## 2 The Phase-Locked Loop and Normalized Fokker-Planck Equation  
In this section, a brief review on the FP equation and the related application issues to PLLs is provided. Detailed discussion can be referred to references [1; 2; 30; 31; 32; 33; 30; 31]. The procedure is computationally efficient, and readily provides numerical results describing the phase-error process, the modulo-2\(\pi\) phase-error process, and the cycle-slipping statistics. The basic PLL configuration contains a phase detector (PD), for detecting the phase error between the output and the input through feedback system; a loop filter; and a voltage-controlled oscillator (VCO), which is the oscillator to be synchronized by adjusting the phase difference. The mathematical block diagram of the phase-locked loop is shown as in Figure 1. The transfer function \(F(s)\) represents the loop filter of a tracking loop. The loop filter is usually a low-pass filter used to suppress the noise and high-frequency signal components from the phase-detector/delay discriminator and provide a dc-controlled signal for the VCO. A first-order tracking loop is obtained when \(F(s)=1\). The equation describing the loop operation is the first-order differential equation given as [1; 2; 3; 4; 10; 30]  
\[\frac{d\phi(t)}{dt}=(\omega-\omega_{o})-K[A\sin\phi(t)+n^{\prime}(t)] \tag{1}\]  
where \(\phi=\Phi_{i}-\Phi\) is the error between the phase angle of the incoming signal and the output of the VCO, \(K\) is the total loop gain, \(A\) is the input signal voltage amplitude, \(\omega_{o}\) is the quiescent frequency of  
Figure 1: Mathematical block diagram of the phase-locked loop (PLL). The basic PLL configuration contains a phase detector (PD), for detecting the phase error between the output and the input through feedback system; a loop filter; and a voltage-controlled oscillator (VCO). The loop filter \(F(s)\) is usually a low-pass filter used to suppress the noise and high-frequency signal components from the phase-detector/delay discriminator and provide a dc-controlled signal for the VCO [1; 2; 3]. the VCO, and \(n^{\prime}(t)\) is a stationary white Gaussian noise (WGN) process. Assume that the input signal to the PLL is a pure sinusoid with the average power of \(A^{2}\) Watt, a constant frequency \(\omega\) rad/s, and that the signal is corrupted by a stationary white Gaussian noise with a single-sided power spectral density \(N_{0}\) Watt/Hz. For a continuous Markov process, the instantaneous probability density function (PDF) of the phase error \(\phi\) at time \(t\), \(p(\phi,t)\), satisfies the partial differential equation  
\[\frac{\partial p(\phi,t)}{\partial t}=-\frac{\partial}{\partial\phi}[A(\phi)p (\phi,t)]+\frac{1}{2}\frac{\partial^{2}}{\partial\phi^{2}}[B(\phi)p(\phi,t)] \tag{2}\]  
with the initial condition  
\[p(\phi,0)=\delta(\phi-\phi_{o}) \tag{3}\]  
where \(\phi_{o}\) is the initial value of \(\phi\); \(A(\phi)\) and \(B(\phi)\) are normalized moments of the infinitesimal increment [10]. The equation that describes the stochastic behavior of the phase error \(\phi(t)\) of the first order PLL is the Fokker-Planck equation  
\[\frac{\partial p(\phi,t)}{\partial t}=\frac{\partial}{\partial\phi}[f(\phi,t )p(\phi,t)]+\frac{K^{2}N_{0}}{4}\frac{\partial^{2}p(\phi,t)}{\partial\phi^{2}} \tag{4}\]  
where  
\[f(\phi,t)=AK\sin\phi(t)+(\omega_{o}-\omega) \tag{5}\]  
After substituting Equation (5) into Equation (4), the Fokker-Planck equation that describes the PDF \(p(\phi,t)\) of the Markov process phase error \(\phi(t)\) can be written as  
\[\frac{\partial p(\phi,t)}{\partial t}=\frac{\partial}{\partial\phi}[(AK\sin \phi+\omega_{o}-\omega)p(\phi,t)]+\frac{K^{2}N_{0}}{4}\frac{\partial^{2}p(\phi,t)}{\partial\phi^{2}} \tag{6}\]  
Although the equation is linear in \(p\), the complete solution for \(p(\phi,t)\) is somewhat complicated by the nonlinear behavior of the variable coefficients [10; 14].