On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation

The problem of extrapolating a band-limited signal in discrete time is viewed as one of solving an underdetermined system of linear equations. Choosing the minimum norm least-squares (MNLS) solution is one criterion for singling out an extrapolation from all the possible solutions to the linear system. Use of the Moore-Penrose inverse yields the MNLS solution, and singular value decomposition (SVD) provides a means for implementing the Moore-Penrose inverse. An expression for the mean-square error incurred in solving a linear system via SVD is derived. This can be used to estimate the number of singular values needed to form the inverse. The error expression also indicates that decimation can be applied in the extrapolation problem to reduce the high computational cost of SVD without degrading the extrapolation. The results developed for the one-dimensional case are extended to higher dimensions. Examples of the SVD approach to extrapolation are given, along with examples using other extrapolation techniques for comparison. The SVD approach compares favorably with known MNLS extrapolation methods.