## 3 Distribution of FLD Output \(C_{2\varphi}\)
### C\({}_{2\varphi}\) Distribution under Frequency Lock  
When frequency locked loop is in lock and \(C/N_{0}\) is high, we can make the following assumptions:  
\[\exp(-\frac{C}{N_{0}}T_{coh})\approx 0 \tag{8}\]  
\[\cos^{2}(\varphi-\Delta\theta)=1-\sin^{2}(\varphi-\Delta\theta) \approx 1-(\varphi-\Delta\theta)^{2} \tag{9}\] \[\int\limits_{-\infty}^{\infty(\varphi-\Delta\theta)\sqrt{2\frac{C}{ N_{0}}T_{coh}}}e^{-\frac{a^{2}}{2}}dx\approx\sqrt{2\pi} \tag{10}\]  
Based on the previous assumptions, (7) can be simplified to  
\[f(\varphi)\approx\frac{1}{\sqrt{2\pi*(\frac{1}{2T_{coh}C/N_{0}})}}*\exp(-\frac {(\varphi-\Delta\theta)^{2}}{2*(\frac{1}{2T_{coh}C/N_{0}})})|\varphi|\leq\pi \tag{11}\]  
where \(\varphi\) follows Gaussian distribution with the mathematical expectation \(\Delta\theta\) and variance \(1/(2T_{coh}\cdot\frac{C}{N_{0}})\) (\(\sigma^{2}=1/(T_{coh}\cdot\frac{C}{N_{0}})\)). Then, we can get the probability distribution function of \(\Delta\varphi\) shown below. When considering noise of \(i\)th and \((i-1)\)th are independent with each other, \(\varphi_{i}\) and \(\varphi_{i-1}\) are independent of each other, while \(\Delta\theta_{i}\) and \(\Delta\theta_{i-1}\) (mathematical expectation of \(\varphi_{i}\) and \(\varphi_{i-1}\)) are approximately equal to zero under frequency lock and interval is small. Thus, we can calculate the mathematical expectation \(E\{\Delta\varphi_{i}\}\) and variance \(VAR\{\Delta\varphi_{i}\}\), and probability distribution function of \(\Delta\varphi\), which are shown as follows:  
\[E\{\Delta\varphi_{i}\}=E\{\varphi_{i}\}-E\{\varphi_{i-1}\}=\Delta\theta_{i}- \Delta\theta_{i-1}\approx 0 \tag{12}\]  
\[VAR\{\Delta\varphi_{i}\}=VAR\{\varphi_{i}-\varphi_{i-1}\}=VAR\{\varphi_{i}\} +VAR\{\varphi_{i-1}\}=\sigma^{2} \tag{13}\]  
\[f(\Delta\varphi_{i})=\int\limits_{-\infty}^{+\infty}f(\Delta\varphi_{i}- \varphi_{i-1},\varphi_{i-1})d\varphi_{i-1}=\frac{1}{\sqrt{2\pi}\sigma}\exp(- \frac{\Delta\varphi_{i}^{2}}{2\sigma^{2}}) \tag{14}\]  
The distribution characterization of FLD output represents a Gaussian stochastic process through a cosine system. We can get the mathematical expectation and variance of C\({}_{2\varphi}\) in (6) as follows:  
\[\begin{array}{ll}E\{C_{2\varphi}\}&=E\{\frac{1}{M}\sum\limits_{i=1}^{M}\cos (2\Delta\varphi_{i})\}\\ &=\frac{1}{M}\sum\limits_{i=1}^{M}\int\limits_{-\infty}^{\infty}\cos(2\Delta \varphi_{i})p(\Delta\varphi_{i})d\Delta\varphi_{i}\\ &\approx\frac{1}{\sqrt{2\pi}\sigma}\int\limits_{-\infty}^{\infty}\cos(2\Delta \varphi_{i})\exp(-\frac{\Delta\varphi_{i}^{2}}{2\sigma^{2}})d\Delta\varphi_{i} \\ &=\exp(-2\sigma^{2})\end{array} \tag{15}\]  
\[\begin{array}{ll}VAR\{C_{2\varphi}\}&=VAR\{\frac{1}{M}\sum\limits_{i=1}^{M} \cos(2\Delta\varphi_{i})\}\\ &=\frac{1}{M^{2}}\sum\limits_{i=1}^{M}VAR\{\cos(2\Delta\varphi_{i})\}\\ &=\frac{E\{\cos^{2}(2\Delta\varphi)\}-\frac{E^{2}\{\cos(2\Delta\varphi)\}}{M} \\ &=\frac{1}{M}\frac{1}{\sqrt{2\pi}\sigma}\int\limits_{-\infty}^{\infty}\cos^{2}(2 \Delta\varphi)\exp(-\frac{\Delta\sigma^{2}}{2\sigma^{2}})d\Delta\varphi-\frac {1}{M}\exp(-4\sigma^{2})\\ &=\frac{1}{2M}(1+e^{-8\sigma^{2}})-\frac{1}{M}e^{-4\sigma^{2}}\end{array} \tag{16}\]  
When the signal is locked, \(\Delta\varphi_{i}\) is stable around zero. Therefore, \(C_{2\varphi}\) approximates to  
\[C_{2\varphi}=\frac{1}{M}\sum\limits_{i=1}^{M}\cos(2\Delta\varphi_{i})\approx 1- \frac{2}{M}\sum\limits_{i=1}^{M}\Delta\varphi_{i}^{2} \tag{17}\]  
For simplification, we can assume a new variable \(Y\) related with \(C_{2\varphi}\), which nearly follows the chi-square distribution  
\[Y=\frac{(1-C_{2\varphi})M}{2\sigma^{2}}=\sum\limits_{i=1}^{M}{(\frac{\Delta \varphi_{i}}{\sigma})}^{2}\sim\chi^{2}(M) \tag{18}\]  
According to (18), the probability distribution function of FLD output is shown as:  
\[f_{C_{2\varphi}}(c_{2\varphi})=\frac{M}{2\sigma^{2}}f_{Y}(\frac{M-Mc_{2\varphi }}{2\sigma^{2}})\quad|c_{2\varphi}|\leq 1 \tag{19}\]  
where \(f_{Y}(y)\) is the probability distribution function of the random variable \(Y(Y\sim\chi^{2}(M))\).