# Analysis of MTF in TDI-CCD Subpixel Dynamic Super-Resolution Imaging by Beam Splitter
## Appendix A Derivation Details of Function (2)
### Derivation Details of \(MTF_{mT}\)  
Wigner-Seitz function of TDI-CCD \(w(x,y)\) is equal to \(\frac{4}{p^{2}}\) when intervals of \(x\) direction are:  
\[\begin{array}{ll}-(p-\sum\limits_{i=1}^{2n-1}a_{i})/2\,\sim\,a_{1}/2,-a_{1}/2 \,\sim\,a_{2}/2,-a_{2}/2\,\sim\,a_{3}/2,\cdots,-a_{2n-1}/2\,\sim(p-\sum\limits _{i=1}^{2n-1}a_{i})/2\end{array}\]  
The intervals of y direction are:  
\[\begin{array}{ll}-(p-\sum\limits_{i=1}^{2n-1}b_{i})/2\,\sim\,b_{1}/2,-b_{1}/2 \,\sim\,b_{2}/2,-b_{2}/2\,\sim\,b_{3}/2,\cdots,-b_{2n-1}/2\,\sim(p-\sum\limits _{i=1}^{2n-1}b_{i})/2\end{array}\]In other cases, it is 0. Therefore, its Fourier transformation is:  
\[\begin{array}{ll}W_{mT}(u,v)&=\int\frac{\pi_{n}}{2}\int\limits_{-(p-\sum\limits_ {i=1}^{2n-1}a_{i})/2}^{\frac{n}{2}}\int\limits_{-(p-\sum\limits_{i=1}^{2n-1}b_{ i})/2}^{\frac{4}{p}}\exp(i2\pi fx)dxdy+\int\limits_{-\frac{\pi_{n}}{2}}^{\frac{n}{2}} \int\limits_{-\frac{\pi_{n}}{2}}^{\frac{n}{2}}\frac{4}{p^{2}}\exp(i2\pi fx)dxdy \\ &+\cdots\int\frac{\pi_{n-1}}{2}\int\limits_{-\frac{\pi_{n}}{2}}^{\frac{n}{2n-1} }\int\limits_{-\frac{\pi_{n}}{2}}^{\frac{n}{2n-1}}\frac{4}{p^{2}}\exp(i2\pi fx )dxdy+\int\limits_{-\frac{\pi_{n}}{2}}^{(p-\sum\limits_{i=1}^{2n-1}a_{i})/2} \int\limits_{-\frac{\pi_{n}}{2}}^{(p-\sum\limits_{i=1}^{2n-1}b_{i})/2}\frac{4 }{p^{2}}\exp(i2\pi fx)dxdy\end{array} \tag{13}\]  
The real part of \(W_{mT}(u,v)\) is:  
\[\begin{array}{ll}\mathrm{Re}(W_{mT}(u,v))&=\{\frac{p-\sum\limits_{i=1}^{2n- 1}a_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1}^{2n-1}a_{i})]+\frac{a_{i}}{p}\sin c [\pi ua_{1}]\}\cdot\\ &\{\frac{p-\sum\limits_{i=1}^{2n-1}b_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1}^{2 n-1}b_{i})]+\frac{b_{i}}{p}\sin c[\pi ub_{1}]\}\\ &+\{\frac{a_{i}}{p}\sin c(\pi ua_{1})+\frac{a_{i}}{p}\sin c(\pi ua_{2})\}\cdot \{\frac{b_{i}}{p}\sin c(\pi ub_{1})+\frac{b_{i}}{p}\sin c(\pi ub_{2})\}+\cdots \\ &+\{\frac{a_{i}p}{p}\sin c(\pi ua_{2n-1})+\frac{a_{i}p-2}{p}\sin c(\pi ua_{2n-2 })\}\\ &\{\frac{b_{i}p-1}{p}\sin c(\pi ub_{2n-1})+\frac{b_{i}p-2}{p}\sin c(\pi ub_{2n -2})\}\\ &+\{\frac{p-\sum\limits_{i=1}^{2n-1}a_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1} ^{2n-1}a_{i})]+\frac{a_{2n-1}}{p}\sin c(\pi ua_{2n-1})\}\cdot\\ &\{\frac{p-\sum\limits_{i=1}^{2n-1}b_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1} ^{2n-1}b_{i})]+\frac{b_{n-1}}{p}\sin c(\pi ub_{2n-1})\}\end{array} \tag{14}\]  
Therefore:  
\[\begin{array}{ll}MTF_{mT}&=F\{w(x,y)*p(x,y)\}\\ &=P(u,v)\cdot\mathrm{Re}(W_{mT}(u,v))\\ &=\sin c(\pi u\cdot pM\tan\theta)\cdot\sin c(\pi v\cdot pMd)\\ &\cdot\{\{\frac{p-\sum\limits_{i=1}^{2n-1}a_{i}}{p}\sin c[\pi u(p-\sum\limits_ {i=1}^{2n-1}a_{i})]+\frac{a_{i}}{p}\sin c[\pi ua_{1}]\}\cdot\{\frac{p-\sum \limits_{i=1}^{2n-1}b_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1}^{2n-1}b_{i})]+ \frac{b_{i}}{p}\sin c[\pi ub_{1}]\}\\ &+\{\frac{a_{i}}{p}\sin c(\pi ua_{1})+\frac{a_{i}}{p}\sin c(\pi ua_{2})\}\cdot \{\frac{b_{i}}{p}\sin c(\pi ub_{1})+\frac{b_{i}}{p}\sin c(\pi ub_{2})\}+\cdots \\ &+\{\frac{a_{i}p-\sum\limits_{i=1}^{2n-1}a_{i}}{p}\sin c(\pi ua_{2n-2})\}\cdot \{\frac{b_{i}}{p}\sin c(\pi ub_{2n-1})+\frac{b_{i}p-2}{p}\sin c(\pi ub_{2n-2})\} \\ &+\{\frac{p-\sum\limits_{i=1}^{2n-1}a_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1} ^{2n-1}a_{i})]+\frac{a_{2n-1}}{p}\sin c(\pi ua_{2n-1})\}\cdot\\ &\{\frac{p-\sum\limits_{i=1}^{2n-1}b_{i}}{p}\sin c[\pi u(p-\sum\limits_{i=1} ^{2n-1}b_{i})]+\frac{b_{2n-1}}{p}\sin c(\pi ub_{2n-1})\}\end{array} \tag{15}\]