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

Abstract

We propose Hilbert transform (HT) and analytic signal (AS) construction for signals over graphs. This is motivated by the popularity of HT, AS, 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 HT and AS use a conjugate-symmetry-like property exhibited by the graph Fourier transform (GFT). We show that a real graph signal (GS) can be represented using smaller number of GFT coefficients than the signal length. We show that the graph HT (GHT) and graph AS (GAS) operations are linear and shift-invariant over graphs. Using the GAS, we define the amplitude, phase, and frequency modulations for a graph signal (GS). Further, we use convex optimization to develop an alternative definition of envelope for a GS. We illustrate the proposed concepts by showing applications to synthesized and real-world signals. For example, we show that the GHT is suitable for anomaly detection/analysis over networks and that GAS reveals complementary information in speech signals.