What is the resulting worst-case error probability

Question

Solve the following problem:

A communication system employs binary antipodal signals with

s₁(t) = {1       0           {0       otherwise

and s₂(t) = -s₁(t). The received signal consists of a direct component, a scattered component, and the additive white Gaussian noise. The scattered component is a delayed version of the basic signal times a random amplification A. In other words, we have r(t) = s(t) + As(t - 1) + n(t), where s(t) is the transmitted message, A is an exponential random variable, and n(t) is a white Gaussian noise with a power spectral density of N₀/2. It is assumed that the time delay of the multipath component is constant (equal to 1) and A and n(t) are independent. The two messages are equiprobable and

ƒ_A(a) =  {e^-a      a>0
             {0        otherwise

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?