DFT and z-transform
Consider a sequence x(n) having the z-transform X(z) with an ROC that includes
the unit circle. If X(z) is sampled at N equally spaced points on the unit circle at
z
k
e
j2pk=N
, k 0, 1, . . . , N À 1, we obtain
Xzj
ze
j2pk=N
I
nÀI
xnz
Àn
ze
j2pk=N
I
nÀI
xne
Àj2pk=Nn
:
7:1:24
This is identical to evaluating the discrete-time Fourier transform X! at the N equally
spaced frequencies !
k
2pk=N, k 0, 1, . . . , N À 1. If the sequence x(n) has a finite
duration of length N, the DFT of a sequence yields its z-transform on the unit circle at a
set of points that are 2p=N radians apart, i.e.,
Xk Xzj
z e
j
2p
N
k
, k 0, 1, . . . , N À 1:
7:1:25
Therefore the DFT is equal to the z-transform of a sequence x(n) of length N, evaluated
at N equally spaced points on the unit circle in the z-plane.
Example 7.4: Consider a finite-length DC signal
xn c, n 0, 1, . . . , N À 1:
From (7.1.3), we obtain
Xk c
NÀ1
n0
W
kn
N
c
1 À W
kN
N
1 À W
k
N
:
Since W
kN
N
e
Àj
2p
N
kN
1for all k and for W
k
N
T
1for k T iN, we have Xk 0
for k 1 , 2, . . . , N À 1. For k 0,
NÀ1
n0
W
kn
N
N. Therefore we obtain
Xk cNdk; k 0, 1, . . . , N À 1:
7.1.3 Circular Convolution
The Fourier transform, the Laplace transform, and the z-transform of the linear con-
volution of two time functions are simply the products of the transforms of the
individual functions. A similar result holds for the DFT, but instead of a linear
convolution of two sequences, we have a circular convolution. If x(n) and h(n) are
real-valued N-periodic sequences, y(n) is the circular convolution of x(n) and h(n)
defined as
yn xn hn, n 0, 1, . . . , N À 1,
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