# Location Information-Assisted Robust Beamforming Design for Ultra-Wideband Communication Systems
## 4 Robust Beamforming  
This section is based on the estimated position of \(k\)th UE for beamforming. The distance between the UE and all BSs is first calculated, and then the closest base station is selected for communication services. For the convenience of representation, we consider the \(j\)th UE and the \(j\)th base station to be the closest. The received signal of the \(j^{th}\) UE is represented by the following equation:  
\[ \begin{array}{l}y_{j}^{\rm(c)}(t)=\sum\limits_{m=1}^{M}\left(\hat{\beta}_{jj,m }+\Delta\beta_{jj,m}\right)*\mathbf{w}_{j}s_{j}^{\rm(c)}(t-\tau_{j,m})\\ \quad\quad+\sum\limits_{i\
eq j}^{M}\sum\limits_{m=1}^{M}\left(\hat{\beta}_{jj, m}+\Delta\beta_{j,m}\right)*\mathbf{w}_{i}s_{i}^{\rm(c)}(t-\tau_{ij,m})+\eta_{j}(t),t \in[T_{\rm p},T_{\rm p}+T_{\rm c}],\end{array}  \tag{28}\]  
where \(s_{j}^{\rm(c)}(t)\) denotes the communication signal of the \(j\)th BS, and \(T_{\rm c}\) is the communication duration. The rate of the \(j\)th UE served by the \(j\)th BS is expressed as follows:  
\[R_{j}=\frac{T_{\rm c}}{T_{\rm p}+T_{\rm c}}\log\left(1+\frac{\left|(\hat{ \mathbf{g}}_{jj}+\Delta\mathbf{g}_{jj})\mathbf{w}_{j}\right|^{2}}{\left|\sum \limits_{i\
eq j}^{J}(\hat{\mathbf{g}}_{ij}+\Delta\mathbf{g}_{ij})\mathbf{w}_ {i}\right|^{2}+\sigma_{j}^{2}}\right), \tag{29}\]  
where \(\hat{\mathbf{g}}_{jj}=\sum\limits_{m=1}^{M}\hat{\beta}_{jj,m}\), \(\Delta\mathbf{g}_{jj}=\sum\limits_{m=1}^{M}\Delta\beta_{jj,m}\). The \(n\)th element of \(\Delta\mathbf{g}_{jj}\) is represented by \(\Delta g_{jj,n}=\mathbf{b}_{jj,n}^{\rm T}\mathbf{e}_{j}\), with:  
\[\mathbf{b}_{j,n}=\left(\frac{\lambda}{4\pi}\left(\frac{(\hat{\mathbf{g}}_{j} -\mathbf{u}_{j,n})}{\left\|\hat{\mathbf{p}}_{j}-\mathbf{u}_{j,n}\right\|_{2}^ {3}}\right)+\sum\limits_{m=2}^{M}\frac{\lambda\sqrt{T_{\rm R}}\left(\hat{ \mathbf{p}}_{j}-\mathbf{u}_{j,n}\right)}{4\pi(\left\|\hat{\mathbf{p}}_{j}- \mathbf{a}_{j}\right\|_{2}+\left\|\mathbf{a}_{j}-\mathbf{u}_{j,n}\right\|_{2} )^{3}}\right), \tag{30}\]  
Each BS is scattered throughout the coverage area according to certain rules. When the BS receives the positioning data signal from the UE, it is connected to the central processing unit through high-speed transmission media such as optical fiber. The robust beamforming optimization problem to minimize the total transmit power is formulated as follows:  
\[ \begin{array}{l}\min\limits_{\{\mathbf{w}_{j}\}_{j=1}^{J}}\quad\quad\quad \sum\limits_{j=1}^{J}\left\|\mathbf{w}_{j}\right\|_{2}^{2}\\ \quad\quad\text{s.t.}\quad\quad\left\|\mathbf{w}_{j}\right\|_{2}^{2}\leq p_{ \rm UWB}^{\rm{T}},j=1,...,J,\\ \quad\quad\quad\quad\quad\quad\Pr\{\text{SINR}_{j}(\Delta\mathbf{g}_{jj}^{\rm H })\leq\gamma_{j}\}\leq\rho_{j},\end{array}  \tag{31}\]  
where \(p_{\rm UWB}^{\rm{max}}\) represents the maximum power, \(\gamma_{j}=2^{\left(T_{\rm p}+T_{\rm c}\right)R}/T_{\rm c}-1\) denotes the minimum SINR constraint of the \(j\)th UE and the maximum outage probability \(\rho_{j}\) of the \(j\)th UE. We define \(\mathbf{W}_{j}\triangleq\mathbf{w}_{j}\mathbf{w}_{j}^{\rm H}\). To simplify the calculation, the SINR of the \(j\)th BS is expressed as follows:  
\[f(\Delta\mathbf{g}_{jj})=\hat{\mathbf{g}}_{jj}^{\rm H}\mathbf{W}_{j}\hat{ \mathbf{g}}_{jj}+2\text{Re}\left\{\Delta\mathbf{g}_{jj}^{\rm H}\mathbf{W}_{j} \mathbf{g}_{jj}\right\}+\Delta\mathbf{g}_{jj}^{\rm H}\mathbf{W}_{j}\Delta \mathbf{g}_{jj}, \tag{32}\]  
Therefore, the constraint (31) of the optimization problem is represented by the following equation:  
\[\Pr\left\{\frac{f(\Delta\mathbf{g}_{jj})}{\gamma_{j}}-\sum\limits_{i=1,J\
eq j }^{J}f(\Delta\mathbf{g}_{ij})\leq\sigma_{j}^{2}\right\}\leq p_{\rm out}, \tag{33}\]  
According to formula (25), the signal amplitude error is expressed as:  
\[\Delta\mathbf{g}_{jj}\sim\mathcal{CN}(0,\Delta\mathbf{B}_{jj}^{\rm T}\Delta \mathbf{B}_{jj}\mathbf{p}_{\rm R}^{-1}), \tag{34}\]where \(\Delta\mathbf{B}_{jj}\in\mathbb{R}^{N\times 2}\), \(\Delta\mathbf{B}_{jj}=\left[b_{jj,1}^{\mathrm{T}},\ldots,b_{jj,N}^{\mathrm{T}} \right]^{\mathrm{T}}\).