On the bandwidth necessary and sufficient for the determination of a periodic quantized signal
K. Sakaniwa · 1st IASTED International Symposium on Signal Processing and its Applications · 1987
A periodic quantized signal treated in this paper is defined as a piece-wise constant signal which has a finite number of jumps in a period and has constant levels between adjacent jumping points. According as the jumping points and/or the height of jumps at the jumping points are, respectively, restricted to integer multiples of a prescribed quantities or not, periodic quantized signals are classified into the four groups. In this paper, we consider 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 r Delta= 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 t 0 integer multiples of a specified step size Delta and r, respectively (Signal Set IV). And we have studied about the smallest number 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 give n by: (1) Up to the [N/2]-th order for signals b elonging to the signal set III, where [x] denotes the maximum integer not exceeding x. (2) Up to the N0 (Delta=N/p1)-th order for signals belonging to the signal set IV, where p1 is the smallest prime factor of N.