Bit allocation and power minimization - Signal Processing and Optimization for Transceiver Systems - page 473

The coding gain (14.56) is therefore given by

M− 1

AM c k
σ h,k

1
M

c k
σ h,k

k =0

GM c k
σ h,k

G = = (14 . 60)

1 /M

M − 1

c k
σ h,k

k =0

where AM and GM stand for arithmetic mean and geometric mean, respectively.
For the special case where the required BER values P e ( k ) are identical for all k ,

c k is identical for all k. In this case the coding gain reduces to

AM 1
σ h,k

G = (14 . 61)

GM 1
σ h,k

Recall here that σ h,k are the first M dominant singular values of the channel
H . For a channel with large variation in singular values, the coding gain is large
whereas for a channel with σ h,k nearly identical for all k, thecodinggainisquite
small.

14.8 Concluding remarks

In the previous chapters we minimized the mean square reconstruction error in
transceivers subject to the power constraint and possibly the zero-forcing con-
straint. In this chapter we showed how the transmitted power can be minimized
for a given set of error probabilities by optimizing the transceiver matrices and bit
allocation. For further interesting work on transceiver optimization with bit al-
location, the reader should study Pandharipande and Dasgupta [2003], Palomar
and Barbarossa [2005], Vemulapalli, Dasgupta, and Pandharipande [2006], and
Weng et al. , [2010a]. In Chap. 15 we will consider a special class of transceivers
which are restricted to have orthonormal precoders. Chapter 16 addresses the
problem of minimizing the average symbol error probability. A one-page sum-
mary of the results of this chapter is included in Appendix I at the end of the
topic.