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412 Chapter 8 IIR FILTER DESIGN
roots of these polynomial, then
where
a = 1 2
[ ]
π (2k +1)π
σ k =(aΩ c ) cos +
2 2N
[ ] k =0,...,N − 1 (8.53)
π (2k +1)π
Ω k =(bΩ c ) sin +
2 2N
( N
α − N√ )
1/α , b = 1 ( N
α + N√ )
1/α ,
2
and α = 1 √1+
ɛ + 1 ɛ 2
(8.54)
These roots fall on an ellipse with major axis bΩ c and minor axis aΩ c .
Now the system function is given by
K
H a (s) = ∏
(s − p k )
k
(8.55)
where K is a normalizing factor chosen to make
⎧
⎪⎨ 1, N odd
H a (j0) = 1
⎪⎩ √ ,Neven (8.56)
1+ɛ
2
8.3.6 MATLAB IMPLEMENTATION
MATLAB provides a function called [z,p,k]=cheb1ap(N,Rp) to design
a normalized Chebyshev-I analog prototype filter of order N and passband
ripple Rp and that returns zeros in z array, poles in p array, and
the gain value k. Weneed an unnormalized Chebyshev-I filter with arbitrary
Ω c . This is achieved by scaling the array p of the normalized filter
by Ω c . Similar to the Butterworth prototype, this filter has no zeros.
The new gain k is determined using (8.56), which is achieved by scaling
the old k by the ratio of the unnormalized to the normalized denominator
polynomials evaluated at s =0.Inthe following function, called
U chb1ap(N,Rp,Omegac), wedesign an unnormalized Chebyshev-I analog
prototype filter that returns H a (s) inthe direct form.
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