# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 8 Tracking Error as a Design Metric
### Probability Distribution of Tracking Error  
To investigate the role of the discriminator in carrier tracking, the probability distribution of the tracking error is determined analytically and examined for varying carrier-to-noise ratios.  
Equation (18) is used to derive the probability distribution of the tracking error. In this exercise, we assume the phase error \(\phi\) is constant in time. Therefore, only \(n_{I}^{\prime}(t)\) and \(n_{Q}^{\prime}(t)\) are random variables. Equation (18) can be written as,  
\[\epsilon(t)=\tan^{-1}\left(\frac{\phi+\left\langle n_{Q}^{\prime}(t)\right\rangle }{1+\left\langle n_{I}^{\prime}(t)\right\rangle}\right) \tag{43}\]  
with the coherent averaging only affecting the additive noise terms. The argument of the arctangent becomes a ratio of two Gaussian random variables with non-zero means,  
\[Z=\tan^{-1}\left(\frac{Y}{X}\right) \tag{44}\]  
with, \(Y\sim N(\phi\), \(\sigma_{n_{Q}^{\prime}}^{2}/N)\) and \(X\sim N(1\), \(\sigma_{n_{I}^{\prime}}^{2}/N)\), where \(N\) is the number of samples being averaged. Random variables \(X\) and \(Y\) have the same variance (\(\sigma_{n_{I}^{\prime}}^{2}=\sigma_{n_{Q}^{\prime}}^{2}\)), which is proportional to the carrier-to-noise ratio. The distribution of \(Z\) is derived in Appendix C using a result from [21], and is non-zero between \(\pm 90\) degrees.  
Figure 12 shows the probability distribution of the tracking error, \(\epsilon\), for a 1 ms coherent averaging time and (an example) true phase error, \(\phi\), of 5 degrees. The distributions corresponding to three different carrier-to-noise ratios are plotted. The figure legend includes the error with respect to the true phase error of five degrees (using the expected value to estimate the phase error), as well as the standard deviation of the distribution.  
In the absence of interference, for example, a carrier-to-noise ratio of 45.5 dB-Hz, the tracking error has a very Gaussian-like distribution. The true phase error can be accurately estimated from this distribution using the expected value. As the carrier-to-noise ratio is decreased, the tracking error distribution approaches a uniform distribution. This makes it increasingly difficult for the loop filter (modeled by the expected value operator in this example) to obtain a reasonable estimate of the  
Figure 12: PDF of tracking error.  
true phase error. The standard deviation of a uniform distribution over \(\pm 90\) degrees is 52 degrees, which matches with the maximum \(\sigma_{e}\) seen from simulation in Figure 10.  
Examination of Figures 10-12 suggests that the discriminator pull-in region plays an important role in cycle-slipping, and that the jitter metric does not adequately consider this effect.  
While the jitter metric threshold is set based on the discriminator's pull-in region, the metric does not account for the effect of the AWGN (see Figure 2) on the discriminator. As wideband interference levels increase, and \(C/N_{0}\) decreases, the AWGN overwhelms the discriminator and the tracking error distribution approaches a uniform distribution as shown in Figure 12.  
This is likely the reason why the phase jitter metric is not a trustworthy design metric. Our proposed metric based on the tracking error uses a linear model that includes coherent averaging. In addition, the threshold for this new metric is based upon the discriminator's characteristics. Therefore, both coherent averaging as well as the discriminator are accounted for in the tracking error metric.