Random Data

JULIUS S. BENDAT, PHD, is President of the J. S. Bendat Company, an independent mathematical consulting firm in Los Angeles, California. An internationally recognized authority in the field, Dr. Bendat has over fifty years of consulting experience in the formulation of mathematical models, the development of statistical error analysis criteria, and the interpretation of engineering results. He is the author of Nonlinear System Techniques and Applications and coauthor of Engineering Applications of Correlation and Spectral Analysis, Second Edition, both published by Wiley. The late ALLAN G. PIERSOL, PE, was president of Piersol Engineering Company. His consulting career spanned over fifty years and focused on a wide range of topics including the development of machinery condition monitoring techniques and the statistical analysis of all types of mechanical shock, vibration, and acoustic data. A Fellow of the Acoustical Society of America and the Institute of Environmental Sciences and Technology, Piersol is the coauthor of Engineering Applications of Correlation and Spectral Analysis, Second Edition.

• Preface xvPreface to the Third Edition xviiGlossary of Symbols xix1. Basic Descriptions and Properties 11.1. Deterministic Versus Random Data 11.2. Classifications of Deterministic Data 31.2.1. Sinusoidal Periodic Data 31.2.2. Complex Periodic Data 41.2.3. Almost-Periodic Data 61.2.4. Transient Nonperiodic Data 71.3. Classifications of Random Data 81.3.1. Stationary Random Data 91.3.2. Ergodic Random Data 111.3.3. Nonstationary Random Data 121.3.4. Stationary Sample Records 121.4. Analysis of Random Data 131.4.1. Basic Descriptive Properties 131.4.2. Input/Output Relations 191.4.3. Error Analysis Criteria 211.4.4. Data Analysis Procedures 232. Linear Physical Systems 252.1. Constant-Parameter Linear Systems 252.2. Basic Dynamic Characteristics 262.3. Frequency Response Functions 282.4. Illustrations of Frequency Response Functions 302.4.1. Mechanical Systems 302.4.2. Electrical Systems 392.4.3. Other Systems 412.5. Practical Considerations 413. Probability Fundamentals 453.1. One Random Variable 453.1.1. Probability Density and Distribution Functions 463.1.2. Expected Values 493.1.3. Change of Variables 503.1.4. Moment-Generating and Characteristic Functions 523.1.5. Chebyshev’s Inequality 533.2. Two Random Variables 543.2.1. Expected Values and Correlation Coefficient 553.2.2. Distribution for Sum of Two Random Variables 563.2.3. Joint Moment-Generating and Characteristic Functions 573.3. Gaussian (Normal) Distribution 593.3.1. Central Limit Theorem 603.3.2. Joint Gaussian (Normal) Distribution 623.3.3. Moment-Generating and Characteristic Functions 633.3.4. N-Dimensional Gaussian (Normal) Distribution 643.4. Rayleigh Distribution 673.4.1. Distribution of Envelope and Phase for Narrow Bandwidth Data 673.4.2. Distribution of Output Record for Narrow Bandwidth Data 713.5. Higher Order Changes of Variables 724. Statistical Principles 794.1. Sample Values and Parameter Estimation 794.2. Important Probability Distribution Functions 824.2.1. Gaussian (Normal) Distribution 824.2.2. Chi-Square Distribution 834.2.3. The t Distribution 844.2.4. The F Distribution 844.3. Sampling Distributions and Illustrations 854.3.1. Distribution of Sample Mean with Known Variance 854.3.2. Distribution of Sample Variance 864.3.3. Distribution of Sample Mean with Unknown Variance 874.3.4. Distribution of Ratio of Two Sample Variances 874.4. Confidence Intervals 884.5. Hypothesis Tests 914.5.1. Chi-Square Goodness-of-Fit Test 944.5.2. Nonparametric Trend Test 964.6. Correlation and Regression Procedures 994.6.1. Linear Correlation Analysis 994.6.2. Linear Regression Analysis 1025. Stationary Random Processes 1095.1. Basic Concepts 1095.1.1. Correlation (Covariance) Functions 1115.1.2. Examples of Autocorrelation Functions 1135.1.3. Correlation Coefficient Functions 1155.1.4. Cross-Correlation Function for Time Delay 1165.2. Spectral Density Functions 1185.2.1. Spectra via Correlation Functions 1185.2.2. Spectra via Finite Fourier Transforms 1265.2.3. Spectra via Filtering–Squaring–Averaging 1295.2.4. Wavenumber Spectra 1325.2.5. Coherence Functions 1345.2.6. Cross-Spectrum for Time Delay 1355.2.7. Location of Peak Value 1375.2.8. Uncertainty Relation 1385.2.9. Uncertainty Principle and Schwartz Inequality 1405.3. Ergodic and Gaussian Random Processes 1425.3.1. Ergodic Random Processes 1425.3.2. Sufficient Condition for Ergodicity 1455.3.3. Gaussian Random Processes 1475.3.4. Linear Transformations of Random Processes 1495.4. Derivative Random Processes 1515.4.1. Correlation Functions 1515.4.2. Spectral Density Functions 1545.5. Level Crossings and Peak Values 1555.5.1. Expected Number of Level Crossings per Unit Time 1555.5.2. Peak Probability Functions for Narrow Bandwidth Data 1595.5.3. Expected Number and Spacing of Positive Peaks 1615.5.4. Peak Probability Functions for Wide Bandwidth Data 1625.5.5. Derivations 1646. Single-Input/Output Relationships 1736.1. Single-Input/Single-Output Models 1736.1.1. Correlation and Spectral Relations 1736.1.2. Ordinary Coherence Functions 1806.1.3. Models with Extraneous Noise 1836.1.4. Optimum Frequency Response Functions 1876.2. Single-Input/Multiple-Output Models 1906.2.1. Single-Input/Two-Output Model 1916.2.2. Single-Input/Multiple-Output Model 1926.2.3. Removal of Extraneous Noise 1947. Multiple-Input/Output Relationships 2017.1. Multiple-Input/Single-Output Models 2017.1.1. General Relationships 2027.1.2. General Case of Arbitrary Inputs 2057.1.3. Special Case of Mutually Uncorrelated Inputs 2067.2. Two-Input/One-Output Models 2077.2.1. Basic Relationships 2077.2.2. Optimum Frequency Response Functions 2107.2.3. Ordinary and Multiple Coherence Functions 2127.2.4. Conditioned Spectral Density Functions 2137.2.5. Partial Coherence Functions 2197.3. General and Conditioned Multiple-Input Models 2217.3.1. Conditioned Fourier Transforms 2237.3.2. Conditioned Spectral Density Functions 2247.3.3. Optimum Systems for Conditioned Inputs 2257.3.4. Algorithm for Conditioned Spectra 2267.3.5. Optimum Systems for Original Inputs 2297.3.6. Partial and Multiple Coherence Functions 2317.4. Modified Procedure to Solve Multiple-Input/Single-Output Models 2327.4.1. Three-Input/Single-Output Models 2347.4.2. Formulas for Three-Input/Single-Output Models 2357.5. Matrix Formulas for Multiple-Input/Multiple-Output Models 2377.5.1. Multiple-Input/Multiple-Output Model 2387.5.2. Multiple-Input/Single-Output Model 2417.5.3. Model with Output Noise 2437.5.4. Single-Input/Single-Output Model 2458. Statistical Errors in Basic Estimates 2498.1. Definition of Errors 2498.2. Mean and Mean Square Value Estimates 2528.2.1. Mean Value Estimates 2528.2.2. Mean Square Value Estimates 2568.2.3. Variance Estimates 2608.3. Probability Density Function Estimates 2618.3.1. Bias of the Estimate 2638.3.2. Variance of the Estimate 2648.3.3. Normalized rms Error 2658.3.4. Joint Probability Density Function Estimates 2658.4. Correlation Function Estimates 2668.4.1. Bandwidth-Limited Gaussian White Noise 2698.4.2. Noise-to-Signal Considerations 2708.4.3. Location Estimates of Peak Correlation Values 2718.5. Autospectral Density Function Estimates 2738.5.1. Bias of the Estimate 2748.5.2. Variance of the Estimate 2788.5.3. Normalized rms Error 2788.5.4. Estimates from Finite Fourier Transforms 2808.5.5. Test for Equivalence of Autospectra 2828.6. Record Length Requirements 2849. Statistical Errors in Advanced Estimates 2899.1. Cross-Spectral Density Function Estimates 2899.1.1. Variance Formulas 2929.1.2. Covariance Formulas 2939.1.3. Phase Angle Estimates 2979.2. Single-Input/Output Model Estimates 2989.2.1. Bias in Frequency Response Function Estimates 3009.2.2. Coherent Output Spectrum Estimates 3039.2.3. Coherence Function Estimates 3059.2.4. Gain Factor Estimates 3089.2.5. Phase Factor Estimates 3109.3. Multiple-Input/Output Model Estimates 31210. Data Acquisition and Processing 31710.1. Data Acquisition 31810.1.1. Transducer and Signal Conditioning 31810.1.2. Data Transmission 32110.1.3. Calibration 32210.1.4. Dynamic Range 32410.2. Data Conversion 32610.2.1. Analog-to-Digital Converters 32610.2.2. Sampling Theorems for Random Records 32810.2.3. Sampling Rates and Aliasing Errors 33010.2.4. Quantization and Other Errors 33310.2.5. Data Storage 33510.3. Data Qualification 33510.3.1. Data Classification 33610.3.2. Data Validation 34010.3.3. Data Editing 34510.4. Data Analysis Procedures 34910.4.1. Procedure for Analyzing Individual Records 34910.4.2. Procedure for Analyzing Multiple Records 35111. Data Analysis 35911.1. Data Preparation 35911.1.1. Data Standardization 36011.1.2. Trend Removal 36111.1.3. Digital Filtering 36311.2. Fourier Series and Fast Fourier Transforms 36611.2.1. Standard Fourier Series Procedure 36611.2.2. Fast Fourier Transforms 36811.2.3. Cooley–Tukey Procedure 37411.2.4. Procedures for Real-Valued Records 37611.2.5. Further Related Formulas 37711.2.6. Other Algorithms 37811.3. Probability Density Functions 37911.4. Autocorrelation Functions 38111.4.1. Autocorrelation Estimates via Direct Computations 38111.4.2. Autocorrelation Estimates via FFT Computations 38111.5. Autospectral Density Functions 38611.5.1. Autospectra Estimates by Ensemble Averaging 38611.5.2. Side-Lobe Leakage Suppression Procedures 38811.5.3. Recommended Computational Steps for Ensemble-Averaged Estimates 39511.5.4. Zoom Transform Procedures 39611.5.5. Autospectra Estimates by Frequency Averaging 39911.5.6. Other Spectral Analysis Procedures 40311.6. Joint Record Functions 40411.6.1. Joint Probability Density Functions 40411.6.2. Cross-Correlation Functions 40511.6.3. Cross-Spectral Density Functions 40611.6.4. Frequency Response Functions 40711.6.5. Unit Impulse Response (Weighting) Functions 40811.6.6. Ordinary Coherence Functions 40811.7. Multiple-Input/Output Functions 40811.7.1. Fourier Transforms and Spectral Functions 40911.7.2. Conditioned Spectral Density Functions 40911.7.3. Three-Input/Single-Output Models 41111.7.4. Functions in Modified Procedure 41412. Nonstationary Data Analysis 41712.1. Classes of Nonstationary Data 41712.2. Probability Structure of Nonstationary Data 41912.2.1. Higher Order Probability Functions 42012.2.2. Time-Averaged Probability Functions 42112.3. Nonstationary Mean Values 42212.3.1. Independent Samples 42412.3.2. Correlated Samples 42512.3.3. Analysis Procedures for Single Records 42712.4. Nonstationary Mean Square Values 42912.4.1. Independent Samples 42912.4.2. Correlated Samples 43112.4.3. Analysis Procedures for Single Records 43212.5. Correlation Structure of Nonstationary Data 43612.5.1. Double-Time Correlation Functions 43612.5.2. Alternative Double-Time Correlation Functions 43712.5.3. Analysis Procedures for Single Records 43912.6. Spectral Structure of Nonstationary Data 44212.6.1. Double-Frequency Spectral Functions 44312.6.2. Alternative Double-Frequency Spectral Functions 44512.6.3. Frequency Time Spectral Functions 44912.6.4. Analysis Procedures for Single Records 45612.7. Input/Output Relations for Nonstationary Data 46212.7.1. Nonstationary Input and Time-Varying Linear System 46312.7.2. Results for Special Cases 46412.7.3. Frequency–Time Spectral Input/Output Relations 46512.7.4. Energy Spectral Input/Output Relations 46713. The Hilbert Transform 47313.1. Hilbert Transforms for General Records 47313.1.1. Computation of Hilbert Transforms 47613.1.2. Examples of Hilbert Transforms 47713.1.3. Properties of Hilbert Transforms 47813.1.4. Relation to Physically Realizable Systems 48013.2. Hilbert Transforms for Correlation Functions 48413.2.1. Correlation and Envelope Definitions 48413.2.2. Hilbert Transform Relations 48613.2.3. Analytic Signals for Correlation Functions 48613.2.4. Nondispersive Propagation Problems 48913.2.5. Dispersive Propagation Problems 49513.3. Envelope Detection Followed by Correlation 49814. Nonlinear System Analysis 50514.1. Zero-Memory and Finite-Memory Nonlinear Systems 50514.2. Square-Law and Cubic Nonlinear Models 50714.3. Volterra Nonlinear Models 50914.4. SI/SO Models with Parallel Linear and Nonlinear Systems 51014.5. SI/SO Models with Nonlinear Feedback 51214.6. Recommended Nonlinear Models and Techniques 51414.7. Duffing SDOF Nonlinear System 51514.7.1. Analysis for SDOF Linear System 51614.7.2. Analysis for Duffing SDOF Nonlinear System 51814.8. Nonlinear Drift Force Model 52014.8.1. Basic Formulas for Proposed Model 52114.8.2. Spectral Decomposition Problem 52314.8.3. System Identification Problem 524Bibliography 527Appendix A: Statistical Tables 533Appendix B: Definitions for Random Data Analysis 545List of Figures 557List of Tables 565List of Examples 567Answers to Problems in Random Data 571Index 599

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