Determine the noise power spectral density

Question

Solve the following problem:

A binary communication system uses two equiprobable messages 1(t) = p(t) and s₂(t) = -p(t). The channel noise is additive white Gaussian with power spectral density N₀/2. Assume that we have designed an optimal receiver for this channel, and let the error probability for the optimal receiver be Pe.

1. Find an expression for Pe.

2. If this receiver is used on an AWGN channel using the same signals but with the noise power spectral density N₁ > N₀, find the resulting error probability P1 and explain how its value compares with Pe.

3. Let Pe1 denote the error probability in part 2 when an optimal receiver is designed for the new noise power spectral density N₁. Find Pe₁ and compare it with P_1.

4. Answer parts 1 and 2 if the two signals are not equiprobable but have prior probabilities p and 1 - p.