## 4 Discussion and Conclusions  
The various experiments we made were aimed at verifying the different aspects that had been highlighted theoretically in the paper. The correspondence between the noise superimposed on the \(g_{i}(t)\) functions and the noise superimposed on the \(h_{k}(t)\) functions, estimated from the \(g_{i}(t)\) functions, was verified through the tests performed in Section 3.1: The estimates made on the noise sequences superimposed on the \(g_{i}(t)\) functions and the \(h_{k}(t)\) functions showed to be in line with what could be theoretically predicted using relation (10), confirming the validity of the proposed technique. The subsequent Sections 3.2 and 3.3 had instead the purpose of verifying the theoretical hypotheses made on a possible improvement in the estimation of the \(h_{k}(t)\) parameters, in the presence of noise, through a low-pass filtering operation. As a preliminary remark, since we wanted to consider the time behavior of the \(h_{k}(t)\) functions, we chose to perform low-pass filtering through linear phase FIR filters, each with a specifically defined cutoff frequency related to the kernel order \(k\), in order to preserve, as far as possible, the time \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(h_{1}(t)\) & \(h_{2}(t)\) & \(h_{3}(t)\) & \(h_{4}(t)\) \\ \hline No processing & 8.05 & 68.60 & 12.21 & 63.62 \\
20 repetitions & 6.05 & 54.28 & 8.72 & 51.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Noise variance estimates on the \(h_{k}(t)\) in the absence of processing and averaging the estimates over 20 repetitions; case of the higher SNR value (31 [dB]). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(h_{1}(t)\) & \(h_{2}(t)\) & \(h_{3}(t)\) & \(h_{4}(t)\) \\ \hline No filter & 8.05 & 68.60 & 12.21 & 63.62 \\ Low pass filter & 5.84 & 35.59 & 7.20 & 45.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Noise variance estimates on the \(h_{k}(t)\) in the absence and presence of low-pass filtering in the case of the higher SNR value (31 [dB]). course of the \(h_{k}(t)\) functions linked to the harmonic contents present in the pass band of each low-pass filter. Both in the case of synthetic data and in the case of real data it was possible to find that the filtering operations significantly improve the quality of the estimate and do not significantly alter the \(h_{k}(t)\) functions. The results in Tables 4 and 5 deserve specific comment. The variance of the noise superimposed on the functions \(h_{k}(t)\), estimated in the initial portion of the sequences and averaged over 50 trials, gives the values reported in the table: The positive effect of filtering is evident in all four cases and for both SNR levels, but especially in the cases of order 2 and 4. This effect, so different among the different orders, can be easily explained: In the calculation of \(h_{2}(t)\) and \(h_{4}(t)\), the transformation (7) implies a strong emphasis (with coefficient \(18\,1\)) of the function \(g_{4}(t)\) (the one of highest order \(N_{H}=4\)) and therefore of the noise associated with it: The high variance value of the noise on \(h_{2}(t)\) and \(h_{4}(t)\) before filtering can be ascribed to this aspect. The same function \(g_{4}(t)\) is the one associated to the fourth harmonic, so the corresponding noise components have an harmonic content that includes high frequencies: They can be easily removed by the low pass filtering operation. This also explains the strong reduction, as a consequence of filtering, of the estimated variance. The very good results we presented encourage us to work on further extensions of the research activity; we plan to use the techniques proposed in this paper in new experimental setups for further verifications. In addition, we want to test how these techniques combine with the filtering techniques proposed in [22] for the elimination of oscillations due to the band-limited swept sine excitation signal.