Characterizing the Hilbert transform by the Bedrosian theorem

Abstract

It is proved that a bounded linear translation invariant operator on L 2 (R d ) satisfies the Bedrosian theorem if and only if it is a linear combination of the compositions of the partial Hilbert transforms and the identity operator. This observation justifies a definition of multidimensional analytic signals in the papers [T. Bulow, G. Sommer, Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Process. 49 (2001) 2844-2852] and [S.L. Hahn, Multidimensional complex signals with single-orthant spectra, Proc. IEEE 80 (1992) 1287-1300].

Multidimensional Analytic Signals and the Bedrosian Identity

“…The following results was proved in [30]. Thus, if we desire that the operator T satisfy the type one Bedrosian theorem then it must be a linear combination of the compositions of the partial Hilbert transforms.…”

“…For some applications, it is desirable to restrict the frequency to a certain d-hyperoctant, especial the first one, [12]. Following the notations of [30], we let ν k , k ∈ N 2 d , be the extreme points of the cube [−1, 1] d . Assume that ν 1 and ν 2 are the point each of whose component is 1 and −1, respectively.…”

A necessary and sufficient condition for a Bedrosian identity

“…The authors would like to remark that there exist already several such results for some special cases (see, e.g. [10,11], among others). In this paper, we present conditions for the Bedrosian identity to hold for a large class of mono-components, which we obtain by considering the boundary-value of a single Blaschke product, i.e.…”

Recent Mathematical Developments on Empirical Mode Decomposition

“…Since it was first introduced in [12], the empirical mode decomposition (EMD) has been found useful applications in many engineering areas. Recently, EMD related mathematical problems attracted much attention from the mathematical community (cf., [7,16,22,23,24,25,26,29,31,32,33,34,36,37,39]). These studies aim at better understanding the mathematical insight of the algorithm, building a reasonable mathematical foundation for the method and improving upon it.…”