Browsing by Subject "Fractional Fourier transform (FRFT)"

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Open Access
Graph fractional Fourier transform
(2024-09) Alikaşifoğlu, Tuna
The fractional Fourier transform (FRFT) parametrically generalizes the Fourier transform (FT) by a transform order, representing signals in intermediate time-frequency domains. The FRFT has multiple but equivalent definitions each offering benefits like derivational ease and computational efficiency. Concurrently, graph signal processing (GSP) extends traditional signal processing to irregular graph structures, enabling concepts like sampling, filtering, and Fourier transform for graph signals. The graph fractional Fourier transform (GFRFT) is recently extended to the GSP domain. However, this extension only generalizes one definition of FRFT based on specific graph structure with limited transform order range. Ideally, the GFRFT extension should be consistent with as many alternative definitions as possible. This work introduces a unified framework for GFRFT that supports multiple definitions with any graph structure and transform order. The proposed approach also allows faster transform matrix computations on large graphs and learnable transform order. Additionally, data sources on each vertex can also continually provide time-series signals such that graph signals are time varying. Joint time-vertex Fourier transform (JFT), with the associated framework of joint time-vertex (JTV) processing, provides spectral analysis tools for such signals. Just as the FRFT generalizes FT, we propose the joint time-vertex fractional Fourier transform (JFRFT) to generalize JFT. JFRFT provides an additional fractional analysis tool for JTV processing by extending temporal and vertex domains to fractional orders. Recently, the classical optimal Wiener filtering problem is introduced to JTV domain. However, the ordinary Fourier domain is not always optimal for separating the signal and noise; one can reach the smaller error in a fractional domain. We generalize the optimal Wiener filtering to the proposed JFRFT domains and provide a theoretical analysis and solution to the problem in the fractional JTV domains. We numerically verify our claims by presenting computational cost analysis and experiments with comprehensive comparisons to state-of-the-art approaches.
Open Access
Graph fractional Fourier transform: a unified theory
(IEEE, 2024) Alikaşifoğlu, Tuna; Kartal, Bünyamin; Koç, Aykut
The fractional Fourier transform (FRFT) parametrically generalizes the Fourier transform (FT) by a transform order, representing signals in intermediate time-frequency domains. The FRFT has multiple but equivalent definitions, including the fractional power of FT, time-frequency plane rotation, hyper-differential operator, and many others, each offering benefits like derivational ease and computational efficiency. Concurrently, graph signal processing (GSP) extends traditional signal processing to data on irregular graph structures, enabling concepts like sampling, filtering, and Fourier transform for graph signals. The graph fractional Fourier transform (GFRFT) is recently extended to the GSP domain. However, this extension only generalizes the fractional power definition of FRFT based on specific graph structures with limited transform order range. Ideally, the GFRFT extension should be consistent with as many alternative definitions as possible. This paper first provides a rigorous fractional power-based GFRFT definition that supports any graph structure and transform order. Then, we introduce the novel hyper-differential operator-based GFRFT definition, allowing faster forward and inverse transform matrix computations on large graphs. Through the proposed definition, we derive a novel approach to select the transform order by learning the optimal value from data. Furthermore, we provide treatments of the core GSP concepts, such as bandlimitedness, filters, and relations to the other transforms in the context of GFRFT. Finally, with comprehensive experiments, including denoising, classification, and sampling tasks, we demonstrate the equivalence of parallel definitions of GFRFT, learnability of the transform order, and the benefits of GFRFT over GFT and other GSP methods.¹¹ The codebase is available at https://github.com/koc-lab/gfrft-unified.
Open Access
Operator theory-based computation of linear canonical transforms
(Elsevier, 2021-08-12) Koç, Aykut; Özaktaş, Haldun M.
Linear canonical transforms (LCTs) are extensively used in many areas of science and engineering with many applications, which requires a satisfactory discrete implementation. Recently, hyperdifferential operators have been proposed as a novel way of defining the discrete LCT (DLCT). Here we first focus on improving the accuracy of this approach by considering alternative discrete coordinate multiplication and differentiation operations. We also consider canonical decompositions of LCTs and compare them with the originally proposed Iwasawa decomposition. We show that accuracy of the approximation of the continuous LCT with the DLCT can be drastically improved. The advantage and elegance of this approach lie in the fact that it reduces the problem of defining sophisticated discrete transforms to merely defining discrete coordinate multiplication and differentiation operations, by reducing the transforms to these operations. As a result of systematic investigation of possible parameters and design choices, we achieve a DLCT that is both theoretically satisfying and highly accurate.