Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 382

The digital filtering is performed as

v 0 ð0Þ¼2v 0 ð1Þv 0 ð2Þþxð0Þ¼0 þ 0 þ 1 ¼ 1

v 0 ð1Þ¼2v 0 ð0Þv 0 ð1Þþxð1Þ¼2 1 þ 0 þ 2 ¼ 4

v 0 ð2Þ¼2v 0 ð1Þv 0 ð0Þþxð2Þ¼2 4 1 þ 3 ¼ 10

v 0 ð3Þ ¼ 2v 0 ð2Þ v 0 ð1Þ þ xð3Þ ¼ 2 10 4 þ 4 ¼ 20

v 0 ð4Þ¼2v 0 ð3Þv 0 ð2Þþxð4Þ¼2 20 10 þ 0 ¼ 30

Then the squared magnitude is determined by

4

v 0

3 jX ð0Þj 2 ¼ v 0 þ v 0 ¼ð30Þ 2 þð20Þ 2 2 30 20 ¼ 100

4 3 2v 0

Thus, the amplitude spectrum is computed as
A 0 ¼ 1
4

r

jX ð0Þj 2

¼ 2:5

A MATLAB function for the Geortzel algorithm is shown in Program 8.20.
Program 8.20. MATLAB function for Geortzel Algorithm.

function [ Xk, Ak] ¼ galg(x,k)
% Geortzel Algorithm
% [ Xk, Ak] ¼ galg(x,k)
%x

¼ input vetcor; k ¼ frequency index

%Xk ¼ kth DFT coeficient; Ak ¼ magnitude of the kth DFT coefficient

N ¼ length(x); x ¼ [x 0];

vk ¼ zeros(1,N þ 3);

for n ¼ 1:N þ 1

vk(n þ 2) ¼ 2*cos(2*pi*k/N)*vk(n þ 1)-vk(n) þ x(n);

end
Xk ¼ vk(N þ 3)-exp(-2*pi*j*k/N)*vk(N þ 2);
Ak ¼ vk(N þ 3)*vk(N þ 3) þ vk(N þ 2)*vk(N þ 2)-2*cos(2*pi*k/N)*vk(N þ 3)*vk(N þ 2);
Ak ¼ sqrt(Ak)/N;
end

EXAMPLE 8.28
Use Program 8.20 to verify the results in Examples 8.26 and 8.27.
Solution:
a. For Example 8.26, we obtain
>> x¼[1 2 3 4]