Let R denote the random variable obtained by observing the output of an envelope detector at some fixed time. Intuitively, the envelope detector is expected to be operating well into the threshold region if the probability that the random variable R exceeds the carrier amplitude A_c is 0.5. On the other hand, of this same probability is only 0.01, the envelope detector is expected to be relatively free of loss of message and the threshold effect.

(a) Assuming that the narrowband noise at the detector input is white, zero-mean, Gaussian with spectral density N₀/2 and the message bandwidth is W, show that the probability of the event R > A_c is P (R > A_c) = exp (-p), where p is the carrier-to-noise ratio: p = A²_c / 4WN₀.

(b) Using the formula for this probability, calculate the carrier-to-noise ratio when (1) the envelope detector is expected to be will into the threshold region, and (2) it is expected to be operating satisfactorily.