Approximated Cramér–Rao bound for estimating the mixing matrix in the two-sensor noisy Sparse Component Analysis (SCA)

In this paper, we address theoretical limitations in estimating the mixing matrix in noisy Sparse Component Analysis (SCA) in the two-sensor case. We obtain the Cramér–Rao Bound (CRB) error estimation of the mixing matrix based on the observation vector x=(x1,x2)T. Using the Bernoulli–Gaussian (BG) sparse distribution for sources, and some reasonable approximations, the Fisher Information Matrix (FIM) is approximated by a diagonal matrix. Then, the effect of off-diagonal terms in computing the CRB is investigated. Moreover, we compute an oracle CRB versus the blind uniform CRB and show that this is only 3 dB better than the blind uniform CRB. Finally, the CRB, the approximated CRB, the uniform CRB and the oracle CRB are compared to each other and to some of the main mixing matrix estimation methods in the literature. Simulation results show that the approximated CRB is close to the CRB for high SNRʼs. They also show that the approximated CRB is approximately equal to the oracle CRB.