## Appendix  
According to Equation (33), if the discriminator is perfectly normalized and the front-end bandwidth is infinite, the variance of the discriminator output is  
\[\sigma^{2}_{EML}=(Var[E^{2}_{I}-L^{2}_{I}]+Var[E^{2}_{Q}-L^{2}_{Q}])/k^{2}_{d} \tag{11}\]  
where \(Var[x]\) is the variance of \(x\). \(k_{d}\) is the gain of the discriminator which is  
\[\frac{\partial}{\partial\tau}[(E^{2}_{I}+E^{2}_{Q})-(L^{2}_{I}+L^{2}_{Q})]=4P( 1-\tau_{b}) \tag{12}\]  
Taking \(Var[E^{2}_{I}-L^{2}_{I}]\) as an example, the variance is  
\[Var[E^{2}_{I}-L^{2}_{I}]=Var[E^{2}_{I}]+Var[L^{2}_{I}]-2 cov(E^{2}_{I},L^{2}_{I}) \tag{13}\]  
where  
\[Var[E^{2}_{I}]=4 PR^{2}(\tau-k\tau_{b}-\tau_{b})cos^{2}\Delta \varphi\cdot E[n^{2}_{I,E}]+2(E[n_{I,E}])^{2}\] \[Var[L^{2}_{I}]=4 PR^{2}(\tau-k\tau_{b}+\tau_{b})cos^{2}\Delta \varphi\cdot E[n^{2}_{I,L}]+2(E[n_{I,L}])^{2}\] \[cov(E^{2}_{I},L^{2}_{I})=4 PR^{2}(2\tau_{b})cos^{2}\Delta\varphi \cdot E[n_{I,E}n_{I,L}]+E[n^{2}_{I,E}n^{2}_{I,L}]-E[n^{2}_{I,E}]E[n^{2}_{I,L}]\]  
\(cov(x,y)\) is the covariance between \(x\) and \(y\). According to Equations (2) and (3), the noise components of the in-phase correlation outputs \(E_{I}\) and \(L_{I}\) are as below. The noise components of quadrature correlation outputs are more or less the same and the only difference is the local carrier multiplied is sine wave instead of the cosine wave of the in-phase branches.  
\[n_{I,E}=\frac{1}{N_{c}}\sum_{i=0}^{N_{c}-1}n(t_{i})c(t_{i}-k\tau_{b}-\tau_{b}) cos(2\pi(f_{IF}+f_{k})t_{i}+\varphi) \tag{14}\]  
\[n_{I,L}=\frac{1}{N_{c}}\sum_{i=0}^{N_{c}-1}n(t_{i})c(t_{i}-k\tau_{b}+\tau_{b}) cos(2\pi(f_{IF}+f_{k})t_{i}+\varphi) \tag{15}\]  
The product of \(n_{I,E}\) and \(n_{I,L}\) is  
\[\begin{split} n_{I,E}n_{I,L}=&\frac{1}{N_{c}^{2}} \sum_{i=0}^{N_{c}-1}\sum_{j=0}^{N_{c}-1}[n(t_{i})c(t_{i}-k\tau_{b}-\tau_{b})n( t_{j})c(t_{j}-k\tau_{b}+\tau_{b})\\ &\times cos(2\pi(f_{IF}+f_{k})t_{i}+\varphi)cos(2\pi(f_{IF}+f_{k} )t_{j}+\varphi)]\end{split} \tag{16}\]  
The mean value of \(n^{2}_{I,E}n^{2}_{I,L}\) is\[E[n_{I,E}^{2}n_{I,L}^{2}]=E[(\frac{1}{N_{c}^{2}}\sum_{i=0}^{N_{c}-1} \sum_{j=0}^{N_{c}-1}n(t_{i})c(t_{i}-k\tau_{b}-\tau_{b})n(t_{j})c(t_{j}-k\tau_{b}+ \tau_{b})\] \[\times cos(2\pi(f_{IF}+f_{k})t_{i}+\varphi)cos(2\pi(f_{IF}+f_{k})t _{j}+\varphi))^{2}]\] \[= E[\frac{1}{N_{c}^{2}}\sum_{i=0}^{N_{c}-1}\sum_{j=0}^{N_{c}-1}[n^ {2}(t_{i})c^{2}(t_{i}-k\tau_{b}-\tau_{b})n^{2}(t_{j})c^{2}(t_{j}-k\tau_{b}+ \tau_{b})\] \[\times cos^{2}(2\pi(f_{IF}+f_{k})t_{i}+\varphi)cos^{2}(2\pi(f_{ IF}+f_{k})t_{j}+\varphi)]]\] \[+2E[\frac{1}{N_{c}^{2}}\sum_{i=0}^{N_{c}-1}\sum_{j=0}^{N_{c}-1}[n ^{2}(t_{i})c(t_{i}-k\tau_{b}-\tau_{b})c(t_{i}-k\tau_{b}+\tau_{b})n^{2}(t_{j})c (t_{j}-k\tau_{b}-\tau_{b})c(t_{j}-k\tau_{b}+\tau_{b})\] \[\times cos^{2}(2\pi(f_{IF}+f_{k})t_{i}+\varphi)cos^{2}(2\pi(f_{ IF}+f_{k})t_{j}+\varphi)]]\] \[= (1+2R^{2}(2\tau_{b}))\cdot N_{0}/2T_{c}\cdot N_{0}/2T_{c} \tag{10}\]  
Supposing that \(\tau\) is equal to \(k\tau_{b}\) which is the best case that the grid parameter of the peak is that of the incoming signal, \(Var[E_{I}^{2}-L_{I}^{2}]\) and \(Var[E_{Q}^{2}-L_{Q}^{2}]\) are  
\[Var[E_{I}^{2}-L_{I}^{2}]=8 PR(\tau_{b})cos^{2}\Delta\varphi(1-R(2\tau_{b}))\cdot N _{0}/2T_{c}+4(1-R^{2}(2\tau_{b}))\cdot N_{0}/2T_{c}\cdot N_{0}/2T_{c} \tag{11}\]  
\[Var[E_{Q}^{2}-L_{Q}^{2}]=8 PR(\tau_{b})sin^{2}\Delta\varphi(1-R(2\tau_{b})) \cdot N_{0}/2T_{c}+4(1-R^{2}(2\tau_{b}))\cdot N_{0}/2T_{c}\cdot N_{0}/2T_{c} \tag{12}\]  
Hence,  
\[\sigma_{EML}^{2}=\frac{\tau_{b}}{2\cdot C/N_{0}\cdot T_{c}}(1+\frac{1}{C/N_{0 }\cdot T_{c}(1-\tau_{b})}) \tag{13}\]