### 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

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

*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