## 2 The CNES PPP-WIZARD Demonstrator
### The User Side of the Demonstrator
#### 2.3.1 Measurement Modeling in the User
 \tag{20}\]  
It is then possible to make the code and phase clocks relative to the frequency \(f_{1}\) appear in the code and phase models, respectively, for the other frequencies. Thus, we define new code and phase biases as:  
\[\tilde{b}_{r,\text{IF},C_{j}}=b_{r,\text{IF},C_{j}}-b_{r,\text{IF},C_{1}}\quad \text{and}\quad\tilde{b}_{r,\text{IF},L_{j}}=b_{r,\text{IF},L_{j}}-b_{r,\text {IF},C_{1}}, \tag{22}\]  
and the measurement model (1) of the code and phase measurements for the frequency \(f_{j\
eq 1}\) is transformed to:  
\[ \begin{split}\tilde{C}_{j}=C_{j}+h_{\text{IF}}^{s}+b_{\text{IF}, C_{j}}^{s}&=\rho_{r}^{s}+h_{r,\text{IF}}+b_{r,\text{IF},C_{j}}+ \gamma_{j}e+m(E^{s})T_{Z},\\ &=\rho_{r}^{s}+\tilde{h}_{r,\text{IF},C_{1}}+\tilde{b}_{r,\text {IF},C_{j}}+\gamma_{j}e+m(E^{s})T_{Z},\\ \lambda_{j}\tilde{L}_{j}=\lambda_{j}L_{j}+h_{\text{IF}}^{s}+b_{ \text{IF},L_{j}}^{s}&=\rho_{r}^{s}+h_{r,\text{IF}}+b_{r,\text{ IF},L_{j}}-\gamma_{j}e+m(E^{s})T_{Z}+\lambda_{j}N_{j}^{s}+\lambda_{j}W,\\ &=\rho_{r}^{s}+\tilde{h}_{r,\text{IF},C_{1}}+\tilde{b}_{r,\text {IF},L_{j}}-\gamma_{j}e+m(E^{s})T_{Z}+\lambda_{j}N_{j}^{s}+\lambda_{j}W.\end{split}  \tag{23}\]  
Since the same clock parameter is shared between all the frequencies, the parameterization proposed by the set of Equations (20), (21), (23) and (24) is called the common clock parameterization. It is performed for each constellation separately, so four receiver clock offsets have to be estimated, one for each constellation. The method for recovering the integer ambiguities is presented in [31] and recalled here for the sake of completeness. Choosing the frequency \(f_{1}\) as a pivoting frequency, the ambiguity \(N_{j}\) of the phase measurement at the frequency \(f_{j}\) can be expressed as a combination of \(N_{1}\) and the widelane ambiguity \(N_{\text{WL},\text{J}}=N_{j}-N_{1}\): \(N_{j}=N_{1}+N_{\text{WL},\text{J}}\). In the same way, the ambiguity \(N_{k}\) of the phase measurement at the frequency \(f_{k}\) can be expressed using the widelane ambiguity \(N_{\text{WL},\text{J}}\) and an extra-widelane ambiguity \(N_{\text{EWL},\text{J}}\)\(N_{k}=N_{1}+N_{\text{WL},1j}+N_{\text{EWL},jk}\). The decomposition of the ambiguities on all the single frequencies for the constellations is thus performed as follows:  
\[\text{for GPS}: \begin{cases}N_{1}=N_{1},\\ N_{2}=N_{1}+N_{\text{WL},12},\\ N_{5}=N_{1}+N_{\text{WL},12}+N_{\text{EWL},25},\end{cases}  \tag{25}\]  
\[\text{for Galileo}: \begin{cases}N_{1}=N_{1},\\ N_{5a}=N_{1}+N_{\text{WL},15a},\\ N_{5b}=N_{1}+N_{\text{WL},15a}+N_{\text{EWL},5a5b},\\ N_{6}=N_{1}+N_{\text{WL},15a}+N_{\text{EWL},5a5b}++N_{\text{EWL},5b6},\end{cases}  \tag{26}\]  
\[\text{for Beidou}: \begin{cases}N_{1I}=N_{1I},\\ N_{3I}=N_{1I}+N_{\text{WL},1I3I},\\ N_{2I}=N_{1I}+N_{\text{WL},1I3I}+N_{\text{EWL},3I2I},\\ N_{2b}=N_{1I}+N_{\text{WL},1I3I}+N_{\text{EWL},3I2b},\\ N_{2a}=N_{1I}+N_{\text{WL},1I3I}+N_{\text{EWL},3I2a},\\ N_{1C}=N_{1I}+N_{\text{WL},1I1C}.\end{cases}  \tag{27}\]  
The widelane \(N_{\text{WL},jj}\) and the extra-widelane ambiguities \(N_{\text{EWL},jk}\) are estimated from the Melbourne-Wubbena combination [28; 29]. It is a linear combination of the carrier-phase widelane and the code narrowlane, whose aim is to build an ionosphere-free geometry-free measurement. In this case, the only remaining quantities are the carrier-phase ambiguity and a combination of the code and phase biases of the involved observables.