(5.10)
_,
Since Pm-1 (w, k) is only known at discrete points, each sample can be approximated as a weighted, shifted window in k-space, i.e.,
(5.11)
Pm-1 (w, k) -W (k - k,)
Pm-1 (w, k) =
n
Substituting this model into the expression for the covariance matrix update
yields:
[Rm]i,j =
Zn Pm-1 (w, kn) ((27)1 fo(k) W(k -
[Rm]i,j =
Zn Pm-1 (w, kn) e-
(Ci-a
(27r) I
kn)eJk (PiPJ)
2(k)
dk) + am_1Ai
W(k)e-kT(PiPJ) dk)
(5.12)
+ am_13)
(5.13)
[Rm]j,,
where Wjj
=
=
(
n
IJ)
Pm-1 (w, kn) e -jkT(pi-pj)
.
147
jj
+
2
6
(5.14)
[Wi,, is the inverse Fourier transform of the window. W is a
covariance matrix taper and can be computed offline. A variety of windows and their
46
inverse Fourier transforms are presented in Appendix B. The effects of the different
windows on the power estimates are presented in the results section of this thesis.
The windows in Appendix B are applicable in cases when 2-D array geometry is used
and when the discrete wavenumber points have a linear grid spacing (this is the case
for the test scenarios in this thesis).
5.3.2
Power Spectrum Update
The PCML power spectrum estimate is updated at each point according to the gradient
a)
,9Pm,,_(w,kn,)
5
Pm (w, kn) =
DL (Rm, IDATA)
,
(Pm-, (Li, kn)
OPm-1 (w, k,,))
0
(5.15)
where 0 (.) is the update function. In general, 0 (.) should be chosen such that:
1. # (Pmi
(wk) ,
(Pm~j~jkn~i
2.
4 ()
0=
PL(RD
Pmj(w,kn)
m-1
(w, kn);
(w, kn) monotonically for
increases the scale of Pmi-
PLRRDATA
9Pmn(wkn)
> 0;
and
3.
# (.)
decreases the scale of Pm-i (w, kn) monotonically for
In this thesis
# (.)
< 0;
was chosen to be a multiplicative update:
{
A-/2~1
eA/2-1
P. (w, kn) = P.-i (w, kn) -
e-5~21i
5
aL (Rm RDATA
(arctan
(a Pm
+
-
fPMi1
(
arctan~p,
-
1
+
iL
_>0
'oth-e1(w,k )
.otherwise
(5.16)
See Appendix A for derivation and interpretation.
47
0.5, P 5
Multiplicativpficative Update (non-dB): A- 125, B 0.8, a
10
IC -
---
- -
- -
-- - -
I6-
L
.. ..
2 -..
0
-0.5
-0.4
-03
-0.2
0
0.1
Slope: aLfdP((o, kn)
-0.1
Additive Update (dB): A = 125, B = 0.8, a
0.2
0.3
0.4
0.
0.3
0.4
0.5
=
0.5, 0 5
L10 -
-0
8
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Slope: aLlaP(wo, k0 )
0.2
Figure 5-3: Sample Multiplicative Update Scale
where A and B are the scale's upper and lower limits respectively, and a and 3
control how quickly and stably the algorithm converges. The multiplicative scale used
in this thesis is depicted in Figure 5-3.
5.3.3
White Noise Update
Similarly to the PCML power spectrum estimate, the PCML white noise power estimate is updated at each point as a function of its gradients
2
m
- ~m-1)
(2
__
02L 2
2u~ (aOra2_)
6.
(5.17)
where -y (.) has similar properties as q (-).
In this thesis, -y (-) was designed to give the white noise estimate an additive
update.
'See Appendix B for derivation and interpretation
48
OL
or
= or2_I + 1
i0
4
(5.18)
"
&2L
5.4
5.4.1
Outputs
Likelihood Function
The likelihood function is calculated at the mIh iteration as:
Lm (Rm,
Monitoring Lm (Rm,
RDATA)
RDATA)
=
-log Rml
-
Tr (RR
DATA)
(5.19)
indicates whether or not the algorithm is converged.
Other useful quantities to monitor are the white noise estimate ao2 and Am, the
eigenvalue spectrum of Rm.
5.4.2
PCML-MVDR Power Estimate
The PCML-MVDR power estimate will be will be computed and compared to the
MVDR power estimates of other reduced-rank ABF methods.
P
(k)MVDR,PCML
=
(v'(k)R vk(k))
where Rm is the PCML covariance matrix estimate at the mth iteration.
49
(5.20)
50
Chapter 6
Results
6.1
6.1.1
Test Environment
Array
The array used in this test environment was a 50-element circular array, with a radius
of 5 wavelengths. A non-linear array geometry is used so that Toeplitz constraints
would not apply. The array and a cross-section of its radially symmetric frequency
wavenumber response
1 steered
to k,
=
ky = 0 with uniform weighting are depicted
in Figures 6-1 and 6-2.
6.1.2
Signal Environment
The simulated signal environment will consist of sensor noise, isotropic noise, and
discrete sources whose parameters are given in Table 6-1. The ensemble covariances
for each environmental component are generated and summed together to form the
total ensemble covariance matrix, R. A sample covariance is formed from R as
'T (w, k)
= wHVk(k) =
w n [Vk(k)],
N-
=
51
_ -jk
T
p.
Circular Array, R =
5
5
4
2
0
0
0
0
3-
1
00
00
0
0
0
0
-
0
o
0
0
0_
0
0 -
00
0
0
0
-2
0
00
0
-3
0
0
0
0
-4
-
0
0
0
00000000
-5
5
0
x/&
-5
Figure 6-1: 50 Element Circular Array, R = 5A
4
Radial Cross Section of Frequency Wavenumber Response,
-............
0
-............
-10
,
,
-20
... ...
-.-.
~
- ~ ~~
-.---
-30
-.-.
.
co-40
-..
.....
......
-..
........-..
.........
...
-.-.
-.
--..
.
-.
--..-.
.
-.
--.. -.-.
.
--.
.
--.-.
.
.
-..
-- -
.-. ..
- --
. - .-..--.--.
-50
..............
................
..... .. ... .... . . .......
... ...
.... ... .
-
-60
---.-.
--.
-- - .-. -.
-.--.-.-.-.-...
-.-- ...
-70 - -.
-
-80L
-1I
0.5
-0.5
r
1
= (u + u )12
x Y
Figure 6-2: Frequency Wavenumber Response of Array
52
discussed in Section 6.1.3 and is input to the various Reduced-Rank ABF processing
schemes.
Sensor Noise
The ensemble covariance matrix of spatially white sensor noise is:
RSN = USN(
where a2
is the true white noise power that will be estimated by the PCML
algorithm.
Isotropic Noise
The ensemble covariance matrix of 3-D isotropic noise projected onto a 2-D array is:
[RINij
where a-2
-
sinc27r(62
U2
IN rn
A
is the isotropic noise power, and
the vector magnitude
1 Pi
-
P'
(6.)
is the operator for computing
2
Discrete Sources
The discrete sources are modeled as plane waves with some amount of directional
spreading. The motivation for including a small amount of source spreading is that
the power spectral density estimate of the PCML algorithm tends to carve out "wells"
in the areas adjacent to a plane wave source, as will be shown in Section 6.5.3. A
Gaussian model of source spreading is used in this simulation to avoid this situation.
The ensemble covariance matrix the the kth discrete source is:
211 all-
v'fa
53
Power
(re 0 dB)
0
20
(dB)
(dB)
12
(dB)
[0.5
9
(dB)
[-0.275
6
(dB)
[-0.275
Sensor Noise
Isotropic Noise
Discrete #1
Discrete #2
Discrete #3
[u, , u,,]
Position
Spread
-N/A-N/A-
=,
-N/A-N/A(27r)
-0.2810
~
, 0.4763]
(25)
~~0.2810
,-0.4763]
(27r)
-0.2810
~
,
0]
Table 6.1: Signal Environment Details
[RSk]i~j
where O-)
=r
S
e2X.e-1(,.-j.,232
I(2*-jY13
[v(k Sk)]i [VH(k Sk)]j
is the source power, ksk is the source wavenumber, and /3 and
(6.3)
#,
are
the standard deviations (in wavenumber) of the source spread. Note that as /3 , and
fy go to zero, Rs, becomes the covariance matrix for a plane wave source.
Ensemble Covariance
The ensemble covariance, R is computed as the sum of the ensemble covariances from
each environmental signal and noise component.
K
R = RSN + RIN ±
RSk
(6.4)
k=1
where K is the total number of discrete sources.
The specific parameters of the environment used in this thesis are summarized
in Table 6-1.
These values were chosen to include "high", "medium" and "low"
level discrete sources relative to the ambient environment. The source positions were
chosen such that they do not necessarily fall on a discrete wavenumber point in the
power computations.
54
6.1.3
Sample Covariance Matrix Generation
The sample covariance matrix is formed by generating "noisy" snapshots from the ensemble covariance matrix. T he eigenvalue decomposition of the ensemble covariance
matrix is:
R=UAUH
The
1 th
(6.5)
snapshot is generated as:
X, = UA1/ 2n1
(6.6)
where
1
/2
0
0
0
0
A2
0
0
0
0
---
(6.7)
A/2
and is a (Nxl) random vector drawn from a Gaussian distribution with zero-mean
and identity covariance.
n ~ N (0, I)
(6.8)
This gives the sample covariance matrix the desired property that its expectation
equals R.
E [LDATA
= E
55
L
,
- 1 XIXH
L =1.
(6.9)
E
[tDATA]
[X X '
=ZE
(6.10)
L=1
E[ADATA]
E [UA1/2ninHA/2UH
(6.11)
E IADATA
UA1/2E [nin] A1/ 2 UH
(6.12)
UA/ 2 IAl/ 2 UH
(6.13)
E
-
RDATA
1
E NIDATA
6.1.4
=
R
(6.14)
Discrete Wavenumber Spacing
The spacing of the sample points follows the 2-D scheme depicted in Figure 5-2. The
spacing between sample points was chosen to be Au
=
0.1, which is approximately
0.1
(6.15)
the resolution of the array.
Aures
~
-
-
Larray
10A
=
The effect of using a coarser and a finer grid spacing on the power estimates will
be shown in Section 6.5.1.
56
6.1.5
Covariance Matrix Taper
A uniform window was applied to each power estimate for the integral computation
in the PCML algorithm. Additional windows are described in Appendix B and the
effect of using a non-uniform window on the power estimates will be shown in Section
6.5.2.
57
6.2
Power Estimates vs. Snapshots
This section presents the detection performance of the PCML ABF algorithm relative to other reduced-rank ABF algorithms in terms of MVDR power estimates for
several values of snapshots (L
=
2, 5, 10, 30, 50, 100, 150, 200). The benchmark
for comparison is the MVDR power estimate formed with the ensemble covariance
matrix, R, in Figure 6-33
For each method of power estimation, the expected power estimate and standard
deviation of the power estimate over 50 trials are presented. The means and standard
deviations are drawn as contour plots, with contours placed every 2 dB. Each discrete
wavenumber grid point is marked to emphasize where the power is estimated 4, and
the discrete source locations are marked with a star.
The PCML algorithm cycled through 50 iterations to generate its final output
estimates. To view the convergence of the PCML algorithm, several quantities are
presented including the final Power Spectral Density (PSD) estimate at the
5 0 th
itera-
tion (mean and standard deviation), the white noise estimate vs. iteration (mean and
standard deviation), the likelihood function vs. iteration (mean and standard deviation), and the eigenvalue spectrum of the PCML estimated covariance vs. iteration
(mean and standard deviation).
3
The discrete source powers of 12 dB, 9 dB and 6 dB are attenuated slightly at the source location
even in the ensemble MVDR power estimate due to the presence of source spreading
4 The contour values outside of the discrete wavenumber grid are artificially forced to non-zero
values for plotting purposes, and should be ignored
58
Ensemble MVDR Output Power
15
1
0.8
10
0.6
0.4
5
0.2
S0
I
I
I
|
|
-0.2
0
-0.4
-0.6
-5
-0.8
-1
-1
-0.5
0.5
0
1
-10
U
Figure 6-3: Ensemble MVDR Power Estimate
59
6.2.1
L = 200, MVDR Power Estimates and Standard Deviations
CBF: E[PMVDR] (dB), L = 200, 50 trials
CBF: StdDeV[PMVDRE[PMVDR] (dB), L = 200, 50 trials
15
D
1
0.8
0.810
0.6
0.6-
0.4
0.4-
0.2
=
5
0.2-
0
-0.2-
0
0-
-0.2
-
x
-10
-0.4-
-0.4
-0.6
-0.6-
-5
-0.8
-1
-5
-0.8
-1
-1
-0.5
0
0.5
1
-10
-
-1
-0.5
U
(a)
0
U
0.5
(b)
Figure 6-4: CBF, L = 200: (a) Mean Output Power (b) Standard Deviation of Output
Power
60
-15
PCML: E[PMVDR] (dB), L = 200, 50 trials
PCML: StdDeV[PMVDRYE[PMVD] (dB), L = 200, 50 trials
0
15
1
0.8
0.810
0.6
0.6-
0.4
0.4-
1.
0.2
5
0
Z3.
-0.2
-
x
I
0.20-0.2-
0
-0.4
-10
-0.4-
-0.6
-5
-0.6-
-5
-0.8
-0.8-
-1
-1
-0.5
0
0.5
-1
-10
1
--
-0.5
-1
U
0
0.5
L.J-15
U
(a)
(b)
Figure 6-5: PCML, L = 200: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMVDR] (dB), L = 200, 50 trials
Diagonal Loading: StdDeV[PMVDRYE[PMVDR] (dB), L =200, 50 trials
15
1
0.8
0.810
0.6
0.6-
0.4
0.4-
0.2
5
0.2-
0
-0.2-
N.0
5
2" 0-
-
-0.2
-0.4
-
-0.6
-0.4-
10
-0.6-
-5
-0.8
-0.8-
-1
-1
-
-10
-1
-0.5
0
0.5
U
(a)
(b)
Figure 6-6: DL, L = 200: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
61
-.15
DMR rank 5: E[PMVD] (dB), L = 200,50 trials
I
1
I
I
I
DMR rank 5: StdDev[PMvD
I
15
0.8-
-
0.6-
-
0.4-
-
5
0.2-
-
0
-0.2 -
-
-0.4 -
-
-0.6 -
-
10
0.4
0.2
00
1 - IIII-
0.8
0.6
E[PMvD] (dB), L =200, 50 trials
-5
ZZ. 0
-0.2
-0.4
-0.6
-5
-0.8 -
-0.8
-1
-1
-0.5
0
0.5
1
-
-10
1
-05
0
(a)
-15
0.5
Ux
U
-10
-1
(b)
Figure 6-7: DMR, L = 200: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
EV rank 5: E[PMvDR] (dB), L = 200,50 trials
EV rank 5: StdDev[PMVDR ]EPMVDR] (dB), L = 200, 50 trials
15
II0
SI
0.8
0.810
0.6
0.6-
0.4
0.2
0.4-
-
5
0.2-
-
0
-0.2 --
-
-0.4-
-
- - --
0
0.5
-5
0-0.2
-0.4
-0.6
-
-
-
-
-
-
-
-
10
-0.6 --
-5
-0.8 --
-0.8
-1
-1
-05
0
0!5
1
-
-10
-1
-0.5
U
U,
(a)
(b)
Figure 6-8: EVF, L = 200: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
62
-15
-1
L = 200, PCML Algorithm Performance
6.2.2
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDlv[PSDyE[PSD] (dB), L = 200, 50 trials
30
10
1
0.8
8
0.8
25
0.6
6
0.6
0.4
4
0.4
0.2
20
zz.
-0.2
-
-0.4
-
-0.6
2
0.2
: C
0
0
-0.2
15
-2
-0.4
-4
-0.6
10
-0.8
-0.8
-1
-1
-6
-8
-1
-0.5
0
0.5
1
5
I
-1
I
III
-0.5
U
(a)
0
U
0.5
(b)
Figure 6-9: L = 200: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate
63
-10
PCML: Average a2 value vs. Iteration, L
*I
6
0
5.
=
200, 50 trials
PCML: StdDev/Mean of a2 value vs. Iteration, L = 200, 50 trials
1
0.95-
0
-o000
0.90
55-
0
0
00
4-0.85-
-
C00.8 4. 5-
0
4-
> 0.7 0
0.65
00
000
00
3.5 -
-
0
-
000
I
0
0.6 0
IIII
I
10
20
30
40
50
I
10
20
Iteration
30
40
50
Iteration
(a)
(b)
Figure 6-10: L = 200: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average Likelihood value vs. Iteration, L = 200, 50 trials
I
-350
I
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 200, 50 trials
I
01 -jaa
-300
-0
0 0 0 0
00 0 0 0 0
noyuuuoouu
'
0u--
-0.051-400
i
-450
ii-0.1
C"
2-550
10.15
-.600
-650
-0.2
-700
0
00 20
n10
30
40
50
10
Iteration
20
40
Iteration
50
(b)
(a)
Figure 6-11: L = 200: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence
64
PCML: Average eigenspectrum vs. Iteration, L = 200, 50 trials
1S.
z
I
I
2
i
10
PCML: StdDev/Mean of eigenspectrum value vs. Iteration, L = 200, 50 trials
-4.
I
I
I
I
I
I
20
I
30
40
bw__
0
-10
-8
-- 12
>
S-14
-5
-16
..10
0
I
10
I
I
20
30
Iteration
I
40
0
0
10
I
50
Iteration
(a)
(b)
Figure 6-12: L = 200: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
65
6.2.3
L = 150, MVDR Power Estimates and Standard Deviations
CBF: E[PMvD,
(dB), L = 150, 50 trials
CBF: StdDeV[PMVDRE[PMVDR] (dB), L = 150, 50 trials
1 --
0
15
0.8
0.8-
0.6
10
0.6-
0.4-
0.4-
0.2-
5
0.2-
0
-0.2-
S0 -~
0
-0.2-0.4-
-0.4-
-0.6
-0.6-
-0.8
-0.8
-1
I
-1
-5
-0.5
II
I
0
0.5
-10
-1
-10
-1
-0.5
Ux
0
0.5
U
x
x
(a)
Figure 6-13: CBF, L
Output Power
-15
I
1
(b)
=
150: (a) Mean Output Power (b) Standard Deviation of
66
-
PCML: StdDev[PMVDRYE[PMVDR] (dB), L = 150, 50 trials
PCML: E[PMVDR] (dB), L = 150, 50 trials
0
15
0.8
0.8
10
0.6
0.6
0.4
0.4
0.2
&~0
23.~
-0.2
-
-
-5
0.2
5
0
-0.2
0
-10
-0.4
-0.4
-0.6
-0.6
-5
-0.8
-0.8
-1
-1
-1
-0.5
0
0.5
-10
1
-1
-0.5
U
0
0.5
-15
1
U
(a)
(b)
Figure 6-14: PCML, L = 150: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMVDR] (dB), L
=
150, 50 trials
Diagonal Loading: StdDe[PMvDRYE[PmVDR] (dB), L = 150, 50 trials
15
0.8
10
0.6
0.4
0.8-
-
0.6-
-
0.4-
0.2
5
0.2-
0
-0.2-
:?.1 0
-5
-
0
-0.2
-
-0.4
-
-0.6
-10
-0.4-0.6-
-5
-0.8
-0.8-
-1
-1
-
-1
-10
-1
-0.5
-1-
0
0.5
U
(b)
(a)
Figure 6-15: DL, L = 150: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
67
-
DMR rank 5: E[PMVDR] (dB), L = 150,50 trials
DMR rank 5: StdDev[PMvDRIEPmvDR] (dB), L = 150, 50 trials
15
0
1
0.8
10
0.6
0.4
-
0.6-
-
0.4-
0.2
&
0.8-
0.2
5
0
Z2
-0.2
-
-U
-
0
-0.2-
0
-0.4
-10
-0.4-
-0.6
-0.6-
-5
-0.8 -1
I,
I P '
-5
-
-.
-0.8
-1
-10
-1
-0.5
0
0.5
-15
U
(a)
(b)
Figure 6-16: DMR, L = 150: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
EV rank 5: E[PMVDR] (dB), L = 150,50 trials
EV rank 5: StdDev[PMVDR]/E[PmvD] (dB), L = 150, 50 trials
15
0
1
1 - IIII
0.8
0.8
10
0.6
0.6-
0.4
0.4-
5
0.2
2.
0
s
-0.2
-5
0.2-
0
0
-0.2-
-0.4
-10
-0.4-
-0.6
-0.6-
-5
-0.8
-0.8
-1
-1-
-1
-0.5
0
0.5
-10
-1
U
-0.5
0
0.5
U
(a)
(b)
Figure 6-17: EVF, L = 150: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
68
-15
L = 150, PCML Algorithm Performance
6.2.4
PCML: E[PSD] (dB), L = 150, 50 trials
PCML: StdDev[PSDYE[PSD] (dB), L = 150, 50 trials
30
1
10
1
0.8
0.6
20
&0
-0.2
15
-
-0.4
6
0.6
0.4
0.2
8
0.8
25
0.4
4
0.2
2
S0
0
-0.2
-
-0.4
-0.6
-0.6
-0.8
-0.8
-
-
-
-
-2
-4
-6
-8
-1
-1
-0.5
0
0.5
5
-
-.
U
(a)
(b)
Figure 6-18: L = 150: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate
In the high snapshot regime of L = 200 and 150 snapshots, the PCML- and DMRMVDR power estimates most closely resemble the ensemble MVDR power estimate
in terms of their shape and their ability to distinguish the peak of the 6 dB source at
k = [-0.275, -0.4763].
Also, these two algorithms have the lowest standard deviation
relative to their mean.
It is also interesting to observe how the PCML algorithm converges in the high
snapshot regime:
The expected value of the likelihood function monotonically increases as the algorithm iterates (as it should), and the standard deviation of the likelihood function tends to decrease with iteration. So that for an increasingly larger number of
iterations, the PCML algorithm will converge to a certain likelihood value with a
decreasing amount of error.
69
-10
2
Iteration, L = 150, 50 trials
PCMVL: Average a 2 value vs.
PCML: StdDev/Mean of a value vs. Iteration, L = 150, 50 trials
7.4
-0
7.24
2.5
0
0-
0
6.
2
6m6.
46.
1.5
0
2-0
0
40
60
0
5. 8-
0
0
0.5[oooooooo
0
6-
5.
0
0
5.4
0
10
30
20
50
40
20
Iteration
a10
Iteration
40
30
50
(b)
(a)
Figure 6-19: L = 150: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
-
PCML: Average Likelihood value vs. Iteration, L = 150, 50 trials
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 150, 50 trials
p
-300 - 0000001--
S
ooee
01
000000000
-350-
-
-400-
-
--
A
-0.05-
-450-
e
-0.1-
-
-500a -550
10.151
-650-
-0.2
0
-700
0
10
30
20
40
-0.25L
0
50
10
30
20
40
50
Iteration
Iteration
(b)
(a)
Figure 6-20: L = 150: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence
70
PCML: Average elgenspectrum vs. Iteration, L = 150, 50 trials
IC
I
.
I
I
I
PCML: StdDev/Mean of eigenspectrum value vs. Iteration, L = 150, 50 trials
-4
111
10-
-6
-10 -2
rA
-140
-5
-
-10
10
20
30
40
0
Iteration
Iteration
(a)
(b)
Figure 6-21: L = 150: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
The white noise estimate converges to a final value (within 5.8 of the ensemble
level of a2 = 1 in these cases), however in neither case does it converge to the
ensemble value. A possible explanation could be that the white noise power estimate
is absorbing some of the isotropic noise power. Also, the convergence of the white
noise estimate for the L = 150 case exhibits curious behavior around the
12 th
iteration.
This is due to a single "wild" trial where the white noise estimate jumped to a very
large value
(&2
~ 127).
The larger eigenvalues associated with the 3 discrete sources are well estimated
by the
5 0 th
iteration. For reference, the ensemble values for these eigenvalues are
marked with stars. The largest eigenvalue estimate has the least bias, but the largest
standard deviation relative to its mean. The second- and third-largest eigenvalues
are biased low (by fractions of a dB) in the high snapshot regime.
71
6.2.5
L
=
100, MVDR Power Estimates and Standard Devi-
ations
CBF: E[PMVDR] (dB), L = 100, 50 trials
CBF: StdDev[PMVDR]E[PMvR] (dB), L = 100, 50 trials
15
0
0.8
10
0.6
0.4
5
0.2
Z2
0
0.8-
-
0.6-
-
0.4-
-
0.2-
-
-5
22. 0-
-0.2
0
-0.4
-0.6
-5
-0.8
-1
-0.2-
-
-0.4-
-
-0.6-
-
-0.8-
-
-10
-1-1
-0.5
0
0.5
-10
-1
U
-0.5
0
0.5
U
(a)
(b)
Figure 6-22: CBF, L = 100: (a) Mean Output Power (b) Standard Deviation of
Output Power
72
-15
PCML: E[PMVDR] (dB), L = 100, 50 trials
I
1
I
I
I
PCML: StdDeV[PMVDRYE[PMVDR] (dB), L = 100, 50 trials
-- -r
15
0
I1-
0.8
IIII-
0.810
0.6
0.6-
0.4
0.4-
0.2
I
5
S0
31"
-0.2
-5
0.20
-0.2-
0
-0.4
-0.4-
-0.6
-0.6-
-5
-0.8
-10
-0.8-
-1
-1-1
-0.5
0
-10
0.5
-0.5
-1
U
(a)
0
U
-15
0.5
(b)
Figure 6-23: PCML, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMVDR] (dB), L = 100, 50 trials
Ii
I
I
I
ii1
Diagonal Loading: StdDev[PMvDRYE[PMVDR] (dB), L = 100, 50 trials
15
1
0
IIII-
I1-
0.8
0.810
0.6
0.6-
0.4
0.45
0.2
5
0.2-
S0
0
-0.2
-0.2
0
-0.4
10
-0.4-
-0.6
-0.6-
-5
-0.8
-0.8
-1
-1
-1
-0.5
0
0.5
-10
-
-1
:-
-0.5
0.5
U
U
(a)
Figure 6-24: DL, L
Power
0
=
(b)
100: (a) Mean MVDR Power (b) Standard Deviation of MVDR
73
15
DMR rank 5: E[PMVDR] (dB), L = 100,50 trials
I
1
I
I
I
DMR rank 5: StdDov[PMVDR]E[PMvD] (dB), L = 100, 50 trials
15
I
0.8
10
0.6
0.4
0.2
5
23. 0
0.8-
-
0.6-
-
0.4-
-
0.2-
-
-5
i~0
-0.2
-0.2-
0
-
-0.4
x
-
-0.4
-0.6
-10
-
-0.6 -
-5
-0.8
-0.8 -
-1
-1
-0.5
0
0.5
-1 -
-10
1
-1
-0.5
U
(a)
0
U
0.5
-
(b)
Figure 6-25: DMR, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
EV rank 5: StdDev[PMVDR]/E[PMVDR] (dB), L = 100, 50 trials
EV rank 5: E[PMVDR] (dB), L = 100,50 trials
0
15
1
I1-
0.8
IIII-
0.8-
10
0.6
0.6-
0.4
0.45
0.2
I
-0.2
I
I
I
I
-
21
-
-5
0.20-
-0.2-
0
-0.4
-10
-0.4-
-0.6
-0.6-
-5
-0.8
-0.8
-1 --
-1
-1
-0.5
0
0.5
1
-10
-1
-0.5
U
0
0.5
U
(a)
(b)
Figure 6-26: EVF, L = 100: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
74
-15
6.2.6
L = 100, PCML Algorithm Performance
PCML: E[PSD] (dB), L = 100, 50 trials
PCML: StdDev[PSDYE[PSD] (dB), L = 100, 50 trials
30
10
0.8
8
0.8
25
0.6
0.4
20
0.2
0
-0.2
-
-0.4
-
-0.6
15
10
0.6
6
0.4
4
0.2
2
S0
0
-0.2
-2
-0.4
-4
-0.8
-0.6
-0.8
-1
-1
-6
-8
-1
-0.5
0
0.5
1
5
-1
-0.5
U
0
0.5
1
U
(a)
(b)
Figure 6-27: L = 100: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate
Here, in the L = 100 case, the PCML- and DMR-MVDR power estimates most
closely resemble the ensemble MVDR power estimate. However, the PCML estimate
does a better job at capturing the power around the periphery of the wavenumber
grid (which is associated with the isotropic noise), and has a clearer peak at the 6 dB
source. Also, the PCML estimate has the lowest standard deviation at any point in
the wavenumber grid relative to its mean.
75
-10
PCML: Average a2 value vs. Iteration, L = 100, 50 trials
PCML: StdDev/Mean of a2 value vs. Iteration, L = 100, 50 trials
0. 581001
-T-- -
6.2
0
6 .0 0
0
-0
5.8
0
0. 54-
0
5.6
c
0. 56 -
0
0
-
52-
5.4-
0.5- -
a 5.2-
0
0.
48 0
0.46 -
4.8-
0
0
4.64.44
20
0
10
20
30
40
0
0
0
0000000000000
0.
50
40
0
10
50
40
30
20
Iteration
Iteration
(a)
(b)
Figure 6-28: L = 100: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average Likelihood value vs. Iteration, L = 100,
50 trials
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 100, 50 trials
01
d aaR
-300
-0.05-
-400 V
S-500-
*j -0.1-
I
I
I
i
-00
10.15-
-700-0.2-
-900
10
20
30
40
-0.251
0
50
10
20
30
40
50
Iteration
Iteration
(b)
(a)
Figure 6-29: L = 100: (a) Mean PCML Likelihood Convergence (b) Standard Deviation of PCML Likelihood Convergence
76
PCML: Average eigenspectrum vs. Iteration, L = 100, 50 trials
10
I
I
I
I
PCML: StdDev/Mean of elgenspectrum value vs. Iteration, L = 100, 50 trials
10
-I
za
6
-6
4
2
0
J -2
S-12
--
~-1o
W-14-
-6
10
20
30
40
50
-0
Iteration
10
20
3
0
5
Iteration
(b)
(a)
Figure 6-30: L = 100: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
77
6.2.7
L = 50, MVDR Power Estimates and Standard Deviations
CBF: StdDev[PMvDR]/E[PvDR] (dB), L = 50, 50 trials
CBF: E[PMVDR] (dB), L = 50, 50 trials
I
I
I
0
15
I
1
1
0.8
0.8
10
0.6
-
0.6
-W
0.4
0.4
0.2
5
0.2
0
-0.2
-5
S0
-0.2
-0.4
-0.8
-1
0
.
-
-0.
-
-
-0.4
-
-10
-0.6
-0.8
-5
-0.0
-1
-1
-10
-1
-
U
(a)
(b)
Figure 6-31: CBF, L = 50: (a) Mean Output Power (b) Standard Deviation of Output
Power
78
-15
PCML: StdDev[PMVDRYE[PMVD] (dB), L = 50, 50 trial 3
PCML: E[PMVDR (dB), L = 50, 50 trials
I
1
I
I
I
15
1
0.8
I
I
I
I
I
0
0.8
10
0.6
0.6
0.4
0.4
0.2
5
0.2
0
-0.2
S0
-5
I.0
-0.2
-0.4
-10
-0.4
-0.6
-0.6
-5
-0.8
-0.8
-1
I
-1
I
I
-0.5
0
U
I
I
0.5
1
-1
-10
-1
(a)
-0.5
0
U
0.5
1
-15
(b)
Figure 6-32: PCML, L = 50: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMvDR] (dB), L = 50, 50 trials
Diagonal Loading: StdDev[P
15
1
1
0.8
0.6
0.4
S0
-0.2
-0.4
-0.6
-0.8
0
0.8
10
0.6
0.2
/R]VE[PMVDR]
(dB), L = 50, 50 trials
-r
0.4
-J
-1
-1
5
-5
0.2
ZZ, 0
I
.................
-0.2
0
.
I
-10
-0.4
-
-0.6
-5
-
-
-
-
-
.--
-
-0.8
I
I
I
-0.5
0
0.5
i
1
-1
-10
-1
-0.5
0.5
1
U
U
(a)
(b)
Figure 6-33: DL, L = 50: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
79
-15
DMR rank 5: E[PMvDR] (dB), L = 50,50 trials
DMR rank 5: Stdlev[PMVD
MVD
(dB), L = 50, 50 trials
15
1
0
1 - IIII-
0.8
10
0.6
0.4
5
0.2
Z
-0.2
0
-0.4
-0.6
-5
-0.8
-1
II
-1
-0.5
I
II
0
0.5
,
1
0.8-
-
0.6-
-
0.4-
-
0.2-
-
0
-0.2-
-
-0.4-
-
-0.6-
-
-0.8
-
-1 -
-10
-5
-1
U
(a)
-10
-0.5
0
U
0.5
-
(b)
Figure 6-34: DMR, L = 50: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
EV rank 5: E[PMVDR] (dB), L = 50,50 trials
EV rank 5: StdDev[PMvDR/E[PMvD] (dB), L = 50, 50 trials
0
15
1
I
I - IIII-
0.8
0.8-
10
0.6
0.6-
0.4
0.4-
5
0.2
-5
0.2S0
-0.2
-0.2-
0
-0.4
-10
-0.4-
-0.6
-0.6-
-5
-0.8
-0.8
-1
-1 --1
-05
0.5
0
U
-10
-1
-0.5
0
0.5
U
(a)
(b)
Figure 6-35: EVF, L = 50: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
80
-15
6.2.8
L
=
50, PCML Algorithm Performance
PCML: E[PSD] (dB), L = 50,50 trials
PCML: StdDov[PSD]/E[PSD] (dB), L = 50, 50 trials
30
10
1
I
0.8
8
0.8
0.6
6
25
0.6
0.4
0.2
-4
4
20
0.2
2
15
-0.2
K.0
0
-0.2
-0.4
-
-
-2
-
-0.4
-0.6
-0.6
-0.8
-0.8
-4
-6
-8
-1
-1
-
-0.5
0
5
0.
-
-0.5
U
0
0.5
1
U
(a)
(b)
Figure 6-36: L = 50: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate
At L = 50, the sample covariance matrix is on the verge of snapshot deficiency.
The PCML- and DL-MVDR power estimates have a peak at the 6 dB source, and
the DMR algorithm does not. The PCML estimate continues to have a lower bias
and standard deviation than the other reduced-rank ABF methods.
It is also interesting to note that the mean PCML PSD estimate at L = 50 is not
drastically different than that at L
=
200. However, its standard deviation increases
with decreasing snapshot support, as expected.
81
-10
PCML: Average a2 value vs. Iteration, L = 50,
6. a
00
50 trials
a2 value vs. Iteration, L = 50, 50 trials
PCML: StdDev/Mean of
0
0
~0 0
0000002
0.8
0
0
0
5.51
0
5-
0
000
0
00
0.7
00
00 0000
4.5
00
0
o
0000
80.6 -
0000
4
00
-
0
0-
2
0
0 00
0.5 0
-50
4
3.5
Io
I
3.0
10
20
30
I
40
Iteration
-o
r
0
0.40
50
00
00
0 00
0
10
50
40
30
20
Iteration
(a)
(b)
Figure 6-37: L = 50: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average LIkelIhood value vs. Iteration, L = 50, 50 trials
-200
1
1
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 50, 50 trials
1
0
-400-
-0.
-600 -0
,-0. 2-
-600
1-0.3-
-)
0
1000-
4
1200-
2-0..5-
-1400 7F-
-0 .6 0
U-
-1600
0
10
30
20
40
.
-
10
50
20
30
40
50
Iteration
Iteration
(b)
(a)
Figure 6-38: L = 50: (a) Mean PCML Likelihood Convergence (b) Standard Deviation
of PCML Likelihood Convergence
82
PCML: Average eigenspectrum vs. Iteration, L = 50, 50 trials
1
1
1
20 1
PCML: StdDev/Mean of elgenspectrum value vs. Iteration, L = 50, 50 trials
15
z
-
-2--
-
10
r-4-
*5
2
0
I
-5
I
-I
-12-
%
10
20
30
40
14
0
50
Iteration
|
10
1
30
1
20
Iteration
(a)
1
40
50
(b)
Figure 6-39: L = 50: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
83
6.2.9
L = 30, MVDR Power Estimates and Standard Deviations
CBF: E[PMVDR] (dB), L = 30,
50 trials
CBF: StdDev[PMmvDyE[PMVDR] (dB), L = 30, 50 trials
0
15
1
0.8
0.8
10
0.6
0.6
0.4
0.4
0.2
Z2
5
0.2
0
-0.2
: 0
0
-0.2
-0.4
-5
-1
-05
0
.
-
-0.4
-U
-0.6
-
-
-10
-0.6
-5
-0.8
-0.8
-1
-1
-10
-1
-0.5
0
0.5
U
(a)
(b)
Figure 6-40: CBF, L = 30: (a) Mean Output Power (b) Standard Deviation of Output
Power
84
-15
PCML: E[PMVDR] (dB), L = 30, 50 trials
I
0
I
I
PCML: StdDev[PMV ]/E[PMVD
I
I
(dB), L = 30, 50 trials
15
0
1
0.8
0.8
10
0.4
0.6
0.4
0.4
0.2
5
-5
0.2
S0
-0.2
0
-0.2
-0.4
-10
-0.4
-0.6
-0.6
-5
-0.8
-0.8
-1
-1
I
-0.5
I
0
I
I
0.5
1
-1
-10
-1
,
U
(a)
-0.5
0
U
0.5
-15
1
(b)
Figure 6-41: PCML, L = 30: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMVDR] (dB), L = 30, 50 trials
Diagonal Loading: StdDev[PMvDRYE[PMVDR] (dB), L = 30, 50 trials
15
1
0
I
0.8
A
0.8
10
0.6
0.6
0.4
0.4
0.2
5
0.2
I.0
2,
A
-0.2
. ...................
.
- -.
- . . .
--
-. -
.-.. -
.
-6
.
.A................
0
-0.2
0
-0.4
-10
-0.4
I..
-0.6
-. . . . . . . . . . . . . . . . . . .
-0.6
-5
-0.8
-
-0.8
-1
-1
-1
-0.5
0
U
0.5
-10
-1
(a)
-0.5
0
U
0.5
1
(b)
Figure 6-42: DL, L = 30: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
85
.-
-15
DMR rank 5: E[PMVD] (dB), L = 30,50 trials
I
I
I
DMR
rank 5: StdDev[PVR./E[PMvD] (dB), L = 30, 50 trials
15
I
1-
0.8
-
0
0.810
0.6
0.6-L1
. . . . . . . .
0.4
0.41-
0.2
5
-5
0.2
S0
0
-0.2
. . . . . . . . .
-0.2
0
-0.4
-10
-0.4
-0.6
-0.6
-5
-0.8
-0.8
-1
-1
-0.5
0
0.5
1
-1
-10
-1
-0.5
(a)
0
-15
0.5
(b)
Figure 6-43: DMR, L = 30: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
EV rank 5: E[PMVDR] (dB), L = 30,50 trials
EV rank 5: StdDe[PMVDR yE[PMVDR] (dB), L = 30, 50 trials
15
I
I
0.8
I
I
I
I
0
0.8
10
0.6
0.6
0.4
0.4
0.2
5
&~0
&
-0.2
-5
0.2
0
-0.2
0
-0.4
-0.4
-0.6
-0.6
-5
-0.8
-10
-0.8
-1
-1
IIII~
-1
-0.5
0
0.5
-10
-1
U
(a)
-0.5
0
U
0.5
1
(b)
Figure 6-44: EVF, L = 30: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
86
-15
6.2.10
L = 30, PCML Algorithm Performance
PCML: E[PSD] (dB), L = 30,50 trials
PCML: StdDev[PSD]/E[PSD] (dB), L = 30, 50 trials
30
1
10
0.8
8
0.8
25
0.6
6
0.4
0.2
0.4
4
20
0.2
2
15
-0.2
~*0
0
-0.2
-0.4
-
-
-0.6
-2
-0.4
-4
-0.6
10
-0.8
-
-0.8
-1
-
-8
-1
-1
-0.5
0
0.5
1
5
-1
U
-0.5
0
0.5
1
U
(a)
(b)
Figure 6-45: L = 30: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate
87
-6
-10
PCML: Average a2 value vs. Iteration, L = 30, 50 trials
PCML: StdDev/Mean of a2 value vs. Iteration, L = 30, 50 trials
079
0
0
6.5
00
0
-
I
*
0
+
-
0.7
0.65- -
0
0000000
%5.5-
0
0.6
0
0
0
-
.50.55
0
0
5
0.5
-
0
0
4 .5-
0
4-
000
0.45 -
0000
4
1000000020000
0.4
0
0
00000
n00001
350
30
20
10
40
50
0
10
50
40
30
20
Iteration
Iteration
(b)
(a)
Figure 6-46: L = 30: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average Likelihood value vs. Iteration, L = 30, 50 trials
-250
1
1
1
-300-
0
0
0
0
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 30, 50 trials
0000000000000000000000000"1
-
ooo~oo~oooooooooo 000o00000000000
-0.02-
-350 -000*
-
-400-
-
-
-0.04- -
-j
-450-
1U
0~
-0.1-
-
-550-
-
k08 -
-
-500-
0
0
10
30
20
40
n
50
20
0
I
10
Iteration
30
20
40
50
Iteration
(b)
(a)
Figure 6-47: L = 30: (a) Mean PCML Likelihood Convergence (b) Standard Deviation
of PCML Likelihood Convergence
88
PCML: Average eigenspectrum vs. Iteration, L = 30, 50 trials
I
I
I
I
z
PCML: StdDev/Mean
10
1
-7
105
of eigenspectrum
value vs. Iteration, L = 30, 50 trials
Z -12
-5
-14
0
10
20
30
Iteration
40
50
0
10
20
30
40
50
Iteration
(a)
(b)
Figure 6-48: L = 30: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
89
L = 10, MVDR Power Estimates and Standard Devi-
6.2.11
ations
CBF: E[PMVDR (dB), L = 10, 50 trials
I
SI
1-
II
CBF: StdDev[PMvDRYE[PMVDR] (dB), L = 10, 50 trials
0
15
1
0.8-
0.8
10
0.6-
0.6
0.4-
0.4
5
0.2-
MD.
............
0-
S0
-0.2
-5
0.2
.
...............
-1
0
.
-05
-0.2
0
-0.4-
-10
-0.4
-0.6-
-0.6
-5
-0.8
-
. .
-. . -
.
.
.
. . .
. . . -
.. ... .... ... ...
-0.8
-1
-1
-1
-0.5
0
0.5
-10
-1
-0.5
U
0
0.5
1
U,
(a)
Figure 6-49: CBF, L
Power
=
(b)
10: (a) Mean Output Power (b) Standard Deviation of Output
90
-15
PCML: E[PmvDR] (dB), L = 10,
50 trials
PCML: StdDev[PMVIIE[PMvDR] (dB), L = 10, 50 trials
15
1
I
1
0.8
I
I
0
I
0.8
10
0.6
0.6
0.4
0.4
0.2
5
0.2
0
-0.2
&~0
-5
::~0
-0.2
-0.4
-10
-0.4
-0.6
-0.6
-5
-0.8
-0.8
-1
I
-1
-0.5
I
0
-1
I
0.5
1
-10
I
I
-1
I
-0.5
U
I
I
I
0
0.5
1
-15
U
(a)
(b)
Figure 6-50: PCML, L = 10: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMvDR) (dB), L = 10, 50 trials
Diagonal Loading: StdDev[PMvD/E[PVDR] (dB), L = 10, 50 trials
P
15
1
1
0.8
0.8
10
0.6
0.6
0.4
-
0.4
0.2
5
-5
0.2
..
...
Z2.
0
-0.2
-0.4
-0.6
-0.8
23
21
-1
i
-1
-0.5
....
0
..
-0.2
0
0
.. .
.. .
-10
-0.4
-0.6
-0.8 I
I
0
0.5
. . .
.
.
.
..
. . .
-1
-10
-1
-0.5
U
(a)
0
0.5
1
(b)
Figure 6-51: DL, L = 10: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
91
-15
DMR rank 5: E[PmvDR] (dB), L = 10,50 trials
DMR rank 5: StdDev[PMvDR E[PMvD] (dB), L = 10,
50 trials
15
0
1
0.8-
0.8
10
0.6-
........
0.6
0.4-
0.4
5
0.2
-5
0.2
S0
-0.2
-
-0.2
0
-0.4
I-
-
-
-
-:
-4:::
-10
-0.4
-0.0
-0.6
-5
-0.8
-0.I
-1
--1
I
-0.5
I
I
0
0.5
I
-1
-10
-15
-I
-0x
U
(a)
5
U
(b)
Figure 6-52: DMR, L = 10: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
EV rank 5: E[PMVDR] (dB), L = 10,50 trials
EV rank 5: StdDev[PMvDIEiPMVDR]
(dB), L = 10, 50 trials
15
1 --
10
1
0.8-
0.8
10
0.6-
0.6
0.4-
0.4
0.2-
5
-5
0.2
0
0
0
-0.2-
0
-0.2
-0.4-
-10
-0.4
-0.6-
-0.6
-5
-0.8-
-0.8
-1 --1
-0.5
0
0.5
-1 - ----
-10
-1
__
I
-0.5
U
I
I
0
0.5
I
U
(a)
Figure 6-53: EVF, L
Power
=
(b)
10: (a) Mean MVDR Power (b) Standard Deviation of MVDR
92
-15
L = 10, PCML Algorithm Performance
6.2.12
PCML: E[PSD] (dB), L = 10, 50 trials
PCML: StdDev[PSD]IE[PSD] (dB), L = 10, 50 trials
30
1
10
25
1.8
0.8
20
0.4
0.2
15
-0.2
0.8
0.6
0.4
0.2
,.
8
6
4
2
... . . . . . . . .
0
0
-0.2
-
-
-2
-0.4
-4
-0.6
-0.6
-0.8
-0.8
-6
- -
-1
-1
-0.5
0
0.5
1
5
-
- - - -0.5
U
- 0
-8
0.5
1
U
(a)
(b)
Figure 6-54: L = 10: (a) Mean PCML PSD Estimate (b) Standard Deviation of
PCML PSD Estimate
Here, at L = 10, the PCML- and EVF-MVDR power estimates are the only power
estimates with clear peaks around the locations of the 12-dB and 9-dB sources. The
PCML estimate continues to have superior performance in terms of bias and variance.
It is interesting to note that even though the 6-dB source does not show up in
the mean PCML-MVDR power estimate, it is clearly visible in the mean PCML-PSD
estimate. This suggests that the PCML-PSD estimate may be more effective for lowSNR source detection in cases of low snapshot support. This is further explored in
Section 6.4.
93
-10
PCML: Average c2 value vs. Iteration, L = 10, 50 trials
2
PCML: StdDev/Mean of a value vs. Iteration, L = 10, 50 trials
5.2
(1575.
0
0.57[-
0
0
5.1-
0.5650
0
0
m0.56 -
0
'60.555
-
0.55
-
5 o00000000000000000000000000000000000,
-
000000
00000
000000000
0 0
0
-
*
0.545
0.54
4I
0
10
20
30
1
40
0"3
0
50
Iteration
10
20
30
40
Iteration
(a)
50
(b)
Figure 6-55: L = 10: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average Likelihood value vs. Iteration, L = 10, 50 trials
'
I
-.
0 0til
s trto,
PCL tovMa o ieiodvlu
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 10, 50 trials
o0o000
-2860
-0
-286
0
0000000000-00600000
0
-282-
00
0
000OOOC
0
0
0000000
0
-J -290
.C
00
0
0 00
0
Eu -292-
-0.01 -
0
0*
0
1
4
0000
0
-2940
-296-
000000
0 0
0
0
-298'
0
10
20
30
40
50
0
Iteration
(a)
10
ZO40
10
20
Iteration
5
30
40
so
(b)
Figure 6-56: L = 10: (a) Mean PCML Likelihood Convergence (b) Standard Deviation
of PCML Likelihood Convergence
94
PCML: Average eigenspectrum vs. Iteration, L = 10, 50 trials
I
I
PCML: StdDev/Mean of elgenspectrum value vs. Iteration, L = 10, 50 trials
8-_
z 7
C' 6
- -
-7-
[
51
4
-
3
00
2
----------------
-1'
0
I
10
1
1
- 113
0
I
20
30
40
50
0
10
Iteration
20
30
40
50
Iteration
(a)
(b)
Figure 6-57: L = 10: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation of PCML Eigenvalue Convergence
95
L = 5, MVDR Power Estimates and Standard Devia-
6.2.13
tions
CBF: E[PmvD] (dB), L = 5, 50 trials
CBF: StdDev[PMvDRYE[PMVDR
(dB), L = 5, 50 trials
0
15
1
0.8
0.8
10
0.6
0.60.4-
0.4
S0
23
-0.2
-5
0.2 -
5
0.2
4-
0-0.2 -
0
-0.4
-0.4-
-0.6
-
-10
-0.6 -
-5
-0.8
-0.8
-1
-1
-0.5
0
0.5
-10
1
-0.5
U
(a)
0
U,
0.5
(b)
Figure 6-58: CBF, L = 5: (a) Mean Output Power (b) Standard Deviation of Output
Power
96
-15
PCML: E[PMVD] (dB), L = 5, 50 trials
I
I
I
PCML: StdDev[PmvRYE[PMvD] (dB), L = 5, 50 trials
I
0.8-
0.8
-
10
0.6-
-
0.6
0.4-
0.4
5
0.2-
22
-
-0.4-
-
-0.6-
-
-5
0.2
S0
-0.2-
0
-- - - .
-
-
.
. . . . . ..
-0.2
0
-10
-0.4
-0.6
-5
-0.8 -1
0
15
I
-0.8
--
I
1
-0.5
I
I
0
0.5
|
1
-1
-10
-1
I
-0.5
I
0
I
0.5
4
-15
1
U,
(a)
(b)
Figure 6-59: PCML, L = 5: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PmvD] (dB), L = 5, 50 trials
Diagonal Loading: StdDOv[PMVDR]/E[PMVDR] (dB), L = 5, 50 trials
15
I
1 I
0.8-
0
I-
0.810
0.60.4
5
0.2&
I
0.60
-
-
0.4-
-1
1
-5
0.2
Z~ 0
0
-0.2-
-0.2
0
-0.4-
-10
-0.4
-0.6-
-0.6
-5
-0.8
-0.8
-1-
-1
-1
-0.5
0
0.5
1
-10
-1
-0.5
0
0.5
1
U
U
(a)
(b)
Figure 6-60: DL, L = 5: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
97
-15
DMR rank 5: StdDevPMVDRYE[PMVDR (dB), L = 5, 50 trials
DMR rank 5: E[PMVDR] (dB), L = 5,50 trials
0
15
1
0.8
0.8
10
0.6
0.6
0.4
0.2
0.4-
-
5
0.2-
-
0
-0.2- -
-5
: 0
S0
-0.2
-0.4
-0.6
-
-0.6 -
-5
I
-0.8
--
-1
-0.5
0.5
0
III
1
-10
I
-
-1-10
-0.8
-1
-10
-
-0.4-
-0.5
U
(a)
0
U
I
0.5
-15
I
-
(b)
Figure 6-61: DMR, L = 5: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
EV rank 5: StdDev[PMVDRE(PMVDR] (dB), L = 5, 50 trials
EV rank 5: E[PmvDR] (dB), L = 5,50 trials
0
15
1
0.8
0.810
0.6
0.6-
0.4
0.45
0.2
Z2.
0
I
-0.2
I
I
I
0
I
-
-
-5
0.2-
-0.2-
0
-10
-0.4-
-0.4
-0.6
-0.6-
-5
-0.8-
-0.8
-1 -
-1
-1
-0.5
0
0.5
1
-10
-1
-0.5
0
0.5
U
U
(a)
Figure 6-62: EVF, L
Power
=
(b)
5: (a) Mean MVDR Power (b) Standard Deviation of MVDR
98
-15
6.2.14
L
=
5, PCML Algorithm Performance
PCML: E[PSD] (dB), L = 5,50 trials
1
-
-
PCML: StdDev[PSDYE[PSD] (dB), L = 5, 50 trials
30
10
1
0.8
0.8
25
0.6
6
0.6
0.4
20
0.2
Z3.
8
0
-0.2
_
_
-0.4
-0.6
15
0.4
4
0.2
2
S0
0
-0.2
-2
-0.4
-4
-0.6
10
-0.8
-6
-0.8
-1
-8
-1
-
~
-
5
I
-1
(a)
I
-0.5
III
0
U
0.5
1
(b)
Figure 6-63: L = 5: (a) Mean PCML PSD Estimate (b) Standard Deviation of PCML
PSD Estimate
At L = 5 the PCML algorithm continues to achieve the least bias and lowest
standard deviation of any of the other reduced-rank ABF algorithms. For this scenario
and level of snapshot support, all algorithms yield peaks in the neighborhoods of the
9-dB and 12-dB sources. The PCML-MVDR power estimate is clearest in terms
of having the most circular contours around the source locations, and having no
extraneous peaks.
As in the case of L = 10, the 6-dB source is visible in the mean PCML PSD
estimate and is not in the mean PCML MVDR power estimate.
99
-10
PCML: Average a2 value vs. Iteration, L = 5, 50 trials
6.5
PCML: StdDev/Mean of a2 value vs. Iteration, L = 5, 50 trials
6 - 000 00
3
"0000**
000
.5
0
3-
0
500
5-0
00
Co-
50
WL2. 50
00000000000
20.
4.5 -
~1. 5-
0
41 -
-
0
3.5-
0. 5 0000000000000000
0
3
0
10
20
30
40
Iteration
0
50
10
20
30
40
50
Iteration
(a)
(b)
Figure 6-64: L = 5: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average Likelihood value vs. Iteration, L = 5, 50 trials
100000009
-20
I
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 5, 50 trials
I*
-290-
0000000
0
-0.02-
00
-3000
-310
0
0
-0.022
0
00
0
-330-
0000
000
x-0.024 -
0
-320-
0
E0
0
.C
-
0
0
00
:3
0
0
0
0
0
0
0
0
L0.026 -
0
0
0
0
0
-3400
0
0
-0.03-
0
00
0
0
10
20
30
40
-0.03
50
Iteration
01
.3000
10
20
30
40
50
Iteration
(a)
(b)
Figure 6-65: L = 5: (a) Mean PCML Likelihood Convergence (b) Standard Deviation
of PCML Likelihood Convergence
100
PCML: Average eigenspectrum vs. Iteration, L
=
14
z
12-
-
0
5, 50 trials
PCML: StdDev/Mean of elgenspectrum value vs. Iteration, L = 5, 50 trials
3
--
-
8
-6 -
-6-C
2-
44
2
0-
--
~
-9 -
0
10
20
30
40
50
0
Iteration
(a)
10
20
Iteration
30
40
50
(b)
Figure 6-66: L = 5: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation
of PCML Eigenvalue Convergence
101
6.2.15
L = 2, MVDR Power Estimates and Standard Deviations
CBF: E[PMVR] (dB), L = 2, 50 trials
-
1
CBF: StdDev[PMVDR/E[PMVD
T -_
0
0.8
0.8
10
0.6
-
0.4
0.6
P
0.4
0.2
Z3.
(dB), L =2, 50 trials
15
-5
5
0.2
0
-0.2
0
0
-0.2
-0.4
-
-0.4
-0.6
-
-10
-0.6
-5
-0.8
-0.8
-1
-1
-1
-0.5
0
0.5
,
1
-10
-1
U
-0.5
0
0.5
U
(a)
(b)
Figure 6-67: CBF, L = 2: (a) Mean Output Power (b) Standard Deviation of Output
Power
102
-15
PCML: E[PMVDR] (dB), L = 2, 50 trials
I
I
I
PCML: StdDev[MVDREIPMVD
I
0
0.8
0.8
10
0.6
0.6
0.4
0.4
5
0.2
23l.
(dB), L = 2, 50 trials
mmml 15
-5
0.2
.
0
-0.2
. - .. - .- - .- .
-0.2
0
-0.4
................
-0.4
-0.6
-0.6
-5
-0.8
. ....
i
-10
.
.. .
-0.8
-1
-1
-1
-0.5
0
0.5
1
-10
-1
-0.5
U
0
-15
0.5
U
(a)
(b)
Figure 6-68: PCML, L = 2: (a) Mean MVDR Power (b) Standard Deviation of
MVDR Power
Diagonal Loading: E[PMVDR] (dB), L = 2, 50 trials
Diagonal Loading: StdDe[PMVDRYEPMVDR] (dB), L =2, 50 trials
15
1-
0.8-
0
0.810
0.6_
0.6-
0.4-
-5
0.41-
0.2-
5
0.2
0
-0.2
0-0.2-0.4-
-10
-0.4
-0.6-
-0.8
-5
-0.8-
-0.8
-1 -1
I
-0.5
I
I
I
0
0.5
1
-1
L I
-10
-1
,
U
I
-0.5
I
II
0
0.5
1
U
(a)
(b)
Figure 6-69: DL, L = 2: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
103
-15
DMR rank 5: E[PmvDR] (dB), L = 2,50 trials
DMR rank 5: StdDev[PMVDRyE[PMVDR] (dB), L =2, 50 trials
15
0
0.8
0.810
0.6
0.6-
0.4
0.4-
0
x
-0.2
-5
0.2-
5
0.2
2
I
IIII-
I1-
0
-0.2-
-
-0.4
-0.4-
-
-0.6
-0.6-
-
0
-5
-0.8
-10
-0.8 -
-1
-1
-0.5
0
0.5
-1 --
-10
1
-1
U
(a)
-0.5
0
U
0.5
-15
(b)
Figure 6-70: DMR, L = 2: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
EV rank 5: E[PMVDR] (dB), L = 2,50 trials
EV rank 5: StdDev[PMVDRYE[PMVDR] (dB), L = 2, 50 trials
15
0
0.8
0.810
0.6
0.6-
0.4
0.4-
1.
5
0.2
s
0
-0.2
-
-
-5
0.2=
0
-0.2-
0
-0.4
-10
-0.4-
-0.6
-0.6-
-5
-0.8
-0.8
-1
-1
-1
-0.5
0
0.5
1
-10
-1
-0.5
U
(a)
0
U
0.5
(b)
Figure 6-71: EVF, L = 2: (a) Mean MVDR Power (b) Standard Deviation of MVDR
Power
104
-15
6.2.16
L = 2, PCML Algorithm Performance
PCML: E[PSD] (dB), L = 2, 50 trials
PCML: StdDev[PSDyE[PSD] (dB), L = 2, 50 trials
30
10
1
0.8
25
0.6
0.4
0.2
20
C
:'
-0.2
15
0.8
8
0.6
6
0.4
4
0.2
2
0
0
-0.2
-0.4
-2
-0.4
-4
-0.6
10
-0.6
-0.8
-0.8
-6
-8
-1
-1
-1
-0.5
0
0.5
5
-1
1
U
-0.5
0
0.5
1
U
(a)
(b)
Figure 6-72: L = 2: (a) Mean PCML PSD Estimate (b) Standard Deviation of PCML
PSD Estimate
With only L = 2 snapshots of sample support, all of the reduced-rank ABF algorithms fail to yield useful output: No method detects any of the discrete sources, and
all outputs are biased high with a large standard deviation.
This is the only case where the white noise estimate, likelihood function and
eigenvalue spectrum did not converge for the PCML algorithm by the
50
th
iteration 5 .
5 The is the only case where the
white noise estimate converges to a non-zero value by the 5 0th
iteration
105
-10
PCML: StdDev/Mean of a2 value vs. Iteration, L = 2, 50 trials
PCML: Average o2 value vs. Iteration, L =2, 50 trials
6
5-
000
7-
-
0
6-
0
4-
V5--
0
0
U
0
3-
00
0
0
4-
0
0
2-
00
0
0
0
0
0
1 -00
0
cL-
0000
O"-.pppppppppp------pppp
20
30
I
0
-
2-
10
50
40
0
Iteration
0
10
20
30
40
50
Iteration
(a)
(b)
Figure 6-73: L = 2: (a) Mean PCML White Noise Estimate (b) Standard Deviation
of PCML White Noise Estimate
PCML: Average Likelihood value vs. Iteration, L
=
-AA'.
2, 50 trials
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 2, 50 trials
-42
I
0
-444
-0.022 0000 0o 0
0
-
0 00000
-446
00
-
00
-- 0.024 -
0000 00000
-"a
00
000 000000000
og-
000
-0.026
00000
-454
0
-456 -
-458
000
0000
0
-
1-0.03
-
000000000
-0.032
I
10
-0.028
-
0000
000
000000
00
00
o
S000000
- -450
-452
0
I
20
30
-
-0.034
-
40
I
50
0
Iteration
-1
10
20
30
40
50
iteration
(a)
(b)
Figure 6-74: L = 2: (a) Mean PCML Likelihood Convergence (b) Standard Deviation
of PCML Likelihood Convergence
106
PCML: Average eigenspectrum vs. Iteration, L = 2, 50 trials
PCML: StdDev/Mean of elgenspectrum value vs. Iteration, L = 2, 50 trials
0
Z25
20
S_
15
V
410
0
I
I
I
I
10
20
30
40
50
-'
Iteration
10
20
Iteration
30
4
50
(b)
(a)
Figure 6-75: L = 2: (a) Mean PCML Eigenvalue Convergence (b) Standard Deviation
of PCML Eigenvalue Convergence
107
6.3
Noise Estimate at k, = ky
=
0 vs. Snapshots
Figure 6-76 illustrates the MVDR power estimates (mean and standard deviation)
from the various reduced-rank ABF methods at k, = ky = 0 as a function of snapshot
support. The environment at k, = ky = 0 contains only noise, and this point is a
good indication of the bias present in the processing.
In the low snapshot regime, the PCML MVDR power estimate is the least biased.
And, for any given number of snapshots, the PCML MVDR power estimate has the
smallest standard deviation, and is therefore the most stable of the reduced-rank
adaptive algorithms tested.
The Capon-Goodman (C-G) bias and standard deviation curves (which are valid
only for full-rank processing) are plotted for reference. The DL method tends to trace
the C-G curve for bias for L > 50, whereas the DMR and PCML curves are closer to
the optimum.
Average MVDR Power Estimates at u = u =0 vs. Snapshots
Ensemble M
40
S
Dev / Mean of MVDR Power Estimates at u = u = 0 vs. Snapshots
-CBF
30
-
--
. -
- -
- -MVDR-DL
CBF
-U vRD
4- MVD
R5
-~MvDR-Evs
MvDR-PCML
2--- -ev -
4-
0 - --
-..
..
---..
---
-
MR5
MvDR-EVS
MVDR-PCML
C- S-d
20
0
1
---.
.
-- .. -
ot
-
-.-- ---.
-- - -.
.--.--.
-..
1 0
0
0
100
150
200
-
-5
-..-.-.-.-.-.-.-
-
50
-L
Snapshots
1
-
150
20
Snapshots
(a)
(b)
Figure 6-76: (a) Mean Power Estimate at u. = uY = 0 (b) Standard Deviation of
Power Estimate at u, = uY = 0
108
6.4
Low-Level Source Detection
Figure 6-77 illustrates the power estimates from the various reduced-rank ABF methods at the point in the wavenumber grid nearest to the low-level (6 dB) source relative
to the local mean. The local mean was calculated to be the average power in the eight
points adjacent to the point nearest to the source. Running the data through such a
normalizer reduces the visual effects bias can have on the power estimates.
The purpose of this figure is to illustrate the ability of each algorithm to detect
the low-level source as a function of snapshot. In order for a discrete source to be
detected at a particular point in the wavenumber grid, that point would have to be
larger than its neighboring points by some amount.
6 dB Source Detection: Power Above Local Mean
7
I:
S5
Ensemble MVDR
CBF
- -
6
-o
-
-
----
---
MVDR-DL
-
-..- .-
.-
d4+
0
MVDR-DMR5
MVDR-EV5
MVDR-PCML
PSD-PCML
-- .
-.
...
-.. ----.
...
3
....--..
.--.
0
0 2
--:
-- -
0
0
-2
-3
0
50
100
Snapshots
150
200
Figure 6-77: Power Near Low Level Source r.e. Local Mean
In the low snapshot regime, the PCML MVDR power estimate has the largest peak
at the 6-dB source out of all the reduced-rank ABF algorithms. Once the number
of snapshots becomes less than 30, this peak is no longer visible in the mean MVDR
109
power plots where a contour is placed every 2 dB.
For the cases of L = 5, 10 and 20 the peak at the 6-dB source was not visible in
the mean PCML-MVDR power estimates, but was visible in the mean PCML-PSD
estimates. This suggests that the PCML-PSD may be a more appropriate data set
to use for source detection. Figure 6-77 also includes the value of the PCML-PSD
estimate at the point nearest to the 6-dB source relative to the local mean. For all
levels of snapshot support, except for L = 2, the PCML-PSD has the most distinct
peak in the area of the low-level source out of all the output power estimates.
110
6.5
Variants of PCML Implementation
6.5.1
Wavenumber Grid Spacing
In order to observe the effect of a wavenumber grid spacing Au
$
0.1 on the output
of the PCML algorithm, plots of the PSD estimate (mean and standard deviation)
are given for Au = 0.2 and Au = 0.05 at L = 200 and L = 30 snapshots.
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDev[PSDYE[PSD] (dB), L = 200, 50 trials
30
10
1
0.8
25
0.6
0.4
0.8-
-
0.6-
-
0.2
-
0.2-
S0
0
-0.2
-
-
15
-0.4
-0.6
10
-0.8
6
4
0.4
20
8
2
0
-0.2--
-2
-0.4 -
-4
-0.6 --0.8 --
-6
-8
-1
-1
-0.5
0
0.5
1
5
-1
-0.5
U
(a)
0
U,
0.5
1
(b)
Figure 6-78: L = 200, Au = 0.2: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
As shown in Figures 6-78 and 6-79, using the coarser wavenumber grid spacing of
Au = 0.2 causes the PCML algorithm to be unable to resolve the peak in the power
spectrum near the weak discrete.
For the grid spacing of Au = 0.1, all three discretes are visible at L = 200 and L
-
30 snapshots.
Using a finer grid spacing of Au = 0.05 does produce a "clearer" PSD estimate at
L = 200 and L = 30 than the grid spacing of Au = 0.1. However, no new information
111
-10
PCML: E[PSD] (dB), L = 30, 50 trials
PCML: StdDev[PSD]/E[PSD] (dB), L = 30,50 trials
30
1
I
0.8
8
0.8
25
0.6
6
0.6
0.4
4
0.4
0.2
20
0.2
15
-0.2
-2
-0.4
-4
0
K
10
I
0
-0.2
-
-0.4
-
-0.6
-
2
0
-0.6
10
-0.8
-6
-0.8
-1
-8
-1
-1
-0.5
0
0.5
5
1
-1
-0.5
U
0
0.5
1
-10
U
(a)
(b)
Figure 6-79: L = 30, Au = 0.2: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDev[PSDyE[PSD] (dB), L = 200,50 trials
30
1
10
0.8
a
0.8
25
0.6
0.4
20
0.2
Z2. 0
-0.2
-
-
6
0.6
0.4
4
0.2
2
0
0
-0.2
15
-0.4
-2
-0.4
-0.6
-4
-0.6
10
-0.8
-6
-0.8
-1
-1
-0.5
0
0.5
1
5
-8
IL I
I
-1
-0.5
U
III
0
0.5
1
U
(a)
(b)
Figure 6-80: L = 200, Au = 0.05: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
112
-10
1-I
PCML: E[PSD] (dB), L = 30, 50 trials
I
I
I
PCML: StdDev[PSD]/E[PSD] (dB), L = 30, 50 trials
30
10
8
0.8
0.8
0.6
0.6
6
0.4
4
0.2
2
0
0
0.4
0.2
20
:&~0
-0.2
-0.2
15
-0.4
-0.4
-0.6
-0.6
10
-0.8
-
-2
-+ -
-4
-6
-0.8
-8
-1
I
SI
-1
-0.5
0
U
0.5
1
5
1
-05
0
05
1
U
(a)
(b)
Figure 6-81: L = 30, Au = 0.05: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
is added, and the computational requirements exponentially grow with the decreased
wavenumber spacing.
113
-10
6.5.2
Covariance Matrix Taper
Plots of the PSD estimates (mean and standard deviation) generated using the Hanning Taper and Triangle Taper are given for L = 200 and L = 30 snapshots in order
to observe the effect of using a non-uniform taper over each point of the PCML PSD
estimate,
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDev[PSDYE[PSD] (dB), L = 200, 50 trials
30
1
10
0.8
8
0.8
25
0.6
0.2
20
~.0
-0.2
-
-
6
0.6
0.4
0.4
4
0.2
2
0
0
-0.2
15
-0.4
-2
-0.4
-0.6
-0.8
-0.8
-1
-1
-1
-0.5
0
0.5
1
-4
-0.6
10
5
-6
-8
-1
-0.5
U
0
0.5
1
U
(a)
(b)
Figure 6-82: L = 200, Hanning Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
At both L = 200 and L = 30, there is no noticeable difference in the mean PSD
estimates generated using different covariance matrix tapers. Although, the PSD
estimate generated using a Triangular taper has a slightly higher standard deviation
in both cases.
114
-10
I
1
PCML: E[PSD] (dB), L = 30, 50 trials
I
I
----= -
- -r
PCML: StdDev[PSD]/E[PSD] (dB), L = 30, 50 trials
30
10
0.8
8
0.8-
25
0.6
0.4
0.2
20
-0.2
15
0.6
6
0.4
4
0.2-
2
-0.2 -
-0.4
-0.4 -
-0.6
-0.6 -
10
-0.8
. .
.
.
.
.
.
-
-2
4
..
-
-0.8-8
-1
-1
-0.5
0.5
0
U
5
1
-1
-0.5
(a)
0
U
0.5
-10
1
(b)
Figure 6-83: L = 30, Hanning Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
PCML: E[PSD] (dB), L = 200, 50 trials
1
-
I
-
PCML: StdDev[PSD]/E[PSD] (dB), L = 200, 50 trials
30
10
0.8
8
0.8
25
0.6
0.4
20
0.2
S0
0.6
6
0.4
4
0.2
2
S0
-0.2
0
-0.2
15
-0.4
-
-
-0.4
-0.6
10
-4
-0.6
-0.8
-0.8
-2
-8
-8
-1
-1
-1
-0.5
0
U
0.5
1
5
-1
-0.5
0
0.5
U
(a)
(b)
Figure 6-84: L = 200, Triangle Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
115
-10
PCML: E[PSD] (dB), L = 30, 50 trials
I
1
I
I
PCML: StdDev[PSD]IE[PSD] (dB), L
30
|I
0.8
-
0.4
20
-0.4
-
0.4
4
0.2
2
-
-0.2
15
-
6
S0
-0.2
8
i:e-
0.6
25
0.2
30, 50 trials
10
0.8
0.6
=
-.....
-..
- -y.:
0
. ..*
-2
-0.4
-0.6
-4
-0.6
10
-0.8
-6
-0.8
-8
-1
I
-1
-0.5
I
iI
0
U
0.5
-1
1
5
-1
-0.5
0
0.5
1
U
(a)
(b)
Figure 6-85: L = 30, Triangle Taper: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
116
-10
6.5.3
Spreading of Discrete Sources
In order to observe the effect of using a non-spread plane wave model for a discrete
source, plots of the PSD estimate (mean and standard deviation) are given for using
plane wave sources at L = 200 and L = 30 snapshots.
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDov[PSDyE[PSD] (dB), L = 200, 50 trials
30
1
10
0.8
0.6
0.2
0.4
4
20
0.2
2
0
0
15
-0.2
0
&
-0.2
6
0.6
0.4
&
8
0.8
25
-0.4
-2
-0.4
-0.6
-4
-0.6
10
-0.8
-6
-0.8
-8
-1
-1
-0.5
0
0.5
-1
5
-1
U
-0.5
0
0.5
1
U
(a)
(b)
Figure 6-86: L = 200, Source Spread . =
= (27r)
i.i0o: (a) Mean PCML PSD
Estimate (b) Standard Deviation of PCML PSD Estimate
As shown in Figures 6-86 and 6-87, modeling the discrete sources as plane waves
with no directional spreading (or very little spreading) causes wells to appear around
the positions of the discretes in the PSD estimates. The wells are visible when L =
200 and L = 30, but are more pronounced when a higher number of snapshots is used.
An immediate physical explanation for this behavior is not known. The derivative
of the PSD estimate at the points adjacent to the discrete sources seem to always
remain negative (therefore, causing the power level at these points to decrease) as the
PCML algorithm iterates. Understanding this behavior remains for future work.
117
-10
PCMVL: E[PSD] (dB), L = 30, 50 trials
PCMLL =30,0
E[SD] tralsPCIVL:
dB)
1
0.8
--
10
-
8
0.825
0.4 -
StdDnvrPSDUIRPS131 IdB) L = 30 50 trils
1
-
0.1
0.6_
0.2
1
330
0.6_
6
.4
0.44
--
0.2
20
0
--
--
2
0
-0.2-
15
-0.2-
-0.4 --.
-2
4
-0.6-
10
-0.6-
-0.8-
-6
-0.8-8
1
-0.5
0
0.5
1
5
-
(a)
-0.5
0
0.5
1
(b)
Figure 6-87: L = 30, Source Spread # =
= (27r)
.0,,i:
Estimate (b) Standard Deviation of PCML PSD Estimate
118
(a) Mean PCML PSD
-10
6.5.4
Constant Estimate of White Noise
Plots of the PSD estimate and likelihood function convergence (mean and standard
deviation) are given for several cases where the white noise estimate is held constant.
That is, the estimate is not updated as the PCML algorithm iterates. The plots were
generated for L = 200 and L = 30 snapshots with white noise levels of 0 dB (ensemble
value), -2 dB, and +2 dB.
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDev[PSD]/E[PSD] (dB), L = 200, 50 trials
30
1
10
0.8
8
0.8
25
0.6
6
0.6
0.4
0.2
20
0
-0.2
4
0.2
2
0
0
-0.2
15
-
-0.4
0.4
-
-2
-0.4
-0.6
-4
-0.6
10
-6
-0.8
-0.8
-1
-1
-8
-1
-0.5
0
0.5
1
5
-1
-0.5
U
0
0.5
U
(a)
(b)
Figure 6-88: L = 200, &PCML = -2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
From the power spectral density plots it appears that holding white noise level
constant does not have a noticeable effect on the final mean value of the PCML PSD
estimate or its standard deviation for any of the levels of white noise used. Also, the
convergence of the likelihood function itself seems to be unaffected.
119
-10
PCML: Average Likelihood value vs. Iteration, L = 200, 50 trials
-ZM.
-300
I-
I
I
PCMII
I
-~-
%dDev/Mean of Likelihood value vs. Iteration, L = 200, 50 trials
-
-3
-400
-
000
00
-4
0
00000000000
0000
0000000000
-500
-600
Vl
-700
Vr
-7
-800
-900
j1 9
-1000
-
-10
-1100
-
-11
0
0
_UUu-
0
10
20
30
40
0
50
Iteration
10
20
30
40
50
Iteration
(a)
(b)
Figure 6-89: L = 200, &2CML = -2 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence
PCML: E[PSD] (dB), L = 30, 50 trials
PCML: StdDev[PSD]/E[PSD] (dB), L
30
1
=
30, 50 trials
10
0.8
8
0.8
25
0.6
0.4
20
0.2
S0
22.
-0.2
-
-
0.6
6
0.4
4
0.2
2
0
0
-0.2
15
-0.4
-2
-0.4
-0.6
10
-4
-0.6
-0.8
-0.8
-6
-8
-1
-1
-1
-0.5
0
0.5
1
5
-1
-0.5
u
(a)
0
u
0.5
'
(b)
Figure 6-90: L = 30, &2CML = -2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
120
-10
PCML: Average Likelihood value vs. Iteration, L = 30, 50 trials
-200.
PCML : StdDev/Mean of Likelihood value vs. Iteration, L = 30, 50 trials
-300
-0.01
0
--0.02
-400 -
0
5 -0.03
-500 -
U-0.04
9-600
-. 05
-700[
-0.06
-
-800-
0
10
20
30
40
-0.07
--.
0
50
10
Iteration
20
30
40
50
Iteration
(a)
(b)
Figure 6-91: L = 30, 6rC'L = -2 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDev[PSDyE[PSD] (dB), L = 200, 50 trials
30
1
10
1
0.8
8
0.8
25
0.6
6
0.6
0.4
0.2
20
-0.2
-
-
0.4
4
0.2
2
:,~ 0
0
-0.2
15
-0.4
-2
-0.4
-0.6
-4
-0.6
10
-0.8
-6
-0.8
-1
-8
-1
-1
-0.5
0
0.5
1
5
-1
-0.5
U
0
0.5
U
(a)
(b)
Figure 6-92: L = 200, cYPCML
0 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
121
-10
PCML: Average Likelihood value vs. Iteration, L = 200, 50 trials
-200 1
1
1
-T
I
PCML;-
dDev/Mean of Likelihood value vs. Iteration, L = 200, 50 trials
o
-300
000
-3
-400
ooooo
I
oooo*
-4
-500
i -6
-600
&3
br -71
-700
-
...
-800
0-9
-900
-1C0
-11 00
-10
10
20
30
40
0
50
10
Iteration
20
30
40
50
Iteration
(a)
(b)
Figure 6-93: L = 200, &2CML = 0 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence
PCML: E[PSD] (dB), L = 30,50 trials
PCML: StdDev[PSD]/E[PSD] (dB), L = 30, 50 trials
30
1
10
0.8
0.6
0.2
0.4
4
20
0.2
2
Sa
0
15
-0.2
0
-0.2
6
0.6
0.4
23
8
0.8
25
-0.4
-2
-0.4
-0.6
-4
-0.6
-0.8
10
-0.8
-6
-8
-1
-1
-1
-0.5
0
0.5
5
-1
u
(a)
-0.5
0
u
0.5
1
(b)
Figure 6-94: L = 30, &2CML = 0 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
122
-10
PCML: Average Likelihood value vs. Iteration, L = 30, 50 trials
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 30, 50 trials
I
-300-
00000
000oooofto00000000
-400-
ooooo
oom
T-0.02
--500-04
-600-
~0.0
-700-
0
0
0
10
20
30
40
-U Im0
50
Iteration
10
20
30
40
50
Iteration
(a)
(b)
Figure 6-95: L = 30, &1,CM = 0 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence
PCML: E[PSD] (dB), L = 200, 50 trials
PCML: StdDev[PSDYE[PSD] (dB), L = 200, 50 trials
30
10
1
1
0.8
8
0.8
25
0.6
6
0.6
0.4
4
0.4
20
0.2
S0
-0.2
15
0.2
2
S0
0
-0.2
-0.4
-
-2
I
-0.4
x-
-0.6
-4
-0.6
10
-0.8
-6
-0.8
-1
-
-1
-1
-0.5
0
0.5
1
5
-1
U
-0.5
0
0.5
U
(a)
(b)
Figure 6-96: L = 200, &PCML = 2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
123
-8
-10
-
20PCML:
0F1
Average Likelihood value vs. Iteration, L = 200, 50 trials
1
1
1
PCMII
-~
-300
%dDev/Mean of Likelihood value vs. Iteration, L = 200, 50 trials
-.
-3 0000 0000
0000000
0
-40C
*
-4
'0
0 0
0
---
---
---
0o
ooo0000
-5
-500 -
~1;
:3
-600 -7
-
-700
.1 -I
-800
-9
-900
-1C
-innni a
0
I
10
I
20
I
30
I
40
50
0
I
I
I
I
10
20
30
40
iteration
50
iteration
(a)
(b)
Figure 6-97: L = 200, &2CML = 2 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence
I
1
PCML: E[PSD] (dB), L = 30,50 trials
I
I
-- 1
PCML: StdDev[PSD]/E[PSD] (dB), L = 30, 50 trials
30
0.8
8
0.8
25
0.6
0.4
0.2
0.6
6
0.4
4
20
0.2
2
S0
0
15
-0.2
-2
&.0
-0.2
-0.4
-0.4
-0.6
10
10
-4
-0.6
-0.8
-0.8
-1
-6
-a
-1
-1
-0.5
0
0.5
5
-1
u
-0.5
0
05
U
(a)
(b)
Figure 6-98: L = 30, d4 CML = 2 dB: (a) Mean PCML PSD Estimate (b) Standard
Deviation of PCML PSD Estimate
124
-10
PCML: Average Likelihood value vs. Iteration, L = 30, 50 trials
PCML: StdDev/Mean of Likelihood value vs. Iteration, L = 30, 50 trials
n
I
III
300 -
I
00000000000000000000004000000000
0
o@@oooo o.oooo00000
oo000000000
400
-0.02
.j
600600
04
-.
700-
-
-
0
Soc
0
10
10
I
20
I
30
I0
40
o6
-0-
50
50
10
Iteration
20
30
40
50
Iteration
(a)
(b)
Figure 6-99: L = 30, &PCML = 2 dB: (a) Mean PCML Likelihood Convergence (b)
Standard Deviation of PCML Likelihood Convergence
125
126
Chapter 7
Conclusions
7.1
The PCML Method as a Reduced Rank Adaptive Processor
The goal of this thesis is to assess the performance of the PCML adaptive processor as a reduced-rank adaptive processor, operating with a limited amount of data,
in comparison with the performance of other current reduced-rank adaptive processing techniques (Diagonal Loading (DL), Dominant Mode Rejection (DMR), and
Eigenvalue Filtering (EVF)). The performance can be judged in terms of i) source
detectability; ii) power estimate bias; and iii) power estimate standard deviation.
It was shown that the PCML-MVDR method achieves the best power estimate
(having the lowest bias and lowest standard deviation) at a given number of snapshots. This holds true even into the snapshot deficient regime where reduced-rank
methods are needed. In terms of source detectability, the PCML-MVDR power output performs as well, or better than the other RR-ABF methods for detecting a peak
in the area of the low level source. The PCML-PSD estimate exhibits a peak in
the area of the low-level source even in cases where the PCML-MVDR output does
127
not. This and the fact that these peaks tend to rise higher above the local mean for
a given number of snapshots suggest that the PCML-PSD estimate may serve as a
more appropriate data set for source detection. The compromise is that the PCML
method is significantly more computationally intensive than the other reduced-rank
methods due to its iterative update.
7.2
Effects of Various Implementation Choices
This section summarizes the observed effects on the PCML PSD Estimate when
different processing or model parameters are changed. The parameters varied were
the wavenumber grid spacing, the amount of discrete source spreading, the covariance
matrix taper, and the method of white noise estimation.
Grid Spacing
It was shown in Section 6.5.1 that using a coarser wavenumber grid spacing than the
nominal resolution of the array causes the PCML algorithm to be unable to resolve
the peak in the power spectrum near the weak discrete. Using a finer grid spacing
than the nominal resolution of the array does produces a "clearer" PSD estimate at
a large computational expense, but does not necessarily add any new information.
Covariance Matrix Taper
Of the three covariance matrix tapers used (Uniform, Hanning, and Triangular), there
was no noticeable difference in the mean PSD estimates in terms of source resolution
or bias. The PSD estimate produced with a Triangular taper had slightly larger
standard deviations than those of produced with the other windows.
128
Discrete Source Model
With simulated data, discrete source models should have some amount of spreading
in order to avoid having wells appear in the PCML-PSD estimate. A physical explanation for this behavior is not known, but it should not pose a problem when the
PCML method is applied to real data, as some amount of source spreading is usually
present.
White Noise Estimation Method
Section 6.5.4 shows that using a single iteration for estimating the white noise power
level does not produce noticeably different results than if the white noise power were
updated at each iterative cycle. Also, the actual value used for the constant white
noise estimate within ± 2 dB of the ensemble value did not seem to effect the final
PCML outputs.
In the high snapshot regime, the white noise power does not necessarily converge
to its ensemble value.
This may suggest that a certain amount of power can be
distributed between the sensor noise and the isotropic noise in the PCML algorithm,
without noticeably affecting the output. So, the computational expense of continually
updating the white noise may not be necessary as part of the convergence process.
129
130
Appendix A
Gradient Derivations
A.1
Gradient of Likelihood Function With Respect
to Directional Power Spectrum
This section provides the derivation of the gradient used in the iterative update of the
power estimate in the PCML algorithm. The gradient is evaluated at each iteration
where P(w, k,) is the current PCML power estimate at wavenumber ka, R is the
covariance matrix obtained from inverse Fourier Transforming P(w, k,), and
is the sample covariance matrix computed directly from the data.
OL (R, RDATA)
OP (w, k,)
aL (R,
a
-
RDATA)
aP (w, k,)
H
OP (w,k.) (10g R
_
(log IRI)
OP (w, k,)
0
131
-
Tr (R-IDATA)
0 (Tr (R-iDATA))
OP (w, k,)
RDATA
DL (R, RDATA)
=
aP (w, k,)
DL (R,DATA)
0
DP (w, kn)
-Tr (R-1
OR
OP (W,
+ Tr R-1 O
R-()nDATA
( P (w, k,,)
kn))
-Tr (R-1vk(kl)v'H(kn)) + Tr (R-vk(kl)vH (kn)R-IDATA
DL (R, DATA)
-vgH(kn)R-vk(kl)
'c
aP (w, kn)
+ VH (k)R~RDATAR-'vk(k,)
aL RA DATA
P(w,kn)
Equation A.1 is a convenient form for computing -
(A.1)
but more insight
can be gained from further rearrangements:
DL (R, RDATA)
OP (w, kn)
1 L
(x
-PijVDR(w, kn) +
DL (R, RDATA)
aP (w, kn)
OP (w, kn
pj1
C -Pj,VDR (wlkf)+
DL (R, RDATA)
VH(kn)R- 1 XlXHR-lvk(k,)
L1=1k1
MVDR
PVDR(w,
MC
kn)
L
(
22
k)
1=1 I WMVDRX1
PMVDR
(w,
MVDR
L1=1I
2
(A.2)
kn)
An interpretation of equation A.2 is that minimizing the likelihood function using
the first gradient with respect to P (w, kn) attempts to:
1. Design an optimal weighting,
WMVDR,
for direction kn;
2. Process the data snapshots for an output power estimate; and
132
3. Compare the output power estimate with
A.2
PMVDR
(w, kn) for consistency.
Gradients of Likelihood Function With Respect to Sensor Noise
A.2.1
First Gradient
DL (R,
DATA)
a
IRI 2
Du (-log
002
DL (R, fDATA)
OU2
aL
Tr (R-DATA))
& (Tr (R-DATA))
D (log IRI)
+R
-2
(R, fDATA)
R-1
=-Tr
O_
DL (R,
DATA)
(R,
-Tr (R-I)
ADATA
a2
+ Tr (R-1 OR RDT)
Oou2
-T
R=
AL
)
-r(=--rR-
+
Tr (R-1IR-1RDATA
+ Tr (R-iDATAR-1)
Equation A.3 is a convenient form for computing
9L (RDATA,
can be gained from further rearrangements:
OL (R,
ADATA)
DL
Tr (RA NDATAR1
133
-
R1)
(A.3)
but more insight
DL (R, fDATA)
= Tr (R- (fTAR--
2u
(A.4)
I))
An interpretation of equation A.4 is that minimizing the likelihood function using
the first gradient with respect to a 2 checks that the inverse of R whitens the sample
covariance matrix,
A.2.2
RDATA-
Second Gradient
02L (RIDATA)- =
a 2 L (R,
r2 (-Tr (R-1) + Tr (R-IDATAR- ))
=-
(a r2) 2
RDATA)
2
2
(Do- )
a L (R, RDATA)
= -Tr
(R2)
+ Tr
OR-'
R-'nDATA
+
R-2
R2 RDATAR)
-
2
(ao2
a
2
L (R,
=Tr R-
ao
R-1)
-Tr
R-nDATAR-1
ao
RR1
+ R- 1 OR-DATAR-1
2
2
D0.
RDATA)
2
D2L (R,
(2 )
DATA)
2
(BO.2)2
= Tr
(R1IR-1) -Tr
(R-'RDATAR-1IR-
+ R-IR-RDATAR1)
= Tr (R-R-1) - Tr (R-1fDATAR-R-- + R~1R-1nDATAR-
134
Appendix B
2-D Windows for Integral
Approximation
The windows presented here are intended for use in the iterative covariance matrix
update equation to better approximate the inverse Fourier transform integral:
[Rm]ij
Pm- 1 (w, k,) e-n(pipj)
n
W,j
+
am
Al
All windows in this section assume the power estimates are evaluated on an equally
spaced grid of points in 2-D wavenumber space, as depicted in Figure 5-2. The spacing
between two points (along the same dimension) is denoted as Ak= -
u, where u
is the normalized wavenumber, or slowness vector.
B.1
Uniform
The 2-D Uniform Window is expressed in 2-D normalized wavenumber space as:
135
W(u) = W(uX, U,) =
1
for JUXI < 4" and juyl < AU
0
otherwise
and the inverse Fourier transform is:
Wij= JW (u) e+juT"(Pi-Pj) du
wi,j
Au,e
j
e/ j X Yu(Pi,Y-Pj,y)
Au
-ux
(Pi,x P,x) du X)
2
Wj,j
B.2
AU sinc
(27r Au
'A 2 (Pi,
(i
Au sinc u27r
( A r2 u P
Pj,X))
dul
Y),
\
'yP
))
Hanning
The 2-D, separable Hanning Window is expressed in 2-D normalized wavenumber
space as:
W(u) = W(uX, uY) =
{
}+
Cos
A
0UX)
(
,for juxI < Au and ju.1 < Au
+ ! cos ,
otherwise
0
and the inverse Fourier transform is:
Wij =
W (u) e+j
136
TUT(pi-P)
du
+Au (1
1
2
-Au
(
+A
cos
7r
Au
uX e+j2'ux(Pix-Pj,x) dux
u (1 + 1-Cos
2
-Au
7r
-U.
Au
e+j2'uy(PiY-Piyv) du,
wij = (Au sinc (+Au(Pi,X - Pj,X)) + - --
Au
-sic
2
Au
27
Pjx) -7r
(Au sinc 2Au(pi,y
+
Au
Au.
2 sinc
27
rAu(pi, Pj,x)
27
-- A U(pi,, - pj,Y) - 7r) +
p
2 sinc
B.3
2
+ 2 sinc
PY)+7r
Triangular
The 2-D separable Triangle Window is expressed in 2-D normalized wavenumber space
as:
W(u) =W(ux, uY) = -y(ux)'Y(uy)
where
137
J
7Y(a) =
a+1
,for -Au < a < 0
-±a+1
,for 0 < a < Au
0
,otherwise
Viewing 'y(a) as the convolution of two boxcars, each of width Au/2 and height
1/
allows one to write the inverse Fourier Transform by inspection:
TAu'
WWi'j= (u)
A
2
27
sinC2 (A
2(2
Au
2 (P'.
j')
138
sinc2
A
(Au
2 (iy-P~)
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