## 3 Experimental Results
### Mitigation of Noise Effects: Ultrasonic System with in-Air Propagation
We observe also in this real data case, that the filtering operation has not altered the temporal trend of the \(h_{k}(t)\) functions, as far as it is possible to appreciate in cases where the noise level is not excessively high. The variance of the noise superimposed on the \(h_{k}(t)\) functions in the case of the lower SNR value, estimated in the initial portion of the sequences and averaged over 50 trials, gives the values reported in Table 4: The positive effect of filtering is evident in all four orders. Improvements of different magnitudes are apparent for different orders: this will be discussed in the following section. Figure 8: Non-contact point focused ultrasonic probes experiment: \(g_{i}(t)\) functions for the four orders considered (expanded representation scale) in the case of the lower SNR value (20 [dB]). Plots **(A–D)**: functions \(g_{1}(t),\ g_{2}(t),\ g_{3}(t),\ g_{4}(t)\), respectively. Figure 9: Estimated \(h_{k}(t)\) kernels in the case of the ultrasonic acquisition system designed for in-air propagation in the case of the lower SNR value (20 [dB]). Plots **(A–D)**: \(h_{1}(t),\ h_{2}(t),\ h_{3}(t),\ h_{4}(t)\) estimates before lowpass filtering. Plots **(E–H)**: \(h_{1}(t),\ h_{2}(t),\ h_{3}(t),\ h_{4}(t)\) estimates after lowpass filtering. The variance of the noise superimposed on the \(g_{i}(t)\) functions in the case of the higher SNR value, estimated in segments of the output signal \(u(t)\) preceding the instants \(\Delta t_{k}\) and averaged over 50 trials, gives the following values:  
\[\left[\sigma^{2}_{g}\right]=\left[ \begin{array}{cccc}1.28&1.14&1.65&3.81\\ \end{array} \right]^{T}\]  
The variance of the noise superimposed on the \(h_{k}(t)\) functions in the case of the higher SNR value, estimated in the initial portion of the sequences and averaged over 50 trials, gives the values reported in Table 5: The positive effect of filtering is evident in all four \begin{table}
\begin{tabular}{c c c c c} \hline  & \(h_{1}(t)\) & \(h_{2}(t)\) & \(h_{3}(t)\) & \(h_{4}(t)\) \\ \hline No filter & 4.69 & 29.15 & 5.25 & 25.95 \\ Low pass filter & 3.70 & 5.31 & 1.22 & 11.91 \\ \hline \end{tabular}
\end{table} 
Table 4: Noise variance estimates on the \(h_{k}(t)\) in the absence and presence of low-pass filtering in the case of the lower SNR value. Figure 10: Non-contact point focused ultrasonic probes experiment: \(g_{i}(t)\) functions for the four orders considered (expanded representation scale) in the case of the higher SNR value (31 [dB]). Plots (**A–D**): functions \(g_{1}(t)\), \(g_{2}(t)\), \(g_{3}(t)\), \(g_{4}(t)\), respectively. Figure 11: Estimated \(h_{k}(t)\) kernels in the case of the ultrasonic acquisition system designed for in-air propagation in the case of the higher SNR value (31 [dB]). Plots (**A–D**): \(h_{1}(t)\), \(h_{2}(t)\), \(h_{3}(t)\), \(h_{4}(t)\) estimates before lowpass filtering. Plots (**E–H**): \(h_{1}(t)\), \(h_{2}(t)\), \(h_{3}(t)\), \(h_{4}(t)\) estimates after lowpass filtering. orders. Improvements of different magnitude are apparent for different orders: This will be discussed in the following section. It can be interesting to compare the results obtained using the method we propose in the present paper with those obtained using the method proposed in [20]. Table 6 shows the results obtained by applying, on the same experimental setup, the noise reduction technique proposed in [20] in the case of 20 repetitions. We first observe that the comparison assumes the applicability of the method proposed in [20], and thus, that the nonlinear system is stationary, at least for the duration of the repetitions. Table 6 shows that in the cases of the estimation of \(h_{1}(t)\) and \(h_{3}(t)\), the results obtained are comparable with those of the method proposed in this paper, but considerably lower than what should be theoretically expected in the presence of Gaussian white noise. In the cases of \(h_{2}(t)\) and \(h_{4}(t)\) estimation, the improvement obtained with the averaging technique is in line with theoretical expectations.