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

h[0] h[l] h[L] 0 0
12.7 OFDM (Multicarrier) Communications 695
0 h[0] h[ I] h[L] w[N]
w[N - l]
0
0 0 h[0] h[l] h[L] w[L]
h[L] 0 h[0] h[L - I] ltJ,
h[l] h[L] 0
Hep : (N X N)
0 h[0]
w[l]
(12.64b)
The critical role of the cyclic prefix is to convert the convolution channel matrix in Eq. ( 12.64a)
into a well-structured N x N cyclic matrix Hep in Eq. (12.64b).
Next, we need to introduce the N-point DFT matrix and the corresponding inverse DFT
matrix. First, it is more convenient to denote
This complex number WN has some useful properties:
Wff = l
W-i - W N-i
N - N
If we take the DFT of the N -dimensional vector
then we have the DFT
N-1 ( ) N-1
V [n] = L Vi exp -j2n ; = L Vi W,v k
k=O
k=O
n = 0, l, . .. , (N - l)
and
(· nk
)
- nk
V [-n] = L Vi exp 12n N
= L Vi W N
k=O
k=O
n = 0, l, ... , (N - 1)
The inverse DFT can also be simplified as
l N - 1
(
nk
)
I N - I
n=O
k=O
V k = N
L V[n] exp j2n N
= N
L V[n]WN nk
k = 0, I, ... , (N - 1)