Preprint Article Version 1 This version is not peer-reviewed

Undirected Graphs: Is the Shift-Enabled Condition Trivial or Necessary?

Version 1 : Received: 25 September 2019 / Approved: 26 September 2019 / Online: 26 September 2019 (11:03:47 CEST)

How to cite: Chen, L.; Cheng, S.; He, K.; Stankovic, L.; Stankovic, V. Undirected Graphs: Is the Shift-Enabled Condition Trivial or Necessary?. Preprints 2019, 2019090298 (doi: 10.20944/preprints201909.0298.v1). Chen, L.; Cheng, S.; He, K.; Stankovic, L.; Stankovic, V. Undirected Graphs: Is the Shift-Enabled Condition Trivial or Necessary?. Preprints 2019, 2019090298 (doi: 10.20944/preprints201909.0298.v1).

Abstract

It has recently been shown that, contrary to the wide belief that a shift-enabled condition (necessary for any shift-invariant filter to be representable by a graph shift matrix) can be ignored because any non-shift-enabled matrix can be converted to a shift-enabled matrix, such a conversion in general may not hold for a directed graph with non-symmetric shift matrix. This paper extends this prior work, focusing on undirected graphs where the shift matrix is generally symmetric. We show that while, in this case, the shift matrix can be converted to satisfy the original shift-enabled condition, the converted matrix is not associated with the original graph, that is, it does not capture anymore the structure of the graph signal. We show via a counterexample, that a non-shift-enabled matrix cannot be converted to a shift-enabled one and still maintain the topological structure of the underlying graph, which is necessary to facilitate localized signal processing.

Subject Areas

graph signal processing; shift-enabled graphs; shift-invariant filter; undirected graph

Comments (0)

We encourage comments and feedback from a broad range of readers. See criteria for comments and our diversity statement.

Leave a public comment
Send a private comment to the author(s)
Views 0
Downloads 0
Comments 0
Metrics 0


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.