Bounds, sampling, and amplitude uncertainty of band-limited functions

This work examines the amplitude fluctuations of band-limited functions and the representation of these functions by piecewise linearly interpolated samples. Such representations are implicitly assumed in many digital signal processing computations, including the estimation of level crossing profiles and the location of extrema in a discrete sequence of samples.

A comprehensive analysis of the average rate of change of bandlimited functions is presented, including a derivation of the least upper bound on functions for which either the zeroth absolute moments (spectral areas) or Fourier amplitude spectra of the functions are bounded. These results are extended to the case of functions that are limited in both frequency and amplitude. The average rate of change of a function over an interval in which one end point of the interval is an extremum of the function is similarly bounded and used to establish a sampling rate which guarantees that, between successive samples of a band-limited function, the function itself does not deviate from either sample by more than some predefined amplitude change. Based on these results, several problems of practical importance, including slope overload error in delta modulation, aperture time and amplitude uncertainty in analog-to-digital conversion, the estimation of level crossing statistics, and the detection and elimination of high frequency pulses from a low frequency signal, are examined.