Digital Audio Processing Fundamentals

Preface
Acknowledgments
Contents
1 Vibrating Systems
	1.1 Introduction
	1.2 Vibrating Systems in One Dimension
		1.2.1 Simple Harmonic Motion
		1.2.2 Damped Harmonic Oscillator
		1.2.3 Phasor Related to Quantities x, v and a
	1.3 Forced Oscillations
		1.3.1 Transfer Function, Frequency, and Impulse Responses
		1.3.2 Transient and Steady-State Response
		1.3.3 Sinusoidal Response, Mobility and Impedance
		1.3.4 Calculation of Complete Response
		1.3.5 Generic Helmholtz Oscillating Systems
	1.4 Electrical Circuit Analogies
		1.4.1 Kirchhoff Laws
		1.4.2 Analogy of Mobility
		1.4.3 Analogy of Impedance
	1.5 Nonlinear Oscillating Systems
		1.5.1 General Notation and State Space Representation
		1.5.2 2nd-Order Nonlinear Oscillating Systems
		1.5.3 Undamped Pendulum
		1.5.4 Van Der Pol and Rayleigh Oscillators
		1.5.5 Duffing Nonlinear Oscillator
		1.5.6 Musical Instrument as a Self-sustained Oscillator
		1.5.7 Multimodal Nonlinear Oscillator
	1.6 Continuous Vibrating Systems
		1.6.1 Ideal String Wave Equation
		1.6.2 Solution of the String Wave Equation
		1.6.3 Strings Vibration and Musical Scales
		1.6.4 Lossy and Dispersive String
		1.6.5 The Vibration of Membranes
	1.7 Sound Waves in the Air
		1.7.1 Plane Wave
		1.7.2 Spherical Waves
		1.7.3 Acoustic Impedance and Characteristic Acoustic Impedance
		1.7.4 Lumped Parameters Acoustic Impedance
		1.7.5 Sound Field and Intensity
		1.7.6 Effects Related to Propagation
	1.8 Acoustic Transducers
		1.8.1 The Microphone and Its Directional Characteristics
		1.8.2 Loudspeakers: Operating Principle and Model
	References
2 Discrete-Time Signals, Circuits, and System Fundamentals
	2.1 Introduction
		2.1.1 Ideal Continuous-Discrete Conversion Process
	2.2 Basic Deterministic Sequences
		2.2.1 Unitary Impulse
		2.2.2 Unit Step
		2.2.3 Real and Complex Exponential Sequences
	2.3 Discrete-Time Circuits
		2.3.1 General DT System Properties and Definitions
		2.3.2 Properties of DT Linear Time-Invariant Circuits
		2.3.3 Basic Elements of DT Circuits
		2.3.4 Frequency Domain Representation of DT Circuits
	2.4 DT Circuits Representation in Transformed Domains
		2.4.1 The z-Transform
		2.4.2 Discrete-Time Fourier Transform
		2.4.3 Discrete Fourier Transform
		2.4.4 Ideal Filters
	2.5 Discrete-Time Signal Representation with Unitary Transformations
		2.5.1 DFT as Unitary Transformation
		2.5.2 Discrete Hartley Transform
		2.5.3 Discrete Sine and Cosine Transforms
		2.5.4 Haar Unitary Transform
		2.5.5 Data Dependent Unitary Transformation
	2.6 Finite Difference Equations
		2.6.1 Transfer Function and Pole–Zero Plot
		2.6.2 BIBO Stability Criterion
	2.7 Finite Impulse Response Filter
		2.7.1 Online Convolution Computation
		2.7.2 Batch Convolution as a Matrix-Vector Product
		2.7.3 Convolutional-Matrix Operator
		2.7.4 FIR Filters Design Methods
	2.8 Infinite Impulse Response Filter
		2.8.1 Digital Resonator
		2.8.2 Anti-Resonant Circuits and Notch Filter
		2.8.3 All-Pass Filters
		2.8.4 Inverse Circuits
		2.8.5 Linear Ordinary Differential Equation Discretization: IIR Filter from an Analog Prototype
	2.9 Multiple-Input Multiple-Output FIR Filter
		2.9.1 MIMO Filter in Composite Notation 1
		2.9.2 MIMO (P, Q) System as Parallel of Q Filters Banks
		2.9.3 MIMO Filter in Composite Notation 2
		2.9.4 MIMO Filter in Snap-Shot or Composite Notation 3
	References
3 Digital Filters for Audio Applications
	3.1 Introduction
	3.2 Analog and Digital Audio Filters
		3.2.1 Classifications of Audio Filters
		3.2.2 Shelving and Peaking Filter Transfer Functions
		3.2.3 Frequency Bandwidth Definitions for Audio Application
		3.2.4 Constant-Q Equalizers
		3.2.5 Digital Audio Signal Equalization
	3.3 IIR Digital Filters for Audio Equalizers
		3.3.1 Bristow-Johnson Second-Order Equalizer
	3.4 Robust IIR Audio Filters
		3.4.1 Limits and Drawbacks of Digital Filters in Direct Forms
		3.4.2 All-Pass Decompositions
		3.4.3 Ladder and Lattice Structures
		3.4.4 Lattice FIR Filters
		3.4.5 Orthogonal Control Shelving Filters: Regalia-Mitra Equalizer
		3.4.6 State-Space Filter with Orthogonal Control
		3.4.7 TF Mapping on Robust Structures
	3.5 Fast Frequency Domain Filtering for Audio Applications
		3.5.1 Block Frequency Domain Convolution
		3.5.2 Low Latency Frequency Domain Filtering
	References
4 Multi-rate Audio Processing and Wavelet Transform
	4.1 Introduction
	4.2 Multirate Audio Processing
		4.2.1 Sampling Rate Reduction by an Integer Factor
		4.2.2 Sampling Rate Increase by an Integer Factor
		4.2.3 Polyphase Representation
		4.2.4 Noble Identity of Multirate Circuits
		4.2.5 Fractional Sampling Ratio Frequency Conversion
	4.3 Filter Banks for Audio Applications
		4.3.1 Generalities on Filter Banks
		4.3.2 Two-Channel Filter Banks
		4.3.3 Filter Bank Design
		4.3.4 Lowpass Prototype Design
		4.3.5 Cosine-Modulated Pseudo-QMF FIR Filter Banks
		4.3.6 Non-uniform Spacing Filter Banks
	4.4 Short-Time Frequency Analysis
		4.4.1 Time–Frequency Measurement Uncertainty
		4.4.2 The Discrete Short-Time Fourier Transform
		4.4.3 Nonparametric Signal Spectral Representations
		4.4.4 Constant-Q Fourier Transform
	4.5 Wavelet Basis and Transforms
		4.5.1 Continuous-Time Wavelet
		4.5.2 Inverse Continuous-Time Wavelet
		4.5.3 Orthogonal Wavelets and the Discrete Wavelet Transform
		4.5.4 Multiresolution Analysis: Axiomatic Approach
		4.5.5 Dilation Equations for Dyadic Wavelets
		4.5.6 Compact Support Orthonormal Wavelet Basis
		4.5.7 Wavelet for Discrete-Time Signals
		4.5.8 Wavelet Examples
	References
5 Special Transfer Functions for DASP
	5.1 Introduction
	5.2 Comb Filters
		5.2.1 FIR Comb Filters
		5.2.2 IIR Comb Filters
		5.2.3 Feedback Delay Networks
		5.2.4 Universal All-Pass Comb Filters
		5.2.5 Nested All-Pass Filters
	5.3 Rational Orthonormal Filter Architecture
		5.3.1 Kautz–Broome Orthogonal Basis Filter Model
		5.3.2 Parameters Estimation of OBF Models
		5.3.3 Laguerre Filters
		5.3.4 Frequency Warped Signal Processing
	5.4 Circular Buffer Delay Lines
		5.4.1 Circular Buffer Addressing
		5.4.2 Delay Lines with Nested All-Pass Filters
	5.5 Fractional Delay Lines
		5.5.1 Problem Formulation of Band-Limited Interpolation
		5.5.2 Approximate FIR Solution
		5.5.3 Approximate All-Pass Solution
		5.5.4 Polynomial Interpolation
		5.5.5 Time-Variant Delay Lines
		5.5.6 Arbitrary Sampling Rate Conversion
		5.5.7 Robust Fractional Delay FIR Filter
		5.5.8 Taylor Expansion of Lagrange Interpolation Filter
	5.6 Digital Oscillators
		5.6.1 Sinusoidal Digital Oscillator
		5.6.2 Wavetable Oscillator
	References
6 Circuits and Algorithms for Physical Modeling
	6.1 Introduction
		6.1.1 Local and Global Approach
		6.1.2 Structural, Functional and Interconnected Models
		6.1.3 Local Approach with Circuit Model
	6.2 Wave Digital Filters
		6.2.1 Representation of CT Circuits with Wave Variables
		6.2.2 Mapping the Electrical Elements from CT to DT
		6.2.3 Connecting DT Circuit Elements
		6.2.4 DT Circuit Corresponding to Given Analog Filter
	6.3 Digital Waveguide Theory
		6.3.1 Lossless Digital Waveguides
		6.3.2 Losses Digital Waveguides
		6.3.3 Terminated Digital Waveguides
		6.3.4 Alternative and Normalized Wave Variables
		6.3.5 Digital Waveguides Connection
	6.4 Finite-Differences Modeling
		6.4.1 FDM Definition
		6.4.2 Derivation of Connection Models for FDTD Simulators
	6.5 Nonlinear WDF and DW Models
		6.5.1 Mapping Memoryless Nonlinear Elements
		6.5.2 Mapping of Nonlinear Elements with Memory
		6.5.3 Impedance Adaptation with Scattering Junction
		6.5.4 Mixed Modeling Methods
	6.6 System Modeling by Modal Decomposition and Impulse Response Estimation
		6.6.1 Theoretical Route of Vibration Analysis
		6.6.2 On the Impulse Response Estimation
		6.6.3 Physical Model Synthesis by Modal Decomposition
	References
7 Digital Audio Effects
	7.1 Introduction
	7.2 Room Acoustic Simulation
		7.2.1 Physical Modeling Versus Perceptual Approach
	7.3 Schroeder's Artificial Reverberator
		7.3.1 Schroeder's First Model
		7.3.2 The Schroeder–Moorer Model
		7.3.3 The Frequency-Dependent Moorer Model
		7.3.4 Selecting Reverberator Parameters
	7.4 The Quality of Artificial Reverberation
		7.4.1 Energy Decay Curves
		7.4.2 Characterization of Diffuse Radiation
		7.4.3 Early Reflections Characterization
	7.5 Reverb Model with Feedback Delay Networks
		7.5.1 Stautner and Puckette's Model
		7.5.2 Jot and Chainge Model
		7.5.3 Choice of Feedback Matrix
		7.5.4 Other Reverberator Models
	7.6 Acoustic Modeling with Digital Waveguides Networks
		7.6.1 Wave Propagation Modeling with Digital Waveguide Networks
		7.6.2 DWN Topologies
	7.7 Dynamic Range Control of Audio Signal
		7.7.1 DRC Static Curves
		7.7.2 Dynamic Gain Control
		7.7.3 Signal Level Calculation
		7.7.4 Constructive Considerations of the DRC
		7.7.5 DRC with Multiband Approach
		7.7.6 Dynamic Range Control Applications
	7.8 Effects Based Time-Variant Fractional-Delay Lines
		7.8.1 Angular Modulation with TV-FDL
		7.8.2 Vibrato and Other TV-FDL-Based Effects
		7.8.3 Amplification Systems with Rotating Speakers (Leslie)
	7.9 Effects Based on Time–Frequency Transformations
		7.9.1 Frequency Transposition and Time Scale Change
		7.9.2 Classification of TFT Algorithms
		7.9.3 Time-Domain FTF Algorithms
		7.9.4 Time–Frequency Domain Algorithms Based Effects
	References
8 Sound Synthesis
	8.1 Introduction
		8.1.1 Synthesis by Recorded Sound
		8.1.2 Abstract Algorithm Synthesis
	8.2 Sampling Instruments
		8.2.1 The ADSR Envelope Control
		8.2.2 Looping Technique
		8.2.3 Tremolo and Vibrato
		8.2.4 Sampling Techniques
		8.2.5 The Cross-Fade Technique
	8.3 Wavetable Synthesizer
		8.3.1 Additive Synthesis and Wavetable Summation
		8.3.2 Wavetable and Parameters Estimation
		8.3.3 Granular Synthesis
		8.3.4 WS Hybrid Models
	8.4 Spectral Representation of the Signal
		8.4.1 Additive Synthesis
		8.4.2 Subtractive Synthesis
	8.5 Discrete-Time Modeling of Analog Synthesizer
		8.5.1 Synthesizer's Units and Signal and Control Paths
		8.5.2 Layout of Simple Custom Synthesizers
		8.5.3 Voltage Control Filter
		8.5.4 Physical Modeling of Analog Synthesizer: The Virtual Analog Music Synthesizers
		8.5.5 Large-Signal DT Model of Moog VCF
	8.6 Frequency Modulation Synthesis
		8.6.1 FM Sound Synthesis Principles
		8.6.2 Concatenated FM with Multiple Operators
	8.7 Nonlinear Distortion Synthesis
		8.7.1 NLD Theoretical Development
		8.7.2 Extensions of the NLD Technique
		8.7.3 Comparison with FM Technique
	8.8 Karplus–Strong Algorithm
	References
9 Physical Modeling
	9.1 Introduction
	9.2 Physical, Mathematical, and Computational Models
	9.3 Vibrating String Model
	9.4 Excitation Models of Vibrating Systems
		9.4.1 Plucked String
		9.4.2 Continuously Excited String
		9.4.3 Generalized Excitation by Data-Driven Pseudo-Physical Model
	9.5 Modeling of Wind Instruments
		9.5.1 Excitation Mechanism Modeling of Wind Instrument
		9.5.2 Wind Instruments Acoustic Tube Modeling
		9.5.3 Modeling of Tonal Holes
	9.6 Woodwinds Physical Modeling
		9.6.1 Single-Reed Woodwinds: Clarinet
		9.6.2 Air-Jet Instruments
	9.7 Brass Physical Modeling
		9.7.1 Shape and Characteristics of the Cup Mouthpiece
		9.7.2 Bell-Shaped Flaring and Radiation Modeling
		9.7.3 Brass Excitation Model
	9.8 Commuted Synthesis
		9.8.1 Generalized Commuted Synthesis Model
	9.9 Guitar Physical Model
		9.9.1 Guitar Body Physical Model
		9.9.2 Soundboard Bracing
		9.9.3 Guitar Overall Physical Model
	9.10 Bowed String Instruments
		9.10.1 Bow Excitation Model
		9.10.2 Helmholtz Motion of Bowed String
		9.10.3 Bow Control Variables: Position, Velocity, and Force
		9.10.4 Body and Bridge Modeling of Bowed String Instruments
		9.10.5 Violin's Impulse Response
		9.10.6 Physical Model of the Bowed String Instrument
	9.11 Piano Physical Model
		9.11.1 Mechanical Principle and Simplified Piano Structure
		9.11.2 Piano String Equation
		9.11.3 Hammer Excitation Mechanism
		9.11.4 Numerical Piano Modeling by Modal Synthesis
		9.11.5 Bridge and Soundboard
	References
Index