MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 423

q ( n )

1

s ( n )

P P

s ( n )

M M

F 1 G 1

H

(a)

q ( n )

1

s ( n ) M M

s ( n )

F 1

U h G 1

V h Σ h

(b)

call this F

call this G

q ( n )

s ( n )

s ( n )

M P P

M

F

Σ h

G

(c)

diagonal channel

Figure 12.10 . (a) Transceiver with channel H , (b) SVD representation of the channel,
and (c) equivalent transceiver with diagonal channel.

12.B.2 ZF-MMSE solution for the diagonal channel

For the system shown in Fig. 12.10(c) consider again the ZF-MMSE problem
discussed in Sec. 12.3. We know that the problem can be reduced to the mini-
mization of Eq. (12.22). That is, find the precoder F such that
E mse = σ q Tr

− 1

F † Σ h F

(12 . 82)

is minimized. The optimal F cannot be assumed to be diagonal because we have
not yet proved this to be the case. However, we know from Sec. 12.3.2 that F
canalwaysbeassumedtobeoftheform
F = U f Σ f
0

,

(12 . 83)

where U f is P × P unitary and Σ f is M × M diagonal. Thus

σ q Tr

− 1

F † Σ h F

E mse =

0 ] U f Σ h U f Σ f

− 1

σ q Tr [ Σ f

=

0