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Sec. 7–1 Error Probabilities for Binary Signaling 495
In our development of a general formula for the BER, assume that the polarity of the
processing circuits of the receiver is such that if the signal only (no noise) were present at the
receiver input, r 0 7 V T when a binary 1 is sent and r 0 6 V T when a binary 0 is sent; V T is the
threshold (voltage) setting of the comparator (threshold device).
When signal plus noise is present at the receiver input, errors can occur in two ways.
An error occurs when r 0 6 V T if a binary 1 is sent:
V r
P(error|s 1 sent) = f(r 0 ƒ s 1 ) dr 0
3
-q
(7–5)
This is illustrated by a shaded area to the left of V T in Fig. 7–2. Similarly, an error occurs
when r 0 7 V T if a binary 0 is sent:
The BER is then
P(error|s 2 sent) =
L
q
(7–6)
(7–7)
This follows from probability theory (see Appendix B), where the probability of an event that
consists of joint events is
P(E) = a
2
i=1
P(E, s i ) = a
2
V T
f(r 0 ƒ s 2 ) dr 0
P e = P(error|s 1 sent) P(s 1 sent) + P(error|s 2 sent) P(s 2 sent)
When we combine Eqs. (7–5), (7–6), and (7–7), the general expression for the BER of any binary
communication system is
i=1
P(E|s i )P(s i )
V T
q
P e = P(s 1 sent) f(r 0 |s 1 ) dr 0 + P(s 2 sent)
L L
-q
V T
f(r 0 |s 2 ) dr 0
(7–8)
P(s 1 sent) and P(s 2 sent) are known as the source statistics or a priori statistics. In most
applications, the source statistics are considered to be equally likely. That is,
P(binary 1 sent) = P(s 1 sent) = 1 2
(7–9a)
P(binary 0 sent) = P(s 2 sent) = 1 2
(7–9b)
In the results that we obtain throughout the remainder of this chapter, we will assume that the
source statistics are equally likely. The conditional PDFs depend on the signaling waveshapes
involved, the channel noise, and the receiver processing circuits used. These are obtained subsequently
for the case of Gaussian channel noise and linear processing circuits.
Results for Gaussian Noise
Assume that the channel noise is a zero-mean wide-sense stationary Gaussian process and
that the receiver processing circuits, except for the threshold device, are linear. Then we know
(see Chapter 6) that for a Gaussian process at the input, the output of the linear processor will