Optimization of cyclic prefix transceivers - Signal Processing and Optimization for Transceiver Systems - page 540

q ( n )

s ( n )

s ( n )

W
M

W
M

Λ a

Λ b

H

(a)

diagonal normalized IDFT circulant channel diagonal

normalized DFT

q ( n )

x ( n )

s ( n ) x ( n ) s ( n )

W
M

W
M

Λ a

Λ b

U H U

(b)

circulant

unitary diagonal diagonal unitary

Figure 17.7 . (a) The MMSE cyclic-prefix transceiver, and (b) the modified version
with unitary matrices U and U † inserted.

Lemma 17.1. Diagonal structure of error covariance . For the MMSE cyclic-
prefix system shown in Fig. 17.7(a) (with or without zero forcing), the error
covariance matrix is diagonal.

♠

♦

Proof. The reconstructed signal in Fig. 17.7(a) is given by

s ( n )= Λ b WHW † Λ a
M

s ( n )+ Λ b W

√ M q ( n ) .

Since H is circulant, WHW † is diagonal (Appendix D). So

s ( n )= Λ 0 s ( n )+ Λ 1 Wq ( n )

for some diagonal matrices Λ 0 and Λ 1 . The reconstruction error is therefore

e ( n )= s ( n ) − s ( n )=( Λ 0 − I ) s ( n )+ Λ 1 Wq ( n )= Λ 2 s ( n )+ Λ 1 Wq ( n ) ,

where Λ 2 is also diagonal. Since s ( n )and q ( n ) are zero-mean uncorrelated
processes with covariances σ s I and σ q I , respectively, the covariance of the
error is

R ee = σ s Λ 2 Λ 2 + σ q Λ 1 WW † Λ 1 = σ s Λ 2 Λ 2 + Mσ q Λ 1 Λ 1 ,

which is diagonal indeed.

As explained in detail in Sec. 16.3, the MMSE system without zero forcing
creates a bias in the reconstruction error. This bias should be removed before the