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Computation

Visualization

Programming

For Use with MATLAB

Higher-Order Spectral

Analysis Toolbox

User’s Guide

Version 2

Ananthram Swami

Jerry M. Mendel

Chrysostomos L. (Max) Nikias

How to Contact the Authors

Please note that support and maintenance is provided by United Signals & Systems, Inc. (the

authors of the toolbox), and is no longer provided by The MathWorks.

a.swami@ieee.org Technical support and product enhancement suggestions

usands@pacbell.net Sales, pricing, and general information

The software described in this document is furnished under a license agreement. The software may be used

or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro-

duced in any form without prior written consent from United Signals & Systems, Inc.

FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by

or for the federal government of the United States. By accepting delivery of the Program, the government

hereby agrees that this software qualifies as "commercial" computer software within the meaning of FAR

Part 12.212, DFARS Part 227.7202-1, DFARS Part 227.7202-3, DFARS Part 252.227-7013, and DFARS Part

252.227-7014. The terms and conditions of United Signals & Systems, Inc. Software License Agreement shall

pertain to the government’s use and disclosure of the Program and Documentation, and shall supersede any

conflicting contractual terms or conditions. If this license fails to meet the government’s minimum needs or

is inconsistent in any respect with federal procurement law, the government agrees to return the Program

and Documentation, unused, to United Signals & Systems, Inc.

Higher-Order Spectral Analysis Toolbox User’s Guide

MATLAB, Simulink, Stateflow, Handle Graphics, and Real-Time Workshop are registered trademarks, and

Target Language Compiler is a trademark of The MathWorks, Inc.

Other product or brand names are trademarks or registered trademarks of their respective holders.

Printing History: May 1993 First printing (The MathWorks)

September 1995 Second printing (The MathWorks)

January 1998 Third printing (The MathWorks)

ii Contents

Parametric Estimators, ARMA Models . . . . . . . . . . . . . . . . . . 1-26

MA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-29

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-30

AR Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-31

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-32

ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-32

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-34

AR Order Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-34

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-35

MA Order Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-36

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-37

Linear Processes: Impulse Response Estimation . . . . . . . . . . . 1-37

The Polycepstral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-38

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-41

The Matsuoka-Ulrych Algorithm . . . . . . . . . . . . . . . . . . . . . 1-41

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-42

Linear Processes: Theoretical Cumulants and Polyspectra . . 1-43

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-43

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-45

Linear Prediction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-47

Levinson Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-47

Trench Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-49

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-50

Deterministic Formulation of FBLS . . . . . . . . . . . . . . . . . . . . . 1-53

Adaptive Linear Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-54

RIV Algorithm: Transversal Form . . . . . . . . . . . . . . . . . . . . 1-56

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-57

RIV Algorithm: Double-Lattice Form . . . . . . . . . . . . . . . . . . 1-58

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-60

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-61

Harmonic Processes and DOA . . . . . . . . . . . . . . . . . . . . . . . . . 1-62

Resolution and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-65

AR and ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-66

Pisarenko’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-67

Multiple Signal Classification (MUSIC) . . . . . . . . . . . . . . . . . . 1-68

Minimum-Norm Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-69

ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-70

iii

Criterion-Based Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-72

Cumulant-Based Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-74

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-75

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-77

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-79

Nonlinear Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-80

Solution Using Cross-Bispectra . . . . . . . . . . . . . . . . . . . . . . . . 1-80

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-82

Solution Using FTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-82

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-83

Quadratic Phase Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-84

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-87

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-88

Time-Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . 1-89

Wigner Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-90

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-93

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-94

Wigner Bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-94

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-96

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-97

Wigner Trispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-98

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-99

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-100

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-100

Time-Delay Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-101

A Cross-Correlation Based Method . . . . . . . . . . . . . . . . . . . . . 1-101

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-103

A Cross-Cumulant Based Method . . . . . . . . . . . . . . . . . . . . . . 1-103

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-105

A Hologram Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-105

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-107

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-107

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