## 4 Simulation Results and Discussion
### Simulation Parameters
Observe that the relative error of the UWB-OOK system is about 24%, of UWB-PPM is about 11%, of UWB-PPM-ATS with \(\zeta=-0.08\) ns is about 7%, with \(\zeta=-0.16\) ns is about 13%; so the UWB-PPM-ATS with \(\zeta=-0.08\) ns performs better for all distance values. This can be explained by comparing the correlation functions which are given in (8), (13) and Figures 7 and 8; we observe that \(R_{1i}(0)\) is significantly greater than \(R_{0i}(0)\) for \(\zeta=-0.08\) ns and \(R_{0i}(0)\) is not the maximum value of \(R_{0i}(\tau)\), which leads to a smaller than average error in the PPM-ATS with \(\zeta=-0.08\) ns when compared to PPM scheme; but with \(\zeta=-0.16\) ns, \(R_{1i}(0)\) is less than \(R_{0i}(0)\) thus the PPM outperforms the PPM-ATS. Besides that, the UWB-OOK is one of the amplitude modulation techniques, thus it is affected by the transmission environment and more difficult to separate the received signal from noise interference in comparison with the PPM techniques which have a constant amplitude. The results in Figure 11 indicate that the time shift \(\zeta\) directly affects the performance of PPM systems. The value of \(\zeta\) is selected so that \((T_{\text{PPM}}+\zeta)\) achieves the optimal value at which \(R_{\text{G4}}(\tau-T_{\text{PPM}}-\zeta)\) gets the minimum value. From (6) and (13), we also know that \(\zeta\) is determined by the pulse shape and the pulse width normalization factor \(\mu_{p}\). Due to the influence of the UWB-PPM system to the choice of the time shift, a suitable value of \(\zeta\) should be chosen according to the specific pulse shape and pulse width employed in a particular UWB application. First the parameters \(\epsilon_{1}\), \(d_{1}\) are estimated based on the propagation time \(\tau_{1i}\) and rotation angle \(\alpha_{i}\) with both UWB-PPM and UWB-PPM-ATS systems. Then, in a similar way, the parameters \(d_{2}\), \(\epsilon_{2}\), and \(x_{T}\) are estimated based on \(\mathbf{o_{2}}\) and \(\mathbf{x_{D}}\); therefore, the depth of buried object \(d_{T}=d_{1}+d_{2}\) is also determined. The estimated results after 30 iterations and errors of UWB systems are listed in Table 3 and Figures 12-14. Figures 12 and 13 show the relationship between \(\tau_{1i}\) and \(\alpha_{i}\) in (15). In those figures, the dashed black line denote the curve of \(\tau_{1i}\) vs. \(\alpha_{i}\) with the true values of \(d_{1}\) and \(\epsilon_{1}\); the blue dots are \(\tau_{1i}\) values measured by simulated UWB systems and the red line denote the curve of \(\tau_{1i}\) vs. \(\alpha_{i}\) with the estimated values \(\widehat{d_{1}}\) and \(\widehat{\epsilon_{1}}\). Figure 14 shows the curve of \((\tau_{2i}-\tau_{10})\) depends on \(x_{Di}\) according to (17). In those figures, the dashed black line denotes the curve of \((\tau_{2i}-\tau_{10})\) vs. \(x_{Di}\) with the true values of \(d_{2}\), \(\epsilon_{2}\) and \(x_{T}\); the blue dots are \((\tau_{2i}-\tau_{10})\) values measured by simulated UWB systems and the red line denotes the curve of \((\tau_{2i}-\tau_{10})\) vs. \(x_{Di}\) with the estimated values \(\widehat{d_{2}}\), \(\widehat{\epsilon_{2}}\) and \(\widehat{x_{T}}\). Table 3 and Figure 15 show the estimated results by the UWB systems. As seen, the modulation techniques for UWB pulses combine with nonlinear estimation method LMFA can be used to determine the thickness of different layers in the nondestructive environments, their relative permittivities and also the position of buried object. We observe that the UWB-PPM-ATS system outperforms the UWB-PPM system for all parameters of the model. This behavior results directly from the features of the correlation functions of different signals shown in Figures 7 and 8. When using UWB-PPM-ATS technique with the optimal value of \(\zeta\), the estimated traveling time values have a smaller error than using UWB-PPM technique, so the results from the LMFA of the PPM-ATS system give the higher accuracy than conventional PPM system.