## 2 System Model  
A distance measurement and positioning system using UWB technology is illustrated in Figure 1, where \(d_{i}\) and \(\varepsilon_{i}\) are the thickness and the relative permittivity of the \(i^{th}\) layer in the nondestructive environment, respectively. \(s(t)\) denotes the transmitted pulse signals, and \(r(t)\) is the received signal. For the environment with three layers and one buried object, the \(r(t)\) is represented by  
\[r(t)= \sum r_{i}(t)+r_{ob}(t)+n(t), \tag{1}\] \[r_{1}(t)= A_{1}s(t-\tau_{1}),\] \[r_{2}(t)= A_{1}A_{2}s(t-\tau_{1}-\tau_{2}),\] \[r_{3}(t)= A_{1}A_{2}A_{3}s(t-\tau_{1}-\tau_{2}-\tau_{3}),\] \[r_{ob}(t)= A_{ob}s(t-\tau_{ob}),\]  
where \(A_{i}\) denotes the amplitude factor which represents the reflection and transmission properties of the propagation environment, and its value depends on the properties of the \(i\)th layers, \(\tau_{i}\) is the traveling time of UWB pulse in the \(i\)th layer. \(A_{ob}\) and \(\tau_{ob}\) are respectively the amplitude and traveling time of the signal which is reflected from the buried object. \(n(t)\) represents the additive white Gaussian noise. At the receiver side, the procedure for determining the position of buried object is indicated in Figure 2. Accordingly, the position of buried object is determined by the distance from it to  
Figure 1: The distance measurement and positioning system in the nondestructive environments. the transceiver. The LMFA method is applied to calculate the relative permittivity of environment and these distances. In the UWB system, \(s(t)\) is generated based on Gaussian [25] or Hermite [26] functions and their derivatives. Compared to Gaussian pulses, the Hermite pulse is useful for parallel data transmission with high data rates, but it is hard to achieve in the real world [27]. Therefore, the Gaussian pulses UWB signals are applied in this work. A typical Gaussian pulse usually takes the form [25]:  
\[g(t)=A_{p}e^{-2\pi(\frac{t}{pp})^{2}}, \tag{2}\]  
where \(A_{p}\) denotes the amplitude of pulse, \(\mu_{p}\) is a factor which influences the amplitude and the width of Gaussian pulse, also called time normalization factor, the width of a pulse becomes narrower when the \(\mu_{p}\) is reduced. The \(\ n^{th}\) derivative of Gaussian pulse, named \(n^{th}\)-order monocycle, is  
\[g_{n}(t)=A_{np}\frac{d^{n}}{dt^{n}}e^{-2\pi(\frac{t}{pp})^{2}}, \tag{3}\]  
where \(A_{np}\) is the amplitude of the \(n^{th}\)-order monocycle. The shapes of different types of Gaussian monocycles are indicated in Figure 3.  
\[r_{1}(t)= A_{1}s(t-\tau_{1}),\]0 The different types of Gaussian monocycles. \[r_{1}(t)= A_{1}s(t-\tau_{1}),\]1 The procedure for determining the position of buried object. The selected Gaussian pulse shape must meet FCC requirements about power and frequency range used. According to Figure 4, it is observed that for the power spectral density (PSD), only the first-order Gaussian monocycle does not comply with the FCC requirements about the effective isotropic radiated power (EIRP) level. In this paper, we restrict our performance analysis of UWB system to the fourth-order Gaussian monocycle in (4). However, our method can be applied for any Gaussian pulse shape. \[r_{1}(t)= A_{1}s(t-\tau_{1}),\]2  
It is observed that with impulse radio UWB (IR-UWB) signal, the correlation and therefore the propagation time can be calculated. However, the IR-UWB signal is not strong enough to be processed at the receiver in the case of multiple layers medium (the value of cross-correlation is lower than the noise lever). Hence, an orthogonal pseudo sequence \(a(n)\) is applied to the UWB signal in pulse position modulation (PPM) to ensure the cross-correlation is obtained when receiving many reflected signals from many surfaces and buried objects.