Imperfect Phase Information

The following binary signal set is used to transmit equally likely messages over an additive white Gaussian noise channel (spectral height $\frac{N_0}{2}$),

$$\begin{array}{cc} s_0(t) = & \left\{ \begin{array}{cl} \sqr... ...begin{array}{cl} 0 & \mbox{for all $t$.} \end{array}\end{array}$$

1. Draw a block diagram of the optimum receiver and compute the probability of error for this receiver.
2. Assume now that the signal $s_0(t)$ experiences an unknown phase shift during transmission, i.e., if $s_0(t)$ is transmitted the received signal is
   
   $$R_t = \sqrt{\frac{2E}{T}} \cos(2\pi f_c t + \phi) + N_t.$$
   
   
   Compute the probability of error that your receiver from part (a) achieves in this situation.
3. Illustrate the effect of the unknown phase shift in a suitably chosen signal space, i.e., sketch the location of signals $s_1(t)$ and $\sqrt{\frac{2E}{T}} \cos(2\pi f_c t + \phi)$ in signal space as a function of $\phi$ for $0 \leq \phi < 2\pi$.
4. Based on your observations from part (c), what is the decision boundary associated with the optimum receiver for distinguishing between $s_1(t)$ and $\sqrt{\frac{2E}{T}} \cos(2\pi f_c t + \phi)$ for unknown $\phi$. It is sufficient to indicate this decision boundary in signal space and describe it in words. You do not have to write down the decision rule.
5. Draw the block diagram of a receiver that implements the decision boundary you determined above.

Hint: $\cos(x+y) = \cos x \cos y - \sin x \sin y$.