- But How Does It Sound?
- What is a Model?
- Overview of Model Types
- Signal Models
- Physical Models
- All We Need is Newton
- Formulations
- ODEs
- PDEs
- Difference Equations (Finite Difference Schemes)
- State Space Models
- Linear State Space Models
- Transfer Functions
- Modal Representation
- Equivalent Circuits
- Impedance Networks
- Wave Digital Filters
- Digital Waveguide Modeling Elements
- General Modeling Procedure
- Our Plan
- Elementary Physical Modeling Problems
- Delay Lines
- Acoustic Wave Propagation Simulation
- Lossy Acoustic Propagation
- Digital Waveguides
- Tapped Delay Line (TDL)
- Comb Filters
- Feedback Delay Networks (FDN)
- Allpass Filters
- Allpass Digital Waveguide Networks
- The Reverberation Problem
- Perceptual Aspects of Reverberation
- Early Reflections
- Late Reverberation Approximations
- Schroeder Reverberators
- Freeverb
- FDN Reverberation
- History of FDNs for Artificial Reverberation
- Choice of Lossless Feedback Matrix
- Choice of Delay Lengths
- Achieving Desired Reverberation Times
- Delay-Line Damping Filter Design
- Spectral Coloration Equalizer
- Tonal Correction Filter
- FDNs as Digital Waveguide Networks
- FDN Reverberators in Faust
- Zita-Rev1
- Further Extensions
- Scattering Delay Networks
- Delay-Line Interpolation
- 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
- 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
- Recent Developments in Lagrange Interpolation
- Relation of Lagrange to Sinc Interpolation
- Thiran Allpass Interpolators
- Windowed Sinc Interpolation
- Delay-Line Interpolation Summary
- Variable Delay Lines
- Doubling and Slap-Back
- Flanging
- Phasing
- Vibrato Simulation
- Doppler Effect
- Doppler Simulation
- Chorus Effect
- The Leslie
- Ideal Vibrating String
- Ideal Acoustic Tube
- Rigid Terminations
- Moving Rigid Termination
- The Ideal Plucked String
- The Ideal Struck String
- The Damped Plucked String
- Frequency-Dependent Damping
- The Stiff String
- The Externally Excited String
- Loop Filter Identification
- String Coupling Effects
- Nonlinear Elements
- Impedance
- One-Port Network Theory
- Digitization of Lumped Models
- More General Finite-Difference Methods
- Summary of Lumped Modeling
- Outline
- Sampling the Impulse Response
- Impulse Invariant Method
- Matched Z Transformation
- Pole Mapping with Optimal Zeros
- Modal Expansion
- General Filter Design Methods
- Commuted Synthesis
- Resonator Factoring
- Virtual Analog Example: Phasing
- Electric Guitars
- 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
- Bridge 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
- Pluck Modeling
- Ideal String
Struck by a Mass
- Piano Synthesis
- Woodwinds
- Bowed Strings
- Brasses
- Other Instruments
- Early Musical Acoustics
- History of Modal Expansion
- Mass-Spring Resonators
- Sampling Theory
- Physical Digital Filters
- Voice Synthesis
- String Models
- Karplus-Strong Algorithms
- Digital Waveguide Models
- Summary
- Newton's Laws of Motion
- Work and Energy
- Momentum
- Rigid-Body Dynamics
- Center of Mass
- Translational Kinetic Energy
- Rotational Kinetic Energy
- Mass Moment of Inertia
- Perpendicular Axis Theorem
- Parallel Axis Theorem
- Stretch Rule
- Area Moment of Inertia
- Radius of Gyration
- Two Masses Connected by a Rod
- Angular Velocity Vector
- Vector Cross Product
- Angular Momentum
- Angular Momentum Vector
- Mass Moment of Inertia Tensor
- Principal Axes of Rotation
- Rotational Kinetic Energy Revisited
- Torque
- Newton's Second Law for Rotations
- Equations of Motion for Rigid Bodies
- Properties of Elastic Solids
- Wave Equation for the Vibrating String
- 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
- The Ideal Vibrating String
- The Finite Difference Approximation
- Traveling-Wave Solution
- Sampled Traveling Waves
- A Lossy 1D Wave Equation
- The Dispersive 1D Wave Equation
- Alternative Wave Variables
- Scattering at Impedance Changes
- Digital Waveguide Filters
- ``Traveling Waves'' in Lumped Systems
- Properties of Passive Impedances
- Loaded Waveguide Junctions
- Two Coupled Strings
- Digital Waveguide Mesh
- FDNs as Digital Waveguide Networks
- Waveguide Transformers and Gyrators
- The Digital Waveguide Oscillator
- Non-Cylindrical Acoustic Tubes
-plane Positive-Real Functions
- Introduction
- State Transformations
- Excitation Examples
- State Space Formulation
- Computational Complexity
- Summary
- Future Work
- Acknowledgments
- Wave Digital Elements
- Adaptors for Wave Digital Elements
- Wave Digital Modeling Examples
- ``Piano hammer in flight''
- Force Driving a Mass
- Force Driving a Spring against a Wall
- Spring and Free Mass
- Mass and Dashpot in Series
- 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
