MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 401

derived from a single user stream s ( n )by parsing as in a DMT system, rather
than by blocking (Secs. 3.5 and 3.6). In this case s k ( n ) would have different
powers. This scenerio allows bit allocation to minimize transmitted power for
fixed error probabilities, as we shall explain in detail in Chap. 14. Some further
remarks for the case of arbitrary diagonal R ss can be found at the end of this
chapter (Sec. 12.5).

12.3 Problem formulation

Under the zero-forcing constraint the reconstruction error in Fig. 12.2 comes
entirely from the channel noise q ( n ) processed through G . Thus

e ( n )= s ( n ) − s ( n )= Gq ( n ) (12 . 8)

so that the error covariance is

R ee = GR qq G † = σ q GG † , (12 . 9)

where we have used the assumption R qq = σ q I . The total mean square error is
therefore 2

E mse = E [ e † ( n ) e ( n )] = Tr( R ee )= σ q Tr( GG † ) . (12 . 10)

Now, the goal is to minimize E mse under the power constraint

M− 1

E [ |x k ( n ) | 2 ]= p 0 ,

(12 . 11)

k =0

where x k ( n ) are the signals at the channel input (Fig. 12.1). We can express
the left-hand side as

| 2 ]=Tr( R xx )=Tr FR ss F † = σ s Tr FF † ,

M− 1

E [ | x k ( n )

k =0

where we have used the assumption R ss = σ s I . So the power constraint can be
rewritten as σ s Tr( FF † )= p 0 . The optimization problem of interest is therefore
as follows:

Problem 1. The ZF-MMSE problem. Find the precoder F and equalizer G to
minimize the total MSE

E mse = σ q Tr( GG † ) (12 . 12)

subject to the power constraint
σ s Tr FF † = p 0 (12 . 13)

2 A more appropriate notation would have been E zfmse where zf is a reminder of the zero-
forcing constraint, but we keep it simple here.