Application of the Mittag-Leffler expansion to sampling discontinuous signals

In applying Shannon´s sampling theorem, evaluation of the sampled signal Fourier spectrum is based on the fact that sampling the continuous-time signal is the result of multiplying the signal by distributions. If the signal has discontinuities, a multiplication of distributions - an undefined operation - is encountered. Such undefined operation has led to errors in the literature which to date accompany the formulation of sampling of signals containing discontinuities. This paper presents an approach to evaluating the product of distributions as a means of sampling discontinuous signals, eliminating such errors. It is shown that the value of the product of distributions may be found by invoking the Mittag-Leffler expansion. As an illustration of errors that have existed for decades and still exist in the digital signal processing literature whenever discontinuous signals are sampled the approach of impulse invariance provides a case in point. It was already noted that this approach has an inherent error. Yet, impulse invariance is still considered as one of the two main approaches for converting analogue to digital filters. In this study, the true spectra of sampled discontinuous signals are evaluated, and a new approach to the transformations between continuous-time and discrete-time systems eliminating the error, is proposed.