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Characteristics of Prototype Analog Filters 413
function [b,a] = u_chb1ap(N,Rp,Omegac);
% Unnormalized Chebyshev-1 Analog Lowpass Filter Prototype
% --------------------------------------------------------
% [b,a] = u_chb1ap(N,Rp,Omegac);
% b = numerator polynomial coefficients
% a = denominator polynomial coefficients
% N = Order of the Elliptic Filter
% Rp = Passband Ripple in dB; Rp > 0
% Omegac = Cutoff frequency in radians/sec
%
[z,p,k] = cheb1ap(N,Rp); a = real(poly(p)); aNn = a(N+1);
p = p*Omegac; a = real(poly(p)); aNu = a(N+1);
k = k*aNu/aNn;
b0 = k; B = real(poly(z)); b = k*B;
8.3.7 DESIGN EQUATIONS
Given Ω p ,Ω s , R p , and A S , three parameters are required to determine
a Chebyshev-I filter: ɛ, Ω c , and N. From equations (8.3) and (8.4), we
obtain
√
ɛ = 10 0.1Rp − 1 and A =10 As/20
From these properties, we have
Ω c =Ω p and Ω r = Ω s
(8.57)
Ω p
The order N is given by
g = √ (A 2 − 1) /ɛ 2 (8.58)
⎡
log 10
[g + √ ] ⎤
g 2 − 1
N =
⎢log 10
[Ω r + √ ]
(8.59)
Ω 2 r − 1 ⎥
Now using (8.54), (8.53), and (8.55), we can determine H a (s).
□ EXAMPLE 8.5 Design a lowpass Chebyshev-I filter to satisfy
Passband cutoff: Ω p =0.2π ;Passband ripple: R p = 1dB
Stopband cutoff: Ω s =0.3π ; Stopband ripple: A s = 16dB
Solution
First compute the necessary parameters.
ɛ = √ 10 0.1(1) − 1=0.5088 A =10 16/20 =6.3096
Ω c =Ω p =0.2π Ω r = 0.3π
0.2π =1.5
g = √ (A 2 − 1) /ɛ 2 =12.2429 N =4
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