GATE EC 2020 Set 1 — Question 58
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Communications → Digital Modulation & Detection → Error Probability
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Question
In a digital communication system, a symbol randomly chosen from the set is transmitted. It is given that and . The received symbol is . is a zero-mean unit-variance Gaussian random variable and is independent of . is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol . The index for which the conditional symbol error probability is the highest is ______.
Correct answer
3 to 3
Solution
In a digital communication system with additive white Gaussian noise (AWGN), the maximum likelihood (ML) decoder uses the minimum distance rule. The decision boundaries are the midpoints between adjacent symbols.Given symbols: .
Distances between adjacent symbols:
The conditional error probability for symbol is the probability that the noise (with variance ) pushes the received signal outside the decision region for .For : Error if . .
For : Error if or . .
For : Error if or . .
For : Error if . .Since the -function is a strictly decreasing function of :
Therefore:
The highest conditional error probability is , so the index is 3.
Distances between adjacent symbols:
The conditional error probability for symbol is the probability that the noise (with variance ) pushes the received signal outside the decision region for .For : Error if . .
For : Error if or . .
For : Error if or . .
For : Error if . .Since the -function is a strictly decreasing function of :
Therefore:
The highest conditional error probability is , so the index is 3.
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