On the Clifford short-time Fourier transform and its properties

TL;DR

This paper extends the short-time Fourier transform into a Clifford algebra framework, establishing its fundamental properties, effects of modulation and translation, and demonstrating an uncertainty principle within this setting.

Contribution

It introduces the Clifford short-time Fourier transform and proves key properties, including orthogonality, reconstruction, and reproducing kernel formula, advancing harmonic analysis in Clifford algebras.

Findings

Established orthogonality relation for the Clifford STFT

Proved the reconstruction property and reproducing kernel formula

Demonstrated Lieb's uncertainty principle in the Clifford setting

Abstract

In this paper we investigate how the short-time Fourier transform can be extended in a Clifford setting. We prove some of the main properties of the Clifford short-time Fourier transform such as the orthogonality relation, the reconstruction property and the reproducing kernel formula. Moreover, we show the effects of modulating and translating the signal and the window function, respectively. Finally, we demonstrate the Lieb's uncertainty principle for the Clifford short-time Fourier transform.