A system block diagram comprising the SIGN and SDGN models is shown in Figure1. It consists of a transmitter (i.e. source, encoder and OOK mapper), channel and receiver (i.e. APD aperture, channel decoder, demapper and sink). The data bits *x* ∈ (0,1) from the source encoder are encoded by the channel encoder, and the coded bits are then sent through the atmospheric turbulent channel using the OOK transmission scheme. The information bits are modulated such that the presence of light pulse transmits bit ‘1’ and the absence transmits bit ‘0’. We considered the atmospheric turbulence condition, background noise from the extraneous noise sources (i.e. stars and sun), and the Gaussian noise (which is considered as the SDGN and SIGN). At the receiver, the incoming optical beam is focused onto the photodetector that converts it into an electrical signal. Channel decoder decodes the received signal using the suggested coding technique and then demapped to get the estimated transmitted data.

### 2.1 SDGN and SIGN models

Following the derivation done by[10, 11], we model the output of the APD photodetector as the Gaussian approximation of the Webb model. The average number of photons absorbed by an APD illuminated with total optical intensity *P* can be expressed as

\stackrel{\u0304}{n}=\left(\frac{\eta}{{h}_{p}\nu}\right)P

(1)

where *h*_{
p
} is Planck’s constant, *ν* is the optical frequency, and *η* is the quantum efficiency, which is defined as the ratio of the absorbed to the incident photons. According to[12], in an optical communication system using APD, the actual number of photons absorbed, *n*, is a Poisson distributed random variable with probability functionp(n)=({\stackrel{\u0304}{n}}^{n}/n!)exp(-\stackrel{\u0304}{n}). The conditional probability density function (pdf)p(k|\stackrel{\u0304}{n}) of generating an avalanche of *k* output photoelectrons for the mean absorbed photons\stackrel{\u0304}{n} is derived by the McIntyre-Conradi (MC) distribution[10]. The pdf obtained by MC gives the exact mathematical analysis which can be approximated by the continuous Webb density[11]. In[9], we propose the SDGN model which is the approximation of the continuous Webb density whose pdf is given by

{p}_{\text{SDGN}}(k)=\frac{1}{\sqrt{2\pi \stackrel{\u0304}{n}{G}^{2}F}}exp\left(\frac{-{(k-\stackrel{\u0304}{n}G)}^{2}}{2\stackrel{\u0304}{n}{G}^{2}F}\right)

(2)

where *F* = *k*_{eff}*G* + (2 - 1/*G*)(1 - *k*_{eff}) is the excess noise factor, *G* is the average APD gain and *k*_{eff} is the ionization ratio constant. As shown in Figure1, the information bit ‘1’ is represented by{\stackrel{\u0304}{n}}_{1} (i.e. assuming *η* = 1,{\stackrel{\u0304}{n}}_{1}=\lambda +\mathit{\text{Ph}}, which is considered to be the background level *λ* plus the signal intensity *P* and *h* shows the effect of scintillation) and bit ‘0’ by{\stackrel{\u0304}{n}}_{0} (i.e.{\stackrel{\u0304}{n}}_{0}=\lambda)[13]. For intensity fluctuations, we adopt the log-normal (LN) distribution that best describes the moderate turbulence conditions[14]. In[14], irradiance fluctuations for the moderate scintillation were shown to be well modelled by a LN distribution, and the pdf of the LN distribution is given by

{p}_{h}(h)=\frac{1}{\sqrt{2\pi {\sigma}_{\mathit{\text{lnh}}}^{2}}h}exp\left[-\frac{{(logh-{\mu}_{\mathit{\text{lnh}}})}^{2}}{2{\sigma}_{\mathit{\text{lnh}}}^{2}}\right]

(3)

where *μ*_{
lnh
} and{\sigma}_{\mathit{\text{lnh}}}^{2} are the mean and variance of the logarithm of *h*. It is assumed that *E*[*h*] = 1 so that the average received optical power remains constant, and from the moments of the LN distribution, it follows that{\mu}_{\mathit{\text{lnh}}}=-\frac{1}{2}{\sigma}_{\mathit{\text{lnh}}}^{2} and{\sigma}_{\mathit{\text{lnh}}}^{2}=log(1+{\sigma}_{I}^{2}), where{\sigma}_{I}^{2} is the scintillation index (SI) defined in[15]. In[9], we relate physical parameters of the APD with statistical parameters of the SDGN model by saying{\mu}_{x}={\stackrel{\u0304}{n}}_{x}G and{\sigma}_{x}^{2}={\stackrel{\u0304}{n}}_{x}{G}^{2}F. The relationship shows the dependence of statistical parameters (i.e. *μ*_{
x
} and{\sigma}_{x}^{2}) on physical parameters (i.e. *G* and *F*) of the APD.

In[9], we exploit the concept of the double Gaussian noise model[13], with the name of SDGN model and did the investigation for the optimum/sub-optimum detectors^{a}. In this paper, we exploit the concept of the SDGN model and compare its performance with the SIGN model. We also implement the LDPC decoder to improve the system performance under turbulent conditions. For the SIGN model, the variance (*σ*^{2}) does not depend on the signal intensity, and its pdf is derived as

{p}_{\text{SIGN}}(k)=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}exp\left(\frac{-{(k-\stackrel{\u0304}{n}G)}^{2}}{2{\sigma}^{2}}\right).

(4)

For an APD detector, parameters used are the APD internal current gain *G*, *k*_{eff} and quantum efficiency *η*. We relate physical parameters of the APD with our signal model for the Gaussian approximation: *μ*_{0} ≈ *λ* *G*, *μ*_{1} ≈ (*λ* + *P* *h*)*G*,{\sigma}_{0}^{2}\approx \lambda {G}^{2}F,{\sigma}_{1}^{2}\approx (\lambda +\mathit{\text{Ph}}){G}^{2}F[9]. In[9], we simulate density functions for the Webb and SDGN model considering the InGaAs APD with *G* = 10, *k*_{eff} = 0.45 and *F* = 5.5[16]. Simulation results in[9] show the agreement between the Webb and SDGN model near peaks of distributions. These results illustrate the simulation results of the effect of varying background level for the SDGN and SIGN models which are referred to in Section 5.

### 2.2 Channel model

The received signal for the SDGN and SIGN models after optical/electrical conversion can be given by (5) and (6), respectively.

{y}_{\text{SDGN}}=\left\{\begin{array}{cc}\mathcal{N}\left({\mu}_{1},{\sigma}_{1}^{2}\right)& \phantom{\rule{1.5em}{0ex}}\text{Signal presence}\\ \mathcal{N}\left({\mu}_{0},{\sigma}_{0}^{2}\right)& \phantom{\rule{1.5em}{0ex}}\text{Signal absence}\end{array}\right.

(5)

{y}_{\text{SIGN}}=\left\{\begin{array}{cc}\mathcal{N}\left({\mu}_{1},{\sigma}^{2}\right)& \phantom{\rule{1.5em}{0ex}}\text{Signal presence}\\ \mathcal{N}\left({\mu}_{0},{\sigma}^{2}\right)& \phantom{\rule{1.5em}{0ex}}\text{Signal absence}\end{array}\right.

(6)

The FSO channel, LLR mappings for the SIGN and SDGN can be calculated as

\mathrm{\Lambda}=log\left(\frac{p(x=0|y,h)}{p(x=1|y,h)}\right)

(7)

where *p*(*x* = 1|*y*,*h*) represents the probability of *x* = 1 given the received symbol *y* (i.e. either the SIGN and SDGN) under a certain channel condition and *p*(*x* = 0|*y*,*h*) represents the probability of *x* = 0 given the received symbol *y* (i.e. either the SIGN and SDGN) under a certain channel condition. By the application of Bayes rule and assuming equi-likely input bits, (7) is replaced by

\mathrm{\Lambda}=log\left(\frac{p(y|x=0,h)}{p(y|x=1,h)}\right).

(8)

For the not equi-likely input bits (i.e. *p*(*x*) ≠ 1/2), we can rewrite (8) as

\mathrm{\Lambda}=log\frac{p(y|x=0,h)}{p(y|x=1,h)}+L(x)

(9)

whereL(x)=log\left(\frac{p(x=0)}{p(x=1)}\right). The LLR mappings are analysed so that we can use the LLR mappings for the calculation of the uncoded and coded BER.