Signals, Sound, and Sensation

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1 Signals, Sound, and Sensation William M. Hartmann Department of Physics and Astronomy Michigan State University East Lansing, Michigan Л1Р

2 Contents Preface xv Chapter 1: Pure Tones 1 Mathematics of the Pure Tone 1 *H. R. Hertz ( ) 4 The Spectrum 5 Importance of the Pure Tone in Psychoacoustics 6 Sound of a Pure Tone 7 Chapter 2: Complex Representation 10 Complex Numbers Rectangular Form 10 Complex Numbers Polar Form 11 Addition of Complex Numbers 14 Multiplication of Complex Numbers 14 Complex Conjugate and Absolute Value 15 Reciprocal of a Complex Number 16 Phasors 17 Chapter 3: Power, Intensity, and Decibels 24 Average Power and RMS Value 25 Crest Factor 27 Decibels 28 Acoustics: Pressure and Intensity 31 Waves in Time and Space 33 *Lord Rayleigh ( ) 34 Measuring Sound Pressure Levels 39 Adding Signals 40 Spectral Density and Spectrum Level 46 Observations on the Decibel Scale 49 Asterisks throughout Contents refer to biographies. VÜ

3 viii CONTENTS A Note on Metric Units of Pressure 50 Chapter 4: Intensity and Loudness 59 Loudness Level 59 Difference Limens 61 Psychophysics 63 Intensity Difference Limens and Loudness 69 *Harvey Fletcher ( ) 72 Loudness Summation Across Frequency 72 Temporal Integration 74 Chapter 5: Fourier Series 81 *J. B. J. Fourier ( ) 82 Definition of Fourier Series 82 Projecting of Functions 84 The Spectrum 89 Symmetry 92 Computer Evaluation of Fourier Series Coefficients 98 Chapter 6: Perception of Periodic Complex Tones 117 Musical Tones and Their Sources 118 Harmonic Analysis by the Auditory Periphery 119 Method: Pulsation Threshold 122 Hearing Out Harmonics 123 Method: Bekesy Tracking 132 Segregation and Integration 133 Pitch and Tone Color 140 *Georg Simon Ohm ( ) 145 Pitch, Chroma, and Shepard Tones 145 Chapter 7: Delta Functions 150 Basic Definition 150 Relation to the Unit Step Function 151 Translation and Selection Properties 151 Application to Fourier Transforms 152 Relation to the Kronecker Delta 153

4 CONTENTS ix Further Properties of the Delta Function 154 Lattice Sum 155 Representations of the Delta Function 156 Chapter 8: Fourier Integral 161 The Transforms Defined 162 Transforms of Sine and Cosine 163 Real Functions, Even and Odd Functions 165 Time Shifting 169 Derivatives and Integrals 170 Convolution 171 Introduction to Correlation Functions 173 Introduction to Filtering 175 Periodic Functions Fourier Series 176 Periodic Functions and the Lattice Sum 179 Large and Small Scales 182 Fourier Transforms of sgn and Theta 184 Chapter 9: Filters 195 Filter Fundamentals 195 First-Order Filters 202 Second-Order Filters 209 Transversal Filters 214 Dispersion Relations 216 All-Pass Filters 218 Minimum-Phase Filters 223 Transfer Function of the Hubert Transformer 225 Measuring Transfer Functions and Coherence 227 Chapter 10: Auditory Filters 238 Cochlear Tuning 238 Excitation Patterns and Critical Bands 242 Critical Bandwidth 249 *Heinrich Barkhausen ( ) 254 Gammatone Filters 254 The Ubiquitous Critical Band 256

5 x CONTENTS Chapter 11: Musical Measures of Frequency 264 Intervals 264 Dividing the Octave 266 Unit of Cents 271 Absolute Musical Scale 273 *H. L. F. von Helmholtz ( ) 274 Stretched and Compressed Tuning 275 Chapter 12: Pitch of Sine Tones 282 Definition of Pitch 283 Dependence of Pitch on Intensity 285 Diplacusis, Threshold Microstructure, and Emissions 288 Noise-Induced Pitch Shift 290 Post-Stimulatory Pitch Shifts 290 Place Theories and Timing Theories 292 *Georg Von Bekesy ( ) 295 Mel Scale 295 Pitches of Sines and Complex Tones 297 Chapter 13: Applications of the Fourier Transform 302 Energy Spectrum and Power Spectrum 302 *S. S. Stevens ( ) 310 Alternating Lattice 310 Forgetful Fourier Transformer 314 Uncertainty Principle 317 Spectral Rakes 322 Caveat on Representations 327 Chapter 14: Correlation Functions and Spectra 332 Finite-Duration Signal Autocorrelation Function 333 Infinite-Duration Signal Autocorrelation Function 335 Autocorrelation for Bands of Noise 342 Symmetry of the Autocorrelation Function 346 Cross-Correlation in Three Easy Cases 346

6 CONTENTS xi Cross-Spectrum 349 Revcor Technique 351 Autocorrelation and Pitch Perception 352 Chapter 15: Delay-and-Add Filtering 361 Impulse Response and Transfer Function for Delay and Add 361 Repetition Pitch 364 Comb Filters 368 Delay-and-Add Filters in Motion 372 Chapter 16: Probability Density Functions 377 Derivation of the PDF 377 PDF for the Sum of Two Functions 380 PDF for Random Events 382 Averages and the PDF 386 Central Moments 386 Chapter 17: Beats and Amplitude Modulation 393 Beats of Equal-Amplitude Sines 393 Beats of Sines with Unequal Amplitudes 396 Amplitude Modulation 399 Balanced Modulation 403 Beats and the Frequency-Domain Grating 404 Chapter 18: The Envelope 412 Formal Evaluation of the Envelope 412 Envelope Rules 415 Calculation of the Envelope 417 The Envelope and the Hubert Transform 421 *David Hubert ( ) 423 More Envelope Rules 423 The Envelope and Perception 424 Ensemble-Average Envelope 426

7 xii CONTENTS Chapter 19: Frequency Modulation 430 Narrow-Band FM (NBFM) 432 Wide-Band FM 437 Detection of Frequency Modulation 442 Chapter 20: Modulation Detection and Perception 449 Mixed Modulation 449 Modulation Detection and the Critical Band 453 FM and AM Detection Unified? 454 Masking by Two Tones 455 Modulation Transfer Function 457 Roughness 462 Chapter 21: Sampled Signals 468 The Digitized Signal 469 The Sampled Signal 470 The Output Signal 477 Sampling-Jitter Noise 481 Discrete Fourier Transform 482 Fast Fourier Transform 484 Oversampling 485 Chapter 22: Nonlinear Distortion 491 Memoryless Representation 491 Dynamical Representation 492 Harmonic Distortion 495 Measurement of Harmonic Distortion 503 Intermodulation Distortion 505 Distortion and the Audibility of Phase 509 Auditory Nonlinearity 511 Dynamic Range Compression 516 Chapter 23: Noise 522 Gaussian and Thermal Noise 522 Making Noise 524

8 CONTENTS xiii Power Spectrum 525 Color of Noise 527 Uniform Exciting Noise 527 *Ernest Glen Wever ( ) 531 Noise Fluctuations 531 Chapter 24: Signal Detection Theory 541 Two-Alternative Forced-Choice 541 (M +1)-Alternative Forced-Choice 545 Yes-No 546 *G. T. Fechner ( ) 548 Trials and Procedures 549 Staircase Methods 549 Efficiency 556 Appendices Appendix A: Greek Alphabet 560 Appendix B: Trigonometric Functions 561 Definitions 561 Angle Sum and Difference Formulas 563 Euler's Formula and Equivalents 566 Appendix C: Series 567 Infinite Series 567 Binomial Series 568 Geometric Series 569 Special Finite Series 571 Special Infinite Series 572 Appendix D: Integrals Even and Odd Functions 575 Appendix E: Integrals in the Complex Plane 578 Impulse Response and the Residue 578

9 xiv CONTENTS Hubert Transform Integrals 580 Appendix F: Hubert Transforms 583 Components of the Analytic Signal 583 Dispersion Relations 586 Minimum-Phase Filters, Gain and Phase Shift 588 Appendix G: Electrical Filters 591 Appendix H: Normal Distribution 597 Error Function 598 Cumulative Normal 600 Central Limit Theorem 602 Appendix I: Rayleigh Distribution 605 Generating Random Numbers Having Gaussian and Rayleigh Distributions 607 Appendix J: Standards 610 ISO Frequencies 610 Mathematical Constants 610 Units of Pressure and Hearing Threshold 610 Physical Constants 611 Appendix K: Calculation of Intermodulation Distortion 613 References 618 Index 633

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