Musimathics The Mathematical Foundations of Music Volume 2. Gareth Loy. Foreword by John Chowning

Transcription

1 Musimathics The Mathematical Foundations of Music Volume 2 Gareth Loy Foreword by John Chowning The MIT Press Cambridge, Massachusetts London, England

2 Foreword by John Chowning Preface Acknowledgments Digital Signals and Sampling Measuring the Ephemeral Analog-to-Digital Conversion Aliasing Digital-to-Analog Conversion Binary Numbers Synchronization Discretization Precision and Accuracy Quantization Noise and Distortion Information Density of Digital Audio Codecs Further Refinements Cultural Impact of Digital Audio Musical Signals Why Imaginary Numbers? Operating with Imaginary Numbers Complex Numbers de Moivre's Theorem Euler's Formula Phasors XI xiii xiv

3 Graphing Complex Signals Spectra of Complex Sampled Signals Multiplying Phasors Graphing Complex Spectra Analytic Signals Spectral Analysis and Synthesis Introduction to the Fourier Transform Discrete Fourier Transform Discrete Fourier Transfom in Action Inverse Discrete Fourier Transform Analyzing Real-World Signals Windowing Fast Fourier Transform Properties of the Discrete Fourier Transform A Practical Hilbert Transform Convolution Rolling Shutter Camera Defining Convolution Numerical Examples of Convolution Convolving Spectra Convolving Signals Convolution and the Fourier Transform Domain Symmetry between Signals and Spectra Convolution and Sampling Theory Convolution and Windowing Correlation Functions Suggested Reading Filtering Tape Recorder as a Model of Filtering Introduction to Filtering A Simple Filter Finding the Frequency Response Linearity and Time Invariance of Filters FIR Filters

4 ix 5.7 IIR Filters Canonical Filter Time Domain Behavior of Filters Filtering as Convolution Z Transform Z Transform of the General Difference Equation Filter Families Resonance The Derivative Differential Equations Mathematics of Resonance The Wave Equation One-Dimensional Wave Equation and String Motion An Example Modeling Vibration with Finite Difference Equations Striking Points, Plucking Points, and Spectra Acoustical Systems Dissipation and Radiation Acoustical Current Linearity of Frictional Force Inertance, Inductive Reactance Compliance, Capacitive Reactance Reactance and Alternating Current Capacitive Reactance and Frequency Inductive Reactance and Frequency Combining Resistance, Reactance, and Alternating Current Resistance and Alternating Current Capacitance and Alternating Current Acoustical Impedance Sound Propagation and Sound Transmission Input Impedance: Fingerprinting a Resonant System Scattering Junctions Suggested Reading 362

5 x Contents 9 Sound Synthesis Forms of Synthesis A Graphical Patch Language for Synthesis Amplitude Modulation Frequency Modulation Vocal Synthesis Synthesizing Concert Hall Acoustics Physical Modeling Source Models and Receiver Models Dynamic Spectra Gabor's Elementary Signal Short-Time Fourier Transform Phase Vocoder Improving on the Fourier Transform Psychoacoustic Audio Encoding Suggested Reading 509 Epilogue 5 Appendix 53 A.I About Algebra 53 A.2 About Trigonometry 54 A.3 Series and Summations 57 A.4 Trigonometric Identities 58 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. 0 Spectral Effects of Ring Modulation 529 A. Derivation of the Reflection Coefficient 530 Notes 533 Glossary 539 References 543 Index of Equations and Mathematical Formulas 547 Subject Index 55