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
2.4.2. Electrical Systems 39
2.4.3. Other Systems 41
2.5. Practical Considerations 41
3. Probability Fundamentals 45
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
4. Statistical Principles 79
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
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
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
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
10.2.3. Sampling Rates and Aliasing Errors 330
10.2.4. Quantization and Other Errors 333
10.2.5. Data Storage 335
10.3. Data Qualification 335
10.3.1. Data Classification 336
10.3.2. Data Validation 340
10.3.3. Data Editing 345
10.4. Data Analysis Procedures 349
10.4.1. Procedure for Analyzing Individual Records 349
10.4.2. Procedure for Analyzing Multiple Records 351
11. Data Analysis 359
11.1. Data Preparation 359
11.1.1. Data Standardization 360
11.1.2. Trend Removal 361
11.1.3. Digital Filtering 363
11.2. Fourier Series and Fast Fourier Transforms 366
11.2.1. Standard Fourier Series Procedure 366
11.2.2. Fast Fourier Transforms 368
11.2.3. Cooley–Tukey Procedure 374
11.2.4. Procedures for Real-Valued Records 376
11.2.5. Further Related Formulas 377
11.2.6. Other Algorithms 378
11.3. Probability Density Functions 379
11.4. Autocorrelation Functions 381
11.4.1. Autocorrelation Estimates via Direct Computations 381
11.4.2. Autocorrelation Estimates via FFT Computations 381
11.5. Autospectral Density Functions 386
11.5.1. Autospectra Estimates by Ensemble Averaging 386
11.5.2. Side-Lobe Leakage Suppression Procedures 388
11.5.3. Recommended Computational Steps for Ensemble-Averaged Estimates 395
11.5.4. Zoom Transform Procedures 396
11.5.5. Autospectra Estimates by Frequency Averaging 399
11.5.6. Other Spectral Analysis Procedures 403
11.6. Joint Record Functions 404
11.6.1. Joint Probability Density Functions 404
11.6.2. Cross-Correlation Functions 405
11.6.3. Cross-Spectral Density Functions 406
11.6.4. Frequency Response Functions 407
11.6.5. Unit Impulse Response (Weighting) Functions 408
11.6.6. Ordinary Coherence Functions 408
11.7. Multiple-Input/Output Functions 408
11.7.1. Fourier Transforms and Spectral Functions 409
11.7.2. Conditioned Spectral Density Functions 409
11.7.3. Three-Input/Single-Output Models 411
11.7.4. Functions in Modified Procedure 414
12. Nonstationary Data Analysis 417
12.1. Classes of Nonstationary Data 417
12.2. Probability Structure of Nonstationary Data 419
12.2.1. Higher Order Probability Functions 420
12.2.2. Time-Averaged Probability Functions 421
12.3. Nonstationary Mean Values 422
12.3.1. Independent Samples 424
12.3.2. Correlated Samples 425
12.3.3. Analysis Procedures for Single Records 427
12.4. Nonstationary Mean Square Values 429
12.4.1. Independent Samples 429
12.4.2. Correlated Samples 431
12.4.3. Analysis Procedures for Single Records 432
12.5. Correlation Structure of Nonstationary Data 436
12.5.1. Double-Time Correlation Functions 436
12.5.2. Alternative Double-Time Correlation Functions 437
12.5.3. Analysis Procedures for Single Records 439
12.6. Spectral Structure of Nonstationary Data 442
12.6.1. Double-Frequency Spectral Functions 443
12.6.2. Alternative Double-Frequency Spectral Functions 445
12.6.3. Frequency Time Spectral Functions 449
12.6.4. Analysis Procedures for Single Records 456
12.7. Input/Output Relations for Nonstationary Data 462
12.7.1. Nonstationary Input and Time-Varying Linear System 463
12.7.2. Results for Special Cases 464
12.7.3. Frequency–Time Spectral Input/Output Relations 465
12.7.4. Energy Spectral Input/Output Relations 467
13. The Hilbert Transform 473
13.1. Hilbert Transforms for General Records 473
13.1.1. Computation of Hilbert Transforms 476
13.1.2. Examples of Hilbert Transforms 477
13.1.3. Properties of Hilbert Transforms 478
13.1.4. Relation to Physically Realizable Systems 480
13.2. Hilbert Transforms for Correlation Functions 484
13.2.1. Correlation and Envelope Definitions 484
13.2.2. Hilbert Transform Relations 486
13.2.3. Analytic Signals for Correlation Functions 486
13.2.4. Nondispersive Propagation Problems 489
13.2.5. Dispersive Propagation Problems 495
13.3. Envelope Detection Followed by Correlation 498
14. Nonlinear System Analysis 505
14.1. Zero-Memory and Finite-Memory Nonlinear Systems 505
14.2. Square-Law and Cubic Nonlinear Models 507
14.3. Volterra Nonlinear Models 509
14.4. SI/SO Models with Parallel Linear and Nonlinear Systems 510
14.5. SI/SO Models with Nonlinear Feedback 512
14.6. Recommended Nonlinear Models and Techniques 514
14.7. Duffing SDOF Nonlinear System 515
14.7.1. Analysis for SDOF Linear System 516
14.7.2. Analysis for Duffing SDOF Nonlinear System 518
14.8. Nonlinear Drift Force Model 520
14.8.1. Basic Formulas for Proposed Model 521
14.8.2. Spectral Decomposition Problem 523
14.8.3. System Identification Problem 524
Bibliography 527
Appendix A: Statistical Tables 533
Appendix B: Definitions for Random Data Analysis 545
List of Figures 557
List of Tables 565
List of Examples 567
Answers to Problems in Random Data 571
Index 599
