Bandlimited extrapolation using time-bandwidth dimension

The problem of extrapolating discrete-index bandlimited signals from a finite number of samples is addressed in this paper. The algorithm presented in this paper exploits the fact that the set of bandlimited signals that are also essentially time-limited is approximated well by a low-dimensional linear subspace. This fact, which is well known for one-dimensional (1-D) signals with contiguous passbands and time-concentration intervals, is established for a more general class of multidimensional (m-D) signals with discontiguous passbands and discontiguous time-concentration regions. A criterion is presented for determining the dimension of the approximating subspace and the minimax optimal subspace itself based on knowledge of the passband, time-concentration regions, energy concentration factor, and bounds on the tolerable extrapolation error. The extrapolation is constrained to lie in this subspace, and parameters characterizing the extrapolation are obtained from the data by solving a linear system of equations. For certain sampling patterns, the system is ill conditioned, and a second rank reduction is needed to reduce the deleterious effects of observation noise and modeling error. A novel criterion for rank selection based on known bounds on noise power and modeling error is presented. The effectiveness of the new algorithm and the rank selection criterion are demonstrated by means of computer simulations