# A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals
## 2 Mathematical Model  
GPS, BDS and Galileo signals are code division multiple access (CDMA) signals. Ignoring the multipath, their double difference (DD) pseudo-range and carrier phase measurements can be formulated as [27]  
\[\
abla\Delta\mathrm{P_{i}}=\
abla\Delta\rho+\ \
abla\Delta\frac{K}{f_{i}^{2}}+\ \
abla\Delta T+\ \
abla\Delta\varepsilon_{i,P} \tag{1}\]  
\[\
abla\Delta\lambda_{i}\phi_{i}=\
abla\Delta\rho+\ \lambda_{i}\
abla\Delta N_{i} -\ \
abla\Delta\frac{K}{f_{i}^{2}}+\ \
abla\Delta T+\ \
abla\Delta\varepsilon_{i,\phi} \tag{2}\]  
where \(\
abla\Delta\)--DD operator;  
\(i\)--the \(i\)-th frequency, e.g., GPS L1, L2 or L5;  
\(f\)--frequency (Hz);  
\(P\)--pseudo-range measurements (meter);  
\(\phi\)--carrier phase measurements (cycle);  
\(\lambda\)--wavelength (meter);  
\(\rho\)--geometric distance from satellite to receiver (meter);  
\(N\)--carrier phase ambiguity;  
\(K\)--the parameter of the first-order ionospheric delay, K = 40.28 TEC; TEC represents the Total Electron Content;  
\(T\)--tropospheric delay (meter);  
\(\varepsilon_{i,P}\),\(\varepsilon_{i,\phi}\)--measurement noise of pseudo range and carrier phase, respectively. In contrast to GPS/BDS/Galileo, GLONASS uses frequency division multiplexing (FDMA--frequency division multiple access) to make the signals from individual satellites distinguishable. Because of FDMA, GLONASS carrier phase and pseudo-range observations suffer from inter-channel biases (ICBs). The effects of receiver ICBs are different from channel to channel of the same receiver and cannot be eliminated by the basic DD technique [28]. Additionally, receiver ICBs in code and carrier phase measurements are different. However, the GLONASS AR process is more complicated compared to GPS because of the FDMA signal structure. At least one GLONASS satellite participates in the DD observation, the wavelengths are no longer identical, and Equation (2) can no longer be simplified in this way without losing the integer characteristic of the ambiguity terms. Therefore, single difference terms remain in the DD observation equation. Forming DD carrier phase observations between users and satellites \(i\) and \(j\), the GLONASS observation equation becomes  
\[\
abla\Delta P_{i}=\
abla\Delta\rho+\ \
abla\Delta\frac{K}{f_{i}^{2}}+\ \
abla\Delta T+\
abla\Delta d_{i,P}+\ \
abla\Delta \varepsilon_{i,P} \tag{3}\]  
\[\phi_{i}=\
abla\Delta\rho+\ \lambda_{i}^{j}\
abla\Delta N+\Big{(}\lambda_{i} ^{j}-\ \lambda_{i}^{r}\Big{)}\Delta N^{r}-\ \
abla\Delta\frac{K}{f_{i}^{2}}+\ \
abla\Delta T+\
abla\Delta d_{i,\phi}+\
abla\Delta \varepsilon_{i,\phi} \tag{4}\]  
In the equations, \(\Delta\)--Single difference operator--ICBs for two receivers on adjacent GLONASS pseudo-range and carrier phase observation (meter). The majority of studies reveal that receivers of the same type have almost the same ICBs. As a consequence, ICBs can be eliminated by a single difference between satellites [28]. However, ICBs should be calibrated before the AR of GLONASS for the relative positioning with different types of receivers [29]. In this study, we focus only on the short-range baseline, and most of the ionospheric delay and the tropospheric delay can be eliminated by DD. Consequently, Equations (1) and (2) can be rewritten as  
\[\
abla\Delta\mathrm{P_{i}}=\
abla\Delta\rho+\
abla\Delta\varepsilon_{i,P} \tag{5}\]\[\
abla\Delta\lambda_{i}\phi_{i}=\
abla\Delta\rho+\lambda_{i}\
abla\Delta N_{i}+ \
abla\Delta\epsilon_{i,\phi} \tag{6}\]  
The general form of the linear observation equation on GNSS carrier phase observations can be re-expressed as [3]  
\[AX+BN+\epsilon=L \tag{7}\]  
where \(L\) denotes the DD observation vector, \(N\) is the DD carrier phase integer ambiguity vector (\(N\in Z^{n}\)), \(X\) is the vector of other unknown parameters (including position coordinates), \(\epsilon\) is the random errors, and the matrices \(A\) and \(B\) are the corresponding design matrices. Note that we only use the carrier phase observation to estimate the float solution of the rover in this paper.