# Amplitude Modeling of Specular Multipath Components for Robust Indoor Localization
## 5 Exploiting the GPR Model for (Multipath-Assisted) Positioning  
To formulate a signal model for position estimation, Equation (3) can be rewritten as  
\[\mathbf{r}=\mathbf{S}(\mathbf{p})\mathbf{\alpha}+\mathbf{w}\sim\mathcal{CN}(\mathbf{S}(\mathbf{p})\mathbf{ \alpha},\mathbf{C}) \tag{10}\]  
where \(\mathbf{r}=[\ldots,r(nT_{s}),\ldots]\in\mathbb{C}^{N}\) is a sampled version of the received signal, \(\mathbf{S}(\mathbf{p})=[\ldots,\mathbf{s}(\tau_{k}(\mathbf{p})),\ldots]\) is the parameterized signal matrix with \(\mathbf{s}(\tau)=[\ldots,s(nT_{s}-\tau),\ldots]^{T}\in\mathbb{C}^{N}\), denoting the SMCs with delays \(\tau_{k}(\mathbf{p})\) that are determined by \(\mathbf{p},\mathbf{\alpha}=[\ldots,\alpha_{k},\ldots]^{T}\in\mathbb{C}^{K}\) are the SMC amplitudes, and \(\mathbf{w}\) is a noise vector with covariance \(\mathbf{C}=\mathbb{E}\{\mathbf{w}\mathbf{w}^{H}\}\). The noise vector \(\mathbf{w}\) may account for AWGN only or for the DM as well. A maximum likelihood (ML) estimation \(\hat{\mathbf{p}}=\arg\max_{\mathbf{p}^{*}}\mathcal{L}(\mathbf{r}|\mathbf{p}^{*})\) considers \(\mathbf{\alpha}\) as nuisance parameters and finds the position by maximizing the concentrated log-likelihood function (LLHF)  
\[\mathcal{L}(\mathbf{r}|\mathbf{p}^{*})=\max_{\mathbf{\alpha}}\mathcal{L}(\mathbf{r}|\mathbf{p}^{*},\mathbf{\alpha}). \tag{11}\]  
The LLHF evaluated at candidate position \(\mathbf{p}^{*}\) follows from (10),  
\[\mathcal{L}(\mathbf{r}|\mathbf{p}^{*},\mathbf{\alpha})=-\log\det(\mathbf{C})-(\mathbf{r}-\mathbf{S}( \mathbf{p})\mathbf{\alpha})^{H}\mathbf{C}^{-1}(\mathbf{r}-\mathbf{S}(\mathbf{p})\mathbf{\alpha}). \tag{12}\]  
The maximization in (11) is solved by a weighted least square solution for \(\|\mathbf{r}-\mathbf{S}(\mathbf{p})\mathbf{\alpha}\|^{2}\) which yields \(\hat{\mathbf{x}}=\big{(}\mathbf{S}^{H}(\mathbf{p})\mathbf{C}^{-1}\mathbf{S}(\mathbf{p})\big{)}^{-1} \mathbf{S}^{H}(\mathbf{p})\mathbf{C}^{-1}\mathbf{r}\). This ML positioning algorithm is our reference algorithm, denoted as \"SALMA-light\" and described in [54; 55] if AWGN is assumed, i.e., \(\mathbf{C}=\sigma_{n}^{2}\mathbf{I}\). The GPM model provides prior knowledge about the absolute value of \(\mathbf{\alpha}\) but not on its phase. We thus expand the amplitude vector as \(\mathbf{\alpha}=\mathbf{\Phi}\mathbf{x}\), where diagonal matrix \(\mathbf{\Phi}=\text{diag}(\mathbf{\phi})\) and \([\mathbf{\phi}]_{k}=\exp(j\angle_{k})=\exp(j\zeta_{k})\). From the GPM, we know that \(\mathbf{x}|\mathbf{p}\sim\mathcal{N}\Big{(}\mathbf{\mu}(\mathbf{p}),\frac{1}{2}\mathbf{\Lambda}( \mathbf{p})\Big{)}\) with \([\mathbf{\mu}(\mathbf{p})]_{k}=\frac{\mathbb{E}\{\psi(\hat{\mathbf{\alpha}}(\mathbf{p}))|\mathcal{ D}_{k}\mathbf{\mu}^{\text{obs}}\}}{\|\mathbf{x}_{k}-\mathbf{p}\|^{2}}\), and the diagonal matrix has elements \([\mathbf{\Lambda}(\mathbf{p})]_{k,k}=2\frac{\mathbb{V}\left\{\hat{\mathbf{\varphi}}(\hat{ \mathbf{\varphi}}(\mathbf{p}))|\mathcal{D}_{k}\mathbf{\mu}^{\text{obs}}\right\}}{\|\mathbf{x}_ {k}-\mathbf{p}\|^{2}}\). The signal model then becomes  
\[\mathbf{r}=\mathbf{S}(\mathbf{p})\mathbf{\Phi}\mathbf{\mu}(\mathbf{p})+\mathbf{S}(\mathbf{p})\mathbf{\tilde{x}}+ \mathbf{w} \tag{13}\]  
with \(\mathbf{\tilde{x}}|\mathbf{p}\sim\mathcal{CN}(0,\mathbf{\Lambda}(\mathbf{p}))\) is introduced to account for the variance of the SMC amplitudes and the uncertainty of the phases in \(\mathbf{\Phi}\).