Digital Signal Processing Reference
In-Depth Information
e
j
θ
.Theprob-
ability of error decreases exponentially with the gain and, therefore, with the
power of the transmitter.
where we have used (12.17) and the change of variable
z
=
ρ
l
g
r
,
y
i
d
.
,
©
,
L
s
·
·
·
G
0
·
·
·
·
·
Figure 12.15
Decision region and its circular approximation.
The concept of “reliability” is quantified by the probability of error that
we are willing to tolerate; note that this probability can never be zero, but
it can be made arbitrarily low - values on the order of
p
e
10
−
6
are usually
taken as a reference. Assume that the transmitter transmits at the maxi-
mum permissible power so that the SNR on the channel is maximized. Un-
der these conditions it is
SNR
=
σ
=
s
σ
2
α
σ
0
=
G
0
σ
2
0
and from (12.18) we have
2
2
α
SNR
=
−
ln
(
p
e
)
σ
(12.19)
For a regular square-lattice constellation we can use (12.15) to determine
the maximum number of bits per symbol which can be transmitted at the
given reliability figure:
log
2
1
3
2
SNR
ln
M
=
−
(12.20)
(
p
e
)
