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Finite Difference Approximation

A finite difference approximation (FDA) approximates derivatives with finite differences, i.e.,

$\displaystyle \frac{d}{dt} x(t) \isdefs \lim_{\delta\to 0} \frac{x(t) - x(t-\delta)}{\delta} \;\approx\; \frac{x(n T)-x[(n-1)T]}{T} \protect$ (8.2)

for sufficiently small $ T$.8.5 Equation (7.2) is also known as the backward difference approximation of differentiation. See §C.2.1 for a discussion of using the FDA to model ideal vibrating strings.

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

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

Delay Operator Notation

It is convenient to think of the FDA in terms of time-domain difference operators using a delay operator notation. The delay operator $ d$ is defined by

$\displaystyle d^k x(n) \eqsp x(n-k).

Thus, the first-order difference (derivative approximation) is represented in the time domain by $ (1-d)/T$. We can think of $ d$ as $ z^{-1}$ since, by the shift theorem for $ z$ transforms, $ z^{-k}
X(z)$ is the $ z$ transform of $ x$ delayed (right shifted) by $ k$ samples. The obvious definition for the second derivative is

$\displaystyle {\hat{\ddot x}}(n) \eqsp \frac{(1-d)^2}{T^2} x(n).$ (8.4)

However, a better definition is the centered finite difference

$\displaystyle {\hat{\ddot x}}(n) \isdefs \frac{(d^{-1}-1)(1-d)}{T^2} x(n) \eqsp \frac{d^{-1}-2+d}{T^2}x(n) \protect$ (8.5)

where $ d^{-1}$ denotes a unit-sample advance. This definition is preferable as long as one sample of look-ahead is available, since it avoids an operator delay of one sample. Equation (7.5) is a zero phase filter, meaning it has no delay at any frequency, while (7.4) is a linear phase filter having a delay of $ 1$ sample at all frequencies.
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