Minimization of error probability in transceivers - Signal Processing and Optimization for Transceiver Systems - page 518

a 2

E

E 0

0

E t

Figure 16.9 . Plot of the left-hand side of Eq. (16.91).

The threshold E t can be calculated from Eq. (16.91). For this we simply calculate

the two roots of the quadratic equation

E 2 −

+ a 2 =0 . The roots are positive
and real (these are the zero crossings in Fig. 16.9), and the smaller root is equal
to

a 1 E

E t . To give a numerical example, consider a QAM constellation. We can

compute c and A from Eq. (16.6), evaluate a 1 and a 2 , and then compute the

smaller root E t . Ths yields

3

2

2 b

E t
σ s

3 × × 2 b 3

=

1) −

0 . 5 −

8(2 b − 4(2 b − 1) 2 b − 1

This ratio is shown in the table below for various choices of b (bits per symbol).
This is the threshold error-to-signal ratio below which convexity holds. The table
also shows the signal-to-error ratio σ s /E t in dB, and the symbol error probability
if the detector were to have this SER at its input. Thus, if the symbol error
probabilities exceed the numbers shown, then the convexity assumption fails. As
long as the error probabilities in practice are smaller than these (rather large)
values, the convexity assumption is therefore valid.

E t /σ s σ s /

Number of bits b E t P e

(QAM) threshold in dB per symbol

2 0.5000 03.0 0.317

4 0.0683 11.6 0.148

6 0.0160 18.0 0.152

8 0.0039 24.1 0.158

For the case of PAM constellations a similar table can be constructed and is
shown below. The conclusions are similar. Note that b -bit PAM and 2 b -bit QAM