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. ^{1 1}

The codebase is available at https://github.com/koc-lab/gfrft-unified.

中文翻译：

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图分数阶傅立叶变换：统一理论


分数阶傅里叶变换（FRFT）通过变换阶数参数化地概括了傅里叶变换（FT），表示中间时频域中的信号。 FRFT 有多个但等效的定义，包括 FT 的分数幂、时频平面旋转、超微分算子等，每个定义都具有推导简便性和计算效率等优点。同时，图信号处理 (GSP) 将传统信号处理扩展到不规则图结构上的数据，从而实现了图信号的采样、滤波和傅里叶变换等概念。图分数阶傅立叶变换 (GFRFT) 最近已扩展到 GSP 域。然而，该扩展仅概括了基于具有有限变换阶数范围的特定图结构的 FRFT 的分数幂定义。理想情况下，GFRFT 扩展应与尽可能多的替代定义一致。本文首先提供了严格的基于分数幂的 GFRFT 定义，支持任何图结构和变换顺序。然后，我们引入了新颖的基于超微分算子的 GFRFT 定义，允许在大型图上进行更快的正向和逆变换矩阵计算。通过所提出的定义，我们得出了一种通过从数据中学习最优值来选择变换顺序的新方法。此外，我们还提供了对核心 GSP 概念的处理，例如带限性、滤波器以及与 GFRFT 背景下其他变换的关系。 最后，通过包括去噪、分类和采样任务在内的综合实验，我们证明了 GFRFT 并行定义的等效性、变换阶数的可学习性以及 GFRFT 相对于 GFT 和其他 GSP 方法的优势。 ^{1 1}

代码库位于 https://github.com/koc-lab/gfrft-unified。