# Location Information-Assisted Robust Beamforming Design for Ultra-Wideband Communication Systems
## 3 CSI Estimation Error Model  
The goal of this paper is to derive an error model for CSI estimation based on the UE location and known environmental parameters. We intend to achieve this in two steps: firstly, the position error bound of UE is derived from the observed signal. Secondly, the channel error model is reconstructed under the assumption that the localization error distribution follows a Gaussian distribution [29]. After the location information of UE is obtained by the location algorithm, in order to analyze the location accuracy of UE, we will derive the CRLB of UEs to construct the location error distribution. Finally, we define the following vectors:  
\[ \begin{array}{l}\mathbf{\tau}_{k}^{\mathrm{T}}=[\mathbf{\tau}_{1k}^{ \mathrm{T}},\mathbf{\tau}_{2k}^{\mathrm{T}},...,\mathbf{\tau}_{Jk}^{\mathrm{ T}}],\\ \mathbf{\hat{\beta}}_{k}^{\mathrm{T}}=[\mathbf{\hat{\beta}}_{1k}^{\mathrm{T}}, \mathbf{\hat{\beta}}_{2k}^{\mathrm{T}},...,\mathbf{\hat{\beta}}_{Jk}^{\mathrm{ T}}],\end{array}  \tag{14}\]where \(\tau_{j,k}=[\tau_{j,k,1},\tau_{j,k,2},...,\tau_{j,k,M}]^{T}\) and \(\beta_{j,k}=[\beta_{j,k,1},\beta_{j,k,2},...,\beta_{j,k,M}]^{T}\). Then, define the parameter vector of the \(k^{th}\) UE as follows:  
\[\mathbf{\varphi}_{k}=[\tau_{k}^{T},R\mathbf{\beta}_{k}^{T},I\mathbf{\beta}_{k}^{T}]^{T}, \tag{15}\]  
where \(R\mathbf{\beta}_{k}\overset{\Delta}{=}\Re\{\mathbf{\beta}_{k}\}\) and \(I\mathbf{\beta}_{k}\overset{\Delta}{=}\Im\{\mathbf{\beta}_{k}\}\) represent the real and imaginary parts of the amplitude between the UE and BS, respectively. The mean square error matrix of unbiased estimation \(\mathbf{\hat{\varphi}}_{k}\) satisfies the information inequality as follows [30]:  
\[\mathbb{E}\Big{\{}(\mathbf{\hat{\varphi}}_{k}-\mathbf{\varphi}_{k})(\mathbf{\hat{\varphi} }_{k}-\mathbf{\varphi}_{k})^{T}\Big{\}}\succeq\mathbf{J}_{k}^{-1}, \tag{16}\]  
The element of the FIM matrix \(\mathbf{J}_{k}\in\mathbb{C}^{3 JM\times 3 JM}\) can be expressed as:  
\[[\mathbf{J}_{k}]_{u,v}\overset{\Delta}{=}\frac{1}{N_{0}}\int_{0}^{T_{\text{P} }}\Re\{\frac{\partial\mathbf{\mu}_{k}^{H}(t)}{\partial\varphi_{k,u}}\frac{ \partial\mathbf{\mu}_{k}(t)}{\partial\varphi_{k,v}}\}dt, \tag{17}\]  
where \(\mathbf{\mu}_{k}(t)\) is the noiseless part of the received signal in (5), given by the following equation:  
\[\mathbf{\mu}_{k}(t)\overset{\Delta}{=}\left[\begin{array}{c}\sum\limits_{n=1}^{ N}\sum\limits_{m=1}^{M}\beta_{1k,m}w_{1,n}s_{1}^{(\text{P})}(t-\tau_{1k,m})\\ \vdots\\ \sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}\beta_{jk,  
where \(\mathbf{B}_{k}\)= diag\(\left\{\beta_{k}\right\}\); for details, see Appendix A. Assuming that the symbols are independent and identically distributed, the matrix \(\mathbf{R}_{i}\) elements are expressed as follows:  
\[[\mathbf{R}_{i}]_{u,\
u}\triangleq\int_{-W/2}^{W/2}{(2\pi f)^{i}|P(f)|^{2}e^{- 2\pi f\Delta\tau_{u,\
u}}}df, \tag{22}\]  
in which \(\Delta\tau_{u,\
u}=\tau_{jk,\mu}-\tau_{jk,\
u}\), \(i=\left\{0,1,2\right\}\), and \(W\) is the signal bandwidth. As can be seen from the formula above, when \(i\) is 0 or 2, the diagonal element of \(\mathbf{R}_{0}\) and \(\mathbf{R}_{2}\) is the largest 1 and \(4\pi^{2}W_{\text{eff}}^{2}\), respectively, and \(W_{\text{eff}}^{2}=\int_{-W/2}^{W/2}{f^{2}|P(f)|^{2}df}\) is the effective bandwidth. In addition, for any bandwidth W, diag\(\left(\mathbf{R}_{1}\right)\) =\(\mathbf{0}_{M}\), \(\underset{W\rightarrow\infty}{\lim}\mathbf{R}_{1}=\mathbf{0}_{M\times M}\).