Optimization of cyclic prefix transceivers - Signal Processing and Optimization for Transceiver Systems - page 539

Λ g =

⎡
⎣

⎤
⎦

√ q 00 | C [0]

e −jθ 0
1+ q 00 | C [0]

|

... 0

| 2

. .

. . .

σ s
σ q ×

. (17 . 31)

√ q M− 1 ,M− 1 | C [ M

e −j

θ M− 1

− 1] |

0 ...

1+ q M− 1 ,M− 1 | C [ M − 1] | 2

Here the angles θ k are the phases of the reordered channel DFT coe cients, that
is, C [ k ]= | C [ k ] e j

θ k .

|

17.4 CP systems with minimum error probability

The cyclic-prefix transceivers with MMSE property are summarized in Fig.
17.7(a) with fresh notations for the diagonal matrices at the transmitter and
receiver. These diagonal matrices are now indicated as Λ a and Λ b .Thematri-
ces are chosen as follows: for the ZF-MMSE system choose

Λ a = α Λ − 1 / 2 Λ b = α − 1 Λ − 1 / 2

, (ZF-MMSE), (17 . 32)

c c

where α is chosen to satisfy the power constraint. And for the pure-MMSE

system choose

Λ a = Σ p , Λ b = Λ p (pure-MMSE), (17 . 33)

where Σ p and Λ p are computed as described in Sec. 17.3.2. In Chap. 16 we
showed that systems with minimum symbol error probability can be obtained
from systems with minimum MSE (with the power constraint and possibly the
ZF constraint) by using a unitary matrix U at the receiver and its inverse U †
at the transmitter. 2 With such matrices inserted the MMSE system of Fig.
17.7(a) takes the form shown in Fig. 17.7(b). The matrix U does not change
the transmitted power or the MSE. The purpose of U is to equalize the mean
square errors of the M component signals at its output (review Sec. 16.2.2 here).

17.4.1 Choice of the unitary matrix U

We begin by proving an important result on error covariances. For this, first
recall that the MMSE cyclic-prefix system shown in Fig. 17.7(a) assumes that
the signal s ( n ) and noise q ( n ) are zero-mean uncorrelated processes with covari-
ances σ s I and σ q I , respectively. The components q k ( n ) of the noise q ( n )are
assumed to be Gaussian with variance σ q .

2 Such systems are also called minimum BER systems, since the bit error rate is directly
related to the symbol error rate for Gray-coded systems (Sec. 2.3.3).