Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 612

where μ k are the eigenvalues of ( U f H † HU f ) M .Deno ingthe M dominant
singular values of the channel as

σ 0 ,h ≥ σ 1 ,h ≥ ...≥ σ h,M− 1 (19 . 84)

we can show, as we did in Sec. 12.4.2, that

σ h, 0 ≥ σ h, 1 ≥ σ h,M− 1 ≥ μ 0 , μ 1 , μ M− 1 . (19 . 85)

Thus,

φ =det

σ q Σ f ( U f H † HU f ) M

M− 1

I + σ s (1 + σ s

σ q

σ f,k σ h,k )

≤ (19 . 86)

k =0

with equality achieved whenever U f is chosen such that U f H † HU f is diago-
nalized, with the top M diagonal elements equal to the M dominant values of
σ h,k . This is readily achieved by choosing

U f = V h , (19 . 87)

where V h is the unitary matrix appearing in the channel SVD (Eq. (19.31)).
Summarizing, we have proved that

1+ σ s
σ q

σ f,k σ h,k − 1 /M

M− 1

Mσ s E mse ≥ . (19 . 88)

k =0

The bound in Eq. (19.88) depends on the elements σ f,k ≥ 0 of the precoder

matrix. These can be optimized subject to the power constraint to minimize
the right-hand side of Eq. (19.88) as shown next. After this we will show how
the minimized bound can actually be achieved by choosing the remaining free
parameters (feedback matrix B , and the unitary part V f of the precoder).

19.4.4 Minimizing the bound (19.88)

For simplicity of notation let

k = σ s
σ q

α k = σ f,k ≥ σ h,k .

0 , (19 . 89)

But the power constraint can be rewritten as Eq. (19.38), that is,

M− 1

p σ s

σ f,k =

k =0

To minimize the right-hand side of Eq. (19.88) we have to maximize

M− 1

(1 + β k α k ) (19 . 90)

k =0