Determine the covariance matrix of the noise

Question

Solve the following problem:

Consider an uncoded MIMO system with NT = NR antennas that transmits over a frequency-nonselective channel in which the channel matrix H has iid complex-valued, zero-mean Gaussian elements. The received signal vector is

y = Hs + η

where the elements of η are iid complex-valued, zero-mean Gaussian. The detector used at the receiver is the inverse channel detector (ICD).

a. Determine the covariance matrix of the noise at the output of the detector.

b. If the detector makes independent decisions on each of the NT transmitted symbols, is this detector optimum (in the sense of minimizing the error probability)?

c. If BPSK modulation is employed, determine the error probability of the detector described in (b).

d. Now, suppose that NR > NT and the decisions made by the detector are based on the signal estimate sˆ = W H y, where W_H = (HH H) -1H_H. Repeat parts (a) and (b).