Matched Z Transformation
The matched z transformation uses the same pole-mapping
Eq.(8.2) as in the impulse-invariant method, but the zeros
are handled differently. Instead of only mapping the poles of the
partial fraction expansion and letting the zeros fall where they may,
the matched z transformation maps both the poles and zeros in the
factored form of the transfer function [362, pp.
224-226].
The factored form [449] of a transfer function
can be written as
The matched z transformation is carried out by replacing each first-order term of the form


to get
Thus, the matched z transformation normally yields different digital zeros than the impulse-invariant method. The impulse-invariant method is generally considered superior to the matched z transformation [343].
Relation to Finite Difference Approximation
The Finite Difference Approximation (FDA) (§7.3.1) is a
special case of the matched transformation applied to the point
. To see this, simply set
in Eq.
(8.5) to obtain
which is the FDA definition in the frequency domain given in Eq.

Since the FDA equals the match z transformation for the point , it maps
analog dc (
) to digital dc (
) exactly. However, that is the
only point on the frequency axis that is perfectly mapped, as shown in
Fig.7.15.
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Pole Mapping with Optimal Zeros
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Impulse Invariant Method