Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 492

transceiver required a special result due to Witsenhausen. Notice that this
result is not required in the derivation for the case where the precoder is
restricted a priori to be unitary.

2. The integer K . In Chap. 13, where we solved the MMSE problem without
imposing the unitary condition on the precoder, the precoder had rank
K M, where K is an appropriately chosen integer. In this section, with

F † F = I M the precoder has rank M and there is no K involved.

≤

3. Laziness of the precoder. In all cases where the precoder is restricted to
be unitary, the equalization (inversion of channel singular values 1 /σ h,k )
actually takes place at the receiver. For the P

M precoder the transmitter
helps in the diagonalization of the channel. For the case where the precoder
is M

×

M the transmitter need not even do this. It can just be the lazy
precoder! This example arises in the zero-padding precoder (Sec. 7.2),
where the precoder is a square matrix. Restricting the precoder to be
unitary is as good as restricting it to be identity. In Chap. 16, where we
minimize symbol error probabilities (rather than just minimize the MSE),
we will show that the best unitary precoder cannot in general be lazy.

×

4. Power control . For power control we can insert α I and α − 1 I as in Fig. 15.2.
By using the same approach that was used to derive Eq. (15.34) we can
verify that the minimized mean square errors for the case of rectangular
precoders are still given by Eq. (15.34).

15.4 Concluding remarks

In this chapter we minimized the mean square error in transceivers under the
constraint that the precoder be orthonormal. The zero-forcing transceiver and
the non-ZF transceiver were both considered. In both cases we found that, if the
precoder is square ( P = M ), then the unitary precoder can be chosen arbitrarily,
with no loss of optimality. For P>M , the precoder should be chosen based on
the SVD of the channel matrix. For all cases, we also derived the expression for
the minimized MSE. In Chap. 16 we turn our attention to the case where the
symbol error rate, rather than the MSE, will be the focus of our attention.