On the minimum sampling rate of signals of non-bandlimited response

Recently, consistently resampling has been proposed to resampling discrete signals without band-limit constraint. In this paper, we study the constraints of the resampling rate such that the input discrete signal can be consistently resampled. We approach it through identifying the rate of innovation (RI) of the signal as introduced by innovation sampling. First, an upper bound on the RI for signals in shift invariant spaces is established. The RI of the signal indicates the globally minimum sampling rate for the signal regardless of the choice of sampling function. Then the properties of the sampling filters such that the signal can be sampled at this rate are specified. Further, we extend the choice of sampling functions to a wider set of general functions from the Hilbert space and the locally minimum sampling rate for the function used accordingly. The results obtained herein are used to obtain the minimum resampling rate for consistent resampling.