A mathematical framework for graph signal processing of time-varying signals

We propose a general framework from which to understand the design of filters for time-series signals supported on graphs. We organize linear, time-invariant filters into three increasingly restrictive classes of operators: linear time-invariant filters, linear time-invariant filters which commute with a graph operator, and linear time-invariant filters which are functions of a graph operator. Using spectral theory, we show that these yield $\mathcal{O}(n^2)$, $\mathcal{O}(n)$, and $\mathcal{O}(1)$ design parameters respectively. We consider arbitrary graph operators as to accommodate non-self-adjoint weight operators and all classes of graph Laplacian-based operators. We provide an example application of each class of filter.