Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 17

The same is true for the imaginary part y. So the average energy of a b -bit QAM
constellation (with M =2 b words) is

1) A 2

= 2(2 b

1) A 2

E ave,QAM = 2( M

− −

. (2 . 5)

3 3

The energy per bit is therefore

1) A 2
3log 2 M

E b,QAM = E ave,QAM
log 2 M

= 2( M

−

. (2 . 6)

2.3 Error probability

In a digital communication system the receiver constructs an approximation
s ( n ) of the transmitted symbol stream s ( n ) . The reconstructed version s ( n )
differs from s ( n ) because of errors introduced by the channel H ( jω )andthe
noise q ( t ). We can write

s ( n )= s ( n )+ e ( n ) , (2 . 7)

where e ( n ) is the reconstruction error, which can often be modeled as random
noise. Thus, even though s ( n ) is a codeword belonging to a constellation such
as a PAM constellation, the reconstructed number s ( n )

is processed by a decision making device, called the detector , which takes the
signal

s ( n ) is not. In practice

s ( n ) and estimates the transmitted codeword s ( n ) (Fig. 2.5). There is
a nonzero probability that this estimated codeword s est ( n ) is different from the
original codeword s ( n ) . This error probability depends on the statistics of the
error term e ( n ) in Eq. (2.7). Figure 2.6 shows how the received symbols s ( n )get

spread out into a “cloud” owing to the noise e ( n ) in a QAM constellation. The
received signal can be anywhere in the shaded areas. If the shaded area for a
symbol overlaps with the corresponding shaded area of an adjacent symbol, there
is nonzero probability of symbol error. In this section we derive mathematical
expressions for error probabilities.

2.3.1 Error probability for PAM signals

First consider the PAM constellation shown in Fig. 2.7 for 3 bits. We have
indicated small vertical lines called decision boundaries. These are placed exactly
midway between every pair of symbols. If s ( n ) falls within a pair of decision

boundaries, then the unique codeword within those boundaries is assumed to
be transmitted because it is the closest codeword. This is the estimated symbol
s est ( n ) corresponding to the transmitted symbol s ( n ) . To demonstrate, Fig.
2.8(a) shows the threshold detector characteristics for 1-bit PAM. This figure
says that the 1-bit PAM detector estimates the symbol according to the rule
s est ( n )= A

if s ( n ) ≥ 0

(2 . 8)

− A if s ( n ) < 0 .