# Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals
## 3 GNSS Receiver Signal Processing
### GNSS Code/Phase-Based RTK Positioning and CRB  
The previous SPP approach is limited by the time-delay MLE precision, which is directly linked to the baseband signal resolution (i.e., the shape of the ACF). Precise positioning techniques rely on the use of the signal's phase information. Unfortunately, exploiting this information implies solving a more complicated problem, mainly because the phase measurement is ambiguous, and the estimation of the unknown number of cycles \(N_{i}\) in (15) is the bottleneck [11]. As already stated, two main approaches are available in the literature: (i) differential techniques such as RTK [2] and (ii) PPP techniques [1]. It is important to notice that PPP techniques require a long convergence time and need high accuracy satellite orbits, while clock and propagation (ionospheric and tropospheric) error corrections are required, which may be obtained from a network broadcasting precise corrections. However, such corrections are not available in real time [39; 40]; thus, PPP is not suitable for safety-critical applications. In contrast, RTK exploits the connection to a reference station at a known position, which if close enough observes the same propagation errors as the receiver. Therefore, the goal is to combine such observations in order to eliminate all nuisance parameters and then estimate the base-to-receiver baseline. The combination of base and receiver observations is obtained by double-differencing, that is subtracting the measurements from the receiver w.r.t. the base station and a pivot (reference) satellite. A key point is that such a combination also eliminates the fractional phase parts \(B_{i}\) and \(B_{r}\); therefore, the unknown ambiguities become integer parameters. The problem of mixed real and integer parameter estimation was pioneered by [PERSON] [41; 42; 43], and its solution typically combines a WLS with an integer least squares (ILS). It is out of the scope to give a complete discussion on RTK, and only a brief description is given in the sequel. The reader can refer to [11] and the references therein for details. If we consider \(M\ +\ 1\) visible satellites being tracked at both the base station and the receiver, the code and phase double difference (DD) observations and the corresponding linearized model are (subscript 0 and superscript \(B\) are used to refer to the pivot reference satellite and the base station, respectively; superscript \(R\) refers to quantities related to the receiver):\[\mathbf{y}_{\mathbf{e}}^{\top} =\left[\widehat{\phi}_{1,0}^{R,B},\ldots,\widehat{\phi}_{M,0}^{R,B} \right],\quad\widehat{\phi}_{i,0}^{R,B}=\widehat{\phi}_{i}^{R}-\widehat{\phi}_{ i}^{B}-\left(\widehat{\phi}_{0}^{R}-\widehat{\phi}_{0}^{B}\right) \tag{19a}\] \[\mathbf{y}_{\Phi}^{\top} =\left[\widehat{\Phi}_{1,0}^{R,B},\ldots,\widehat{\Phi}_{M,0}^{R, B}\right],\quad\widehat{\Phi}_{i,0}^{R,B}=\widehat{\Phi}_{i}^{R}-\widehat{\Phi}_{ i}^{B}-\left(\widehat{\Phi}_{0}^{R}-\widehat{\Phi}_{0}^{B}\right),\] (19b) \[\mathbf{y}\approx\mathbf{Dz}\ +\ \mathbf{n}_{\Phi,\mathbf{e}},\ \mathbf{y}= \begin{bmatrix}\mathbf{y}_{\Phi}\\ \mathbf{y}_{\varrho}\end{bmatrix} ,\ \mathbf{z}= \begin{bmatrix}\mathbf{b}\\ \mathbf{a}\end{bmatrix} ,\ \mathbf{D}= \begin{bmatrix}\mathbf{B}&\mathbf{A}\\ \mathbf{B}&\mathbf{0}\end{bmatrix} ,\ \mathbf{B}= \begin{bmatrix}-\left(\mathbf{u}_{1}( \mathbf{p}_{B})-\mathbf{u}_{0}(\mathbf{p}_{B})\right)^{\top}\\ \vdots\\ -\left(\mathbf{u}_{M}(\mathbf{p}_{B})-\mathbf{u}_{0}(\mathbf{p}_{B})\right)^{ \top}\end{bmatrix} , \tag{19c}\]  
with \(\mathbf{A}=\lambda_{c}\mathbf{I}\), \(\mathbf{B}\) is the double-difference geometry matrix, \(\mathbf{z}\) the set of unknown parameters, \(\mathbf{b}=\mathbf{p}_{R}-\mathbf{p}_{B}\) the baseline vector between the receiver and base station, and \(\mathbf{a}\) the vector of DD integer ambiguities. The designed matrix \(\mathbf{D}\) relates the state estimate \(\mathbf{z}\) to the vector of observations \(\mathbf{y}\). Finally, the noise is:  
\[\mathbf{n}_{\Phi,\mathbf{e}}=