PHYSICAL AUDIO SIGNAL PROCESSING
FOR VIRTUAL MUSICAL INSTRUMENTS AND AUDIO EFFECTS

JULIUS O. SMITH III

Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University, Stanford, California 94305 USA

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• Preface
  • Organization
  • Book Series Overview
  • Acknowledgments
  
  
• Physical Signal Modeling Intro
  • But How Does It Sound?
  • What is a Model?
    • The Basic Science Loop
    • Models for Music and Audio
  • Overview of Model Types
  • Signal Models
    • Recordings (Samples)
    • Structured Sampling
    • Spectral Models
    • Virtual Analog
  • Physical Models
    • All We Need is Newton
    • Formulations
    • ODEs
    • PDEs
    • Difference Equations (Finite Difference Schemes)
    • State Space Models
      • Forming Outputs
      • State-Space Model of a Force-Driven Mass
      • Numerical Integration of General State-Space Models
      • State Definition
    • Linear State Space Models
      • Impulse Response of State Space Models
      • Zero-Input Response of State Space Models
    • Transfer Functions
    • Modal Representation
      • State Space to Modal Synthesis
      • Force-Driven-Mass Diagonalization Example
      • Typical State-Space Diagonalization Procedure
      • Efficiency of Diagonalized State-Space Models
    • Equivalent Circuits
    • Impedance Networks
    • Wave Digital Filters
    • Digital Waveguide Modeling Elements
    • General Modeling Procedure
  • Our Plan
    • A Simplified Starting Theory
  • Elementary Physical Modeling Problems
  
  
• Acoustic Modeling with Delay
  • Delay Lines
    • A Software Delay Line
  • Acoustic Wave Propagation Simulation
    • Traveling Waves
    • Damped Traveling Waves
    • Dispersive Traveling Waves
    • Converting Propagation Distance to Delay Length
    • Spherical Waves from a Point Source
    • Reflection of Spherical or Plane Waves
    • An Acoustic Echo Simulator
    • Program for Acoustic Echo Simulation
  • Lossy Acoustic Propagation
    • Exponentially Decaying Traveling Waves
    • Frequency-Dependent Air-Absorption Filtering
    • Dispersive Traveling Waves
    • Summary
  • Digital Waveguides
    • Physical Outputs
    • Physical Inputs
    • Interacting Physical Input
  • Tapped Delay Line (TDL)
    • Example Tapped Delay Line
    • Transposed Tapped Delay Line
    • TDL for Parallel Processing
    • General Causal FIR Filters
  • Comb Filters
    • Feedforward Comb Filters
    • Feedback Comb Filters
    • Feedforward Comb Filter Amplitude Response
    • Feedback Comb Filter Amplitude Response
    • Filtered-Feedback Comb Filters
    • Equivalence of Parallel Combs to TDLs
    • Equivalence of Series Combs to TDLs
    • Time Varying Comb Filters
  • Feedback Delay Networks (FDN)
    • FDN and State Space Descriptions
    • Single-Input, Single-Output (SISO) FDN
    • FDN Stability
  • Allpass Filters
    • Allpass from Two Combs
    • Nested Allpass Filters
    • More General Allpass Filters
    • Example Allpass Filters
    • Gerzon Nested MIMO Allpass
  • Allpass Digital Waveguide Networks
    • Signal Scattering
    • Digital Waveguide Networks
  
  
• Artificial Reverberation
  • The Reverberation Problem
    • Exact Reverb via Transfer-Function Modeling
    • Complexity of Exact Reverberation
    • Possibility of a Physical Reverb Model
  • Perceptual Aspects of Reverberation
    • Perception of Echo Density and Mode Density
    • Perceptual Metrics for Ideal Reverberation
      • Energy Decay Curve
      • Energy Decay Relief
  • Early Reflections
  • Late Reverberation Approximations
    • Desired Qualities in Late Reverberation
    • Schroeder Allpass Sections
    • Nested Allpass Filters
  • Schroeder Reverberators
    • Example Schroeder Reverberators
  • Freeverb
    • Freeverb Main Loop
    • Lowpass-Feedback Comb Filter
    • Freeverb Allpass Approximation
    • Conclusions
  • FDN Reverberation
    • History of FDNs for Artificial Reverberation
    • Choice of Lossless Feedback Matrix
      • Hadamard Matrix
      • Householder Feedback Matrix
      • Householder Reflections
      • Most General Lossless Feedback Matrices
      • Triangular Feedback Matrices
    • Choice of Delay Lengths
      • Mean Free Path
      • Mode Density Requirement
      • Prime Power Delay-Line Lengths
    • Achieving Desired Reverberation Times
      • Conformal Map Interpretation of Damping Substitution
      • Damping Filters for Reverberation Delay Lines
    • Delay-Line Damping Filter Design
      • First-Order Delay-Filter Design
      • Orthogonalized First-Order Delay-Filter Design
      • Multiband Delay-Filter Design
    • Spectral Coloration Equalizer
    • Tonal Correction Filter
    • FDNs as Digital Waveguide Networks
    • FDN Reverberators in Faust
    • Zita-Rev1
      • Zita-Rev1 Delay-Line Filters
    • Further Extensions
      • Spatialization of Reverberant Reflections
      • Distribution of Mode Frequencies
      • Digital Waveguide Reverberators
      • The Digital Waveguide Mesh for Reverberation
      • Time Varying Reverberators
  
  
• Delay/Signal Interpolation
  • Delay-Line Interpolation
    • Linear Interpolation
      • One-Multiply Linear Interpolation
      • Fractional Delay Filtering by Linear Interpolation
    • First-Order Allpass Interpolation
      • Minimizing First-Order Allpass Transient Response
    • Linear Interpolation as Resampling
      • Convolution Interpretation
      • Frequency Response of Linear Interpolation
      • Triangular Pulse as Convolution of Two Rectangular Pulses
      • Linear Interpolation Frequency Response
      • Special Cases
    • Large Delay Changes
  • Lagrange Interpolation
    • Interpolation of Uniformly Spaced Samples
    • Fractional Delay Filters
    • Lagrange Interpolation Optimality
    • Explicit Lagrange Coefficient Formulas
    • Lagrange Interpolation Coefficient Symmetry
    • Matlab Code for Lagrange Interpolation
    • Maxima Code for Lagrange Interpolation
    • Faust Code for Lagrange Interpolation
    • Lagrange Frequency Response Examples
      • Orders 1 to 5 on a fractional delay of 0.4 samples
      • Order 4 over a range of fractional delays
      • Order 5 over a range of fractional delays
    • Avoiding Discontinuities When Changing Delay
    • Lagrange Frequency Response Magnitude Bound
    • Even-Order Lagrange Interpolation Summary
    • Odd-Order Lagrange Interpolation Summary
    • Proof of Maximum Flatness at DC
    • Variable Filter Parametrizations
      • Table Look-Up
      • Polynomials in the Delay
      • Farrow Structure
      • Farrow Structure Coefficients
      • Differentiator Filter Bank
    • Recent Developments in Lagrange Interpolation
    • Relation of Lagrange to Sinc Interpolation
  • Thiran Allpass Interpolators
    • Thiran Allpass Interpolation in Matlab
    • Group Delays of Thiran Allpass Interpolators
  • Windowed Sinc Interpolation
    • Theory of Ideal Bandlimited Interpolation
    • From Theory to Practice
    • Implementation
      • Choice of Table Size and Word Lengths
    • Summary of Windowed Sinc Interpolation
  • Delay-Line Interpolation Summary
  
  
• Time-Varying Delay Effects
  • Variable Delay Lines
  • Doubling and Slap-Back
  • Flanging
    • Flanger Speed and Excursion
    • Flanger Depth Control
    • Flanger Inverted Mode
    • Flanger Feedback Control
    • Summary of Flanging
  • Phasing
  • Vibrato Simulation
  • Doppler Effect
    • Vector Formulation
  • Doppler Simulation
    • Doppler Simulation via Delay Lines
    • Time-Varying Delay-Line Reads
    • Multiple Read Pointers
    • Multiple Write Pointers
    • Stereo Processing
    • System Block Diagram
  • Chorus Effect
  • The Leslie
    • Rotating Horn Simulation
    • Rotating Woofer-Port and Cabinet
    • Recent Research Modeling the Leslie
  
  
• Digital Waveguide Models
  • Ideal Vibrating String
    • Wave Equation
    • Wave Equation Applications
    • Traveling-Wave Solution
    • Sampled Traveling-Wave Solution
    • Wave Impedance
  • Ideal Acoustic Tube
  • Rigid Terminations
    • Velocity Waves at a Rigid Termination
    • Force or Pressure Waves at a Rigid Termination
  • Moving Rigid Termination
    • Digital Waveguide Equivalent Circuits
    • Animation of Moving String Termination
    • Terminated String Impedance
  • The Ideal Plucked String
  • The Ideal Struck String
  • The Damped Plucked String
    • Computational Savings
  • Frequency-Dependent Damping
  • The Stiff String
    • Stiff String Synthesis Models
  • The Externally Excited String
    • Equivalent Forms
      • Algebraic derivation
      • Related Forms
      • Summary
  • Loop Filter Identification
    • General Loop-Filter Design
    • Damping Filter Design
    • Dispersion Filter Design
    • Fundamental Frequency Estimation
      • Approximate Maximum Likelihood F0 Estimation
      • References on F0 Estimation
      • Extension to Stiff Strings
    • EDR-Based Loop-Filter Design
  • String Coupling Effects
    • Horizontal and Vertical Transverse Waves
    • Coupled Horizontal and Vertical Waves
    • Asymmetry of Horizontal/Vertical Terminations
    • Coupled Strings
    • Longitudinal Waves
  • Nonlinear Elements
    • Memoryless Nonlinearities
      • Clipping Nonlinearity
      • Arctangent Nonlinearity
      • Cubic Soft Clipper
      • Series Expansions
      • Arctangent Series Expansion
      • Spectrum of a Memoryless Nonlinearities
      • Square Law Series Expansion
      • Power Law Spectrum
      • Arctangent Spectrum
      • Cubic Soft-Clipper Spectrum
      • Stability of Nonlinear Feedback Loops
      • Practical Advice
  
  
• Lumped Models
  • Impedance
    • Dashpot
    • Ideal Mass
    • Ideal Spring
  • One-Port Network Theory
    • Series Combination of One-Ports
    • Mass-Spring-Wall System
    • Parallel Combination of One-Ports
    • Spring-Mass System
    • Mechanical Impedance Analysis
    • General One-Ports
    • Passive One-Ports
  • Digitization of Lumped Models
    • Finite Difference Approximation
      • FDA in the Frequency Domain
      • Delay Operator Notation
    • Bilinear Transformation
      • FDA vs. Bilinear Transform
    • Application of the Bilinear Transform
      • Practical Considerations
    • Limitations of Lumped Element Digitization
  • More General Finite-Difference Methods
    • General Nonlinear ODE
    • Forward Euler Method
    • Backward Euler Method
    • Trapezoidal Rule
    • Newton's Method of Nonlinear Minimization
    • Semi-Implicit Methods
    • Semi-Implicit Backward Euler
    • Semi-Implicit Trapezoidal Rule
    • Summary
    • Further Reading in Nonlinear Methods
  • Summary of Lumped Modeling
  
  
• Transfer Function Models
  • Outline
  • Sampling the Impulse Response
  • Impulse Invariant Method
  • Matched Z Transformation
    • Relation to Finite Difference Approximation
  • Pole Mapping with Optimal Zeros
  • Modal Expansion
    • State Space Approach to Modal Expansions
    • Delay Loop Expansion
  • General Filter Design Methods
    • Ideal Differentiator (Spring Admittance)
    • Digital Filter Design Overview
    • Digital Differentiator Design
    • Fitting Filters to Measured Amplitude Responses
      • Measured Amplitude Response
      • Desired Impulse Response
      • Converting the Desired Amplitude Response to Minimum Phase
    • Further Reading on Digital Filter Design
  • Commuted Synthesis
    • Body-Model Factoring
    • Further Reading in Commuted Synthesis
  • Resonator Factoring
    • Mode Extraction Techniques
      • Amplitude response peak measurement
      • Weighted digital filter design
      • Linear prediction
      • Sinusoidal modeling
      • Late impulse-response analysis
    • Inverse Filtering
      • Empirical Notes on Inverse Filtering
      • Matlab Code for Inverse Filtering
    • Sinusoidal Modeling of Mode Decays
    • Parallel Body Filterbank Design
    • Excitation Noise Substitution
    • Body Factoring Example
  • Virtual Analog Example: Phasing
    • Phasing with First-Order Allpass Filters
      • Classic Analog Phase Shifters
      • Classic Virtual Analog Phase Shifters
    • Phasing with 2nd-Order Allpass Filters
      • Phaser Notch Parameters
      • Phaser Notch Distribution
  
  
• Virtual Musical Instruments
  • Electric Guitars
    • Length Three FIR Loop Filter
    • Brightness and Sustain Control
    • One-Zero Loop Filter
    • The Karplus-Strong Algorithm
    • The Extended Karplus-Strong Algorithm
    • Nonlinear Distortion
      • Tension Modulation
      • String Length Modulation
      • Hard Clipping
      • Soft Clipping
      • Enhancing Even Harmonics
      • Software for Cubic Nonlinear Distortion
    • Amplifier Feedback
    • Cabinet Filtering
    • Duty-Cycle Modulation
    • Vacuum Tube Modeling
  • Acoustic Guitars
    • Bridge Modeling
      • Passive String Terminations
      • A Terminating Resonator
      • Bridge Reflectance
      • Bridge Transmittance
      • Digitizing Bridge Reflectance
      • A Two-Resonance Guitar Bridge
      • Measured Guitar-Bridge Admittance
      • Building a Synthetic Guitar Bridge Admittance
      • Passive Reflectance Synthesis--Method 1
      • Passive Reflectance Synthesis--Method 2
      • Matlab for Passive Reflectance Synthesis Method 1
      • Matlab for Passive Reflectance Synthesis Method 2
      • Matrix Bridge Impedance
    • Body Modeling
  • String Excitation
    • Ideal String Struck by a Mass
      • Mass Termination Model
      • Mass Reflectance from Either String
      • Simplified Impedance Analysis
      • Mass Transmittance from String to String
      • Force Wave Mass-String Model
      • Summary of Mass-String Scattering Junction
      • Digital Waveguide Mass-String Model
      • Displacement-Wave Simulation
    • Piano Hammer Modeling
      • Nonlinear Spring Model
      • Including Hysteresis
      • Piano Hammer Mass
    • Pluck Modeling
      • Digital Waveguide Plucked-String Model
      • Incorporating Control Motion
      • Successive Pluck Collision Detection
      • Plectrum Damping
      • Digitization of the Damped-Spring Plectrum
      • Feathering
  • Piano Synthesis
    • Stiff Piano Strings
      • Piano String Wave Equation
      • Damping-Filter Design
      • Dispersion Filter-Design
    • Nonlinear Piano Strings
      • Nonlinear Piano-String Synthesis
      • Regimes of Piano-String Vibration
      • Efficient Waveguide Synthesis of Nonlinear Piano Strings
      • Checking the Approximations
    • High-Accuracy Piano-String Modeling
      • A Stiff Mass-Spring String Model
      • Nonlinear Piano-String Equations of Motion in State-Space Form
      • Finite Difference Implementation
    • Commuted Piano Synthesis
      • Force-Pulse Synthesis
      • Multiple Force-Pulse Synthesis
      • Commuted Piano Synthesis Architecture
      • Progressive Filtering
      • Excitation Factoring
      • Excitation Synthesis
      • Coupled Piano Strings
      • Commuted Synthesis of String Reverberation
      • High Piano Key Numbers
      • Force-Pulse Filter Design
    • Literature on Piano Acoustics and Synthesis
  • Woodwinds
    • Single-Reed Instruments
      • Digital Waveguide Single-Reed Implementation
    • A View of Single-Reed Oscillation
    • Single-Reed Theory
      • Scattering-Theoretic Formulation
      • Computational Methods
      • Clarinet Synthesis Implementation Details
    • Tonehole Modeling
      • The Clarinet Tonehole as a Two-Port Junction
      • Tonehole Filter Design
      • The Tonehole as a Two-Port Loaded Junction
  • Bowed Strings
    • Digital Waveguide Bowed-String
    • The Bow-String Scattering Junction
    • Bowed String Synthesis Extensions
    • Linear Commuted Bowed Strings
      • Bowing as Periodic Plucking
      • More General Quasi-Periodic Synthesis
      • Stochastic Excitation for Quasi-Periodic Synthesis
  • Brasses
    • Modeling the Lips and Mouthpiece
    • Bell Models
      • Literature Relevant to Brasses
  • Other Instruments
    • Singing Voice
    • Flutes, Recorders, and Pipe Organs
    • Percussion Instruments
  
  
• Conclusion
• History of Enabling Ideas
  • Early Musical Acoustics
  • History of Modal Expansion
  • Mass-Spring Resonators
  • Sampling Theory
  • Physical Digital Filters
    • Analog Computers
    • Finite Difference Methods
    • Transfer Function Models
    • Wave Digital Filter Models
  • Voice Synthesis
    • Dudley's Vocoder
    • Vocal Tract Analog Models
    • Singing Kelly-Lochbaum Vocal Tract
    • Linear Predictive Coding of Speech
    • Formant Synthesis Models
    • Further Reading in Speech Synthesis
  • String Models
  • Karplus-Strong Algorithms
  • Digital Waveguide Models
  • Summary
  
  
• Physics, Mechanics, and Acoustics
  • Newton's Laws of Motion
    • Newton's Three Laws of Motion
    • Mass
    • Gravitational Force
    • Hooke's Law
    • Applying Newton's Laws of Motion
  • Work and Energy
    • Potential Energy in a Spring
    • Kinetic Energy of a Mass
    • Mass Kinetic Energy from Virtual Work
    • Energy in the Mass-Spring Oscillator
    • Energy Conservation
    • Energy Conservation in the Mass-Spring System
  • Momentum
    • Conservation of Momentum
  • Rigid-Body Dynamics
    • Center of Mass
      • Linear Momentum of the Center of Mass
      • Whoops, No Angular Momentum!
    • Translational Kinetic Energy
    • Rotational Kinetic Energy
    • Mass Moment of Inertia
      • Circular Disk Rotating in Its Own Plane
      • Circular Disk Rotating About Its Diameter
    • Perpendicular Axis Theorem
    • Parallel Axis Theorem
    • Stretch Rule
    • Area Moment of Inertia
    • Radius of Gyration
      • Rectangular Cross-Section
      • Circular Cross-Section
    • Two Masses Connected by a Rod
      • Striking the Rod in the Middle
      • Striking One of the Masses
    • Angular Velocity Vector
    • Vector Cross Product
      • Cross-Product Magnitude
      • Mass Moment of Inertia as a Cross Product
      • Tangential Velocity as a Cross Product
    • Angular Momentum
      • Relation of Angular to Linear Momentum
    • Angular Momentum Vector
      • Angular Momentum Vector in Matrix Form
    • Mass Moment of Inertia Tensor
      • Simple Example
      • Example with Coupled Rotations
      • Off-Diagonal Terms in Moment of Inertia Tensor
    • Principal Axes of Rotation
      • Positive Definiteness of the Moment of Inertia Tensor
    • Rotational Kinetic Energy Revisited
    • Torque
    • Newton's Second Law for Rotations
    • Equations of Motion for Rigid Bodies
      • Body-Fixed and Space-Fixed Frames of Reference
      • Angular Motion in the Space-Fixed Frame
      • Euler's Equations for Rotations in the Body-Fixed Frame
      • Examples
  • Properties of Elastic Solids
    • Young's Modulus
      • Young's Modulus as a Spring Constant
    • String Tension
  • Wave Equation for the Vibrating String
    • Non-Stiff String
    • Wave Momentum
  • Properties of Gases
    • Particle Velocity of a Gas
    • Volume Velocity of a Gas
    • Pressure is Confined Kinetic Energy
    • Bernoulli Equation
    • Bernoulli Effect
    • Air Jets
    • Acoustic Intensity
    • Acoustic Energy Density
    • Energy Decay through Lossy Boundaries
    • Ideal Gas Law
    • Isothermal versus Isentropic
    • Adiabatic Gas Constant
    • Heat Capacity of Ideal Gases
    • Speed of Sound in Air
    • Air Absorption
  • Wave Equation in Higher Dimensions
    • Plane Waves in Air
    • Vector Wavenumber
    • Solving the 2D Wave Equation
    • 2D Boundary Conditions
    • 3D Sound
  
  
• Digital Waveguide Theory
  • The Ideal Vibrating String
  • The Finite Difference Approximation
    • FDA of the Ideal String
  • Traveling-Wave Solution
    • Traveling-Wave Partial Derivatives
    • Use of the Chain Rule
    • String Slope from Velocity Waves
    • Wave Velocity
    • D'Alembert Derived
    • Converting String-State to Traveling-Waves
  • Sampled Traveling Waves
    • Digital Waveguide Model
    • Digital Waveguide Interpolation
    • Relation to the Finite Difference Recursion
  • A Lossy 1D Wave Equation
    • Loss Consolidation
    • Frequency-Dependent Losses
    • Well Posed PDEs for Modeling Damped Strings
    • Digital Filter Models of Damped Strings
    • Lossy Finite Difference Recursion
      • Frequency-Dependent Losses
  • The Dispersive 1D Wave Equation
    • Higher Order Terms
  • Alternative Wave Variables
    • Spatial Derivatives
    • Force Waves
    • Wave Impedance
    • State Conversions
    • Power Waves
    • Energy Density Waves
    • Root-Power Waves
    • Total Energy in a Rigidly Terminated String
  • Scattering at Impedance Changes
    • Plane-Wave Scattering
      • Scattering Solution
      • Reflection Coefficient
    • Plane-Wave Scattering at an Angle
      • Reflection and Refraction
      • Evanescent Wave due to Total Internal Reflection
    • Longitudinal Waves in Rods
    • Kelly-Lochbaum Scattering Junctions
    • One-Multiply Scattering Junctions
    • Normalized Scattering Junctions
    • Junction Passivity
  • Digital Waveguide Filters
    • Ladder Waveguide Filters
    • Reflectively Terminated Waveguide Filters
    • Half-Rate Ladder Waveguide Filters
    • Conventional Ladder Filters
    • Power-Normalized Waveguide Filters
  • ``Traveling Waves'' in Lumped Systems
    • Reflectance of an Impedance
  • Properties of Passive Impedances
    • Passive Reflectances
      • Reflectance and Transmittance of a Yielding String Termination
      • Power-Complementary Reflection and Transmission
    • Positive Real Functions
      • Relation to Stochastic Processes
      • Relation to Schur Functions
      • Relation to -plane Positive-Real Functions
      • Special cases and examples
      • Minimum Phase (MP) polynomials in
      • Miscellaneous Properties
  • Loaded Waveguide Junctions
  • Two Coupled Strings
    • Two Ideal Strings Coupled at an Impedance
    • Coupled Strings Eigenanalysis
  • Digital Waveguide Mesh
    • The Rectilinear 2D Mesh
    • Dispersion
    • Recent Developments
    • 2D Mesh and the Wave Equation
    • The Lossy 2D Mesh
    • Diffuse Reflections in the Waveguide Mesh
  • FDNs as Digital Waveguide Networks
    • Lossless Scattering
    • Normalized Scattering
    • General Conditions for Losslessness
  • Waveguide Transformers and Gyrators
    • Gyrators
  • The Digital Waveguide Oscillator
    • Additive Synthesis
    • Digital Sinusoid Generators
    • The Second-Order Waveguide Filter
    • Application to FM Synthesis
    • Digital Waveguide Resonator
    • State-Space Analysis
    • Eigenstructure
      • Damping and Tuning Parameters
      • Eigenvalues in the Undamped Case
    • Summary
    • Matlab Sinusoidal Oscillator Implementations
    • Faust Implementation
  • Non-Cylindrical Acoustic Tubes
    • Horns as Waveguides
      • Overview of Methods
      • Back to the Cone
    • Conical Acoustic Tubes
      • Digital Simulation
    • Momentum Conservation in Nonuniform Tubes
      • Cylindrical Tubes
    • Wave Impedance in a Cone
    • More General One-Parameter Waves
    • Generalized Wave Impedance
    • Generalized Scattering Coefficients
    • Cylinder with Conical Cap
      • Scattering Filters at the Cylinder-Cone Junction
      • Reflectance of the Conical Cap
      • Stability Proof
      • Reflectance Magnitude
      • Poles at
  
  
• Finite-Difference Schemes
  • Finite-Difference Schemes
  • Convergence
    • Consistency
    • Well Posed Initial-Value Problem
      • A Class of Well Posed Damped PDEs
      • Proof that the Third-Order Time Derivative is Ill Posed
    • Stability of a Finite-Difference Scheme
    • Lax-Richtmyer equivalence theorem
    • Passivity of a Finite-Difference Scheme
    • Summary
    • Convergence in Audio Applications
  • Characteristic Polynomial Equation
  • Von Neumann Analysis
  
  
• Waveguide and FDTD Equivalence
  • Introduction
    • Finite Difference Time Domain (FDTD) Scheme
    • Digital Waveguide (DW) Scheme
    • FDTD and DW Equivalence
  • State Transformations
  • Excitation Examples
    • Localized Displacement Excitations
    • Localized Velocity Excitations
    • More General Velocity Excitations
    • Additive Inputs
    • Interpretation of the Time-Domain KW Converter
  • State Space Formulation
    • FDTD State Space Model
    • DW State Space Model
      • DW Displacement Inputs
      • DW Non-Displacement Inputs
      • Input Locality
    • Boundary Conditions
      • Resistive Terminations
      • Boundary Conditions as Perturbations
      • Reactive Terminations
      • Interior Scattering Junctions
    • Lossy Vibration
    • State Space Summary
  • Computational Complexity
  • Summary
  • Future Work
  • Acknowledgments
  
  
• Wave Digital Filters
  • Wave Digital Elements
    • A Physical Derivation of Wave Digital Elements
      • Reflectance of a General Lumped Waveguide Termination
      • Reflectances of Elementary Impedances
      • Capacitor Reflectance
      • Inductor Reflectance
      • Resistor Reflectance
      • Choosing Impedance to Simplify Element Reflectance
      • Digitizing Elementary Reflectances by Bilinear Transform
    • Summary of Wave Digital Elements
    • Wave Digital Mass
    • Wave Digital Spring
    • Wave Digital Dashpot
    • Limiting Cases
    • Unit Elements
  • Adaptors for Wave Digital Elements
    • Two-Port Parallel Adaptor for Force Waves
      • Compatible Port Connections
    • General Parallel Adaptor for Force Waves
      • Alpha Parameters
      • Reflection Coefficient, Parallel Case
      • Physical Derivation of Reflection Coefficient
      • Reflection Free Port
    • Two-Port Series Adaptor for Force Waves
    • General Series Adaptor for Force Waves
      • Beta Parameters
      • Reflection Coefficient, Series Case
      • Physical Derivation of Series Reflection Coefficient
      • Series Reflection Free Port
  • Wave Digital Modeling Examples
    • ``Piano hammer in flight''
      • Extracting Physical Quantities
    • Force Driving a Mass
      • A More Formal Derivation of the Wave Digital Force-Driven Mass
    • Force Driving a Spring against a Wall
    • Spring and Free Mass
    • Mass and Dashpot in Series
      • Checking the WDF against the Analog Equivalent Circuit
    • Wave Digital Mass-Spring Oscillator
      • Oscillation Frequency
      • DC Analysis of the WD Mass-Spring Oscillator
      • WD Mass-Spring Oscillator at Half the Sampling Rate
      • Linearly Growing State Variables in WD Mass-Spring Oscillator
      • A Signal Processing Perspective on Repeated Mass-Spring Poles
      • Physical Perspective on Repeated Poles in the Mass-Spring System
      • Mass-Spring Boundedness in Reality
      • Energy-Preserving Parameter Changes (Mass-Spring Oscillator)
      • Exercises in Wave Digital Modeling
  
  
• Resources on the Internet
• Sound Examples
• Bibliography
• Index for this Document
• About this document ...

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.