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

Abstract

Communicated by T. QianThe analytic signal method via the circular Hilbert transform Q H is a critical tool in the time-frequency analysis of signals of finite duration. The circular Bedrosian identity Q H.fg/ D f Q Hg is of major theoretical and practical value in the method. The identity holds whenever the Fourier coefficients of f, g 2 L 2 .OE , / are respectively supported on A D OE n, m and B D Z n OE m, n for some non-negative integers 0 Ä n, m Ä C1. In this note, we investigate the existence of such an identity for a general-bounded linear translation-invariant operator on L 2 .OE , d / and for general support sets A, B Â Z d . We give an insightful geometric characterization of the support sets for the existence. In addition, we find all the support sets for the partial Hilbert transforms.