Performance Analysis of Source Image Estimators in Blind Source Separation

Blind methods often separate or identify signals or signal subspaces up to an unknown scaling factor. Sometimes it is necessary to cope with the scaling ambiguity, which can be done through reconstructing signals as they are received by sensors, because scales of the sensor responses (images) have known physical interpretations. In this paper, we analyze two approaches that are widely used for computing the sensor responses, especially, in frequency-domain independent component analysis. One approach is the least-squares projection, whereas the other one assumes a regular mixing matrix and computes its inverse. Both estimators are invariant to the unknown scaling. Although frequently used, their differences were not studied yet. A goal of this paper is to fill this gap. The estimators are compared through a theoretical study, perturbation analysis, and simulations. We point to the fact that the estimators are equivalent when the separated signal subspaces are orthogonal, and vice versa. Two applications are shown, one of which demonstrates a case where the estimators yield substantially different results.