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Consider a discrete memoryless source having a set of 5 letters each occurring with equal probability. If three letters are encoded at a time into a binary sequence, then what will be the efficiency of the binary code? (Answer up to three decimal place)
Correct answer is '0.994'. Can you explain this answer?
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Consider a discrete memoryless source having a set of 5 letters each ...
Calculation of Efficiency of Binary Code for Discrete Memoryless Source with 5 Letters
Step 1: Calculate the number of possible combinations for 3 letters to be encoded into a binary sequence.
- Since each letter occurs with equal probability, the probability of choosing any one letter is 1/5.
- Therefore, the number of possible combinations for 3 letters is 5 x 5 x 5 = 125.
Step 2: Calculate the number of bits required to represent each combination.
- Since there are 2 possible values for each bit (0 or 1), the number of bits required to represent each combination is log2(125) ≈ 6.98.
- Since we can't use fractional bits, we need to round up to 7 bits.
Step 3: Calculate the total number of bits required to encode all possible combinations.
- There are 125 possible combinations, and each combination requires 7 bits.
- Therefore, the total number of bits required is 125 x 7 = 875 bits.
Step 4: Calculate the average number of bits per letter.
- Since we are encoding 3 letters at a time, the average number of bits per letter is 875 / (3 x 5) = 58.33 bits/letter.
Step 5: Calculate the efficiency of the binary code.
- Efficiency = (number of bits in original message) / (number of bits in encoded message)
- Since each letter is equally likely, the entropy of the source is log2(5) ≈ 2.32 bits/letter.
- Therefore, the number of bits in the original message is 3 x 2.32 = 6.96 bits.
- The efficiency of the binary code is 6.96 / 58.33 ≈ 0.994.
Conclusion: The efficiency of the binary code for a discrete memoryless source with 5 letters each occurring with equal probability, when three letters are encoded at a time into a binary sequence, is 0.994.
Step 1: Calculate the number of possible combinations for 3 letters to be encoded into a binary sequence.
- Since each letter occurs with equal probability, the probability of choosing any one letter is 1/5.
- Therefore, the number of possible combinations for 3 letters is 5 x 5 x 5 = 125.
Step 2: Calculate the number of bits required to represent each combination.
- Since there are 2 possible values for each bit (0 or 1), the number of bits required to represent each combination is log2(125) ≈ 6.98.
- Since we can't use fractional bits, we need to round up to 7 bits.
Step 3: Calculate the total number of bits required to encode all possible combinations.
- There are 125 possible combinations, and each combination requires 7 bits.
- Therefore, the total number of bits required is 125 x 7 = 875 bits.
Step 4: Calculate the average number of bits per letter.
- Since we are encoding 3 letters at a time, the average number of bits per letter is 875 / (3 x 5) = 58.33 bits/letter.
Step 5: Calculate the efficiency of the binary code.
- Efficiency = (number of bits in original message) / (number of bits in encoded message)
- Since each letter is equally likely, the entropy of the source is log2(5) ≈ 2.32 bits/letter.
- Therefore, the number of bits in the original message is 3 x 2.32 = 6.96 bits.
- The efficiency of the binary code is 6.96 / 58.33 ≈ 0.994.
Conclusion: The efficiency of the binary code for a discrete memoryless source with 5 letters each occurring with equal probability, when three letters are encoded at a time into a binary sequence, is 0.994.
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Community Answer
Consider a discrete memoryless source having a set of 5 letters each ...
The source entropy is
In case of encoding three letters at a time, we have 53 = 125 possible sequences.
We need [log2 125] + 1 = [6.96] + 1 = 6 + 1 = 7 bits/3 letter
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Consider a discrete memoryless source having a set of 5 letters each occurring with equal probability. If three letters are encoded at a time into a binary sequence, then what will be the efficiency of the binary code? (Answer up to three decimal place)Correct answer is '0.994'. Can you explain this answer?
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