## 2 Principle
### Analysis of Anti-Dispersion Ability
\[g^{\prime}_{cp}(0,t)=\sqrt{\frac{\pi}{2}}\cdot exp(-\frac{\pi^{2}t^{2}}{2T_{s} ^{2}}) \tag{27}\]  
\[g^{\prime}_{cp}(z,t)=\sqrt{\frac{\pi}{2}}\cdot(\frac{T_{s}^{2}}{T_{s}^{2}+j \beta^{\prime\prime}\pi^{2}z})^{\frac{1}{2}}exp(-\frac{\pi^{2}t^{2}}{2(T_{s} ^{2}+j\beta^{\prime\prime}\pi^{2}z)}) \tag{28}\]  
\[g^{\prime}_{cp}(z,t)=(1+{z^{2}}/{L_{D}}^{2})^{\frac{1}{2}}exp[-\frac{\pi^{2}t ^{2}}{2T_{s}^{2}(1+{z^{2}}/{L_{D}}^{2})^{2}}]exp[j\varphi(z,t)]\]  
\[L_{D}=T_{S}^{2}/(|\beta^{\prime\prime}|\pi^{2})\]  
\[\varphi(z,t)=\frac{\beta^{\prime\prime}\pi^{2}z}{2T_{s}^{2}(1+{z^{2}}/{L_{D}} ^{2})^{2}}\cdot\frac{\pi^{2}t^{2}}{T_{s}^{2}}-\frac{1}{2}\text{arctan}(\frac{ \beta^{\prime\prime}\pi^{2}z}{T_{s}^{2}}) \tag{29}\]  
After completing Equation (28), the intensity and phase of the pulse-shaping function can be obtained, where \(L_{D}\) is the dispersion length. We found that the pulse-shaping function produced by the Gauss-shaped filter was still a Gaussian function, even after transmission through the optical fiber, with only broadening and phase changes, and the degree of distortion was much smaller than the pulse-shaping function produced by the Sinc-shaped filter. When the transmission distance was much smaller than the dispersion length, there would be no inter-symbol crosstalk due to dispersion. The pulse-shaping function produced by the Sinc-shaped filter was affected by dispersion. The function is difficult to express with mathematical formulas. We will analyze it from Figures 5 and 6. Figure 5 shows the pulse-shaping function \(g(t)\) of an AO-OFDM signal based on a Sinc-shaped filter affected by dispersion after being transmitted in optical fibers of different lengths. The symbol period was 100 ps, \(\beta^{\prime\prime}=-20\text{ps}^{2}/\text{km}\). We can see from the change in the pulse-shaping function after being affected by different accumulative dispersions that chromatic dispersion had a great influence on the pulse-shaping function of the symbol, and these distortions were irregular. At \(t\) = 100 ps, the amplitude of the optimal sampling point in the adjacent period changed drastically. Even if the accumulative dispersion was 200 ps/nm, the ISI was already close to 0.1. And the size of ISI did not change regularly with the transmission distance. We can see from Figure 5 that, as the influence of the dispersion on the system became greater and greater, the energy in each symbol period leaked from the current period to other periods more and more. Figure 6 is the pulse-shaping function based on the Gauss-shaped filter as the comparison group of Figure 5 under the influence of chromatic dispersion, which transmitted different distances in the optical fiber. From Equation (29) and Figure 6, we can see that the intensity of the pulse-shaping function generated by the Gauss-shaped filter expanded as the influence of the dispersion increased. Compared with the pulse-shaping function based on the Sinc-shaped filter, the influence of the dispersion at the three sampling points of \(t\) = 0, \(-T_{s}\)(\(-\)100 ps), and \(T_{s}\)(100 ps)\({}_{\text{s}}\) was small, and the pulse-shaping function did not have a large distortion. Figure 7 shows the pulse-shaping function of a certain sub-carrier through the Sinc-shaped filter at the receiving end after being affected by 800 ps/nm accumulative dispersion and the pulse-shaping function of adjacent carrier crosstalk. Similar to Figure 4, we normalized the received signal. The symbol period was 100 ps, CP = 0.33, and the second-order dispersion coefficient was \(\beta^{\prime\prime}=-20\)ps\({}^{2}\)/km. The red triangle to the right and the blue triangle to the left in Figure 7 represent the ICIs of the best sampling points of the two AO-OFDMs in the current symbol period and adjacent cycles. We can see that the crosstalk between the carriers of the AO-OFDM system based on the Sinc-shaped filter was much greater than the crosstalk between the carriers of the AO-OFDM system based on the Gauss-shaped filter.