# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 5 Phase Jitter Performance Metric  
Up until this point, we have derived the linear model and defined PSDs for both AWGN and clock phase-noise. Now, we introduce the traditionally used jitter performance metric, and some issues in experimental results in prior work.  
A commonly used test for analyzing tracking performance in the presence of dominant sources of error is often referred to as the total phase jitter rule. This rule states that the \"three-sigma jitter must not exceed one-fourth of the phase pull-in range of the PLL discriminator\" ([12], p. 184). For an inverse-tangent discriminator with an pull-in range of 180 degrees, this metric can be expressed (in units of degrees) as,  
\[3\sigma_{PLL}=3\sigma_{\phi}+\theta_{e}\leq 45\quad\text{degrees} \tag{38}\]  
where \(\sigma_{PLL}\) is the total phase jitter, \(\sigma_{\phi}\) is the standard deviation of phase error (from all sources, including clocks and thermal noise), and \(\theta_{e}\) is the phase error caused by dynamic stresses. Since this work focuses on stationary receivers, dynamic stresses will be neglected.  
Nominal contributions to phase jitter typically include satellite and receiver clock errors, and thermal noise [5; 18]. The variance of phase error, \(\sigma_{\phi}^{2}\), for a stationary receiver is defined in Equation (28). Again, note that this variance corresponds to the phase error, \(\phi\), in Figure 2.  
The contribution due to additive white noise, \(\sigma_{\phi,n}^{2}\), derived in [20], can be expressed as,  
\[\sigma_{\phi,n}^{2}=\frac{B_{n}}{(C/N_{0})}\left[1+\frac{1}{2T_{co}(C/N_{0})}\right] \tag{39}\]  
where, \(B_{n}\) is the single sided PLL bandwidth, \(C/N_{0}\) is the carrier-to-noise ratio, and \(T_{co}\) is the coherent integration time. Equation (39) would be equivalent to Equation (29), except for the second term in brackets (in Equation (39)) which accounts for squaring loss. In fact, by using Equations (27) and (36),  
Figure 3: Receiver clock power spectral density (PSD) comparison.  
Equation (29) can be reduced to the ratio of \(B_{n}\) to \(C/N_{0}\). Therefore, there is no squaring loss when an arctangent discriminator is used. This agrees with the result in [6].  
The phase noise due to oscillator instabilities, \(\sigma^{2}_{\phi,\text{clk}}\), is split into components due to satellite and receiver clocks as,  
\[\sigma^{2}_{\phi,\text{clk}}=\sigma^{2}_{\phi,n_{\text{tr}}}+\sigma^{2}_{\phi, n_{\text{tr}}} \tag{40}\]  
with,  
\[\sigma^{2}_{\phi,n_{\text{tr}}} =\int_{0}^{\infty}|1-H(j2\pi f)|^{2}S_{\text{sv}}(f)\,df \tag{41}\] \[\sigma^{2}_{\phi,n_{\text{tr}}} =\int_{0}^{\infty}|1-H(j2\pi f)|^{2}S_{\text{rx}}(f)\,df \tag{42}\]  
where \(S_{\text{sv}}(f)\) and \(S_{\text{rx}}(f)\) are the single-sided phase error PSDs for the satellite and receiver clocks, respectively [18].  
The difference between the standard deviation of phase jitter and the standard deviation of tracking error (our proposed metric) can be seen in Figure 2. The phase jitter corresponds to the true phase error \(\phi\), while the tracking error, denoted as \(\epsilon\), is defined as the discriminator's estimate of the phase error.