# 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 pulse-compression identification technique has allowed excellent results to be obtained in numerous applications. For example, the results obtained in the estimation of intermodulation distortion carried out by means of a double exponential swept-sine signal [20], or in increasing the sensitivity of defect detection systems in materials by means of non-destructive techniques [21]. A careful analysis of the results, however, shows limitations that, while they do not affect the characteristics of the result in the case of integral-type estimator as, for example, in [21; 22], they become significant in the case where there is interest in the time course of the response of the system represented through the nonlinear model. A verification of this statement can be achieved by analyzing the results of a laboratory experiment, the bench of which is shown in Figure 4. The experiment was performed to verify the presence of nonlinear effects in a couple of identical Tx and Rx air-coupled ultrasonic transducers. Figure 4: The measurement bench (**a**) and one of the transducers (**b**) of the air-coupled ultrasonic transducers experiment. Figure 3: Schematic description of the PuC processing procedure for Hammerstein model identification. The system configuration, shown in Figure 4, consists, in sequence, of a personal computer for measurement management and supervision, a TiePie handyscope HS5 [28] used as an arbitrary waveform generator, a Falco System power amplifier [29], a pair of identical Ultran brand transmit and receive transducers designed to emit in air [30], a TiePie handyscope HS5 used as a data logger, and a personal computer used for data storage. The ultrasonic probes operate at a center frequency of 200 kHz and allow a maximum input voltage of 150 volts; both probes are of the focused type, with nominal focal distance F = 10 cm. The beam emitted in air by the transmitting piezoelectric transducer is focused at a single point (the focal point); the receiving system detects the signal emitted from a point on the axis of the transducer and placed at a focal distance of 10 cm. The Falco System signal amplification system is used on the signal generation side and has an amplification factor of 50\(\times\) in the useful band. The resolution adopted in the measurement was 14 bits. The ultrasonic probes were mounted on precision mounts and positioned at a distance of 21 cm, slightly more than twice the focal distance, which proved to be the distance at which the received signal level was found to be maximum. Figure 5 shows the signal obtained at different stages of processing, starting from the data measured in the laboratory with the measuring bench described in Figure 4. Figure 5a shows the signal picked up by the receiving probe; Figure 5b plots the output of the matched filter, the figure also reports the positions in which the pulses corresponding to the different harmonics deriving from nonlinearity -if present- can be expected (dotted lines at \(\Delta\delta_{k}=L\,ln(k)\)); Figure 5c,d report a magnification of the curve in Figure 5b in the neighborhood of the time where the peaks corresponding to the fundamental frequency and to the second harmonic are expected. Figure 5b,c highlight the aspect described in the Introduction, i.e., the presence of artifacts in the response over time, which manifest themselves in the form of spurious oscillations that are particularly evident in the instants preceding the ideal response attack. Possible causes of such limitations in the identification process will be analyzed in the following section. 2.2 PuC Identification Procedure Reformulated in the Frequency Domain to Highlight Its Limitations and Their Causes  
The PuC technique of Hammerstein model identification relys on the functions \(g_{i}(t)\) extracted from the response of the matched filter to the exponential swept-sine signal input to the nonlinear system.