$(2)$ The trellis diagram of an even parity check code of code length $4$ is given below. Let

$x=(x_1,x_2,x_3,x_4)$ be the transmit codeword and $Y=(y_1,y_2,y_3,y_4)$ be the receive

codeword, find the posterior probabilities $P(y_i=1|Y)$ and $P(y_i=0|Y),i=1,2,3,4$ per symbol

by the full $($brute$-$force$)$ search and BCJR algorithm, respectively. Also, find the

probability that transmitted codewords are $0011,1100,$ and $1010,$ respectively. Show the

calculation process in detail.

where $P(x_i=1)=0\mathrm{.}5,P(x_i=0)=0\mathrm{.}5$

$P(y_1|x_1=0)=0\mathrm{.}8,P(y_2|x_2=0)=0\mathrm{.}9,P(y_3|x_3=0)=0\mathrm{.}4,P(y_4|x_4=0)=0\mathrm{.}9$

$P(y_1|x_1=1)=0\mathrm{.}2,P(y_2|x_2=1)=0\mathrm{.}1,P(y_3|x_3=1)=0\mathrm{.}6,P(y_4|x_4=1)=0\mathrm{.}1$