Diversity Reception

One method by which transmission errors can be reduced is diversity reception. Simply stated, diversity reception means that the transmitted information can be received more than once, thereby enabling the receiver to have more opportunities to determine what was transmitted.

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

$$\begin{array}{ccll} s_1(t) & = & \sqrt{E/T} & \mbox{for $0 \... ... & & \\ s_0(t) & = & 0 & \mbox{for $0 \leq t < T$} \end{array}$$

to transmit equally-likely messages. When the transmitter sends the signal $s_i(t)$, the signal received at the $j$-th antenna is

$$r_j(t) = s_i(t) + n_j(t),\;\; j=1,2,3,$$

where the noise processes $n_j(t)$ are white and Gaussian with spectral height $\frac{N_0}{2}$. Furthermore, the noise processes at different antennas are statistically independent from each other.

1. Compute the minimum probability of error if the receiver considers only the signal received at one antenna.
2. 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 $r(t) = \sum_{j=1}^3 r_j(t)$ is taken and processed by the optimum receiver for $r(t)$. Find this optimum receiver and determine the resulting probability of error.
3. 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?
4. One of the preceding receivers is optimum. Determine which is optimum and prove it to be so.