Bandwidth necessary and sufficient for the determination of a periodic quantized signal

Abstract

AbstractThe periodic quantized signal in this paper is defined as a piecewise constant signal which has a finite number of jumps in a period and has constant levels between adjacent jumping points. Periodic quantized signals are classified into four groups according to whether the jumping points and/or the height of jumps at the jumping points are, respectively, restricted to integer multiples of prescribed quantities or not.This paper considers the two groups among them: (1) the set of signals whose levels {aK} between adjacent jumping points are allowed to take arbitrary real values but the jumping points {tk} are restricted to integer multiples of T/N, where T is the time interval of a period and N is the number of jumping points in a period (signal set III); and (2) the set of signals whose levels aK} between adjacent jumping points and jumping points {tK} are restricted to integer multiples of a specified step size δ and respectively (signal set IV). We have studied the smallest amount of Fourier coefficients required for the determination of the original periodic quantized signal having N jumping points in a period. Our results assert that the Fourier coefficients necessary and sufficient for the determination of the original periodic quantized signal are given by: Up to the [N/2]th order for signals belonging to the signal set III, where [x] denotes the maximum integer not exceeding x. Up to the N0 ( N/p1)th order for signals belonging to the signal set IV, where p1 is the smallest prime factor of N.