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  of a transfer function
can be written as
The matched z transformation is carried out by replacing each first-order term of the form by its digital equivalent , i.e.,
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 .
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.(7.3).
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.
Pole Mapping with Optimal Zeros
Impulse Invariant Method