A Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform

TL;DR

This paper surveys arithmetic transform algorithms for spectrum analysis, introduces a new arithmetic Hartley transform, and emphasizes the importance of interpolation in arithmetic transform theory.

Contribution

It provides a comprehensive survey of existing arithmetic Fourier transform algorithms and introduces a novel arithmetic Hartley transform.

Findings

Arithmetic transforms reduce multiplications in spectrum computation

The interpolation process is crucial in arithmetic transform theory

The new arithmetic Hartley transform offers an alternative for discrete Hartley transform computation

Abstract

Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the proposed algorithms for the arithmetic Fourier transform are surveyed. A new arithmetic transform for computing the discrete Hartley transform is introduced: the Arithmetic Hartley transform. The interpolation process is shown to be the key element of the arithmetic transform theory.