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

700 DIGITAL COMMUNICATIONS UNDER LIN EARLY DISTORTIVE CHANNELS
N-l N-l
N
N L, L, N 2
= _!_ '°"' '°"' W k 2•j- k 1• i . -8[k1 - k2]
k1=0 k2 =0
N-l
=N/2N L w 1
1 ( i
-i J
k1=0
= N/2N . {
N
= 2
8[i - j]
i = j
i f= j
(12.69)
Because {w[n]} are zero mean with zero correlation, they are uncorrelated according to
Eq. (12.69). Moreover, {w[n]} are also Gaussian noises. Since uncorrelated Gaussian random
variables are also independent, {w[nl} are independent Gaussian noises with zero mean and
identical variance of N /2. The independence of the N channel noises demonstrates that OFDM
converts an FIR channel with ISi and order up to L into N parallel, independent, and A WGN
channels as shown in Fig. 12.11.
12.7.3 Zero-Padded OFDM
We have shown that by introducing a cyclic prefix of length L, a circular convolution channel
matrix can be established. Because any circular matrix of size N x N can be diagonalized by
IDFT and DFT (Prob. 12.7-2), the ISi channel of order less than or equal to L is transformed
into N parallel independent subchannels.
There is also an alternative approach to the use of cyclic prefix. This method is known
as zero padding. The transmitter first perform an IDFT on the N input data. Then, instead of
repeating the last L symbols as in Eq. ( 12.64b) to transmit
S[
SN
SN-L+l
we can simply replace the cyclic prefix with L zeros and transmit
SJ
0
(N + L) x l
0