## 4 Numerical Implementation and Analysis of the Phase-Locked Loop Statistics
As shown in this example, \(g\) had a left-skewed distribution, while \(p\) had a right-skewed distribution. The density \(p\) shows that the peak values appeared at the phase error near by \(2\pi\) rad. As \(\alpha\) decreased, \(p\) tended to be flatter, and the variance of the phase error increased. Three-dimensional plots of information geometry were provided such that more comprehensive understanding could be gained on the relationship among the parameters, \(p/q\), \(\phi\), and \(\alpha\). Figure 8: Nonmodulo-2\(\pi\) phase-error (**a**) density function \(g(\phi,\tau)\) and (**b**) PDF \(p(\phi,\tau)\), at \(\tau=10\) for various SNR values (\(\gamma=0\)). Table 1.0 Three-dimensional plots for nonmodulo-\(2\pi\,g\) and \(p\), respectively, at \(\tau=10\) for various SNR values: front view (**left**) and back view (**right**; \(\gamma=0\)). Table 1.1 Nonmodulo-\(2\pi\) phase-error cumulative probability distribution at \(\tau=10\) for various SNR values (\(\gamma=0\)). Table 1.2 Nonmodulo-2\(\pi\) phase-error (**a**) density function \(g(\varphi,\tau)\) and (**b**) PDF \(p(\phi,\tau)\), at \(\tau=10\) for various SNR values (\(\gamma=\sin(\pi/4)\)). Table 1.3 Nonmodulo-\(2\pi\) phase-error cumulative probability distribution at \(\tau=10\) for various SNR values (\(\gamma=\sin(\pi/4)\)). Table 1.4 Three-dimensional plots for nonmodulo-\(2\pi\)\(g\) and \(p\), respectively, at \(\tau=10\) for various SNR values: front view (**left**) and back view (**right**; \(\gamma=\sin(\pi/4)\)). Table 1.5 Nonmodulo-2\(\pi\) phase-error (**a**) density function \(g(\phi,\tau)\) and (**b**) PDF \(p(\phi,\tau)\), at \(\tau=10\) for various detuning factor \(\gamma\) when SNR \(\alpha=1\). In the following discussion, the modulo-\(2\pi\) PDF, \(P(\phi,\tau)\) (denoted by a capital letter \(P\)) as well as the cumulative probability distribution, and the cycle slipping density function \(q(\phi,\tau)\) of the phase error for the first-order PLL are presented. The time-evolution of modulo-\(2\pi\) phase-error PDF \(P(\phi,\tau)\)  
Table 1.6 Three-dimensional plots for nonmodulo-\(2\pi\)\(g\) and \(p\), respectively, at \(\tau=10\) for various detuning factors \(\gamma\) when SNR \(\alpha=1\): front view (**left**) and back view (**right**). Table 1.7 Nonmodulo-\(2\pi\) phase-error cumulative probability distribution at \(\tau=10\) for various detuning factors \(\gamma\) when SNR \(\alpha=1\). and cumulative probability distribution is shown in Table 1.8 In this illustrative example, the SNR \(\alpha=1\) and detuning factor \(\gamma=0\) were utilized. The numbers beside individual curves indicate the normalized times, switching \(\tau\) from 0.625 to 5. The steady state was reached when the normalized time \(\tau\geq 5\). Since the total time steps for simulation were set as 1000, the grid spacing in the time marching direction was \(\Delta\tau=\tau/1000\), which is dependent on the normalized time \(\tau\). As time \(\tau\) progressed, the peak value of \(P(\phi,\tau)\) at zero phase error (\(\phi=0\)) was decreasing, while remaining symmetric with respect to the phase-error axis. Figure 18 provides the comparison on variation of modulo-\(2\pi\) phase-error steady state PDF \(P(\phi)\) and cumulative probability distribution function for various detuning factor \(\gamma\) where SNR \(\alpha=1\) was utilized. The numbers beside individual curves indicate the detuning factors in the range 0-\(\sin(\pi/2)\). It can be seen that, initially with a symmetric shape, \(P(\phi)\) became the left-skewed distribution. The peak moved rightwards with its amplitude decreasing as \(\gamma\) increased. Figure 19 provides the variation of modulo-\(2\pi\) phase-error steady state PDF and cumulative probability distribution function, respectively, of the phase error at \(\tau=0.625\) for the SNR values \(\alpha\) in the range 0-1 in the case of \(\gamma=0\). The numbers beside individual curves indicate the SNR values \(\alpha\).