## 2 Principle
### Analysis of Anti-Dispersion Ability
When \(m\) is equal to \(k\), that is, when the target subcarrier passes through the corresponding Sinc-shaped filter with the same central frequency, the pulse-shaping function is \(s_{kk}(t)_{(m=k)}\). The other sub-carriers will also pass through the filter, and the pulse-shaping function is \(s_{mk}(t)_{(m\
eq k)}\) after filtering. \(e\!f\! (\cdot)\) is the error function and \(e\!f\! (\cdot)\) is the imaginary error function. The sub-carriers that are different from the center frequency of the target filter will inevitably cause inter-carrier crosstalk to the target sub-carrier. Figure 4 shows the pulse-shaping functions \(s_{kk}(t)_{(m=k)}\), \(s_{mk}(t)_{(m\
eq k)}\), \(s^{\prime}_{kk}(t)_{(m=k)}\), and \(s^{\prime}_{mk}(t)_{(m\
eq k)}\) of the two systems with 0 CP inserted and the same subcarrier symbol rate at the receiving end after passing the same Sinc-shaped filter. These four pulse-shaping functions can clearly reflect the crosstalk between carriers after passing the filter. For convenience of comparison, we normalized the four pulse-shaping functions. We discuss the situation of transmitting \"1\" in an ideal state and receiving \"1\" after scaling at the receiving end. In the following analysis of the influence of chromatic dispersion on the AO-OFDM system, we also performed the same analysis on these four pulse-shaping functions. The red and blue dashed lines in Figure 4 respectively indicate the pulse-shaping function of a certain sub-carrier of the AO-OFDM system using two filters after passing through the corresponding Sinc-shaped filter with the same center sub-carrier. The red and blue dashed lines respectively indicate the crosstalk of adjacent subcarriers suffered by a single period of a certain subcarrier of the AO-OFDM system using two filters. The red lines represent using a Sinc-shaped filter and the blue lines represent the use of a Gauss-shaped filter. When \(t=0\), that is, at the optimal sampling point in each symbol period, the filtered AO-OFDM system was sampled, the symbol \(C_{kk}\) carried by the target subcarrier was obtained, and the crosstalk \(e_{(m-k)}\) from other carriers was also obtained. \[e_{(m-k)}=\frac{j}{2}\cdot exp\left\{-\left[2(m-k)^{2}{f_{b}}^{2}\right]/{f_{s }}^{2}\right\}\left\{erfi\left[\frac{4(m-k)^{2}{f_{b}}^{2}-j\pi{f_{s}}}{2\sqrt {2}{f_{s}}}\right]-erfi\left[\frac{4(m-k)^{2}{f_{b}}^{2}+j\pi{f_{s}}}{2\sqrt{ 2}{f_{s}}}\right]\right\} \tag{21}\]  
The carrier interference (ICI) generated by the \(m\)th sub-carrier passing through the \(k\)th filter, which is the target filter, is shown as the red and blue solid lines. The red and blue triangles in Figure 4 indicate the ICI of the best sampling point in the current symbol period and the best sampling point in the adjacent symbol period. It can be clearly seen in Figure 4 that the AO-OFDM system using a Sinc-shaped filter was not subject to crosstalk between adjacent carriers, while the AO-OFDM system using a Gauss-shaped filter was  
Figure 4: Under ideal conditions, the two AO-OFDM systems received the pulse-shaping functions of the corresponding sub-carrier and adjacent sub-carriers at the receiving end of the filter. subject to crosstalk between carriers. Table 1 shows the influence of CP size and ICI on the current symbol period and adjacent symbol period of the AO-OFDM system based on a Gauss-shaped filter. The CP size in the first row of Table 1 corresponds to the ratio of the sub-carrier spacing to the symbol rate after the CP is inserted in the second row. The third row in the table indicates the crosstalk of adjacent sub-carriers to the current sub-carrier at the optimal sampling point \(t=0\). The fourth row represents the crosstalk of adjacent subcarriers to the current subcarrier for the two adjacent sampling points \(t=\pm T_{s}\) of the optimal sampling point. We can find from Table 1 that the crosstalk between carriers was the smallest when the inserted CP size was 0.23. The size of ICI was not directly proportional to the size of CP.