GATE 2014 EC (EC4) – Question 62
A discrete-time channel has Y = X + Z and X ∈ {−a,+a}. Conditional on X, independent noise Z is Gaussian with mean βX and unit variance. A zero-threshold detector gives BER $Q(a)=10^{-8}$ when β = 0. With $Q(v)=(2\pi)^{-1/2}\int_v^\infty e^{-u^2/2}du$ and approximation $Q(v)\approx e^{-v^2/2}/2$ for v > 1, find the closest BER for β = −0.3.
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Correct answer: (C) $10^{-4}$
Explanation
Conditional on X, Yhas mean(1+β)Xand unit variance. At β=−0.3the decision distance becomes0.7a, so BER=Q(0.7a). Using the supplied approximation Q(v)≈e^(−v²/2)/2and Q(a)=10⁻⁸gives Q(0.7a)≈(1/2)(2×10⁻⁸)^0.49≈8.44×10⁻⁵. The closest listed order is **10⁻⁴,C**.