Infinite Sequences, Series Convergence and the Discrete Time Fourier Transform over Finite Fields

TL;DR

This paper introduces finite field versions of the discrete time Fourier transform and the finite field Fourier transform, enabling their application to digital signal processing over finite algebraic structures.

Contribution

It presents a finite field adaptation of the DTFT and redefines the FFFT with a complex kernel, enhancing their suitability for finite algebraic applications.

Findings

Finite field DTFT introduced for the first time.

Redefinition of FFFT with a complex kernel.

Transforms applicable to FIR and IIR filters over finite fields.

Abstract

Digital Transforms have important applications on subjects such as channel coding, cryptography and digital signal processing. In this paper, two Fourier Transforms are considered, the discrete time Fourier transform (DTFT) and the finite field Fourier transform (FFFT). A finite field version of the DTFT is introduced and the FFFT is redefined with a complex kernel, which makes it a more appropriate finite field version of the Discrete Fourier Transform. These transforms can handle FIR and IIR filters defined over finite algebraic structures.