Bit allocation and power minimization - Signal Processing and Optimization for Transceiver Systems - page 470

all k because they feed directly into the channel eigenmodes (Fig. 14.4(b),

equivalently Fig. 14.5(b)).

14.6.4 Comparison with MMSE transceivers

Figure 14.6(a) shows the general form of the optimal transceivers derived in
Chaps. 12 and 13 for minimizing the mean square error under a power con-
straint. The diagonal matrices Σ f and Σ g depend on whether there is zero
forcing or not. Compare this with the optimal transceiver configuration in Fig.
14.6(b), which minimizes the transmitted power under optimal bit allocation.
The MMSE transceiver has input covariance σ s I by assumption, and the power-
minimizing transceiver also has input covariance σ s I (as shown in Ex. 14.1).
The MMSE transceiver has diagonal matrix Σ f before the unitary matrix V h .
This diagonal matrix performs part of the equalization, and the matrix Σ g at
the receiver performs part of the equalization. We cannot arbitrarily trade off
the equalizers at the transmitter and receiver of the MMSE system. This is
unlike the situation in the power minimizer of Fig. 14.4(a). Since the precoder
of the MMSE transceiver has the form
F = V h Σ f
0

,

(14 . 51)

we have

F † F = Σ f . (14 . 52)
That is, the columns of the precoder are orthogonal, though not orthonormal
(i.e., the columns do not have unit norm). Compare this with the power mini-
mizer configuration shown in Fig. 14.6(b), which has the precoder
F = V h I M
0

.

(14 . 53)

This precoder is orthonormal (or unitary) because it satisfies 4

F † F = I M . (14 . 54)

Thus, in the power-minimization problem the optimal solution can be assumed
to have an orthonormal precoder without loss of generality. The same is not
true for the MMSE problem.
For further clarity, Fig. 14.7 shows the diagonal equivalent structures for the
two systems. Here we have used the fact that

U h HV h = Σ h . (14 . 55)

The first M dominant singular values σ h,k of the channel are shown in the
figure. We see that, in the MMSE system, the optimal multipliers σ f,k at the
transmitter determine the power allocation into the different eigenmodes σ h,k
of the channel. This power allocation is a crucial feature of the MMSE system,
since the multipliers σ f,k and σ g,k cannot be traded arbitrarily in Fig. 14.7(a).

4 Note that FF † = I P unless P = M .