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In this chapter, we will look at a variety of ways to digitize macroscopic point-to-point transfer functions $ \Gamma (s)$ corresponding to a desired impulse response $ \gamma(t)$:
  1. Sampling $ \gamma(t)$ to get $ \gamma(nT), n = 0,1,2,\ldots$
  2. Pole mappings (such as $ z_i = e^{s_i T}$ followed by Prony's method)
  3. Modal expansion
  4. Frequency-response matching using digital filter design methods

Next, we'll look at the more specialized technique known as commuted synthesis, in which computational efficiency may be greatly increased by interchanging (``commuting'') the series order of component transfer functions. Commuted synthesis delivers large gains in efficiency for systems with a short excitation and high-order resonators, such plucked and struck strings. In Chapter 9, commuted synthesis is applied to piano modeling. Extracting the least-damped modes of a transfer function for separate parametric implementation is often used in commuted synthesis. We look at a number of ways to accomplish this goal toward the end of this chapter. We close the chapter with a simple example of transfer-function modeling applied to the digital phase shifter audio effect. This example classifies as virtual analog modeling, in which a valued analog device is converted to digital form in a way that preserves all valued features of the original. Further examples of transfer-function models appear in Chapter 9.
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State Space Approach to Modal Expansions
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Practical Considerations