Designing an optimal matched filter receiver

Question

Solve the following problem:

A binary communication scheme uses two equiprobable messages m = 1, 2 corresponding to signals s₁(t) and s₂(t), where

s₁(t)= x(t)

s₂(t)= x(t-1)

and x(t) is shown Figure.

The power spectral density of the noise is N₀/2.

1. Design an optimal matched filter receiver for this system. Carefully label the diagram and determine all the required parameters.

2. Determine the error probability for this communication system.

3. Show that the receiver can be implemented using only one matched filter.

4. Now assume that s1(t) = x(t) and

s₂(t) ={x(t-1)     with probability 0.5
          {x(t)        with probability 0.5

In other words, in this case for m = 1 the transmitter always sends x(t), but for m = 2 it is equally likely to send either x(t) or x(t - 1). Determine the optimal detection rule for this case, and find the corresponding error probability.