4.2 IIR filter design
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4. Find the transfer function Ha (s) of the analogue filter corresponding to the frequency response Ha (Ω). 5. Find the transfer function H(z) of the digital filter by applying a proper transformation from the s-domain to the z-domain. Before describing this procedure in detail, let us review some important concepts related to analogue filters. Consider an analogue system function PM
Ha (s) = Pk=0 N
dk sk
k=0 ck
=
sk
Ya (s) Xa (s)
(4.3)
where xa (t) is the input and ya (t) is the output and Xa (s) and Ya (s) are their respective Laplace transforms. An alternative representation of the same system is obtained by means of the differential equation N X k=0
M
dk ya (t) X dk xa (t) dk = ck dtk dtk
(4.4)
k=0
The corresponding rational system function for digital filters has the form PM
k=0 H(z) = PN k=0
bk z −k ak
z −k
=
Y (z) X(z)
(4.5)
or, equivalently, by the difference equation N X k=0
ak y(n − k) =
M X
bk x(n − k)
(4.6)
k=0
We are interested in finding a transformation that allows us to find a digital system from its analogue counterpart and vice-versa. Specifically, in transforming an analogue system to a digital system we must obtain h(n) or H(z) from the analogue filter design. In such transformations we generally require that the essential properties of the analogue frequency response be preserved in the frequency response of the resulting digital filter. Loosely speaking, this implies that we want the imaginary axis of the s-plane to map into the unit circle of the z-plane. A second condition is that a stable analogue filter should be transformed to a stable digital filter. That is, if the analogue system has poles only in the left-half s-plane, then the digital filter must have poles only inside the unit circle. In the following, we consider two transformations that attain this goal: bilinear transformation and impulse response invariance. We will show that only the former is adequate as far as filter design is concerned.