Q$4$ Consider a zero$-$memory source with $7$ symbols $\{a1,a2,a3,a4,a5,a6,a7\},$ and the corresponding source symbol probabilities are $\{0\mathrm{.}12,0\mathrm{.}15,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}05,0\mathrm{.}08\}$ respectively.

$($a$)$ Compute the entropy of the source $H=-{\displaystyle \underset{i=1}{\overset{L}{}}}{P}_{i}lo{g}_2{P}_i.$

$5$

$($b$)$ Develop a data encoding method following Huffman's procedure.

$($i$)$ Create a series of source reductions.

$5$

$($ii$)$ Code each reduced source.

$5$

$($iii$)$ Discuss about the advantages and disadvantages of this method.

$4$

$($c$)$ Evaluate the above encoding method using average code length $R={\displaystyle \underset{i}{\overset{?}{}}}{P}_i{L}_i,$ compression ratio $c=\frac{b_1}{b_2}$ and coding efficiency $\frac{H}{R}$

$6$