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A digital communication system uses a repetition code for channel encoding/decoding. During
transmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).
It is assumed that the source puts out symbols independently and with equal probability . The
decoder operates as follows: In a block of three received bits, if the number of zeros exceeds the
number of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the number
of zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossover
probability p = 0.1, the average probability of error is ________.
transmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).
It is assumed that the source puts out symbols independently and with equal probability . The
decoder operates as follows: In a block of three received bits, if the number of zeros exceeds the
number of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the number
of zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossover
probability p = 0.1, the average probability of error is ________.
Correct answer is '0.028'. Can you explain this answer?
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Verified Answer
A digital communication system uses a repetition code for channel enco...
Pe = P3 + 3P2 (1 –P)
P = 0.1
Pe = (0.1)3 + 3 x (0.1)2 (1–0.1) = 0.001 + 3 x 0.01 x 0.9 = 0.001 + 0.027 = 0.028
P = 0.1
Pe = (0.1)3 + 3 x (0.1)2 (1–0.1) = 0.001 + 3 x 0.01 x 0.9 = 0.001 + 0.027 = 0.028
Most Upvoted Answer
A digital communication system uses a repetition code for channel enco...
To calculate the average probability of error in a digital communication system that uses a repetition code for channel encoding/decoding, we need to consider the probability of different error scenarios.
Given Information:
- Each bit is repeated three times during transmission (0 is transmitted as 000, and 1 is transmitted as 111).
- The source puts out symbols independently and with equal probability.
- The decoder decides in favor of a 0 if the number of zeros exceeds the number of ones and in favor of a 1 if the number of ones exceeds the number of zeros.
- The binary symmetric channel has a crossover probability p = 0.1.
Calculating the Error Scenarios:
To calculate the average probability of error, we need to consider the different error scenarios and their probabilities.
Scenario 1: Error in Transmitting 0
In this scenario, the transmitter sends 000 (0) but the receiver receives either 111 (1) or some combination with at least two 1s. The probability of this error scenario occurring is given by:
P(error in transmitting 0) = P(111) + P(1XX) + P(X1X) + P(XX1)
= p^3 + 3p^2(1-p)
= 0.1^3 + 3(0.1^2)(0.9)
= 0.009 + 0.027
= 0.036
Scenario 2: Error in Transmitting 1
In this scenario, the transmitter sends 111 (1) but the receiver receives either 000 (0) or some combination with at least two 0s. The probability of this error scenario occurring is given by:
P(error in transmitting 1) = P(000) + P(0XX) + P(X0X) + P(XX0)
= p^3 + 3p^2(1-p)
= 0.1^3 + 3(0.1^2)(0.9)
= 0.009 + 0.027
= 0.036
Scenario 3: No Error
In this scenario, the transmitter sends either 000 (0) or 111 (1), and the receiver receives the same sequence without any errors. The probability of this scenario occurring is given by:
P(no error) = P(000) + P(111)
= (1-p)^3 + p^3
= 0.9^3 + 0.1^3
= 0.729 + 0.001
= 0.73
Calculating the Average Probability of Error:
The average probability of error is calculated as the weighted sum of the probabilities of the error scenarios:
Average probability of error = P(error in transmitting 0) * P(0) + P(error in transmitting 1) * P(1)
= 0.036 * 0.5 + 0.036 * 0.5
= 0.018 + 0.018
= 0.036
Final Answer:
The average probability of error in the given digital communication system is 0.036 or 3.6%.
Given Information:
- Each bit is repeated three times during transmission (0 is transmitted as 000, and 1 is transmitted as 111).
- The source puts out symbols independently and with equal probability.
- The decoder decides in favor of a 0 if the number of zeros exceeds the number of ones and in favor of a 1 if the number of ones exceeds the number of zeros.
- The binary symmetric channel has a crossover probability p = 0.1.
Calculating the Error Scenarios:
To calculate the average probability of error, we need to consider the different error scenarios and their probabilities.
Scenario 1: Error in Transmitting 0
In this scenario, the transmitter sends 000 (0) but the receiver receives either 111 (1) or some combination with at least two 1s. The probability of this error scenario occurring is given by:
P(error in transmitting 0) = P(111) + P(1XX) + P(X1X) + P(XX1)
= p^3 + 3p^2(1-p)
= 0.1^3 + 3(0.1^2)(0.9)
= 0.009 + 0.027
= 0.036
Scenario 2: Error in Transmitting 1
In this scenario, the transmitter sends 111 (1) but the receiver receives either 000 (0) or some combination with at least two 0s. The probability of this error scenario occurring is given by:
P(error in transmitting 1) = P(000) + P(0XX) + P(X0X) + P(XX0)
= p^3 + 3p^2(1-p)
= 0.1^3 + 3(0.1^2)(0.9)
= 0.009 + 0.027
= 0.036
Scenario 3: No Error
In this scenario, the transmitter sends either 000 (0) or 111 (1), and the receiver receives the same sequence without any errors. The probability of this scenario occurring is given by:
P(no error) = P(000) + P(111)
= (1-p)^3 + p^3
= 0.9^3 + 0.1^3
= 0.729 + 0.001
= 0.73
Calculating the Average Probability of Error:
The average probability of error is calculated as the weighted sum of the probabilities of the error scenarios:
Average probability of error = P(error in transmitting 0) * P(0) + P(error in transmitting 1) * P(1)
= 0.036 * 0.5 + 0.036 * 0.5
= 0.018 + 0.018
= 0.036
Final Answer:
The average probability of error in the given digital communication system is 0.036 or 3.6%.
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A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer?
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A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer?.
A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A digital communication system uses a repetition code for channel encoding/decoding. Duringtransmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111).It is assumed that the source puts out symbols independently and with equal probability . Thedecoder operates as follows: In a block of three received bits, if the number of zeros exceeds thenumber of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the numberof zeros, the decoder decides in favor of a 1,Assuming a binary symmetric channel with crossoverprobability p = 0.1, the average probability of error is ________.Correct answer is '0.028'. Can you explain this answer?.
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