## 2 System Model
### Signal Model for Incident Spherical Wavefronts  
In the uplink, \(M\) users operate on orthogonal frequencies but share the same configurations of RISs. Given the carrier frequency \(f_{c}\), subcarrier spacing \(\Delta f\), the frequency of the \(m\)-th user can be expressed with \(f_{m}=f_{c}+m\Delta f-B/2\), and the signal bandwidth is \(B=M\Delta f\). Each user transmits a single data symbol \(p[n]=1\), which meets the criteria of \(\mathbb{E}\{p[n]p^{*}[n]\}=1\). The LoS paths between the BS and users are presumed to be blocked, while VLoS paths reflected by RISs always exist. The received signal from the \(m\)-th user at the BS, \(\mathbf{y}_{m}\in\mathbb{C}^{N_{B}\times 1}\), is  
\[\mathbf{y}_{m}=\sqrt{P}\sum_{k=1}^{K}\mathbf{H}_{k}\mathbf{\Omega}_{k}\mathbf{ h}_{k,m}+\mathbf{n}_{m}, \tag{2}\]  
where \(P\) denotes the transmit signal power, \(\mathbf{H}_{k}=\{h_{k;b,p}\}\in\mathbb{C}^{N_{B}\times N_{R}}\) and \(\mathbf{h}_{k,m}=\{h_{k,m;p}\}\in\mathbb{C}^{N_{B}\times 1}\) are the channel matrices of BS-(\(k\)-th)RIS and (\(k\)-th)RIS-(\(m\)-th)user links, respectively.  
Figure 2: Geometry of tracking system.  
Here, we put the normalized power radiation pattern and gain of each RIS element into the channel matrices. Thus, \(\mathbf{\Omega}_{k}=\text{diag}(c_{k,0},\ldots,c_{k,p},\ldots,c_{k,N_{k}-1})\in\mathbb{ C}^{N_{k}\times N_{R}}\) only contains the unit-norm load coefficient of each RIS element, i.e., \(c_{k,p}\) denotes the unit-norm load coefficient of the \(p\)-th element of the \(k\)-th RIS and \(c_{k,p}=e^{j\Phi_{k,p}}\). \(\mathbf{\mathrm{n}}_{m}\in\mathbb{C}^{N_{k}\times 1}\) denotes the additive white Gaussian noise (AWGN) sequences with variance \(\sigma_{m}^{2}\). BS estimates the users' positions by exploiting the spherical waveform model. The components of the channel matrix are modelled as  
\[h_{k;b,p}=\rho_{k;b,p}e^{-j\frac{2\pi}{\lambda_{m}}d_{k;b,p}}, \tag{3}\]  
\[h_{k,m;p}=\sqrt{\frac{\kappa_{h}}{\kappa_{h}+1}}\rho_{k,m;p}e^{-j\frac{2\pi}{ \lambda_{m}}d_{k;m;p}}+\sqrt{\frac{1}{\kappa_{h}+1}}\delta_{k,m;p}, \tag{4}\]  
where \(\rho_{k;b,p}\triangleq\frac{\lambda_{m}}{4\pi}\frac{\sqrt{C_{b}C_{p}F(\Phi_{ k,m;p})}}{d_{k}}\), \(\rho_{k,m;p}\triangleq\frac{\lambda_{m}}{4\pi}\frac{\sqrt{C_{p}C_{m}F(\Phi_{ k,m;p})}}{d_{k}}\) are the attenuation due to propagation of the LoS components, and \(\lambda_{m}\) satisfies the relation of \(\lambda_{m}f_{m}=c\), where \(c\) is the speed of light. We adopt a simple Rice channel model to describe the links between RISs and users, and \(\kappa_{h}\geq 0\) are the Ricean factors for (\(k\)-th)RIS-(\(m\)-th)user links. We assume that the LoS paths always exist, as the probability of the LoS condition is close to 1 in the near-field region [37]. \(\delta_{k,m;p}\sim\mathcal{CN}(0,\rho_{k,m;p}^{2})\) denotes the random complex fading coefficients of the NLoS components. The received signal at the \(b\)-th antenna on the BS from the \(m\)-th user results in  
\[y_{m;b}=\sqrt{\frac{\kappa_{h}P}{\kappa_{h}+1}}\sum_{k=0}^{K-1}\sum_{p=0}^{N_ {k}-1}\rho_{k;b,p}\rho_{k,m;p}c_{k,p}e^{-j\frac{2\pi}{\lambda_{m}}(d_{k;b,p}+ d_{k,m;p})}+w_{m;b}, \tag{5}\]  
where \(w_{m;b}\sim\mathcal{CN}\big{(}0,\sigma_{m;b}^{2}\big{)}\) is the interference-plus-noise term containing the AWGN noise and the multipath components and is given by  
\[w_{m;b}=n_{m;b}+\sqrt{\frac{P}{\kappa_{h}+1}}\sum_{k=0}^{K-1}\sum_{p=0}^{N_ {k}-1}\rho_{k;b,p}\delta_{k,m;p}c_{k,p}e^{-j\frac{2\pi}{\lambda_{m}}d_{k;b,p}}. \tag{6}\]