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The Discrete Fourier Series 147
DFS of Sq. wave: L=5, N=20
DFS of Sq. wave: L=5, N=40
|Xtilde(k)|
|Xtilde(k)|
5
4
3
2
1
0
−10 −5 0 5 10
k
DFS of Sq. wave: L=5, N=60
5
4
3
2
1
|Xtilde(k)|
|Xtilde(k)|
5
4
3
2
1
0
−20 −10 0 10 20
k
DFS of Sq. wave: L=7, N=60
6
4
2
0
0
−20 0 20
−20 0 20
k
k
FIGURE 5.2 The DFS plots of a periodic square wave for various L and N
Then we can compute its z-transform:
X(z) =
N−1
∑
n=0
x(n)z −n (5.9)
Now we construct a periodic sequence ˜x(n) byperiodically repeating x(n)
with period N, that is,
x(n) =
{˜x(n), 0 ≤ n ≤ N − 1
0, Elsewhere
(5.10)
The DFS of ˜x(n) isgiven by
˜X(k) =
N−1
∑
n=0
˜x(n)e −j 2π N nk =
Comparing it with (5.9), we have
N−1
∑
n=0
x(n)
[e j 2π k] −n
N
(5.11)
˜X(k) =X(z)| z=e
j 2π N k (5.12)
which means that the DFS ˜X(k) represents N evenly spaced samples of
the z-transform X(z) around the unit circle.
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