Random Data: Analysis and Measurement Procedures

RANDOM DATA A TIMELY UPDATE OF THE CLASSIC BOOK ON THE THEORY AND APPLICATION OF RANDOM DATA ANALYSIS First published in 1971, Random Data served as an authoritative book on the analysis of experimental physical data for engineering and scientific applications. This Fourth Edition features coverage of new developments in random data management and analysis procedures that are applicable to a broad range of applied fields, from the aerospace and automotive industries to oceanographic and biomedical research. This new edition continues to maintain a balance of classic theory and novel techniques. The authors expand on the treatment of random data analysis theory, including derivations of key relationships in probability and random process theory. The book remains unique in its practical treatment of nonstationary data analysis and nonlinear system analysis, presenting the latest techniques on modern data acquisition, storage, conversion, and qualification of random data prior to its digital analysis. The Fourth Edition also includes: A new chapter on frequency domain techniques to model and identify nonlinear systems from measured input/output random data New material on the analysis of multiple-input/single-output linear models The latest recommended methods for data acquisition and processing of random data Important mathematical formulas to design experiments and evaluate results of random data analysis and measurement procedures Answers to the problem in each chapter Comprehensive and self-contained, Random Data, Fourth Edition is an indispensible book for courses on random data analysis theory and applications at the upper-under-graduate and graduate level. It is also an insightful reference for engineers and scientists who use statistical methods to investigate and solve problems with dynamic data.

Table of 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 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