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170 Chapter 5 THE DISCRETE FOURIER TRANSFORM
end
N = length(x); n = 0:(N-1);
xec = 0.5*(x + x(mod(-n,N)+1));
xoc = 0.5*(x - x(mod(-n,N)+1));
□ EXAMPLE 5.10 Let x(n) =10(0.8) n , 0 ≤ n ≤ 10 as in Example 5.9.
a. Decompose and plot the x ec(n) and x oc(n) components of x(n).
b. Verify the property in (5.34).
Solution
a. MATLAB script:
>> n = 0:10; x = 10*(0.8) .^ n;
>> [xec,xoc] = circevod(x);
>> subplot(2,1,1); stem(n,xec); title(’Circular-even component’)
>> xlabel(’n’); ylabel(’xec(n)’); axis([-0.5,10.5,-1,11])
>> subplot(2,1,2); stem(n,xoc); title(’Circular-odd component’)
>> xlabel(’n’); ylabel(’xoc(n)’); axis([-0.5,10.5,-4,4])
The plots in Figure 5.14 show the circularly symmetric components of x(n).
10
8
Circular-even component
xec(n)
6
4
2
0
4
0 1 2 3 4 5 6 7 8 9 10
n
Circular-odd component
2
xoc(n)
0
−2
−4
0 1 2 3 4 5 6 7 8 9 10
n
FIGURE 5.14 Circular-even and circular-odd components of the sequence in
Example 5.10a
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