**3. Residual Errors**

Due to approximations, the considered methods may exhibit a bias between the estimated and actual values of the signal tones, even when noise is not superimposed to the signal, and their expected values are not equal to their actual values. Such behaviour can be associated with interharmonic interference; as for IFFTc and IWPA, the behaviour can be associated with inadequate knowledge of the values required by parametric methods or the finite word length of the data processing.

To evaluate the proposed methods and produce a clear comparison of their performance, the multi-frequency signals described by (1) were considered; the tests are made for different values of the number of tones (*Ns*), the number of samples (*N*), the frequency (*fi*), the amplitude (*Ai*), and the phase (*φi*). All the simulations have been made supposing an observation window longer than two periods of the signal. The major effects analysed are the frequency resolution, the signal dynamic range, and the harmonic interference [10].

$$f\_i \quad = \quad (k\_i + \delta\_i) \Delta f \tag{12a}$$

$$A\_i \quad = \ \beta\_i \cdot A\_0 \tag{12b}$$

$$d\_{\rm ij} = \begin{array}{c} f\_{\rm j} - f\_{\rm i} \\ \Delta f \end{array} \tag{12c}$$

Given Equations (12a)–(12c), where *ki* is the frequency bin index corresponding to *fi* and *δ<sup>i</sup>* is the fractional bin deviation, the simulations are made at changing values of *δi*, *dij*, *βi*, *φi*, and *N* in order to analyse the dependence of the harmonic interference effects on the signal characteristics and the measurement system settings. In order to evaluate the interference effects on the different methods, tests with only two tones with the same amplitude (*A*<sup>1</sup> = *A*2), corresponding to more substantial interference on both tones, refs. [3–5] are carried out at changing distances between tones, *d*12; with *d*<sup>1</sup> always greater than 20.

The logarithms of the absolute errors on the fractional bin deviations are calculated as the difference between the measured ( ˆ *δi*) and the real value (*δi*), as follows:

$$E\_{\delta\_{\hat{i}}} = |\hat{\delta}\_{\hat{i}} - \delta \dot{\imath}| \tag{13}$$

In Figure 1 the estimation error (13) for the first tone versus the distance between tones (*d*12) is reported for the considered methods; similar results are obtained with the second tone. Interpolated FFT algorithms use the Hanning window, while for MUSIC and ESPRIT, *M* = *N*/4 was posed, and the matrix covariance was calculated using the samples with no noise added.

**Figure 1.** Absolute errors on *δ* obtained for a two-tone signal versus the distance, *d*12, between tones.

Some considerations can be outlined, as follows:


Since parametric algorithms require the knowledge of the number of spectral components, but the information can not be obtained in some applications, a characterization of all the algorithms will be reported for the case in which a different and generally wrong number of spectral components (*Ns*<sup>0</sup> ) is specified. For instance, Figure 2 shows the errors on *δ* versus the specified number of tones, *Ns*<sup>0</sup> , in the case of a five-tone signal (*Ns* = 5) for the considered algorithms. The results refer to a signal with all the tones at the same amplitude (*Ai* = 1) and uniformly spaced with *di*,*i*−<sup>1</sup> = 3. For the cases where *Ns*<sup>0</sup> < 5, the error on *δ* for a non-detected tone is evaluated with respect to the closest detected tone.

As expected, the algorithms based on IFFT, being non-parametric algorithms, are not influenced by *Ns*<sup>0</sup> , and the residual errors are quite similar for each tone. Parametric methods MUSIC and ESPRIT manifest a different behaviour: errors are very high for each tone as long as *Ns*<sup>0</sup> is lower than the actual number of tones. In other words, if *Ns*<sup>0</sup> is lower than *Ns*, then the estimated frequencies are significantly different (at least Δ*f* /2) from the actual frequencies of each of the five tones. When *Ns*<sup>0</sup> ≥ *Ns*, the ESPRIT method gives the best performance: it does not show residual errors, and small differences (less than 10−15) are only caused by the finite word length of the precessing unit (CPU); MU-SIC shows greater errors (about 10−7), but these are negligible with respect to the other

methods. IWPA is less sensitive to an underestimated number of tones (*Ns*<sup>0</sup> ≥ *Ns*): in these cases, the frequency estimations are better than the other parametric methods, while for a *Ns*<sup>0</sup> ≥ *Ns*, its estimation deteriorates, since noise components are considered erroneously as signal tones until *Ns*<sup>0</sup> components are detected. In the results of Figure 2, when *Ns*<sup>0</sup> is less than *Ns* = 5, the errors for the undetected components are evaluated as the absolute difference between the actual value for that component and the estimated value for the closest component.

**Figure 2.** Errors on *δ* or a 5-tone signal (*Ns* = 5), versus the specified number of tones, *Ns*<sup>0</sup> . Each figure refers to a single tone starting from tone 1 (on the left) to tone 5 (on the right).

Further tests were carried out to highlight the sensitivity of the different methods to the number of processed samples; in particular, the trends (not reported here for the sake of brevity) of the errors on *δi*, versus the bin distance, and versus the tone amplitudes, do not change when the number of acquired samples changes from 128 to 2048. This is expected for the error on *δi*, which is a kind of relative error and is different from the error on the frequency. Once the sampling frequency has been set, the greater the number of samples, the lower the spectral resolution, and, consequentially, the lower the error on frequency will be. However, a small reduction in the residual errors is measured only for the IWPA and IFFT methods when *<sup>N</sup>* increases (from about *<sup>E</sup><sup>δ</sup>* = <sup>3</sup> × <sup>10</sup>−<sup>3</sup> with *<sup>N</sup>* = 128 to *<sup>E</sup><sup>δ</sup>* = <sup>2</sup> × <sup>10</sup>−<sup>4</sup> for *<sup>N</sup>* = 4096).

#### **4. Repeatability under Noisy Conditions**

Some amount of noise always corrupts real-life signals, so the considered methods have to be evaluated when applied with noisy signals, since their performance may worsen significantly. The tests are carried out by changing the signal characteristics to estimate each method's sensitivity to the tone composition; only two-tone signals are considered. Once the signal and the measurement parameters have been fixed, a Gaussian noise is added, noisy signal samples are generated, and the algorithms process these points in order to estimate the signal characteristics. For each signal, configuration, and noise level, the tests are repeated 1000 times; the mean and the standard deviation of the results of the algorithms are calculated. For the three algorithms, based on the FFT interpolation, a Hanning window is used.

## *4.1. Sensitivity to the First Tone Distance*

Figure 3 reports the behaviour of the algorithms respect to a signal composed of two tones very close in frequency (*d*<sup>12</sup> = 3 bins) and with the same null phase. The measured mean square error (MSE) versus the signal-to-noise ratio (SNR) for the different methods are reported, where the Cramér–Rao bound (CRB) [14] is also reported, since it gives information about the best theoretical performance (minimum variance of the quantity of interest) achievable with an ideal estimator, versus the level of superimposed noise. It has to be highlighted that the MSE considers both the random variability and the systematic effects [13]. The adopted CRB values are obtained with relationships valid in the specific case of a single-tone signal. However, the CRB estimation can be considered a kind of lower limit, and the goodness of the estimation of a proposed method can be evaluated through the closeness of the resulting MSE to the CRB.

**Figure 3.** Mean square errors (MSE) versus the SNR for a two-tone signal with *A*<sup>1</sup> = *A*<sup>2</sup> = 1, *N* = 256, *f*1/Δ*f* = 40.2 bins, *d*<sup>12</sup> = 3 bins, and zero phase difference.

Analysing these results, it is possible to state that IFFT and IFFTc algorithms are less sensitive to a high noise level than the other algorithms. In particular, IFFTc shows an MSE on *δ* less than 0 dB for SNR less than zero, while the errors can reach 20 dB for the other algorithms. For higher SNR, MUSIC, and ESPRIT show the best performance, but the results of IFFTc and IWPA are comparable with those of the other two methods when the phases are equal to zero. In presence of phase difference, not reported here for simplicity, the performance of ESPRIT and MUSIC does not change while IFFT deteriorates slightly (about 2 dB) for SNR values between 0 dB and 20 dB; the MSE on *δ* of the IFFTc algorithm declines of about 3 dB for high SNR (greater than 40 dB) when residual systematic effects on the phase estimation become predominant, and IWPA remarkably loses its estimation capability at the point that it can hardly be adopted.

Figure 4 reports the MSE on *δ* versus the relative distance between the two tones of a signal. The improvement of interpolation of the IFFTc over IFFTs is evident since IFFTc keeps good performance from *d*<sup>12</sup> equal to 3 onwards. However, the lowest values of MSE*<sup>δ</sup>* are reached by ESPRIT and MUSIC.

**Figure 4.** MSE of *δ* for the first tone versus the normalized tone distance *d*12, with *A*<sup>1</sup> = *A*<sup>2</sup> = 1, *N* = 256, *f*1/Δ*f* = 40.2, random phases, and SNR = 40 dB.

#### *4.2. Sensitivity to the Tone–Amplitude Ratio*

In Figure 5 the trends of the MSE in the estimation of the bin deviations for a two-tone signal with very close frequencies (*d*<sup>12</sup> = 3) and with random phases are reported, versus the amplitude of the second tone (*β*<sup>2</sup> changes in the range [0.1, 2], while *β*<sup>1</sup> = 1) for two different SNRs (5 dB and 40 dB). The figures show only the performance of the parametric method ESPRIT and the non-parametric algorithm based on IFFT, since the results of MUSIC are very similar to those of ESPRIT, while IWPA introduces very high errors in presence of phase variations.

**Figure 5.** MSE of *δ* versus the amplitude of the second tone with *β*<sup>1</sup> = 1 fixed at two different SNR values 5 dB (on the left) and 40 dB (on the right). *N* = 256, *f*1/Δ*f* = 40.2, *d*<sup>12</sup> = 3, and random phases.

ESPRIT algorithm exhibits worse performance in the estimation of the the second tone frequency when *β*<sup>2</sup> is low, due to the low values of SNR at the second tone especially in the case of the lowest of the two SNR values (5 dB), while the MES value decreases for increasing amplitudes of the second tone. The estimation of the highest tone is not influenced by the amplitude of the lowest one. On the other hand, the IFFTc method is slightly influenced by the change in amplitude. Moreover, the variability obtained with all the methods on *δ<sup>i</sup>* is comparable (IFFTc is characterized by a standard deviation *σδ<sup>i</sup>* a bit greater than the others) and the same behaviour is observed when the second tone amplitude becomes significantly greater than the noise (*β*<sup>2</sup> > 0.5).
