In this paper, we present a novel convolution theorem which encompasses the well known convolution theorem in (graph) signal processing as well as the one related to time-varying filters. Specifically, we show how a node-wise convolution for signals supported on a graph can be expressed as another node-wise convolution in a frequency domain graph, different from the original graph. This is achieved through a parameterization of the filter coefficients following a basis expansion model. After showing how the presented theorem is consistent with the already existing body of literature, we discuss its implications in terms of non-stationarity. Finally, we propose a data-driven algorithm based on subspace fitting to learn the frequency domain graph, which is then corroborated by experimental results on synthetic and real data.

中文翻译：

---

卷积定理的推广及其与非平稳性和图频域的联系


在本文中，我们提出了一种新颖的卷积定理，它包含（图）信号处理中众所周知的卷积定理以及与时变滤波器相关的定理。具体来说，我们展示了图上支持的信号的节点卷积如何可以表示为频域图中的另一个节点卷积，与原始图不同。这是通过遵循基扩展模型对滤波器系数进行参数化来实现的。在展示了所提出的定理与现有文献的一致性之后，我们讨论了它在非平稳性方面的含义。最后，我们提出了一种基于子空间拟合的数据驱动算法来学习频域图，然后通过合成数据和真实数据的实验结果得到证实。