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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
![$\displaystyle H(s) \isdef \frac{B(s)}{A(s)} \isdef \frac{b_M s^M + \cdots b_1 s + b_0}{a_N s^N + \cdots a_1 s + a_0} \protect$](img1772.png) |
(9.3) |
can be written as
![$\displaystyle H(s) = \left(\frac{b_M}{a_N}\right) \frac{\prod_{i=1}^M (s-\xi_i) }{\prod_{i=1}^N (s-p_i) } \protect$](img1773.png) |
(9.4) |
The matched z transformation is carried out by replacing each first-order
term of the form
by its digital equivalent
, i.e.,
![$\displaystyle \zbox {s+a \;\to\; 1 - e^{-aT}z^{-1}} \protect$](img1776.png) |
(9.5) |
to get
![$\displaystyle H_d(z) = g\left(\frac{b_M}{a_N}\right) \frac{ \prod_{i=1}^M (1 - e^{\xi_iT}z^{-1})}{ \prod_{i=1}^N (1 - e^{p_iT}z^{-1}}), \protect$](img1777.png) |
(9.6) |
where the free gain
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].
Subsections
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