## 4 Numerical Implementation and Analysis of the Phase-Locked Loop Statistics
The histogram skewed in such a way that its right side (or \"tail\") was longer than its left side. Again, for the density function \(p\), there were local peaks appearing with a period of \(2\pi\) rad. It should be noticed that the position of global peak of \(p\), which initially appeared at the phase error near by zero, moved rightwards as time progressed. As can be seen, the peak position appeared at \(2\pi\) rad away from its original position when \(\tau\) exceeded a certain value, e.g., \(\tau=10\) in this illustration. According to Equation (10), if the solution is properly resolved, the total probability remains constant (=1) for all time. The cumulative probability distribution can be used to check the result. As compared to the actual total probability, the numerically calculated cumulative probabilities  
for various normalized time are summarized in Table 1. The errors using the numerical approach was of the order of 0.02% to 0.5% depending on the various \(\tau\), for \(\gamma=0\) and \(\gamma=\sin(\pi/4)\), respectively. For better resolution that is required, a smaller grid can be utilized. Figure 2: The time-evolution of the nonmodulo-\(2\pi\) (**a**) density function \(g(\phi,\tau)\) and (**b**) PDF \(p(\phi,\tau)\), of the phase error for the signal-to-noise ratio (SNR) \(\alpha=1\) (\(\gamma=0\)). Figure 4: Three-dimensional plots for nonmodulo-\(2\pi\) transition \(g\) and \(p\), respectively, for SNR \(\alpha=1\) as time progresses: front view (**left**) and back view (**right**) (\(\gamma=0\)). Figure 3: The time-evolution of the nonmodulo-\(2\pi\) cumulative probability distribution of the phase error for SNR \(\alpha=1\) (\(\gamma=0\)). Figure 5: The time-evolution of the nonmodulo-\(2\pi\) (**a**) density function \(g(\phi,\tau)\) and (**b**) PDF \(p(\phi,\tau)\), of the phase error for SNR \(\alpha=1\) (\(\gamma=\sin(\pi/4)\)). Figure 6: The time-evolution of the nonmodulo-\(2\pi\) cumulative probability distribution of the phase error for SNR \(\alpha=1\) (\(y=\sin(\pi/4)\)). Figure 7: Three-dimensional plots for the nonmodulo-\(2\pi\) transition \(g\) and \(p\), respectively, for SNR \(\alpha=1\) as time progresses: front view **(left)** and back view **(right)**\(\gamma=\sin(\pi/4)\)). In the next example, an illustration of how the SNR \(\alpha\) influences the information geometry of the phase-error statistics is presented. Figures 8-10 provide the results of the nonmodulo-\(2\pi\) statistics at normalized time \(\tau=10\) for SNR \(\alpha\) in the range 0.3-1 where the detuning factor \(\gamma=0\) was utilized. The results are shown to capture the effect of different SNR values and understand the information changes on the error-phase statistics of PLL. Figure 8 presents the phase-error density function \(g(\phi,\tau)\) and the PDF \(p(\phi,\tau)\) and Figure 9 shows the corresponding cumulative probability distribution of \(p(\phi,\tau)\). The numbers beside individual curves indicate the various SNR values. Again, three-dimensional plots for nonmodulo-\(2\pi\)\(g\) and \(p\), respectively, are given in Figure 10. As in the presented case where \(\gamma=0\) was utilized, both the density functions \(g\) and \(p\) remained symmetric with respect to the phase-error axis as \(\alpha\) varied. As \(\alpha\) decreased, the global peak values of the density functions at \(\phi=0\) decreased and thus the initial narrow density functions became 'flatter', implying that the variance the phase error increased, and the possibility of a larger phase error increased. As functions of \(\phi\), the local peak positions appeared at a period of \(2\pi\) rad when moving outwards, with the peak amplitudes monotonically nonincreasing. Figures 11-13 elucidated information similar to that of Figures 8-10, and the parameter values were identical except that the detuning factor \(\gamma=\sin(\pi/4)\) was utilized. It can be seen that the density functions \(g\) and \(p\) virtually stretched (or expanded) to one direction where they stretched to opposite directions.