Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

Abstract

Abstract-This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper (Part I) and studies M -channel polynomial filter banks on graphs. The behavior of such graph filter banks differs from that of classical filter banks in many ways, the precise details depending on the eigenstructure of the adjacency matrix A. It is shown that graph filter banks represent (linear and) periodically shift-variant systems only when A satisfies the noble identity conditions developed in Part I. It is then shown that perfect reconstruction graph filter banks can always be developed when A satisfies the eigenvector structure satisfied by M -block cyclic graphs and has distinct eigenvalues (further restrictions on eigenvalues being unnecessary for this). If A is actually M -block cyclic then these PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages. In this case the PR condition is identical to PR in classical filter banks -any classical PR example can be converted to a graph PR filter bank on an M -block cyclic graph. It is shown that for M -block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks. Polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction, again for graphs with specific eigenstructures. It is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms.

Extending classical multirate signal processing theory to graphs

“…17,18 This eigenvalue relation of block cyclic matrices has also been observed in earlier studies. [19][20][21][22] This property is as follows: Theorem 2 (Eigen-families of M -Block cyclic graphs).…”

“…Graphs that satisfy only the condition in (15) are referred to as Ω-graphs. 18 To be consistent with the double indexing of the eigenvectors, the elements of p x and p y will be indexed in accordance with the scheme in (16). That is to say,…”

“…While this idea is still not easy to develop for arbitrary graphs, the theory can be developed under some further assumptions on the graph, namely that A be M -Block cyclic. As long as the underlying graph is M -Block cyclic, it is possible to design polynomial filters that do not depend on the underlying eigenvalues of the adjacency matrix such that overall response of the filter bank is T pAq " A n for some integer n. 18 This result is as follows:…”

“…In Sections 2-4, we will first review some recent results on the multirate processing of graph signals, 17,18 which consider the concepts such as decimation, expansion, noble identities and filter banks for an arbitrary rate M . In general these concepts do not extend to graphs in a straightforward manner.…”

“…17,18 With more restrictive conditions on the graph, it is possible to generalize the classical multirate theory to graph signals for arbitrary M . For this purpose the following graph structure is considered.…”