Topic: Electronics Engineering
A source produces 4 symbols with probabilities $1/2$, $1/4$ $1/8$ $1/8$ For this source a practical coding scheme has an average code word length of 2 bits/symbol. The efficiency of the code is
- A. 1
- B. $7/8$
- C. $1/2$
- D. $1/4$
Answer: B. $7/8$
Explanation: Efficiency $(\eta)=\frac{H(s)}{\overline{L}}$
Where, $H(s)=$ entropy and $\overline{L}=$ Average code word length
$H(s)=\sum_{i=0}^{N-1}P_{i}log_{2}^{(\frac{1}{P_{i}})}$
$=\frac{1}{2}log_{2}^{2}+\frac{1}{4}log_{2}^{4}+\frac{1}{8}log_{2}^{8}+\frac{1}{8}log_{2}^{8} = \frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{3}{8}=\frac{7}{4}.$
So, efficiency $(\eta)=\frac{\frac{7}{4}}{2}=\frac{7}{8}.$