Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 602

where

σ f,k =[ Σ f ] kk
The right-hand side of Eq. (19.37) can be further simplified by using the power
constraint (19.17). From (19.30) we have

⎡ ⎤

0 ] U f = U f Σ f 0

Σ f

⎣ ⎦ V f V f [ Σ f

U f , FF † = U f

00

0

where we have used the unitarity of V f . Taking traces we have

Tr ( FF † )=Tr U f Σ f 0

00

U f

=Tr U f U f Σ f 0

00

M− 1

=Tr Σ f = σ f,k ,

k =0

where we have used the facts that Tr ( AB )=Tr( BA )and U f U f = I . Thus
the power constraint (19.17) can be rewritten as

M− 1

p σ s

σ f,k = (19 . 38)

k =0

Now, from the AM-GM inequality we know that

σ f,k 1 /M

M− 1 M− 1

1
M

σ f,k ≤ (19 . 39)

k =0 k =0

with equality if and only if σ f,k are identical. From Eq. (19.37) we therefore
obtain
det

1
M

σ f,k M

M− 1 M− 1 M− 1

F † H † HF σ f,k σ h,k ≤ σ h,k .

≤ (19 . 40)

k =0 k =0 k =0

In view of Eq. (19.38) this finally yields

det

p 0
Mσ s

M

M− 1

F † H † HF σ h,k . ≤ (19 . 41)

k =0

Substituting into Eq. (19.24), the inequality (19.25) follows immediately.

19.3.4 Achieving the lower bound (19.25)

Since the bound (19.25) was derived using several inequalities, we can achieve the
bound by satisfying each of these inequalities with equality. We now summarize
the inequalities that have been used: