Fundamentals of Statistical Signal Processing, Volume II Detection Theory

by Kay, Steven M.

• ISBN13: 9780135041352
• ISBN10: 013504135X
• Edition: 1st
• Format: Hardcover
• Copyright: 1998-01-27
• Publisher: Pearson

Summary

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Author Biography

STEVEN M. KAY is Professor of Electrical Engineering at the University of Rhode Island and a leading expert in signal processing.

Table of Contents

1 Introduction	1	(19)
1.1 Detection Theory in Signal Processing	1	(6)
1.2 The Detection Problem	7	(1)
1.3 The Mathematical Detection Problem	8	(5)
1.4 Hierarchy of Detection Problems	13	(1)
1.5 Role of Asymptotics	14	(1)
1.6 Some Notes to the Reader	15	(5)
2 Summary of Important PDFs	20	(40)
2.1 Introduction	20	(1)
2.2 Fundamental Probability Density Functions and Properties	20	(12)
2.2.1 Gaussian (Normal)	20	(4)
2.2.2 Chi-Squared (Central)	24	(2)
2.2.3 Chi-Squared (Noncentral)	26	(2)
2.2.4 F (Central)	28	(1)
2.2.5 F (Noncentral)	29	(1)
2.2.6 Rayleigh	30	(1)
2.2.7 Rician	31	(1)
2.3 Quadratic Forms of Gaussian Random Variables	32	(1)
2.4 Asymptotic Gaussian PDF	33	(3)
2.5 Monte Carlo Performance Evaluation	36	(9)
2A Number of Required Monte Carlo Trials	45	(2)
2B Normal Probability Paper	47	(3)
2C MATLAB Program to Compute Gaussian Right-Tail Probability and its Inverse	50	(2)
2D MATLAB Program to Compute Central and Noncentral X(2) Right-Tail Probability	52	(6)
2E MATLAB Program for Monte Carlo Computer Simulation	58	(2)
3 Statistical Decision Theory I	60	(34)
3.1 Introduction	60	(1)
3.2 Summary	60	(1)
3.3 Neyman-Pearson Theorem	61	(13)
3.4 Receiver Operating Characteristics	74	(1)
3.5 Irrelevant Data	75	(2)
3.6 Minimum Probability of Error	77	(3)
3.7 Bayes Risk	80	(1)
3.8 Multiple Hypothesis Testing	81	(8)
3A Neyman-Pearson Theorem	89	(1)
3B Minimum Bayes Risk Detector-Binary Hypothesis	90	(2)
3C Minimum Bayes Risk Detector-Multiple Hypotheses	92	(2)
4 Deterministic Signals	94	(47)
4.1 Introduction	94	(1)
4.2 Summary	94	(1)
4.3 Matched Filters	95	(10)
4.3.1 Development of Detector	95	(6)
4.3.2 Performance of Matched Filter	101	(4)
4.4 Generalized Matched Filters	105	(7)
4.4.1 Performance of Generalized Matched Filter	108	(4)
4.5 Multiple Signals	112	(10)
4.5.1 Binary Case	112	(2)
4.5.2 Performance for Binary Case	114	(5)
4.5.3 M-ary Case	119	(3)
4.6 Linear Model	122	(3)
4.7 Signal Processing Examples	125	(14)
4.8 Reduced Form of the Linear Model	139	(2)
5 Random Signals	141	(45)
5.1 Introduction	141	(1)
5.2 Summary	141	(1)
5.3 Estimator-Correlator	142	(12)
5.4 Linear Model	154	(11)
5.5 Estimator-Correlator for Large Data Records	165	(2)
5.6 General Gaussian Detection	167	(2)
5.7 Signal Processing Example	169	(14)
5.7.1 Tapped Delay Line Channel Model	169	(14)
5A Detection Performance of the Estimator-Correlator	183	(3)
6 Statistical Decision Theory II	186	(62)
6.1 Introduction	186	(1)
6.2 Summary	186	(5)
6.2.1 Summary of Composite Hypothesis Testing	187	(4)
6.3 Composite Hypothesis Testing	191	(6)
6.4 Composite Hypothesis Testing Approaches	197	(8)
6.4.1 Bayesian Approach	198	(2)
6.4.2 Generalized Likelihood Ratio Test	200	(5)
6.5 Performance of GLRT for Large Data Records	205	(3)
6.6 Equivalent Large Data Records Tests	208	(9)
6.7 Locally Most Powerful Detectors	217	(4)
6.8 Multiple Hypothesis Testing	221	(11)
6A Asymptotically Equivalent Tests -- No Nuisance Parameters	232	(3)
6B Asymptotically Equivalent Tests -- Nuisance Parameters	235	(4)
6C Asymptotic PDF of GLRT	239	(2)
6D Asymptotic Detection Performance of LMP Test	241	(2)
6E Alternate Derivation of Locally Most Powerful Test	243	(2)
6F Derivation of Generalized ML Rule	245	(3)
7 Deterministic Signals with Unknown Parameters	248	(54)
7.1 Introduction	248	(1)
7.2 Summary	248	(1)
7.3 Signal Modeling and Detection Performance	249	(4)
7.4 Unknown Amplitude	253	(5)
7.4.1 GLRT	254	(3)
7.4.2 Bayesian Approach	257	(1)
7.5 Unknown Arrival Time	258	(3)
7.6 Sinusoidal Detection	261	(11)
7.6.1 Amplitude Unknown	261	(1)
7.6.2 Amplitude and Phase Unknown	262	(6)
7.6.3 Amplitude, Phase, and Frequency Unknown	268	(1)
7.6.4 Amplitude, Phase, Frequency, and Arrival Time Unknown	269	(3)
7.7 Classical Linear Model	272	(7)
7.8 Signal Processing Examples	279	(18)
7A Asymptotic Performance of the Energy Detector	297	(2)
7B Derivation of GLRT for Classical Linear Model	299	(3)
8 Random Signals with Unknown Parameters	302	(34)
8.1 Introduction	302	(1)
8.2 Summary	302	(1)
8.3 Incompletely Known Signal Covariance	303	(8)
8.4 Large Data Record Approximations	311	(3)
8.5 Weak Signal Detection	314	(1)
8.6 Signal Processing Example	315	(17)
8A Derivation of PDF for Periodic Gaussian Random Process	332	(4)
9 Unknown Noise Parameters	336	(45)
9.1 Introduction	336	(1)
9.2 Summary	336	(1)
9.3 General Considerations	337	(4)
9.4 White Gaussian Noise	341	(9)
9.4.1 Known Deterministic Signal	341	(2)
9.4.2 Random Signal with Known PDF	343	(2)
9.4.3 Deterministic Signal with Unknown Parameters	345	(4)
9.4.4 Random Signal with Unknown PDF Parameters	349	(1)
9.5 Colored WSS Gaussian Noise	350	(8)
9.5.1 Known Deterministic Signals	350	(3)
9.5.2 Deterministic Signals with Unknown Parameters	353	(5)
9.6 Signal Processing Example	358	(13)
9A Derivation of GLRT for Classical Linear Model for XXX(2) Unknown	371	(4)
9B Rao Test for General Linear Model with Unknown Noise Parameters	375	(2)
9C Asymptotically Equivalent Rao Test for Signal Processing Example	377	(4)
10 NonGaussian Noise	381	(35)
10.1 Introduction	381	(1)
10.2 Summary	381	(1)
10.3 NonGaussian Noise Characteristics	382	(3)
10.4 Known Deterministic Signals	385	(7)
10.5 Deterministic Signals with Unknown Parameters	392	(8)
10.6 Signal Processing Example	400	(10)
10A Asymptotic Performance of NP Detector for Weak Signals	410	(3)
10B Rao Test for Linear Model Signal with IID NonGaussian Noise	413	(3)
11 Summary of Detectors	416	(23)
11.1 Introduction	416	(1)
11.2 Detection Approaches	416	(11)
11.3 Linear Model	427	(6)
11.4 Choosing a Detector	433	(4)
11.5 Other Approaches and Other Texts	437	(2)
12 Model Change Detection	439	(34)
12.1 Introduction	439	(1)
12.2 Summary	439	(1)
12.3 Description of Problem	440	(5)
12.4 Extensions to the Basic Problem	445	(4)
12.5 Multiple Change Times	449	(6)
12.6 Signal Processing Examples	455	(14)
12.6.1 Maneuver Detection	455	(5)
12.6.2 Time Varying PSD Detection	460	(9)
12A General Dynamic Programming Approach to Segmentation	469	(2)
12B MATLAB Program for Dynamic Programming	471	(2)
13 Complex/Vector Extensions, and Array Processing	473	(56)
13.1 Introduction	473	(1)
13.2 Summary	473	(1)
13.3 Known PDFs	474	(10)
13.3.1 Matched Filter	474	(4)
13.3.2 Generalized Matched Filter	478	(1)
13.3.3 Estimator-Correlator	479	(5)
13.4 PDFs with Unknown Parameters	484	(2)
13.4.1 Deterministic Signal	484	(2)
13.4.2 Random Signal	486	(1)
13.5 Vector Observations and PDFs	486	(6)
13.5.1 General Covariance Matrix	490	(1)
13.5.2 Scaled Identity Matrix	491	(1)
13.5.3 Uncorrelated from Temporal Sample to Sample	491	(1)
13.5.4 Uncorrelated from Spatial Sample to Sample	492	(1)
13.6 Detectors for Vector Observations	492	(9)
13.6.1 Known Deterministic Signal in CWGN	492	(3)
13.6.2 Known Deterministic Signal and General Noise Covariance	495	(1)
13.6.3 Known Deterministic Signal in Temporally Uncorrelated Noise	495	(1)
13.6.4 Known Deterministic Signal in Spatially Uncorrelated Noise	496	(1)
13.6.5 Random Signal in CWGN	496	(3)
13.6.6 Deterministic Signal with Unknown Parameters in CWGN	499	(2)
13.7 Estimator-Correlator for Large Data Records	501	(7)
13.8 Signal Processing Examples	508	(18)
13.8.1 Active Sonar/Radar	510	(5)
13.8.2 Broadband Passive Sonar	515	(11)
13A PDF of GLRT for Complex Linear Model	526	(3)
A1 Review of Important Concepts	529	(16)
A1.1 Linear and Matrix Algebra	529	(8)
A1.1.1 Definitions	529	(2)
A1.1.2 Special Matrices	531	(2)
A1.1.3 Matrix Manipulation and Formulas	533	(2)
A1.1.4 Theorems	535	(1)
A1.1.5 Eigendecompostion of Matrices	536	(1)
A1.1.6 Inequalities	537	(1)
A1.2 Random Processes and Time Series Modeling	537	(8)
A1.2.1 Random Process Characterization	538	(2)
A1.2.2 Gaussian Random Process	540	(1)
A1.2.3 Time Series Models	541	(4)
A2 Glossary of Symbols and Abbreviations (Vols. I & II)	545

Excerpts

Preface This text is the second volume of a series of books addressing statistical signal processing. The first volume, Fundamentals of Statistical Signal Processing: Estimation Theory, was published in 1993 by Prentice-Hall, Inc. Henceforth, it will be referred to as Kay-I 1993. This second volume, entitled Fundamentals of Statistical Signal Processing: Detection Theory, is the application of statistical hypothesis testing to the detection of signals in noise. The series has been written to provide the reader with a broad introduction to the theory and application of statistical signal processing. Hypothesis testing is a subject that is standard fare in the many books available dealing with statistics. These books range from the highly theoretical expositions written by statisticians to the more practical treatments contributed by the many users of applied statistics. This text is an attempt to strike a balance between these two extremes. The particular audience we have in mind is the community involved in the design and implementation of signal processing algorithms. As such, the primary focus is on obtaining optimal detection algorithms that may be implemented on a digital computer. The data sets are therefore assumed to be samples of a continuous-time waveform or a sequence of data points. The choice of topics reflects what we believe to be the important approaches to obtaining an optimal detector and analyzing its performance. As a consequence, some of the deeper theoretical issues have been omitted with references given instead. It is the author''s opinion that the best way to assimilate the material on detection theory is by exposure to and working with good examples. Consequently, there are numerous examples that illustrate the theory and others that apply the theory to actual detection problems of current interest. We have made extensive use of the MATLAB scientific programming language (Version 4.2b) Footnote: MATLAB is a registered trademark of The MathWorks, Inc. for all computer-generated results. In some cases, actual MATLAB programs have been listed where a program was deemed to be of sufficient utility to the reader. Additionally, an abundance of homework problems has been included. They range from simple applications of the theory to extensions of the basic concepts. A solutions manual is available from the author. To aid the reader, summary sections have been provided at the beginning of each chapter. Also, an overview of all the principal detection approaches and the rationale for choosing a particular method can be found in Chapter 11. Detection based on simple hypothesis testing is described in Chapters 3--5, while that based on composite hypothesis testing (to accomodate unknown parameters) is the subject of Chapters 6--9. Other chapters address detection in nonGaussian noise (Chapter 10), detection of model changes (Chapter 12), and extensions for complex/vector data useful in array processing (Chapter 13). This book is an outgrowth of a one-semester graduate level course on detection theory given at the University of Rhode Island. It includes somewhat more material than can actually be covered in one semester. We typically cover most of Chapters 1--10, leaving the subjects of model change detection and complex data/vector data extensions to the student. It is also possible to combine the subjects of estimation and detection into a single semester course by a judicious choice of material from Volumes I and II. The necessary background that has been assumed is an exposure to the basic theory of digital signal processing, probability and random processes, and linear and matrix algebra. This book can also be used for self-study and so should be useful to the practicing engineer as well as the student. The author would like to acknowledge the contributions of the many people who over the years have provided stimulating discussions of research problems, opportunities to apply the results of that research, and support for conducting research. Thanks are due to my colleagues L. Jackson, R. Kumaresan, L. Pakula, and P. Swaszek of the University of Rhode Island, and L. Scharf of the University of Colorado. Exposure to practical problems, leading to new research directions, has been provided by H. Woodsum of Sonetech, Bedford, New Hampshire, and by D. Mook and S. Lang of Sanders, a Lockheed-Martin Co., Nashua, New Hampshire. The opportunity to apply detection theory to sonar and the research support of J. Kelly of the Naval Undersea Warfare Center, J. Salisbury, formerly of the Naval Undersea Warfare Center, and D. Sheldon of the Naval Undersea Warfare Center, Newport, Rhode Island are also greatly appreciated. Thanks are due to J. Sjogren of the Air Force Office of Scientific Research, whose support has allowed the author to investigate the field of statistical signal processing. A debt of gratitude is owed to all my current and former graduate students. They have contributed to the final manuscript through many hours of pedagogical and research discussions as well as by their specific comments and questions. In particular, P. DjuriO{c} of the State University of New York proofread much of the manuscript, and S. Talwalkar of Motorola, Plantation, Florida proofread parts of the manuscript and helped with the finer points of MATLAB. Steven M. Kay University of Rhode Island Kingston, RI 02881 Email: [e-mail]