Solve the following problem:
Two equiprobable messages m1 and m2 are to be transmitted through a channel with input X and output Y related by Y = ρX + N, where N is a zero-mean Gaussian noise with variance σ2 and ρ is a random variable independent of the noise.
1. Assuming an antipodal signaling scheme (X = ±A) and a constant ρ = 1, what is the optimal decision rule and the resulting error probability?
2. With antipodal signaling, if ρ takes ±1 with equal probability, what will be the optimal decision rule and the resulting error probability?
3. With antipodal signaling, if ρ takes 0 and 1 with equal probability, what will be the optimal decision rule and the resulting error probability?
4. Assuming an on-off signaling (X = 0 or A) and ρ taking ±1 with equal probability, what will be the optimal decision rule?