List of Examples

2.1 Illustration of nonlinear system 26 2.2 Illustration of unstable system 27 2.3 Illustration of resonant system 34 3.1 Discrete distribution 46 3.2 Uniform (rectangular) distribution 48 3.3 Sine wave distribution with fixed amplitude 51 3.4 Illustration of probability intervals 54 3.5 Sum of two independent uniformly distributed variables 57 3.6 Sine wave distribution with random amplitudes 73 3.7 Sine wave in Gaussian noise 74 4.1 Illustration of confidence intervals 90 4.2 Illustration of hypothesis test design 93 4.3 Illustration of test for normality 96 4.4 Illustration of reverse arrangement test 98 4.5 Illustration of linear correlation analysis 101 4.6 Illustration of linear regression analysis 104 5.1 Autocorrelation function of sine wave process 113 5.2 Autocorrelation function of rectangular wave process 113 5.3 Autocorrelation function of sum of two processes 114 5.4 Uncorrelated-dependent random variables 114 5.5 Autospectral density function of sine wave process 124 5.6 Autospectral density function of rectangular wave process 126 5.7 Autospectral density function of sum of two processes 126 5.8 Low-pass white noise 139 5.9 Gaussian spectrum noise 139 5.10 Exponential autocorrelation function noise 140 5.11 Nonergodic stationary random process 144 5.12 Zero crossings of low-pass Gaussian white noise 158 5.13 Zero crossings of bandwidth-limited Gaussian white noise 158 5.14 Peak probability for narrow bandwidth Gaussian data 160

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Random Data: Analysis and Measurement Procedures, Fourth Edition. By Julius S. Bendat and Allan G. Piersol Copyright © 2010 John Wiley & Sons, Inc.

568 LIST OF EXAMPLES

5.15 Expected number of positive peaks for narrow bandwidth Gaussian data 162

6.1 Response properties of low-pass filter to white noise 177 6.2 Response properties of low-pass filter to sine wave 178 6.3 Force-input/displacement-output system 178 6.4 Displacement-input/displacement-output system 179 6.5 Physical illustration of coherence measurement 181 6.6 Illustration of noisy measurements 197 7.1 Multiple coherence function for two uncorrelated inputs 213 7.2 Illustration of erroneous high coherence 214 8.1 Approximate 95% confidence intervals for mean square

and rms value estimates 252 8.2 Random error in mean value estimate 255 8.3 Bandwidths for random noise with a nonuniform autospectrum 259 8.4 Bias in probability density estimate of Gaussian random data 263 8.5 Time-delay estimate from cross-correlation calculation 272 8.6 Illustration of bias error in autospectrum estimate 275 8.7 Illustration of optimum resolution bandwidth selection 279 9.1 Illustration of random errors in cross-spectral density estimate 298 9.2 Illustration of bias and random errors in coherent output

spectrum estimate 303 9.3 Illustration of confidence interval for coherence function estimate 306 9.4 Illustration of random errors in frequency response function

estimate 311 10.1 Illustration of piezoelectric accelerometer 319 10.2 Illustration of digitization 334 10.3 Test for stationarity 338 10.4 Autospectrum of sine wave in noise 339 11.1 Illustration of recursive digital filter 365 11.2 FFT speed ratio for powers of two 370 11.3 Spectral errors due to side-lobe leakage 391 11.4 Parameter selections for zoom transform 398 11.5 Number of computations for frequency- versus

ensemble-averaged autospectra estimates 402 12.1 Illustration of time-averaged probability density function 422 12.2 Variance of mean value estimate for exponential correlation

between samples 427 12.3 Variance of mean square value estimate for exponential

correlation between samples 431 12.4 Illustration of nonstationary mean square value estimates

using a moving average 433 12.5 Illustration of time interval bias error determination 435 12.6 Instantaneous autocorrelation function of modulated

random data 439 12.7 Double-frequency autospectrum of modulated random data 448

LIST OF EXAMPLES 569

12.8 Instantaneous autospectrum of modulated random data 454 12.9 Selection of optimum analysis parameters for nonstationary data 459 12.10 Illustration of energy spectrum estimate 486 13.1 Digital formulas for x(t) and k\t) 477 13.2 Exponential causal function 483 13.3 Exponential-cosine causal function 484 13.4 Digital formulas for R^x) and Rxy(x) 488 13.5 Low-Pass White Noise 491 13.6 Nondispersive propagation through multiple paths 492 13.7 Nondispersive propagation from multiple uncorrelated sources 493 13.8 Nondispersive propagation from an unmeasured single source

to measured multiple outputs 494 13.9 Exponential-cosine cross-correlation function 500