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Performance of Communication Systems Corrupted by Noise Chap. 7
The binary signal plus noise at the receiver input produces a baseband analog waveform at
the output of the processing circuits that is denoted by
r 0 (t) = e r 01(t), 0 6 t … T, for a binary 1 sent
r 02 (t), 0 6 t … T, for a binary 0 sent
(7–2)
where r 01 (t) is the output signal that is corrupted by noise for a binary 1 transmission and
r 02 (t) is the output for the binary 0 transmission. (Note that if the receiver uses nonlinear
processing circuits, such as an envelope detector, the superposition of the signal plus noise
outputs are not valid operations.) This analog voltage waveform r 0 (t) is sampled at some time
t 0 during the bit interval. That is, 0 t 0 T. For matched-filter processing circuits, t 0 is
usually T. The resulting sample is
r 0 (t 0 ) = e r 01(t 0 ),
r 02 (t 0 ),
for a binary 1 sent
for a binary 0 sent
(7–3)
It is realized that r 0 (t 0 ) is a random variable that has a continuous distribution because the
channel noise has corrupted the signal. To shorten the notation, we will denote r 0 (t 0 ) simply
by r 0 . That is,
r 0 = r 0 (t 0 ) = e r 01,
r 02 ,
for a binary 1 sent
for a binary 0 sent
(7–4)
We call r 0 the test statistic.
For the moment, let us assume that we can evaluate the PDFs for the two random variables
r 0 = r 01 and r 0 = r 02 . These PDFs are actually conditional PDFs, since they depend,
respectively, on a binary 1 or a binary 0 being transmitted. That is, when r 0 = r 01 , the PDF is
f(r 0 |s 1 sent), and when r 0 = r 02 , the PDF is f(r 0 |s 2 sent). These conditional PDFs are shown in
Fig. 7–2. For illustrative purposes, Gaussian shapes are shown. The actual shapes of the PDFs
depend on the characteristics of the channel noise, the specific types of filter and detector
circuits used, and the types of binary signals transmitted. (In later sections, we obtain specific
PDFs using the theory developed in Chapter 6.)
f(r 0 |s 2 sent)
f
f(r 0 |s 1 sent)
P (error | s 1 sent)
P (error | s 2 sent)
m r 02
m r
0
01
V T
Figure 7–2 Error probability for binary signaling.
r0