This work defines the unique instantaneous frequency (IF) of an arbitrary time signal to be the circular instantaneous frequency (cIF) of the curvature radius of the signal's trajectory on the phase plane, where the signal's conjugate part is obtained from Hilbert transform (HT). Because a general signal of a dynamical system of multiple degrees of freedom contains multiple modal vibrations, its cIF varies dramatically and is not useful for system identification and other applications. If the signal is decomposed into modal vibration components without moving average, each component has no local extrema within each fundamental period and no local loops on the phase plane, each component's referred instantaneous frequency (rIF) with respect to the origin on the phase plane may be non-circular but is always non-negative, and the time-varying rIF and referred instantaneous amplitude (rIA) are convenient for combining the use of perturbation analysis for system identification. The empirical mode decomposition (EMD) of Hilbert–Huang transform (HHT) is valuable for decomposing a general nonlinear nonstationary signal into zero-mean intrinsic mode functions (IMFs), and HT enables accurate calculation of rIF and rIA of each IMF. Although the concept of circular frequency cannot be used for signal decomposition, it enables the development of time-domain-only techniques for online frequency tracking. A 5-point frequency tracking method is developed to eliminate the incapability of the original 4-point Teager–Kaiser algorithm (TKA) for frequency tracking of signals with moving averages. Moreover, a 3-point conjugate-pair decomposition (CPD) method is derived based on circle-fitting using a pair of conjugate harmonic functions. It is shown that both CPD and TKA are based on the concept of circle fitting, but TKA uses finite difference and CPD uses curve fitting in numerical implementation. However, the accuracy of TKA is easily destroyed by noise because of the use of finite difference. On the other hand, because CPD is based on curve fitting, noise filtering is an implicit capability and its accuracy increases with the number of processed data points. The rIF from HHT and the cIF from CPD and TKA are different by definition. Moreover, because the instantaneous frequency and amplitude are assumed to be constant in CPD and TKA, the cIF from CPD and TKA also deviates from the exact cIF.
Instantaneous frequency tracking of harmonic distortions for grid impedance identification based on Kalman filtering