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FDA in the Frequency Domain

Viewing Eq.(7.2) in the frequency domain, the ideal differentiator transfer-function is $ H(s)=s$ , which can be viewed as the Laplace transform of the operator $ d/dt$ (left-hand side of Eq.(7.2)). Moving to the right-hand side, the z transform of the first-order difference operator is $ (1-z^{-1})/T$ . Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution

$\displaystyle s \eqsp \frac{1-z^{-1}}{T} \protect$ (8.3)

in any continuous-time transfer function (Laplace transform of an integro-differential operator) to obtain a discrete-time transfer function (z transform of a finite-difference operator).

The inverse of substitution Eq.(7.3) is

$\displaystyle z \eqsp \frac{1}{1 - sT} \eqsp 1 + sT+ (sT)^2 + \cdots \, .

As discussed in §8.3.1, the FDA is a special case of the matched $ z$ transformation applied to the point $ s=0$ .

Note that the FDA does not alias, since the conformal mapping $ s = {1
- z^{-1}}$ is one to one. However, it does warp the poles and zeros in a way which may not be desirable, as discussed further on p. [*] below.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University