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Principles of Signal Detection and Parameter Estimation.pdf
Preface
Contents
A Note to Instructors
Introduction
Book Organization
Complementary Readings
References
Binary and M-ary Hypothesis Testing
Introduction
Bayesian Binary Hypothesis Testing
Sufficient Statistics
Receiver Operating Characteristic
Neyman-Pearson Tests
ROC Properties
Minimax Hypothesis Testing
Gaussian Detection
Known Signals in Gaussian Noise
Detection of a Zero-Mean Gaussian Signal in Noise
M-ary Hypothesis Testing
Bayesian M-ary Tests
Sufficient Statistics for M-ary Tests
Performance Analysis
Bounds Based on Pairwise Error Probability
Bibliographical Notes
Problems
References
Tests with Repeated Observations
Introduction
Asymptotic Performance of Likelihood Ratio Tests
Bayesian Sequential Hypothesis Testing
Sequential Probability Ratio Tests
Optimality of SPRTs
Bibliographical Notes
Problems
Proof of Cramér's Theorem
References
Parameter Estimation Theory
Introduction
Bayesian Estimation
Optimum Bayesian Estimator
Properties of the MSE Estimator
Linear Least-squares Estimation
Estimation of Nonrandom Parameters
Bias
Sufficient Statistic
Cramér-Rao Lower Bound
Uniform Minimum Variance Unbiased Estimates
Asymptotic Behavior of ML Estimates
Consistency
Asymptotic Distribution of the ML Estimate
Bibliographical Notes
Problems
Derivation of the RBLS Theorem
References
Composite Hypothesis Testing
Introduction
Uniformly Most Powerful Tests
Invariant Tests
Linear Detection with Interfering Sources
Generalized Likelihood Ratio Tests
Asymptotic Optimality of the GLRT
Multinomial Distributions
Exponential Families
Bibliographical Notes
Problems
Proof of Sanov's Theorem
References
Robust Detection
Introduction
Measures of Model Proximity
Robust Hypothesis Testing
Robust Bayesian and NP Tests
Clipped LR Tests
Asymptotic Robustness
Least Favorable Densities
Robust Asymptotic Test
Robust Signal Detection
Least-Favorable Densities
Receiver Structure
Bibliographical Notes
Problems
References
Karhunen-Loève Expansion of Gaussian Processes
Introduction
Orthonormal Expansions of Deterministic Signals
Eigenfunction Expansion of Covariance Kernels
Properties of Covariance Kernels
Decomposition of Covariance Matrices/Kernels
Differential Characterization of the Eigenfunctions
Gaussian Reciprocal Processes
Partially Observed Gaussian Reciprocal/Markov Processes
Rational Stationary Gaussian Processes
Karhunen-Loève Decomposition
Asymptotic Expansion of Stationary Gaussian Processes
Bibliographical Notes
Problems
References
Detection of Known Signals in Gaussian Noise
Introduction
Binary Detection of Known Signals in WGN
Detection of a Single Signal
General Binary Detection Problem
M-ary Detection of Known Signals in WGN
Detection of Known Signals in Colored Gaussian Noise
Singular and Nonsingular CT Detection
Generalized Matched Filter Implementation
Computation of the Distorted Signal g(t)
Noise Whitening Receiver
Bibliographical Notes
Problems
References
Detection of Signals with Unknown Parameters
Introduction
Detection of Signals with Unknown Phase
Signal Space Representation
Bayesian Formulation
GLR Test
Detector Implementation
Detection of DPSK Signals
Detection of Signals with Unknown Amplitude and Phase
Bayesian Formulation
GLR Test
Detection with Arbitrary Unknown Parameters
Waveform Parameter Estimation
Detection of Radar Signals
Equivalent Baseband Detection Problem
Cramér-Rao Bound
ML Estimates and GLR Detector
Ambiguity Function Properties
Bibliographical Notes
Problems
References
Detection of Gaussian Signals in WGN
Introduction
Noncausal Receiver
Receiver Structure
Smoother Implementation
Causal Receiver
Asymptotic Stationary Gaussian Test Performance
Asymptotic Equivalence of Toeplitz and Circulant Matrices
Mean-square Convergence of ST
Large Deviations Analysis of the LRT
Detection in WGN
Bibliographical Notes
Problems
References
EM Estimation and Detection of Gaussian Signals with Unknown Parameters
Introduction
Gaussian Signal of Unknown Amplitude in WGN of Unknown Power
EM Parameter Estimation Method
Motonicity Property
Example
Convergence Rate
Large-Sample Covariance Matrix
Parameter Estimation of Hidden Gauss-Markov Models
EM iteration
Double-sweep smoother
Example
GLRT Implementation
Bibliographical Notes
Problems
References
Detection of Markov Chains with Known Parameters
Introduction
Detection of Completely Observed Markov Chains
Notation and Background
Binary Hypothesis Testing
Asymptotic Performance
Detection of Partially Observed Markov Chains
MAP Sequence Detection
Pointwise MAP Detection
Example: Channel Equalization
Markov Chain Model
Performance Analysis
Bibliographical Notes
Problems
References
Detection of Markov Chains with Unknown Parameters
Introduction
GLR Detector
Model
GLR Test
Per Survivor Processing
Path Extension
Parameter Vector Update
EM Detector
Forward-backward EM
EM Viterbi Detector
Example: Blind Equalization
Convergence Analysis
Convergence Rate
Bibliographical Notes
Problems
References
Index
Principles of Signal Detection
and Parameter Estimation
Bernard C. Levy
Principles of Signal Detection
and Parameter Estimation
123
Bernard C. Levy
Dept. of Electrical and
Computer Engineering
University of California
1 Shields Avenue
Davis, CA 95616
ISBN: 978-0-387-76542-6
DOI: 10.1007/978-0-387-76544-0
e-ISBN: 978-0-387-76544-0
Library of Congress Control Number: 2008921987
c 2008 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY
10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection
with any form of information storage and retrieval, electronic adaptation, computer software, or by similar
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The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
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to proprietary rights.
Printed on acid-free paper
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springer.com
In these matters the only certainty is that nothing is certain.
Pliny the Elder
Preface
As a discipline, signal detection has evolved significantly over the last 40 years.
Some changes have been caused by technical advances, like the development
of robust detection methods, or the use of the theory of large deviations to
characterize the asymptotic performance of tests, but most changes have been
caused by transformations in the engineering systems to which detection tech-
niques are applied. While early applications of signal detection focused on
radar and sonar signal processing or the design of digital communication re-
ceivers, newer areas of application include image analysis and interpretation,
document authentification, biometrics, and sensor or actuator failure detec-
tion. This expanded scope of application has required some adjustment in
standard ways of formulating detection problems. For example, image process-
ing applications typically combine parameter estimation and detection tasks,
so the separation of parameter estimation and detection in distinct operations
typical of early communication systems, where parameter estimation was ac-
complished through the use of training signals, needs to be abandoned. Other
changes have occured in the design of communication systems which make
it increasingly difficult to treat the detection of communications signals and
of radar/sonar signals in a unified manner. This common framework assumes
implicitly that intersymbol interference is not present and that channel cod-
ing and modulation are implemented separately, since in this case modulated
signals can be detected one symbol at a time. But modern communication
systems are typically designed to operate over bandlimited channels where in-
tersymbol interference is present, and starting with the introduction of trellis
coded modulation, modulation and coding have become intertwined. In this
context, the detection of modulated signals can no longer be treated on a
symbol-by-symbol basis but needs to be viewed as a sequence detection prob-
lem, where the sequence is generated by a Markov chain. Another feature of
modern radar and communication systems, in particular wireless systems, is
that they often need to operate in a rapidly changing environment. So even
if training or calibration signals are available to estimate the system param-
eters, because parameters may change quickly, it is desirable to constantly
VIII
Preface
refresh estimates while at the same time performing detection tasks on re-
ceived signals. In other words, detection and estimation need to be performed
simultaneously and can no longer be viewed as separate tasks. Finally, an-
other feature of modern engineering systems to which detection algorithms
are applied is that due to modelling errors, imperfect calibration, changes in
the environment, as well as the presence of interfering signals, it is not en-
tirely realistic to assume that accurate models are available, and thus robust
detection techniques need to be applied.
The objective of this book is to give a modern presentation of signal de-
tection which incorporates new technical advances, while at the same time
addressing issues that reflect the evolution of contemporary detection sys-
tems. Recent advances which are covered include the use of the theory of
large deviations to characterize the asymptotic performance of detectors, not
only for the case of independent identically distributed observations, but also
for detection problems involving Gaussian processes or Markov chains. In ad-
dition, a chapter discusses robust signal detection, and another the application
of the EM algorithm to parameter estimation problems where ML estimates
cannot be evaluated in closed form. At the same time, changes in modern
communications technology are addressed by examining the detection of par-
tially observed Markov chains, both for the case when the Markov chain model
is known, or when the model includes unknown parameters that need to be
estimated. To accommodate the need for joint estimation and detection in
modern communication systems, particular attention is given to the general-
ized likelihood ratio test (GLRT), since it explicitly implements detection and
estimation as a combined task, and because of its attractive invariance and
asymptotic properties.
This book is primarily intended for use in signal detection courses di-
rected at first or second year graduate electrical engineering students. Thus,
even though the material presented has been abstracted from actual engi-
neering systems, the emphasis is on fundamental detection principles, rather
than on implementation details targeted at specific applications. It is expected
that after mastering the concepts discussed here, a student or practicing en-
gineer will be able to analyze a specific detection problem, read the available
literature, and design a detector meeting applicable specifications. Since the
book is addressed at engineeering students, certain compromises have been
made concerning the level of precision applied to mathematical arguments. In
particular, no formal exposure to measure theory, modern real analysis, and
the theory of operators in Hilbert spaces is assumed. As a consequence, even
though derivations are conceptually accurate, they often leave some technical
details out. This relatively casual presentation style has for objective to ensure
that most students will be able to benefit from the material presented, regard-
less of preparation. On the other hand, it is expected that readers will have a
solid background in the areas of random processes, linear algebra, and convex
optimization, which in the aggregate form the common frame of reference of
the statistical signal processing community.
Preface
IX
Another aspect of this book that may be controversial is that it does not
follow a theorem/proof format. To explain this choice, I would like to point
out that whereas the hypothesis testing and parameter estimation techniques
used in signal detection lend themselves naturally to a formal presentation
style, because of its applied nature, signal detection consists primarily of a
methodology for converting an observed signal model and some specifications
for the detector to be constructed into first a formulation of the problem in
hypothesis testing format, followed by a solution meeting the given specifica-
tions. In this context, the most important skills needed are first the ability
to think geometrically in higher dimensional spaces, and second the capac-
ity to reason in a manner consistent with the assumed observation model.
For example, if the parameters appearing in the signal model admit proba-
bility distributions, a Bayesian framework needs to be employed to construct
a detector, whereas when parameters are unknown but nonrandom, the pa-
rameters need to be estimated as part of the detector construction. Slightly
different modelling assumptions for the same problem may lead to different
detector structures. Accordingly, signal detection cannot really be reduced to
a collection of mathematical results. Instead, it is primarily a methodology
that can be best explained by employing a continuous presentation flow, with-
out attempting to slice the material into elementary pieces. The continuous
flow approach has also the advantage that it makes it easier to connect ideas
presented in different parts of the book without having to wait until each
analytical derivation is complete.
Obviously, since the field of signal detection covers a vast range of sub-
jects, it has been necessary to leave out certain topics that are either covered
elsewhere or that are too advanced or complex to be presented concisely in
an introductory text. Accordingly, although Kalman and Wiener filters are
employed in the discussion of Gaussian signal detection in Chapter 10, it is
assumed that optimal filtering is covered elsewhere as part of a stand-alone
course, possibly in combination with adaptive filtering, as is the case at UC
Davis. In any case, several excellent presentations of optimal and adaptive
filtering are currently available in textbook form, so it makes little sense to
duplicate these efforts. Two other topics that have been left out, but for en-
tirely different reasons, are change detection/failure detection, and iterative
detection. To explain this choice, let me indicate first that change detection
and failure detection represent one of the most interesting and challenging
fields of application of the methods presented in this book, since in addi-
tion to detecting whether a change occurs, it is necessary to detect when the
change occurred, and for safety critical applications, to do so as quickly as
possible. However, important advances have occurred in this area over the
last 20 years, and it does not appear possible to give a concise presentation
of these results in a manner that would do justice to this topic. As for the
iterative detection techniques introduced recently for iterative decoding and
equalization, it was felt that these results are probably best presented in the
context of the communications applications for which they were developed.
X
Preface
The scope of the material presented in this book is sufficiently broad to
allow different course organizations depending on length (quarter or semester)
and on the intended audience. At UC Davis, within the context of a one
quarter course, I usually cover Chapter 2 (hypothesis testing), followed by
Chapter 4 (parameter estimation), and the first half of Chapter 5 (composite
hypothesis testing). Then I move on to Chapter 7 presenting the Karhunen-
Lo`eve decomposition of Gaussian processes, followed by Chapters 8 and 9
discussing the detection of known signals, possibly with unknown parameters,
in white and colored Gaussian noise. A semester length presentation directed
at a statistical signal processing audience would allow coverage of the second
half of Chapter 3 on sequential hypothesis testing, as well as Chapters 10 and
11 on the detection of Gaussian signals, possibly with unknown parameters.
On the other hand, a semester course focusing on communications applications
would probably add Chapters 12, 13 and parts of Chapter 11 to the one-
quarter version of the course outlined above.
The idea of writing this book originated with a lunch conversation I had
with a UC Davis colleague, Prof. Zhi Ding about four years ago. I was com-
plaining that available textbooks on signal detection did not include several
topics that I thought were essential for a modern presentation of the material,
and after listening politely, Zhi pointed out that since I had all these bright
ideas, maybe I should write my own book. Against my better judgement,
I decided to follow Zhi’s suggestion when I became eligible for a sabbatical
year in 2004–2005. In spite of the hard work involved, this has been a re-
warding experience, since it gave me an opportunity to express my views on
signal detection and parameter estimation in a coherent manner. Along the
way, I realized how much my understanding of this field had been impacted
by teachers, mentors, friends, collaborators, and students. Among the many
individuals to whom I am indebted, I would like to start with my teachers
Pierre Faurre and Pierre Bernhard at the Ecole des Mines in Paris, who got
me interested in optimal filtering and encouraged me to go to Stanford to
pursue graduate studies. As soon as I arrived at Stanford, I knew this was the
right choice, since in addition to the expert guidance and scientific insights
provided by my advisors, Tom Kailath and Martin Morf, I was very fortunate
to interact with an unusually talented and lively group of classmates includ-
ing Sun-Yuan Kung, George Verghese, and Erik Verriest. Later, during my
professional life at MIT and UC Davis, I benefited greatly from the mentor-
ship and advice provided by Alan Willsky, Sanjoy Mitter, and Art Krener. I
am particularly grateful to Art for showing me through example that good
research and fun are not mutually exclusive. In addition, I would like to thank
Albert Benveniste and Ramine Nikoukhah for fruitful research collaborations
during and after sabbatical visits at INRIA in France. Like most professors,
I have learnt a lot from my students, and among those whose research was
directly related to the topic of this book, I would like to acknowledge Ahmed
Tewfik, Mutlu Koca, Hoang Nguyen and Yongfang Guo. A number of volun-
teers have helped me in the preparation of this book. Yongfang Guo helped
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