## 3 Proposal of Positioning Approach in Nondestructive Environments  
### Proposal of UWB-PPM-ATS  
From (1), (5), and (6), it can be seen that the time shift \(T_{\text{PPM}}\) directly affects the quality of UWB-PPM systems in positioning applications. The performance of the system can be improved by selecting the appropriate value of the time shift. To increase the accuracy in the estimated distance and the position of buried objects by using the UWB-PPM system, the time shift in this system should be selected so that the ability to detect the received UWB pulse is the best. To accomplish this task, in UWB-PPM systems with the time shift \(T_{\text{PPM}}\) is invariant, we recommend adjusting this time shift with a certain time constant to achieve its optimal value. The optimal value of \(T_{\text{PPM}}\) defined in this paper is the value at which the UWB-PPM system gives the smallest error in distance estimation. So we propose an UWB pulse position modulation with an additional time shift (UWB-PPM-ATS). In the proposed technique, the pulse position will be changed with a time constant denoted by \(\zeta\). The signal of UWB-PPM-ATS is given as:  
\[s(t)=\sum_{n=1}^{N}g_{4}(t-nT_{r}-a_{n}.(T_{PPM}+\zeta)). \tag{12}\]  
The \"+\" sign in (12) means that algebraic additions, so \(\zeta\), can take either positive or negative values. The effect of \(\zeta\) on the quality of the system will be evaluated later. An example of the conventional UWB, UWB-PPM, and UWB-PPM-ATS signal shapes are illustrated in Figure 6. In Figure 6, there is an example of a seven pulses sequence carrying seven bits 1010101. The pulse width is 0.28 ns, and it adopts the fourth derivative of the Gaussian function described by the blue line; the conventional PPM modulated pulses sequence is denoted by the red line. Whereby, when the bit value is 0, the transmitter will send out a pulse \(g_{40}(t)\) as the original pulse (without any change from the unmodulated pulse) when the value bit is 1, the transmitter will send out a pulse \(g_{41}(t)\) which is the original pulse with a time shift of \(T_{\text{PPM}}\). With the proposed UWB-PPM-ATS (black line), instead of the normal \(T_{\text{PPM}}\) time shift, the new shift level is set to \((T_{\text{PPM}}+\zeta)\). The procedure for estimating distances in UWB-PPM-ATS system is the same as in the conventional system (UWB-PPM) and in (7); the template waveform is \((g_{4}(t)-g_{4}(t-T_{\text{PPM}}-\zeta))\). The correlation of the proposed system denoted by \(R_{1i}\) and determined as:  
Figure 6: The ultra-wide band (UWB) pulse shapes with pulse position modulation (PPM) and PPM-ATS. \[R_{1i}(\tau)=\int\limits_{-\infty}^{\infty}r_{i}(t)[g_{4}(t)-g_{4}(t-T_{\text{ PPM}}-\zeta)]dt. \tag{13}\]  
So, we have:  
\[R_{1i}(\tau)= R_{G4}(\tau)-R_{G4}(\tau-T_{\text{PPM}}-\zeta)]dt. \tag{14}\]  
The shapes of \(R_{G4}(\tau)\),\(R_{0i}(\tau)\) and \(R_{1i}(\tau)\) are shown in Figure 7.  
Figure 7 indicates the autocorrelation function of the fourth-order Gaussian monocycle \(R_{G4}\) and correlation functions of UWB-PPM \(R_{0i}\) and UWB-PPM-ATS \(R_{1i}\) systems (with additional time \(\zeta=-0.08\) ns and \(\zeta=0.08\) ns); those functions are compared at the value of \(\tau=0\). In the figure below inside Figure 7, the blue line with \"+\" sign denotes the value of \(R_{0i}(\tau)\), the red line denotes the value of \(R_{1i}(\tau)\) with \(\zeta\) gets a negative value of \(-0.08\) ns, and the dashed black line denotes the value of \(R_{1i}(\tau)\) with \(\zeta\) gets a positive value of \(0.08\) ns. We can observe that the choice of a negative value of \(\zeta\) makes \(R_{1i}(\tau)\) get the maximum value at \(\tau=0\) and this value is greater than \(R_{1i}(0)\)- with a positive value of \(\zeta\) and also \(R_{0i}(0)\) (both of these functions are not maximized at \(\tau=0\)).