Physical Models

We now turn to the main subject of this book, physical models of musical instruments and audio effects. In contrast to the non-physical signal models mentioned above, we will consider a signal model to be a physical signal model when there is an explicit representation of the relevant physical state of the sound source. For example, a string physical model must offer the possibility of exciting the string at any point along its length.

We begin with a review of physical models in general, followed by an overview of computational subtypes, with some indication of their relative merits, and what is and is not addressed in this book.

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Subsections

• 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

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Next: All We Need is Newton

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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.