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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

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

can be written as

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

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

to get

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

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2018-01-17 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University