2.1.1. IFFT-2p

Interpolated FFT algorithms [10,11] have been known in the literature for several years, and those based on a two-point interpolation are the most common. The frequency, *fi*, of the *i*-th tone can be evaluated as: *fi* = (*ki* + *δi*)Δ*f* , where Δ*f* is the DFT frequency resolution (Δ*f* = *fs*/*N*), *k* is the integer part of the bin (*f* /Δ*f*), and *δ<sup>i</sup>* ∈ [−1/2, +1/2] is the fractional bin deviation. The fractional bin deviation, *δi*, is evaluated from the ratio between the two largest samples closest to the peak: *<sup>α</sup><sup>i</sup>* <sup>=</sup> <sup>|</sup>*X*(*ki*+*i*)<sup>|</sup> <sup>|</sup>*X*(*ki*)<sup>|</sup> , where: *<sup>i</sup>* <sup>=</sup> *sign*(|*X*(*ki* <sup>+</sup> <sup>1</sup>)|−|*X*(*ki* <sup>−</sup> <sup>1</sup>)|). Considering the sampled spectrum of the window function, *W*(*k*), the following is obtained [9]:

$$\alpha\_{i} = \frac{|\mathcal{W}(\varepsilon\_{i} - \delta\_{i})|}{|\mathcal{W}(-\delta\_{i})|} = \frac{|\mathcal{W}(k\_{i} - \delta\_{i})|}{|\mathcal{W}(k\_{i})|} \tag{4}$$

The value of *δ<sup>i</sup>* can be evaluated from the latter relationship, given the window function and its analytical expression.

#### 2.1.2. IFFT-3p

The interpolated three-point DFT algorithm [12–15] is based on an interpolation of the DFT results of the signal, windowed by cosine windows, and using three points for each tone peak. Considering the multi-tone signal, with *Ns* spectral components of (1), like the IFFT-2p, the frequency *fi* of the *i*-th tone is evaluated as *fi* = (*ki* + *δ*3*i*)Δ*f* ; in this case, *δ*3*<sup>i</sup>* is evaluated considering the three largest samples of the peak:

$$\alpha\_{3i} = \frac{|X(k\_i - 1)| + |X(k\_i + 1)|}{|X(k\_i - 1)| + 2|X(k\_i)| + |X(k\_i + 1)|} \tag{5}$$

$$
\delta\_{\mathfrak{H}} = \mathbb{K} \* \mathfrak{a}\_{\mathfrak{H}}.\tag{6}
$$

where *K* is a proportional factor that depends on the used windowing function; in the case of an Hanning window, this is *K* = 2.
