Preface ......................................................... v
Acknowledgements ............................................... ix
About the Book Cover Design .................................... xi
1 Acoustics, Hearing Limitations, and Sampling ................. 1
1 Introduction .............................................. 1
2 The Sine Tone ............................................. 2
3 Human Hearing and Its Limitations ......................... 4
3.1 Duration ............................................. 4
3.2 Pitch ................................................ 5
3.3 Amplitude and sound levels ........................... 6
3.3.1 Sound intensity level (SIL) ................... 6
3.3.2 Sound pressure level (SPL) .................... 8
3.3.3 Just noticeable difference (JND) .............. 8
3.3.4 Equal loudness curve .......................... 8
3.4 Auditory masking .................................... 11
4 Sampling: The Art of Being Discrete ...................... 13
4.1 Sampling theorem .................................... 18
4.2 Aliasing ............................................ 20
5 Quantization and Pulse Code Modulation (PCM) ............. 26
5.1 SNR and QSNR ........................................ 28
6 DC Component ............................................. 29
7 Distortion and Square Waves .............................. 31
7.1 Dithering ........................................... 33
8 Musical Examples ......................................... 37
References and Further Reading ........................... 38
2 Time-Domain Signal Processing I ............................. 40
1 Introduction ............................................. 40
2 Amplitude Envelope and ADSR .............................. 41
3 Wavetable Synthesis ...................................... 43
4 Windowing, RMS, and Amplitude Envelope ................... 46
4.1 Windowing: More details ............................. 46
4.2 RMS and amplitude envelope .......................... 50
5 Time-Domain Fundamental Frequency Computation ............ 53
5.1 Zero-crossing rate .................................. 54
5.2 Autocorrelation ..................................... 58
5.3 Cross-correlation ................................... 64
6 Sample Rate Conversion ................................... 65
6.1 Up-sampling ......................................... 65
6.2 Down-sampling ....................................... 66
7 Overlap and Add (OLA) .................................... 68
7.1 OLA: Problems and solutions ......................... 71
8 Musical Examples ......................................... 72
References and Further Reading ........................... 74
3 Time-Domain Processes II .................................... 75
1 Introduction ............................................. 75
2 Granular Synthesis ....................................... 75
2.1 Basic granular synthesis parameters ................. 76
2.2 Asynchronous granular synthesis ..................... 79
2.3 Pitch shifting and time stretching/compression ...... 80
2.4 Sound morphing with granular synthesis .............. 81
3 Amplitude Distortion and Waveshaping ..................... 81
3.1 Dynamic compressor .................................. 83
3.2 Distortion .......................................... 85
3.3 Dynamic expander .................................... 86
3.4 Tape saturation ..................................... 87
3.5 Waveshaping synthesis ............................... 87
3.5.1 Chebychev polynomials of the 1st kind ........ 88
4 Some Familiar Time-Domain DSP Effects .................... 91
4.1 Equal power panning ................................. 91
4.2 Delays .............................................. 92
4.2.1 Echo, chorus, and flanging ................... 92
5 Musical Examples ......................................... 95
References and Further Reading ........................... 96
4 Sine Waves .................................................. 97
1 Introduction ............................................. 97
2 Sinusoids Revisited ...................................... 98
3 Imaginary, Complex Numbers, and Euler's Formula .......... 99
3.1 Euler's formula .................................... 101
4 Sinusoidal Modulation Techniques I: Amplitude ........... 102
4.1 Beating ............................................ 102
4.2 Amplitude modulation and ring modulation ........... 105
4.3 Amplitude modulation (AM) .......................... 106
4.4 Ring modulation .................................... 108
4.4.1 Ring modulation with complex signals ........ 108
5 Sinusoidal Modulation Techniques II: Frequency .......... 110
5.1 FM: Sidebands and the Bessel function .............. 1ll
5.2 Modulation index ................................... 115
5.3 General topics in FM control parameters ............ 116
6 Musical Examples ........................................ 118
References and Further Reading .......................... 121
5 Linear Time-Invariant Systems .............................. 122
1 Introduction ............................................ 122
2 Difference Equations: Starting with the Moving
Average Algorithm ....................................... 123
2.1 Causality .......................................... 125
2.2 Difference equations: General form ................. 125
3 Linear-Time Invariant (LTI) Systems ..................... 127
3.1 Linearity property: Scalability and
superposition ...................................... 128
3.2 Time-invariance property: Time-shift invariance .... 129
3.3 Importance of LTI systems in DSP ................... 131
4 Impulse Response ........................................ 132
4.1 Finite impulse response (FIR) and infinite
impulse response (IIR) ............................. 135
4.2 Stability and IIR systems .......................... 136
5 Convolution ............................................. 138
5.1 Convolution "Need to Knows" ........................ 140
6 System Diagrams and Digital Building Blocks ............. 141
7 Musical Examples ........................................ 143
6 Frequency Response ......................................... 145
1 Introduction ............................................ 145
2 The Frequency Response .................................. 146
2.1 Characteristics and properties of H(ejθ) ........... 149
2.1.1 Frequency range and Nyquist revisited ....... 149
2.1.2 H(ejθ) and periodicity property ............. 154
2.1.3 Symmetry .................................... 155
2.2 More stuff on the frequency response ............... 156
3 Phase Response and Phase Distortion ..................... 160
3.1 Phase delay ........................................ 163
3.2 Linearity and phase ................................ 165
3.3 Phase response and continuous phase ................ 167
3.4 Group delay ........................................ 168
4 The (Almost) Magical Z-Transform ........................ 172
4.1 What does all this mean? Part I .................... 174
4.2 What does all this mean? Part II: poles and
zeros .............................................. 175
4.3 What does all this mean? Part III: the unit
circle and the z-plane ............................. 176
4.4 More "complex" systems ............................. 181
5 Region of Convergence (ROC) ............................. 181
5.1 Causal system ROC .................................. 186
5.2 Mixed-causality systems ............................ 187
5.3 ROC summary ........................................ 188
6 Stability and the Unit Circle ........................... 188
7 The Inverse Z-Transform ................................. 190
7.1 Long division method ............................... 191
7.2 Taylor series expansion method ..................... 193
7.3 Contour integration/residue theorem method ......... 195
8 Useful Tools in MATLAB® ................................. 197
9 Musical Examples ........................................ 198
References and Further Reading .......................... 199
7 Filters .................................................... 200
1 Introduction ............................................ 200
2 Low/High/Band-Pass and Band-Stop Filters ................ 201
2.1 Filter design specifications ....................... 203
2.2 Passband, stopband, and transition band ............ 204
2.3 Cutoff frequency ................................... 205
2.4 Filter order, filter sharpness, and ripple ......... 205
2.5 MATLAB® filter design tools ........................ 219
3 Filter Examples ......................................... 222
3.1 Subtractive synthesis and filters .................. 223
3.2 Bi-quadratic filter (a.k.a. Bi-quad filter) and
the Wah-Wah filter ................................. 225
3.3 The Comb-filter .................................... 233
3.3.1 Comb-filter interpretation .................. 235
3.3.2 Comb-filter examples ........................ 238
3.4 String vibration and standing waves ................ 242
3.5 Physical modeling synthesis and the plucked
string model ....................................... 245
3.5.1 Direct implementation of difference
equations ................................... 253
3.6 Phase as a filtering application ................... 254
3.6.1 The chorus effect ........................... 254
3.6.2 Multi-tap filters and filter banks .......... 256
3.6.3 Fractional delay ............................ 259
3.6.4 The flanger effect .......................... 261
3.6.5 The all-pass filter ......................... 262
3.6.6 Very basic all-pass filter reverb ........... 266
4 Musical Examples ........................................ 272
References and Further Reading .......................... 274
8 Frequency-Domain and the Fourier Transform ................. 276
1 Introduction ............................................ 276
2 Additive Synthesis ...................................... 277
3 The Fourier Transform ................................... 280
4 The Discrete-Time Fourier Transform (DTFT) and the
Discrete Fourier Transform (DFT) ........................ 281
4.1 Magnitude, Phase, and Other Basic Properties of
the DFT ............................................ 288
4.2 Time Resolution vs. Frequency Resolution in DFTs ... 289
5 Short-Time Fourier Transform (STFT) ..................... 292
6 Zero Padding ............................................ 293
7 Aliasing Revisited ...................................... 295
8 Another Look: Down-Sampling and Up-Sampling Revisited ... 298
8.1 Down-Sampling ...................................... 298
8.2 Up-Sampling ........................................ 302
9 Windowing Revisited: A View from the Frequency-Domain
Side .................................................... 304
9.1 Rectangular Window ................................. 310
9.2 Hann Window ........................................ 310
9.3 Hamming Window ..................................... 311
9.4 Blackman Window .................................... 312
9.5 Chebychev and Kaiser Windows ....................... 312
9.6 Not Just More Windowing Stuff ...................... 313
10 The Fast Fourier Transform (FFT) ........................ 319
11 Convolution (also) Revisited ............................ 321
11.1 Circular Convolution and Time-Aliasing ............. 322
12 One More Look at Dithering .............................. 325
13 Spectrogram ............................................. 327
14 Fourier Transform Properties and Summary ................ 329
15 MATLAB® and Fourier Transform ........................... 331
16 Musical Examples ........................................ 331
References and Further Reading .......................... 332
9 Spectral Analysis, Vocoders, and other Goodies ............. 333
1 Introduction ............................................ 333
1.1 Musical signals and important nomenclatures ........ 333
2 Spectral Analysis ....................................... 335
2.1 Long-term average spectrum (LTAS) .................. 336
2.2 Log vs. linear ..................................... 336
2.3 Spectral peaks, valleys, and spectral envelope ..... 340
2.4 Extraction of fundamental frequency and
harmonics .......................................... 344
2.4.1 Inverse comb-filtering ...................... 345
2.4.2 Cepstrum analysis ........................... 349
2.4.3 Harmonic product spectrum ................... 350
3 Vocoders (Voice Coders) ................................. 352
3.1 Channel-vocoder .................................... 352
3.1.1 Filter banks, envelope followers, and the
encoder ..................................... 354
3.1.2 Voiced and unvoiced analysis and the
decoder ..................................... 356
3.1.3 Voiced and unvoiced decision-making ......... 359
3.1.3.1 Zero-crossing analysis ............. 359
3.1.3.2 Pre-emphasized energy ratio ........ 363
3.1.3.3 Low-band to full-band energy
ratio .............................. 363
3.1.3.4 Spectral flatness measure .......... 364
3.2 Linear predictive coding (LPC) ..................... 366
3.3 LPC coefficient computation ........................ 371
3.4 The phase vocoder .................................. 373
3.4.1 Estimation of instantaneous frequency ....... 375
3.4.2 Phase unwrapping ............................ 377
3.4.3 Phase vocoder: Filter-bank interpretation ... 379
3.4.4 Phase vocoder: Fourier transform
interpretation .............................. 385
3.4.5 Phase vocoder basics: Time and
pitch-shifting .............................. 390
3.4.5.1 Time-shifting ...................... 391
3.4.5.2 Pitch-shifting ..................... 396
4 Research Topics in Computer Music ....................... 397
4.1 Salient feature extraction ......................... 397
4.1.1 Spectral envelope ........................... 398
4.1.2 Spectral centroid ........................... 399
4.1.3 Shimmer and Jitter .......................... 400
4.1.4 Spectral flux ............................... 400
4.1.5 Log spectral spread ......................... 401
4.1.6 Roll-off .................................... 401
4.1.7 Attack time (rise time) ..................... 402
4.1.8 Amplitude modulation (Tremolo) .............. 402
4.1.9 Temporal centroid ........................... 403
4.2 MIR (Music information retrieval) .................. 403
4.2.1 Query-by-humming (QbH) ...................... 404
4.2.2 Automatic beat detection and rhythm
analysis .................................... 405
4.2.3 Automatic timbre recognition ................ 406
4.3 FMS (feature modulation synthesis) ................. 407
5 Musical Examples ........................................ 409
References and Further Reading ............................. 411
Appendix ...................................................... 415
1 To Scale or Not to Scale: That is the Question .......... 415
1.1 Equal temperament scale ............................ 415
1.2 Just intonation .................................... 416
1.3 Bark scale ......................................... 416
1.4 Mel scale .......................................... 417
2 MATLAB® Programs Used in This Book ...................... 418
Index ......................................................... 423
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