# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 4 Receiver Clock Comparison  
The clock phase noise power spectral density (PSD), needed for satellite and receiver clocks in Equations (26) and (29), is modeled using a power law as [5; 13],  
\[S(f)=h_{0}f^{0}+h_{1}f^{-1}+h_{2}f^{-2}+h_{3}f^{-3}+h_{4}f^{-4} \tag{37}\]  
where \(S(f)\) is in units of \(\mathrm{dB}/\mathrm{Hz}\), \(f\) is in Hz, and \(h\) coefficients for several common crystal and atomic oscillators are given in Table 1. Power law coefficents for the temperature compensated crystal oscillator (TCXO) and oven controlled crystal oscillator (OCXO) are taken from [13]. The rubidium model corresponds to the SpectraTime LPFRS-01/AV1 oscillator used when collecting experimental data, and the coefficient computation is detailed in Appendix A.  
The phase noise power spectral densities (PSDs) for these clocks are shown in Figure 3. Note that all PSDs coincide at high frequencies. This is because frequency synthesis is taken into account, and the synthesizer's voltage controlled oscillator (VCO) dominates at higher frequencies. More information on phase noise in frequency synthesizers is available in [16; 17]. While a rubidium clock remains stable (i.e., does not drift) over longer periods of time relative to a crystal oscillator, it does not necessarily have lower phase-noise. This is evident in Figure 3, where the PSD of our rubidium oscillator lies between the TCXO and OCXO. When choosing a reference oscillator, priority should be placed on frequency stability as well as low phase-noise to reduce its effect on the standard deviation of tracking error.  
Prior work [6; 18] has used very conservative methods to compute power law model \(h\) coefficients for space vehicle (SV) clocks. The authors essentially shifted the TXCO PSD curve such that it passes through a single point specified by the GPS Standard Positioning Service (SPS) document [19]. The result is a PSD curve that completely upper bounds that of the TCXO clock. This is extremely conservative considering that satellites are equipped with rubidium (or cesium) atomic oscillators [19]. Therefore, the authors divided the resulting PSD by a factor of 100 to reduce the conservatism. However, this would create a PSD which is lower than the TCXO/OCXO PSD for high frequencies. This is not a realistic PSD curve because, as previously discussed, high-frequency phase-noise is dominated by the VCO in the frequency synthesizer.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \(\mathbf{d}\) & **TCXO** & **OCXO** & **Rubidium** \\ \hline \(h_{0}\) & \(5.0\times 10^{-8}\) & \(5.5\times 10^{-8}\) & \(5.0\times 10^{-8}\) \\ \(h_{1}\) & \(6.2\times 10^{-5}\) & \(5.0\times 10^{-5}\) & \(6.2\times 10^{-5}\) \\ \(h_{2}\) & \(9.6\times 10^{-4}\) & \(6.5\  
Since GPS satellites are equipped with either ceisium or rubidium atomic clocks, we assume that the SV clocks have a PSD representative of a rubidium atomic oscillator. In addition, in this work, we consider atomic clocks for use as receiver reference oscillators because they are becoming increasingly affordable and compact, and have high frequency stability compared to crystal oscillators. Even though this work focuses on stationary users, chip scale atomic clocks (CSACs) are available, and well suited for mobile navigation applications [4; 5].