# 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  
All the considerations made so far refer to the functional relationship between the harmonics of the input signal and its integer powers and assume that the signal at the input is characterized, at each time instant, by a single harmonic component, both in the case of the single sine wave and in that of the chirp signal. The link between the powers of the input signal generated in the polynomial model, and its harmonics, is governed by the matrix \([A_{c}]\) appearing in (3), directly related to the coefficients of Chebyshev's polynomials of the first kind. This relationship must be reformulated in the case where the input signal has characteristics of a different type. For any input signal there will still be a linear relationship between integer powers and the harmonics of the input signal, as a consequence of [PERSON]'s model linearity in its parameters, but if the input signal becomes more complex, the harmonic components that are generated by a model of order \(N_{H}\) will be different from the simple \(N_{H}\) harmonics of a sine wave, and thus the matrix connecting the signal powers with the harmonics will no longer be the \([A_{c}]\) seen previously. In the case of multitone signals at the input, the number of harmonic components at the output of the nonlinear system will be larger than the order \(N_{H}\) of the Hammerstein model, and the square matrix of order \(N_{H}\) that identifies the linear link between input and output to the Hammerstein model will have to be calculated in relation to both the specific signal chosen as input and to which harmonics of the signal input we want to consider. Consequently, also the matrix connecting the kernels \(h_{k}(t)\) to the peaks \(g_{k}(t)\) associated with the harmonics will be directly related to the specific signal adopted as the input. The number of harmonics generated as a consequence of a multi-frequency input signal will depend not only on the order \(N_{H}\), but also on the complexity of the input signal and on the characteristics of the nonlinear system, since all possible combinations of intermodulation components up to order \(N_{H}\) can be generated in principle. We carry out our considerations with practical objectives in mind, such as that of estimating parameters for early defect detection, or that of estimating the frequency behavior of harmonic or intermodulation distortions from the Hammerstein model [14; 15]. For both of these aspects we are interested in harmonic components of the type \(f_{2}-f_{1}\), \(2f_{2}-f_{1}\), and \(2f_{2}-2f_{1}\) which are generated when the signal at the input consists of the sum of two sinusoidal tones, i.e., it is of the type \(x(t)=R\{Cos[\omega_{1}t]+Cos[\omega_{2}t]\}\). To achieve this, it is necessary to reformulate the PuC technique by including the case where, at each instant of time, two sinusoidal tones are input. The signal we will consider consists of the superposition of two exponential swept sine signals, which share the same parameters, but are shifted in time between them by a quantity \(\triangle t\). It is an extension of what we have seen above for the formulation of the Hammerstein model identification technique based on a single chirp: at each time instant the signal at the input of the nonlinear system will therefore contain two harmonic components, whose frequencies evolve exponentially. Because of the way the signal is constructed, the ratio of the two instantaneous frequencies is constant throughout the duration of the signal. At the output of the matched filter we will have, in addition to the peaks associated with the harmonics of each of the two chirps, the peaks associated with the intermodulation harmonics that are generated by the simultaneous presence, at each instant, of two tones.