Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 477

Transceivers with
orthonormal precoders

15.1 Introduction

In the preceding three chapters we addressed three different optimization prob-
lems for transceivers. Namely, MMSE transceivers under the zero-forcing (ZF)
constraint (Chap. 12), MMSE transceivers without the ZF constraint (Chap.
13), and transceivers which minimize transmitted power under optimal bit allo-
cation and the ZF constraint (Chap. 14). We found that the optimal precoder
in each case turns out to be orthogonal, that is,

F † F = Σ f (15 . 1)

for some diagonal Σ f . Furthermore, for the transceiver which minimizes power
under bit allocation, we found that the precoder can be assumed to be orthonor-
mal without loss of generality:

F † F = I M . (15 . 2)

Thus, for power minimization, orthonormality is not a loss of generality, but for
the MMSE problem, orthonormality is a loss of generality. The reader should
review Sec. 14.6.4 for a detailed comparison of these systems.
In this chapter we consider the MMSE problem with the precoders restricted
to satisfy orthonormality (15.2). We call these orthonormal precoders. We also
say that the P

M matrix F is “unitary” even though it should be remembered
that when P>M, FF † = I P .

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