GATE EC 2023 Set 1 — Question 65
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Communications → Information Theory → Information Measure & Entropy
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Question
The frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of “yes/no” questions which are assumed to be truthfully answered. The average number of questions when asked in the most efficient sequence, to determine the chosen symbol, is _______ (rounded off to two decimal places).
| Symbols | a | b | c | d | e | f | g | h |
|---|---|---|---|---|---|---|---|---|
| Frequency of occurrence |
Correct answer
1.97 to 1.99
Solution
1.The most efficient sequence of questions to identify a symbol corresponds to an optimal prefix-free code, such as Huffman coding.
2.The average number of questions is equivalent to the average code length , where is the number of questions (bits) for symbol .
3.For probabilities that are powers of 2, the optimal code length for each symbol is , and the average length equals the entropy .
4.Calculate the entropy:
.
5.Rounding to two decimal places, we get .
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