# The BER Performance of the LDPC-Coded MPPM over Turbulence UWOC Channels
## 2 System and Channel Models  
Figure 1 shows a simplified configuration of an LDPC-coded MPPM UWOC system over turbulent channels. At the transmitter, pseudorandom binary streams are parsed into groups of \(k\)-bits and encoded by a QC-LDPC encoder following IEEE.802.16 standard,  
generating corresponding groups of codewords with \(m\)-bits, where \(k\) and \(m\) are the information length and code length of LDPC, respectively. Further, the codewords are grouped into blocks with a length of \(L\), and then the blocks are mapped into symbol constellations on the basis of \(w\)-pulse \(M\)-slot \((M,\ w)\) MPPM format, producing a modulated electrical signal, where \(L=\left\lfloor\log_{2}\right(\begin{array}{c}M\\ w\end{array})\left\rfloor\right\). Assuming no interslot interference and perfect slot synchronization, the system performance is not affected by the bit-to-symbol mapping. The electrical signal modulates the laser, resulting in an optical signal \(x\) with an average optical power of \(P_{av}\). After propagating through the turbulent underwater channel, the optical signal is received and converted to an electrical signal \(y\) by a photodetector (PD). Subsequently, the signal is demodulated, based on soft-decision with log-likelihood ratio information, which is further fed to the decoder for iterative decoding.  
The received electrical signal is expressed as [31]:  
\[y=Rhx+n, \tag{1}\]  
where \(R\) is the photodetector responsivity, \(x\) is equal to zero and \(MP_{av}/w\) in the non-signal time slot and signal time slot, respectively. The AWGN with zero mean is represented by \(n\), single-sided power spectral density by \(N_{0}\), and the variance of \(\sigma_{n}^{2}=N_{0}/2\). The channel gain is \(h\), which can be expressed as \(h=h_{l}h_{f}\), where \(h_{l}\) is the path loss and is assumed to be unity, \(h_{f}\) represents the slow-fading coefficient. Here, we use GG distribution to model received optical intensity fluctuation.  
The PDF of GG distribution is expressed as [7]:  
\[f_{\text{GG}}(s)=\frac{p}{a^{d}\Gamma(d/p)}s^{d-1}\exp\left(-(s/a)^{p}\right), \ s>0 \tag{2}\]  
where \(d,\ a\) and \(p\) are fading, scaling and shape parameters, respectively. The gamma function is represented by \(\Gamma(\cdot)\). The fading coefficient is described as \(h_{f}\sim GG(m,v,\Omega)\), the CDF of which is given by:  
\[F_{\text{GG}}(s)=\frac{\text{Y}\left(\frac{d}{p},\ \ \left(\frac{s}{a}\right)^{p} \right)}{\Gamma(d/p)}, \tag{3}\]  
where \(\text{Y}(a,x)\) denotes the lower incomplete gamma function.  
At the receiver, slot-by-slot detection is applied. Thus, the received electrical signal in one slot can be written as:  
\[y=\left\{\begin{array}{cc}Rh_{l}h_{f}MP_{av}/w+n&\text{signal time slot}\\ n&\text{non-signal time slot}\end{array}\right. \tag{4}\]  
Further, the instantaneous electrical signal-to-noise ratio (SNR) is given by:  
\[\gamma=\left(\frac{Rh_{l}MP_{av}}{\sqrt{2}w\sigma_{n}}\right)^{2}h_{f}^{2}= \widetilde{\gamma}h_{f}^{2} \tag{5}\]  
Figure 1: Schematic of the LDPC-coded MPPM UWOC system over turbulent channels.  
where \(\widetilde{\gamma}=\left(\frac{Rh_{l}MP_{\mu\
u}}{\sqrt{2w}w_{n}}\right)^{2}\) represents the average SNR in the absence of fading.