Fundamentals of Statistical Signal Processing Volume II Detection Theory

Transcription

1 Fundamentals of Statistical Signal Processing Volume II Detection Theory Steven M. Kay University of Rhode Island PH PTR Prentice Hall PTR Upper Saddle River, New Jersey

2 Contents 1 Introduction Detection Theory in Signal Processing The Detection Problem The Mathematical Detection Problem Hierarchy of Detection Problems Role of Asymptotics Some Notes to the Reader 15 2 Summary of Important PDFs Introduction Fundamental Probability Density Functions and Properties Gaussian (Normal) Chi-Squared (Central) Chi-Squared (Noncentral) F (Central) F (Noncentral) Rayleigh Rician Quadratic Forms of Gaussian Random Variables Asymptotic Gaussian PDF Monte Carlo Performance Evaluation 36 2A Number of Required Monte Carlo Trials 45 2B Normal Probability Paper 47 2C MATLAB Program to Compute Gaussian Right-Tail Probability and its Inverse 50 2D MATLAB Program to Compute Central and Noncentral \ 2 Right- Tail Probability 52 2E MATLAB Program for Monte Carlo Computer Simulation 58 vn

3 Vlll 3 Statistical Decision Theory I Introduction Summary Neyman-Pearson Theorem Receiver Operating Characteristics Irrelevant Data Minimum Probability of Error Bayes Risk Multiple Hypothesis Testing 81 ЗА Neyman-Pearson Theorem 89 3B Minimum Bayes Risk Detector - Binary Hypothesis 90 3C Minimum Bayes Risk Detector - Multiple Hypotheses 92 4 Deterministic Signals Introduction Summary Matched Filters Development of Detector Performance of Matched Filter Generalized Matched Filters Performance of Generalized Matched Filter Multiple Signals Binary Case Performance for Binary Case M-ary Case Linear Model Signal Processing Examples 125 4A Reduced Form of the Linear Model Random Signals Introduction Summary Estimator-Correlator Linear Model Estimator-Correlator for Large Data Records General Gaussian Detection Signal Processing Example Tapped Delay Line Channel Model 169 5A Detection Performance of the Estimator-Correlator 183

4 CONTENTS ix 6 Statistical Decision Theory II Introduction Summary Summary of Composite Hypothesis Testing Composite Hypothesis Testing Composite Hypothesis Testing Approaches Bayesian Approach Generalized Likelihood Ratio Test Performance of GLRT for Large Data Records Equivalent Large Data Records Tests Locally Most Powerful Detectors Multiple Hypothesis Testing 221 6A Asymptotically Equivalent Tests - No Nuisance Parameters 232 6B Asymptotically Equivalent Tests - Nuisance Parameters 235 6C Asymptotic PDF of GLRT 239 6D Asymptotic Detection Performance of LMP Test 241 6E Alternate Derivation of Locally Most Powerful Test 243 6F Derivation of Generalized ML Rule Deterministic Signals with Unknown Parameters Introduction Summary Signal Modeling and Detection Performance Unknown Amplitude GLRT Bayesian Approach Unknown Arrival Time Sinusoidal Detection Amplitude Unknown Amplitude and Phase Unknown Amplitude, Phase, and Frequency Unknown Amplitude, Phase, Frequency, and Arrival Time Unknown Classical Linear Model Signal Processing Examples 279 7A Asymptotic Performance of the Energy Detector 297 7B Derivation of GLRT for Classical Linear Model 299

5 X 8 Random Signals with Unknown Parameters Introduction Summary Incompletely Known Signal Covariance Large Data Record Approximations Weak Signal Detection Signal Processing Example 315 8A Derivation of PDF for Periodic Gaussian Random Process Unknown Noise Parameters Introduction Summary General Considerations White Gaussian Noise Known Deterministic Signal Random Signal with Known PDF Deterministic Signal with Unknown Parameters Random Signal with Unknown PDF Parameters Colored WSS Gaussian Noise Known Deterministic Signals Deterministic Signals with Unknown Parameters Signal Processing Example 358 9A Derivation of GLRT for Classical Linear Model for a 1 Unknown B Rao Test for General Linear Model with Unknown Noise Parameters 375 9C Asymptotically Equivalent Rao Test for Signal Processing Example NonGaussian Noise Introduction Summary NonGaussian Noise Characteristics Known Deterministic Signals Deterministic Signals with Unknown Parameters Signal Processing Example A Asymptotic Performance of NP Detector for Weak Signals B Rao Test for Linear Model Signal with IID NonGaussian Noise

6 CONTENTS xi 11 Summary of Detectors Introduction Detection Approaches Linear Model Choosing a Detector Other Approaches and Other Texts Model Change Detection Introduction Summary Description of Problem Extensions to the Basic Problem Multiple Change Times Signal Processing Examples Maneuver Detection Time Varying PSD Detection A General Dynamic Programming Approach to Segmentation B MATLAB Program for Dynamic Programming Complex/Vector Extensions, and Array Processing Introduction Summary Known PDFs Matched Filter Generalized Matched Filter Estimator-Correlator PDFs with Unknown Parameters Deterministic Signal Random Signal Vector Observations and PDFs General Covariance Matrix Scaled Identity Matrix Uncorrelated from Temporal Sample to Sample Uncorrelated from Spatial Sample to Sample Detectors for Vector Observations Known Deterministic Signal in CWGN Known Deterministic Signal and General Noise Covariance. 495

7 XII CONTENTS Known Deterministic Signal in Temporally Uncorrelated Noise Known Deterministic Signal in Spatially Uncorrelated Noise Random Signal in CWGN Deterministic Signal with Unknown Parameters in CWGN Estimator-Correlator for Large Data Records Signal Processing Examples Active Sonar/Radar Broadband Passive Sonar A PDF of GLRT for Complex Linear Model 526 Al Review of Important Concepts 529 Al.l Linear and Matrix Algebra 529 Al.1.1 Definitions 529 Al.1.2 Special Matrices 531 Al. 1.3 Matrix Manipulation and Formulas 533 Al.1.4 Theorems 535 Al. 1.5 Eigendecompostion of Matrices 536 Al. 1.6 Inequalities 537 A 1.2 Random Processes and Time Series Modeling 537 A Random Process Characterization 538 Al.2.2 Gaussian Random Process 540 Al.2.3 Time Series Models 541 A2 Glossary of Symbols and Abbreviations (Vols. I & II) 545