B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

696 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
Thus, the N -point DFT of v can be written in the matrix form
-
V
[
V[O]
V [l]
V [N '.- II
] [
wz-o
w-o
w o-1
N
N
-
w)f"-no
W l I
W (N- 1 )-1
N
w o - c N-1 )
N
W l - (N-1 )
N
W (N-{) ·(N-1)
N
l
·V (12.65)
If we denote the N x N DFT matrix as
1
W l
N
wr
,
,
l
(12.66a)
W v
£
[j
W (N- 1 )
N
W (N-1) 2
N
then W N also has an inverse
1
l 1 w-l
N
w-1 =
N
N :
[' l
W - (N-1)
N
This can be verified (Prob. 12.7-1) by showing that
WN ' " l
W - (N-1)2
N
(12.66b)
Given this notation, we have the relationship of
V = WN . V
V = W,v 1 . V
An amazing property of the cyclic matrix H ep can be established by applying the DFT and
IDFT matrices.
h[O] h[l] h[L] 0 0
0 h[O] h[l] h[L]
0
Hep · W,,v 1 = 0 0 h[O] h[ l] h[L]
h[L] 0 h[O] h[L - 1]
h[l] h[L] 0 0 h[O]