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?. Preprints2019, 2019090298. https://doi.org/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. https://doi.org/10.20944/preprints201909.0298.v1.
Cite as:
Chen, L.; Cheng, S.; He, K.; Stankovic, L.; Stankovic, V. Undirected Graphs: Is the Shift-Enabled Condition Trivial or Necessary?. Preprints2019, 2019090298. https://doi.org/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. https://doi.org/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.
Keywords
graph signal processing; shift-enabled graphs; shift-invariant filter; undirected graph
Subject
MATHEMATICS & COMPUTER SCIENCE, Information Technology & Data Management
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
