# Analysis of 5G and LTE Signals for Opportunistic Navigation and Time Holdover
## 3 Results
### Scale-Dependent Stability  
While it is typical to express uncertainty levels in a navigation observable using a standard deviation at an arbitrary observation interval, within the domain of timing and time stability, it is nearly universal that the instability should be expressed at a specific measurement interval or period. The reason for this is that clocks are systems from which measurements may be used or could be useful over several orders of magnitude of measurement intervals, from precision interval measurement at fractions of a second to extended time hold overs for periods of days or longer. Depending on which noise processes are present in the device under test, their relative intensity, the observation interval, and the  
Figure 4: (**a**) The delta between the two Telenor towers was bounded to a 1.25-m peak-to-peak range over the course of one hour, but exhibited step changes; (**b**) the delta between a Telenor and Telia tower reached a 3.5 m peak-to-peak range over the hour. level of timing uncertainty can also span orders of magnitude--even when expressed as a fraction of the interval rather than as an absolute time error. Typically, over'short' time scales, a given timing source will be dominated by white measurement noise, meaning that measuring for longer or averaging a longer interval results in a monotonically decreasing uncertainty with a slope near \(-1\):1 when plotted on a log-log graph of instability versus time. Within some system-dependent averaging intervals, however, one or more other noise processes will come to dominate the result, typically starting with flicker noise from system electronics, or in random walk phases or frequency noises, which make further averaging counterproductive. One of the traditional ways of expressing this time-scale-dependent uncertainty is the Allan variance [31], which is calculated using Equation (1):  
\[\sigma_{y}^{2}(\tau)=\sum_{i=1}^{M-1}[x_{i+1}-x_{i}]^{2}, \tag{1}\]  
The non-overlapped Allan variance or deviation has been generally superseded by the overlapped modified Allan deviation or the Hadamard deviation due to their better use of data, as well as due to their insensitivity to common time-keeping device errors, which are trivially compensated such as in constant linear drifts. Here, we have elected to use the modified Allan deviation [14] as it is better suited to exposing the transition between flicker phase noise and other processes. It was implemented via Equation (2) and applied to the phase measurements in units of meters. \[\text{Mod}\ \sigma_{y}^{2}(\tau)=\frac{1}{2m^{2}\tau^{2}(N-3m+1)}\sum_{j=1}^{N-3m+1 }\Bigl{(}\sum_{t=j}^{j+m-1}[x_{t+2m}-2x_{t+m}+x_{i}]\Bigr{)}^{2} \tag{2}\]  
When the differential cell measurement time series, shown in Figure 4a,b, were converted to the modified Allan deviations plotted in Figure 5, the time domain variation was converted into a time-scale-dependent uncertainty parameter, where different noise processes manifested at different time scales and in different proportions. A counterintuitive result immediately emerged showing that the pair that produced the far worse (larger) peak-to-peak variation was, in fact, more stable at most time scales than the less variable station pair. By inspecting these curves, we can extract very useful information about the relative stabilities of the measurements produced, as well as how these vary with increasing observation intervals or measurement averaging. In Figure 5, the series that had the worse peak-to-peak variation (Figures 4b and 5b) was statistically more stable over all measurement time scales that were less than 9 s of interrogation, except for a small region around a 1 Hz measurement. Meanwhile, both stations showed a 'knee' in their noise  
Figure 5: (**a**) Phase measurement instability between two Telenor cells; (**b**) phase measurement instability between one Telenor cell and one Telia cell. performances around 10\({}^{-2}\) s or 100 Hz update rates.