## 3 PPM Probability of Error  
Following previous works [24; 25], the electrical signal that arrives at the PPM demodulator follows a central \(\chi^{2}\) distribution with \(M\) degrees of freedom  
\[p_{0}\left(x;M\right)=\frac{x^{M-1}}{\left(M-1\right)!}\,e^{-x}\,, \tag{1}\]  
during empty slots, and a non-central \(\chi^{2}\) distribution  
\[p_{1}\left(x;M,\lambda\right)=e^{-\left(x+\lambda\right)}\,\left(\frac{x}{ \lambda}\right)^{\frac{M-1}{2}}\,\,I_{M-1}\left(2\sqrt{x\,\lambda}\right)\,, \tag{2}\]  
during the slot that carries the symbol energy. In the last equation \(I_{n}(\cdot)\) denotes the modified Bessel function of the first kind ([29], Equation (8.445)), \(\lambda=E_{s}/N_{0}=E_{b}/N_{0}\,\log_{2}Q\) and \(E_{b}\) is the optical energy per bit at the amplifier output. A symbol is decoded incorrectly if the signal in one or more empty slots is higher than the signal in the energy slot, and the symbol error probability (SEP) equals ([30], Equation (1.1))  
\[SEP=1-\int_{0}^{\infty}\,\left(\int_{v=0}^{x}\,p_{0}\left(v;M\right)\,dv\right) ^{Q-1}\,p_{1}\left(x;M,\lambda\right)\,dx\,. \tag{3}\]  
We use the relation ([29], Equation (8.352/6))  
\[\int_{v=0}^{x}\,p_{0}\left(v;M\right)\,dv=1\,-\,e^{-x}\,\sum_{i=0}^{M-1}\frac{ x^{i}}{i!} \tag{4}\]  
and the SEP becomes  
\[ \begin{split} SEP&=1-\int_{0}^{\infty}\,\left(1\,- \,e^{-x}\,\sum_{i=0}^{M-1}\,\frac{x^{i}}{i!}\right)^{Q-1}\,p_{1}\left(x;M, \lambda\right)\,dx\\ &=\sum_{q=1}^{Q-1}\,\left(\begin{matrix}Q-1\\ q\end{matrix}\right)\left(-1\right)^{q+1}\,\int_{0}^{\infty}\,e^{-q\,x}\, \left(\sum_{i=0}^{M-1}\,\frac{x^{i}}{i!}\right)^{q}\,p_{1}\left(x;M,\lambda \right)\,dx\,.\end{split}  \tag{5}\]  
We now expand the inner sum  
\[\left(\sum_{i=0}^{M-1}\,\frac{x^{i}}{i!}\right)^{q}=\sum_{n=0}^{q\,\left(M-1 \right)}b_{n}^{q}\,x^{n}\,, \tag{6}\]  
where the \(b_{n}^{q}\) coefficients are calculated from the recursive relation  
\[b_{n}^{q}=\sum_{\mu=0}^{\min\{n,M-1\}}\frac{b_{n-\mu}^{q-1}}{\mu!}\,. \tag{7}\]  
The SEP is then given by  
\[SEP\,=\,\sum_{q=1}^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+1}\sum_{n=0}^{q\,\left(M-1 \right)}b_{n}^{q}\,\int_{0}^{\infty}\,e^{-q\,x}\,x^{n}\,p_{1}\left(x;M,\lambda \right)\,dx \tag{8}\]  
and the appearing integral is evaluated from the \(n\)-th derivative of the Laplace transform of the non-central \(\chi^{2}\) distribution ([24], Equation (2))  
\[P_{1}\left(s;M,\lambda\right)=\frac{\exp(-\,\frac{s\,\lambda}{1+s})}{(1+s)^{M }}\,. \tag{9}\]Using the generating function for the Laguerre polynomials \(L_{a}^{b}\left(\cdot\right)\) ([29], Equation (8.975/1)) and the multiplication theorem ([31], Equation (8)), we find that  
\[ \begin{split}\frac{d^{n}}{ds^{n}}\left(\frac{\exp(-\frac{s\lambda}{ 1+s})}{(1+s)^{M}}\right)&=(-1)^{n}\,n!\,\sum_{m=n}^{\infty} \binom{m}{n}\,(-1)^{m-n}\,L_{m}^{M-1}\,(-\lambda)\,\,s^{m-n}\\ &=(-1)^{n}\,n!\,\frac{\exp\left(-\frac{s\,\lambda}{1+s}\right)}{( 1+s)^{n+M}}\,L_{n}^{M-1}\left(\frac{-\lambda}{1+s}\right)\end{split}  \tag{10}\]  
and as a result  
\[\int_{0}^{\infty}\,e^{-q\,x}\,x^{n}\,p_{1}\left(x;M,\lambda\right)\,dx=n!\, \frac{\exp\left(-\frac{q\,\lambda}{1+q}\right)}{(1+q)^{n+M}}\,L_{n}^{M-1}\left( \frac{-\lambda}{1+q}\right)\,. \tag{11}\]  
A finite sum expression for the SEP is now given by  
\[SEP\,=\,\sum_{q=1}^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+1}\sum_{n=0}^{(M-1)}b_{n}^{ q}\,n!\,\frac{\exp\left(-\frac{q\,\lambda}{1+q}\right)}{(1+q)^{n+M}}\,L_{n}^{M-1} \left(\frac{-\lambda}{1+q}\right)\,. \tag{12}\]  
For a more stable numerical calculation we define \(c_{n}^{q}=n!\,b_{n}^{q}\), where the new coefficients obey the recursive relation  
\[c_{n}^{q}=n!\,\sum_{\mu=0}^{\min\{n,M-1\}}\,\frac{c_{n-\mu}^{q-1}}{(n-\mu)!\,n!}\,=\,\sum_{\mu=0}^{\min\{n,M-1\}}\,\binom{n}{\mu}\,c_{n-\mu}^{q-1}\,. \tag{13}\]  
We also expand the Laguerre polynomials as ([29], Equation (8.970/1))  
\[L_{n}^{M-1}\left(\frac{-\lambda}{1+q}\right)=\sum_{i=0}^{n}\,\frac{(n+M-1)!}{ (i+M-1)!\,(n-i)!\,i!}\,\left(\frac{\lambda}{1+q}\right)^{i} \tag{14}\]  
to find that the PPM PER is calculated as  
\[