Introduction to Physical Signal Models

This book is about techniques for building real-time computational physical models of musical instruments and audio effects. So, why would anyone want to do this, and what exactly is a ``computational physical model''?

There are several reasons one might prefer a computational model in place of its real-world counterpart:

• A virtual musical instrument (or audio effect) is typically much less expensive than the corresponding real instrument (effect). Consider, for example, the relative expense of a piano versus its simulation in software. (We discuss real-time piano modeling in §9.4.)

• Different instruments can share common controllers (such as keyboards, wheels, pedals, etc.). Any number of virtual instruments and/or effects can be quickly loaded as ``presets''.

• Sound quality (``signal to noise ratio'') can far exceed what is possible with a recording. This is because we can use any number of bits per sample, rendering the ``noise floor'' completely inaudible at all times.

• Software implementations are exactly repeatable. They never need to be ``tuned'' or ``calibrated'' like real-world devices.

• It is useful to be able to ``archive'' and periodically revive rare or obsolete devices in virtual form.

• The future evolution of virtual devices is less constrained than that of real devices.

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Subsections

• 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

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Previous: Acknowledgments
Next: But How Does It Sound?

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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.