## 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.

### MSC:

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

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