Multidimensional Analytic Signals and the Bedrosian Identity

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 (f g) = f T g 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.Let H 1 (R) be the Banach space of all the functions φ ∈ L 1 (R) such that Hφ ∈ L 1 (R) equipped with the norm φ H 1 (R) := φ L 1 (R) + Hφ L 1 (R) .

Multidimensional Analytic Signal with Application on Graphs

“…2(b). Then corresponding Scheffers algebra-valued analytic signal f h (x) was constructed, by first restricting the spectrumf (ω) to positive frequencies according to (19) and applying inverse formula (15). All computations were performed in the diffusion map domain.…”

“…Appropriate generalizations of analytic signals to two or more dimensions and its connection with complex and hypercomplex analysis were studied in the recent decades. This was motivated by the development of signal processing methods for image analysis [3,4,5,6] and analysis of multivariate signals [7,8,9,10,11,12,13,14,15].…”

The Bedrosian identity for generalized functions

“…However, most of the previous works are focused on one‐dimensional case. Recently, Zhang 17 studied Bedrosian identity for partial Hilbert transform and established characterizations and several necessity theorems for functions in space . The main goal of this paper is to discuss Bedrosian identity on space .…”

The circular Bedrosian identity for translation‐invariant operators: existence and characterization

“…The great success of empirical mode decomposition in the signal analysis has stimulated considerable attention to the time‐frequency analysis from the mathematics community. Especially, the important Bedrosian identity has been extensively studied . So far, most studies have been focused on the continuous case, namely, the functions considered therein live in .…”

“…Set for . The partial Hilbert transforms on L 2 ([− π , π ] d ) are defined through Fourier multipliers as follows: The partial Hilbert transforms are ideal for the time‐frequency analysis of multi‐dimensional periodic signals . The following result answers question (B) for this class of translation‐invariant operators.…”