Algorithm 1: Sinusoidal data fitting algorithm for parameters estimation

The signal $s\left[n\right]$in its general form can be represented as sum of sinusoids as follows: $$s\left[n\right]={{\displaystyle \sum }}_{l=1}^p\left({\alpha }_l\sin \left({\omega }_{l}n+{\phi }_l\right)+C_l\right)$$ (3) In Equation (3), ${\alpha }_l$is real valued constant describing the amplitude and ${\omega }_l$represent angular frequency, ${\phi }_l$is the initial phase. The constant $C_l$shows the mean value other than zero. It is convenient to write the Equation (3) as follows. $$s\left[n\right]={{\displaystyle \sum }}_{l=1}^p(A_l\cos \left({\omega }_{l}n\right)+B_l\sin \left({\omega }_{l}n\right)+C_l)$$ (4) It is convenient to use a vector representation of Equation (4). Let’s stack the samples in column vector as follows: $$s={\displaystyle \sum }_{l=1}^p{H}_l{\theta }_l$$ (5) In the above Equation (5); $$H_l=\left[\begin{array}{c}\cos \left(\omega l\right) \sin \left(\omega l\right) 1\\ \cos \left(2\omega l\right) \sin \left(2\omega l\right) 1\\ .\\ .\\ .\\ \cos \left(N\omega l\right) \sin \left(N\omega l\right) 1\end{array}\right] {\theta }_l=\left[\begin{array}{c}{A}_l\\ B_l\\ C_l\end{array}\right]$$ (6) In our case, the model given by Equation (4) is simplified as; $p=1$and $C=0.$ Consider a signal s[n] with unknown parameters $A,B,C$ and $\omega$ i.e., Equation (4) with$p=1$. The measured signal is then given by $s\left[n\right]$ deteriorated by additive noise$w\left[n\right]$ as shown in Equation (7): $$x\left[n\right]=s\left[n\right]+w\left[n\right] n=1,2,3, \dots \mathrm{..} , N$$ (7) The noise is assumed zero mean white Gaussian noise. The estimation problem is to estimate the signal parameters using the measured samples of the input data $x=\left[x\left(1\right), x\left(2\right), \dots . x\left(N\right)\right].$ The probability density function (pdf) $$p\left(x;\theta ,\omega \right)=\frac{1}{\left(2\pi {\sigma }^2\right)}\exp \left[\frac{-1}{2{\sigma }^2}{\left(x-H\theta \right)}^T\left(x-H\theta \right)\right]$$ (8) The above equation describes the probability per infinitesimal volume of receiving the data samples x given a set of parameters$\left\{\theta ,\omega \right\}$. The maximum likelihood estimator (MLE) tries to maximize the pdf with respect to unknown parameters for given values of x and use those parameters as estimates i.e., $$\left[\widehat{\theta },\widehat{\omega }\right]={\arg }_{\theta ,\omega }^{max}p\left(x;\theta ,\omega \right)$$ (9) Finally, the estimate of $\theta$ is given by least-squares solution as follows $$\widehat{\theta }={\left(H^{T}H\right)}^{-1}{H}^{T}x$$ (10) By using the least-squares solution, the MLE criterion function can be concentrated to one parameter as follows $$g\left(w\right)=x^{T}H{\left(H^{T}H\right)}^{-1}{H}^{T}x$$ (11) The frequency estimate is then obtained from maximizing the function g(w), that is $$\widehat{\omega }={arg}_{\omega }^{max}g\left(w\right)$$ (12) The Equation (12) can be solved either by using iterative step method i.e., Gauss-Newton iteration [42] or by a non-linear search.