The Arithmetic Cosine Transform: Exact and Approximate Algorithms

TL;DR

This paper introduces the arithmetic cosine transform (ACT), a new transform method utilizing analytic number theory, with exact and approximate algorithms, demonstrating its potential for digital signal processing applications.

Contribution

The paper presents the mathematical foundation, exact and approximate algorithms, and potential applications of the novel arithmetic cosine transform (ACT).

Findings

Exact signal interpolation is achievable for any block-length.

Numerical examples demonstrate the potential of ACT in digital signal processing.

The use of analytic number theory enhances the mathematical properties of the transform.

Abstract

In this paper, we introduce a new class of transform method --- the arithmetic cosine transform (ACT). We provide the central mathematical properties of the ACT, necessary in designing efficient and accurate implementations of the new transform method. The key mathematical tools used in the paper come from analytic number theory, in particular the properties of the Riemann zeta function. Additionally, we demonstrate that an exact signal interpolation is achievable for any block-length. Approximate calculations were also considered. The numerical examples provided show the potential of the ACT for various digital signal processing applications.