Minimization of error probability in transceivers - Signal Processing and Optimization for Transceiver Systems - page 499

The reasoning based on majorization and Schur convexity can also be summa-
rized directly without using U in the statement as follows:

Lemma 16.2. Bound on average error probability. Given any transceiver
{ F , G }

♠

for a fixed channel and fixed transmitted power, let the mean square
errors of the components be E s,k . Then the average symbol error probability

(16.14) is lower bounded as follows:

A
√ E mse

,

P e ≥ c Q (16 . 24)

E mse = k E s,k /M. Equality is achieved when where E s,k = E mse for all k. ♦

Proof. The vector E mse [1

1 ... 1 ] is majorized by

y =[ E s, 0 E s, 1 ... E s,M− 1 ]

as shown in Lemma 21.1 (Chap. 21). Since Eq. (16.14) is Schur-convex in
y , the result follows immediately.

16.2.3 ZF transceiver with minimum error probability

We are now ready for the main result. Recall again the assumption that the
signal-to-noise ratio is large enough to satisfy the convexity condition in Eq.
(16.16). The signal and noise statistics are as in Eq. (16.1).

Theorem 16.1. Minimum error probability with zero-forcing. For a fixed
channel H and fixed transmitted power p 0 ,let

♠

E mmse be the minimum achievable
average mean square error among all zero-forcing transceivers { F , G } . Then the

minimum possible average symbol error probability is

A
√ E mmse

.

P e,min = c Q (16 . 25)

This minimum can be achieved as follows:

1. First design a ZF-MMSE transceiver by joint optimization of F

and G . This evidently has MSE equal to E mmse .
2. Insert unitary matrices U , U † as in Fig. 16.2 so that the mean square
errors E s,k are equalized.

{ F , G }

{ FU † , UG } Then the resulting transceiver achieves the minimum error proba-

bility (16.25). ♦

Proof. This follows from Lemmas 16.1 and 16.2. Lemma 16.2 says that,
for any transceiver

{ F , G } , the average symbol error probability cannot