The GATE Grind

GATE 2014 EC (EC4) – Question 59

Communications · Digital Communications · 2 marks · Multiple choice

A binary signal X has P(X = +1) = 0.75 and P(X = −1) = 0.25. Y = X + Z, with independent zero-mean Gaussian noise of variance σ². A detector outputs +1 if Y > τ and −1 otherwise. To minimize error probability, τ should be ______.

  1. strictly positive
  2. zero
  3. strictly negative
  4. strictly positive, zero or strictly negative depending on nonzero σ²

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Correct answer: (C) strictly negative

Explanation

MAP detection compares p(y|+1)/p(y|−1) with prior ratio0.25/0.75=1/3. For equal Gaussian variance, the log-likelihood ratio is2y/σ². Hence thresholdτ=(σ²/2)ln(1/3)=−(σ²/2)ln3, which is **strictly negative,C** for positive noise variance. The more likely+1symbol receives a larger decision region.