Strictry Proper Digital Filters with Hypercomplex Coefficients and Reduction of Multiplications in Hypercomplex Algebra

Abstract

In this paper, we consider strictly proper digital filters with hypercomplex coefficients, and we also consider reduction of multiplications in hypercomplex algebra.
As for the former problem, the first-order digital filter with hypercomplex coefficients can realize arbitrary-order strictly proper digital filters with real coefficients less than 4, whose order of numerator is smaller than that of denominator by 1.
As for the latter problem, if we apply Quadratic Residue Number System to hypercomplex algebra, we can reduce the number of multiplications to 1/4 of the original one. And the most important thing of this result is that we only need 4 multiplications per multiplier and that these 4 multiplications can be done at the same time.