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274 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
6.8.2 EFFECT ON POLE-ZERO LOCATIONS
One aspect can be reasonably analyzed, which is the movement of filter
poles when a k is changed to â k . This can be used to check the stability
of IIR filters. A similar movement of zeros to changes in numerator
coefficients can also be analyzed.
To evaluate this movement, consider the denominator polynomial of
H(z) in(6.65)
N∑
N∏
D(z) =1+
△ a k z −k (
= 1 − pl z −1) (6.67)
k=1
where {p l }s are the poles of H(z). We will regard D(z) asafunction
D(p 1 ,...,p N )ofpoles {p 1 ,...,p N } where each pole p l is a function of the
filter coefficients {a 1 ,...,a N }—that is, p l = f(a 1 ,...,a N ), l =1,...N.
Then the change in the denominator D(z) due to a change in the kth
coefficient a k is given by
( ) ( )( ) ( )( ) ( )( )
∂D(z) ∂D(z) ∂p1 ∂D(z) ∂p2 ∂D(z) ∂pN
=
+
+···+
∂a k ∂p 1 ∂a k ∂p 2 ∂a k ∂p N ∂a k
(6.68)
where from (6.67)
( ) [
∂D(z)
= ∂ ∏ N
(
1 − pl z −1)] = −z ∏ −1 (
1 − pl z −1) (6.69)
∂p i ∂p i
l=1 l≠i
( )∣
From (6.69), note that ∂D(z) ∣∣z=pl
∂p i
=0for l ̸=i. Hence from (6.68) we
obtain
l=1
( )∣ ( )∣ ( )
∂D(z) ∣∣∣z=pl ∂D(z) ∣∣∣z=pl ∂pl
=
∂a k ∂p l ∂a k
or
( ) ∂pl
=
∂a k
( )∣
∂D(z) ∣∣z=pl
∂a k
( )∣
∂D(z) ∣∣z=pl
∂p l
Now
( ∂D(z)
∂a k
) ∣
∣∣∣z=pl
= ∂
∂a k
(
1+
N∑
i=1
(6.70)
a i z −i )∣ ∣∣∣∣z=pl
= z −k∣ ∣
z=pl
= p −k
l
(6.71)
From (6.69), (6.70) and (6.71), we obtain
( ) ∂pl
p −k
l
=
∂a k −z ∏ −1 i≠l (1 − p i z −1 ) ∣ z=pl
p N−k
l
= −∏
i≠l (p l − p i )
(6.72)
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