The interpolation of signals sampled at less than the Nyquist rate

To obtain analog information by digitally processing uniformly spaced samples depends on the interpolation scheme. The constraints for the usual Shannon-Whittaker interpolation is to sample such signals at or above the Nyquist rate of twice the highest frequency present. This interpolation scheme produces poor reconstruction for signals with no highest frequency. In this dissertation, a new sampling theorem is developed based on the Newton-Gregory interpolation formula which is applicable to signals produced by stable physical systems with rational transfer functions. According to this theorem, such a signal is uniquely specified by its samples whenever the poles of the system transfer function all have frequencies less than one-sixth the sampling rate. A direct application of the Newton-Gregory interpolation scheme is made to numerically evaluate the spectra of poorly bandlimited signals at frequencies well above the sampling rate.