Phase Shift Keying (PSK)

The signal set

$$\begin{array}{ccll} s_1(t)& =& \cos(2\pi f_ct + \frac{\pi}{... ...pi f_ct+\frac{\pi}{4}) & \mbox{for $0 \leq t < T$,} \end{array}$$

where $f_c T$ is an integer, is used to transmit one of two equally-likely messages over an additive, white Gaussian noise channel with spectral height $\frac{N_0}{2}$.

1. Draw a block diagram of the optimum receiver. Can this receiver be implemented using only one correlator.
2. Compute the probability of error for this receiver.
3. Now, the signals
   
   $$\begin{array}{ccll} s_3(t)& =& \sin(2\pi f_ct + \frac{\pi}{... ...pi f_ct+\frac{\pi}{4}) & \mbox{for $0 \leq t < T$,} \end{array}$$
   
   
   are added to the above signal set. Each signal is then used to transmit one of four equally likely messages. Draw a block diagram of the optimum receiver for this quaternary signal set in additive white Gaussian noise.
4. Compute the resulting probability of error.

Hint: Remember that the likelihood ratio test

$$\Lambda(\underline{R}) = \frac{p_{\underline{R}\vert H_1}(\u... ...}\vert H_0)} \stackrel{\displaystyle >}{<} \frac{\Pi_0}{\Pi_1}$$

can also be written as

$$\Pi_1 p_{\underline{R}\vert H_1}(\underline{R}\vert H_1) \st... ...}{<} \Pi_0 p_{\underline{R}\vert H_0}(\underline{R}\vert H_0).$$