Next:
Contents
Contents
Digital Sound Modeling
lecture notes for Com Sci 295
Michael J. O'Donnell
Date:
10 March 1994--revised 13 May 2004
Contents
List of Figures
1. Physical and Mathematical Foundations of Sound Modeling
1.1 Physics: What Is Sound?
1.1.1 Vibrating Springs
1.2 Mathematics: How Do We Model Sound?
1.2.1 Vibration as the Circular Movement of a Rotor
1.2.2 Rotor State as a Complex Number
1.2.2.1 Review of Complex Arithmetic
1.2.3 Sound Signals in the Time Domain
1.3 Exercises
2. Perceptual Foundations of Sound
2.1 The Ear as a Frequency Analyzer
2.2 Sound Imaging--What is ``a Sound''?
2.3 Perceptual Parameters of a Sound
2.3.1 Pitch
2.3.2 Loudness
2.3.3 Timbre
2.3.4 Structural Interaction of Pitch and Timbre
2.3.5 Transient Effects
2.3.6 Sound Events
2.3.7 Perception of Error in a Sound
2.3.7.1 Noise
2.3.7.2 Distortion
3. Digital Sampled Sound
3.1 Discrete time
3.2 Quantized Vibration State
3.2.1 Real vs. Complex Signals
3.3 Other Ways to Digitize a Sound Signal
3.4 Direct Manipulation of Digital Sampled Sound
4. The Frequency Spectrum
4.1 Pure Helical, Periodic, and Quasiperiodic Signals
4.2 Idealizing the Ear as a Spectral Analyzer
4.3 Mathematical Spectral Analysis with the Fourier Transform
4.3.1 Discrete Spectra
4.3.1.1 Generalized Functions
4.3.2 Continuous Spectra and Noise
4.3.2.1 Random Functions
4.3.2.2 Combinations of Deterministic Signals and Noise
4.3.3 A Calculus of Fourier Transforms
4.3.3.1 Basic Functions and Their Transforms
4.3.3.2 Functional Operators
4.4 The Meaning of Multiplication and Convolution
4.4.1 Pointwise Multiplication
4.4.2 Two Views of Convolution
4.4.2.1 Convolution as Shifting and Scaling
4.4.2.2 Convolution as Melting Together
4.4.3 Multiplication and Convolution in Signal Processing
4.5 Magnitude and Phase in Fourier Spectra
4.6 Analyzing Sampled Sound
4.6.1 The Finite Discrete Fourier Transform
4.6.2 Spectral Impact of Conversion Between Analog and Digital
4.6.3 Transforms of Real-Valued Signals
5. Additive Spectral Synthesis
5.1 Steady-State Sound
5.2 Amplitude Modulation, Enveloping
5.2.1 Enveloping a Multifrequency Sound
5.2.2 Enveloping Individual Spectral Components
5.3 Frequency Modulation
5.4 Noisy Components of Sound
5.4.1 Helical Partials Plus Noise
5.4.2 Noisy Partials
5.4.2.1 Generating Noisy Partials by Modulation
5.4.2.2 Independence/Correlation of Noise in Different Partials
6. Time-Varying Spectral Analysis
6.1 Shortcomings of the Fourier Transform
6.2 Time-Varying Spectral Analysis with the Continuous Wavelet Transform
7. Synthesis by Resonant Modes
7.1 Filters
7.1.1 Unit Resonances and Antiresonances
7.1.1.1 Unit Resonances
7.1.1.2 Unit Antiresonances
7.1.1.3 Identity, Null, Time Shifting
7.1.1.4 Combining Filters in Sequence and In Parallel
7.1.2 Forms for Representing Filters
7.1.2.1 Impulse Response and the Convolution Form
7.1.2.2 Integral and Differential Equation Forms
7.1.3 Analyzing Filters with the Laplace Transform
7.1.3.1 Tranforming the Impulse Response
7.1.3.2 Poles and Zeroes in the Laplace Transform
7.1.3.3 Laplace Transforms and Differential Equations
7.1.3.4 Filtering Real-Valued Signals
7.1.4 Discrete Filters and the
Transform
7.1.4.1 Difference Equation Forms
7.1.4.2 Analyzing Discrete Filters with the Laplace and
Transforms
7.2 Creating Sounds by Modal Synthesis
7.2.1 Continuously Driven Resonators
7.2.2 Ringing Resonators
7.3 Sound Modification with Formant Filters
About this document ...
Mike O'Donnell 2004-05-13
Transform