Analysis of the chinese remainder theorem and cyclotomic polynomials-based algorithms for cyclic convolution - Part I: Rational number system

Abstract

The Chinese remainder theorem plays a central role in the design of fast algorithms for computing cyclic and acyclic convolution of data sequences. A mathematical analysis of the theorem and related aspects form the topic of this investigation. The focus is almost exclusively on cyclic convolution algorithms. The number system that is studied is the field of rational numbers. Several properties related to the mathematical structure of the algorithms are derived. Note: This work is being reported in two parts. In Part I, we analyze the structure of the cyclic convolution algorithms over the rational number system. In Part II, we describe complex cyclotomic polynomials and the structure of the resulting cyclic convolution algorithms over the complex rational number system.