Minimization of error probability in transceivers - Signal Processing and Optimization for Transceiver Systems - page 502

q ( n )
0

s ( n )
0

s ( n )
0
s ( n )
1

s ( n )
1

F H

G

s ( n )
M − 1

q ( n )
P

s ( n )
M − 1

− 1

channel
noise

equalizer

precoder channel

Figure 16.3 . A linear transceiver with precoder F and equalizer G .

q ( n )
0

s ( n )
0
s ( n )
1

s ( n )
0 , br
s ( n )
1 , br

γ 0
γ 1

s ( n )
0
s ( n )
1

F H G

s ( n )
M − 1

s ( n )
M −1 , br

q ( n )
P − 1

s ( n )
M − 1

γ M − 1
bias-removing
multipliers

channel
noise

equalizer

precoder channel

Figure 16.4 . A linear transceiver with precoder F , equalizer G , and conceptual bias-
removing multipliers γ k .

16.3.2 Error probability after bias removal from an MMSE estimate

For the MMSE estimate, the reconstruction errors before and after bias removal
are, respectively,

e k = s k − s k and e k,br = s k,br − s k . (16 . 34)

Here the subscript “ br ” is a reminder for “bias-removed.” Denote the corre-
sponding mean square errors as

E k and E k,br so that the signal-to-reconstruction