Coherent Reception with Imperfect Phase Information

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

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

1.

Draw a block diagram of the optimum receiver and compute the probability of error for this receiver.

2.

Assume now that the signal s₀(t) experiences a phase shift during transmission, i.e., if s₀(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.

If the phase error is large, the performance of the coherent receiver is unacceptable. Draw a block diagram of a receiver that performs well for all values of $\phi$.

4.

If it is known that the phase error is small, i.e., $\vert\phi\vert < \frac{\pi}{6}$, which receiver performs better: the coherent receiver from part (a) or the non-coherent receiver from part (c). Explain.

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