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Tight Frames Generated By A Graph Short-Time Fourier Transform

Martin Buck, Kasso Okoudjou

Abstract

A graph short-time Fourier transform is defined using the eigenvectors of the graph Laplacian and a graph heat kernel as a window parametrized by a non-negative time parameter $t$. We show that the corresponding Gabor-like system forms a frame for $\mathbb{C}^d$ and give a description of the spectrum of the corresponding frame operator in terms of the graph heat kernel and the spectrum of the underlying graph Laplacian. For two classes of algebraic graphs, we prove the frame is tight and independent of the window parameter $t$.

Tight Frames Generated By A Graph Short-Time Fourier Transform

Abstract

A graph short-time Fourier transform is defined using the eigenvectors of the graph Laplacian and a graph heat kernel as a window parametrized by a non-negative time parameter . We show that the corresponding Gabor-like system forms a frame for and give a description of the spectrum of the corresponding frame operator in terms of the graph heat kernel and the spectrum of the underlying graph Laplacian. For two classes of algebraic graphs, we prove the frame is tight and independent of the window parameter .
Paper Structure (6 sections, 6 theorems, 52 equations, 4 figures)

This paper contains 6 sections, 6 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction and Background
  2. The STFT and Gabor Frames on $\mathbb{C}^N$
  3. A Short-time Fourier Transform and Gabor Frames on Graphs
  4. Tight Frames on Vertex-transitive and Strongly Regular Graphs
  5. Vertex-transitive Graphs
  6. Strongly Regular Graphs

Key Result

Theorem 3.1

The set of graph Gabor atoms $\{\psi_{ij}(t)\}_{i,j=1}^N$ forms a frame for $\mathbb{C}^N$ for all $t \geq 0$. Furthermore, the frame operator $S(t)$ is a diagonal matrix and has the following form:

Figures (4)

  • Figure 1: A piecewise cosine $f(n)$ and the magnitude of the DFT $|\hat{f}(m)|^2$. The magnitude $|\hat{f}(m)|^2$ localizes around the modulus of the constituent frequencies but provides no information on when the frequency change occurs.
  • Figure 2: Spectrograms visualize the magnitude of the DSTFT $|V_{g}f(k, l)|^2$ for two different sized windows. The windowing allows the DSTFT to determine when the frequency change occurs in $f(n)$ from Figure as well as the constituent frequencies \ref{['fig:cos']}.
  • Figure 3: Difference between the maximum and minimum eigenvalue of the graph Gabor frame operator $S(t)$ as a function of time. Regular graphs with larger Fiedler values see faster decay towards a tight frame.
  • Figure 4: The Shrikande graph is a vertex-transitive and strongly regular graph with parameters $(16,6,2,2)$; it is the Cayley graph of $\mathbb{Z}_4 \times \mathbb{Z}_4$. The Chang graphs are a family of strongly regular graphs with parameters $(28,12,6,4)$

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 1 more