Influence of Phase Errors

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

$$\begin{array} {cl} s_0(t) = & \left\{ \begin{array} {cl} \s... ... \leq t \leq T$}\\ 0 & \mbox{else}\end{array}\right.\end{array}$$

1.

Draw a block diagram of the optimum receiver which uses only a single correlator.

2.

Compute the probability of error for this receiver.

3.

Assume now that the local oscillator at the receiver is not in-phase with the received signal. In other words, the receiver correlates with $\sin(2\pi f_c t + \phi)$instead of $\sin(2\pi f_c t)$.Find the probability of error as a function of the phase error $\phi$.

4.

Compute the probability of error for the case $\phi=\frac{\pi}{2}$and illustrate this case in a suitably chosen signal space.

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