Fundamentals of multirate graph signal processing

In this work, the fundamental blocks of multirate signal processing on graphs are analyzed. First the decimator is defined, and expander is solved accordingly. Then, noble identities and lazy filter bank for graph signals are constructed. After decimation, the length of the signal changes and the original adjacency matrix is not applicable. Therefore, such equations on graphs do not exist in general. For noble identities to exist and lazy filter bank to provide perfect reconstruction, the necessary and sufficient conditions on the graph are derived. Some graph examples, on which the conditions are satisfied, are also provided.