GATE EC 2022 Set 1 — Question 61
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Communications → Information Theory → Channel Capacity Theorem
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Question
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 lies in the interval . The value of for which the capacity of this channel is maximized, is ________ (rounded off to two decimal places).
Correct answer
1 to 1
Solution
The given channel is a symmetric channel with 3 input and 3 output symbols. Based on the diagram, the transition matrix is:For a symmetric channel, the capacity is achieved with a uniform input distribution and is calculated as:where is the number of output symbols () and is the entropy of any row of the transition matrix. where is the binary entropy function. Thus, the capacity as a function of is:To maximize the capacity , we must minimize the entropy . The binary entropy function is minimized when or . Given the constraint , the minimum value of in this interval occurs at .At , , and the capacity is bits/symbol. This corresponds to a noiseless permutation channel ().Therefore, the value of that maximizes the capacity is .
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