Harmonic Modeling
The Harmonic Estimation is based upon the least-squares fitting of sinusoids or damped sinusoids in the time-domain.
The fitting proceeds in two distinct steps:
1.The component count and frequencies are taken from the spectral estimation procedure (this will be specific to the algorithm)
2.Amplitudes and phases are then estimated using a linear least-squares fit of the model set up for the spectral components found.
The model can be one of the following:
Sinusoidal: Y=Ampl*sin(2*π*Freq*X+Phase)
Damped sinusoidal: Y=Ampl*exp(-k*X)*sin(2*π*Freq*X+Phase)
Suboptimal Least-Squares
A linear sinusoidal fit is a suboptimal regression minimization because the frequencies are locked at the values determined by the spectral procedure. Only the amplitudes and phases (and damping factors for damped sinusoids) are allowed to vary.
Although the fit is suboptimal in the statistical sense, a better characterization of harmonics generally occurs. This is particularly true in the instance where the harmonics vary significantly in power. The manner in which noise impacts the largest harmonic in least-squares modeling can introduce a bias into the model fit that distorts the estimated parameters for lower power sinusoids. For this analysis to be meaningful, it must be possible to model the data with a combination of narrowband components.
The data must be wide-sense stationary where sinusoids serve as valid narrowband models. For the fit to be useful, the component count must be also correct and the frequencies must be accurately determined. The count of spectral components and the set of spectral frequencies automatically come from the spectral procedure or can be provided manually as fixed values.
The accuracy of the parametric model depends on the two-step procedure. The native spectral algorithm is used only to determine the frequencies and component count. The linear fit estimates amplitudes and phases as reliably as these original frequencies permit. If the estimates for the frequencies are poor or if the component count is incorrect, the parametric estimates will not represent the true optimum.