# 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
If we had adopted a chirp of amplitude \(R_{\beta}\)-different from \(R_{\alpha}\)-(i.e., a different unit of measurement), we would have obtained functions \(g_{\beta}(t)\) that, in general, are different from \(g_{\alpha}(t)\); consequently, by adopting the Chebyshev transformations, again related to unit amplitude, we would obtain functions \(h_{\beta}(t)\) different from the previous \(h_{\alpha}(t)\). Again, the model represents the nonlinear system for an input signal of amplitude up to \(R_{\beta}\) of the chirp used. The \(h_{\alpha}(t)\) and \(h_{\beta}(t)\) are different, but the physical system is the same, so we look for the link between the representation obtained for the amplitude of chirp \(R_{\beta}\) and that obtained for the amplitude of chirp \(R_{\alpha}\): for this purpose, we modify the Chebyshev model representation to take into account the chirp amplitude. Let us start from the representation obtained with the chirp of higher amplitude: let it be \(R_{\alpha}>R_{\beta}\). We can express the output of the Hammerstein model as a function of \(h_{\alpha}(t)\), as already shown in (5):  
\[ \begin{array}{rl}y_{H-\alpha}(t)&=\Big{[}[Cos[\phi(t)]]^{k}\Big{]}^{T}\otimes [h_{\alpha}(t)]=\Big{[}[A_{c}]^{-1}Cos[k\phi(t)]_{c}\Big{]}^{T}\otimes[h_{ \alpha}(t)]=\\ &=[Cos[k\phi(t)]_{c}]^{T}\Big{[}[A_{c}]^{-1}\Big{]}^{T}\otimes[h_{\alpha}(t)]= \\ &=[Cos[k\phi(t)]_{c}]^{T}\otimes\Big{(}\Big{[}[A_{c}]^{-1}\Big{]}^{T}\Big{)}[h_ {\alpha}(t)]=\\ &=[Cos[k\phi(t)]_{c}]^{T}\otimes[g_{\alpha}(t)]\end{array}  \tag{9}\]  
Now imagine that we provide, as input to the Hammerstein model characterized by the \([h_{\alpha},(t)]\) kernels (referred to the amplitude \(R_{\alpha}\) as unit of measurement, so \(R_{\alpha}=1\) ), a chirp of amplitude \(R_{\beta}=\text{R}\times R_{\alpha}=\text{R}<1\), where \(R=R_{\beta}\) is the ratio the amplitudes of the two chirp signals: \(R=\frac{R_{\beta}}{R_{\alpha}}\). The input-output relationship becomes the following:  
\[ \begin{array}{rl}y_{H\_\beta}(t)&=\Big{[}[RCos[\phi(t)]]^{k}\Big{]}^{T} \otimes[h_{\alpha}(t)]=\Big{[}[R_{c}][Cos[\phi(t)]]^{k}\Big{]}^{T}\otimes[h_{ \alpha}(t)]=\\ &=\Big{[}[R_{c}][A_{c}]^{-1}Cos[k\phi(t)]_{c}\Big{]}^{T}\otimes[h_{\alpha}(t) ]=[Cos[k\phi(t)]_{c}]^{T}\Big{[}[R_{c}][A_{c}]^{-1}\Big{]}^{T}\otimes[h_{ \alpha}(t)]=\\ &=[Cos[k\phi(t)]_{c}]^{T}\otimes\Big{(}\Big{[}[A_{c}]^{-1}\Big{]}^{T}[R_{c}][ h_{\alpha}(t)]\Big{)}=\\ &=[Cos[k\phi(t)]_{c}]^{T}\otimes[g_{\beta}(t)]\end{array}  \tag{10}\]  
where \([R_{c}]\) is the diagonal matrix whose element \(R_{i,\,i}\) is the \(i\) -th power \(R^{i}\). Expression (10) highlights how we can use the model based on the kernels \(h_{\alpha}(t)\) to estimate the \(g_{\beta}(t)\) functions that we would obtain using a chirp of amplitude \(R_{\beta}=\text{R}\times R_{\alpha}\). We will see in the following sections that the functions \(g(t)\) are the ones that underlie the estimation of both the distortion and damage indexes, so having a method available to estimate the functions \(g(t)\) that would be obtained under stresses of amplitudes different from the one really adopted for the experiments allows us to estimate the distortions that would be obtained under a range of stresses of different amplitudes. It is interesting to note that we can directly compute the functions \(g_{\beta}(t)\) related to chirps of amplitude \(R_{\beta}\) from the knowledge of the \(g_{\alpha}(t)\) associated with chirps of amplitude \(R_{\alpha}\).