# Location Information-Assisted Robust Beamforming Design for Ultra-Wideband Communication Systems
## 4 Robust Beamforming
The signal amplitude error is rewritten as:  
\[\Delta\mathbf{g}_{jj}=\mathbf{E}_{jj}\mathbf{v}_{j}, \tag{35}\]  
where \(\mathbf{v}_{j}\sim\mathcal{N}(0,\mathbf{I})\), \(\mathbf{E}_{jj}=\sqrt{\Delta\mathbf{B}_{jj}^{\mathrm{H}}\Delta\mathbf{B}_{jj} \mathbf{J}_{\mathrm{P}_{j}}^{-1}}\). In order to simplify the equation, we defined \(\mathbf{E}_{j}=\mathrm{diag}\{\mathbf{E}_{1,j},...,\mathbf{E}_{j,j}\}\) and \(\mathbf{\tilde{g}}_{j}=\left[\mathbf{g}_{1,j}^{\mathrm{T}},...,\mathbf{g}_{j, j}^{\mathrm{T}}\right]^{\mathrm{T}}\). Then, we can obtain the following equation:  
\[\Delta\mathbf{g}_{j}=\mathbf{\bar{E}}_{j}\mathbf{\tilde{g}}_{j}, \tag{36}\]  
where  
\[\Delta\mathbf{\tilde{g}}_{j}=[\Delta\mathbf{g}_{1,j}^{\mathrm{T}},...,\Delta \mathbf{g}_{j,j}^{\mathrm{T}}]^{\mathrm{T}}, \tag{37}\]  
then  
\[\frac{f(\Delta\beta_{jj})}{\gamma_{j}}-\sum_{i=1,j\
eq j}^{J}f(\Delta\beta_{ ij})=\mathbf{\bar{v}}_{j}^{\mathrm{T}}\mathbf{\tilde{A}}_{j}\mathbf{\bar{v}}_{j}^{ \ast}+2\mathrm{Re}\{\mathbf{\bar{v}}_{j}^{\mathrm{T}}\mathbf{\tilde{a}}_{j} \}+\mathbf{\tilde{g}}_{j}^{\mathrm{T}}\mathbf{\tilde{W}}_{j}\mathbf{\tilde{g} }_{j}^{\ast}, \tag{38}\]  
where  
\[\mathbf{\tilde{A}}_{j}\mathbf{=}\mathbf{\bar{E}}_{j}\odot\mathbf{\tilde{W}}_{ j}\odot\mathbf{\bar{E}}_{j},\mathbf{\tilde{a}}_{j}=\mathbf{\bar{E}}_{j}\odot \mathbf{\tilde{W}}_{j}\mathbf{\tilde{g}}_{j}, \tag{39}\]  
where \(\odot\) denotes the Hadamard product. Therefore, the constraint item (31) can be expressed as:  
\[\Pr\{\mathbf{\bar{v}}_{j}^{\mathrm{T}}\mathbf{\tilde{A}}_{j}\mathbf{\bar{v}}_ {j}^{\ast}+2\mathrm{Re}\{\mathbf{\bar{v}}_{j}^{\mathrm{T}}\mathbf{\tilde{a}}_{ j}\}\leq c_{j}\}\leq\rho_{j}, \tag{40}\]  
where \(c_{j}=\sigma_{j}^{2}-\mathbf{\bar{g}}_{j}^{\mathrm{T}}\mathbf{\tilde{W}}_{j} \mathbf{\tilde{g}}_{j}^{\ast}\). In order to transform probabilistic constraints into deterministic constraints, this paper uses the following Bernstein-type inequalities: \(\mathbf{G}\mathbf{=}\mathbf{v}^{\mathrm{H}}\mathbf{A}\mathbf{v}+2\mathbf{v}^ {\mathrm{H}}\mathbf{a}\), \(\mathbf{A}\in\mathbb{H}^{N}\), \(\mathbf{a}\in\mathbb{C}^{N}\), \(\mathbf{v}\sim\mathcal{CN}(0,\mathbf{I})\). For any \(\varsigma>0\), the following inequality holds:  
\[\Pr\{\mathbf{G}\leq\mathrm{Tr}(\mathbf{A})-\sqrt{2\varsigma}\sqrt{\|\mathrm{ vec}(\mathbf{A})\|^{2}+2\|\mathbf{a}\|^{2}}-\varsigma\mathbf{s}^{-}(\mathbf{A}) \}\leq exp(-\varsigma), \tag{41}\]  
where \(s^{-}(\mathbf{A})=\max\{\lambda_{max}(-\mathbf{A}),0\}\), \(s^{+}(\mathbf{A})=\max\{\lambda_{max}(\mathbf{A}),0\}\), \(\lambda_{\max}(\mathbf{A})\) is the largest eigenvalue of matrix \(\mathbf{A}\). According to the Bernstein-type inequality, the constraint term (33) is the following deterministic constraint:  
\[\Pr\left\{\frac{f(\Delta\mathbf{g}_{jj})}{\gamma_{j}}-\sum\limits_{i=1,J\
eq j }^{J}f(\Delta\mathbf{g}_{ij})\leq\sigma_{j}^{2}\right\}\leq p_{\rm out}, \tag{33}\]0  
where \(\varsigma_{j}=-\ln(\rho_{j})\). Furthermore, the constraints of (42) can be reformulated as:  
\[\Pr\left\{\frac{f(\Delta\mathbf{g}_{jj})}{\gamma_{j}}-\sum\limits_{i=1,J\
eq j }^{J}f(\Delta\mathbf{g}_{ij})\leq\sigma_{j}^{2}\right\}\leq p_{\rm out}, \tag{33}\]1 \[\Pr\left\{\frac{f(\Delta\mathbf{g}_{jj})}{\gamma_{j}}-\sum\limits_{i=1,J\
eq j }^{J}f(\Delta\mathbf{g}_{ij})\leq\sigma_{j}^{2}\right\}\leq p_{\rm out}, \tag{33}\]2 (43b) \[\Pr\left\{\frac{f(\Delta\mathbf{g}_{jj})}{\gamma_{j}}-\sum\limits_{i=1,J\
eq j }^{J}f(\Delta\mathbf{g}_{ij})\leq\sigma_{j}^{2}\right\}\leq p_{\rm out}, \tag{33}\]3  
where \(\mu_{j}\) and \(v_{j}\) represent a relaxation variable. We have transformed the second non-convex constraint of formula (31) into the convex constraints 43a, 43b, 43c, so that problem (31) can be effectively solved by the convex optimization tool: CVX.