## 4 Results and Discussion
### Eigen-Frequency Fitting and Regression Analysis  
The fitting analysis of the acquired impact frequency responses started from a scrap diameter of 14-23 mm. The fitting model of impact eigen-frequency response is introduced by Equation (9). The resuling curves are shown as follows in Figure 6 for PP, ABS, and SMA scraps with different color, respectively.  
The goodness of fit \(R^{2}\) for PP, ABS, and SMA is 0.873, 0.911, and 0.866, respectively.  
In Figure 6, the impact eigen-frequency responses of PP scraps were fitted to a unique curve with model parameter \(k_{\mathrm{C}}=242.78\) m\({}^{2}\)/s. The impact eigen-frequency responses of ABS scraps were fitted to a unique curve with model parameter \(k_{\mathrm{C}}=564.23\) m\({}^{2}\)/s. Additionally, the impact eigen-frequency responses of SMA scraps were fitted to a unique curve with model parameter \(k_{\mathrm{C}}=353.33\) m\({}^{2}\)/s. In this figure, a solid line illustrates the curve fitted by experimental data, and the dotted line means the expectation of impact AE responses, extracted from other sizes of scraps which currently cannot be observed with actual devices. Here we expanded the diameter range to 8-29 mm.  
The curves of impact eigen-frequency responses of ABS scraps shows that the \(f_{\mathrm{nom}}\) of ABS scrap impacts were expected to extend across the upper hearing threshold of 22 kHz, when scrap diameters were less than 10 mm, whose impact AE signals could no longer be acquired by a traditional soundcard, and professional ultrasonic signal acquisition hardware was required. The PP scraps with diameter less than 6.64 mm and SMA scraps with diameters less than 8 mm are in the same situation.  
Figure 6 also shows that the curves of impact eigen-frequency response were expected to get closer with increasing scrap diameters, and with decreasing scrap diameters, the curves moved further apart from each other. The vertical distance between the curves of corresponding materials depends on the variation of \(k_{\mathrm{C}}\), _i.e._, the farther the vertical distance between adjacent curves, the higher the separation  
Figure 6: Fitting results of all three kinds of material.  
rate of the corresponding plastic scraps could be obtained. Obviously here the PP scraps could be distinguished more accurately from ABS than SMA.  
By fitting residue distribution analysis of the PP curve, we found that with a confidence level of 100% for PP, the fitting residues located within a range of \(\pm\)400 Hz. Also, the SMA scraps have a relative wider residue distribution, which covers the range of \(\pm\)600 Hz with a confidence level of 100%. ABS fitting residues are distributed in the range from \(-\)600 Hz to \(+\)400 Hz with confidence level of 100%. The 100% confidence intervals of the three materials did not have overlap, which meant that scraps of all three tested plastics could be completely distinguished.  
For other plastic material mixtures with more similar values of \(k_{\mathrm{C}}\), their corresponding 100% confidence interval overlapped, which meant the scraps, whose impact eigen-frequency responses were located in the overlapped area could be miss-characterized and further influence the purity of the separation products. Therefore, with really closed curves, in order to avoid the confusion of overlapping we needed to narrow the confidence intervals, _i.e._, reduce the characterization/separation rate to insure competent purity of the separation products. The curves that have an adequate distance between each other do not have such a problem and theoretically the corresponding scraps are expected to be sorted accurately and completely.  
Finally through this experiment, 64.6% of PP scraps, 61.7% of PP scraps, and 63.9% of SMA scraps in mass were successfully characterized and further sorted.