# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 6 Bandwidth and Averaging Time Effects  
This section explains how PLL bandwidth and coherent averaging time affect the magnitude of the input-output and tracking error transfer functions used in Equations (26) and (29).  
Figure 4 shows the effect of PLL bandwidth on the _input-output_ transfer function for a given coherent averaging time \(T_{co}=1\) ms. The two curves represent PLL bandwidths of 1 and 10 Hz. At higher frequencies, the effect of coherent averaging (represented by the sinc function) is unmistakable, with the first lobe of the sinc function visible at about 1000 Hz. However, since the input-output transfer function is already a low-pass filter, there is little additional attenuation due to the coherent averaging (because the PLL noise bandwidth is much smaller than the bandwidth of the coherent averaging operation). In contrast, tightening the PLL bandwidth has a large impact on the transfer function magnitude curve, and results in a reduction in the integral of the transfer function (provided in the figure legend). Furthermore, note that Equation (29) relates the transfer function integral to the variance of phase error due to AWGN, \(\sigma^{2}_{\phi,n}\). Therefore, tightening PLL bandwidth will reduce \(\sigma^{2}_{\phi,n}\). Figure 5 shows the same curves, but corresponding to a 20 ms coherent averaging time.  
Again, a reduction in PLL bandwidth greatly affects the transfer function and its integral, and therefore the phase error variance, \(\sigma_{\phi,n}^{2}\).  
Now, compare Figures 4 and 5 to one another. For a given PLL bandwidth, increasing the coherent average time from 1 to 20 ms produces very little effect on the transfer function integral (again, given in figure legends) in comparison to the the effect of tightening bandwidth.  
These figures illustrate the conclusions reached using the total phase jitter metric. When considering the effect of only AWGN, \(\sigma_{\phi,n}^{2}\) is the only component of, and thus equivalent to, the total phase jitter metric. Tightening PLL bandwidth should reduce the impact of AWGN on the phase error variance, thus improving the PLL's ability to continuously track carrier phase. Another conclusion, which is stated in [6], is that increasing coherent averaging time would not help improve tracking ability/performance, since it does not significantly affect the phase error variance. While the latter is true, we will show in later sections of this paper that increasing coherent averaging time will definitely improve the PLL's ability to continuously track carrier phase in the presence of AWGN.  
Now, we look at the same pair of figures, but for the _tracking error_ transfer function. These are Figures 6 and 7. The trends are reversed in this case. For a given PLL bandwidth, the averaging time  
Figure 4: Effect of \(T_{co}\) and PLL bandwidth on input–output transfer function, 1 ms.  
Figure 5: Effect of \(T_{co}\) and PLL bandwidth on input–output transfer function, 20 ms.  
significantly affects the transfer function integration, and thus the outcome of Equation (26). However, for a given \(T_{co}\), reducing the bandwidth does not have a significant impact in comparison.  
At higher frequencies, the coherent averaging is attenuating what would otherwise be a high-pass filter (if the standard linear model was used). Again, the first lobe of the sinc function is visible at about 1000 Hz in Figure 6. It is because of the addition of coherent averaging in the linear model that Equation (26) can now be used to compute a realistic value for the tracking error variance due to AWGN.  
The next section shows these insights in graphical form, and validates that the tracking error variance is being accurately computed using the augmented linear model.