Scale-Invariant Reconstruction of Separated Sources

Consider an array processing system that separates or identifies a signal or a signal subspace up to an unknown scaling factor. Sometimes it is necessary to cope with the scaling ambiguity, which can be done through reconstructing the signal as it is received by sensors, because scales of the sensor responses have known physical interpretations. In this paper, we propose computing the sensor responses using a scale-invariant formula that is derived based on the assumption that the signal (subspace) of interest is uncorrelated with other components in the original mixture. This approach is compared with a widely used one that assumes a regular mixing matrix and computes its inverse. We show, through a theoretical perturbation analysis and simulations, that the former approach is less sensitive to identification errors and is more practical, because the whole mixing matrix need not be identified. Moreover, in an underdetermined case, the approach is optimal in the mean-squared error sense. Applications from the area of noise reduction in speech and de-noising of signals from electrocardiogram are demonstrated.