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

X __!:_ [ :
N
l
1
H[O]
H[O]
=
N __!:_ :
H[O]
= W,v 1 ·DH
1
w- 1
N
W - (N- 1 )
N
1
w- 1 N
W - (N- 1)
N
H[-1]
H[-l]WN I
12.7 OFDM (Multicarrier) Communications 697
w;;<-,,
l
W N
- (N - 1 ) 2
H[-N + l ]
H[-N + l ] W; (N-I)
H r-1 1w;; 1 N - I) H r-N + 1 1w;;< N - l ) (N-I)
. . .
w ; )• - 1 1
] l
H[OJ
. . . W - (N- 1 ) 2
N
H[-lJ
where we have defined the diagonal matrix with the channel DFT entries as
DH =
lH[O]
H[- 1]
· . .
H[-N + I ]
]
-
H[N]
l
H[N - l]
]
H[-N + J
(12.67a)
The last equality follows from the periodic nature of H[n] given in Eq. (12.626). We leave it as
homework to show that any cyclic matrix of size N x N can be diagonalized by premultiplication
with W N and postmultiplication with W,v 1 (Prob. 12.7-2).
Based on Eq. (12.67a) we have established the following very important relationship for
OFDM:
( 12.676)
Recall that after the cyclic prefix has been added, the channel input-output relationship is
reduced to Eq. (12.646). As a result,
l z[i ]
l]
] - ( 1 ) - 1 (
1 ) l
s: 1
] l
w_J 1]
]
. - -WN ·DH· -WN . +
: ,.Jii ,.Jii : :
z[l] s1 w[I]
This means that if we put the information source data into