### Finite Difference Approximation

A*finite difference approximation*(FDA) approximates derivatives with finite differences,

*i.e.*,

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

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

#### 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

*shift theorem*for transforms, is the transform of delayed (right shifted) by samples. The obvious definition for the second derivative is

However, a better definition is the

*centered finite difference*

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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Bilinear Transformation

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Passive One-Ports