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As introduced in Appendix N, the finite difference approximation (FDA) amounts to replacing derivatives by finite differences, or
See §H.2.1 for a discussion of using the FDA to model ideal vibrating strings.
Viewing Eq.
(L.2) in the frequency domain, the transfer function
of an ideal differentiator is
, which can be viewed as the
Laplace transform of the operator
(left-hand side of
Eq.
(L.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
The inverse of substitution Eq.
(L.3) is
The FDA is a special case of the matched
transformation
[371] applied to the point
. In general, the
matched
transformation maps a pole at
to the point
, where
is the sampling period. Thus, each pole and
zero are mapped according to
Since the FDA is the matched
transformation for poles and zeros at
the origin of the
plane, it follows that it maps analog dc (
)
to digital dc (
). However, that is the only ideal mapping in the
frequency domain, as discussed further below.