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Problems 377
where coefficients ˜c(n) are given by
c(1) = ˜c(0) − 1 2 ˜c(1),
c(n) = 1 M − 3
[˜c(n − 1) − ˜c(n)] , 2 ≤ n ≤ ,
2 2
( ) M − 1
c
= 1 ( ) M − 3
2 2 ˜c .
2
P7.5 The Type-4 linear-phase FIR filter is characterized by
h(n) =−h(M − 1 − n), 0 ≤ n ≤ M − 1, M even
1. Show that its amplitude response H r(ω) isgiven by
M/2
∑
H r(ω) = d(n) sin { ω ( )}
n − 1 2
n=1
where coefficients {d(n)} are obtained as defined in (7.16).
2. Show that the above H r(ω) can be further expressed as
( ) ω ∑ L
H r(ω) =sin
2
where coefficients ˜d(n) are given by
n=0
d(1) = ˜d(0) − 1 2 ˜d(1),
d(n) = 1 2
)
( M
d
2
˜d(n) cos(ωn), L = M 2 − 1
[ ˜d(n − 1) − ˜d(n)
] , 2 ≤ n ≤
M
2 − 1,
= 1 2 ˜d ( M
2 − 1) .
P7.6 Write a MATLAB function to compute the amplitude response H r(ω) given a linear phase
impulse response h(n). The format of this function should be
function [Hr,w,P,L] = Ampl_Res(h);
% Computes Amplitude response Hr(w) and its polynomial P of order L,
% given a linear-phase FIR filter impulse response h.
% The type of filter is determined automatically by the subroutine.
%
% [Hr,w,P,L] = Ampl_Res(h)
% Hr = Amplitude Response
% w = frequencies between [0 pi] over which Hr is computed
% P = Polynomial coefficients
% L = Order of P
% h = Linear Phase filter impulse response
The function should first determine the type of the linear-phase FIR filter and then use the
appropriate Hr Type# function discussed in this chapter. It should also check if the given
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