# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 2 Materials and Methods
### Amplitude of the Swept Sine Signal and Hammerstein Model  
The response of a physical system depends on the system itself and on the characteristics of the excitation used to stimulate it: if the system is linear, the response can be derived from the knowledge of the excitation (described in the time or in the frequency domain) and the input-output relationship of the system; if the system is nonlinear, it becomes essential, in addition to the above data, to take into account the amplitude of the signal at the input. In the case of nonlinear systems whose Hammerstein model is identified by the PuC technique, there is no a priori rule for defining the amplitude of the swept sine signal to be adopted as test signal for the identification phase: what we can check experimentally is, given the chirp amplitude, whether peaks related to harmonics from the second order onward are detectable at the output of the compression filter. If the system was ideal and therefore not noisy, regardless of the amplitude of the input signal, we could detect all the peaks \(g(t)\) associated with the harmonics being generated, however small. From these, we could derive the corresponding trends of the \(h(t)\) functions which identify the Hammerstein model. Where the physical system is non-ideal, some level of noise will be present, which will tend to obscure the peaks of more limited amplitude and alter the tails of those of larger amplitude, making them less closely adhere to their ideal trends. The presence of noise limits the number of peaks that can actually be detected and thus the maximum order of the pattern that can be identified. In fact, there are several tools available to avoid noise-related problems: the identification technique we refer to in this work is based on a pulse compression procedure. This processing procedure involves, among other characteristics, an increase in the signal-to-noise ratio of the post-processing signal compared to the pre-processing signal. This peculiarity allows, in theory, to modify the signal-to-noise ratio of the signal obtained after processing without any limitation. In practice, the maximum values of the signal-to-noise ratio are linked to the maximum length and maximum amplitude level of the signal that can practically be managed by the measurement system. Furthermore, in very particular cases, situations may arise in which the presence of unwanted reflections in the propagation path between transmitter and receiver place limits on the maximum duration of the signal that can be used as excitation, and this translates into a limitation of the maximum increase in the signal-to-noise ratio that can practically be obtained in those particular situations. Although not frequently, limitations related to the available hardware or particular signal propagation conditions can lead, even in the case of this technique based on pulse compression, to limitations in the practically obtainable value of the signal-to-noise ratio. This problem has been addressed in the work [13], in which it is shown how filtering the different portions of the signal obtained at the output to the matched compression filter in appropriate frequency bands can significantly improve the signal-to-noise ratio. Once the amplitude of the input signal has been chosen, it constitutes the reference amplitude of our experiment; in a sense, it becomes the unit of measurement of the amplitude of the input signal. Let us now consider two different amplitudes of the input chirp signal, amplitudes that we denote as \(R_{\alpha}\) and \(R_{\beta}\). By processing the response to the chirp signal of amplitude \(R_{\alpha}\) with the matched filter, we obtain the functions \(g(t)\) associated with the different harmonics; from these, we obtain the trends of the functions \(h(t)\) through the well-known Chebyshev transformations (8). As they are obtained in correspondence to an input of amplitude \(R_{\alpha}\) we denote these functions as \(g_{\alpha}(t)\) and \(h_{\alpha}(t)\).