## 3 Performance Analysis
### Analysis for Multipath Fading Channel in the Presence of In-Band Narrow Band Interferences
Commonly, the above reception NBI component can be approximated by a tone with the frequency \(f_{m}^{\rm N}\), so the NBIs can be written as:  
\[s^{\rm N}(t)\approx\sqrt{\frac{2E_{\rm N}}{T_{c}^{\rm N}}}\sum_{m=1}^{M}cos(2 \pi f_{m}^{\rm N}(t-\tau_{m})+\theta_{m}), \tag{18}\]  
where \(T_{c}^{\rm N}\) denotes the average symbol duration of in-band narrow band interference and \(\tau_{m}\) denotes the random time delay of the \(m\)-th NBI. The total received signal of the ultra-wide band wireless sensors can be written as:  
\[E\{\left|\alpha_{k,l}\right|^{2}\}=\Omega_{l}\frac{1}{\gamma_{l}[(1-\beta) \lambda_{1}+\beta\lambda_{2}+1]}\text{exp}(-\tau_{k,l}/\tau_{l}), \tag{11}\]0  
One of the key advantages of UWB signals is immunity to fading; the Rake combining receiver can exploit the high degree of diversity that results from a large number of multipath components (MPCs) [51]. Combining all resolvable paths as in the all-Rake (ARake) receiver provides the optimal performance. However, the number of MPCs that can be utilized in a typical Rake combiner is limited by power consumption constraints, complexity considerations and the availability of channel estimates. To reduce the computation complexity, selective Rake (SRake) and partial Rake (PRake) are also widely utilized. In this subsection, we consider a Rake receiver to combine a set of multipath components of the hybrid signal. Without loss of generality, we assume User 1 is the primary user of interest. To simplify the discussion, only symbol synchronization and the chip synchronization case are considered in this work. The local template signal for the \(k\)-th information bit can be expressed as:  
\[E\{\left|\alpha_{k,l}\right|^{2}\}=\Omega_{l}\frac{1}{\gamma_{l}[(1-\beta) \lambda_{1}+\beta\lambda_{2}+1]}\text{exp}(-\tau_{k,l}/\tau_{l}), \tag{11}\]1  
where:  
\[E\{\left|\alpha_{k,l}\right|^{2}\}=\Omega_{l}\frac{1}{\gamma_{l}[(1-\beta) \lambda_{1}+\beta\lambda_{2}+1]}\text{exp}(-\tau_{k,l}/\tau_{l}), \tag{11}\]2  
with \(\Psi=[\psi_{1},\psi_{2},...,\psi_{L_{r}}]\) being the Rake combining weight coefficients [54; 55; 48; 56]. The local template described by (20) and (21) can represent different multipath diversity combining schemes through assigning a proper value for the weight coefficients vector \(\Psi\). Commonly, the fingers number \(L_{r}\) of a practical Rake receiver should be smaller than the number of resolvable multipath \(L_{p}\), with \(L_{r}\leq L_{p}\). In an \(L_{r}\) finger Rake, the weight coefficients for \(L_{p}-L_{r}\) multipath components not used in the Rake receiver are set to zero while the remaining \(L_{r}\) weight coefficients are determined according to a combining scheme, typically including maximal ratio combining (MRC) [55], equal gain combining (EGC) [51] and selection combining (SC) [54]. The output of the Rake receiver is given by:  
\[E\{\left|\alpha_{k,l}\right|^{2}\}=\Omega_{l}\frac{1}{\gamma_{l}[(1-\beta) \lambda_{1}+\beta\lambda_{2}+1]}\text{exp}(-\tau_{k,l}/\tau_{l}), \tag{11}\]3  
where the first term \(U_{\text{des}}\) is due to the desired ultra-wide band signal:  
\[E\{\left|\alpha_{k,l}\right|^{2}\}=\Omega_{l}\frac{1}{\gamma_{l}[(1-\beta) \lambda_{1}+\beta\lambda_{2}+1]}\text{exp}(-\tau_{k,l}/\tau_{l}), \tag{11}\]4  
with \(R_{u_{[q]}v_{[q]}}(x)=\int u_{[q]}(t-x)v_{[q]}(x)dt\) denoting the cross-correlation between \(u_{[q]}(t)\) of (16) and \(v_{[q]}(t)\) of (21), \(I_{\text{mai}}\) is the MAI, \(I_{\text{nbi}}\) is the interference term caused by the narrow band interferences and \(N\) is the output noise, which is approximate to a normal distribution \(N\sim\mathcal{N}(0,N_{f}\sigma_{n}^{2}E_{v_{[1]}})\) with \(E_{v_{[1]}}=\int(v_{[1]}(t))^{2}dt\).