# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 2 Materials and Methods
### Hammerstein Model Identification and Complexity of the Excitation Signal
The number of peaks present in the signal at the output of the matched filter will therefore be higher than the \(N_{H}\): we should therefore choose \(N_{H}\) among the peaks present at the output of the matched filter, and identify the matrix that links the chosen peaks \(g_{h,k}(t)\), associated with the intermodulation frequency \(hf_{1}+kf_{2}\); \(\{h,k\}\in\mathbb{Z}\), to the \(N_{H}\) kernels \(h_{k}(t)\) that identify the Hammerstein model. There is no specific rule for identifying which peaks present in the output signals to the matched filter are of interest to us: in making such a choice, it is essential to keep in mind the purpose of what is being conducted as well as the presence of noise and numerical alterations; for our reference applications, it is necessary that the peaks associated with the intermodulation components \(f_{2}-f_{1}\), \(2f_{2}-f_{1}\), and \(2f_{2}-2f_{1}\) be present; additional peaks must be chosen to reach the number \(N_{H}\), and it is convenient to choose those with higher energy content, which are typically those of lower order. It should also be kept in mind that the linear independence of the resulting equations must be ensured to guarantee the compatibility of the system of equations and the possibility of inverting the resultant coefficient matrix. Following these directions, we calculated the coefficient matrix, denoted as \([A_{\text{IMD}}]\), up to order \(N_{H}\) = 9 [14]. It relates the kernels \(h_{k}(t)\) to the peaks associated with the harmonics of one of the two frequencies (in our case: \(f_{1}\), \(2f_{1}\), \(3f_{1}\), \(4f_{1}\), \(5f_{1}\)), and the intermodulation harmonics, combinations of the two frequencies \(f_{1}\) and \(f_{2}\) at the input: \(hf_{1}+kf_{2}\) (where \(\{h,k\}\in\mathbb{Z}\) and such that the intermodulation frequency \(hf_{1}+kf_{2}>0\)); in our case we considered the intermodulation components: \(f_{2}-f_{1}\), \(2f_{2}-f_{1}\), \(2f_{2}-2f_{1}\), \(3f_{2}\) - \(2f_{1}\). The matrix \([A_{\text{IMD}}]\) was obtained by developing the integer powers of the function \(R\{Cos[\omega_{1}t]+Cos[\omega_{2}t]\}\) raised up to the 9 th power, and grouping the coefficients in the  
The total harmonic distortion (\(THD\)) and harmonic distortion of order k (\(HD_{k}\)) are defined through the following relations:  
\[ \begin{split} THD(f_{0})=\sqrt{\frac{\sum_{k=2}^{\infty}R_{k}^{2}}{ R^{2}}}=\frac{\sqrt{\sum_{k=2}^{\infty}R_{k}^{2}}}{R}=\sqrt{\frac{\sum_{k=2}^{ \infty}R_{k}^{2}}{\sum_{k=1}^{\infty}R_{k}^{2}}}\\ HD_{k}(f_{0})=\sqrt{\frac{R_{k}^{2}}{R^{2}}}=\frac{R_{k}}{R}\end{split}  \tag{15}\]  
where \(R\) denotes the amplitude of the fundamental component and \(R_{k}\) is the amplitude of the k-th harmonic component. By repeating this measurement and calculating the above values for a number of fundamental values \(f_{0}\), we obtain the frequency trends of \(THD\) and \(HD_{k}\), i.e., the trends \(THD(f)\) and \(HD_{k}(f)\). Within the validity limits of the polynomial-type model, we know from (3) that there is a close relationship between the amplitude \([g]\) of the harmonics and the parameters \([h]\) of the polynomial model. We also know that if we adopt the PuC identification technique of the model, we can obtain the functions \([g(t)]\) associated with the different harmonics of the signal at the input, through sectioning the matched filter response when we have an exponential swept sine signal at the input to the system. The Fourier transform of the functions \([g(t)]\) then directly gives us the terms to be included in the \(THD(f)\) and \(HD_{k}(f)\) formulation: the relations for estimating the harmonic distortions from the results of the PuC identification procedure of the Hammerstein model are thus:  
\[ \begin{split} THD(f)=\sqrt{\frac{\sum_{h=2}^{N_{H}}\abs{G_{h}(hf, \ R)}^{2}}{\sum_{h=1}^{N_{H}}\abs{G_{k}(kf,\ R)}^{2}}}\\ HD_{n}(f)=\frac{\abs{G_{n}(hf,\ R)}}{\sqrt{\sum_{k=1}^{N_{H}}\abs{G_{ k}(kf,\ R)}^{2}}}\end{split}