On Hilbert Transform, Analytic Signal, and Modulation Analysis for Signals over Graphs Arun Venkitaraman, Saikat Chatterjee, Peter Handel¨ Department of Information Science and Engineering, School of Electrical Engineering and ACCESS Linnaeus Center KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden . Abstract We propose Hilbert transform and analytic signal construction for signals over graphs. This is motivated by the popularity of Hilbert transform, analytic signal, and mod- ulation analysis in conventional signal processing, and the observation that comple- mentary insight is often obtained by viewing conventional signals in the graph setting. Our definitions of Hilbert transform and analytic signal use a conjugate-symmetry-like property exhibited by the graph Fourier transform (GFT), resulting in a ’one-sided’ spectrum for the graph analytic signal. The resulting graph Hilbert transform is shown to possess many interesting mathematical properties and also exhibit the ability to high- light anomalies/discontinuities in the graph signal and the nodes across which signal discontinuities occur. Using the graph analytic signal, we further define amplitude, phase, and frequency modulations for a graph signal. We illustrate the proposed con- cepts by showing applications to synthesized and real-world signals. For example, we show that the graph Hilbert transform can indicate presence of anomalies and that arXiv:1611.05269v3 [cs.IT] 29 Jan 2018 graph analytic signal, and associated amplitude and frequency modulations reveal com- plementary information in speech signals. Keywords: Graph signal, analytic signal, Hilbert transform, demodulation, anomaly detection. Email addresses:
[email protected] (Arun Venkitaraman),
[email protected] (Saikat Chatterjee),
[email protected] (Peter Handel)¨ Preprint submitted to Signal Processing 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 (a) (b) Figure 1: Anomaly highlighting behavior of the graph Hilbert transform for 2D image signal graph.