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Sampling and Reconstruction in the z-Domain 149
5.2 SAMPLING AND RECONSTRUCTION IN THE z-DOMAIN
Let x(n) beanarbitrary absolutely summable sequence, which may be of
infinite duration. Its z-transform is given by
X(z) =
∞∑
m=−∞
x(m)z −m
and we assume that the ROC of X (z) includes the unit circle. We sample
X(z) onthe unit circle at equispaced points separated in angle by ω 1 =
2π/N and call it a DFS sequence,
˜X(k) = △ X(z)| j z=e 2π , k =0, ±1, ±2,...
N k
∞∑
∞∑
= x(m)e −j 2π N km = x(m)WN km (5.15)
m=−∞
m=−∞
which is periodic with period N. Finally, we compute the IDFS of ˜X(k),
˜x(n) =IDFS [ ˜X(k)
]
which is also periodic with period N. Clearly, there must be a relationship
between the arbitrary x(n) and the periodic ˜x(n). This is an important
issue. In order to compute the inverse DTFT or the inverse z-transform
numerically, we must deal with a finite number of samples of X(z) around
the unit circle. Therefore we must know the effect of such sampling on
the time-domain sequence. This relationship is easy to obtain.
or
˜x(n) =
=
˜x(n) = 1 N
∞∑
m=−∞
∞∑
= 1 N
x(m)
∞∑
N−1
∑
k=0
N−1
∑
k=0
r=−∞ m=−∞
˜X(k)W −kn
N
[from (5.2)]
{ ∞
∑
m=−∞
N−1
∑
x(m)W km
N
1
W −k(n−m)
N
N
0
} {{ }
{ 1, n− m = rN
=
0, elsewhere
x(m)δ(n − m − rN)
=
}
W −kn
N
[from (5.15)]
∞∑
m=−∞
x(m)
∞∑
r=−∞
δ(n−m−rN)
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