Generalized Distributions, Sampling Theorem Revisited and an End to an Impulse Invariance Error

摘要

Recently proposed generalized Functions of a complex variable extend the domains of existence of Laplace and z-transform. In this paper, basic properties of generalized distributions are extended. In particular, properties of generalized distributions in the context of sampling functions containing discontinuities are explored. Laplace and z-transforms of one-sided, two-sided periodic and exponentially modulated periodic impulses, hitherto nonexistent, and of which the Fourier transform does not exist, are evaluated using generalized distributions. Applications to the formulation of the sampling theorem are explored. The generalized distributions are shown to reveal an unusual anomaly in the well-known digital filtering approach of impulse invariance. The anomaly arises when Laplace and z-transform spectra are compared with the Mittag-Leffler Expansion. It is shown that the transformation, as is presently applied, does not produce the stated Fourier spectrum of the sampled signal. In fact it produces more spectral aliasing than claimed, and not the minimum desired. In converting an analog filter of even a small order to a digital filter, an appreciable spectral deviation error is produced. A new approach to impulse invariance, eliminating the error, is proposed. Matlab still uses the erroneous approach. A generalization is effected to cover a wide class of one-sided, two sided and finite duration signals.