Bit allocation and power minimization - Signal Processing and Optimization for Transceiver Systems - page 469

14.6.3.B User power allocation

Equation (14.45) can be rewritten as

P k = σ g,k σ h,k 2 D , (14 . 47)

where we have substituted from Eq. (14.42). This quantity is in general depen-
dent on k . Thus the powers allocated to different signals s k ( n ) are in general
different. But this allocation depends on the specific configuration chosen for the
optimal solution. For example, it is not the same for Fig. 14.4(a) and 14.4(b).

Example 14.1: User power allocation with orthonormal precoding

Consider the specific optimal transceiver configuration where the free pa-
rameter σ g,k is chosen as σ g,k =1 /σ h,k . This corresponds to the optimal
transceiver configuration shown in Fig. 14.4(b) (equivalently Fig. 14.5(b)),
which has all equalization taking place at the receiver.

In this case the

precoder is orthonormal ,thatis,

F † F = I M , (14 . 48)

because

F = V h I M
0

.

Since σ g,k =1 /σ h,k we see from Eq. (14.47) that

P k =2 D = constant for all k (14 . 49)

in this case. Thus, when the free parameters in the optimal transceiver solu-
tion are chosen such that the precoder is orthonormal, the input covariance
matrix which satisfies the QoS requirements has the form

Λ s = α I (14 . 50)

for constant α. The equivalent diagonal structure of Fig. 14.5(b) gives fur-
ther insight as to why the user powers are identical. Since the powers fed
into all the eigenmodes are identical for any optimal solution (Theorem
14.3), and since s k ( n ) feeds directly into the k th eigenmode σ h,k when the
precoder is orthonormal, all the user powers P k have to be identical.

Summarizing this section, we see that the only nontrivial thing that is happening
in the optimal transceiver is optimal bit allocation. There is no eigenmode power
allocation going into the channel. The largest eigenmode σ h, 0 and the smallest
eigenmode σ h,M− 1 receive the same power no matter how different they are (they
are nonzero because the channel is assumed to have rank M ). The user power
allocation changes with the specific configuration chosen for the nonunique

optimal solution. In particular, when the optimal solution is chosen such that
the precoder is orthonormal ( F † F

{ P k }

= I ), the user powers P k are identical for