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146 Chapter 5 THE DISCRETE FOURIER TRANSFORM
a. By applying the analysis equation (5.3),
˜X(k) =
N−1
∑
n=0
L−1
L−1
∑ ∑ ) n
˜x(n)e −j 2π N nk = e −j 2π N nk =
(e −j 2π N k
n=0
⎧
⎨ L, k =0, ±N,±2N,...
=
⎩ 1 − e −j2πLk/N
, otherwise
1 − e−j2πk/N The last step follows from the sum of the geometric terms formula (2.7) in
Chapter 2. The last expression can be simplified to
n=0
1 − e −j2πLk/N e−jπLk/N e jπLk/N − e −jπLk/N
=
1 − e−j2πk/N e −jπk/N e jπk/N − e −jπk/N
= e −jπ(L−1)k/N sin (πkL/N)
sin (πk/N)
or the magnitude of ˜X(k) isgiven by
⎧
⎪⎨ L, k =0, ±N,±2N,...
∣
∣ ˜X(k) = ⎪ ⎩ sin (πkL/N)
∣ sin (πk/N) ∣ , otherwise
b. The MATLAB script for L =5and N = 20:
>> L = 5; N = 20; k = [-N/2:N/2]; % Sq wave parameters
>> xn = [ones(1,L), zeros(1,N-L)]; % Sq wave x(n)
>> Xk = dfs(xn,N); % DFS
>> magXk = abs([Xk(N/2+1:N) Xk(1:N/2+1)]); % DFS magnitude
>> subplot(2,2,1); stem(k,magXk); axis([-N/2,N/2,-0.5,5.5])
>> xlabel(’k’); ylabel(’Xtilde(k)’)
>> title(’DFS of SQ. wave: L=5, N=20’)
The plots for this and all other cases are shown in Figure 5.2. Note that
since ˜X(k) isperiodic, the plots are shown from −N/2 toN/2.
c. Several interesting observations can be made from plots in Figure 5.2. The
envelopes of the DFS coefficients of square waves look like “sinc” functions.
The amplitude at k =0is equal to L, while the zeros of the functions are
at multiples of N/L, which is the reciprocal of the duty cycle. We will study
these functions later in this chapter.
□
5.1.2 RELATION TO THE z-TRANSFORM
Let x(n) beafinite-duration sequence of duration N such that
{
Nonzero, 0 ≤ n ≤ N − 1
x(n) =
0, Elsewhere
(5.8)
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