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)eiϕ(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)eiP(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)=a0+a1t there holds a1⩽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.