## 2 Theoretical Aspects
### Noise Power on the Output \(y_{H}(t)\) and on Functions \(g(t)\) and \(h(t)\)  
We have seen that the swept sine identification procedure of the Hammerstein model starts by considering the functions \(g_{i}(t)\), obtained by time-windowing at appropriate positions the response \(u(t)\) of the matched filter when its input is \(y_{H}(t)\). The response \(y_{H}(t)\) is the one that we have at the output of the nonlinear system under consideration when it has an exponential swept sine signal as input. From the functions \(g_{i}(t)\) obtained with this procedure, through an appropriate linear transformation we can derive the functions \(h_{i}(t)\) that characterize the Hammerstein model; the coefficients of this linear transformation are the coefficients of Chebyshev polynomials of the first type. To obtain the functions \(h_{i}(t)\), a combination of the functions \(g_{i}(t)\) is performed according to (7). The combination is performed through coefficients given by matrix \([R_{c}]^{-1}[A_{c}]^{T}\). For example, for a model of order 4, we will have:  
\[\left[ \begin{array}{c}h_{1}(t)\\ h_{2}(t)\\ h_{3}(t)\\ h_{4}(t)\end{array} \right]=\left[ \begin{array}{cccc}\frac{1}{R}&0&0&0\\ 0&\frac{1}{R^{2}}&0&0\\ 0&0&\frac{1}{R^{3}}&0\\ 0&0&0&\frac{1}{R^{4}}\end{array} \right]\left[ \begin{array}{cccc}1&0&-3&0\\ 0&2&0&-8\\ 0&0&4&0\\ 0&0&0&8\end{array} \right]=\left[ \begin{array}{cccc}\frac{1}{R}&0&\frac{-3}{ R}&0\\ 0&\frac{2}{R^{2}}&0&\frac{-8}{R^{2}}\\ 0&0&\frac{4}{R^{3}}&0\\ 0&0&0&\frac{8}{R^{4}}\end{array} \right].\left[ \begin{array}{c}g_{1}(t)\\ g_{2}(t)\\ g_{3}(t)\\ g_{4}(t)\end{array} \right] \tag{8}\]  
Assume that additive noise, superimposed on the useful signal, affects the acquisition of the output to the nonlinear system. The ideal output \(y_{H}(t)\) of the nonlinear system will  
Figure 2: Description of the processing procedure for the identification of the Hammerstein model. be superimposed by the noise sequence \(N(t)\), supposed to be white Gaussian with power spectral density \(P_{N}(f)=\
icefrac{{N_{0}}}{{2}}\). The noise-affected output will be \(y_{H\ N}(t)=y_{H}(t)+N(t)\). According to the identification procedure, the output signal \(y_{H\ N}(t)\) is filtered through the matched filter \(\psi(t)\): the noise sequence \(N(t)\) will thus be filtered giving rise, at the output of the matched filter, to the noise sequence \(n_{u}(t)\) superimposed on the signal \(u(t)\). If we say \(\Psi(f)\) is the Fourier transform of the impulse response of the matched filter: \(\Psi(f)=\mathcal{F}\{\psi(t)\}\), the power of noise signal \(n_{u}(t)\) at the output of the matched filter can be evaluated by the:  
\[P_{n_{u}}=\sigma_{n_{u}}^{2}=\int\limits_{-\infty}^{\infty}P_{N}(f)\ |\Psi(f)|^{2}df=\frac{N_{0}}{2}\int\limits_{-\infty}^{\infty}|\Psi(f)|^{2}df \tag{9}\]  
We denote with \(\sigma_{n_{u}}^{2}\) the variance of noise process \(n_{u}(t)\) superimposed on \(u(t)\): all functions \(g_{i}(t)\), are extracted from \(u(t)\) at different positions, and will be thus affected by superimposed noise signals \(n_{g_{i}}(t)\) whose variance will be \(\sigma_{n_{u}}^{2}\) for all of them: \(\sigma_{n_{g_{i}}}^{2}=\sigma_{n_{u}}^{2}\). This common variance will be denoted in the following as \(\sigma^{2}g\); furthermore, the overall noise sequence \(n_{u}(t)\) has an impulsive-like correlation function; consequently, the noise sequences \(n_{g_{i}}(t)\), extracted at different positions from \(n_{u}(t)\), will be uncorrelated with each other, as they refer to different portions of the overall noise sequence \(n_{u}(t)\).