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Consider a communication scheme where the binary valued signal X satisfies P{X = +1} = 0.75 and P{X = −1} = 0.25. The received signal Y = X + Z, where Z is a Gaussian random variable with zero mean and variance σ². The received signal Y is fed to the threshold detector. The output of the threshold detector X̂ is: X̂ = { +1, Y > τ −1, Y ≤ τ. To achieve a minimum probability of error P{X̂ ≠ X}, the threshold should be

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

Correct answer: strictly negative

Solution

A strictly negative threshold is optimal because it accounts for the higher probability of receiving +1 signals, allowing the detector to correctly classify more received signals as +1, thus minimizing the probability of error.

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