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Consider communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. Each transmitted bit is received correctly with probability (1 - ε) , and flipped with probability ε . For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error. For ε = 0.1, the probability that a transmitted codeword is decoded correctly is _________ (round off to two decimal places).
- a)0.84
- b)0.86
Correct answer is between '0.84,0.86'. Can you explain this answer?
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Most Upvoted Answer
Consider communication over a memoryless binary symmetric channel usi...
Given that the bits are transmitted using (7, 4) hamming code
∴ Total number of bits n = 7 .
Given that the transmitted bit are received correctly with probability 1- ε .
So, probability of receiving the bits wrongly will be 1-(1 - ε) = ε
Let, p = ε and q = 1 - ε
p = 0.1 and q = 1 - 0.1 = 0.9 (∵ ε = 0.1)
If X is the event of transmitting the bits, then the probability of at most one bit error will be,
Hence, the correct answer is 0.8502.
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Consider communication over a memoryless binary symmetric channel usi...
Introduction:
In this problem, we are considering communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. The channel introduces bit errors with a probability of ε and the receiver performs minimum Hamming distance decoding to decode the message bits. We need to calculate the probability that a transmitted codeword is decoded correctly when ε = 0.1.
Hamming Code:
The (7, 4) Hamming code is a linear error-correcting code that adds 3 parity bits to a 4-bit message to create a 7-bit codeword. The parity bits are calculated such that the Hamming distance between any two codewords is at least 3. This allows the receiver to detect and correct single-bit errors.
Probability of Decoding Correctly:
To calculate the probability that a transmitted codeword is decoded correctly, we need to consider the different error patterns that can occur and the corresponding decoding outcomes.
Error Patterns:
1. No errors: The transmitted codeword is received correctly.
2. Single-bit error: One bit in the transmitted codeword is flipped.
3. Double-bit error: Two bits in the transmitted codeword are flipped.
Decoding Outcomes:
1. No errors: The received codeword matches one of the valid codewords. It is correctly decoded.
2. Single-bit error: The received codeword is at a Hamming distance of 1 from one of the valid codewords. It is correctly decoded.
3. Double-bit error: The received codeword is at a Hamming distance of 2 from the valid codewords. It is incorrectly decoded.
Calculation:
To calculate the probability of decoding correctly, we need to sum the probabilities of the decoding outcomes that result in correct decoding.
1. No errors: The probability of no errors is (1-ε)^7.
2. Single-bit error: The probability of a single-bit error is 7ε(1-ε)^6. The factor of 7 represents the 7 possible positions for the error.
3. Double-bit error: The probability of a double-bit error is 7C2 * ε^2 * (1-ε)^5. The factor of 7C2 represents the number of ways to choose 2 positions for the errors.
The probability of decoding correctly is the sum of the probabilities of the first two outcomes:
P(correct) = (1-ε)^7 + 7ε(1-ε)^6
Substituting ε = 0.1, we can calculate the probability:
P(correct) = (0.9)^7 + 7(0.1)(0.9)^6
P(correct) ≈ 0.86
Therefore, the probability that a transmitted codeword is decoded correctly when ε = 0.1 is approximately 0.86.
In this problem, we are considering communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. The channel introduces bit errors with a probability of ε and the receiver performs minimum Hamming distance decoding to decode the message bits. We need to calculate the probability that a transmitted codeword is decoded correctly when ε = 0.1.
Hamming Code:
The (7, 4) Hamming code is a linear error-correcting code that adds 3 parity bits to a 4-bit message to create a 7-bit codeword. The parity bits are calculated such that the Hamming distance between any two codewords is at least 3. This allows the receiver to detect and correct single-bit errors.
Probability of Decoding Correctly:
To calculate the probability that a transmitted codeword is decoded correctly, we need to consider the different error patterns that can occur and the corresponding decoding outcomes.
Error Patterns:
1. No errors: The transmitted codeword is received correctly.
2. Single-bit error: One bit in the transmitted codeword is flipped.
3. Double-bit error: Two bits in the transmitted codeword are flipped.
Decoding Outcomes:
1. No errors: The received codeword matches one of the valid codewords. It is correctly decoded.
2. Single-bit error: The received codeword is at a Hamming distance of 1 from one of the valid codewords. It is correctly decoded.
3. Double-bit error: The received codeword is at a Hamming distance of 2 from the valid codewords. It is incorrectly decoded.
Calculation:
To calculate the probability of decoding correctly, we need to sum the probabilities of the decoding outcomes that result in correct decoding.
1. No errors: The probability of no errors is (1-ε)^7.
2. Single-bit error: The probability of a single-bit error is 7ε(1-ε)^6. The factor of 7 represents the 7 possible positions for the error.
3. Double-bit error: The probability of a double-bit error is 7C2 * ε^2 * (1-ε)^5. The factor of 7C2 represents the number of ways to choose 2 positions for the errors.
The probability of decoding correctly is the sum of the probabilities of the first two outcomes:
P(correct) = (1-ε)^7 + 7ε(1-ε)^6
Substituting ε = 0.1, we can calculate the probability:
P(correct) = (0.9)^7 + 7(0.1)(0.9)^6
P(correct) ≈ 0.86
Therefore, the probability that a transmitted codeword is decoded correctly when ε = 0.1 is approximately 0.86.
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Consider communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. Each transmitted bit is received correctly with probability (1 - ε) , and flipped with probability ε . For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error. For ε = 0.1, the probability that a transmitted codeword is decoded correctly is _________ (round off to two decimal places).a)0.84b)0.86Correct answer is between '0.84,0.86'. Can you explain this answer?
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