MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 407

1. The optimal F has to satisfy the stationarity condition (12.32), where H
is the channel matrix.

2. Consequently both F † H † HF and F † F are diagonal matrices.
3. As a result, ( U f H † HU f ) M can be assumed to be diagonal as well, where
U f is the unitary matrix appearing in Eq. (12.24).

♦

12.4 Solution to the ZF-MMSE problem

With F restricted to be of the form (12.24), where U f is unitary and Σ f diagonal,
the objective function (12.22) becomes

Tr

− 1

σ q F † H † HF

E mse =

0 ] U f H † HU f Σ f

− 1

Tr [ Σ f σ q

=

0

Tr Σ f ( U f H † HU f ) M Σ f − 1 , σ q

=

where the notation ( A ) M denotes the M M leading principal submatrix of A

(Sec. B.2.1, Appendix B). The power constraint (12.23) can also be rewritten
as

×

σ s Tr( FF † ) p 0 =

U f Σ f
0

[ Σ f

Tr 0 ] U f σ s

=

U f U f Σ f

[ Σ f

σ s Tr 0 ]

=

0

σ s Tr( Σ f ) ,

=

where we have used the trace identity Tr( AB )= Tr( BA )togetthethird
equality, and the unitarity of U f to get the last equality. Thus the optimization
problem described by Eqs. (12.22) and (12.23) reduces to the following:

Problem 3. Find a P × P unitary matrix U f and an M × M diagonal matrix Σ f
with positive diagonal elements σ f,k > 0 , to minimize

mse = σ q Tr Σ f ( U f H † HU f ) M Σ f − 1 E (12 . 34)

subject to the constraint

σ s Tr( Σ f )= p 0 ,

(12 . 35)

where the channel H and the quantities σ q , σ s ,and p 0 are fixed.

It turns out that the solution to the problem automatically satisfies the con-
straint σ f,k > 0, so we shall ignore it for simplicity. The goal of optimization