Algebra linerar

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 07458 http://www.phptr.com

Contents
1 Introduction 1

1.1 1.2 1.3 1.4 1.5 1.6
2

Detection Theory in Signal Processing The Detection Problem The Mathematical Detection Problem Hierarchy of Detection Problems Role of Asymptotics Some Notes to the Reader

1 7 8 13 14 15
20

Summary of Important PDFs

2.1 2.2

2.3 2.5 2A 2B 2C 2D 2E

Introduction Fundamental Probability Density Functions and Properties 2.2.1 Gaussian (Normal) 2.2.2 Chi-Squared (Central) 2.2.3 Chi-Squared (Noncentral) 2.2.4 F (Central) 2.2.5 F (Noncentral) 2.2.6 Rayleigh 2.2.7 Rician Quadratic Forms of Gaussian Random Variables Asymptotic Gaussian PDF Monte Carlo Performance Evaluation Number of Required Monte Carlo Trials Normal Probability Paper MATLAB Program to Compute Gaussian Right-Tail Probability and its Inverse MATLAB Program to Compute Central and Noncentral x 2 RightTail Probability MATLAB Program for Monte Carlo Computer Simulation vii

20 20 20 24 26 28 29 30 31 32 33 36 45 47 50 52 58

viii 3 Statistical Decision Theory I 3.1 Introduction 3.2 Summary 3.3 Neyman-Pearson Theorem 3.4 Receiver Operating Characteristics 3.5 Irrelevant Data 3.6 Minimum Probability of Error 3.7 Bayes Risk 3.8 Multiple Hypothesis Testing 3A Neyman-Pearson Theorem 3B Minimum Bayes Risk Detector - Binary Hypothesis 3C Minimum Bayes Risk Detector - Multiple Hypotheses

CONTENTS 60 60 60 61 74 75 77 80 81 89 90 92 94 94 94 95 95 101 105 108 112 112 114 119 122 125 139 141 141 141 142 154 165 167 169 169 183

4 Deterministic Signals 4.1 Introduction 4.2 Summary 4.3 Matched Filters 4.3.1 Development of Detector 4.3.2 Performance of Matched Filter 4.4 Generalized Matched Filters 4.4.1 Performance of Generalized