  The processing of signals on simplicial and cellular complexes defined by
nodes, edges, and higher-order cells has recently emerged as a principled
extension of graph signal processing for signals supported on more general
topological spaces. However, most works so far have considered signal
processing problems for signals associated to only a single type of cell such
as the processing of node signals, or edge signals, by considering an
appropriately defined shift operator, like the graph Laplacian or the Hodge
Laplacian. Here we introduce the Dirac operator as a novel kind of shift
operator for signal processing on complexes. We discuss how the Dirac operator
has close relations but is distinct from the Hodge-Laplacian and examine its
spectral properties. Importantly, the Dirac operator couples signals defined on
cells of neighboring dimensions in a principled fashion. We demonstrate how
this enables us, e.g., to leverage node signals for the processing of edge
flows.
