Coupled Fractional Fourier Transform: Properties and Uncertainty Principle

Mawardi Bahri

doi:10.22342/jims.v32i2.1961

Mawardi Bahri ⁽¹⁾

(1) Department of Mathematics, Hasanuddin University, Indonesia

https://doi.org/10.22342/jims.v32i2.1961

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
March 4, 2025

Accepted
June 13, 2025

Published
April 23, 2026

Keywords:

coupled fractional Fourier transform, uncertainty principle, fractional Fourier transform

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Abstract

The coupled fractional Fourier transform is a general form of the fractional Fourier transform. In the present work, we demonstrate basic properties of the proposed transformation, such as linearity, shifting and modulation. We also propose convolution and correlation theorems and then explore a generalized uncertainty principle for the coupled fractional Fourier transform. These crucial results are modifications of the corresponding properties pertaining to the fractional Fourier transform and the conventional Fourier transform.

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References

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Authors

Mawardi Bahri

Department of Mathematics, Hasanuddin University

mawardibahri@gmail.com (Primary Contact)

Bahri, M. (2026). Coupled Fractional Fourier Transform: Properties and Uncertainty Principle. Journal of the Indonesian Mathematical Society, 32(2), 1961. https://doi.org/10.22342/jims.v32i2.1961

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1.INTRODUCTION

The fractional Fourier transform (FrFT) [1,2,3,4] is an efective mathematical tool that has been applied in quantum mechanics, neural networks, diferential equations, optics, radar, sonar, and other communication systems [5,6,7,8,9]. It can be viewed as an expansion of the Fourier transform (FT). In the recent work, the authors of [10,11] have proposed an extended form of the FrFT to the coupled fractional Fourier transform (CFrFT). The utility of the CFrFT kernel in constructing the other general transformations, such as the short-time coupled fractional Fourier transform and the coupled fractional Wigner-Ville distribution were discussed in [12,13,14,15]. Especially, the authors of[10]studied in detail various uncertainty inequalities associated with the CFrFT. However, some properties of the CFrFT like convolution and correlation theorems have not yet studied.

Therefore, in the present research we first establish some basic results pertaining to the CFrFT, which are missing in the existing literature. We then study convolution and correlation operators related to the generalized transformation. Finally, we also explore an uncertainty inequality associated with the proposed CFrFT.

The paper is outlined as follows. In Section2we recall the definition of the FrFT and some existing results related to the transformation. The definition of the CFrFT is also introduced in this section. Section3is devoted to definition and properties of the CFrFT. We also present convolution and correlation theorems related to the CFrFT. Section4deals with the derivation of an uncertainty principles associated with the CFrFT. Lastly, Section5contains a brief conclusion of the work.

2. Preliminaries

First of all, we introduce the basic results related to the FrFT and the CFrFT. We begin with the following definition.

Definition 2.1.The space\(L ^ { r } ( \mathbb { R } ^ { 2 } )\)is the set of measurable functions on\(\mathbb { R } ^ { 2 }\)satisfying

Here \(\pmb { t } = ( t _ { 1 } , t _ { 2 } ) \in \mathbb { R } ^ { 2 } , d \pmb { t } = d t _ { 1 } d t _ { 2 }\). Also for every \(f , g \in L ^ { r } ( \mathbb { R } ^ { 2 } )\) with \(f = g\) if and only if \(f = g \ a . e\).

In particular, when \(r = \infty\) we define \(L ^ { \infty } ( \mathbb { R } ^ { 2 } )\)-norm

and iffis continuous, then

The inner product of \(L ^ { 2 } ( \mathbb { R } ^ { 2 } )\) is defined by

Definition 2.2.Let function\(f \in L ^ { 1 } ( \mathbb { R } ^ { 2 } )\). The 2-D FrFT with parameter ϑ is defined by

where \(K _ { \vartheta } ( \omega , t )\) is given by

Here, \(\delta ( t - \omega ) = \delta ( t _ { 1 } - \omega _ { 1 } ) \delta ( t _ { 2 } - \omega _ { 2 } )\) and

It is not dificult to check that the FrFT kernel fulfills the following basic properties:

and

where \(\overline { { K _ { \vartheta } ( \omega , t ) } }\) is the conjugation of \(K _ { \vartheta } ( \omega , t )\).

Definition 2.3.Let\(f \in L ^ { 1 } ( \mathbb { R } ^ { 2 } )\)and\(\mathfrak { F } _ { \vartheta } \{ f \} \in L ^ { 1 } ( \mathbb { R } ^ { 2 } )\). The inverse formula of the 2-D FrFT for the function\(f\)is given by

Formula (8) tells us how to generate the original function from its FrFT. Next, we introduce a definition of the CFrFT.

Definition 2.4.The CFrFT of any function\(f\)belongs to\(L ^ { 1 } ( \mathbb { R } ^ { 2 } ) \cap L ^ { 2 } ( \mathbb { R } ^ { 2 } )\)is defined as[10,11]

where

In this case,

with \(\alpha , \beta \in \mathbb { R }\) and \(\alpha + \beta \not \in 2 \pi \mathbb { Z }\).

Relation (9) may be expressed in the form

Denoting

we easily get

Now formula(11)may be represented as

which explains the direct interaction between the CFrFT and the 2-D Fourier transformation. Here \(\mathfrak{F}\{ f \} ( \omega )\) denotes the 2-D Fourier transform of \(f \in L ^ { 2 } ( \mathbb { R } ^ { 2 } )\) given by (see [16,17,18])

where \(\begin{array} { r } { \pmb { t } , \omega \in \mathbb { R } ^ { 2 } , \pmb { t } \cdot \omega = t _ { 1 } \omega _ { 1 } + t _ { 2 } \omega _ { 2 } . } \end{array}\)

The original function can be obtained in terms of the CFrFT using the

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