Time-Frequency Aspects of Nonlinear Fourier Atoms

Abstract

In the standard Fourier analysis one uses the linear Fourier atoms e ^int : n ∈ ℤ. With only the linear phases nt Fourier analysis can not expose the essence of time-varying frequencies of nonlinear and non-stationary signals. In this note we study time-frequency properties of a new family of atoms e ^{inθa (t)} : n ∈ ℤ, non-linear Fourier atoms, where a is any but fixed complex number with |a| < 1, and dθ_a (t) a harmonic measure on the unit circle parameterized by t. The nonlinear Fourier atoms e ^inθ _a ^(t) : n ∈ ℤ were first noted in [12] with some examples and theoretically studied in [8]. In this note we show that the real parts cos θ _a (t), |a| < 1, form a family of intrinsic mode functions introduced in the HHT theory [5]. We prove that for a fixed a the set e ^{inθa (t)} : n ∈ ℤ, constitutes a Riesz basis in the space L ²([0, 2π]). Some miscellaneous results including Shannon type sampling theorems are obtained.

Qiuhui Chen is supported in part by NSFC under grant 10201034 and the Project-sponsored by SRF for ROCS, SEM. Luoqing Li is supported in part by NSFC under grant 10371033. Tao Qian is supported by University of Macau under research grant RG065/03-04S/QT/FST and Macao Science and Technology Development Fund (FDCT) 051/2005/A.