Hilbert Transform, Analytic Signal, and Modulation Analysis for Graph Signal Processing

We propose Hilbert transform and analytic signal construction for signals over graphs. This is motivated by the popularity of Hilbert transform, analytic signal, and modulation analysis in conventional signal processing, and the observation that complementary insight is often obtained by viewing conventional signals in the graph setting. Our definitions of Hilbert transform and analytic signal use a conjugate-symmetrylike property exhibited by the graph Fourier transform (GFT), resulting in a ’one-sided’ spectrum for the graph analytic signal. Using the graph analytic signal, we define amplitude, phase, and frequency modulations for a graph signal. Further, we use convex optimization to develop an alternative definition of envelope for a graph signal that explicitly enforces smoothness. We illustrate the proposed concepts 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 graph analytic signal, and associated amplitude and frequency modulations reveal complementary information in speech signals.