Generalizations of the Sampling Theorem
Robert J. Marks · Oxford University Press eBooks · 2009
There have been numerous interesting and useful generalizations of the sampling theorem. Some are straightforward variations on the fundamental cardinal series. Oversampling, for example, results in dependent samples and allows much greater flexibility in the choice of interpolation functions. In Chapter 7, we will see that it can also result in better performance in the presence of sample data noise. Bandlimited signal restoration from samples of various filtered versions of the signal is the topic addressed in Papoulis’ generalization [1086, 1087] of the sampling theorem. Included as special cases are recurrent nonuniform sampling and simultaneously sampling a signal and one or more of its derivatives. Kramer [772] generalized the sampling theorem to signals that were bandlimited in other than the Fourier sense. We also demonstrate that the cardinal series is a special case of Lagrangian polynomial interpolation. Sampling in two or more dimensions is the topic of Section 8.9. There are a number of functions other than the sinc which can be used to weight a signal’s samples in such a manner as to uniquely characterize the signal. Use of these generalized interpolation functions allows greater flexibility in dealing with sampling theorem type characterizations. If a bandlimited signal has bandwidth B, then it can also be considered to have bandwidthW ≥ B.