## 3 Experimental Results
### Mitigation of Noise Effects: Simulation Tests  
We begin testing the effectiveness of the procedure proposed for mitigating the effects of noise in a simulation experiment: The importance of performing simulation tests is to have a complete feedback of the improvement we achieve in the estimation of the kernels \(h_{i}(t)\): Their reference trends are known, since they have been analytically defined to be included in the synthetic system. Consequently, a direct comparison with the result of the identification procedure in the presence of noise is possible, and we can quantify the benefits arising from the proposed mitigation procedure. We have generated a synthetic nonlinear system of the fourth order, defined according to the Hammerstein model. In the different branches of the model we adopted as kernels \(h_{i}(t)\), the four functions are represented in Figure 4. They all have typical behavior of the impulsive response of a filter of the second order and differ for the central frequency and for the attack delay time, which both increase as the order increases. The values adopted in our simulations to characterize the four kernels of the synthetic model are reported in Table 1.  
Figure 3: Correspondence between noise sequences superimposed on \(g(t)\) and \(h(t)\). Plots (**A–D**): noise sequences on \(g_{1}(t)\), \(g_{2}(t)\), \(g_{3}(t)\), \(g_{4}(t)\), respectively. Plots (**E–H**): noise sequences on \(h_{1}(t)\), \(h_{2}(t)\), \(h_{3}(t)\), \(h_{4}(t)\), respectively. For the purpose of verifying the accuracy in model identification, the system is elicited with an exponential swept sine signal of amplitude R = 1 whose frequency range \(f_{MIN}\) = 40 [kHz] to \(f_{MAX}=1600\) [kHz] contains the frequency bands that characterize all the \(h_{i}(t)\) kernels. A Gaussian white noise is added to the response \(y_{H}(t)\); the value of its standard deviation is such that the signal-to-noise ratio (SNR) is brought to the desired level: The values \(SNR_{1}=10\) [dB] and \(SNR_{2}=5\) [dB] were considered. Let us say \(y_{H\ N}(t)\) is the noisy response. It is convolved with the filter matched to the input signal to obtain the output signal \(u(t)\). The functions \(g_{i}(t)\) are obtained by taking portions of \(u(t)\) at the time instants \(\Delta t_{i}\). From \(g_{i}(t)\), the estimates of the kernels \(h_{k}(t)\) are obtained. To verify the effectiveness of the noise reduction technique, we report both the results of the estimates in the absence of lowpass filtering and the results obtained after the additional lowpass filtering step of each of the estimated \(h_{k}(t)\) kernels. The impulse response of the FIR (Finite Impulse Response) filter is defined using a frequency sampling method, starting with the amplitude values that define the required low-pass amplitude response in the frequency domain. Figures 5 and 6 show the superposition between the estimated \(h_{k}(t)\) kernels and the corresponding reference trends, respectively, for the case \(SNR_{1}=10\) [dB], and for the case \(SNR_{2}=5\) [dB]: The upper row of each figure refers to the estimate without filtering and the lower row shows the trend of the estimated \(h_{k}(t)\) after the additional lowpass filtering step. The figures clearly show that the lowpass filters defined ad hoc for each order of the kernel greatly reduce the effects of superimposed noise without deteriorating the time course of the function we want to estimate.