Existence and Uniqueness of Band-Limited, Positive Semidefinite Extrapolations

摘要

This correspondence investigates the extrapolation of a one-dimensional sequencesubject to the constraints that its spectrum be band-limited to a given frequency band and be real-valued and nonnegative on that band. An easily computable matrix test for the existence of a band-limited, positive semidefinite extrapolation is derived, and an algorithm is presented for the recursive computation of such an extrapolation. Positive semidefinite extrapolation of incompletely specified sequences lies at the heart of many signal processing applications of practical interest. This extrapolation problem is equivalent to recovering a band-limited power spectrum from measurements of correlation lags and prior knowledge of spectral support based on the physical properties of sources, sensors, medium, or problem geometry. Also, the extrapolation problem generalizes the well-known band-limited extrapolation problem (1)-(3) to incorporate a positive semidefiniteness constraint (4). The theorem presented here provides an easily computed answer to the questions of existence and uniqueness of such extrapolations. In addition, an algorithm for constructing an extrapolation follows from the constructive proof of the theorem. Keywords: Reprints, Statistical analysis, Acoustics, Speech, Signal processing. (kr)