III-posed sampling theorems

Abstract

There are a number of innocent appearing sampling theorems that are ill-posed, i.e., a small amount of noise superimposed on the data can render the interpolation unstable. Using Papoulis' Generalized Sampling Theorem, we show that a sufficient condition for a sampling theorem to be ill-posed is that an interpolation function has infinite energy. Specific examples include the case where (a) the signal and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2 n)</tex> th derivative of the signal are both simultaneously sampled at half the Nyquist rate and (b) the signal's and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2 n + 1)</tex> th derivative's samples are interlaced at Nyquist intervals.