A decision on whether a signal is present or absent has to be made based on the observation of a random variable Y. Specifically, if the signal is absent, Y=Z, and if the signal is present, Y=A+Z, where A represents the constant (deterministic) signal level, and Z represents zero-mean Gaussian noise of variance σ
2
Z
. The decision is based on the maximum likelihood (ML) principle. If the signal-to-noise ratio (SNR) is A2/σ
2
Z
=5 dB, which of the following are the correct values of the probability of false alarm, Pfa=P{decide present | absent} and probability of correct detection, Pcd=P{decide present | present}? (Numbers are rounded.)

Question

A decision on whether a signal is present or absent has to be made based on the observation of a random variable Y. Specifically, if the signal is absent, Y=Z, and if the signal is present, Y=A+Z, where A represents the constant (deterministic) signal level, and Z represents zero-mean Gaussian noise of variance σ
2
Z
 . The decision is based on the maximum likelihood (ML) principle. If the signal-to-noise ratio (SNR) is A2/σ
2
Z
=5 dB, which of the following are the correct values of the probability of false alarm, Pfa=P{decide present | absent} and probability of correct detection, Pcd=P{decide present | present}? (Numbers are rounded.)

BlackTom AI · Accepted Answer

The problem describes a binary hypothesis test based on observing Y. When the signal is absent, Y ~ N(0, σ^2). When the signal is present, Y ~ N(A, σ^2). With equal priors and equal costs, the ML likelihood ratio test compares the likelihoods f_Y|present(Y) and f_Y|absent(Y) and yields the threshold Y > A/2 for deciding that the signal is present.

To evaluate Pfa, the probability of deciding present given absent, we compute P(Y > A/2 | Y ~ N(0, σ^2)). This is the tail probability of a N(0, σ^2) variable beyond A/2. Standardizing, (A/2)/σ = (1/2)√(A^2/σ^2). The problem gives SNR = A^2/σ^2 = 5 dB. Converting 5 dB to linear: SNR = 10^(5/10) ≈ 3.1623. Then (A/2)/σ = √3.1623 / 2 ≈ 1.778 / 2 ≈ 0.889. The false-alarm probability is Pfa = Q(0.889) ≈ 0.187, i.e., about 18.7%.

Next, for Pcd, the probability of correct detection, we compute P(Y > A/2 | Y ~ N(A, σ^2)). Standardize: (Y - A)/σ is N(0,1), and the threshold Y > A/2 corresponds to (Y - A)/σ > (A/2 - A)/σ = -A/(2σ) = - (1/2)√(A^2/σ^2) = -0.889. Therefore Pcd = P(Z > -0.889) for Z ~ N(0,1), which equals Φ(0.889) ≈ 0.813. So Pcd is about 81.3%.

Thus, the given option Pfa=18.7%, Pcd=81.3% aligns with the ML thresholding result for the specified SNR.

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