Fourier Transforms

Andi Topan, Andi Topan, Mawardi Bahri, Nasrullah, Aidawayati Rangkuti, Muh Nur

The coupled fractional Fourier transform is known to be a generalized version of the fractional Fourier transform. This research paper aims to propose two inequalities related to the coupled fractional Fourier transform. The results are generalizations of uncertainty principles concerning the Fourier transform, emphasizing that the essential principles have not yet been published in the academic literature. The results show that the uncertainty principles were obtained using the properties of the coupled fractional Fourier transform and its relation to the classical Fourier transform. Keywords: Coupled Fractional Fourier Transform, Fourier Transform, Uncertainty Principle DOI： https://doi.org/10.35741/issn.0258-2724.58.2.32

Ajoy Kumar Ghatak, S. Lokanathan

Bernd Thaller

Steffen Weimann, Armando Pérez-Leija, Maxime Lebugle +9 more

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

J. M. Bijvoet, Wilhelm Gerard Burgers, Gunnar Hägg

Tieyu Zhao, Tianyu Yang, Yingying Chi

Quantum Fourier transform (QFT) is an important part of many quantum algorithms. However, there are few reports on quantum fractional Fourier transform (QFRFT). The main reason is that the definitions of fractional Fourier transform (FRFT) are diverse, while some definitions do not include unitarity, which leads to some studies pointing out that there is no QFRFT. In this paper, we first present a reformulation of the weighted fractional Fourier transform (WFRFT) and prove its unitarity, thereby proposing a quantum weighted fractional Fourier transform (QWFRFT). The proposal of QWFRFT provides the possibility for many quantum implementations of signal processing.