**5. Uncertainty Evaluation**

As evidenced in Section 3, all the analysed methods present a residual error, that can be negligible or not function depending on the signal characteristics and the processing parameters. The residual contribution cannot be corrected since it strictly depends on the signal characteristics and the uncertainty evaluation has to be taken into account. To this aim, it is possible to write the following:

$$
\delta = \delta\_m + \mathbb{C}\_{\delta \prime} \tag{14}
$$

where *δ* is the corrected bin value, *δ<sup>m</sup>* is the evaluated bin value, and *C<sup>δ</sup>* is the correction that can be modelled as random variable with mean value equal to zero and standard deviation, *σC*, different from zero. Applying the ISO GUM [27] the measurement uncertainty is equal to:

$$
\mu\_{\delta} = \sqrt{\sigma\_{\delta}^{2} + \sigma\_{\mathbb{C}}^{2}}.\tag{15}
$$

As for the standard deviation of the correction value, the measurement uncertainty can be estimated considering group of signals with similar characteristics. In Table 1 the measurement uncertainty *u<sup>δ</sup>* on the tone frequencies evaluated for a two-tone signal with tones at varying distance (used as the index of the table) and for different values of *β*<sup>12</sup> is reported. The uncertainty is evaluated considering for each configuration 1000 simulations with random phase and varying *d*<sup>12</sup> between the 2 tones, and the FFT is made on 256 samples. The measurement uncertainty is reported for the six considered methods. By looking at these data, it is possible to have an idea of the order of magnitude of the uncertainty, given the signal characteristics *d*1, *d*2, *β*<sup>12</sup> for a given algorithm; the uncertainty of the second tone for *d*<sup>12</sup> between 3 and 4 when *β*<sup>12</sup> is equal to 0.01 are not reported because it is not ever correctly detected with the algorithms based on FFT.

**Table 1.** Measurement uncertainty, *uδ*, evaluated for two-tone signals with tones at varying distance for different values of *β*12, changing *d*<sup>12</sup> from *d*<sup>1</sup> to *d*2. The simulations were repeated 1000 times, randomizing the tone phases with a 256-sample signal.




In order to verify the proposed approach, Table 1 is used to evaluate the expected uncertainty for three different signals that is compared with the measured one, evaluated with a type B approach. The analysed signals refer to different conditions: close-frequency tones (*d*12), one of these with significantly lower amplitude (*β*12); low-noise (Case 1 and Case 2); tones of the same amplitude with high noise (Case 3); tones with a high enough SNR (Case 4). In the first case, a two-tone signal with *β*<sup>12</sup> = 0.1, *d*<sup>12</sup> = 3.6, and SNR = 40 dB has been used; Case 2 reports the same kind of signal with *β*<sup>12</sup> = 0.1, *d*<sup>12</sup> = 4.5, and SNR = 80 dB; for Case 3, *β*<sup>12</sup> = 1, *d*<sup>12</sup> = 5.2, and SNR = 10 dB; meanwhile, in the last case, the signal uses the parameters *β*<sup>12</sup> = 1, *d*<sup>12</sup> = 7.9, and SNR = 60 dB.

In Table 2 the uncertainty of both tones is synthesized for the three algorithms—IFFTc, IFFT3p, and ESPRIT. Generally, one or two digits are enough to express the uncertainty value; however, in Table 2, more digits are used to clearly highlight the differences between the reported methods. It can be seen that there is, for all the signals, high similarity between the measured and the expected uncertainties. Even under different conditions, where the uncertainty components—due to the residual error and the noise—have different contributions, in all cases, the estimation of the uncertainty is accurate and can be an a priori alternative to the measured value. A little overestimation for the IFFT3p algorithm is

*uδ*1

*uδ*2

observed for Case 2, when the contribution—due to the residual error—is prevalent; this is due to the high dependence of the residual error on the tone frequency value, but in our estimation, a medium value is considered.


exp. 1.03 × <sup>10</sup>−<sup>2</sup> 1.72 × <sup>10</sup>−<sup>2</sup> 6.28 × <sup>10</sup>−<sup>3</sup> 6.99 × <sup>10</sup>−<sup>3</sup> 7.52 × <sup>10</sup>−<sup>3</sup> 4.24 × <sup>10</sup>−<sup>3</sup> 8.97 × <sup>10</sup>−<sup>3</sup> 1.03 × <sup>10</sup>−<sup>2</sup> 5.43 × <sup>10</sup>−<sup>3</sup> 2.84 × <sup>10</sup>−<sup>2</sup> 3.16 × <sup>10</sup>−<sup>2</sup> 1.70 × <sup>10</sup>−<sup>2</sup>

meas. 2.96 × <sup>10</sup>−<sup>3</sup> 3.52 × <sup>10</sup>−<sup>2</sup> 1.74 × <sup>10</sup>−<sup>3</sup> 1.58 × <sup>10</sup>−<sup>3</sup> 2.05 × <sup>10</sup>−<sup>3</sup> 1.01 × <sup>10</sup>−<sup>3</sup> 1.28 × <sup>10</sup>−<sup>3</sup> 1.44 × <sup>10</sup>−<sup>3</sup> 6.92 × <sup>10</sup>−<sup>4</sup> 5.26 × <sup>10</sup>−<sup>3</sup> 6.52 × <sup>10</sup>−<sup>3</sup> 3.37 × <sup>10</sup>−<sup>3</sup> exp. 2.98 × <sup>10</sup>−<sup>3</sup> 3.27 × <sup>10</sup>−<sup>3</sup> 1.74 × <sup>10</sup>−<sup>3</sup> 1.58 × <sup>10</sup>−<sup>3</sup> 1.22 × <sup>10</sup>−<sup>2</sup> 1.01 × <sup>10</sup>−<sup>3</sup> 1.28 × <sup>10</sup>−<sup>3</sup> 1.20 × <sup>10</sup>−<sup>3</sup> 6.93 × <sup>10</sup>−<sup>4</sup> 5.26 × <sup>10</sup>−<sup>3</sup> 6.50 × <sup>10</sup>−<sup>3</sup> 3.37 × <sup>10</sup>−<sup>3</sup>

**Table 2.** Comparison of the expected uncertainty and the measured uncertainty for three different cases of a two-tone signal with changing parameters: *β*12, *d*12, and SNR.

It is almost possible to observe an invariability of the uncertainty on both the first and second tone frequencies at the various conditions for the IWPA, ESPRIT, and MUSIC algorithms, with the same order of magnitude for both the tones for a given algorithm. The ESPRIT algorithm shows again the lowest uncertainty compared with the other parametric algorithms; the IWPA shows the worst performance in all cases. The IWPA shows better performance compared with non-parametric algorithms in almost no cases. For higher ratio of *d*<sup>12</sup> the performance in terms of measurement uncertainty on the second tone of the non-parametric algorithms starts to be two orders of magnitude better than the IWPA algorithm. Only in the case of a low ration *β*<sup>12</sup> and low *d*<sup>12</sup> IWPA could be considered a good choice with respect to a non-parametric algorithm. Comparing the algorithms based on FFT, the IFFTc is able to correct the effect of the interfering tone almost in all cases (see tone 2 uncertainty with *β*<sup>12</sup> less than 1).
