A binary communication system transmits the same information on two diversity channels. The two received signals are given by:
r1 = E + n1
r2 = E + n2
where n1 and n2 are uncorrelated Gaussian variables with mean zero and variance. The detector bases its decision on the linear combination of r1 and r2, given by:
r = a1r1 + a2r2
where a1 and a2 are constants.
a) Determine the value of a2 that minimizes the probability of error.
b) Plot the probability of error for a1 = 1 and either a2 = 1 or a2 is the optimum value found in part (a). Compare the results.