B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

402 FUNDAMNTALS OF PROBABILITY THEORY
The reception of each digit is independent of the other digits. This is an example of a
Bernoulli trial with the probability of success p = P e ("success" here is receiving a digit
in error). Clearly, the probability of k successes in n trials (k errors in n digits) is
P(receiving k out of n digits in error) = G) P! (l - P e )"-k
For example, if P e = 10- 5 , the probability of receiving two digits wrong in a sequence
of eight digits is
Example 8.7
PCM Repeater Error Probability
In pulse code modulation, regenerative repeaters are used to detect pulses (before they are lost
in noise) and retransmit new, clean pulses. This combats the accumulation of noise and pulse
distortion.
A certain PCM channel consists of n identical links in tandem (Fig. 8.3). The pulses are
detected at the end of each link and clean new pulses are transmitted over the next link. If P e
is the probability of error in detecting a pulse over any one link, show that PE, the probability
of error in detecting a pulse over the entire channel (over the n links in tandem), is
nP e « 1
Figure 8.3
A PCM repeater.
1st link 2st link kth link
In ----------.-.---+----e--- - - - -.-.-------------- Out
The probabilities of detecting a pulse correctly over one link and over the entire channel
(n links in tandem) are 1 - P e and 1 - PE , respectively. A pulse can be detected correctly
over the entire channel if either the pulse is detected correctly over every link or errors
are made over an even number of links only.
1 - PE = P(correct detection over all links)
+ P(error over two links only)
+ P(error over four links only) + •••
+ P(error over 2 lJ links only)
where LaJ denotes the largest integer less than or equal to a.
Because pulse detection over each link is independent of the other links (see Example 8.6),
P(correct detection over all n links) = (1 - P et
and
n!
P(error over k links only) = P! (1 - P e ) n -k
k!(n - k)!