  In this paper, we present a signal processing framework for directed graphs.
Unlike undirected graphs, a graph shift operator such as the adjacency matrix
associated with a directed graph usually does not admit an orthogonal
eigenbasis. This makes it challenging to define the Fourier transform. Our
methodology leverages the polar decomposition to define two distinct
eigendecompositions, each associated with different matrices derived from this
decomposition. We propose to extend the frequency domain and introduce a
Fourier transform that jointly encodes the spectral response of a signal for
the two eigenbases from the polar decomposition. This allows us to define
convolution following a standard routine. Our approach has two features: it is
lossless as the shift operator can be fully recovered from factors of the polar
decomposition. Moreover, it subsumes the traditional graph signal processing if
the graph is directed. We present numerical results to show how the framework
can be applied.
