Bit allocation and power minimization - Signal Processing and Optimization for Transceiver Systems - page 464

which is diagonal. Thus the M columns of F are orthogonal. The same
property was also observed for the ZF-MMSE transceiver and the MMSE
transceiver described in Chaps. 12 and 13, respectively. So the optimal
system is such that the M user codes (columns of F ) can be assumed to
be orthogonal.

4. Orthonormal precoder system. The presence of Σ g does give us some flexi-
bility in choosing receiver structure. For example, suppose we choose

Σ g = Σ − 1 .

(14 . 38)

Then at the transmitter the inverses of Σ g and Σ cancel each other and
we get the equivalent optimal transceiver shown in Fig. 14.4(b). Thus
the diagonal equalizer has been moved to the receiver side, which is more
traditional. This is called an “orthonormal precoder” system, where the
term orthonormal actually means that the precoder matrix F is unitary,
that is, F † F = I M .

5. Orthonormality of precoder is not a loss of generality. Notice in particular
that the optimal system can be assumed to have a unitary precoder without
loss of generality. In Chaps. 12 and 13, where we optimized for MSE, the
precoder was shown to be orthogonal but not unitary, that is, the columns
of the precoder matrix did not have unit norm. In those chapters, the
input covariance was of the form σ s I . In this chapter, however, additional
flexibility is offered by the diagonal elements of the input covariance Λ s .
The reader might wonder whether this explains intuitively why the columns
of the optimal precoder are allowed to have unit norm because Λ s makes
up for this loss of generality. But it turns out that when optimal solution
is chosen to have unitary F , the diagonal matrix Λ s (which depends on the
specific optimum solution chosen) actually takes the trivial form Λ s = σ s I .
This will be elaborated in Sec. 14.6.3 (Ex. 14.1, to be specific).

14.6 Further properties of the optimal solutions

In this chapter we optimized the transceiver to minimize the transmitted

power under optimum bit allocation (and under the high bit rate assumption).
Observe again that the solution is not unique; in fact, there are infinitely many
solutions: given an optimal solution

{ F , G }

{ F , G } , suppose we define

G 1 = Λ − 1 G ,

F 1 = FΛ , (14 . 39)

where Λ is an arbitrary M × M diagonal matrix with diagonal elements λ k > 0 .

Then continues to be an optimal solution. This follows from the fact

that we are allowed to insert the arbitrary diagonal matrix Σ g in Fig. 14.4(a).
In fact, all solutions of the form in Fig. 14.4(a) can be obtained in this way.

{ F 1 , G 1 }