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Frequency-band Transformations 455
Now in practice we will know the desired highpass frequencies ω s and
ω p , and we are required to find the prototype lowpass cutoff frequencies
ω ′ s and ω ′ p.Wecan choose the passband frequency ω ′ p with a reasonable
value, say ω ′ p =0.2π, and determine α from ω p using the formula from
Table 8.2. Now ω ′ s can be determined (for our highpass filter example)
from α and
Z = − z−1 + α
1+αz −1
where Z = e jω′ s
and z = e
jω s
,or
ω ′ s = ̸
(
)
− e−jωs + α
1+αe −jωs
(8.71)
Continuing our highpass filter example, let ω p =0.6π and ω s =0.4586π be
the band-edge frequencies. Let us choose ω ′ p =0.2π. Then α = −0.38197
from (8.70), and from (8.71)
ω ′ s = ̸
(
)
− e−j0.4586π − 0.38197
1 − 0.38197e −j−0.38197 =0.3π
as expected. Now we can design a digital lowpass filter and transform
it into a highpass filter using the zmapping function to complete our
design procedure. For designing a highpass Chebyshev-I digital filter, the
above procedure can be incorporated into a MATLAB function called the
cheb1hpf function shown here.
function [b,a] = cheb1hpf(wp,ws,Rp,As)
% IIR Highpass filter design using Chebyshev-1 prototype
% function [b,a] = cheb1hpf(wp,ws,Rp,As)
% b = Numerator polynomial of the highpass filter
% a = Denominator polynomial of the highpass filter
% wp = Passband frequency in radians
% ws = Stopband frequency in radians
% Rp = Passband ripple in dB
% As = Stopband attenuation in dB
%
% Determine the digital lowpass cutoff frequencies:
wplp = 0.2*pi;
alpha = -(cos((wplp+wp)/2))/(cos((wplp-wp)/2));
wslp = angle(-(exp(-j*ws)+alpha)/(1+alpha*exp(-j*ws)));
%
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