Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 25

25

20

15

10

5

0

0.5

0.5

0

0

-0.5

-0.5

Figure 2.13 . The pdf of Eq. (2.21) plotted for σ e =0 . 01.

The probability of error in detection of the QAM symbol is therefore

−P e,P AM ( b/ 2)) 2 , P e,QAM ( b )=1 − (1 (2 . 24)

where we have used the functional arguments to indicate the number of bits.
Thus P e,P AM ( b/ 2) is the error probability for a ( b/ 2)-bit PAM constellation

with energy E ave and noise variance σ e . For small errors the preceding equation
can be approximated as

1

P e,P AM ( b/ 2) P e,QAM ( b )=1 − − 2 P e,P AM ( b/ 2) + ≈ 2 P e,P AM ( b/ 2) , (2 . 25)

where we have neglected P e,P AM ( b/ 2). This approximation is quite reasonable

in practice. For example, even if P e,P AM ( b/ 2) = 10 − 3 (a rather large value), its

square is 10 − 6 , which can be neglected. Thus the error probability for the QAM
constellation can be approximated by

3 E ave
(2 b

,

2 −b/ 2 ) P e,QAM ( b ) ≈ 2 P e,P AM ( b/ 2)=4(1 − Q (2 . 26)

− 1) σ e

where b isthenumberofbits, E ave is the average energy of the constellation,
and σ e is the variance of the complex Gaussian error term e ( n ) at the input of
the detector. For comparsion, recall that a b -bit PAM constellation with the
same energy E ave and noise variance σ e

would have error probability

3 E ave
(2 2 b

.

2 −b )

P e,P AM ( b ) = 2(1 − Q (2 . 27)

− 1) σ e