  The goal of this paper is to establish the fundamental tools to analyze
signals defined over a topological space, i.e. a set of points along with a set
of neighborhood relations. This setup does not require the definition of a
metric and then it is especially useful to deal with signals defined over
non-metric spaces. We focus on signals defined over simplicial complexes. Graph
Signal Processing (GSP) represents a special case of Topological Signal
Processing (TSP), referring to the situation where the signals are associated
only with the vertices of a graph. Even though the theory can be applied to
signals of any order, we focus on signals defined over the edges of a graph and
show how building a simplicial complex of order two, i.e. including triangles,
yields benefits in the analysis of edge signals. After reviewing the basic
principles of algebraic topology, we derive a sampling theory for signals of
any order and emphasize the interplay between signals of different order. Then
we propose a method to infer the topology of a simplicial complex from data. We
conclude with applications to real edge signals and to the analysis of discrete
vector fields to illustrate the benefits of the proposed methodologies.
