### Finite Difference Approximation

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

 (8.2)

for sufficiently small .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 , which can be viewed as the Laplace transform of the operator (left-hand side of Eq.(7.2)). Moving to the right-hand side, the z transform of the first-order difference operator is . Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution

 (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

As discussed in §8.3.1, the FDA is a special case of the matched transformation applied to the point .

Note that the FDA does not alias, since the conformal mapping 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 is defined by

Thus, the first-order difference (derivative approximation) is represented in the time domain by . We can think of as since, by the shift theorem for transforms, is the transform of delayed (right shifted) by samples.

The obvious definition for the second derivative is

 (8.4)

However, a better definition is the centered finite difference

 (8.5)

where 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 sample at all frequencies.

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