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Sec. 7–1 Error Probabilities for Binary Signaling 493
7–1 ERROR PROBABILITIES FOR BINARY SIGNALING
General Results
Figure 7–1 shows a general block diagram for a binary communication system. The receiver
input r(t) consists of the transmitted signal s(t) plus channel noise n(t). For baseband signaling,
the processing circuits in the receiver consist of low-pass filtering with appropriate amplification.
For bandpass signaling, such as OOK, BPSK, and FSK, the processing circuits normally
consist of a superheterodyne receiver containing a mixer, an IF amplifier, and a detector. These
circuits produce a baseband analog output r 0 (t). (For example, when BPSK signaling is used,
the detector might consist of a product detector and an integrator as described in Sec. 6–8 and
illustrated in Fig. 6–19.)
The analog baseband waveform r 0 (t) is sampled at the clocking time t = t 0 + nT to produce
the samples r 0 (t 0 + nT), which are fed into a threshold device (a comparator). The
threshold device produces the binary serial-data waveform m ' (t).
In this subsection, we develop a general technique for evaluating the probability of
encountering a bit error, also called the bit-error rate (BER), for binary signaling. In later
sections, this technique will be used to obtain specific expressions for the BER of various
binary signaling schemes, such as OOK, BPSK, and FSK.
To develop a general formula for the BER of a detected binary signal, let T be the duration
of time it takes to transmit one bit of data. The transmitted signal over a bit interval (0, T) is
s(t) = e s 1(t), 0 6 t … T, for a binary 1
(7–1)
s 2 (t), 0 6 t … T, for a binary 0
where s 1 (t) is the waveform that is used if a binary 1 is transmitted and s 2 (t) is the waveform
that is used if a binary 0 is transmitted. If s 1 (t) = -s 2 (t), s(t) is called an antipodal signal.
Channel
Noise
Digital
input
m
Transmitter
s(t)
n(t)
Receiver
r(t)=s(t)+n(t)
Processing
circuits
Baseband
analog
output
r 0 (t)
Sample
and hold
~ m
at t 0
r 0 (t 0 )
0
Threshold
device
Digital
output
~ m
V T
r 0
Clock
Figure 7–1
General binary communication system.