# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 7 Validating Linear Model and Averaging Effects  
In this section, we validate the augmented linear model tracking error calculation, and the effects of PLL bandwidth and coherent averaging time discussed in Section 6, using numerically simulated data. A simulation is used because we can isolate the effect of AWGN (and not have to worry about clock phase-noise).  
Figure 6: Effect of \(T_{co}\) and PLL bandwidth on tracking error transfer function, 1 ms.  
Figure 7: Effect of \(T_{co}\) and PLL bandwidth on tracking error transfer function, 20 ms.  
An intermediate frequency (IF) signal with AWGN (similar to that which would be output from a receiver front-end) is numerically generated and used as the input to the software defined receiver (SDR). The generated IF is 5 MHz and is sampled at 20 MHz, which makes it identical to the experimental data which will be analyzed in Section 9; however, it does not contain the C/A code and navigation data. The SDR used to track the numerically generated signal is based on that provided with [14].  
Figure 8 shows the standard deviation of tracking error versus PLL bandwidth for both theory and simulation with a carrier-to-noise (\(C/N_{0}\)) ratio of 45.5 dB-Hz. A relatively high \(C/N_{0}\) is used because the linear model was derived assuming small levels of noise. However, in the following section, it will be shown that the tracking error computation is valid for much smaller \(C/N_{0}\), corresponding to interference conditions. Coherent average times of 1 ms and 20 ms are shown. Theory matches simulation very well at this relatively high \(C/N_{0}\). However when \(T_{co}=20\) ms, there is a slight deviation between the two as \(\sigma_{e}\) begins to increase with increasing bandwidth. This increase in standard deviation has been noted in prior work [6], and it does not affect our theoretical conclusions since it is not caused by AWGN. It is caused by the PLL bandwidth approaching \(\frac{1}{2T_{co}}\), the bandwidth of the coherent averaging, which is leading to loop stability issues. The results in Figure 8 clearly show that for a given \(T_{co}\) the tracking error variance, \(\sigma_{e}^{2}\), is not a function of the PLL bandwidth. However, increasing the coherent average time significantly reduces \(\sigma_{e}^{2}\). Since only AWGN is considered, this shows the best performance possible, i.e., with a perfect oscillator.  
Now, we can look at the standard deviation of the phase error, \(\sigma_{\phi}\). The true phase error can easily be computed by differencing the PLL output from the simulated IF phase, which allows a cycle slip to be easily detected. A cycle slip occurs if the true phase error reaches one cycle, or 360 degrees.  
Figure 9 shows the same curves as Figure 8, except with the phase error standard deviation, \(\sigma_{\phi}\), on the y-axis. In contrast to the previous results, \(\sigma_{\phi}\) is a strong function of PLL bandwidth. On the other hand, coherent averaging time produces little effect (ignoring the bandwidth phenomenon associated with a 20 ms coherent averaging time). Additionally, the thermal noise term from the jitter metric (Equation (39)) is plotted for 1 ms and 20 ms averaging times. Note that the jitter metric curves coincide with our theoretical calculation using a 1 ms averaging time.  
Figure 8: Sigma tracking error vs. PLL bandwidth.  
Figures 8 and 9 have validated the augmented PLL linear model derived in Section 3, and confirmed a non-obvious effect indicated from our study of the theory (Section 6); tightening bandwidth does reject noise on the phase error, but has no effect on the tracking error. This turns out to be an important issue, as we will discuss next.