# Algebra linerar

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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 DetectionProblems 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 RicianQuadratic 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 2628 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 GeneralizedMatched Filter 4.5 Multiple Signals 4.5.1 Binary Case 4.5.2 Performance for Binary Case 4.5.3 M-ary Case 4.6 Linear Model 4.7 Signal Processing Examples 4A Reduced Form of the Linear Model 5 Random Signals 5.1 Introduction 5.2 Summary 5.3 Estimator-Correlator 5.4 Linear Model 5.5 Estimator-Correlator for Large Data Records 5.6 General Gaussian Detection 5.7 SignalProcessing Example 5.7.1 Tapped Delay Line Channel Model 5A Detection Performance of the Estimator-Correlator

CONTENTS

ix

6

Statistical Decision Theory II

186

Introduction Summary 6.2.1 Summary of Composite Hypothesis Testing 6.3 Composite Hypothesis Testing 6.4 Composite Hypothesis Testing Approaches 6.4.1 Bayesian Approach 6.4.2 Generalized LikelihoodRatio Test 6.5 Performance of GLRT for Large Data Records 6.6 Equivalent Large Data Records Tests 6.7 Locally Most Powerful Detectors 6.8 Multiple Hypothesis Testing 6A Asymptotically Equivalent Tests - No Nuisance Parameters 6B Asymptotically Equivalent Tests - Nuisance Parameters 6C Asymptotic PDF of GLRT 6D Asymptotic Detection Performance of LMP Test 6E AlternateDerivation of Locally Most Powerful Test 6F Derivation of Generalized ML Rule
7

6.1 6.2

186 186 187 191 197 198 200 205 208 217 221 232 235 239 241
243 245 248 248

Deterministic Signals with Unknown Parameters

Introduction Summary Signal Modeling and Detection Performance Unknown Amplitude 7.4.1 GLRT 7.4.2 Bayesian Approach 7.5 Unknown Arrival Time 7.6Sinusoidal Detection 7.6.1 Amplitude Unknown 7.6.2 Amplitude and Phase Unknown 7.6.3 Amplitude, Phase, and Frequency Unknown 7.6.4 Amplitude, Phase, Frequency, and Arrival Time Unknown 7.7 Classical Linear Model 7.8 Signal Processing Examples 7A Asymptotic Performance of the Energy Detector 713 Derivation of GLRT for Classical Linear Model
7.1

7.2 7.3 7.4

248 249...