A necessary and sufficient condition for a Bedrosian identity

Abstract

We present a sufficient and necessary condition for the Bedrosian identity to hold for a large class of mono-components based on a generalized Sinc-function.

Amplitudes of mono-component signals and the generalized sampling functions

“…The following lemma appeared in [4]. However, a completely new proof by direct construction is given here for both sufficiency and necessity.…”

Bedrosian Identity in Blaschke Product Case

“…In the case of a single Blaschke product, we used the fact that the Fourier transform of the generalized Sinc function is a ladder shape filter which can be calculated exactly. But using the same idea here leads to a matrix expression of such a filter, so that a direct application of the method given in [3] is not possible. The way out is to rewrite the matrix expression of the filter in terms of a Vandermonde matrix and show that ρ satisfying Bedrosian identity (1.4) is equivalent to the fact that its Fourier transform fulfils a recursive equation involving the matrix expression of the filter.…”

“…We want to show that form (1.5) is necessary for an amplitude function to satisfy Bedrosian identity (1.4). The outline is borrowed from the one parameter case [3]. To facilitate our discussion, we introduce two subspaces of L 2 (R)…”

“…While the part of showing that condition (1.5) is sufficient follows more or less the same lines as in [3], i.e. studying the corresponding generalized Sinc function and its behaviour under the Hilbert transform, the part of being necessary is much more complicated.…”

“…Recently, the authors gave a sufficient and necessary condition for Bedrosian identity [3] on amplitude functions in the special case that the phase θ is defined via the boundary value of a Möbius transformation τ a for a scalar parameter a ∈ (−1, 1), i.e.…”