GATE 2022 EC – Question 61
The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure. The transition probabilities are as marked.
The parameter $\alpha$ lies in the interval $[0.25,1]$. The value of $\alpha$ for which the capacity of this channel is maximized, is ________ (rounded off to two decimal places).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 1
Explanation
Each input goes to its own output with probability $1-\alpha$ and to the next output (cyclically) with probability $\alpha$, so the channel is symmetric. Its capacity is $C=\log_23-H(\alpha)$, where $H(\alpha)=-\alpha\log_2\alpha-(1-\alpha)\log_2(1-\alpha)$. On $[0.25,1]$ the entropy $H(\alpha)$ is smallest at $\alpha=1$ (where it is 0), so the capacity is maximised at $\alpha=1.00$.