# Practical Use of Starlink Downlink Tones for Positioning
## 2 Signal Detection and Tracking
### Tones Detection and Tracking
In order to reduce noise and produce an accurate estimate of the phase and frequency of the input signal, the phase error is then filtered by a second order loop filter, \(F_{pll}(s)=\left(\omega_{0p}^{3}+1.1\omega_{0p}^{2}s+2.4\omega_{0p}s^{2}\right) /s^{2}\), where \(\omega_{0p}\) is the natural frequency of the PLL that is related to the PLL equivalent noise bandwidth \(\omega_{0p}=\left(0.7845\right)^{-1}\)\(B_{0p}\); and the frequency error is filtered by a first order loop filter \(F_{pll}(s)=\left(\omega_{0f}^{2}+1.41\omega_{0f}s\right)/s\), where \(\omega_{0f}\) is the natural frequency of the FLL that is related to the FL  
Figure 4: Ratio between the tone spectrum peak and the detection threshold as a function of the duration of the FFT burst, T, and for different C/No levels. A tone dynamic of 5 kHz/s is assumed. Figure 3: Principle of the tone detection and tracking. \(\omega_{0f}=(0.53)^{-1}\)\(B_{0f}\). In the implementation, both loops share the same integrators. The FLL frequency error is injected one integrator back from the PLL, since the frequency error is in units of hertz while the phase error is in unit of cycles. The filter output is then integrated into the NCO (\(F_{nco}(s)=1/s\)), which adjusts the phase and frequency of the local carrier so that they match the phase and the frequency of the incoming tone. The NCO output is then mapped to _cosine_ and _sine_ functions, which represent an accurate replica of the incoming signal carrier. In general, since it has a large pull-in range (i.e., \(1/T\)), the FLL is used at the beginning of the tracking process when the frequency error is large; then, after achieving frequency lock, both loops are activated until phase lock is achieved. At this time, the FLL is deactivated and only the PLL is used to keep precise carrier phase tracking. However, for applications with high dynamic stress, the FLL must be maintained over the whole tracking time. The FPLL is traditionally used to track L-band GNSS signal carriers that have a Doppler shift rate of 1 Hz/s (order of magnitude for a static user). We will show that it works for tracking LEO satellite signals in the Ku-band, which have a Doppler shift rate of a few kHz/s. Successful locking of the tracking loops is achieved by keeping the resulting tracking error of the PLL and FLL within the operational ranges of their detectors. The tracking error is caused by thermal noise, dynamic stress, and for PLLs, oscillator-related errors. The discussion below will focus on the first two errors. The error due to thermal noise is inversely proportional to both the integration time, T, and to the received signal carrier-to-noise-density ratio C/No. It is also directly proportional to the equivalent noise bandwidth \(B_{0}\), while the dynamic stress depends only on \(B_{0}\). Thus, \(B_{0}\) and T have to be designed such that the tracking loop remains stable in the intended application environment (i.e., dynamic stress and C/No). Below, we provide the values of \(B_{0}\) and T and explain the reasoning behind the chosen values. First, a third-order-PLL and a second-order-FLL are sensitive to jerk (i.e., \(1/\lambda\times d^{3}R/dt^{3}\) (Hz/s\({}^{2}\)), where \(R\) is the range between the satellite and the user antennas). For a jerk of 70 Hz/s\({}^{2}\) (Table 1), the dynamic stress error of a third-order-PLL is \(70/\left(B_{0f}/0.7845\right)^{3}\) [cycle], and that of a second-order-FLL is \(70/\left(B_{0f}/0.53\right)^{2}\) [Hz]. When we use signals of opportunity of LEO satellites, and appropriate values of \(B_{0p}\) and \(B_{0f}\), such tracking errors are small compared to the error induced by the inaccurate orbit and clock, as we will see later. Note that by increasing \(B_{0}\), the dynamic stress error is reduced, however the phase and frequency thermal noise errors increase. On the other hand, the integration time T must satisfy a compromise between noise reduction (i.e., large value of T) and the ability to capture high frequency errors (i.e., short value of T).