  The paper presents the graph signal processing (GSP) companion model that
naturally replicates the basic tenets of classical signal processing (DSP) for
GSP. The companion model shows that GSP can be made equivalent to DSP 'plus'
appropriate boundary conditions (bc) - this is shown under broad conditions and
holds for arbitrary undirected or directed graphs. This equivalence suggests
how to broaden GSP - extend naturally a DSP concept to the GSP companion model
and then transfer it back to the common graph vertex and graph Fourier domains.
The paper shows that GSP unrolls as two distinct models that coincide in DSP,
the companion model based on (Hadamard or pointwise) powers of what we will
introduce as the spectral frequency vector $\lambda$, and the traditional graph
vertex model, based on the adjacency matrix and its eigenvectors. The paper
expands GSP in several directions, including showing that convolution in the
graph companion model can be achieved with the FFT and that GSP modulation with
appropriate choice of carriers exhibits the DSP translation effect that enables
multiplexing by modulation of graph signals.
