## Matched Z Transformation

The*matched*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*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 by its digital equivalent ,

*i.e.*,

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 obtainwhich 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.

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Pole Mapping with Optimal Zeros

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Impulse Invariant Method