## 2 PPM Reception
### Ppm Per  
In soft-decision decoding, the demodulator selects the transmitted PPM symbol from the slot with the highest energy value. The PPM symbol error probability (SER) and PER are then given by  
\[SER\left(k,\lambda\right)=1-\int\limits_{0}^{\infty}p_{1}\left(x;k,\lambda \right)\times\,\left[\int\limits_{0}^{x}p_{0}\left(z;k\right)\,dz\right]^{Q-1} \,dx\,, \tag{3}\]  
and  
\[P_{e}\left(k,\lambda\right)=\frac{Q/2}{Q-1}\,SER\left(k,\lambda\right)\,, \tag{4}\]  
respectively. An analytical solution to the integral of (3) is not available to our knowledge for \(\chi^{2}\) noise statistics, except for binary PPM. As it has already been discussed in [6], binary PPM performs worse than on-off keying and higher modulation orders must be considered in pre-amplified systems, which calls for numerical integration or approximation of (3). Numerical integration is feasible, but is also time consuming and therefore may find limited applicability the performance evaluation of OWC systems, where additional integrations are required to take into account the impact of turbulence in the received OSNR. Within this context, an approximation of (3) is more appealing for calculating the average PER in OWC systems. Typically, a union bound approximation is used; however, it proves inaccurate in low OSNRs and we make use of a tighter approximation that has been proposed for Gaussian noise detection [12]. Following the results presented therein, the SER (and therefore PER) is bound by  
\[SER\left(k,\lambda\right)\leq 1-\,\left(\int\limits_{0}^{\infty}p_{1}\left(x;k, \lambda\right)\times\,\left[\int\limits_{0}^{x}p_{0}\left(z;k\right)\,dz \right]\,dx\right)^{Q-1}=1-(1-P_{e,bppm}\left(k,\lambda\right))^{Q-1}. \tag{5}\]  
The expression inside the parenthesis equals the binary PPM PER and formulas for calculating it have been previously reported by [6; 10]. An equivalent result can be obtained by noting than the soft-decision binary PPM demodulator compares two \(\chi^{2}\) RVs, one non-central and one central, with equal degrees of freedom. An error occurs when the non-central RV is less than the central one, or whenever their ratio \(f=\chi_{k,\lambda}^{2}/\chi_{k,0}^{2}\) is less than unity. The ratio \(f\) follows the non-central \(F\)-distribution ([13], eq. (26.620)) with a cdf equal to \[ \begin{split} P(f\,\leq\,F)\,=&\,P(F|v_{1},v_{2},\lambda) =e^{-\lambda}\,\sum_{i=0}^{\infty}\frac{\lambda^{i}}{i!}\,I_{p}(\frac{v_{1}}{2} +i,\frac{v_{2}}{2})\,,\\ p=&\frac{v_{1}\,F}{v_{1}\,F+v_{2}}\,,\end{split}  \tag{6}\]  
where \(I_{p}(\cdot)\) denotes the regularized incomplete beta function. Thus, the binary PPM PER is calculated from the \(F\)-distribution for \(F=1\), \(v_{1}=v_{2}=2\,k\)  
\[P_{e,bppm}\left(k,\lambda\right)=e^{-\lambda}\,\sum_{i=0}^{\infty}\frac{ \lambda^{i}}{i!}\,I_{1/2}(k+i,k)\,. \tag{7}\]  
For an integer \(k\) it holds that ([13], eq. (26.5.7))  
\[I_{1/2}(k+i,k)=(1/2)^{2k+i-1}\,\sum_{j=0}^{k-1}\binom{2k+i-1}{j}\,, \tag{8}\]  
and it is straightforward to show that  
\[ \begin{split} P_{e,bppm}\left(k,\lambda\right)=& e^{-\lambda}\,\sum_{j=0}^{\infty}\frac{\lambda^{i}}{i!}\,(1/2)^{2k+i-1 }\,\sum_{j=0}^{k-1}\binom{2k+i-1}{j}\\ =&(1/2)^{2k-1}\,e^{-\lambda}\,\sum_{j=0}^{k-1}\sum_ {i=0}^{\infty}\binom{2k+i-1}{j}\,\frac{(\lambda/2)^{i}}{i!}\\ =&(1/2)^{2k-1}\,e^{-\lambda}\,\sum_{j=0}^{k-1}\binom {2k-1}{j}\,_{1}F_{1}\left(2k;2k-j;\lambda/2\right)\\ =&(1/2)^{2k-1}\,e^{-\lambda/2}\,\sum_{j=0}^{k-1} \binom{2k-1}{j}\,_{1}F_{1}\left(-j;2k-j;-\lambda/2\right)\,,\end{split}  \tag{9}\]  
where \({}_{1}F_{1}\left(\cdot;\cdot;\cdot\right)\) denotes the hypergeometric function. The last relation only requires a finite summation, since the hypergeometric function reduces to a polynomial for the given representation, and it enables the efficient and fast calculation of the PPM PER in standard mathematical software that implements \({}_{1}F_{1}\left(\cdot;\cdot;\cdot\right)\).