Channel capacity - Signal Processing and Optimization for Transceiver Systems - page 238

q

y

P P J

s x

F H

channel

Figure 6.7 . A MIMO channel H with precoder F .

and furthermore

( x ; y )=l gd t

σ q HFF † H †

1

I I +

log det

σ q U h Σ h V h U f Σ f U f V h Σ h U h

1

= I +

log det U h

σ q Σ h V h U f Σ f U f V h Σ h

1

U h = I +

log det

σ q Σ h V h U f Σ f U f V h Σ h

1

= I +

log det

σ q Σ h UΣ f U † Σ h ,

1

= I +

where U = V h U f .Since V h is invertible and U f is free to choose, we can
regard U as a unitary matrix that can be freely chosen in the maximization
process. So the goal is to choose the P P unitary matrix U and the diagonal

matrix Σ f with diagonal elements σ f,k ≥

×

0 such that the preceding expression

is maximized. Defining the P × P matrix

P = UΣ f U † , (6 . 91)

we have

( x ; y )=logdet

σ q Σ h PΣ h .

1

I I + (6 . 92)

Since the matrix within the brackets is positive definite, we have, from Hadamard's
inequality(AppendixB,Sec. B.6)

det

σ q Σ h PΣ h

1+ 1
σ q

σ h,k P kk

P− 1

1

I + ≤ (6 . 93)

k =0

with equality if and only if the matrix is diagonal, that is, if and only if P is
diagonal. That is, given any positive semidefinite P , if we replace it with a
diagonal matrix having the diagonal elements P kk , then the above determinant
can only increase (while the power constraint continues to be satisfied). We can