
Foreword
John Chowning 
xiii 


Preface 
xv 


Acknowledgments 
xvi 

1 
Digital Signals and Sampling 
1 

1.1 
Measuring the Ephemeral 
1 

1.2 
AnalogtoDigital Conversion 
9 

1.3 
Aliasing 
11 

1.4 
DigitaltoAnalog 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 Comlpex 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 
Introduction to the Fourier Transform 
103 

3.2 
Discrete Fourier Transform 
103 

3.3 
Discrete Fourier Transform in Action 
125 

3.4 
Inverse Discrete Fourier Transform 
134 

3.5 
Analyzing RealWorld 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 Sigals 
172 

4.6 
Convolution and the Fourier Transform 
180 

4.7 
Domain Symmetry between Signals and Spectra 
180 

4.8 
Convolution and Sampling Theory 
187 

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 Sample 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 
OneDimensional 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 
339 

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 
The ShortTime 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.1 
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 
WalshHadamard Transform 
525 

A.8 
Sampling, Reconstruction, and Sinc Function 
526 

A.9 
Fourier Shift Theorem 
528 

A.10 
Spectral Effects of Ring Modulation 
529 

A.11 
Derivation of the Reflection Coefficient 
530 


Notes 
533 


Glossary 
539 


References 
543 


Equation Index 
547 
