Determine the optimal error probability

Question

Solve the following problem:

A binary communication system uses equiprobable signals s₁(t) and s₂(t)

s₁(t) = √2ε_bΦ₁(t) cos(2πf_ct)

s₂(t) = √2ε_bΦ₂(t) cos(2πf_ct)

for transmission of two equiprobable messages. It is assumed that φ₁(t) and φ₂(t) are orthonormal. The channel is AWGN with noise power spectral density of N₀/2.

1. Determine the optimal error probability for this system, using a coherent detector.

2. Assuming that the demodulator has a phase ambiguity between 0 and θ (0 ≤ θ ≤ π) in carrier recovery, and employs the same detector as in part 1, what is the resulting worst-case error probability?

3. What is the answer to part 2 in the special case where θ = π/2?