PHYSICAL AUDIO SIGNAL PROCESSING
FOR VIRTUAL MUSICAL INSTRUMENTS AND AUDIO EFFECTS

JULIUS O. SMITH III
Center for Computer Research in Music and Acoustics (CCRMA)

• Preface
  • Organization
  • Book Series Overview
  • Acknowledgments
  • Errata
  
  
• 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
  • Scattering Delay Networks
  
  
• 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 Nonlinearity
      • 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
      • One-Multiply Form for the 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
    • Loaded Scattering Implementation Notes^C.11
      • Parallel Impedance Biquads
      • Parallel Admittance Biquads
    • Unloaded Junctions and Alpha Parameters
  • Two Coupled Strings
    • Two Ideal Strings Coupled at an Impedance
    • Coupled Strings Eigenanalysis
  • Digital Waveguide Mesh
    • The Rectilinear 2D Mesh
    • Dispersion
    • Further 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 of Waveguide Oscillator
    • 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
    • Binary Connection Tree Parallel Adaptor
      • One-Multiply Parallel Reflection-Free Three-Port Adaptor
    • 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
    • Binary Connection Tree Series Adaptor
      • One-Multiply Series Reflection-Free Three-Port Adaptor
  • 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 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 ...