Polarized Signal Unmixing Utilizing Homology Groups Representing Momentum States

A null inner product is successfully applied as an orthogonality constraint for scalar-sensor arrays. This analysis derives a signal processing method optimized for unmixing the signals collected by vector-sensor arrays. Vector-sensor arrays, such as polarization-diverse planar arrays, simultaneously collect multiple polarized signal components from each source. Instead of projecting the collected signals onto a basis set of orthogonal unit polar vectors, this method projects the signals onto a basis set of unitary matrices enabling joint representation of not only polar but also axial symmetries. This method exploits the fact that unmixed solutions preserve a homology group representing a time-invariant momentum state unique to each source. A mixture of multiple elliptically polarized signals cannot be completely unmixed by a signal processing algorithm utilizing a complex vector representation as they do not account for the momentum contribution to the signal´s Hamiltonian induced by interaction with elliptically polarized photons. This group-theoretic unmixing relation is demonstrated to possess a signal processing gain with respect to complex vector-algebra approaches when there are multiple elliptically-polarized signals in the mixture. This method delivers optimal polarized signal unmixing for vector-sensor systems under real-world conditions, where the polarization diversities of signals need not be assumed to be constrained in any manner. Homology group matched filtering will find application in signal processing systems which need to optimally exploit elliptically polarized signals.