Channel equalization with transmitter redundancy - Signal Processing and Optimization for Transceiver Systems - page 288

s ( n )
m, 0

s ( n )
m

F ( z )
m ,0

C ( z )
m

K

P

z

s ( n )
m, 1

channel

F ( z )
m ,1

K

P

z

s ( n )
m,K −

z

1

F ( z )
m , K −1

K

P

new transmitter filters

blocking into K -vector

s ( n )
m s ( n )

m, 0

H ( z )
m ,0

K

+

P

s ( n )
m, 1

z −1

H ( z )
m ,1

K

P

from other
users z −1

s ( n )

z −1

m,K −1

H ( z )
m , K −1

K

P

new receiver filters

unblocking

Figure P7.22 .

7.23. Null resistance. The cyclic-prefix system described in Sec. 7.3.2 uses the
equalizer multipliers 1 /C [ k ] , 0 1 , at the receiver. Here C [ k ]

are the M DFT coe cients of the FIR channel (assumed to be of order
L ). In general, some of these can be very small, or even equal to zero.
This creates practical di culties in the implementation, as the noise can
be severely amplified by 1 /C [ k ] . There are two practical ways to overcome
this problem. One is to replace the zero-forcing equalizers 1 /C [ k ]with
MMSE equalizers. The other is to use extra redundancy at the transmitter
to incorporate what is called null resistance. In this problem we describe
the second approach [Liang and Tran, 2002]. Consider the modified cyclic-
prefix system shown in Fig. P7.23. Here the blocked version s ( n )ofsize
M is transformed into a vector r ( n )ofsize N = M + L using a matrix Q .
The vector r ( n ) is treated as if this is to be transmitted. Thus, we perform
N -point IDFT, and insert cyclic prefix, and so forth, as shown.

≤ k ≤ M −

1. Express the N N transfer matrix from s 1 ( n )to y 1 ( n )intermsof

the channel coe cients c ( n ) .
2. Show that the transfer function from r ( n )to v ( n ) is a diagonal matrix
with elements C ( e j 2 πk/N ) , 0

×

≤ k ≤ N − 1 . These are the N -point DFT

coe cients of the channel.