One of the cornerstones of the field of signal processing on graphs are graphfilters, direct analogues of classical filters, but intended for signalsdefined on graphs. This work brings forth new insights on the distributed graphfiltering problem. We design a family of autoregressive moving average (ARMA)recursions, which (i) are able to approximate any desired graph frequencyresponse, and (ii) give exact solutions for tasks such as graph signaldenoising and interpolation. The design philosophy, which allows us to designthe ARMA coefficients independently from the underlying graph, renders the ARMAgraph filters suitable in static and, particularly, time-varying settings. Thelatter occur when the graph signal and/or graph are changing over time. We showthat in case of a time-varying graph signal our approach extends naturally to atwo-dimensional filter, operating concurrently in the graph and regular timedomains. We also derive sufficient conditions for filter stability when thegraph and signal are time-varying. The analytical and numerical resultspresented in this paper illustrate that ARMA graph filters are practicallyappealing for static and time-varying settings, as predicted by theoreticalderivations.
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