Towards a Number Theoretic Discrete Hilbert Transform

TL;DR

This paper introduces a number theoretic discrete Hilbert transform, providing a 16-point implementation and preliminary inverse, aiming to develop a transform with potential applications in signal processing.

Contribution

It proposes a novel number theoretic approach to the discrete Hilbert transform, including a specific 16-point transform and initial inverse transform formulation.

Findings

Implemented a 16-point number theoretic DHT

Provided example results for selected signals

Identified the inverse transform as provisional and not identical to the forward

Abstract

This paper presents an approach for the development of a number theoretic discrete Hilbert transform. The forward transformation has been applied by taking the odd reciprocals that occur in the DHT matrix with respect to a power of 2. Specifically, the expression for a 16-point transform is provided and results of a few representative signals are provided. The inverse transform is the inverse of the forward 16-point matrix. But at this time the inverse transform is not identical to the forward transform and, therefore, our proposed number theoretic transform must be taken as a provisional result.