## 2 Principle
### Analysis of Anti-Dispersion Ability
However, this situation will become complicated under the influence of chromatic dispersion. According to the simulation results, the larger the CP inserted, the better the anti-dispersion ability of the AO-OFDM based on the Gauss-shaped filter and Sinc-shaped filter. In Ref. [8], the ICI and Inter-Symbol Interference (ISI) of the AO-OFDM system based on the Gauss-shaped filter were not affected by dispersion, and the frequency domain waveform of the Gauss-shaped filter is discussed in this article. Compared with Ref. [8], it was closer to a Sinc-shaped filter, and we already made a formula derivation, so we will not discuss the relationship between the ICI, ISI, and CP of the system without dispersion. We will mainly discuss why the AO-OFDM system based on the Gauss-shaped filter was more resistant to dispersion. The spatio-temporal complex envelope (STCE) of a single symbol period of the kth subcarrier of the AO-OFDM signal along the fiber at position \(z\) and STCE at time \(t\) satisfies NLSE [18]:  
\[\frac{\partial g_{k}(z,t)}{\partial z}-j\frac{\beta^{\prime\prime}}{2}\frac{ \partial^{2}g_{k}(z,t)}{\partial t^{2}}+\frac{\alpha}{2}g(z,t)=j\gamma|g_{k}( z,t)|^{2}g_{k}(z,t),t\to t-\beta^{\prime}z \tag{22}\]  
where \(\alpha\) is the loss coefficient, \(\gamma\) is the nonlinear coefficient, and \(\beta^{\prime}\equiv\partial\beta(\omega)/\partial\omega\) is the first derivative of the transmission constant, which reflects that the difference in group velocity between sub-carriers caused the walk-off between carriers. As the sub-carrier spacing in the AO-OFDM system was small and the deviation was not obvious, the envelope was time-shifted with the group velocity during analysis, and only the second derivative of the transmission constant, that is, the influence of the dispersion on the system, was considered. \(\beta^{\prime\prime}\equiv\partial^{2}\beta(\omega)/\partial\omega^{2}\) is the second derivative of the transmission constant, also known as the second-order dispersion effect, which is one of the main causes of signal envelope distortion and is also the main factor affecting the signal transmission quality of the AO-OFDM system mentioned in Section 1. If only considering the influence of chromatic dispersion on the signal, Equation (22) can be expressed as:  
\[\frac{\partial g_{k}(z,t)}{\partial z}-j\frac{\beta^{\prime\prime}}{2}\frac{ \partial^{2}g_{k}(z,t)}{\partial t^{2}}=0,t\to t-\beta^{\prime}z \tag{23}\]  
Transforming Equation (23) into the frequency domain can be expressed as:  
\[j\frac{\partial G_{k}(z,\Omega_{k})}{\partial z}=\frac{1}{2}\beta^{\prime \prime}\Omega^{2}G_{k}(z,\Omega_{k}) \tag{24}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**CP** & **0.09** & **0.17** & **0.23** & **0.29** & **0.33** \\ \hline _f\({}_{b}\)/f\({}_{s}\)_ & 1.1 & 1.2 & 1.3 & 1.4 & 1.5 \\ \hline ICI(0) & 0.144 & 0.071 & 0.014 & 0.028 & 0.054 \\ ICI(\(\pm T_{s}\)) & 0.049 & 0.047 & 0.044 & 0.042 & 0.040 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relationship between CP and adjacent sub-carrier ICI of the AO-OFDM system based on a Gauss-shaped filter. In Equation (24), \(\Omega_{k}=2\pi(f-f_{k})\), then the frequency domain waveform of the signal after the transmission distance \(z\) is solved as:  
\[G_{k}(z,\Omega_{k})=G_{k}(0,\Omega_{k})exp(-\frac{\dot{j}}{2}\beta^{\prime\prime }{\Omega_{k}}^{2}z) \tag{25}\]  
\[g_{k}(z,t)=\int_{-\infty}^{\infty}g_{k}(0,\Omega_{k})exp(-\frac{\dot{j}}{2} \beta^{\prime\prime}{\Omega_{k}}^{2}z)exp(j2\pi ft)df \tag{26}\]  
The AO-OFDM system pulse-shaping function \(g^{\prime}{}_{cp}\)(0,\(t\)) based on Gauss-shaped filtering is substituted into Equation (26) to obtain the pulse-shaping function \(g^{\prime}{}_{cp}\)(z,\(t\)) after its pulse-shaping symbol is subjected to dispersion.