Contents: Musimathics: (Volume 2)

Contents
Foreword by John Chowning xi
Preface xiii
Acknowledgments xiv

1 Digital Signals and Sampling 1

  • 1.1 Measuring the Ephemeral 1
  • 1.2 Analog-to-Digital Conversion 9
  • 1.3 Aliasing 11
  • 1.4 Digital-to-Analog Conversion 20
  • 1.5 Binary Numbers 22
  • 1.6 Synchronization 28
  • 1.7 Discretization 28
  • 1.8 Precision and Accuracy 29
  • 1.9 Quantization 29
  • 1.10 Noise and Distortion 33
  • 1.11 Information Density of Digital Audio 38
  • 1.12 Codecs 40
  • 1.13 Further Refinements 42
  • 1.14 Cultural Impact of Digital Audio 46 Summary 47

2 Musical Signals 49

  • 2.1 Why Imaginary Numbers? 49
  • 2.2 Operating with Imaginary Numbers 51
  • 2.3 Complex Numbers 52
  • 2.4 de Moivre's Theorem 62
  • 2.5 Euler's Formula 64
  • 2.6 Phasors 68
  • 2.7 Graphing Complex Signals 86
  • 2.8 Spectra of Complex Sampled Signals 87
  • 2.9 Multiplying Phasors 89
  • 2.10 Graphing Complex Spectra 92
  • 2.11 Analytic Signals 95
  • Summary 100

3 Spectral Analysis and Synthesis 103

  • 3.1 ntroduction to the Fourier Transform 103
  • 3.2 Discrete Fourier Transform 114
  • 3.3 Discrete Fourier Transfom in Action 125
  • 3.4 Inverse Discrete Fourier Transform 134
  • 3.5 Analyzing Real-World Signals 138
  • 3.6 Windowing 141
  • 3.7 Fast Fourier Transform 145
  • 3.8 Properties of the Discrete Fourier Transform 147
  • 3.9 A Practical Hilbert Transform 154
  • Summary 156

4 Convolution 159

  • 4.1 Rolling Shutter Camera 159
  • 4.2 Defining Convolution 161
  • 4.3 Numerical Examples of Convolution 163
  • 4.4 Convolving Spectra 168
  • 4.5 Convolving Signals 172
  • 4.6 Convolution and the Fourier Transform 180
  • 4.7 Domain Symmetry between Signals and Spectra 180
  • 4.8 Convolution and Sampling Theory 185
  • 4.9 Convolution and Windowing 187
  • 4.10 Correlation Functions 191
  • Summary 193
  • Suggested Reading 194

5 Filtering 195

  • 5.1 Tape Recorder as a Model of Filtering 195
  • 5.2 Introduction to Filtering 199
  • 5.3 A Simple Filter 201
  • 5.4 Finding the Frequency Response 203
  • 5.5 Linearity and Time Invariance of Filters 217
  • 5.6 FIR Filters 218
  • 5.7 IIR Filters 218
  • 5.8 Canonical Filter 219
  • 5.9 Time Domain Behavior of Filters 219
  • 5.10 Filtering as Convolution 222
  • 5.11 Z Transform 224
  • 5.12 Z Transform of the General Difference Equation 232
  • 5.13 Filter Families 244 Summary 261

6 Resonance 263

  • 6.1 The Derivative 263
  • 6.2 Differential Equations 276
  • 6.3 Mathematics of Resonance 280 Summary 297

7 The Wave Equation 299

  • 7.1 One-Dimensional Wave Equation and String Motion 299
  • 7.2 An Example 307
  • 7.3 Modeling Vibration with Finite Difference Equations 310
  • 7.4 Striking Points, Plucking Points, and Spectra 319 Summary 324

8 Acoustical Systems 325

  • 8.1 Dissipation and Radiation 325
  • 8.2 Acoustical Current 326
  • 8.3 Linearity of Frictional Force 329
  • 8.4 Inertance, Inductive Reactance 332
  • 8.5 Compliance, Capacitive Reactance 333
  • 8.6 Reactance and Alternating Current 334
  • 8.7 Capacitive Reactance and Frequency 335
  • 8.8 Inductive Reactance and Frequency 336
  • 8.9 Combining Resistance, Reactance, and Alternating Current 336
  • 8.10 Resistance and Alternating Current 337
  • 8.11 Capacitance and Alternating Current 337
  • 8.12 Acoustical Impedance 340
  • 8.13 Sound Propagation and Sound Transmission 344
  • 8.14 Input Impedance: Fingerprinting a Resonant System 351
  • 8.15 Scattering Junctions 357
  • Summary 360
  • Suggested Reading 362

9 Sound Synthesis 363

  • 9.1 Forms of Synthesis 363
  • 9.2 A Graphical Patch Language for Synthesis 365
  • 9.3 Amplitude Modulation 384
  • 9.4 Frequency Modulation 389
  • 9.5 Vocal Synthesis 409
  • 9.6 Synthesizing Concert Hall Acoustics 425
  • 9.7 Physical Modeling 433
  • 9.8 Source Models and Receiver Models 449
  • Summary 450

10 Dynamic Spectra 453

  • 10.1 Gabor's Elementary Signal 454
  • 10.2 Short-Time Fourier Transform 459
  • 10.3 Phase Vocoder 486
  • 10.4 Improving on the Fourier Transform 496
  • 10.5 Psychoacoustic Audio Encoding 502
  • Summary 507
  • Suggested Reading 509
  • Epilogue 511

Appendix 513

  • A.l About Algebra 513
  • A.2 About Trigonometry 514
  • A.3 Series and Summations 517
  • A.4 Trigonometric Identities 518
  • A.5 Modulo Arithmetic and Congruence 522
  • A.6 Finite Difference Approximations 523
  • A.7 Walsh-Hadamard Transform 525
  • A.8 Sampling, Reconstruction, and the Sine Function 526
  • A.9 Fourier Shift Theorem 528
  • A.10 Spectral Effects of Ring Modulation 529
  • A.11Derivation of the Reflection Coefficient 530

    Notes 533
    Glossary 539
    References 543
    Index of Equations and Mathematical Formulas 547
    Subject Index 551
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