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Abstract
In digital signal processing, a shiftinvariant filter can be represented as a polynomial expansion of a shift operation, that is, the Ztransform representation. When extended to Graph Signal Processing (GSP), this would mean that a shiftinvariant graph filter can be represented as a polynomial of the shift matrix of the graph. Prior work shows that this holds under the shiftenabled condition that the characteristic and minimum polynomials of the shift matrix are identical. While the shiftenabled condition is often ignored in the literature, this letter shows that this condition is essential for the following reasons. First, we prove that this condition is not just sufficient but also
necessary for any shiftinvariant filter to be representable by the shift matrix. Moreover, we provide a counterexample showing that given a filter that commutes with a nonshiftenabled graph, it is generally impossible to convert the graph to a shiftenabled graph with a shift matrix still commuting with the original filter. The result provides a deeper understanding of shiftinvariant
filters when applied in GSP and shows that further investigation of shiftenabled graphs is needed to make them applicable to practical scenarios.
necessary for any shiftinvariant filter to be representable by the shift matrix. Moreover, we provide a counterexample showing that given a filter that commutes with a nonshiftenabled graph, it is generally impossible to convert the graph to a shiftenabled graph with a shift matrix still commuting with the original filter. The result provides a deeper understanding of shiftinvariant
filters when applied in GSP and shows that further investigation of shiftenabled graphs is needed to make them applicable to practical scenarios.
Original language  English 

Pages (fromto)  13051309 
Number of pages  5 
Journal  IEEE Signal Processing Letters 
Volume  25 
Issue number  9 
Early online date  22 Jun 2018 
DOIs  
Publication status  Published  30 Sep 2018 
Keywords
 graph signal processing
 shiftinvariant filter
 polynomial
 digital signal processing (DSP)
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Projects
 1 Finished

SENSIBLE: SENSors and Intelligence in BuiLt Environment (SENSIBLE) MSCA RISE
Stankovic, L., Glesk, I., Gleskova, H. & Stankovic, V.
European Commission  Horizon 2020
1/01/17 → 31/12/20
Project: Research