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In a digital communication system, a symbol S randomly chosen from the set (s1, s2, s3, s4) is transmitted. It is given that s1 = -3, s2 = -1, s3 = +1 and s4 = +2. The received symbol is Y = S W/. W is a zero-mean unit-variance Gaussian random variable and is independent of S. Pi is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol S = si. The index i for which the conditional symbol error probability Pi is the highest is _____ .
Correct answer is '3'. Can you explain this answer?
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In a digital communication system, a symbol S randomly chosen from the...
Since the noise variable is Gaussian with zero mean and ML decoding is used, the decision boundary between two adjacent signal points will be their arithmetic mean. In the following graphs, the shaded area indicates the conditional probability of decoding a symbol correctly when it is transmitted.
By comparing the above graphs, we can conclude that P3 is larger among the four.
By comparing the above graphs, we can conclude that P3 is larger among the four.
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In a digital communication system, a symbol S randomly chosen from the...
Understanding the problem
In this problem, we have a digital communication system where a symbol S is randomly chosen from a set of four symbols (s1, s2, s3, s4). The symbols are transmitted and the received symbol is denoted as Y. Y is a random variable that depends on the transmitted symbol S and a zero-mean unit-variance Gaussian random variable W, which is independent of S.
We are given the values of the symbols: s1 = -3, s2 = -1, s3 = 1, and s4 = 2. The task is to find the index i for which the conditional symbol error probability Pi is the highest.
Conditional Symbol Error Probability (Pi)
The conditional symbol error probability Pi is the probability that the ML decoder makes an error in decoding the transmitted symbol S when it was actually si. In other words, it is the probability of symbol error given that the transmitted symbol is si.
Maximum Likelihood (ML) Decoding
In ML decoding, the received symbol Y is compared with all possible transmitted symbols S in order to find the most likely transmitted symbol. The ML decoder chooses the symbol that maximizes the likelihood of the received symbol Y given the transmitted symbol S.
Analysis
To find the symbol with the highest conditional symbol error probability Pi, we need to analyze the ML decoding process for each possible transmitted symbol si.
For si = s1 = -3:
- The ML decoder compares the received symbol Y with -3 and calculates the likelihood of Y given -3.
- Since W is a zero-mean unit-variance Gaussian random variable, the likelihood of Y given -3 can be expressed as the probability density function (PDF) of W evaluated at (Y - (-3)).
- The ML decoder calculates the likelihood for all four symbols and chooses the symbol with the maximum likelihood.
Similarly, we repeat this process for si = s2, si = s3, and si = s4.
Evaluating the Conditional Symbol Error Probabilities
The conditional symbol error probability Pi can be calculated as 1 - probability of choosing the correct symbol si. Since the symbols are equiprobable, the probability of choosing the correct symbol si is 1/4.
For si = s1 = -3:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s2 = -1.
- The conditional symbol error probability for si = s1 is 1 - (1/4) = 3/4.
Similarly, we calculate the conditional symbol error probabilities for si = s2, si = s3, and si = s4.
For si = s2 = -1:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s3 = 1.
- The conditional symbol error probability for si = s2 is 1 - (1/4) = 3/4.
For si = s3 = 1:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s4 = 2.
- The conditional symbol error probability for si = s3 is 1 - (1/4) = 3/4.
For si = s4 = 2:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s3 = 1.
-
In this problem, we have a digital communication system where a symbol S is randomly chosen from a set of four symbols (s1, s2, s3, s4). The symbols are transmitted and the received symbol is denoted as Y. Y is a random variable that depends on the transmitted symbol S and a zero-mean unit-variance Gaussian random variable W, which is independent of S.
We are given the values of the symbols: s1 = -3, s2 = -1, s3 = 1, and s4 = 2. The task is to find the index i for which the conditional symbol error probability Pi is the highest.
Conditional Symbol Error Probability (Pi)
The conditional symbol error probability Pi is the probability that the ML decoder makes an error in decoding the transmitted symbol S when it was actually si. In other words, it is the probability of symbol error given that the transmitted symbol is si.
Maximum Likelihood (ML) Decoding
In ML decoding, the received symbol Y is compared with all possible transmitted symbols S in order to find the most likely transmitted symbol. The ML decoder chooses the symbol that maximizes the likelihood of the received symbol Y given the transmitted symbol S.
Analysis
To find the symbol with the highest conditional symbol error probability Pi, we need to analyze the ML decoding process for each possible transmitted symbol si.
For si = s1 = -3:
- The ML decoder compares the received symbol Y with -3 and calculates the likelihood of Y given -3.
- Since W is a zero-mean unit-variance Gaussian random variable, the likelihood of Y given -3 can be expressed as the probability density function (PDF) of W evaluated at (Y - (-3)).
- The ML decoder calculates the likelihood for all four symbols and chooses the symbol with the maximum likelihood.
Similarly, we repeat this process for si = s2, si = s3, and si = s4.
Evaluating the Conditional Symbol Error Probabilities
The conditional symbol error probability Pi can be calculated as 1 - probability of choosing the correct symbol si. Since the symbols are equiprobable, the probability of choosing the correct symbol si is 1/4.
For si = s1 = -3:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s2 = -1.
- The conditional symbol error probability for si = s1 is 1 - (1/4) = 3/4.
Similarly, we calculate the conditional symbol error probabilities for si = s2, si = s3, and si = s4.
For si = s2 = -1:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s3 = 1.
- The conditional symbol error probability for si = s2 is 1 - (1/4) = 3/4.
For si = s3 = 1:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s4 = 2.
- The conditional symbol error probability for si = s3 is 1 - (1/4) = 3/4.
For si = s4 = 2:
- The ML decoder chooses the symbol with the maximum likelihood. Let's assume it chooses s3 = 1.
-
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In a digital communication system, a symbol S randomly chosen from the set (s1, s2, s3, s4) is transmitted. It is given that s1 = -3, s2 = -1, s3 = +1 and s4 = +2. The received symbol is Y = S W/. W is a zero-mean unit-variance Gaussian random variable and is independent of S. Piis the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol S = si. The index i for which the conditional symbol error probability Pi is the highest is _____ .Correct answer is '3'. Can you explain this answer?
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In a digital communication system, a symbol S randomly chosen from the set (s1, s2, s3, s4) is transmitted. It is given that s1 = -3, s2 = -1, s3 = +1 and s4 = +2. The received symbol is Y = S W/. W is a zero-mean unit-variance Gaussian random variable and is independent of S. Piis the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol S = si. The index i for which the conditional symbol error probability Pi is the highest is _____ .Correct answer is '3'. 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 In a digital communication system, a symbol S randomly chosen from the set (s1, s2, s3, s4) is transmitted. It is given that s1 = -3, s2 = -1, s3 = +1 and s4 = +2. The received symbol is Y = S W/. W is a zero-mean unit-variance Gaussian random variable and is independent of S. Piis the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol S = si. The index i for which the conditional symbol error probability Pi is the highest is _____ .Correct answer is '3'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a digital communication system, a symbol S randomly chosen from the set (s1, s2, s3, s4) is transmitted. It is given that s1 = -3, s2 = -1, s3 = +1 and s4 = +2. The received symbol is Y = S W/. W is a zero-mean unit-variance Gaussian random variable and is independent of S. Piis the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol S = si. The index i for which the conditional symbol error probability Pi is the highest is _____ .Correct answer is '3'. Can you explain this answer?.
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