Random Signals Detection, Estimation and Data Analysis

, by Shanmugan, K. Sam; Breipohl, Arthur M.

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ISBN: 9780471815556 | 0471815551
Cover: Paperback
Copyright: 1/16/1991

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Random Signals, Noise and Filtering develops the theory of random processes and its application to the study of systems and analysis of random data. The text covers three important areas: (1) fundamentals and examples of random process models, (2) applications of probabilistic models: signal detection, and filtering, and (3) statistical estimation--measurement and analysis of random data to determine the structure and parameter values of probabilistic models. This volume by Breipohl and Shanmugan offers the only one-volume treatment of the fundamentals of random process models, their applications, and data analysis.

Introduction

Historical Perspective

(1)

Outline of the Book

(3)

References

(1)

Review of Probability and Random Variables

Introduction

(1)

Probability

(12)

Set Definitions

(3)

Sample Space

(1)

Probabilities of Random Events

(2)

Useful Laws of Probability

(1)

Joint, Marginal, and Conditional Probabilities

(6)

Random Variables

(12)

Distribution Functions

(2)

Discrete Random Variables and Probability Mass Function

(2)

Expected Values or Averages

(3)

Examples of Probability Mass Functions

(4)

Continuous Random Variables

(14)

Probability Density Functions

(10)

Examples of Probability Density Functions

(3)

Complex Random Variables

(1)

Random Vectors

(8)

Multivariate Gaussian Distribution

(1)

Properties of the Multivariate Gaussian Distribution

(3)

Moments of Multivariate Gaussian pdf

(2)

Transformations (Functions) of Random Variables

(21)

Scalar Valued Function of One Random Variable

(4)

Functions of Several Random Variables

(15)

Bounds and Approximations

(12)

Thebycheff Inequality

(1)

Chernoff Bound

(1)

Union Bound

(2)

Approximating the Distribution of Y = g(X1...,Xn)

(2)

Series Approximation of Probability Density Functions

(4)

Approximations of Gaussian Probabilities

(1)

Sequences of Random Variables and Convergence

(7)

Convergence Everywhere and Almost Everywhere

(1)

Convergence in Distribution and Central Limit Theorem

(4)

Convergence in Probability (in Measure) and the Law of Large Numbers

(1)

Convergence in Mean Square

(1)

Relationship Between Different Forms of Convergence

(1)

Summary

(1)

References

(1)

Problems

(14)

Random Processes and Sequences

Introduction

111

(2)

Definition of Random Processes

113

(6)

Concept of Random Processes

113

(1)

Notation

114

(2)

Probabilistic Structure

116

(1)

Classification of Random Processes

117

(2)

Formal Definition of Random Processes

119

(1)

Methods of Description

119

(6)

Joint Distribution

119

(2)

Analytical Description Using Random Variables

121

(1)

Average Values

121

(3)

Two or More Random Processes

124

(1)

Special Classes of Random Processes

125

(10)

More Definitions

126

(1)

Random Walk and Wiener Process

127

(4)

Poisson Process

131

(1)

Random Binary Waveform

132

(3)

Stationarity

135

(7)

Strict-sense Stationarity

135

(1)

Wide-sense Stationarity

136

(1)

Examples

137

(4)

Other Forms of Stationarity

141

(1)

Tests for Stationarity

142

(1)

Autocorrelation and Power Spectral Density Functions of Real WSS Random Processes

142

(18)

Autocorrelation Function of a Real WSS Random Process and Its Properties

143

(1)

Cross correlation Function and its Properties

144

(1)

Power Spectral Density Function of a WSS Random Process and Its Properties

145

(3)

Cross-power Spectral Density Function and Its Properties

148

(1)

Power Spectral Density Function of Random Sequences

149

(11)

Continuity, Differentiation, and Integration

160

(6)

Continuity

161

(1)

Differentiation

162

(3)

Integration

165

(1)

Time Averaging and Ergodicity

166

(19)

Time Averages

168

(8)

Ergodicity

176

(9)

Spectral Decomposition and Series Expansion of Random Processes

185

(4)

Ordinary Fourier Series Expansion

185

(2)

Modified Fourier Series for Aperiodic Random Signals

187

(1)

Karhunen-Loeve (K-L) Series Expansion

188

(1)

Sampling and Quantization of Random Signals

189

(13)

Sampling of Lowpass Random Signals

190

(6)

Quantization

196

(1)

Uniform Quantizing

197

(3)

Nonuniform Quantizing

200

(2)

Summary

202

(1)

References

203

(1)

Problems

204

(12)

Response of Linear Systems to Random Inputs

Classification of Systems

216

(2)

Lumped Linear Time-invariant Causal (LLTIVC) System

216

(1)

Memoryless Nonlinear Systems

217

(1)

Response of LTIVC Discrete Time Systems

218

(9)

Review of Deterministic System Analysis

218

(3)

Mean and Autocorrelation of the Output

221

(1)

Distribution Functions

222

(1)

Stationarity of the Output

222

(1)

Correlation and Power Spectral Density of the Output

223

(4)

Response of LTIVC Continuous Time Systems

227

(15)

Mean and Autocorrelation Function of the Output

228

(1)

Stationarity of the Output

229

(1)

Power Spectral Density of the Output

230

(4)

Mean-square Value of the Output

234

(4)

Multiple Input-Output Systems

238

(1)

Filters

239

(3)

Summary

242

(1)

References

243

(1)

Problems

244

(5)

Special Classes of Random Processes

Introduction

249

(1)

Discrete Linear Models

250

(26)

Autoregressive Processes

250

(12)

Partial Autocorrelation Coefficient

262

(3)

Moving Average Models

265

(6)

Autoregressive Moving Average Models

271

(4)

Summary of Discrete Linear Models

275

(1)

Markov Sequences and Processes

276

(19)

Analysis of Discrete-time Markov Chains

278

(11)

Continuous-time Markov Chains

289

(6)

Summary of Markov Models

295

(1)

Point Processes

295

(17)

Poisson Process

298

(5)

Application of Poisson Process---Analysis of Queues

303

(4)

Shot Noise

307

(5)

Summary of Point Processes

312

(1)

Gaussian Processes

312

(19)

Definition of Gaussian Process

313

(1)

Models of White and Band-limited Noise

314

(3)

Response of Linear Time-invariant Systems

317

(1)

Quadrature Representation of Narrowband (Gaussian) Processes

317

(5)

Effects of Noise in Analog Communication Systems

322

(8)

Noise in Digital Communication Systems

330

(1)

Summary of Noise Models

331

(1)

Summary

331

(1)

References

332

(1)

Problems

333

(8)

Signal Detection

Introduction

341

(2)

Binary Detection with a Single Observation

343

(9)

Decision Theory and Hypothesis Testing

344

(1)

MAP Decision Rule and Types of Errors

345

(3)

Bayes' Decision Rule---Costs of Errors

348

(3)

Other Decision Rules

351

(1)

Binary Detection with Multiple Observations

352

(12)

Independent Noise Samples

353

(2)

White Noise and Continuous Observations

355

(6)

Colored Noise

361

(3)

Detection of Signals with Unknown Parameters

364

(2)

M-ary Detection

366

(3)

Summary

369

(1)

References

370

(1)

Problems

370

(7)

Linear Minimum Mean-Square Error Filtering

Introduction

377

(2)

Linear Minimum Mean Squared Error Estimators

379

(18)

Estimating a Random Variable with a Constant

379

(1)

Estimating S with One Observation X

379

(4)

Vector Space Representation

383

(1)

Multivariable Linear Mean Squared Error Estimation

384

(7)

Limitations of Linear Estimators

391

(2)

Nonlinear Minimum Mean Squared Error Estimators

393

(2)

Jointly Gaussian Random Variables

395

(2)

Innovations

397

(9)

Multivariate Estimator Using Innovations

400

(1)

Matrix Definition of Innovations

401

(5)

Review

406

(1)

Digital Wiener Filters

407

(12)

Digital Wiener Filters with Stored Data

407

(4)

Real-time Digital Wiener Filters

411

(8)

Kalman Filters

419

(23)

Recursive Estimators

420

(1)

Scalar Kalman Filter

421

(11)

Vector Kalman Filter

432

(10)

Wiener Filters

442

(23)

Stored Data (Unrealizable Filters)

444

(4)

Real-time or Realizable Filters

448

(17)

Summary

465

(1)

References

466

(1)

Problems

467

(8)

Statistics

Introduction

475

(1)

Measurements

476

(3)

Definition of a Statistic

478

(1)

Parametric and Nonparametric Estimators

478

(1)

Nonparametric Estimators of Probability Distribution and Density Functions

479

(6)

Definition of the Empirical Distribution Function

479

(1)

Joint Empirical Distribution Functions

480

(1)

Histograms

481

(3)

Parzen's Estimator for a pdf

484

(1)

Point Estimators of Parameters

485

(8)

Estimators of the Mean

486

(1)

Estimators of the Variance

486

(1)

An Estimator of Probability

487

(1)

Estimators of the Covariance

487

(1)

Notation for Estimators

487

(1)

Maximum Likelihood Estimators

488

(3)

Bayesian Estimators

491

(2)

Measures of the Quality of Estimators

493

(10)

Bias

493

(3)

Minimum Variance, Mean Squared Error, RMS Error, and Normalized Errors

496

(1)

The Bias, Variance, and Normalized RMS Errors of Histograms

497

(4)

Bias and Variance of Parzen's Estimator

501

(1)

Consistent Estimators

502

(1)

Efficient Estimators

502

(1)

Brief Introduction to Interval Estimates

503

(1)

Distribution of Estimators

504

(9)

Distribution of X with Known Variance

504

(1)

Chi-square Distribution

505

(3)

(Student's) t Distribution

508

(2)

Distribution of S2 and X with Unknown Variance

510

(1)

F Distribution

511

(2)

Tests of Hypotheses

513

(16)

Binary Detection

514

(3)

Composite Alternative Hypothesis

517

(1)

Tests of the Mean of a Normal Random Variable

518

(2)

Tests of the Equality of Two Means

520

(2)

Tests of Variances

522

(1)

Chi-Square Tests

523

(5)

Summary of Hypothesis Testing

528

(1)

Simple Linear Regression

529

(11)

Analyzing the Estimated Regression

536

(2)

Goodness of Fit Test

538

(2)

Multiple Linear Regression

540

(7)

Two Controlled Variables

541

(1)

Simple Linear Regression in Matrix Form

542

(1)

General Linear Regression

543

(2)

Goodness of Fit Test

545

(1)

More General Linear Models

545

(2)

Summary

547

(1)

References

548

(1)

Appendix 8-A

549

(3)

Problems

552

(8)

Estimating the Parameters of Random Processes from Data

Introduction

560

(1)

Tests for Stationarity and Ergodicity

561

(4)

Stationarity Tests

562

(1)

Run Test for Stationarity

562

(3)

Model-free Estimation

565

(19)

Mean Value Estimation

565

(1)

Autocorrelation Function Estimation

566

(3)

Estimation of the Power Spectral Density (psd) Functions

569

(10)

Smoothing of Spectral Estimates

579

(5)

Bias and Variance of Smoothed Estimators

584

(1)

Model-based Estimation of Autocorrelation Functions and Power Spectral Density Functions

584

(29)

Preprocessing (Differencing)

587

(3)

Order Identification

590

(4)

Estimating the Parameters of Autoregressive Processes

594

(6)

Estimating the Parameters of Moving Average Processes

600

(5)

Estimating the Parameters of ARMA (p, q) Processes

605

(1)

ARIMA Preliminary Parameter Estimation

606

(2)

Diagnostic Checking

608

(5)

Summary

613

(1)

References

614

(1)

Problems

615

(11)

APPENDIXES

A. Fourier Transforms

626

(2)

B. Discrete Fourier Transforms

628

(2)

C. Z Transforms

630

(2)

D. Gaussian Probabilities

632

(1)

E. Table of Chi-Square Distributions

633

(4)

F. Table of Student's t Distribution

637

(2)

G. Table of F Distributions

639

(10)

H. Percentage Points of Run Distribution

649

(1)

I. Critical Values of the Durbin-Watson Statistic

650

(1)

Index

651

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Random Signals Detection, Estimation and Data Analysis

Summary

Author Biography

Table of Contents

Supplemental Materials