Unified Theory: Sampling and Interpolation

Abstract

An I→J down-sampling, where J⊂I, performs a restriction of the signal domain, typically from a continuous domain I to a discrete domain J (lattice), with the consequent loss of signal values. Interpolation goes in the opposite direction, with the creation of new signal values, starting from the surviving samples. In this context, the basic problem is the recovery of the original signal from its samples. The structure of the recovery tf is potentially given by a J→I interpolator. The possibility of recovery must be investigated in the frequency domain and will be stated in terms of the band-limitation of the signal to be sampled. This leads to the celebrated Sampling Theorem, which represents a milestone of today’s digital signal processing and, more generally, of the information age. With the Unified Signal Theory, it is possible to formulate a Unified Sampling Theorem, which includes a variety of results as corollaries. Band-limitation is formulated in terms of a cell of the frequency domain, which must contain the support of the Fourier transform, and the interpolator too must be defined in terms of this cell.