# 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
 \tag{16}\]  
If the signal input to the system is not characterized by a single instantaneous frequency, such as \(\ Cos[\phi(t)]\), but is a signal of a more complex type, we have seen that the link between its powers and its harmonic components will be governed by a relation different from (5). We have examined in particular the case of an input signal consisting of the superposition of two harmonic signals. In the case where these are two exponential chirps offset from each other by a delay \(\triangle t\), the relationship that binds the \(N_{H}\) kernels \([h(t)]\) defining the Hammerstein model and the trends of the \([g(t)]\) identified in the output of the matched filter is given by relation (13). At each time instant the signal input to the nonlinear system will contain two harmonic components: the frequency of each component evolves exponentially and the ratio of the two frequencies is constant throughout the duration of the double exponential swept sine signal. If the identified Hammerstein model is of order \(N_{H}\) we will have as many kernels in the vector \([h(t)]\): we need to find the relationships that bind the elements of the vector of \(N_{H}\) kernels \(g_{h,k}(t)\) associated with the intermodulation frequencies \(hf_{1}+kf_{2}\); (with \(\{h,k\}\in\mathbb{Z}\) and such that the intermodulation frequency \(hf_{1}+kf_{2}>0\)), to the \(N_{H}\) elements of the vector \([h(t)]\) that identify the model. Specifically, if we are, for example, interested in estimating the distortion related to the intermodulation harmonic \(2f_{2}-f_{1}\), we will have to make use of the component \(g_{2f_{2}-f_{1}}(t)\), in addition to the \(g_{f_{1}}(t)\) and \(g_{f_{2}}(t)\) that describe, instant by instant, the amplitudes of the two fundamentals at frequencies \(f_{1}(t)\) and \(f_{2}(t)\). In fact, by saying \(f=\frac{f_{1}+f_{2}}{2}\) the reference frequency, the arithmetic mean between those of the two tones at the input, we have for intermodulation distortion \(IMD_{2f_{2}-f_{1}}(f)\) associated with the intermodulation harmonic component \(2f_{2}-f_{1}\):  
\[IMD_{2f_{2}-f_{1}}(f)=\frac{G_{2f_{2}-f_{1}}(2f_{2}-f_{1})}{G_{f_{1}}(f_{1})+ G_{f_{2}}(f_{2})}*100 \tag{17}\]where the functions \(G_{h,k}(f)=G_{hf_{1}+kf_{2}}(f)\) are the Fourier transforms of the time functions \(g_{h,k}(t)=g_{hf_{1}+kf_{2}}(t)\). In general, for estimating the frequency trend of the intermodulation distortion \(IMD_{hf_{1}+kf_{2}}(f)\) associated with any component \(hf_{1}+kf_{2}>0\) of interest, we can identify the relative trend of the function \(g_{hf_{1}+kf_{2}}(t)\) in the output of the matched filter, operate its Fourier transform to obtain \(G_{hf_{1}+kf_{2}}(f)\), and use a formulation of the following type:  
\[IMD_{hf_{1}+kf_{2}}(f)=\frac{G_{hf_{1}+kf_{2}}(hf_{1}+kf_{2})}{G_{f_{1}}(f_{1} )+G_{f_{2}}(f_{2})}*100 \tag{18}\]  
Similarly, the damage index HI proposed in [15], called the \"Harmonic Index\" to evaluate the entity of damages in the structure, can be calculated from the functions \([g(t)]\) obtained during the PuC procedure of Hammerstein model identification:  
\[HI=\frac{\sum_{k=2}^{N_{H}}\int(g_{k}(t))^{2}dt}{\int(g_{1}(t))^{2}dt} \tag{19}\]  
in which each of the integrations is extended to the entire time axis. It is important to emphasize that, once the Hammerstein model of the nonlinear system has been identified; that is, the parameters \([h(t)]\) that characterize the model have been identified, the \([g(t)]\) functions can be obtained using the relations that we have seen bind them to the \([h(t)]\) functions. In particular, if we want to derive the \(g_{k}(t)\) functions associated with harmonic distortion components, we can use relation (5), which binds the \(g_{k}(t)\) functions to the \([h(t)]\) that identifies the Hammerstein model; similarly, if we are interested in estimating the functions \(g_{hf_{1}+kf_{2}}(t)\) associated with intermodulation components \(hf_{1}+kf_{2}\), we can derive them using relation (13), again from the functions \([h(t)]\) that identify the Hammerstein model.