For this problem, consider a radio transmitter and three receive
antennas. The transmitter uses the signals

to transmit equally-likely messages. When the transmitter sends the signal , the signal received at the -th antenna is

where the noise processes are white and Gaussian with spectral height . Furthermore, the noise processes at different antennas are statistically independent from each other.

- Compute the minimum probability of error if the receiver considers only the signal received at one antenna.
- One receiver structure is to simply add the signals from the three antennas and use the optimum receiver for this signal. I.e., the signal is taken and processed by the optimum receiver for . Find this optimum receiver and determine the resulting probability of error.
- Another receiver structure is to operate on each antenna output separately with an optimum receiver and use a majority vote to determine the receiver's output. What is the resulting performance of this scheme?
- One of the preceding receivers is optimum. Determine which is optimum and prove it to be so.

2003-01-28