An application of entire function theory to analytic signals

Analytic signals of finite energy in signal analysis are identical with non-tangential boundary limits of functions in the related Hardy spaces. With this identification this paper studies a subclass of the analytic signals that, with the amplitude-phase representation s ( t ) = ρ ( t ) e i ϕ ( t ) , ρ ( t ) ⩾ 0 , satisfy the relation ρ ′ ( t ) ⩾ 0 a.e., signals in this subclass are called mono-components, and, in that case, the phase derivative ϕ ′ ( t ) is called the analytic instantaneous frequency of s. This paper proves that when s ( t ) = A ( t ) e i P ( t ) , where A ( t ) is real-valued, band-limited with minimal bandwidth B and P ( t ) is real-valued, as the restriction on the real line of some entire function, then s is an analytic signal if and only if P ( t ) is a linear function, and with P ( t ) = a 0 + a 1 t there holds a 1 ⩽ B . In the case s is a mono-component. This generalizes the corresponding result obtained by Xia and Cohen in 1999 in which P ( t ) is assumed to be a real-valued polynomial.