Random data : analysis and measurement procedures

621 Pages·2010·10.242 MB·English

by  Julius S Bendat, Allan G Piersol

Random Data
WILEY SERIES IN PROBABILITY AND STATISTICS
Established by WALTER A. SHEWHART and SAMUEL S. WILKS
Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice,
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Ruey S. Tsay, Sanford Weisberg
Editors Emeriti: Vic Barnett, J. Stuart Hunter, JozefL. Teugels
A complete list of the titles in this series appears at the end of this volume.
Random Data
Analysis and Measurement Procedures
Fourth Edition
JULIUS S. BENDAT
ALLAN G. PffiRSOL
WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
ISBN 978-0-470-24877-5
Printed in the United States of America
10 98765432 1
To Allan G. Piersol
1930-2009
Contents
Preface
xv
Preface to the Third Edition xvii
Glossary of Symbols xix
1. Basic Descriptions and Properties 1
1.1. Deterministic Versus Random Data, 1
1.2. Classifications of Deterministic Data, 3
1.2.1. Sinusoidal Periodic Data, 3
1.2.2. Complex Periodic Data, 4
1.2.3. Almost-Periodic Data, 6
1.2.4. Transient Nonperiodic Data, 7
1.3. Classifications of Random Data, 8
1.3.1. Stationary Random Data, 9
1.3.2. Ergodic Random Data, 11
1.3.3. Nonstationary Random Data, 12
1.3.4. Stationary Sample Records, 12
1.4. Analysis of Random Data, 13
1.4.1. Basic Descriptive Properties, 13
1.4.2. Input/Output Relations, 19
1.4.3. Error Analysis Criteria, 21
1.4.4. Data Analysis Procedures, 23
2. Linear Physical Systems 25
2.1. Constant-Parameter Linear Systems, 25
2.2. Basic Dynamic Characteristics, 26
2.3. Frequency Response Functions, 28
2.4. Illustrations of Frequency Response Functions, 30
2.4.1. Mechanical Systems, 30
vii
2.4.2. Electrical Systems, 39
2.4.3. Other Systems, 41
2.5. Practical Considerations, 41
Probability Fundamentals
3.1. One Random Variable, 45
3.1.1. Probability Density and Distribution Functions, 46
3.1.2. Expected Values, 49
3.1.3. Change of Variables, 50
3.1.4. Moment-Generating and Characteristic Functions, 52
3.1.5. Chebyshev's Inequality, 53
3.2. Two Random Variables, 54
3.2.1. Expected Values and Correlation Coefficient, 55
3.2.2. Distribution for Sum of Two Random Variables, 56
3.2.3. Joint Moment-Generating and
Characteristic Functions, 57
3.3. Gaussian (Normal) Distribution, 59
3.3.1. Central Limit Theorem, 60
3.3.2. Joint Gaussian (Normal) Distribution, 62
3.3.3. Moment-Generating and Characteristic Functions, 63
3.3.4. N-Dimensional Gaussian (Normal) Distribution, 64
3.4. Rayleigh Distribution, 67
3.4.1. Distribution of Envelope and Phase for
Narrow Bandwidth Data, 67
3.4.2. Distribution of Output Record for Narrow
Bandwidth Data, 71
3.5. Higher Order Changes of Variables, 72
Statistical Principles
4.1. Sample Values and Parameter Estimation, 79
4.2. Important Probability Distribution Functions, 82
4.2.1. Gaussian (Normal) Distribution, 82
4.2.2. Chi-Square Distribution, 83
4.2.3. The t Distribution, 84
4.2.4. The F Distribution, 84
4.3. Sampling Distributions and Illustrations, 85
4.3.1. Distribution of Sample Mean with
Known Variance, 85
4.3.2. Distribution of Sample Variance, 86
4.3.3. Distribution of Sample Mean with Unknown Variance, 87
4.3.4. Distribution of Ratio of Two Sample Variances, 87
CONTENTS ix
4.4. Confidence Intervals, 88
4.5. Hypothesis Tests, 91
4.5.1. Chi-Square Goodness-of-Fit Test, 94
4.5.2. Nonparametric Trend Test, 96
4.6. Correlation and Regression Procedures, 99
4.6.1. Linear Correlation Analysis, 99
4.6.2. Linear Regression Analysis, 102
5. Stationary Random Processes 109
5.1. Basic Concepts, 109
5.1.1. Correlation (Covariance) Functions, 111
5.1.2. Examples of Autocorrelation Functions, 113
5.1.3. Correlation Coefficient Functions, 115
5.1.4. Cross-Correlation Function for Time Delay, 116
5.2. Spectral Density Functions, 118
5.2.1. Spectra via Correlation Functions, 118
5.2.2. Spectra via Finite Fourier Transforms, 126
5.2.3. Spectra via Filtering-Squaring-Averaging, 129
5.2.4. Wavenumber Spectra, 132
5.2.5. Coherence Functions, 134
5.2.6. Cross-Spectrum for Time Delay, 135
5.2.7. Location of Peak Value, 137
5.2.8. Uncertainty Relation, 138
5.2.9. Uncertainty Principle and Schwartz Inequality, 140
5.3. Ergodic and Gaussian Random Processes, 142
5.3.1. Ergodic Random Processes, 142
5.3.2. Sufficient Condition for Ergodicity, 145
5.3.3. Gaussian Random Processes, 147
5.3.4. Linear Transformations of Random Processes, 149
5.4. Derivative Random Processes, 151
5.4.1. Correlation Functions, 151
5.4.2. Spectral Density Functions, 154
5.5. Level Crossings and Peak Values, 155
5.5.1. Expected Number of Level Crossings per Unit Time, 155
5.5.2. Peak Probability Functions for Narrow
Bandwidth Data, 159
5.5.3. Expected Number and Spacing of Positive Peaks, 161
5.5.4. Peak Probability Functions for Wide Bandwidth Data, 162
5.5.5. Derivations, 164
6. Single-Input/Output Relationships 173
6.1. Single-Input/Single-Output Models, 173
χ CONTENTS
6.1.1. Correlation and Spectral Relations, 173
6.1.2. Ordinary Coherence Functions, 180
6.1.3. Models with Extraneous Noise, 183
6.1.4. Optimum Frequency Response Functions, 187
6.2. Single-Input/Multiple-Output Models, 190
6.2.1. Single-Input/Two-Output Model, 191
6.2.2. Single-Input/Multiple-Output Model, 192
6.2.3. Removal of Extraneous Noise, 194
7. Multiple-Input/Output Relationships 201
7.1. Multiple-Input/Single-Output Models, 201
7.1.1. General Relationships, 202
7.1.2. General Case of Arbitrary Inputs, 205
7.1.3. Special Case of Mutually Uncorrelated Inputs, 206
7.2. Two-Input/One-Output Models, 207
7.2.1. Basic Relationships, 207
7.2.2. Optimum Frequency Response Functions, 210
7.2.3. Ordinary and Multiple Coherence Functions, 212
7.2.4. Conditioned Spectral Density Functions, 213
7.2.5. Partial Coherence Functions, 219
7.3. General and Conditioned Multiple-Input Models, 221
7.3.1. Conditioned Fourier Transforms, 223
7.3.2. Conditioned Spectral Density Functions, 224
7.3.3. Optimum Systems for Conditioned Inputs, 225
7.3.4. Algorithm for Conditioned Spectra, 226
7.3.5. Optimum Systems for Original Inputs, 229
7.3.6. Partial and Multiple Coherence Functions, 231
7.4. Modified Procedure to Solve Multiple-Input/Single-Output
Models, 232
7.4.1. Three-Input/Single-Output Models, 234
7.4.2. Formulas for Three-Input/Single-Output Models, 235
7.5. Matrix Formulas for Multiple-Input/Multiple-Output
Models, 237
7.5.1. Multiple-Input/Multiple-Output Model, 238
7.5.2. Multiple-Input/Single-Output Model, 241
7.5.3. Model with Output Noise, 243
7.5.4. Single-Input/Single-Output Model, 245
8. Statistical Errors in Basic Estimates 249
8.1. Definition of Errors, 249
8.2. Mean and Mean Square Value Estimates, 252
8.2.1. Mean Value Estimates, 252
CONTENTS xi
8.2.2. Mean Square Value Estimates, 256
8.2.3. Variance Estimates, 260
8.3. Probability Density Function Estimates, 261
8.3.1. Bias of the Estimate, 263
8.3.2. Variance of the Estimate, 264
8.3.3. Normalized rms Error, 265
8.3.4. Joint Probability Density Function Estimates, 265
8.4. Correlation Function Estimates, 266
8.4.1. Bandwidth-Limited Gaussian White Noise, 269
8.4.2. Noise-to-Signal Considerations, 270
8.4.3. Location Estimates of Peak Correlation Values, 271
8.5. Autospectral Density Function Estimates, 273
8.5.1. Bias of the Estimate, 274
8.5.2. Variance of the Estimate, 278
8.5.3. Normalized rms Error, 278
8.5.4. Estimates from Finite Fourier Transforms, 280
8.5.5. Test for Equivalence of Autospectra, 282
8.6. Record Length Requirements, 284
9. Statistical Errors in Advanced Estimates 289
9.1. Cross-Spectral Density Function Estimates, 289
9.1.1. Variance Formulas, 292
9.1.2. Covariance Formulas, 293
9.1.3. Phase Angle Estimates, 297
9.2. Single-Input/Output Model Estimates, 298
9.2.1. Bias in Frequency Response Function Estimates, 300
9.2.2. Coherent Output Spectrum Estimates, 303
9.2.3. Coherence Function Estimates, 305
9.2.4. Gain Factor Estimates, 308
9.2.5. Phase Factor Estimates, 310
9.3. Multiple-Input/Output Model Estimates, 312
10. Data Acquisition and Processing 317
10.1. Data Acquisition, 318
10.1.1. Transducer and Signal Conditioning, 318
10.1.2. Data Transmission, 321
10.1.3. Calibration, 322
10.1.4. Dynamic Range, 324
10.2. Data Conversion, 326
10.2.1. Analog-to-Digital Converters, 326
10.2.2. Sampling Theorems for Random Records, 328