In digital signal processing, shift-invariant filters can be represented as apolynomial expansion of a shift operation,that is, the Z-transformrepresentation. When extended to graph signal processing (GSP), this would meanthat a shift-invariant graph filter can be represented as a polynomial of theadjacency (shift) matrix of the graph. However, the characteristic and minimumpolynomials of the adjacency matrix must be identical for the property to hold.While it has been suggested that this condition might be ignored as it isalways possible to find a polynomial transform to represent the originaladjacency matrix by another adjacency matrix that satisfies the condition, thisletter shows that a filter that is shift invariant in terms of the originalgraph may not be shift invariant anymore under the modified graph and viceversa. We introduce the notion of "shift-enabled graph" for graphs that satisfythe aforementioned condition, and present a concrete example of a graph that isnot "shift-enabled" and a shift-invariant filter that is not a polynomial ofthe shift operation matrix. The result provides a deeper understanding ofshift-invariant filters when applied in GSP and shows that furtherinvestigation of shift-enabled graphs is needed to make it applicable topractical scenarios.
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