In order to increase the reliability of binary communication chan
A. <span style=" line-height: 115%; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">p<sup>2</sup></span>
B. 2 p<sup>2</sup>
C. 3 p<sup>2</sup>
D. (1/3) p<sup>2</sup>
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
Given: The transmitted digits are 000 for 0 and 111 for 1. At receiving end, the decision is made by majority rule, that is, if at least two of three digits are 0, decision is taken in favour of 0.
Bit error probability in the repetition of bits in BSC is given by:
\({p_e} = \mathop \sum \nolimits_{k = 0}^3 \left( {\begin{array}{*{20}{c}} 3\\ k \end{array}} \right){p^k}{\left( {1 - p} \right)^{3 - 4}}\)
\({p_e} = \left( {\begin{array}{*{20}{c}} 3\\ 2 \end{array}} \right){p^2}\left( {1 - p} \right) + \left( {\begin{array}{*{20}{c}} 3\\ 3 \end{array}} \right){p^3}{\left( {1 - p} \right)^{3 - 3}}\)
= 3p2 (1 - p) + p3 ≅ 3p2 ; p ≪ 1 and term containing p3 may be ignored
Hence, the correct option (C).