MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 417

precoder F equalizer G

q ( n )
0

3 /2

1/ c

c

1/ 2

s ( n )
0

s ( n )
0

1/2
−1/2

1/ 2

q ( n )
1

H

1/ 2

4 c

4/ c

s ( n )
1

s ( n )
1

− 1/ 2

3 /2

channel

V h U h

Figure 12.7 . Example 12.2. The precoder and equalizer for the optimal (ZF-
MMSE) transceiver for the channel H .

The SVD of this channel is given by

11
1

1 0
0( / 16)

√ 31

1
√ 2

1
2

− 1 √ 3

H = × × .

− 1

U h Σ h

V h

This determines the optimal linear transceiver completely, as shown in Fig.
12.7, except for the constant c which is determined from the power con-
straint. More extensive examples will be presented in Chaps. 17 and 18
where we compare several optimal transceivers on the basis of symbol error
probability.

12.5 Optimizing the noise-to-signal ratio

In this chapter we have so far assumed a simple signal covariance matrix R ss =
σ s I and minimized the trace of R ee = σ q GG † subject to the power constraint
p 0 = σ s Tr ( FF † ) and the zero-forcing constraint GHF = I .If R ss is a more
general diagonal matrix Λ s of the form (12.6) it makes more sense to minimize
the sum of error-to-signal ratios rather than the sum of errors. For, it is the error-
to-signal ratios at the detectors that determine the performance of the receiver.
Thus, it is more appropriate to minimize the trace of the matrix defined as

R new = σ q Λ − 1 / 2

GG † Λ − 1 / 2
s

.

s

With R ss = Λ s the power constraint is modified to

p 0 = σ s Tr ( FΛ s F † ) .

(12 . 68)

Define

G 1 = Λ − 1 / 2 F 1 = FΛ 1 / 2

G and . (12 . 69)

s s