# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 2 Materials and Methods
### Discussion on the Pulse-Compression Identification Procedure
#### 2.2.1 Limitations in the Identification Procedure
The fact that the matrices \([\mathrm{A}^{-1}]\) and \([\mathrm{A}_{\mathrm{C}}^{-1}]^{T}\), which appear in Equations (10) and (11) and link the functions \(H_{k}(\omega)\) and \(G_{i}(\omega)\), are upper triangular, has relevant consequences from the examined perspective, as discussed in the following. All the above considerations apply both in the case where the input signal is a simple cosine function and in the case where the signal is an exponential swept-sine signal, or any other signal characterized by containing, for each instant of time, a single harmonic component. In the PuC identification technique of the Hammerstein model, the functions \(g_{i}(t)\) are obtained by the convolution between the matched filter and the exponential swept-sine signal at the input, or with its harmonics generated by the nonlinear system. The band in which each \(g_{i}(t)\) is identified is the frequency band common to the i-th harmonic of the exponential swept-sine input signal and the matched filter. The i-th harmonic of the input signal, under the ideal assumption of product T x B tending to infinity, has components between i\(*\)\(f_{MIN}\) and i\(*\)\(f_{MAX}\). The frequency band in which the \(g_{i}(t)\) is defined is, thus, related to the choices made for the bands of the input signal and those of the corresponding matched filter. Figures 6-8 consider some possible cases of choosing the frequency bands of the input signal and of the matched filter, and graphically depict how these choices affect the frequency band in which the \(g_{i}(t)\) functions are identified, in the case where they are estimated by the pulse compression technique. First, let us consider the case in which both the swept-sine signal and the matched filter are defined between \(f_{MIN}\) and \(f_{MAX}\); the situation will be that which, in the case of model order 4, is described in Figure 6. The matched filter, defined between \(f_{MIN}\) and \(f_{MAX}\), is represented by the dotted curve in the four correlation lags in which it fits the fundamental frequency, and the second, third, and fourth harmonics. It is evident that, in this manner, the function \(g_{i}(t)\) is represented only up to the frequency \(f_{MAX}\) whatever harmonic \(i\) is considered. It can then be assumed that, if the function \(g_{i}(t)\) being approximated has significant components of the amplitude response beyond \(f_{MAX}\), they will not be identified; by adopting this implementation of the PuC technique, there will be a sudden reduction in the amplitude of the identification result obtained as an approximation of the function \(G_{i}(\omega)\). Figure 7 represents the superposition between the harmonics and the matched filter in the case where the swept-sine signal is defined between \(f_{MIN}\) and \(f_{MAX}\) and the matched filter is defined in the frequency range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\). Figure 7 evidences that the frequency bands in which the different \(g_{i}(t)\) functions can be identified are different from each other, with each having the frequency \(i*f_{MAX}\) as its upper extreme. It follows that, as shown below, when these \(g_{i}(t)\) functions are adopted to reconstruct the \(h_{k}(t)\) functions that identify the Hammerstein model, the \(g_{i}(t)\) functions combine only partially in the higher frequency ranges, and this is a limitation in the quality of the reconstruction of the \(h_{k}(t)\) functions. This aspect is all the more significant the lower the order k of the \(h_{k}(t)\) function, since, as noted before, the fact that the reconstruction matrices are upper triangular implies that the lower the order k considered for the \(H_{k}(\omega)\), the greater the number of \(g_{i}(t)\) functions that must compose, and in the case that their frequency bands are not the same, it leads to severe limitations in the quality of the result.