In this paper, we study linear filters to process signals defined on simplicial complexes, i.e., signals defined on nodes, edges, triangles, etc. of a simplicial complex, thereby generalizing filtering operations for graph signals. We propose a finite impulse response filter based on the Hodge Laplacian, and demonstrate how this filter can be designed to amplify or attenuate certain spectral components of simplicial signals. Specifically, we discuss how, unlike in the case of node signals, the Fourier transform in the context of edge signals can be understood in terms of two orthogonal subspaces corresponding to the gradient-flow signals and curl-flow signals arising from the Hodge decomposition. By assigning different filter coefficients to the associated terms of the Hodge Laplacian, we develop a subspace-varying filter which enables more nuanced control over these signal types. Numerical experiments are conducted to show the potential of simplicial filters for sub-component extraction, denoising and model approximation.

翻译：在本文中,我们研究线性过滤器,以处理在简单复合物上界定的信号,即简化复合物节点、边缘、三角等上界定的信号,从而对图形信号进行一般过滤操作。我们提议基于Hodge Laplacecian的有限脉冲反应过滤器,并演示如何设计这一过滤器,以扩大或减少简化信号的某些光谱组成部分。具体地说,我们讨论了如何与节点信号不同,从与坡度信号相对应的两个或数方位子空间和Hodge分解后产生的曲流信号的角度理解Fourier在边缘信号方面的变异。我们为Hodge Laplaceian的相关术语分配了不同的过滤系数,我们开发了一个子空间变异过滤器,使这些信号类型的控制更加细微。进行了数字实验,以显示精度过滤器在次构件提取、分解和模型近近化方面的潜力。