Finite Difference Approximation
A finite difference approximation (FDA) approximates derivatives with finite differences, i.e.,
for sufficiently small

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









The obvious definition for the second derivative is
However, a better definition is the centered finite difference
where


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