## 4 Theoretical Analysis of the Influence of AL on Coherent/Non-Coherent OOK Reception
As the optical bandwidths of the ASE and AL are often larger than the optical bandwidth of the system, it is considered that the ASE and AL noise have the same FWHM after filtering. The light output by the EDFA enters the PIN and is converted into the current signal, as is shown in the following  
\[ \begin{split}& i_{proxok}(t)=RGs^{2}(t)P_{S}+RGS_{altn}\delta \overset{M}{\underset{i=-M}{\sum}}\overset{M}{\underset{j=-M}{\sum}}\cos(2\pi( i-j)\delta vt+\varphi_{i}-\varphi_{j})+\\ & RS_{ase}\delta\overset{M}{\underset{l=-M}{\sum}}\overset{M}{ \underset{n=-M}{\sum}}\cos(2\pi(l-n)\delta vt+\varphi_{l}-\varphi_{n})+\\ & 2 RGs(t)\sqrt{P_{S}S_{alln}\delta\overset{M}{\underset{i=-M}{\sum}}} \overset{M}{\underset{i=-M}{\sum}}\cos([2\pi i\delta v)t+\Delta\varphi_{iS}]+ \\ & 2R(t)\sqrt{P_{S}S_{ase}\delta\overset{M}{\underset{l=-M}{\sum}}} \overset{M}{\underset{l=-M}{\sum}}\overset{M}{\underset{l=-M}{\sum}}\cos( 2\pi(i-l)\delta vt+\varphi_{i}-\varphi_{l})+i_{S}+i_{T}\end{split} , \tag{37}\]  
where \(i_{S}\) represents the shot noise current; \(i_{T}\) represents the thermal noise current; \(n\) represents the serial number of each frequency interval; and \(\Delta\varphi_{Sase}=\varphi_{S}(t)-\varphi_{l}\). The first term in Equation (37) represents the communication signal that directly transmits the information. The main noise was caused by the cross terms. The fourth term in Equation (37) represents the signal light-AL beat noise and its current variance can be obtained by its SSPSD and the detector noise bandwidth as expressed as  
\[\sigma_{IS-altn}^{2}=2R^{2}G^{2}P_{S}S_{altn}B_{e}, \tag{38}\]  
The fifth term in Equation (37) represents the signal light-ASE beat noise and its current variance can also be obtained by its SSPSD and the detector noise bandwidth as expressed as  
\[\sigma_{IS-ase}^{2}=2R^{2}GP_{S}S_{ase}B_{e}, \tag{39}\]  
The second term in Equation (37) represents the AL-AL beat noise. When \(i=j\), a total of \(2M\) DC terms are added to obtain Equation (40). \[i_{alt-altn}^{DC}=RGS_{altn}\delta\overset{M}{\underset{l}{\sum}}2M=RGS_{ aln}B_{o}=RGP_{altn}, \tag{40}\]  
where \(P_{altn}\) is the AL power. For each negative frequency component, there is always a corresponding positive frequency component: all negative frequencies correspond to the positive frequency range \((0,B_{o})\). The corresponding positive frequency component multiplied by two is equivalent to the PSD multiplied by four. According to probability statistics, the PSD of the AL-AL beat noise decreases linearly from zero frequency to \(B_{o}\), and its shape is a triangle. The SSPSD close to zero frequency is expressed as  
\[S_{altn-altn}(\delta\overset{M}{\underset{l}{\sum}})=4\times(RGS_{altn} \delta\overset{M}{\underset{l}{\sum}}-1)\times\frac{1}{2}/\delta\overset{M}{ \underset{l}{\sum}}=2R^{2}G^{2}S_{altn}^{2}B_{o}=2R^{2}G^{2}S_{altn}P_{altn}, \tag{41}\]  
The SSPSD of other frequencies can be expressed as  
\[S_{altn-altn}(f)=2R^{2}G^{2}S_{altn}^{2}(B_{0}-f), \tag{42}\]  
The SSPSD of AL-AL beat noise is shown in Figure 11.  
According to the noise bandwidth \(B_{e}\) of the detector and the SSPSD of the AL-AL light beat noise, the AL-AL noise current variance is expressed as  
\[\sigma^{2}_{alth-altn}=\int\limits_{0}^{B_{e}}S_{alth-altn}(f)df=R^{2}G^{2}S_{alth} ^{2}B_{e}(2B_{o}-B_{e}), \tag{43}\]  
The third term in Equation (37) represents the ASE-ASE beat noise. According to the analysis of the SSPSD of the AL-AL beat noise, the ASE-ASE beat noise current variance is expressed as  
\[\sigma^{2}_{lse-ase}=R^{2}S_{ase}^{2}B_{e}(2B_{o}-B_{e}), \tag{44}\]  
The sixth term in Equation (37) represents the AL-ASE beat noise. According to the analysis of the SSPSD of the AL-AL beat noise, the AL-ASE beat noise current variance is expressed  
\[\sigma^{2}_{lahn-ase}=4R^{2}GS_{alth}S_{ase}B_{e}(2B_{o}-B_{e}), \tag{45}\]  
The SSPSD and current variances of different noise for OOK signals containing AL received non-coherently with the preamplifier have been presented in a previous study.