  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.
