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Physical Audio Signal Processing
Introduction to Lumped Models
Finite Difference Approximation

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Finite Difference Approximation

As introduced in Appendix N, the finite difference approximation (FDA) amounts to replacing derivatives by finite differences, or

$\displaystyle \frac{d}{dt} x(t) \isdef \lim_{\delta\to 0} \frac{x(t) - x(t-\delta)}{\delta} \approx \frac{x(n T)-x[(n-1)T]}{T} \protect$

(L.2)

for sufficiently small

.^L.5

See §H.2.1 for a discussion of using the FDA to model ideal vibrating strings.

Viewing Eq. $\,$ (L.2) in the frequency domain, the transfer function of an ideal differentiator is , which can be viewed as the Laplace transform of the operator (left-hand side of Eq. $\,$ (L.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

$\displaystyle s \leftarrow \frac{1-z^{-1}}{T} \protect$

(L.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. $\,$ (L.3) is

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

The FDA is a special case of the matched transformation [371] applied to the point . In general, the matched transformation maps a pole at to the point $z = e^{-aT}$ , where is the sampling period. Thus, each pole and zero are mapped according to

$\displaystyle z_i = e^{s_i T} = 1 + {s_i T} + \frac{(s_i T)^2}{2} + \cdots\, .$

The actual transformation is carried out by factoring

into a product of first-order terms such as

, and substituting

$\displaystyle s+a \to 1 - z^{-1}e^{-aT}.$

Setting

gives the FDA for

.^L.6

Since the FDA is the matched transformation for poles and zeros at the origin of the plane, it follows that it maps analog dc () to digital dc (). However, that is the only ideal mapping in the frequency domain, as discussed further below.