Extending Classical Multirate Signal Processing Theory to GraphsźPart I: Fundamentals

Signal processing on graphs finds applications in many areas. In recent years, renewed interest on this topic was kindled by two groups of researchers. Narang and Ortega constructed two-channel filter banks on bipartitie graphs described by Laplacians. Sandryhaila and Moura developed the theory of linear systems, filtering, and frequency responses for the case of graphs with arbitrary adjacency matrices, and showed applications in signal compression, prediction, etc. Inspired by these contributions, this paper extends classical multirate signal processing ideas to graphs. The graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix. The paper revisits ideas, such as noble identities, aliasing, and polyphase decompositions in graph multirate systems. Drawing such a parallel to classical systems allows one to design filter banks with polynomial filters, with lower complexity than arbitrary graph filters. It is shown that the extension of classical multirate theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph. Thus, classical noble identities cannot be taken for granted. Similarly, one cannot claim that the so-called delay chain system is a perfect reconstruction system (as in classical filter banks). It will also be shown that \($\)-partite extensions of the bipartite filter bank results will not work for \($\)-channel filter banks, but a more restrictive condition called \($\)-block cyclic property should be imposed. Such graphs are studied in detail. A detailed theory for \($\)-channel filter banks is developed in a companion paper.