Optimization with equality and inequality constraints - Signal Processing and Optimization for Transceiver Systems - page 745

where A =( p 0 + K− 1
=0

σ q ) / K− 1 A ,andthe

power is taken to be zero for larger σ q k . Note that ( k )firstincreasesandthen
decreases with σ q k , as shown in Fig. 22.5. Since the derivative

σ q . This is positive for σ q k ≤

=0

∂ ( k )
∂σ q k

= A − 2 σ q k

is zero for σ q k = A/ 2 , the maximum value A 2 / 4 occurs at σ q k = A/ 2 . Thus,
with

σ q 0 ≤ σ q 1 ≤ ... ≤ σ q K− 1

we see that the power need not increase monotonically with k. Typically it
increases first and then decreases, as shown in the example of Fig. 22.6.

2

A / 4

ε ( k )

σ q k

0

A
Figure 22.5 . Dependence of allocated power on noise variance (MMSE design).

A /2

0.3

0.2

0.1

0

0 5 10

channel number k

Figure 22.6 . Example of allocated power as a function of channel number (MMSE
design). Here N =12 ,p 0 =3 , and A =1 . 1134 , so that A 2 / 4=0 . 3099 .