  We study linear filters for processing signals supported on abstract
topological spaces modeled as simplicial complexes, which may be interpreted as
generalizations of graphs that account for nodes, edges, triangular faces etc.
To process such signals, we develop simplicial convolutional filters defined as
matrix polynomials of the lower and upper Hodge Laplacians. First, we study the
properties of these filters and show that they are linear and shift-invariant,
as well as permutation and orientation equivariant. These filters can also be
implemented in a distributed fashion with a low computational complexity, as
they involve only (multiple rounds of) simplicial shifting between upper and
lower adjacent simplices. Second, focusing on edge-flows, we study the
frequency responses of these filters and examine how we can use the
Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We
discuss how these frequencies correspond to the lower- and the upper-adjacent
couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned
independently by our filter designs. Third, we study different procedures for
designing simplicial convolutional filters and discuss their relative
advantages. Finally, we corroborate our simplicial filters in several
applications: to extract different frequency components of a simplicial signal,
to denoise edge flows, and to analyze financial markets and traffic networks.
