# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 2 Signal Model  
We consider the transmission of a band-limited GNSS signal \(c(t)\) (bandwidth \(B\)), so-called Pseudo-Random Noise (PRN) code in the GNSS terminology, over a carrier frequency \(f_{c}\) (\(\lambda_{c}=\frac{c}{f_{c}}\)), from a transmitter (satellite) T to a receiver R. During the observation time, both transmitter and receiver are static, that is, their respective positions is constant \(\mathbf{p}_{T}(t)=\mathbf{p}_{T}\) and \(\mathbf{p}_{R}(t)=\mathbf{p}_{R}\). The complex analytic signal at the output of the receiver's antenna can be written as \(x_{A}(t)=\alpha_{R}c_{R}(t)+n_{A}(t)\), with \(n_{A}(t)\) a zero-mean white complex Gaussian noise, and where the gain \(\alpha_{R}\) depends on the transmitted signal power, the transmitter/receiver antenna gains and polarization vectors, and the radial distance between T and R, \(\mathbf{p}_{TR}\)[17; 18]. In that perspective, the propagation delay \(\tau\left(t\right)\) is constant \(\tau\left(t\right)=\tau=\frac{\left\|\mathbf{p}_{TR}\right\|}{c}\) and the baseband output of the receiver's Hilbert filter is \(x\left(t\right)=\alpha c\left(t-\tau\right)+n\left(t\right)\), with \(n(t)\) a complex white Gaussian noise within the filter bandwidth with unknown variance \(\sigma_{n}^{2}\), and \(\alpha=\alpha_{R}e^{-j2\pi f_{c}\tau}\). The discrete vector signal model is build from \(N=N_{2}-N_{1}+1\) samples at \(T_{s}=\frac{1}{T_{s}}\),  
\[\mathbf{x}=\alpha\mathbf{c}\left(\tau\right)+\mathbf{n}, \tag{1}\]  
where \(\mathbf{n}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{n}^{2}\mathbf{I}_{N}\right)\) and \(\mathbf{x}=\left(x\left(N_{1}T_{s}\right),\ldots,x\left(N_{2}T_{s}\right) \right)^{\top}\), \(\mathbf{n}=\left(n\left(N_{1}T_{s}\right),\ldots,n\left(N_{2}T_{s}\right) \right)^{\top}\), \(\mathbf{c}\left(\tau\right)=\left(c\left(N_{1}T_{s}-\tau\right),\ldots,c\left( N_{2}T_{s}-\tau\right)\right)^{\top}\). Since the transmitter/receiver antenna gains and polarization vectors are in general unknown, \(\alpha\) is assumed to be an unknown complex parameter as well [1; 2; 3; 18; 19]. Thus, the unknown deterministic parameters [20] can be gathered in vector \(\underline{\epsilon}=\left(\sigma_{n}^{2},\tau,\alpha,\alpha^{*}\right)^{\top}\), where \(\alpha^{*}\) is the complex conjugate of \(\alpha\). Notice that \(c(t)\) can be directly a PRN code with a Binary Phase Shift Keying (BPSK) modulation where there is no subcarrier, as in the case of GPS L1 C/A, or a subcarrier modulated PRN, i.e., using a Binary Offset Carrier (BOC) [21] type modulation. The subcarrier has a direct impact on the correlation function, therefore on the estimation performance. On top of that, the signal may have data bits or not, depending on whether it belongs to a data component or a pilot component. A generic multilayer GNSS signal structure is shown in Figure 1 (left) and the autocorrelation function (ACF) for a standard BPSK(1) and BOC(1,1) in Figure 1 (right). Notice that the correlation peak is narrower for the BOC(1,1), which will in turn lead to a better time-delay estimation. Notice that the model in Equation (1) can be reparameterized as  
\[\mathbf{x}=\rho\mathbf{c}^{\prime}\left(\boldsymbol{\theta}\right)+\mathbf{n},\quad\mathbf{c}^{\prime}\left(\boldsymbol{\theta}\right)=\mathbf{c}\left( \tau\right)e^{j\varphi},\;\rho\in\mathbb{R}^{+},\boldsymbol{\theta}^{\top}= \left(\varphi,\tau\right), \tag{2}\]  
and then the unknown deterministic parameters are \(\underline{\epsilon}=\left(\sigma_{n}^{2},\rho,\boldsymbol{\theta}^{\top} \right)^{\top}\), which are used to derive the new joint time-delay and phase estimation CRB.