Solve the following problem:
Consider a digital communication system that uses two transmitting antennas and one receiving antenna. The two transmitting antennas are sufficiently separated so as to provide dual spatial diversity in the transmission of the signal. The transmission scheme is as follows:
If s1 and s2 represent a pair of symbols from either a one-dimensional or a two-dimensional signal constellation, which are to be transmitted by the two antennas, the signal from the first antenna over two signal intervals is (s1,s∗2 ) and from the second antenna the transmitted signal is (s2, -s∗1 ). The signal received by the single receiving antenna over the two signal intervals is
r1 = h1s1 + h2s2 + n1
r2 = h1s*2 + h2s*1 + n2
where (h1, h2) represent the complex-valued channel path gains, which may be assumed to be zero-mean, complex Gaussian with unit variance and statistically independent. The channel path gains (h1, h2) are assumed to be constant over the two signal intervals and known to the receiver. The terms (n1, n2) represent additive white Gaussian noise terms that have zero-mean and variance σ2 and uncorrelated.
a. Show how to recover the transmitted symbols (s1,s2) from (r1,r2) and achieve dual diversity reception.
b. If the energy in the pair (s1,s2) is (Es, Es) and the modulation is binary PSK, determine the probability of error.
c. Repeat (b) if the modulation is QPSK.