## 3 Experimental Results
### Mitigation of Noise Effects: Ultrasonic System with in-Air Propagation  
A verification of the proposed procedures was also carried out in the case of laboratory-acquired data. The measurements were performed on an ultrasonic acquisition system designed for in-air propagation, as shown in Figure 7. The measurement setup consists, in sequence, of a personal computer (PC) for management and supervision; an HS5 TiePie handyscope used as a generator of arbitrary waveforms; a Falco-System power amplifier; a pair of identical non-contact point focused ultrasonic transducers by Ultran, used in transmission and reception; again the TiePie, used this time as a data logger; and the PC for data storage. The ultrasonic probes operate at a center frequency of 200 KHz, and the emitted beam is focused at a single point (F = 10 cm). The probes allow a maximum input voltage of 150 V. The Falco-System power amplifier was used with an amplification factor of 50\(\times\). The probes, a pair of ULTRAN NCG 200-D25 P100 focused probes, were mounted in a through-transmission configuration (through air alone) on precision mounts and placed at a distance of approximately 21 cm, the distance at which the received signal was maximized. The identification step was performed using an exponential swept sine signal of amplitude R = 1. It was assumed that the system could be usefully characterized by a swept sine signal of duration 9.3 [ms] operating in the range between 50 kHz and 400 kHz, sampled at 5 Ms/s. The order adopted for the Hammerstein model was \(N_{H}=4\). Several measurements were carried out by slightly modifying the distance of the probes in order to change the level of useful signal received, and therefore the SNR ratio through slight defocusing. We report in the following the results of two experiments, which differ only in the value of the SNR ratio that characterizes them. Figure 8 shows, in an expanded scale, the trend of the \(g_{i}(t)\) functions for the four orders considered in the case of the lower SNR value (estimated Peak Signal to Noise ratio on \(y_{H\,N}(t)=20\) [dB]). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline _SNR_ = 5 dB & \(PI_{h_{1}}\) & \(PI_{h_{2}}\) & \(PI_{h_{3}}\) & \(PI_{h_{4}}\) & \(PI_{3}\) \\ \hline pre filter & \(-0.99\) & \(2.02\) & \(2.80\) & \(4.07\) & \(2.33\) \\ filtered & \(-8.88\) & \(-2.78\) & \(-4.58\) & \(0.10\) & \(-2.95\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values [dB] of the performance indexes defined by (12): case of SNR = 5 [dB]. Figure 7: Schematic representation (**A**) and image (**B**) of the Ultrasonic non-contact bench. They were obtained by taking the appropriate sections of the signal at the output to the matched filter. A first observation concerns the fact that even in the real data case, the hypothesis seems to be verified--the noise power on the function \(g_{i}(t)\) is independent of the order of the function. The variance of the noise superimposed on the \(g_{i}(t)\) functions, estimated in segments of the output signal \(u(t)\) preceding the instants \(\Delta\mathfrak{t}_{\text{k}}\) and averaged over 50 trials, gives the following values:  
\[\left[\sigma^{2}_{g}\right]=\left[ \begin{array}{cccc}0.47&0.49&0.43&0.44\\ \end{array} \right]^{T}\]  
The corresponding functions \(h_{k}(t)\) are shown in Figure 9. In analogy with Figures 6 and 7, the first line of Figure 9 shows the trend of the functions \(h_{k}(t)\) before the lowpass filtering operation. The second line shows the trends of the same functions after they have been lowpass filtered: Even at visual inspection, the improvement of the signal on the first three functions clearly appears, while it is also evident from \(h_{4}(t)\), that if the noise level exceeds the useful signal, the filtering procedure can remove part of the noise, but cannot fully highlight the trend of the desired function, where present.