Analytic unit quadrature signals with nonlinear phase

The notion of intrinsic mode functions (IMFs) in the algorithm of Hilbert-Huang transform (HHT) [N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.-C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. London A 454 (1998) 903-995] is essentially an engineering description in relation to mono-components of nonlinear and non-stationary signals. In this note we prove a version of Bedrosian's theorem on the unit circle. We give a sufficient condition together with an example for nonlinear phases θ(t) that make the unit quadrature signals eiθ(t) to be analytic. We also establish a corresponding relationship between the periodic and non-periodic signals on the whole time range.