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Filter Design for Two-Channel Filter Banks on Directed Bipartite Graphs

Graph signal processing on directed graphs is more challenging that on undirected graph, primarily because the graph matrices, e.g., adjacency or Laplacian, associated with the latter is non-symmetric. The eigenvalues that represent the spectral frequencies are therefore complex in general, and the design of spectral filters for complex frequencies presents more challenges. In this work we consider the design of two-channel filter banks for directed bipartite graphs. By decomposing any graph into a series of bipartite graphs, the basic two-channel system can be applied in cascade for arbitrary graphs. The graph filters are constructed using ladder structures. The design of the kernels in the ladder structures is achieved via a least-squares formulation. Analytical formulas are derived for the design of the kernel coefficients.