$5.$

Lists of convolutional codes with maximum free distances are available in many textbooks, for example $[$Sodha$,2015].$ The rate of a convolutional code is the number of input bits divided by the number of output bits. A table of the rate$-\backslash (1/2\backslash )$ and rate $\backslash (-1/3\backslash )$ codes is provided below. The free distance $\backslash ($ d$\_\{$f r o$\}\backslash )$ of a convolutional code is equivalent to the minimum distance $\backslash ($ d$\_\{\backslash$min $\}\backslash )$ of a block code and the error$-$correcting capability is equal to $\backslash (\backslash$left$\backslash$lfloor$\backslash$frac$\{$d$\_\{$f r e e$\}-1\}\{2\}\backslash$right$\backslash$rfloor $\backslash ),$ where $\backslash (\backslash$lfloor x$\backslash$rfloor $\backslash )$ is the integer part of $\backslash ($ x $\backslash ).$ The constraint length K is the number of cells of the shift$-$register. Consider the convolutional encoder shown in Fig. $1.$

a$)$ Determine the constraint length K $,$ generators $($in octal notation$)$ and the coding rate of the encoder in Fig. $1.$

b$)$ Sketch the state diagram of this code.

c$)$ Determine the output of the encoder if the message is: $\backslash (1,0,1,0\backslash ).$

d$)$ Assume the first two coded bits of the sequence obtained in c$)$ are received erroneously. Determine the decoded sequence using the Viterbi algorithm and then, the decoded message.

e$)$ Compare the decoded message with the original message $(\backslash (1,0,1,0\backslash )).$ Is the result expected?

f$)$ Design of coding scheme: Assume that two different codes, whose generators and dfee are given in Table $1,$ are considered to replace this one: the $\backslash (\backslash$mathrm$\{$K$\}=3,\backslash$mathrm$\{$R$\}=1/2\backslash )$ code and the $\backslash (\backslash$mathrm$\{$K$\}=7,\backslash$mathrm$\{$R$\}=1/3\backslash )$ code. Discuss how that will affect the spectral requirements$/$data rate, error performance, and complexity.