## 3 Performance Analysis
### Analysis for Multipath Fading Channel in the Presence of In-Band Narrow Band Interferences  
The practical UWB propagation environment is a dense multipath fading channel with the impulse response of the Saleh-Valenzuela (SV) model [12; 50; 51], which can be expressed in general as:  
\[h(t)=\sum_{l=1}^{L}\sum_{k=1}^{K}\alpha_{k,l}\text{exp}\left(j\phi_{k,l}\right) \delta(t-T_{l}-\tau_{k,l}), \tag{10}\]  
where \(\alpha_{k,l}\) denotes the coefficient of the \(k\)-th component in the \(l\)-th cluster, \(T_{l}\) denotes the time delay of the \(l\)-th cluster, \(\tau_{k,l}\) denotes the time delay of the \(k\)-th MPC relative to the \(l\)-th cluster arrival time \(T_{l}\) and \(\phi_{k,l}\) denotes the random phase, which is a uniform distribution random variable. The distributions of the cluster arrival times \(T_{l}\) can be approximated by a basic Poisson process, and the ray arrival times \(\tau_{k,l}\) can be given by a hybrid Poisson random process. The power delay profile is exponential within each cluster:  
\[E\{\left|\alpha_{k,l}\right|^{2}\}=\Omega_{l}\frac{1}{\gamma_{l}[(1-\beta) \lambda_{1}+\beta\lambda_{2}+1]}\text{exp}(-\tau_{k,l}/\tau_{l}), \tag{11}\]  
where \(\Omega_{l}\) denotes the integrated energy of the \(l\)-th cluster and \(\gamma_{l}\) denotes the intra-cluster decay time constant. Commonly, the cluster decay rates depend linearly on the arrival time of the cluster:  
\[r_{l}\propto k_{\gamma}T_{l}+\gamma_{0}, \tag{12}\]  
where \(k_{\gamma}\) describes the increase of the decay constant with delay. The mean energy of the \(l\)-th cluster follows in general an exponential decay:  
\[10\text{log}(\Omega_{l})=10\text{log}(\text{exp}(-T_{l}/\Gamma))+M_{\text{ cluster}}, \tag{13}\]  
where \(M_{cluster}\) is a normally-distributed variable with standard deviation \(\sigma_{cluster}\) around it. For the simplicity of the analysis, we use the modified multi-path channels according to (10); then, the channel propagation model can be considered as [50; 52]:  
\[h_{\text{MP}}(t)=\sum_{l_{p}=1}^{L_{p}}\alpha_{l_{p}}\delta(t-\tau_{l_{p}}), \tag{14}\]  
where we consider \(L_{p}\) resolvable multi-path components, \(\alpha_{l_{p}}\) denotes the fading coefficient of the \(l_{p}\)-th multi-path and \(\tau_{l_{p}}\) denotes the time delay of the \(l_{p}\)-th multi-path. Therefore, the received UWB signal through the multi-path channel can be written as:  
\[s^{\text{U}}(t) =\sum_{q=1}^{N_{q}}s_{[q]}(t)\otimes h_{\text{MP}}(t)\] \[=\sum_{q=1}^{N_{q}}\sqrt{\frac{E_{[q]}}{N_{f}}}\sum_{i=0}^{\infty }\Xi_{[q]}\left(\left\lfloor\frac{i}{N_{f}}\right\rfloor\right)d_{[q]}(i)u_{[ q]}\Big{(}t-iT_{f}-c_{[q]}^{\text{TH}}(i)T_{c}-\tau_{[q]}\Big{)}, \tag{15}\]  
where:  
\[u_{[q]}(t)=\sum_{l_{p}=1}^{L_{p}}\alpha_{[q],l_{p}}p_{\text{rx}}(t-\tau_{[q],l _{p}}+\tau_{[q],0}), \tag{16}\]  
with \(p_{\text{rx}}(t)\) being the received UWB pulse with unit energy. The impact of interferences in UWB-based heterogeneous wireless sensor networks can be modeled by stochastic geometry, and the signal \(s_{m}^{\rm N}(t)\) transmitted by the \(m\)-th narrow band interference [53] can be written as:  
\[s_{m}^{\rm N}(t)=\sqrt{2E_{\rm N}}\sum_{n=-\infty}^{\infty}a_{m,n}^{\rm N}g(t-nT _{c}^{\rm N}-\tau_{m})\times cos(2\pi f_{m}^{\rm N}(t-\tau_{m})+\theta_{m,n}^{ \rm N}), \tag{17}\]  
where \(E_{\rm N}\) denotes the average energy per transmitted symbol, \(g(t)\) denotes the normalized ultra-wide band baseband pulse waveform satisfying the Nyquist criterion, \(a_{m,n}^{\rm N}\) denotes the \(m\)-th transmitted baseband symbol of the \(m\)-th NBI, \(f_{m}^{\rm N}\) denotes the central frequency of the \(m\)-th in-band NBI component and \(\theta_{m,n}^{\rm N}\) denotes the random phase.