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316 Chapter 7 FIR FILTER DESIGN
FIGURE 7.3
A general zero constellation
In the following examples, we illustrate the preceding properties of
linear-phase FIR filters.
□ EXAMPLE 7.4 Let h(n) ={−4, 1, −1, −2, 5, 6, 5, −2, −1, 1, −4}. Determine the amplitude response
↑
H r (ω) and the locations of the zeros of H
(z).
Solution Since M =11, which is odd, and since h(n) issymmetric about α = (11−1)/2 =
5, this is a Type-1 linear-phase FIR filter. From (7.7) we have
a(0) = h (α) =h(5) =6,a(1) = 2h(5 − 1) = 10, a(2) = 2h(5 − 2) = −4
a (3) =2h (5 − 3) = −2, a(4) =2h (5 − 4) = 2, a(5) = 2h (5 − 5) = −8
From (7.8), we obtain
H r(ω) =a(0) + a(1) cos ω + a(2) cos 2ω + a(3) cos 3ω + a(4) cos 4ω + a(5) cos 5ω
= 6+10cosω − 4 cos 2ω − 2 cos 3ω +2cos 4ω − 8 cos 5ω
MATLAB script:
>> h = [-4,1,-1,-2,5,6,5,-2,-1,1,-4];
>> M = length(h); n = 0:M-1;
>> [Hr,w,a,L] = Hr_Type1(h);
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