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442 Chapter 8 IIR FILTER DESIGN
1
0.8913
Magnitude Response
1
Phase Response
|H|
π units
0
0.1778
decibels
0
0 0.2 0.3 1
frequency in π units
0
1
15
Magnitude in dB
Samples
−1
0 0.2 0.3 1
frequency in π units
15
10
5
Group Delay
0 0.2 0.3 1
frequency in π units
0
0 0.2 0.3 1
frequency in π units
FIGURE 8.29
Digital elliptic lowpass filter using bilinear transformation
The advantages of this mapping are that (a) it is a stable design,
(b) there is no aliasing, and (c) there is no restriction on the type of filter
that can be transformed. Therefore this method is used exclusively in
computer programs including MATLAB, as we shall see next.
8.4.5 MATCHED-z TRANSFORMATION
In this method of filter transformation, zeros and poles of H a (s) are directly
mapped into zeros and poles in the z-plane using an exponential
function. Given a root (zero or pole) at the location s = a in the s-plane,
we map it in the z-plane at z = e aT where T is a sampling interval. Thus,
the system function H a (s) with zeros {z k } and poles {p l } is mapped into
the digital filter system function H(z) as
∏ M
k=1
H a (s) =
(s − z ∏ M
(
k)
k=1 1 − e
z k T z −1)
∏ N
l=1 (s − p → H(z) = ∏ N
l)
l=1 (s − (8.69)
ep lT
z −1 )
Clearly the z-transform system function is “matched” to the s-domain
system function.
Note that this technique appears to be similar to the impulse invariance
mapping in that the pole locations are identical and aliasing is unavoidable.
However, these two techniques differ in zero locations. Also the
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