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The paper presents the graph signal processing (GSP) companion model thatnaturally replicates the basic tenets of classical signal processing (DSP) forGSP. The companion model shows that GSP can be made equivalent to DSP 'plus'appropriate boundary conditions (bc) - this is shown under broad conditions andholds for arbitrary undirected or directed graphs. This equivalence suggestshow to broaden GSP - extend naturally a DSP concept to the GSP companion modeland 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 willintroduce as the spectral frequency vector $\lambda$, and the traditional graphvertex model, based on the adjacency matrix and its eigenvectors. The paperexpands GSP in several directions, including showing that convolution in thegraph companion model can be achieved with the FFT and that GSP modulation withappropriate choice of carriers exhibits the DSP translation effect that enablesmultiplexing by modulation of graph signals.

GSP = DSP + Boundary Conditions -- The Graph Signal Processing Companion Model