Extrapolation of discrete-time band-limited signals using singular value decomposition with decimation

The problem of extrapolating a band-limited signal in discrete-time is viewed as one of solving an underdetermined system of linear equations. The matrix to be inverted is also generally ill-conditioned. Singular value decomposition (SVD) provides both a means for implementing the inverse and a method for improving the numerical stability of the problem. An expression for the mean-square error incurred in solving a system of linear equations via SVD is derived. This expression can be used to estimate the number of singular values needed to form the inverse. Further examination of the expression indicates that, in the case of an oversampled signal, decimation can be applied without significantly degrading the extrapolation. The results developed for the one-dimensional case can be extended to higher dimensions, as illustrated by the two-dimensional case. Examples of the SVD approach to extrapolation are given, along with examples using other extrapolation techniques for comparison. The SVD approach is seen to yield the best extrapolation of the known minimum-norm least-squares methods.