# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 2 Materials and Methods
### Swept Sine Identification of the Hammerstein Model
Assume that the input signal is of the harmonic type, \(x(t)=Cos[\phi(t)]\), where \(\phi(t)\) is the function defining the time course of the argument of the cosine function. \(Cos[\phi(t)]\) thus has an instantaneous time-varying angular frequency \(\omega(t)=\frac{d}{d}\frac{\phi(t)}{t}\). The output \(y_{H}(t)\) of the model, the sum of the convolutions of the signal powers with the elements of \([h(t)]\), in this case can be expressed as follows:  
\[y_{H}(t) =\left[\left[Cos[\phi(t)]\right]^{T}\otimes[h(t)]=\left[\left[A_{ c}\right]^{-1}cos[k\phi(t)]_{c}\right]^{T}\otimes[h(t)]=\right. \tag{5}\] \[=\left[cos[k\phi(t)]_{c}\right]^{T}\otimes\left(\left[\left[A_{ c}\right]^{-1}\right]^{T}\right)[h(t)]=\] \[=\left[cos[k\phi(t)]_{c}\right]^{T}\otimes[g(t)]\] \[\qquad\qquad\left.\text{where}\qquad[g(t)]=\left(\left[\left[A_{ c}\right]^{-1}\right]^{T}\right)[h(t)]\]  
The above observations form the basis of the procedure for identifying the Hammerstein model in the case where the excitation signals are exponential swept sine functions [8; 9; 10; 11; 12]. The identification step involves using the filter matched to the function \(x(t)=Cos[\phi(t)]\), whose impulse response \(\psi(t)\) is such that its convolution with the exponential swept sine signal \(x(t)\) at the input produces a band-limited approximation of Dirac's delta function \(\delta(t)\): \(x(t)\otimes\psi(t)=\delta(t)\). By passing the output \(y_{H}(t)\) of the Hammerstein model through the matched filter \(\psi(t)\), it can be shown that we obtain the following, for the output \(u(t)\) of the matched filter [13]:  
\[\mathfrak{u}(t)=y_{H}(t)\otimes\left.\psi(t)=\left[cos[k\phi(t)]_{c}\right]^{T }\otimes\left[g(t)\right]\otimes\left.\psi(t)=\left\{\left[\delta(t+\Delta t_{ k})\right]^{T}\right\}\otimes[g(t)]\right. \tag{6}\]  
Figure 1: The Hammerstein model. where each element \(\delta(t+\Delta t_{k})\) is the band-limited approximation of the Dirac \(\delta(t)\) function, delayed of a quantity \(\Delta t_{k}\) associated with the order of the k-th harmonic [14]:  
\[\big{[}\hat{\delta}(t+\Delta t_{k})\big{]}=\left[ \begin{array}{c}\hat{\delta_ {1}}(t+\Delta t_{1})\\ \hat{\delta_{2}}(t+\Delta t_{2})\\ \vdots\\ \hat{\delta_{N_{H}}^{*}}(t+\Delta t_{N_{H}})\end{array} \right]\ ;\ \Delta t_{k}=L\ ln(k) \tag{7}\]  
and \(L=ln(f_{MAX}/f_{MIN})/T\) defines the variation rapidity in time of the instantaneous frequency of the exponential swept sine signal which sweeps between \(f_{MIN}\) and \(f_{MAX}\) in a time \(T\)[11; 12]. We can then say that the k-th element of the vector \([g(t)]\) of (6) is associated with the k-th harmonic of the exponential swept sine signal input to the system. If the parameter L is large enough to keep the different \(g_{i}(t)\) apart, each of the \(g_{i}(t)\) functions can be obtained by simply extracting an appropriate section of the output signal \(u(t)\), starting at a time instant identified by \(\Delta t_{i}\). Once the \([g(t)]\) functions have been obtained by sectioning the output \(u(t)\) of the matched filter, from the \([g(t)]\) it is possible to derive the \([h(t)]\) functions that identify the Hammerstein model through a simple linear transformation:  
\[[h(t)]=[A_{c}]^{T}[g(t)] \tag{8}\]  
In the following, we will denote the Hammerstein model identification technique based on processing the response to the exponential swept sine signal as the pulse compression (PuC) technique.