# Practical Use of Starlink Downlink Tones for Positioning
## 3 Tones Frequencies Aggregation  
In this section, we aggregate the tone measurements of the same satellite in order to cope with the frequent interruption of the measurement on each single tone, and to reduce the noise effect. For a tone transmitted at frequency \(Fo\), the frequency shift measured at the user's location is written  
\[\widetilde{f}^{k}{}_{Fo}\cong f^{k}_{doppler}+f_{rx}-f^{k}_{sv}+f^{k}_{Tr}+ \eta^{k} \tag{1}\]  
\(f^{k}_{doppler}\) is the Doppler shift of the \(k\)th-satellite relative to \(Fo\),  
\(f_{rx}\) is the receiver clock drift (\(\dot{\delta t}\)) multiplied by \(Fo\),  
\(f^{k}_{sv}\) is the \(k\)th-satellite clock drift (\(\dot{\delta t}\)) multiplied by \(Fo\),  
\(f^{k}_{Tr}\) is the troposphere induced frequency shift of the \(k\)th-satellite relative to \(Fo\),  
\(\eta^{k}\) is the measurement error due to thermal noise, interference, dynamic stress and signal propagation through the ionosphere. It is dominated by the thermal noise component if the interference effect is negligible. For a tone transmitted at frequency \(Fo+N\times\Delta F\), the frequency shift measured at the user's location is written:  
\[\widetilde{f}^{k}\cong\left(f^{k}_{doppler}+f_{rx}-f^{k}_{sv}+f^{k}_{Tr} \right)\times\left(1+\frac{N\times\Delta F}{Fo}\right)+N\times\Delta F+\eta^{k} \tag{2}\]  
\(\Delta F=44\) kHz is the frequency spacing between successive tones,  
\(N\) is an integer which can take one of the nine values: \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\),  
\(\eta^{k}\) is the measurement noise. The goal is to estimate N in order to restore the frequency shift of each tone around Fo before aggregating the frequency shifts of all tones of the same satellite by taking their mean value at each epoch. Since \(N\times\Delta F/Fo<1.6\times 10^{-5}\), when we deal with the estimation of N, we can simplify the frequency measurement (2) to:  
\[\widetilde{f^{k}}\cong f^{k}_{doppler}+f_{rx}-f^{k}_{sv}+f^{k}_{Tr}+N\times \Delta F+\eta^{k} \tag{3}\]  
Figure 9: Probability density function of the frequency error of the tones. On the other hand, the predicted Doppler shift relative to measurement \(\widetilde{f}^{k}_{F_{O}}\) can be written:  
\[\widetilde{f}^{k}_{doppler}\cong f^{k}_{doppler}+f^{k}_{O}+f^{k}_{u}+f^{k}_{ \Delta T} \tag{4}\]  
where:  
* \(f^{k}_{doppler}\) is the \(k\)th-satellite Doppler shift (Hz) \[f^{k}_{doppler}=-\frac{1}{\lambda}\ \frac{\left(r-r^{k}\right)^{T}}{\|r-r^{k}\|} \left(v-v^{k}\right)\] (5) \(\lambda=\frac{c}{F_{0}}\) is the carrier wavelength. \(r\) and \(v\) are the user's position and velocity, respectively, in (m) and (m/s). \(r^{k}\) and \(v^{k}\) are \(k\)th-satellite position and velocity, respectively, in (m) and (m/s). \((. )^{T}\)designates the transposition operator. * \(f^{k}_{O}\) is the frequency shift induced by the orbit error for the \(k\)th-satellite (Hz). A rough approximation is \[f^{k}_{O}\approx\frac{1}{\lambda}\ \left(\frac{\left(r-r^{k}\right)^{T}}{\|r-r^{k} \|}\Delta v^{k}+\frac{\left(v-v^{k}\right)^{T}}{\|r-r^{k}\|}\Delta r^{k}\right)\] (6) where \(\Delta r^{k}\) and \(\Delta v^{k}\) are the \(k\)th-satellite position and velocity errors, respectively, in (m) and (m/s). * \(f^{k}_{u}\) is the frequency shift induced by the user position and velocity error for the \(k\)th-satellite. A rough approximation is \[f^{k}_{u}\approx-\frac{1}{\lambda}\ \left(\frac{\left(r-r^{k}\right)^{T}}{\|r-r^{k} \|}\Delta v+\frac{\left(v-v^{k}\right)^{T}}{\|r-r^{k}\|}\Delta r\right)\] (7) where \(\Delta r\) and \(\Delta v\) are the user's position and velocity errors, respectively, in (m) and (m/s). * \(f^{k}_{\Delta T}\) is the frequency shift induced by the timing error for the \(k\)th-satellite (Hz) \[f^{k}_{\Delta T}=\Delta T\times\dot{f}^{k}_{doppler}\] (8) where \(\Delta T\) is the measurement timestamp error (user clock bias), and \(\dot{f}^{k}_{doppler}\) is the Doppler shift rate (Hz/s).