Fast algorithms for signal subspace fitting with toeplitz matrices and applications to exponential data modeling

Publisher Summary This chapter presents fast methods for solving the following key problem: "Given an L x M Toeplitz matrix T of effective rank K ≪ min {L,M}, find an estimate of its K-dimensional signal subspace". This arises in signal processing and system identification problems, such as exponential data modeling, direction-of-arrival estimation and subspace tracking. The singular value decomposition (SVD), proven to be a valuable and reliable tool for solving this problem, is computationally too expensive. Replacing this SVD by a Low Rank-Revealing (LRR) two-sided orthogonal decomposition is shown to speed up the computations, up to 10 times. This chapter presents properties of this factorization, showing that the obtained subspace is exactly the same as the space generated from the start vectors used. The LRR factorization does not further improve the subspace accuracy and, therefore, iterative methods are more efficient when the rank is fixed. An iterative method, based on Lanczos recursions and block power iterations, is presented and shown to be up to 30 times faster than the SVD computation in an exponential modeling problem encountered in Nuclear Magnetic Resonance (NMR) data quantification.