Application of degenerate perturbation theory to subspace tracking

Abstract

First-order perturbation theory has been applied to subspace tracking. This previous work did not solve the problem of signal subspace near-degeneracy. The signal subspace is nearly degenerate when two or more signal subspace eigenvalues are nearly equal, which can occur under a number of circumstances. Failure to correctly handle the near-degeneracy leads to errors in the subspace decomposition. We demonstrate here the application of degenerate perturbation theory to signal subspace tracking under conditions where the eigenvalues may be degenerate. The problems of noise subspace degeneracy and signal subspace near-degeneracy are both solved through diagonalization of the perturbation matrix over the degenerate subspace. The degenerate subspace diagonalization markedly reduces the subspace error with a minimal increase in computational complexity. Similarly, it is shown that using a second-order perturbation update reduces the subspace error by a substantial margin, again with a minimal increase in computational complexity. The effect of increasing the integration time is also examined. By using the second-order perturbation update and the degenerate subspace diagonalization, the resulting subspace decomposition is extremely accurate, robust, and fast.