Problem $4$

Consider a rate $\frac{1}{2}$ convolutional code. Let $A_1,A_2,$dots denote the encoded output. At each time $i,$ the input produces a new bit, $A_i,$ and the encoder produces two bits, $x_i$ and $Y_i.$ The encoding rule is:

$x_i=A_{i}o+A_{i-1}o+A_{i-2}$

$Y_i=A_{i}o+A_{i-2}$

Consider $A_0$ and $A_{-1}$ to be $0.$ At each point in time, a new bit enters the encoder and two bits leave.

$($a$)$ Draw the trellis diagram for this code.

$($b$)$ Find the minimum Hamming distance between the all$-$zero codeword and the other codewords. Hint: The minimizing codeword must start and end in state $00.$

$($c$)$ Suppose at the output of the channel, each coded bit is flipped with probability $lon$ and kept the same with probability $1-lon,$ independently from all the other bits. Suppose that the output of the channel is $(00)(00)(01)(11)(10)(01)(11)(00)(00).$ Simulate the operation of a Viterbi decoder in decoding this sequence. Hint: You can save yourself some numerical aggravation by simply keeping track of Hamming distances, rather than probabilities.