Multidimensional Analytic Signals and the Bedrosian Identity

TL;DR

This paper extends the analytic signal concept to multidimensional signals using partial Hilbert transforms, explores the Bedrosian identity in this context, and develops basis functions for advanced time-frequency analysis.

Contribution

It provides a mathematical justification for multidimensional analytic signals via partial Hilbert transforms and characterizes the Bedrosian identity in this setting.

Findings

Characterization of the Bedrosian identity for partial Hilbert transforms

Necessity theorems for the Bedrosian identity in multiple dimensions

Construction of basis functions for multidimensional time-frequency analysis

Abstract

The analytic signal method via the Hilbert transform is a key tool in signal analysis and processing, especially in the time-frquency analysis. Imaging and other applications to multidimensional signals call for extension of the method to higher dimensions. We justify the usage of partial Hilbert transforms to define multidimensional analytic signals from both engineering and mathematical perspectives. The important associated Bedrosian identity $T(fg)=fTg$ for partial Hilbert transforms $T$ are then studied. Characterizations and several necessity theorems are established. We also make use of the identity to construct basis functions for the time-frequency analysis.