Optimal transceivers for diagonal channels - Signal Processing and Optimization for Transceiver Systems - page 372

the solutions form an integral part of the solutions for nondiagonal channels.
Since F , H , and G are memoryless, the time-argument ( n ) on the signals will be
dropped whenever convenient.

Chapter outline

Section 11.2 considers the case where the total reconstruction error is minimized
(in the mean square sense) under the zero-forcing (ZF) constraint β k =1 / [ H k α k ] .
This is called the ZF-MMSE transceiver. In Sec. 11.3 the ZF condition is
removed and the multipliers α k and β k are jointly optimized, resulting in the
MMSE transceiver, also called the pure-MMSE transceiver (to differentiate from
ZF-MMSE). Both of these problems are solved under a fixed power constraint.
In Sec. 11.4 we consider a totally different problem, namely that of maximizing
the capacity of the set of parallel independent channels under a power constraint.
Even for a diagonal channel such as the one mentioned above, it is not in
general true that the precoder and equalizer matrices can be restricted to be
diagonal matrices without loss of generality. It depends on what objective func-
tion we want to optimize. For the ZF-MMSE and the pure-MMSE problems it
turns out that the optimal matrices are diagonal as seen from the more general
analysis of Chaps. 12 and 13. For the case where the objective function to be
minimized is the average symbol error rate, the best precoder and equalizer for
diagonal H are not diagonal matrices, as we shall see in Sec. 11.5 of this chapter.

11.2 Minimizing MSE under the ZF constraint

We now consider the system shown in Fig. 11.3 where

1
H k α k

β k = (11 . 4)

This choice of β k makes it a zero-forcing (ZF) system. In this case the recon-
struction error is entirely due to the noise sources q k ( n ) amplified through 1 /H k
and 1 /α k . We assume that the signals s k ( n ) and noise sources q k ( n ) are zero-
mean WSS processes. To state the statistical assumptions concisely, let s ( n )
and q ( n ) be column vectors whose components are s k ( n )and q k ( n ). Then the
covariance matrices are assumed to be as follows:

R ss = σ s I , R qq =diag[ σ q 0 ,σ q 1 ,...,σ q M− 1 ] ,

and R sq = 0 . (11 . 5)

Note in particular that the signal variance is assumed identical for all k ,thatis,

σ s = E | 2 .

| s k ( n ) (11 . 6)

With these assumptions, the reconstruction error for the k th signal has variance

σ q k

E k = (11 . 7)

α k | 2 | H k | 2

|