## 2 Principles of the Proposed Method
### CD Estimation Principle Based on FrFT
#### 2.2.3 CD and Fractional Fourier Transform  
It is well known that CD in the optical fiber link will cause the optical signal to become a chirp signal, that is to say, the CD is understood as a chirp signal in the frequency domain, so the parameter estimation method introduced above can be analogous to the CD estimation.  
According to the EC effect of the chirp signal in the fractional domain, we can calculate the CD value of the optical fiber communication system under different working conditions. For the effect of CD, there is also a specific transform order Q, which can be expressed as follows, by analogy with the chirp parameter estimation formula:  
\[\beta_{2}z=-\frac{(N-1)dt^{2}}{2\pi}\tan(Q\frac{\pi}{2}) \tag{14}\]  
where \(z\) is the optical fiber transmission length, and \(N\) is the optical signal length.  
Combined with the CD coefficient expression, Equation (14) can be transformed into:  
\[D_{\lambda}z=(N-1)dt^{2}\frac{c}{\lambda^{2}}\tan(Q\frac{\pi}{2}) \tag{15}\]  
where \(D_{\lambda}z\) is the cumulative CD of the fiber link at different operating wavelengths.  
Table 2 lists the comparison between the chirp signal in the time domain and the CD signal in the frequency domain, including expression, specific FrFT transform order and the relationship between order and parameters to be estimated.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Search Interval**} & **Sampling Points** & \multirow{2}{*}{**Optimal Order**} & **Estimation Error (Hz/s)** & **Average Error Rate** \\ \hline \multirow{4}{*}{0.01} & 256 & 1.51 & 0.6268 & \multirow{4}{*}{1.6991} \\  & 512 & 1.71 & 0.0246 & \\  & 1024 & 1.85 & 0.7564 & \\  & 2048 & 1.92 & \(-\)1.3108 & \\ \hline \multirow{4}{*}{0.001} & 256 & 1.506 & 0.1193 & \multirow{4}{*}{0.2686} \\  & 512 & 1.71 & 0.0246 & \\ \cline{1-1}  & 1024 & 1.847 & \(-\)0.0735 & \\ \cline{1-1}  & 2048 & 1.923 & 0.2123 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance of the chirp parameter estimation under different sampling points and search intervals.  
After the FrFT with order \(Q\) is performed on the signal, the inter symbol interference (ISI) originally caused by CD can be minimized in the transformed fractional domain. This shows that the group velocity delay introduced by CD does not exist in the specific fractional transform domain, so the signal characteristics in the fractional domain are similar to those in the time domain without CD. In order to find a specific fractional domain transform order suitable for CD monitoring, the EC function mentioned above is still used in this section.  
For optical fiber signals with different modulation formats, their EC may be slightly different. For example, Figure 4 shows the EC performance of PM-QPSK and PM-16 QAM signals under different CD. It can be seen from the Figure that the PM-QPSK and PM-16 QAM signals have downward spikes under the action of FrFT, and different CD values in the optical fiber link correspond to different transform orders. Therefore, the CD values of different optical fiber signals can be monitored according to this characteristic.