Transfer Function Models

For linear time-invariant systems, rather than build an explicit discrete-time model as in §7.3 for each mass, spring, and dashpot, (or inductor, capacitor, and resistor for virtual analog models), we may instead choose to model only the transfer function between selected inputs and outputs of the physical system. It should be emphasized that this is an option only when the relevant portion of the system is linear and time invariant (LTI), or at least sufficiently close to LTI. Transfer-function modeling can be considered a kind of ``large-scale'' or ``macroscopic'' modeling in which an entire physical subsystem, such as a guitar body, is modeled by a single transfer function relating specific inputs and outputs. (A transfer function can also of course be a matrix relating a vector of inputs to a vector of outputs [220].) Such models are used extensively in the field of control system design [150].^9.1

Transfer-function modeling is often the most cost-effective way to incorporate LTI lumped elements (Ch. 7) in an otherwise physical computational model. For wave-propagating distributed systems, on the other hand, such as vibrating strings and acoustic tubes, digital waveguides models (Ch. 6) are more efficient than transfer-function models, in addition to having a precise physical interpretation that transfer-function coefficients lack. In models containing lumped elements, or distributed components that are not characterized by wave propagation, maximum computational efficiency is typically obtained by deciding which LTI portions of the model can be ``frozen'' as ``black boxes'' characterized only by their transfer functions. In return for increased computational efficiency, we sacrifice the ability to access the interior of the black box in a physically meaningful way.

An example where such ``macroscopic'' transfer-function modeling is normally applied is the trumpet bell (§9.7.2). A fine-grained model might use a piecewise cylindrical or piecewise conical approximation to the flaring bell [71]. However, there is normally no need for an explicit bell model in a practical virtual instrument when the bell is assumed to be LTI and spatial directivity variations are neglected. In such cases, the transmittance and reflectance of the bell can be accurately summarized by digital filters having frequency responses that are optimal approximations to the measured (or theoretical) bell responses (§9.7). However, it is then not so easy to insert a moveable virtual ``mute'' into the bell reflectance/transmittance filters. This is an example of the general trade-off between physical extensibilty and computational efficiency/parsimony.

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Subsections

• • • 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

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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.