Extracting Parametric Resonators from a Nonparametric Impulse Response

Commuted synthesis is a technique in which the body resonator is commuted with the string model, as shown in Fig.4.30, in order to avoid having to implement a large body filter at all [449,235,232].^S.1In commuted synthesis, the excitation (e.g., plucking force versus time) can be convolved with the resonator impulse response to provide a single aggregate excitation signal. This signal is short enough to store in a look-up table, and a note is played by simply summing it into the string.

A valuable way of shortening the excitation table in commuted synthesis is to factor the body resonator into its most-damped and least-damped modes. The most-damped modes are then commuted and combined with the excitation in impulse-response form. The least-damped modes can be left in parametric form as recursive digital filter sections. Advantages of this factoring include the following:

• The excitation table is shortened.
  
• The excitation table signal-to-quantization-noise ratio is improved.
  
• The most important resonances remain parametric, facilitating real-time control.
  
• Multiple body outputs become available.
  
• Resonators may be already available in a separate effects unit, making them ``free.''
  
• A memory vs. computation trade-off is available for cost optimization.

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Subsections

• 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 Experiments
  • Matlab Code for Inverse Filtering
  
  
• Sinusoidal Modeling of Mode Decays
• Parallel Body Filterbank Design
• Excitation Noise Substitution
• Body Factoring Example

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written by 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.