Tracking and Estimation of Frequency, Amplitude, and Form Factor of a Harmonic Time Series [Lecture Notes]

Formulas for estimating and tracking the (time-dependent) frequency, form factor, and amplitude of harmonic time series are presented in this lecture note; in particular, sine-dominant signals, where the harmonics follow roughly the dominant first harmonic, such as photoplethysmography (PPG) and breathing signals. Special attention is paid to the convergence behavior of the algorithm for stationary signals and the dynamic behavior in case of a transition to another stationary state. The latter issue is considered to be important for assessing the tracking abilities for nonstationary signals. We will discuss special cases of sine-dominant signals that will converge to an approximate or the same value as for a single sinusoid. The presented formulas are recursive in nature and use only the instantaneous values of the signal, in a low-cost and low-complexity manner. In particular, there is no need to use square roots or trigonometric or matrix operations; therefore, these formulas are suitable for low-power ambulatory measurements and computationally demanding machine learning algorithms. Furthermore, there appears to be an interesting connection between the form factor and Geary's test of normality.