Channel capacity - Signal Processing and Optimization for Transceiver Systems - page 237

From Eq. (6.66) it therefore follows that

( y )=lndet( πe C yy )=lndet πe ( HC xx H † + σ q I ) H (6 . 84)

and, furthermore,

( q )=lndet( πeσ q I ) .

H (6 . 85)

Substituting into Eq. (6.81), the result (6.80) follows.

The problem of maximizing the mutual information (6.80) can be reformulated
by considering Fig. 6.7. Here the vector s is restricted to be a zero-mean circu-
larly symmetric complex Gaussian with covariance matrix C ss = I . The vector
x , which is the output of the linear transformation F , is also circularly symmet-
ric complex Gaussian for any F (Lemma 6.1, Sec. 6.6). Since the covariance of
x is

C xx = FF † ,

(6 . 86)

we can realize any covariance matrix by appropriate choice of F . So the problem
of maximizing the mutual information

( x , y ) between x and y can be solved
by fixing s to be a zero-mean circularly symmetric complex Gaussian with co-
variance I , and optimizing F . Since the mutual information can now be written
as

I

( x ; y )=logdet

σ q HFF † H † ,

1

I I + (6 . 87)

we only have to maximize this by optimizing F subject to the power constraint
which now becomes

Tr ( FF † )= p 0 . (6 . 88)
The same optimization problem also arises in a different context, namely that
of optimizing the precoder in a decision feedback transceiver without the zero-
forcing constraint (Sec. 19.4).

6.7.2 Solution to the maximum mutual information problem

At this point it is convenient to represent the channel H and the precoder F
using their singular value decompositions (Appendix C):

F = U f Σ f V f H = U h Σ h V h , and (6 . 89)

where U f , V f , U h , and V h are unitary matrices, and Σ f and Σ h are diagonal
matrices with non-negative diagonal elements (singular values of F and H ). Note
that H and Σ h are rectangular matrices; all other matrices are square. Since
FF † = U f Σ f U f the power constraint (6.88) becomes

P− 1

σ f,k = p 0 , (6 . 90)

k =0