  One of the cornerstones of the field of signal processing on graphs are graph
filters, direct analogues of classical filters, but intended for signals
defined on graphs. This work brings forth new insights on the distributed graph
filtering problem. We design a family of autoregressive moving average (ARMA)
recursions, which (i) are able to approximate any desired graph frequency
response, and (ii) give exact solutions for tasks such as graph signal
denoising and interpolation. The design philosophy, which allows us to design
the ARMA coefficients independently from the underlying graph, renders the ARMA
graph filters suitable in static and, particularly, time-varying settings. The
latter occur when the graph signal and/or graph are changing over time. We show
that in case of a time-varying graph signal our approach extends naturally to a
two-dimensional filter, operating concurrently in the graph and regular time
domains. We also derive sufficient conditions for filter stability when the
graph and signal are time-varying. The analytical and numerical results
presented in this paper illustrate that ARMA graph filters are practically
appealing for static and time-varying settings, as predicted by theoretical
derivations.
