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Signal Processing on Weighted Line Graphs

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Excursions in Harmonic Analysis, Volume 4

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

This chapter describes a signal processing framework for signals that are represented, or indexed, by weighted line graphs, which are a generalization of directed line graphs used for representation of time signals in classical signal processing theory. The presented framework is based on the theory of discrete signal processing on graphs and on algebraic signal processing theory. It defines fundamental signal processing concepts, such as signals and filters,  z-transform, frequency and spectrum, Fourier transform and others, in a principled way. The framework also illustrates a strong connection between signal processing on weighted line graphs and signal representation based on orthogonal polynomials.

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Notes

  1. 1.

    Filters are  linear if for a linear combination of inputs they produce the same linear combination of outputs. Filters are  shift-invariant if the result of consecutive processing of a signal by multiple graph filters does not depend on the order of processing; that is, shift-invariant filters commute with each other.

  2. 2.

    The minimal polynomial of  A is the unique monic polynomial of the smallest degree that annihilates  A, that is,  m  A ( A) = 0 [7].

  3. 3.

    The  characteristic polynomial of a matrix  A is defined as \(p_{\mathbf{A}}(x) =\det (x\mathbf{I} -\mathbf{A}) =\prod _{ n=0}^{N-1}\) ( x − λ  n ) [7].

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Author information

Authors and Affiliations

  1. Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Aliaksei Sandryhaila

  2. Electrical and Computer Engineering, Biomedical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Jelena Kovačević

Authors
  1. Aliaksei Sandryhaila
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  2. Jelena Kovačević
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Corresponding author

Correspondence to Jelena Kovačević .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Radu Balan

  2. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Matthew Begué

  3. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Wojciech Czaja

  5. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Kasso A. Okoudjou

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Sandryhaila, A., Kovačević, J. (2015). Signal Processing on Weighted Line Graphs. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 4. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20188-7_10

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