Characterization of analytic phase signals

In many cases, a real-valued signal χ(t) may be associated with a complex-valued signal a(t)eiθ(t), the analytic signal associated with χ(t) with the characteristic properties χ(t) = a(t) cosθ(t) and H(a(·)cosθ(·))(t) = a(t)sinθ(t). Using such obtained amplitude-frequency modulation the instantaneous frequency of χ(t) at the time t0 may be defined to be θ′(t0), provided θ′(t0) ≥ 0. The purpose of this note is to characterize, in terms of analytic functions, the unimodular functions F(t) = C(t) + iS(t),C2(t) + S2 (t) = 1, a.e., that satisfy HC(t) = S(t). This corresponds to the case a(t) ≡ 1 in the above formulation. We show that a unimodular function satisfies the required condition if and only if it is the boundary value of a so called inner function in the upper-half complex plane. We also give, through an explicit formula, a large class of functions of which the parametrization C(t) = cosθ(t) is available and the extra condition θ′(t) ≥ 0, a.e. is enjoyed. This class of functions contains Blaschke products in the upper-half complex plane as a proper subclass studied by Picinbono in [1].