Binary Signal Sets

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

$$\begin{array} {cll} s_0(t) & = \sqrt{\frac{2E}{T}} \cos(2 \... ...(2 \pi f_0 t + \phi) & \mbox{for $0 \leq t \leq T$}\end{array}$$

The value of the phase $\phi$ is known at the receiver. You may assume that f₀T is an integer.

1.

Draw and accurately label a block diagram of the optimum receiver.

2.

Compute the probability of error P_e that your receiver from part (a) achieves as a function of the phase $\phi$.

3.

For which value of $\phi$ is the probability of error minimized? For which value of $\phi$ is the probability of error maximized?

4.

Draw and accurately label separate signal space diagrams indicating the location of the two signals and the optimum decision regions for the the following three cases:

• $\phi = 0$
• $\phi = \frac{\pi}{2}$
• $\phi = \pi$

5.

Discuss your results from part (c) in light of the diagrams obtained in part (d).

Hint: The following identity may be useful.

$$\cos x \cos y = \frac{1}{2}(\cos(x-y) + \cos(x+y))$$