Transfer Function Models

For *linear time-invariant* (LTI) systems, instead of building 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 *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 of course be a matrix relating a vector of
inputs to a vector of outputs [221].) Such models are used
extensively in the field of *control system design*
[151].^{9.1} We considered this approach for artificial
reverberation in §3.1.1.

Transfer-function modeling is often the most cost-effective way to
incorporate LTI *lumped* elements (Ch. 7) in an otherwise
physical computational model. For *distributed-parameter*
systems, on the other hand, such as vibrating strings and acoustic
tubes propagating waves along one dimension,
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.

- Outline
- Sampling the Impulse Response
- Impulse Invariant Method
- Matched Z Transformation

- Pole Mapping with Optimal Zeros
- Modal Expansion

- General Filter Design Methods
- Ideal Differentiator (Spring Admittance)
- Digital Filter Design Overview
- Digital Differentiator Design
- Fitting Filters to Measured Amplitude Responses
- Further Reading on Digital Filter Design

- Commuted Synthesis

- Resonator Factoring
- Mode Extraction Techniques
- Inverse Filtering
- Sinusoidal Modeling of Mode Decays
- Parallel Body Filterbank Design
- Excitation Noise Substitution
- Body Factoring Example

- Virtual Analog Example: Phasing

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University