Fundamentals of Signal Processing for Sound and Vibration Engineers

Fundamentals of Signal Processing for Sound and Vibration Engineers......Page 4
Contents......Page 8
Preface......Page 12
About the Authors......Page 14
1 Introduction to Signal Processing......Page 16
1.1 Descriptions of Physical Data (Signals)......Page 21
1.2 Classification of Data......Page 22
Part I Deterministic Signals......Page 32
2.1 Periodic Signals......Page 34
2.2 Almost Periodic Signals......Page 36
2.4 Brief Summary and Concluding Remarks......Page 39
2.5 MATLAB Examples......Page 41
3.1 Periodic Signals and Fourier Series......Page 46
3.2 The Delta Function......Page 53
3.3 Fourier Series and the Delta Function......Page 56
3.4 The Complex Form of the Fourier Series......Page 57
3.5 Spectra......Page 58
3.6 Some Computational Considerations......Page 61
3.8 MATLAB Examples......Page 67
4.1 The Fourier Integral......Page 72
4.2 Energy Spectra......Page 76
4.3 Some Examples of Fourier Transforms......Page 77
4.4 Properties of Fourier Transforms......Page 82
4.5 The Importance of Phase......Page 86
4.6 Echoes......Page 87
4.7 Continuous-Time Linear Time-Invariant Systems and Convolution......Page 88
4.8 Group Delay (Dispersion)......Page 97
4.9 Minimum and Non-Minimum Phase Systems......Page 100
4.10 The Hilbert Transform......Page 105
4.11 The Effect of Data Truncation (Windowing)......Page 109
4.12 Brief Summary......Page 117
4.13 MATLAB Examples......Page 118
5.1 The Fourier Transform of an Ideal Sampled Signal......Page 134
5.2 Aliasing and Anti-Aliasing Filters......Page 141
5.3 Analogue-to-Digital Conversion and Dynamic Range......Page 146
5.4 Some Other Considerations in Signal Acquisition......Page 149
5.5 Shannon’s Sampling Theorem (Signal Reconstruction)......Page 152
5.6 Brief Summary......Page 154
5.7 MATLAB Examples......Page 155
6.1 Sequences and Linear Filters......Page 160
6.2 Frequency Domain Representation of Discrete Systems and Signals......Page 165
6.3 The Discrete Fourier Transform......Page 168
6.4 Properties of the DFT......Page 175
6.5 Convolution of Periodic Sequences......Page 177
6.6 The Fast Fourier Transform......Page 179
6.7 Brief Summary......Page 181
6.8 MATLAB Examples......Page 185
Part II Introduction to Random Processes......Page 206
7.1 Basic Probability Theory......Page 208
7.2 Random Variables and Probability Distributions......Page 213
7.3 Expectations of Functions of a Random Variable......Page 217
7.4 Brief Summary......Page 226
7.5 MATLAB Examples......Page 227
8 Stochastic Processes; Correlation Functions and Spectra......Page 234
8.1 Probability Distribution Associated with a Stochastic Process......Page 235
8.2 Moments of a Stochastic Process......Page 237
8.3 Stationarity......Page 239
8.4 The Second Moments of a Stochastic Process; Covariance (Correlation) Functions......Page 240
8.5 Ergodicity and Time Averages......Page 244
8.6 Examples......Page 247
8.7 Spectra......Page 257
8.8 Brief Summary......Page 266
8.9 MATLAB Examples......Page 268
9.1 Single-Input Single-Output Systems......Page 292
9.2 The Ordinary Coherence Function......Page 299
9.3 System Identification......Page 302
9.4 Brief Summary......Page 312
9.5 MATLAB Examples......Page 313
10.1 Estimator Errors and Accuracy......Page 332
10.2 Mean Value and Mean Square Value......Page 335
10.3 Correlation and Covariance Functions......Page 338
10.4 Power Spectral Density Function......Page 342
10.5 Cross-spectral Density Function......Page 362
10.6 Coherence Function......Page 364
10.7 Frequency Response Function......Page 365
10.8 Brief Summary......Page 367
10.9 MATLAB Examples......Page 369
11.1 Description of Multiple-Input, Multiple-Output (MIMO) Systems......Page 378
11.2 Residual Random Variables, Partial and Multiple Coherence Functions......Page 379
11.3 Principal Component Analysis......Page 385
Appendix A Proof of • ••2M sin 2 aM 2 aM da =1......Page 390
Appendix B Proof of |Sxy( f ) | 2 •Sxx( f )Syy( f )......Page 394
Appendix C Wave Number Spectra and an Application......Page 396
Appendix D Some Comments on the Ordinary Coherence Function • 2 xy ( f )......Page 400
Appendix E Least Squares Optimization: Complex-Valued Problem......Page 402
Appendix F Proof of HW( f ) •H1( f )as ( f ) ••......Page 404
Appendix G Justification of the Joint Gaussianity of X( f )......Page 406
Appendix H Some Comments on Digital Filtering......Page 408
References......Page 410
Index......Page 414