# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 2 Materials and Methods
### Swept Sine Identification of the Hammerstein Model  
Let us consider, to begin with, the case in which the nonlinear system is memoryless and can be adequately represented by the following polynomial expression:  
\[y(t)=h_{1}x(t)+h_{2}x^{2}(t)+\cdots=\left[x(t)^{k}\right]^{T}[h] \tag{1}\]  
where \([\text{h}]\) is a vector of constant coefficients, \(h_{k}\in\mathbb{R}\), and \(\left[x(t)^{k}\right]\) is the vector of the powers of the input signal. If the input \(x(t)\) consists of a harmonic signal of unitary amplitude, \(x(t)=\text{Cos}(\omega_{1}\text{t})\), we can express its powers as functions of its harmonics through the matrix of coefficients of Chebyshev polynomials [12]:  
\[ \begin{bmatrix}\text{Cos}[0\;\omega_{1}t]=1\\ \text{Cos}[\omega_{1}t]\\ \text{Cos}[2\;\omega_{1}t]\\ \text{Cos}[3\;\omega_{1}t]\\ \vdots\end{bmatrix} =\left[\text{A}\right] \begin{bmatrix}\left[\text{Cos}[ \omega_{1}\text{t}]\right]^{0}=1\\ \text{Cos}[\omega_{1}t]\\ \left[\text{Cos}[\omega_{1}\text{t}]\right]^{2}\\ \left[\text{Cos}[\omega_{1}\text{t}]\right]^{3}\\ \vdots\end{bmatrix}  \tag{2}\]  
\[\left[\text{A}\right]= \begin{bmatrix}1&0&0&0&0&0&\cdots\\ 0&1&0&0&0&0&\cdots\\ -1&0&2&0&0&0&\cdots\\ 0&-3&0&4&0&0&\cdots\\ 1&0&-8&0&8&0&\cdots\\ 0&5&0&-20&0&16&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix} \]  
where \([\text{A}]\) is the matrix of coefficient of Chebyshev polynomials of the first type: Chebyshev polynomials of the first type make it possible to express the harmonics of the unit of amplitude \(\text{Cos}[\omega_{1}\text{t}]\) function as functions of their integer powers \(\left[\text{Cos}[\omega_{1}\text{t}]\right]^{k}\) ; \(k\in\mathbb{N}\). In the case where we are not interested in the continuous component (for example, if we are analyzing a zero mean signal), we can neglect the first elements of both vectors of powers and harmonics, i.e., the constant elements over time. The resulting matrix, which we call \([A_{C}]\), equals the matrix \([\text{A}]\) that has been deprived of the first row and column. The above relationships can be used to obtain, for example, the amplitude of the spectral components at the output of a memoryless nonlinear polynomial-type system, when a sinusoidal signal \(x(t)=\text{Cos}[\omega_{1}t]\) is input:  
\[ \begin{split}\text{y}(t)&=\left[\left[\text{Cos}[\omega_{1}t] \right]^{k}\right]^{T}[h]=\left[\left[A_{c}\right]^{-1}\text{Cos}[k\;\omega_{1} t]_{c}\right]^{T}[h]=\\ &=\left[\text{Cos}[k\;\omega_{1}t]_{c}\right]^{T}\left[\left[A_{c} \right]^{-1}\right]^{T}[h]=\\ &=\left[\text{Cos}[k\;\omega_{1}t]_{c}\right]^{T}[g]\end{split}  \tag{3}\]  
where \([g]\) is the vector of the constants denoting the amplitude of the respective harmonics \([g]=\left[\left[A_{c}\right]^{-1}\right]^{T}[h]\). We note that the above relationship can be inverted to provide the vector \([\text{h}]\), which identifies the model of the nonlinear system from the knowledge of the amplitude \([g]\) of the harmonics of the sinusoidal signal: \([h]=\left[A_{c}\right]^{T}[g]\). All the above considerations can be easily extended to the case where the system is with memory, i.e., the polynomial model of the nonlinear system is not characterized by the multiplicative numeric coefficients \([\text{h}]\) associated with the powers of the signal, but by the kernels \(h_{k}(t)\), organized in the vector \([h(t)]\) and associated, for each index \(\text{k}\), with the k-th power of the input signal. This is the Hammerstein model of order \(N_{H}\) that weconsider, whose structure is the one shown in Figure 1; the input-output relation of such a model can be expressed through the following equation:  
\[y_{H}(t)=x(t)\otimes h_{1}(t)+x(t)^{2}\otimes h_{2}(t)+\ldots+x(t)^{N_{H}}\otimes h _{N_{H}}(t)=\left[x(t)^{k}\right]^{T}\otimes[h(t)] \tag{4}\]  
in which the symbol \(\otimes\) denotes convolution and \(h_{k}(t)\) are the kernels identifying the model of the linear system; \(\left[x(t)^{k}\right]\) is the \(N_{H}\)-dimension vector of the ordered powers of the input signal.