# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 3 Results
### Synthetic Experiments
Panel (b) of the same figure allows us to verify the quality of the estimate obtained by extrapolation using the technique proposed in this paper: we started from the parameters of the Hammerstein model identified with an input signal of amplitude R = 1, and using the technique proposed in this paper--and in particular formula (12)--we obtained by extrapolation the parameters of the model associated with the amplitude of the excitation signal R = 0.5. Using these extrapolated parameters, we estimated the distortion trends that are reported in panel (b) of Figure 2. Also, the values obtained by extrapolation from a measurement made with a different amplitude of the input signal, are shown to be very good estimates of the distortion parameters, the values of which are given in formula (21). In Section 2.3, we have shown that it is possible to obtain estimates of the \(g_{IMD}(t)\) functions associated with intermodulation components starting from the \(h(t)\) kernels that identify the Hammerstein model of the nonlinear system. We estimated the \(g_{THD}(t)\) functions that were needed to estimate \(THD\), \(HD_{2}\) and \(HD_{3}\). From these, we estimated the kernels \(h(t)\) and thus the functions \(g_{IMD}(t)\) associated with the intermodulation components. We preliminarily calculated analytically, for both harmonic and intermodulation distortions, the reference values with which to compare the results of the PuC estimation. Figure 2: PuC estimates of \(THD\), \(HD_{2}\) and \(HD_{3}\) for the synthetic experiment. **(a)** Estimates related to the excitation signal amplitude R = 1 obtained from the measurements made for R = 1; **(b)** estimates related to the excitation signal R = 0.5 obtained by extrapolation from the measurements made for R = 1. Let us then consider the superposition of two sinusoidal signals of equal amplitude: \(x(t)=R\left\{\text{\em Cos}[\omega_{1}t]+\text{\em Cos}[\omega_{2}t]\right\}\) develop and recombine among themselves the powers of the signal placed at the input; thus, it is possible to calculate the values of intermodulation distortions of various types, such as the intermodulation distortion related to the frequency component \(f_{2}-f_{1}\). Considering the same parameter values used previously (R = 1; Coef = 0.125), we obtained from the analytical calculation of intermodulation distortion the value \(IMD_{f_{2}-f_{1}}=4.87\%\). To compare this value with that obtained from the PuC procedures proposed in this paper, we started from the impulsive functions \(g_{THD}(t)\) used for estimating the harmonic distortion; through (8) we obtain the trends of the kernels \(h(t)\) that identify the Hammerstein model of the nonlinear system. From these trends, we obtained an estimate of the impulsive functions associated with intermodulation distortion: in particular, we used expression (13) to estimate the first elements of the vector of functions associated, this time, with the intermodulation harmonic components. In our experiment we used \(g_{f_{2}-f_{1}}(t)\), the second trend of the intermodulation vector \(g(t)\) in (13), to estimate that intermodulation component. We were thus able to compare the result of the estimation obtained by starting from a single sine wave excitation, and then move on to the estimation of the parameters \(g(t)\) related to harmonic distortion; from these you have kernels identifying the Hammerstein model, and from these, through the \([A_{\text{IMD}}]\) matrix, the intermodulation distortion related to the frequency component \(f_{2}-f_{1}\), with the result calculated analytically. The result is as shown in Figure 3: It is evident that, despite the complexity of the procedure, the estimated value approximates what had been calculated analytically. Moreover, it is also evident in this case that the fact that the nonlinear system is memoryless results in constant frequency trends of the quantities estimated through the PuC technique.