Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 24

of the transmitted symbol stream s ( n ), of the form

s ( n )= s ( n )+ e ( n ) , (2 . 20)

where e ( n ) is the reconstruction error. Recall here that s ( n ) , and hence s ( n ) ,

are complex numbers. It is usually reasonable to assume that the error e ( n )is
complex and has the form

e ( n )= e r ( n )+ je i ( n ) ,

where e r ( n )and e i ( n ) are independent zero-mean Gaussian random variables
with identical variance 0 . 5 σ e . In this case the total variance of the complex error
e ( n )is

0 . 5 σ e +0 . 5 σ e = σ e .
The joint pdf of the variables [ e r ( n ) ,e i ( n )] is given by
f E ( e r ,e i )= e −e r /σ e

e −e i /σ e
πσ e = e − ( e r + e i ) /σ e

πσ e

πσ e × . (2 . 21)

Figure 2.13 demonstrates this for σ e =0 . 01. This is a special case of a so-called
circularly symmetric complex Gaussian random variable. 4 Next, the complex
symbol s ( n ) is of the form

s r ( n )+ js i ( n ) ,
where s r ( n )and s i ( n ) are PAM symbols. Since
|

| 2 = s r ( n )+ s i ( n ) ,
it follows that the average energy of the constellation is the sum of the average
energies of the real and imaginary parts. Thus, for a b -bit QAM constellation
with average energy E ave , the real and imaginary parts are ( b/ 2)-bit PAM con-
stellations with average energy E ave / 2 , and each of these PAM constellations
sees an error source with variance σ e / 2. For the real-part PAM the probability
of error can be obtained from Eq. (2.17) by replacing b, E ave ,and σ e

s ( n )

with half

their values:

3 E ave
(2 b

.

2 −b/ 2 ) P e,re =2(1 − Q (2 . 22)

− 1) σ e

Since the factor of one-half cancels out in the ratio E ave /σ e , this is nothing but
the error probability for a ( b/ 2)-bit PAM constellation with energy E ave and
noise variance σ e . Similarly for the imaginary part

3 E ave
(2 b

.

2 −b/ 2 ) P e,im =2(1 − Q (2 . 23)

− 1) σ e

The QAM symbol is detected correctly if the real part and imaginary part are
both detected correctly. The probability for this is

(1 −P e,re ) 2 .

4 A detailed discussion of circularly symmetric complex random variables can be found in
Sec. 6.6.