Two families of unit analytic signals with nonlinear phase. (English) Zbl 1104.94004

Summary: This paper focuses on constructing two families of unit analytic signals with nonlinear phase. The first is the \(2\pi\)-periodic extension of the nonlinear Fourier atoms, viz. \(\{e^{i\theta_a(t)}:|a|<1\), \(t\in \mathbb{R}\}\), where \(\theta_a' (t)\) is the Poisson kernel of the unit circle associated with \(a\) in the unit disc in the complex plane and satisfies \(\theta_a(t+2\pi)=\theta_a(t)+2\pi\); and the second consists of \(\{e^{i\varphi_a(t)}:|a|<1\), \(t\in\mathbb{R}\}\), that are the images of the nonlinear Fourier atoms under Cayley transform. These unit analytic signals are mono-components based on which one can define meaningful instantaneous frequency. The pairs \((1,\theta_a(t))\) and \((1,\varphi_a (t))\) form canonical pairs. The real signals \(\cos\theta_a(t)\) corresponding to the first family coincide with the notion of normalized intrinsic mode functions. We finally point out that, starting from nonlinear Fourier atoms, the Gram-Schmidt procedure leads to Laguerre bases.


94A12 Signal theory (characterization, reconstruction, filtering, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
Full Text: DOI


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