M-channel graph filter banks: polyphase analysis and structures

Two channel critically sampled filter banks for signal over graph domains were first proposed for undirected bipartite graphs by Narang and Ortega. Extension to the M-channel critically sampled case for balanced M-block cyclic graphs was then proposed by Teke and Vaidynathan but the filter bank does not achieve strict perfect reconstruction (PR), only generalized PR. In this letter, we consider the more general case of filter banks on unbalanced M-block cyclic graphs where strict PR is achieved. A polyphase analysis to derive the implementation structures in the downsampled domain is presented here. The relevant system/filter matrices have interesting cyclic properties and projection operators are needed to map signals between subgraphs.