*2.1. Non-Parametric Methods*

Considering the multi-tone signal in (1), sampled with a *Ts* sampling period, the obtained signal is described by:

$$\mathbf{x}(n) = \sum\_{i=1}^{N\_s} A\_i \sin(2\pi f\_i n T\_s + \phi\_i) \quad n = 1 \ldots N. \tag{2}$$

Non-parametric methods are based on the DFT algorithm, where the spectrum samples are evaluated as follows:

$$X(k) = \frac{1}{G} \sum\_{n=0}^{N-1} w(n)\mathbf{x}(n)e^{-\frac{2j\pi}{N}kn} \quad k = 0 \ldots N-1,\tag{3}$$

where *x*(*n*) is the sampled signal (2) and *w*(*n*) are the window samples with gain *G*, and *k* is the spectral bin index, also known as the bin number. If the sampled signal is coherent with the module of the sampled sequence DFT (3), then *M*(*k*) = |*X*(*k*)| presents *Ns* peaks, corresponding to the *Ns* tone frequencies; the *i*-th peak is located exactly at index *ki*.

When coherent sampling conditions are not assured, a quantization error arises in the frequency estimation [26]; the tone module is underestimated because of the spectral leakage. Moreover, harmonic interference is present, causing an error in parameters estimation when the sampled signal presents two tones with a small frequency difference compared with the frequency resolution, or when it has only one tone but the frequency is less than two times the frequency resolution Δ*f* .

To correct errors on frequency estimation, phase, and amplitude estimation, several non parametric methods have been exploited in the literature [10]. In the following sections, some non parametric methods will be briefly treated, in particular the interpolated FFT (IFFT) on two points and three points, and the corrected IFFT.
