Channel capacity - Signal Processing and Optimization for Transceiver Systems - page 239

therefore carry out the maximization of ( x ; y ) under the constraint that P be

diagonal. That is, we can choose the unitary matrix U = I , or equivalently

I

U f = V h (6 . 94)

so that

1+ 1
σ q

σ h,k P kk = log 1+ 1

σ q

σ h,k σ f,k .

P− 1 P− 1

I ( x ; y )=log

k =0 k =0

For a fixed set of numbers σ h,k /σ q , the set of numbers { σ f,k } should therefore

be optimized to maximize

log 1+ 1
σ q

σ h,k σ f,k

P− 1

I ( x ; y )= (6 . 95)

k =0

subject to the power constraint

P− 1

σ f,k = p 0 .

(6 . 96)

k =0

This is exactly the problem we solve in Chap. 22 (Sec. 22.3) when we demon-
strate the applications of the KKT technique for constrained optimization. We
show there that the optimum solution is a water-filling solution, taking the form

⎧
⎨

σ q
σ h,k

1
λ −

if this is non-negative

σ f,k =

(6 . 97)

⎩

0 otherwise,

where λ depends on the total available power p 0 .

Discussion

1. The above is precisely also the solution we shall obtain in Sec. 19.4 for
the optimum precoder when the receiver uses an optimum DFE equalizer
without zero forcing.

2. Note that the matrix V f in the optimal F can be taken to be identity.
Only U f and Σ f matter as far as maximizing the mutual information is
concerned.

3. The maximized mutual information (6.95) depends only on channel singu-
lar values and transmitted power, and not on the matrices U h and V h .

4. Note in particular that if the channel is diagonal, that is

H = Σ h ,

so that U h = I and V h = I , the optimal precoder F can also be taken to
be diagonal.