Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 478

Scope and outline In Sec. 15.2 we consider the MMSE problem for the case of
orthonormal precoders restricted to be square matrices. This is generalized in
Sec. 15.3 for the case of rectangular matrix precoders. While orthonormality is
a loss of generality it offers some simplifications in the design of the preocoder,
as we shall see. One example is the case of OFDM systems which use the
cyclic prefix. In this case the optimal unitary precoder is independent of the
channel as we shall show. Thus the channel state information for equalization
is required only at the receiver in these cases. The main results of the chapter
are summarized in Theorems 15.1, 15.2, and 15.3.

Assumptions and reminders

1. Covariances . We assume the signal and noise covariances to be

R ss = σ s I M , R qq = σ q I J . (15 . 3)

2. Channel SVD . The channel H is a J × P matrix, and its SVD will be

written in the form

V h H = U h

J×J

Σ h
J×P

, (15 . 4)

P×P

where U h and V h are unitary.

3. Power constraint .Since F † F = I M we see that the transmitted power is

p 0 = σ s Tr ( FF † )= σ s Tr ( F † F )= Mσ s .

(15 . 5)

We assume that p 0 is fixed during all optimization.

15.2 Orthonormal precoders restricted to be square

We shall first consider the case where P = M , that is, the precoder is a square
matrix with size M × M. So we have a non-redundant precoder and a J × M

channel H .

15.2.1 ZF-MMSE transceivers

First we consider the case where the zero-forcing constraint GHF = I is in place.
(This automatically assumes the channel has rank M .) Recall from Sec. 12.3.1
that under the ZF constraint the MSE is

E mse = σ q Tr ( F † H † HF ) − 1 . (15 . 6)

M unitary, we have F † F = FF † = I , and the preceding equation
can be simplified to
E mse = σ q Tr

Since F is M ×

= σ q Tr

FF † ( H † H ) − 1 = σ q Tr ( H † H ) − 1 , F † ( H † H ) − 1 F