UploadFile_6417

152 Chapter 5 THE DISCRETE FOURIER TRANSFORM
is sufficiently small to result in any appreciable amount of aliasing in practice.
Such information is useful in effectively truncating an infinite-duration sequence
prior to taking its transform.
□
5.2.1 THE z-TRANSFORM RECONSTRUCTION FORMULA
Let x(n) betime-limited to [0,N − 1]. Then from Theorem 1 we should
be able to recover the z-transform X(z) using its samples ˜X(k). This is
given by
X(z) =Z [x(n)] = Z [˜x(n)R N (n)]
= Z[ IDFS{ ˜X(k) }R N (n)]
} {{ }
Samples of X(z)
This approach results in the z-domain reconstruction formula.
Since W −kN
N
X(z) =
=
N−1
∑
0
N−1
∑
0
= 1 N
= 1 N
= 1 N
=1,wehave
x(n)z −n =
{
N−1
∑
1
N
k=0
N−1
∑
k=0
N−1
∑
k=0
N−1
∑
0
˜X(k)
˜X(k)
˜X(k)
X(z) = 1 − z−N
N
N−1
∑
0
˜x(n)z −n
}
˜X(k)W −kn
N
z −n
{ N−1
}
∑
W −kn
N z−n
0
{ N−1
}
∑ (
W
−k
N
z−1) n
0
{
}
1 − W −kN
N
z −N
1 − W −k
N z−1
N−1
∑
k=0
˜X(k)
1 − W −k
N z−1 (5.17)
5.2.2 THE DTFT INTERPOLATION FORMULA
The reconstruction formula (5.17) can be specialized for the discrete-time
Fourier transform by evaluating it on the unit circle z = e jω . Then
X(e jω )= 1 − e−jωN
N
=
N−1
∑
k=0
N−1
∑
k=0
˜X(k)
1 − e j2πk/N e −jω
1 − e
˜X(k)
−jωN
N { 1 − e j2πk/N e −jω}
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.