Jamming and Football

The secret to the Washington Redskins success this year stems from the installation of a new digital communication system for relaying messages from the press box to the field. A former ECE 460 student designed the following binary signal set:

$$\begin{array} {cc} s_0(t) = \sqrt{2f_0} \cos(2 \pi f_0t) & s_1(t) = - \sqrt{2f_0} \cos(2 \pi f_0t). \end{array}$$

The duration of each transmission interval is 1/f₀ and the frequency f₀ is 1 MHz. The communication channel is modeled as an additive, white Gaussian noise channel with spectral amplitude 1. Assume the signals are equally likely.

1.

What is the minimum probability of error that any receiver can achieve when this presumably well designed communication system is used. (2 pts.)

2.

When visiting RFK stadium, the Houston Oilers decided to jam this system and nearly managed to beat the Redskins. What they did is to transmit a constant amplitude cosine wave of frequency f₀. Thus, in the presence of the jammer, the received signal can be modeled as

$$R_t = s_i(t) + \sqrt{2f_0} \cos (2 \pi f_0 t) + N_t.$$

The noise has the same characteristics as described above. What is the probability of error when the receiver from part (a) is used. (3 pts)

3.

In this game, the famous Redskins' half-time adjustments included a redesign of the receiver for the communication system. Find the optimum receiver for communication in the presence of the jamming signal and the corresponding probability of error. (3 pts.)

4.

Illustrate the effecct of the jammer in signal space. I.e., draw and accurately label the positions of the signals (with and without jammer) in signal space and indicate the decision boundaries of the receivers from parts (a) and (c). (2 pts)

Hint: $\cos^2 x = \frac{1}{2}(1+\cos 2x).$