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 [365, pp. 224-226].

The factored form [452] of a transfer function

$$H(s) \isdef \frac{B(s)}{A(s)} \isdef \frac{b_M s^M + \cdots b_1 s + b_0}{a_N s^N + \cdots a_1 s + a_0} \protect$$ (9.3)

can be written as

$$H(s) = \left(\frac{b_M}{a_N}\right) \frac{\prod_{i=1}^M (s-\xi_i) }{\prod_{i=1}^N (s-p_i) } \protect$$ (9.4)

The matched z transformation is carried out by replacing each first-order term of the form $(s+a)$ by its digital equivalent $1 - e^{-aT}z^{-1}$ , i.e.,

$$\zbox {s+a \;\to\; 1 - e^{-aT}z^{-1}} \protect$$ (9.5)

to get

$$H_d(z) = g\left(\frac{b_M}{a_N}\right) \frac{ \prod_{i=1}^M (1 - e^{\xi_iT}z^{-1})}{ \prod_{i=1}^N (1 - e^{p_iT}z^{-1}}), \protect$$ (9.6)

where the free gain $g$ is introduced to implement the desired normalization, such as matching dc gain. Note that 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 [346].