Topological IIR Filters Over Simplicial Topologies via Sheaves

Topology offers a means to formally generalize digital filtering methods based on digital linear translation-invariant (LTI) filters while also, in principle, incorporating translation-variant and nonlinear methods as well as studying large scale (global) properties of filter problems. In this letter we show how the full content of LTI digital filter theory can be incorporated into the formalism of topological filters as introduced by Robinson. In particular, we will give the feedback filter construction associated with infinite impulse responses (IIR). The result allows for direct translation of LTI filters into topological filters, which are sheaves of finite vector spaces and suitably constructed linear maps over simplicial topologies.