# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 4 Discussion  
The present work addressed the problem of the quality of the estimation of kernels characterizing the different branches of a Hammerstein model of a nonlinear system, in the case in which such a model is identified through the PuC technique. First, the work was concerned with verifying the real existence of the problem and, in particular, the presence of spurious oscillations at transitions in the time response of the system. To do this, and to verify that this problem is present in real physical devices, an attempt was made to verify the existence of the problem through a laboratory experiment. The experiment involved an ultrasonic system with probes designed to operate in air. An analysis of the results obtained by modeling the real physical system through a Hammerstein model showed that the impulsive responses of the various filters entering into the model's characterization do indeed exhibit precursors, in the form of oscillations that anticipate the theoretically calculated instant of attack. It was hypothesized that the presence of such oscillations was associated with the Gibbs phenomenon, which motivated us to analyze possible limitations  
Figure 13: Function \(h_{3}(t)\): Comparison of (**a**) the reference trend with those obtained by (**b**) identification Procedure 1 (swept-sine range from \(f_{MIN}\) to \(f_{MAX}\) and matched filter range from \(f_{MIN}\) to \(f_{MAX}\)); (**c**) identification Procedure 2 (swept-sine range from \(f_{MIN}\) to \(f_{MAX}\) and matched filter range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\)); (**d**) identification Procedure 3 (swept-sine range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\) and matched filter range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\)). of the frequency band covered by the characterization of the different impulse responses of the filters that constitute the kernels of the Hammerstein model. This analysis of the frequency bands was carried out by rewriting the PuC procedure in the frequency domain and observing how the final result, i.e., the \(H_{k}(\omega)\) functions, were obtained by linearly combining the \(G_{i}(\omega)\) functions, which, in turn, were obtained through the convolution between the response of the nonlinear system to an exponential swept-sine signal and the filter matched to that signal. Because of the way the matched filter was defined in the present case (optimized for additive white Gaussian noise), the convolution with the matched filter was equivalent to the correlation function of the response of the nonlinear system with the signal at its input. If the signal at the input and the matched filter both had infinite bands, the correlation would extend over the entire frequency range; since the bands are limited, the frequency band in which the correlating signals overlap, i.e., the band in which the procedure allows the definition of the \(G_{i}(\omega)\) functions, depends on the frequency bands in which the input signal and the matched filter are defined. In addition, it can be observed that the combination of \(G_{i}(\omega)\) functions occurs through matrices whose upper triangular structure means that the \(H_{k}(\omega)\) functions of low \(K\) index, which are in most cases energetically more significant, are obtained by combining more \(G_{i}(\omega)\) functions than for the \(H_{k}(\omega)\) functions of high \(K\) index. This implies that if \(G_{i}(\omega)\) functions are defined in frequency bands that are inconsistent with each other, the \(H_{k}(\omega)\) functions that will be most affected will be those that are energetically more significant. Figures 6-8 graphically represent the effect of frequency band limitations in the estimation of \(G_{i}(\omega)\) functions.