# Location Information-Assisted Robust Beamforming Design for Ultra-Wideband Communication Systems
## 2 System Model
### TDOA Correction Method  
In this paper, we will transmit communication and positioning signals simultaneously in the TDMA mode. Considering that the distance between the antennas is much smaller than the distance between the BS and the UE, it is reasonable to assume that the delay in arriving at each antenna of the base station is the same \(\tau_{jk,m}^{\rm{T}}\)=\(\tau_{jk,m}\). For the \(k\)th UE, the received signal vector \(\mathbf{y}_{k}^{\rm{(P)}}(t)=[y_{1,k}^{\rm{(P)}}(t),\ldots,y_{J,k}^{\rm{(P)}}( t)]\) is expressed as follows:  
\[y_{jk}^{\rm{(P)}}(t)=\sum_{m=1}^{M}\mathbf{h}_{jk,m}*\mathbf{w}_{j}s_{j}^{\rm{( P)}}(t-\tau_{jk,m})+n_{k}(t),t\in\left[0,T_{\rm{P}}\right], \tag{5}\]  
where the positioning signal of the \(j\)th BS is \(s_{j}^{\rm{(P)}}(t),t\in\left[0,T_{\rm{P}}\right],\ \mathbf{w}_{j}\in\mathbb{C}^{N}\) represents the beamforming vector of the \(j\)th BS, and \(\mathbf{w}_{j}=[w_{j,1},\ldots,w_{j,N}]^{\rm{T}}\). \(\mathbf{h}_{jk,m}\in\mathbb{C}^{1\times N}=\left[\beta_{jk,m}^{1},\ldots,\beta _{jk,m}^{N}\right]\) represents the channel model between the \(j\)th BS and the \(k\)th UE, and \(\tau_{jk,m}\) represents the TOA of the \(m\)th path. In addition, \(T_{\rm{P}}=N_{\rm{P}}T_{s}\) indicates the duration of the positioning signal, and \(N_{\rm{P}}\) denotes the number of positioning frames. For the receiver of the UWB, the first path component (FPC) plays a very important role in finding out the receiving time of the message [25]. The received signal power is likely to be more concentrated in the FPC than in the other multipath components (MPC). In reality, UE localization is subject to equipment installation error of BS, continuous start-up errors of the UWB, and clock synchronization errors related to the external environment such as the propagation distance of pulse signals and temperature [26]. To obtain accurate real-time localization of the UE, we design a distance-corrected localization algorithm that uses RN to correct the frequency and time differences between base stations. In addition, this paper will adopt the optional bilateral two-way (AltDS-TWR) ranging mode [27]. The \(j\)th BS measures the distance from the \(k\)th UE as [28]:  
\[\tilde{d}_{jk}(t)=\left(1+e_{j}\right)d_{jk}(t)+B_{j}+\omega_{jk}, \tag{6}\]  
where \(d_{jk}(t)\) denotes the real distance between the \(j^{th}\) BS and \(k^{th}\) UE, \(e_{j}\) is the proportional error of the \(j^{th}\) BS, \(B_{j}\) is the measurement deviation of the \(j^{th}\) BS, and \(\omega_{j,k}\) is a zero-mean  
Figure 2: Illustration of multipath propagation using an image-source model. white Gaussian noise with a power spectral density \(N_{0}\). The TOA value measured by the \(j^{th}\) BS from the RN is as follows:  
\[\tilde{d}_{j,f}(t)=\big{(}1+\epsilon_{j}\big{)}d_{j,f}(t)+B_{j}+\omega_{j,f}, \tag{7}\]  
where \(d_{j,f}(t)\) denotes the distance between the \(j^{th}\) BS and RN. The unknown error vectors \(\delta_{j}=[ \begin{array}{cc}e_{j}&B_{j}\end{array} ]^{\mathrm{T}}\) of the \(j^{th}\) BS are estimated by the classical LS algorithm using the known coordinates of the BS and RN combined with the measured TOA data, which as follow:  
\[\delta_{j}=\Big{(}\mathbf{F}_{j}^{\mathrm{T}}\mathbf{F}_{j}\Big{)}^{-1} \mathbf{F}_{j}^{\mathrm{T}}\mathbf{f}_{j}, \tag{8}\]  
with  
\[\mathbf{F}_{j}=\left[ \begin{array}{cc}d_{j,f}(t)&1\\ d_{j,f}(t-\Delta T)&1\\ d_{j,f}(t-2\Delta T)&1\end{array} \right],\mathbf{f}_{j}=\left[ \begin{array}{ cc}\tilde{d}_{j,f}(t)-d_{j,f}(t)\\ \tilde{d}_{j,f}(t-\Delta T)-d_{j,f}(t-\Delta T)\\ \tilde{d}_{j,f}(t-2\Delta T)-d_{j,f}(t-2\Delta T)\end{array} \right], \tag{9}\]  
where \(\Delta T\) denotes the sample period.