On instantaneous amplitude and phase of signals
IEEE Transactions on Signal Processing
A sampling theorem for non-bandlimited signals using generalized Sinc functions
Computers & Mathematics with Applications
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In many cases, a real-valued signal @g(t) may be associated with a complex-valued signal a(t)e^i^@q^(^t^), the analytic signal associated with @g(t) with the characteristic properties @g(t) = a(t) cos@q(t) and H(a(.)cos@q(.))(t) = a(t)sin@q(t). Using such obtained amplitude-frequency modulation the instantaneous frequency of @g(t) at the time t"0 may be defined to be @q'(t"0), provided @q'(t"0) = 0. The purpose of this note is to characterize, in terms of analytic functions, the unimodular functions F(t) = C(t) + iS(t),C^2(t) + S^2 (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) = cos@q(t) is available and the extra condition @q'(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].