Beyond Single-Window Graph Fourier Analysis

TL;DR

This paper introduces a multi-windowed graph Fourier transform (MWGFT) for enhanced joint vertex-frequency analysis of graph signals, providing exact reconstruction, stability, and improved localization especially on irregular graphs.

Contribution

It extends the single-window graph Fourier transform to a multi-window framework with theoretical guarantees and practical advantages.

Findings

Exact reconstruction of graph signals achieved.

Improved stability and localization demonstrated.

Effective on both synthetic and real-world graphs.

Abstract

We introduce a multi-windowed graph Fourier transform (MWGFT) for the joint vertex-frequency analysis of signals defined on graphs. Building on generalized translation and modulation induced by the graph Laplacian, the proposed framework extends the windowed graph Fourier transform by allowing multiple analysis and synthesis windows. Exact reconstruction formulas are derived for complex-valued graph signals, together with sufficient and computable conditions guaranteeing stable invertibility. The associated families of windowed graph Fourier atoms are shown to form frames for the space of graph signals. Numerical experiments on synthetic and real world graphs confirm exact reconstruction up to machine precision and demonstrate improved stability and vertex-frequency localization compared to single-window constructions, particularly on irregular graph topologies.