Optimization of cyclic prefix transceivers - Signal Processing and Optimization for Transceiver Systems - page 541

threshold detector is used at the receiver to identify the transmitted symbols.
This was explained in detail in Sec. 16.3.1. The following result implicitly
assumes that in the non zero-forcing case the bias is removed.

Theorem 17.2. Minimum error probability in cyclic-prefix systems .The
MMSE cyclic-prefix system shown in Fig. 17.7(b) has minimum average error
probability (with or without zero forcing) for any unitary matrix U satisfying
the property

♠

1
√ M | U km | = (17 . 34)

for all k, m. In particular, therefore, the normalized IDFT matrix

U = W † √ M (17 . 35)

also provides a solution with minimum error probability. ♦

Proof. Since U † is unitary the covariance of x ( n ) is the same as that of
s ( n )(namely σ s I ). It therefore follows from application of Lemma 17.1

x ( n )

that the error − x ( n ) has a diagonal covariance matrix. Next, since

s ( n ) x ( n )

− s ( n )= U [ − x ( n )] it follows that the covariances of the errors

s ( n ) − s ( n )and x ( n ) − x ( n ) are related as

R ee , s = UR ee , x U † .

Since R ee , x is diagonal according to Lemma 17.1, it follows that if we choose
U to satisfy Eq. (17.34) then R ee , s has identical diagonal elements (Sec.
21.5.1.B). This ensures that the system in Fig. 17.7(b) has minimum BER
(for reasons explained in detail in Chap. 16). Since W haselementswith
unit magnitude, the last part of the theorem follows readily.

The unitary property of U ensures that the MMSE property continues to hold
for any choice of U and furthermore the transmitted power remains unchanged.
Thus t he minimum-BER system continues to be an MMSE system. With U =

W † / √ M as in the theorem statement, the minimum-BER cyclic-prefix system
takes the specific form shown in Fig. 17.8. With

E mmse denoting the minimized

MSE, the average MSE per symbol is E ave = E mmse /M. For the zero-forcing

system, with E mmse given by Eq. (17.19), the minimized average symbol error

probability is given by

A
√ E ave

(ZF-MMSE case).

P e,min = c Q (17 . 36)

See Sec. 16.2. Here the constants c and A depend on the constellation used. For
example, in a b -bit PAM system,

2 −b )and A = 3 σ s
2 2 b

c =2(1 −

1 .

(17 . 37)

−