## Appendix B Correlator Output Variance Derivations
### Data Bits Misaligned by an Integer Number of Code Chips  
In this case, the variance at the output of correlators will be:  
\[ \begin{array}{l}E\Big{\{}\left|y_{k}\right|^{2}\Big{\}}\\ =\int_{s=0}^{\Delta}\int_{t=0}^{T_{b}}\sum\limits_{p=0}^{19}p_{T_{c}}(t- \Delta-pT+T_{b}-vT_{c})e^{j2\pi f(s-t)}dtds\\ +\int_{s=\Delta}^{T}\int_{t=0}^{T_{b}}\sum\limits_{p=0}^{19}p_{T_{c}}(t- \Delta-pT-vT_{c})e^{j2\pi f(s-t)}dtds\end{array}  \tag{126}\]  
whose integration area will be divided into four smaller regions and be dealt with separately to facilitate derivation:  
\[ \begin{array}{l}D_{1}=\{s:\;0\leq s<\Delta,\;\text{mod}(s,T_{c})<CT_{c}\}\\ D_{2}=\{s:\;0\leq s<\Delta,\;\text{mod}(s,T_{c})\geq CT_{c}\}\\ D_{3}=\{s:\;\Delta\leq s<T_{b},\;\text{mod}(s,T_{c})<CT_{c}\}\\ D_{4}=\{s:\;\Delta\leq s<T_{b},\;\text{mod}(s,T_{c})\geq CT_{c}\}\end{array}  \tag{127}\]  
This leads to a four-part integration:\[ \begin{array}{l}E\Big{\{}|y_{k}|^{2}\Big{\}}\\ =\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{1}}}}^{\Lambda}\int_{t=0}^{T_{b}}\sum_{p=0 }^{19}p_{T_{c}}(t-\Delta-pT+T_{b}-vT_{c})e^{j2\pi f(s-t)}dtds\\ +\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{2}}}}^{\Lambda}\int_{t=0}^{T_{b}}\sum_{p= 0}^{19}p_{T_{c}}(t-\Delta-pT+T_{b}-vT_{c})e^{j2\pi f(s-t)}dtds\\ +\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{3}}}}^{T_{s=\,\mathrm{D_{3}}}}\int_{t= 0}^{T_{b}}\sum_{p=0}^{19}p_{T_{c}}(t-\Delta-pT-vT_{c})e^{j2\pi f(s-t)}dtds\\ +\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{4}}}}^{T_{s=\,\mathrm{D_{4}}}}\int_{t= 0}^{T_{b}}\sum_{p=0}^{19}p_{T_{c}}(t-\Delta-pT-vT_{c})e^{j2\pi f(s-t)}dtds\end{array}  \tag{15}\]  
which becomes:  
\[ \begin{array}{l}E\Big{\{}|y_{k}|^{2}\Big{\}}\\ =\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{1}}}}^{\Lambda}\int_{t=0}^{e^{j2\pi f}s }\sum_{p=19-K}^{19}e^{-j2\pi f(\Delta+pT-T_{b}+vT_{c})}ds\\ +\int_{{}_{s<\,\mathrm{D_{2}}}}^{\Lambda}\int_{p=20-K}^{\Delta}e^{-2\pi f( \Delta+pT-T_{b}+vT_{c})}ds\\ +\int_{{}_{s=\Delta}\atop{}_{s<\,\mathrm{D_{3}}}}^{T_{s=\,\mathrm{D_{3}}}}e^{ j2\pi fs}\sum_{p=0}^{18-K}e^{-j2\pi f(\Delta+pT+vT_{c})}ds\\ +\int_{{}_{s<\,\mathrm{D_{4}}}}^{T_{s=\,\mathrm{D_{3}}}}e^{j2\pi fs}\sum_{p=0} ^{19-K}e^{-j2\pi f(\Delta+pT+vT_{c})}ds\end{array}  \tag{16}\]  
The first integration of Equation (16) is:  
\[ \begin{array}{l}\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{1}}}}^{\Lambda}e^{j2 \pi fs}\sum_{p=19-K}^{19}e^{-j2\pi f(\Delta+pT-T_{b}+vT_{c})}ds\\ =e^{j2\pi f/T_{b}}\sum_{p=19-K}^{19}e^{-j2\pi fpT}\int_{s=0}^{\Lambda}e^{j2\pi f (s-\Delta-vT_{c})}ds\\ s\in D_{1}\\ =e^{j2\pi f/T_{b}}\sum_{p=19-K}^{19}e^{-j2\pi fpT}\sum_{n=0}^{K}\int_{nT}^{ \Lambda}e^{j2\pi f(s-\Delta-vT_{c})}ds\\ =e^{j2\pi f/T_{b}}\sum_{p=19-K}^{19}e^{-j2\pi fpT}\sum_{n=0}^{K}e^{j2\pi f(n- K-1)T}\sum_{m=0}^{C-1}\int_{(m-C)T_{c}}^{(m-C+1)T_{c}}e^{j2\pi f(\Delta- \lfloor\alpha/T_{c}\rfloor T_{c})}d\alpha\\ =CT_{c}\frac{\sin(\pi f/T_{c})}{(\pi f/T_{c})}e^{j\pi fT_{c}}\frac{\sin^{2}( \pi f(K+1)T)}{\sin^{2}(\pi f/T)}\end{array}  \tag{17}\]  
The second integration of Equation (16) is:  
\[ \begin{array}{l}\int_{{}_{s=0}\atop{}_{s<\,\mathrm{D_{2}}}}^{\Lambda}e^{j2 \pi fs}\sum_{p=20-K}^{19}e^{-j2\pi f(\Delta+pT-T_{b}+vT_{c})}ds\\ =e^{j2\pi f/T_{b}}\sum_{p=20-K}^{19}e^{-j2\pi fpT}\int_{s=0}^{\Lambda}e^{j2\pi f (s-\Delta-vT_{c})}ds\\ s\in D_{2}\\ =e^{j2\pi f/T_{b}}\sum_{p=20-K}^{19}e^{-j2\pi fpT}\sum_{n=0}^{K-1}\int_{C_{c}+ nT}^{(n+1)T}e^{j2\pi f(s-\Delta-vT_{c})}ds\\ =e^{j2\pi f/T_{b}}\sum_{p=20-K}^{19}e^{-j2\pi fpT}\sum_{n=0}^{K-1}e^{j2\pi f(n -K)T}\int_{(n-K)T}^{(n-K+1)T-CT_{c}}e^{j2\pi f(\Delta-\lfloor\alpha/T_{c} \rfloor T_{c})}d\alpha\\ =e^{j2\pi f/T_{b}}\sum_{p=20-K}^{19}e^{-j2\pi fpT}\sum_{n=0}^{K-1}e^{j2\pi f(n -K-1)T}\cdot(2046-C)\int_{0}^{T_{c}}e^{j2\pi f/\alpha}d\alpha\\ =(2046-C)T_{c}\frac{\sin(\pi f/T_{c})}{(\pi f/T_{c})}e^{j\pi f/T_{c}}\frac{\sin^ {2}(\pi fkT)}{\sin^{2}(\pi fT)}\end{array}  \tag{18}\]The third integration of Equation (A16) is:  
\[