Question

SANDUO Example 8.7 Db PCM Repeater Error Probability this 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. se 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 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 Fe Se Pe – npe npe « 1 1st link 2st link Figure 8.3 A PCM repeater. kth link - In Out

Answer 1

The probabilities of detecting a pulse correctly over one link and over the entire channel (n links in tandem) are 1 - Pe 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) + Perror over four links only) +:.. + P(error over 26 | links only) 1 where [a] denotes the largest integer less than or equal to a. Because pulse detection over each link is independent of the other links P(correct detection over all n links) = (1 - P.)" and philidade de P(error over k links only) = P(1 - Pen-k

Hence, 1-PE = (1-P)" + r n ! skrap in-k k!(n**pk (1 – Pe)nk ,4,6,.. In practice. Pe « 1, so only the first two terms on the right-hand side of this of significance. Also, (1 – Pe)n--* ~ 1, and on the right-hand side of this equation are n! 1- Pe = (1 – Pe)" + 1" o pa | 2!(n - 2)! S9 = (1 - P.)n + n(n − 1) p2 2 If nP, « 1, then the second term can also be neglected, and 1-PE (1 – Pe)" ~1-nPe ( ni .. Pe<1 and Penpe We can explain this result heuristically by considering the transmission of N (N −0). pulses. Each link makes NP errors, and the total number of errors is approximately nNP. (approximately, because some of the erroneous pulses over a link will be erroneous ove other links). Thus the overall error probability is nPe.