Minimization of error probability in transceivers - Signal Processing and Optimization for Transceiver Systems - page 506

In Appendix 16.B at the end of the chapter we show that

A 1

1
σ s

E k )=

f ( Q

E k −

(16 . 54)

is a convex function of E k (MSE before bias removal), as long as the error E k is

smaller than a threshold E t . In fact, for QPSK constellations, we will see that

this is true 5 for all σ s . Thus the sum (16.53) is Schur-convex (Chap. 21)

E k ≤

in the vector

E M− 1 ] , (16 . 55)
which is the vector of mean square errors before bias removal . Now consider Fig.
16.5, which shows the transceiver

y =[ E 0 E 1 ...

, and the modified transceiver which
has the unitary matrix U and its inverse U † inserted. Recall from Sec. 16.2.1
that this insertion of U and U † does not affect either the transmitted power or
the average mean square error

{ F , G }

M− 1

1
M

E mse = E k . (16 . 56)

k =0

Since P br in Eq. (16.53) is a Schur-convex function of the vector y , it follows

that the best unitary U which minimizes P br is such that the mean square errors

E k are equalized. The proof is precisely as in Sec. 16.2.2. Thus, for any precoder
F and channel H , the best U is such that

E 0 = E 1 = ... = E M− 1 = E mse , (16 . 57)

where E mse is the average MSE with or without U . In Sec. 16.2.2 we explained

how such an U can be designed (see the remarks after Eq. (16.19)). With U so
chosen, the average error probability is clearly given by

A 1 .

1
σ s

P e,min = c Q

E mse −

(16 . 58)

From the preceding discussion it follows that if a transceiver { F o , G o } minimizes

the average error probability then the following should be true:

1. G o is an MMSE equalizer for the precoder F o . As a result the expression

for error probability is as in Eq. (16.53).

2. With no loss of generality, the mean square errors E k can be assumed to be

equal, as in Eq. (16.55), for otherwise we can append U and U † to achieve
this without increasing the average error probability. So the expression for
the average error probability actually reduces to the simpler form (16.58).

3. Since the function (16.58) decreases with decreasing E mse , the optimal

precoder equalizer pair { F o , G o } which minimizes the average error prob-

ability should be such that E mse itself is minimized!

5 Since the mean square error of an MMSE estimate of s k ( n ) cannot exceed the mean square
value of s k ( n ), the condition E k ≤ σ s is always satisfied!