Optimization of cyclic prefix transceivers - Signal Processing and Optimization for Transceiver Systems - page 535

1

α Σ − 1 / 2

Λ − 1
θ

= (using Eq. (17.18))

h

Substituting from Eq. (17.10) we therefore have

1
α Λ − 1 / 2

Λ − 1 / 2
θ

Λ g = . (17 . 20)

c

Now, the diagonal matrix Λ − 1 / θ is unitary because the diagonal elements
have the form e −jθ k / 2 . Thus, if we replace Σ f and Λ g in Fig. 17.3(a) as
follows:

Σ f → Σ f × Λ − 1 / 2
θ

Λ g → Λ g × Λ 1 / 2
θ

and ,

the system performance is unaffected, that is, neither the transmitted power
nor the MSE is changed. From Eq. (17.20) we have

1
α Λ − 1 / 2

Λ g × Λ 1 / 2
θ

= ,

c

whereas from Eqs. (17.17) and (17.10) we have

Σ f × Λ − 1 / 2
θ

= α Λ − 1 / 2
c

.

This shows that the optimal system of Fig. 17.3(a) can be replaced with the
system shown in Fig. 17.4. Equation (17.19) is obtained by replacing σ h,k
with

| C [ k ] | in Eq. (12.66).

17.3.2 Pure-MMSE transceiver

For the MMSE transceiver without zero-forcing constraint the diagonal matrices
Σ f and Σ g were derived in Sec. 13.6. These are reproduced below with the
substitution P = J = M. First, the diagonal matrix Σ f at the transmitter is

⎡
⎣

⎤
⎦

√ q 00

0 ... 0

√ q 11

0 ... 0

Σ f = σ q
σ s

. (17 . 21)

. . .

. . .

√ q M− 1 ,M− 1

0 0 ...

Here

⎧
⎨

D 1

σ h,k

σ h,k −

0 ≤ k ≤ K − 1

q kk = (17 . 22)

⎩

0 otherwise,

where

K− 1

p σ q 1

σ h,k

+

k =0

D = (17 . 23)

K− 1

1
σ h,k

k =0