Amplitude, phase, frequency - Fundamental concepts in the theory of oscillations

The paper investigates the use of the analytic signal in the theory of oscillations. Mathematical properties of the Hilbert transform and the analytic signal are studied, which enable one to define amplitude, phase, and frequency of an arbitrary oscillation at each moment of time. It is shown that a number of radio components (mixers, frequency modulators, detectors, frequency discriminators, etc.) transform the oscillation parameters defined by the analytic signal. The connection between the adiabatic invariant and the oscillation equation for the analytic signal is established.