Two families of unit analytic signals with nonlinear phase

This paper focuses on constructing two families of unit analytic signals with nonlinear phase. The first is the 2 π -periodic extension of the nonlinear Fourier atoms, viz. { e i θ a ( t ) : | a | < 1 , t ∈ R } , where θ a ′ ( t ) is the Poisson kernel of the unit circle associated with a in the unit disc in the complex plane and satisfies θ a ( t + 2 π ) = θ a ( t ) + 2 π ; and the second consists of { e i ϕ a ( t ) : | a | < 1 , t ∈ 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 , θ a ( t ) ) and ( 1 , ϕ a ( t ) ) form canonical pairs. The real signals cos θ 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.