## Abstract

In this paper, a new and simple rate-adaptive transmission scheme for free-space optical (FSO) communication systems with intensity modulation and direct detection (IM/DD) over atmospheric turbulence channels is analyzed. This scheme is based on the joint use of repetition coding and variable silence periods, exploiting the potential time-diversity order (TDO) available in the turbulent channel as well as allowing the increase of the peak-to-average optical power ratio (PAOPR). Here, repetition coding is firstly used in order to accomodate the transmission rate to the channel conditions until the whole time diversity order available in the turbulent channel by interleaving is exploited. Then, once no more diversity gain is available, the rate reduction can be increased by using variable silence periods in order to increase the PAOPR. Novel closed-form expressions for the average bit-error rate (BER) as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam follows negative exponential and gamma-gamma distributions, covering a wide range of atmospheric turbulence conditions. Obtained results show a diversity order as in the corresponding rate-adaptive transmission scheme only based on repetition codes but providing a relevant improvement in coding gain. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also OOK formats with any pulse shape, corroborating the advantage of using pulses with high PAOPR, such as Gaussian or squared hyperbolic secant pulses. We also determine the achievable information rate for the rate-adaptive transmission schemes here analized.

© 2010 Optical Society of America

## 1. Introduction

Optical wireless communications using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications [1], providing an unregulated spectral segment and high security. Recently, the use of atmospheric free-space optical (FSO) transmission is being specially interesting to solve the *“last mile” problem*, as well as a supplement to radio-frequency (RF) links [2, 3]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance [4, 5]. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [6–10]. In particular, heuristic space-time code (STC) designs such as repetition codes (RCs) [11–13] and orthogonal space-time block codes (OSTBCs) [14–16] have been proposed for FSO systems with IM/DD. In [17], a closed-form expression has recently been derived for the asymptotic pairwise error probability (PEP) of general FSO STCs for two lasers and an arbitrary number of photodetectors for channels suffering from Gamma-Gamma fading, showing the quasi-optimality of STC designs based on repetition codes and their superiority compared to conventional orthogonal space-time block codes. In [18, 19], selection transmit diversity is proposed for FSO links over strong turbulence channels, where the transmit diversity technique based on the selection of the optical path with a greater value of scintillation has shown to be able to extract full diversity as well as providing better performance compared to general FSO STC designs.

An alternative approach to improving the performance in this turbulence FSO scenario is the employment of rate-adaptive transmission in order to make suitable the communication to the adverse channel conditions, depending on the available SNR until a sufficiently low error probability can be attained. Various FSO systems using adaptive modulation have been proposed [20–23]. In [20], a variable rate FSO system employing adaptive Turbo-based coding schemes with on-off keying (OOK) formats was investigated. In [21, 22], an adaptive transmission scheme that varied both the power and the modulation order of a FSO system with M-ary pulse amplitude modulation (MPAM) has been studied. In [23], an adaptive transmission technique employing subcarrier phase shift keying (S-PSK) intensity modulation has been proposed. Another solution is the employment of adaptive transmission based on repetition coding, a well known technique employed in RF systems [24]. Based on the concept of temporal-domain diversity reception (TD-DR), this idea has been applied for FSO links in [25–27], where two separate channels over the same transmit and receive path are implemented. Both channels carry the same data, but one of the channels is delayed by the expected fade duration.

In this paper, a new and simple rate-adaptive transmission scheme for FSO communication systems with intensity modulation and direct detection over atmospheric turbulence channels is analyzed. This scheme is based on the joint use of repetition coding and variable silence periods, exploiting the potential time-diversity order (TDO) available in the turbulent channel as well as allowing the increase of the peak-to-average optical power ratio (PAOPR), which has shown to be a favorable characteristic in IM/DD FSO links [15, 18, 19]. Here, no adaptive signal constellation is required, proposing the use of OOK signaling due to its simplicity and low implementation cost and, hence, providing a simple rate-adaptive transmission scheme with a low-complexity hardware implementation. Novel closed-form expressions for the average bit-error rate (BER) as well as their corresponding asymptotic expressions are presented when the scintillation follows negative exponential and gamma-gamma distributions, covering a wide range of atmospheric turbulence conditions. Unlike rate-adaptive transmission only based on variable silence periods where diversity gain is not achieved, obtained results show a diversity gain as in the corresponding rate-adaptive transmission scheme only based on repetition codes, i.e. with the same slope of the BER versus average signal-to-noise ratio (SNR), but providing a relevant improvement in coding gain, i.e. a significant horizontal shift in the BER performance in the limit of large SNR [28]. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also OOK formats with any pulse shape, corroborating the advantage of using pulses with high PAOPR, such as Gaussian or squared hyperbolic secant pulses. Moreover, an analysis of the achievable information rate for the different rate-adaptive transmission schemes is presented.

## 2. Atmospheric turbulence channel model

The use of infrared technologies based on IM/DD links is considered, where the instantaneous current in the receiving photodetector, *y*(*t*), can be written as

*η*is the detector responsivity, assumed hereinafter to be the unity,

*X*≜

*x*(

*t*) represents the optical power supplied by the source, and

*I*≜

*i*(

*t*) the scintillation at the optical path;

*Z*≜

*z*(

*t*) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance

*σ*

^{2}=

*N*

_{0}/2, i.e.

*Z*∼

*N*(0,

*N*

_{0}/2), independent of the on/off state of the received bit [1]. Since the transmitted signal is an intensity,

*X*must satisfy ∀

*t*

*x*(

*t*) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of

*X*is limited. Although limits are placed on both the average and peak optical power transmitted, in the case of most practical modulated optical sources, it is the average optical power constraint that dominates [29]. The received electrical signal

*Y*≜

*y*(

*t*), however, can assume negative amplitude values. In this fashion, the atmospheric turbulence channel model consists of a multiplicative noise model, where the optical signal is multiplied by the channel irradiance. Here, we firstly consider the gamma-gamma turbulence model proposed in [4, 30], where the normalized irradiance

*I*is defined as the product of two independent random variables, i.e.

*I*=

*I*, representing

_{x}I_{y}*I*and

_{x}*I*large scale and small scale turbulent eddies and each of them following a gamma distribution. This leads to the so-called gamma-gamma distribution, i.e.

_{y}*I*∼

*GG*(

*α,β,μ*), whose probability density function (PDF) is given by

*μ*represents the mean value

*μ*=

*E*[

*I*], Γ(·) is the well-known Gamma function,

*K*(·) is the

_{ν}*ν*th-order modified Bessel function of the second kind [31, Eq. (8.43)]. Assuming spherical wave propagation, the parameters

*α*and

*β*are related to the atmospheric conditions through the following expresions [7, 30]:

*d*= (

*kD*

^{2}/4

*L*)

^{1/2}. Here,

*k*= 2

*π/λ*is the optical wave number,

*λ*is the wavelength,

*D*is the diameter of the receiver collecting lens aperture and

*L*is the link distance in meters. ${C}_{n}^{2}$ stands for the altitude-dependent index of the refractive structure parameter and varies from 10

^{−13}

*m*

^{−2/3}for strong turbulence to 10

^{−17}

*m*

^{−2/3}for weak turbulence [4, 7]. Since the mean value of this turbulence model here considered is

*μ*= 1 and the second moment is given by

*E*[

*I*

^{2}] = (1+1/

*α*)(1+1/

*β*), the scintillation index (SI), a parameter of interest used to describe the strength of atmospheric fading, is defined as

Together with this distribution and considering a limiting case of strong turbulence conditions [4, 15], a negative exponential model with PDF given by

is also adopted to describe turbulence-induced fading, leading to an easier mathematical treatment to evaluate error rate performance for different values of rate reduction and time-diversity order. In this case, since the mean value of this turbulence model is*E*[

*I*] = 1 and the second moment is given by

*E*[

*I*

^{2}] = 2, the scintillation index is

*SI*=

*E*[

*I*

^{2}]/(

*E*[

*I*])

^{2}– 1 = 1. This distribution can be seen as the gamma-gamma distributed turbulence model in Eq. (2) when the channel parameters are

*β*= 1 and

*α*→ ∞. From the point of view of scintillation index, it is easy to deduce the fact that the strength of atmospheric fading represented by the gamma-gamma distributed turbulence model with channel parameters

*β*= 1 and increasing

*α*tends to be closer and closer to that corresponding to the negative exponential distributed turbulence model.

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [15, 18]. In spite of the atmosphere can cause pulse distortion and broadening during propagation at extremely high signalling rates and, especially, at high levels of turbulence strength when increasing either
${C}_{n}^{2}$ or the path length *L*, or both, pulse shape distortion is assumed to be negligible for the typical FSO scenario here analyzed [32, 33]. A new basis function *ϕ*(*t*) is defined as
$\varphi (t)=g(t)/\sqrt{{E}_{g}}$ where *g*(*t*) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤ *g*(*t*) ≤ 1 in the bit period and 0 otherwise, and
${E}_{g}={\int}_{-\infty}^{\infty}{g}^{2}(t)\mathit{dt}$ is the electrical energy. In this way, an expression for the optical intensity can be written as

*G*(

*f*= 0) represents the Fourier transform of

*g*(

*t*) evaluated at frequency

*f*= 0, i.e. the area of the employed pulse shape, and

*T*parameter is the bit period. The random variable (RV)

_{b}*a*follows a Bernoulli distribution with parameter

_{k}*p*= 1/2, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is

*P*, defining a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of $d=2P\sqrt{{T}_{b}\xi}$ where

*ξ*=

*T*/

_{b}E_{g}*G*

^{2}(

*f*= 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse.

Since atmospheric scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, we consider the time variations according to the theoretical block-fading model, where the channel fade remains constant during a block (corresponding to the channel coherence interval) and changes to a new independent value from one block to next. In other words, channel fades are assumed to be independent and identically distributed (i.i.d.). This temporal correlation can be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [6,7,34]. However, as in [10], we here assume that the interleaver depth can not be infinite and, hence, we can potentially benefit from a degree of time diversity limited equal to TDO. This consideration is justified from the fact that the latency introduced by the interleaver is not an inconvenience for the required application. For example, for repetition coding and a time diversity order available of TDO=2, i.e. two channel fades *i*_{1} and *i*_{2} per frame, perfect interleaving can be done by simply sending the same information delayed by the expected fade duration, as shown experimentally in [27] for a rate reduction of 2.

## 3. Proposed rate-adaptive transmission scheme

Adaptive transmission here considered is based on reducing the initial bit rate, *R _{b}* = 1/

*T*, for a rate reduction (

_{b}*RR*) parameter as

*R*/

_{b}*RR*in order to satisfy a predefined BER requirement. In this section, we firstly explain the case corresponding to the rate-adaptive transmission only based on variable silence periods; secondly, we present the case corresponding to the rate-adaptive transmission only based on repetition coding and, finally, we consider the adaptive transmission scheme here proposed and based on the joint use of repetition coding and variable silence periods, exploiting the potential time-diversity order available in the turbulent channel as well as allowing the increase of the PAOPR, which has shown to be a favorable characteristic in IM/DD links.

A rate-adaptive transmission scheme based on variable silence periods has already been proposed in [35] for indoor optical wireless communication systems as a consequence of the good performance of signaling techniques having a reduced duty cycle [36] and, hence, providing a high PAOPR. The advantage of using pulses with high PAOPR has also been corroborated in FSO links over atmospheric turbulence channels by closed-form expressions corresponding to the average BER performance [15, 18, 19], suggesting its application to the turbulent FSO scenario. Recently, following this idea, a simple rate-adaptive transmission scheme based on the use of variable silence periods has been proposed in [37], evaluating by simulation the performance in terms of burst error rate when the lognormal channel model presented in [38] is adopted for the atmospheric channel under the assumption of weak turbulence regime. Unfortunately, no closed-form expressions for the error rate performance are presented, not allowing us to obtain conclusions about diversity and coding gain as well as about the influence of the pulse shape used on the performance of the rate-adaptive transmission scheme. Next, in order to rightly analyze this rate-adaptive scheme, a similar expression to Eq. (7) for the optical intensity corresponding to this adaptive technique only based on variable silence periods can be written as

*RR*– 1 is the number of silence bit periods added in order to accomodate the transmission rate to the channel conditions, so that the higher rate reduction (

_{s}*RR*=

*RR*), the larger silence time. From this expression, it is easy to deduce that an increase of the PAOPR is required in order to maintain the average optical power at the same constant level of

_{s}*P*.

In relation to the rate-adaptive transmission only based on repetition coding, the expression for the optical intensity can be written as

*RR*represents the repetition of each information bit and, hence, a rate reduction of

_{rc}*RR*=

*RR*is considered. From this expression, it is easy to deduce that an increase of the PAOPR is not achieved; however, unlike rate-adaptive transmission only based on variable silence periods, a potential diversity gain can be exploited based on the concept of temporal-domain diversity reception [27]. To the best of the authors’ knowledge, closed-form expressions for the error rate performance corresponding to this rate-adaptive scheme in the context proper to FSO systems where a limited time-diversity order is only available have not been reported in the open literature.

_{rc}Next, we combine the best of previous approaches, proposing a novel adaptive transmission scheme based on the joint use of repetition coding and variable silence periods and, this way, exploiting the potential time-diversity order available in the turbulent channel as well as allowing the increase of the PAOPR. In this fashion, the expression for the optical intensity can be written as

*RR*=

*RR*·

_{rc}*RR*. In this adaptive transmission technique, repetition coding is firstly used in order to accomodate the transmission rate to the channel conditions until the whole time diversity order available in the turbulent channel by interleaving is exploited, i.e.

_{s}*RR*≤ TDO and

_{rc}*RR*= 1. Then, once no more diversity gain is available, the rate reduction can be increased by using variable silence periods in order to increase the PAOPR, i.e.

_{s}*RR*= TDO and

_{rc}*RR*> 1. For the sake of simplicity, we here consider values multiples of 2 for

_{s}*RR*when rate reduction is applied, i.e

*RR*= {1,2,4,8, ···} as well as for the time-diversity order effective TDO which is provided by interleaving, i.e. TDO = {1,2,4}, in this fashion, allowing to satisfy different latency requirements in the system [10].

## 4. BER performance analysis

In this section, assuming channel side information at the receiver, we present closed-form expressions for the average BER when the scintillation follows negative exponential and gamma-gamma distributions, which cover a wide range of atmospheric turbulence conditions.

For the adaptive scheme in Eq. (8) wherein only variable silence periods are used for implementing the rate reduction and the information is detected each *RR _{s}* bit periods, the conditional BER is given by

*Q*(·) is the Gaussian-

*Q*function defined as $Q(x)=\frac{1}{\sqrt{2\pi}}{\int}_{x}^{\infty}{d}^{-\frac{{t}^{2}}{2}}\mathit{dt}$. Substituting the value of the Euclidean distance

*d*gives ${P}_{{b}_{s}}(E|I)=Q\left(\sqrt{2\gamma \xi}R{R}_{s}i\right)=Q\left(\sqrt{2\gamma \xi}{\Omega}_{c}^{s}i\right)$ where

*γ*=

*P*

^{2}

*T*/

_{b}*N*

_{0}is the average receiver electrical signal-to-noise spectral density ratio (SNR) in the presence of the turbulence [5], knowing that PDF in Eqs. (2) or (6) is normalized. It can be noted that the parameter ${\Omega}_{c}^{s}=R{R}_{s}$ is related to the coding gain corresponding to this rate-adaptive scheme. Hence, the average BER,

*P*(

_{bs}*E*), can be obtained by averaging

*P*

_{bs}*E*|

*I*) over the turbulence PDF as follows

In relation to the adaptive scheme in Eq. (9) wherein only repetition coding is used for implementing the rate reduction, the information is detected each bit period, combining with the same weight *RR _{rc}* noisy faded signals in a similar manner to a single-input multiple-output (SIMO) FSO scheme with equal gain combining (EGC) [13] and, this way, achieving a diversity gain, never greater than TDO. When TDO >

*RR*, considering that the variance of the noise of the combined signal is

_{rc}*RR*

_{rc}N_{0}/2 and substituting the value of

*d*, the conditional BER is given by

*RR*, assuming for simplicity in this analysis that

_{rc}*RR*is multiple of TDO, the conditional BER can be written as

_{rc}*RR*and ${\Omega}_{c}^{rc}=\sqrt{R{R}_{rc}}/\text{TDO}$ when TDO <

_{rc}*RR*. Hence, the average BER,

_{rc}*P*(

_{brc}*E*), can be obtained by averaging ${P}_{{b}_{rc}}\left(E|{\left\{{I}_{k}\right\}}_{k=1}^{{\Omega}_{d}^{rc}}\right)$ over the turbulence PDF as follows

From the expressions in Eqs. (12) and (16), it can be deduced that a greater value for the parameter related to the coding gain using silence periods can be achieved once the whole time diversity order available in the turbulent channel by interleaving is exploited in order to achieve as much diversity gain as possible. In this way, repetition coding is firstly used in the rate-adaptive transmission proposed in Eq. (10) and then, once no more diversity gain is available, *RR* is increased by using variable silence periods in order to improve the coding gain. A better performance can be obtained when TDO < *RR*, being *RR* = TDO · *RR _{s}*. In this way, the conditional BER is given by

*RR*, as in repetition coding, and ${\Omega}_{c}^{\mathit{rcs}}=R{R}_{s}/\sqrt{\text{TDO}}$ when TDO <

*RR*, a greater value than ${\Omega}_{c}^{rc}$. Hence, the average BER,

*P*(

_{brcs}*E*), can be obtained by averaging ${P}_{{b}_{\mathit{rcs}}}\left(E|{\left\{{I}_{k}\right\}}_{k=1}^{{\Omega}_{d}^{\mathit{rcs}}}\right)$ over the turbulence PDF as follows

Finally, the average BER corresponding to the three previous rate-adaptive transmission schemes can be written in a unified way as

where*I*represents the sum of variates ${I}_{T}=\sum _{k=1}^{{\Omega}_{d}}{I}_{k}$ whose PDF is given by

_{T}*f*(

_{IT}*i*), Ω

*represents the parameter related to the coding gain, being equal to ${\Omega}_{c}^{s}$, ${\Omega}_{c}^{rc}$ and ${\Omega}_{c}^{\mathit{rcs}}$, respectively, and Ω*

_{c}*represents the diversity gain, i.e. the slope of the BER versus average SNR, being equal to 1, ${\Omega}_{d}^{rc}$ and ${\Omega}_{d}^{\mathit{rcs}}$, respectively.*

_{d}#### 4.1. Gamma-gamma atmospheric turbulence channel

Particularizing with the gamma-gamma distribution in Eq. (2), we assume that the probability of fade associated with the summed output of Ω* _{d}* samples by using direct detection can still be reasonably approximated by a gamma-gamma PDF [4, 22] as

*I*∼

_{T}*GG*(

*α*,

_{T}*β*, Ω

_{T}*), where ${\alpha}_{T}=(1+{\Omega}_{d}\beta )/({\Omega}_{d}\beta S{I}_{T}^{2}-1)$,*

_{d}*β*= Ω

_{T}*and the equivalent scintillation index,*

_{d}β*SI*, presents a good approximation as

_{T}*SI*≅

_{T}*SI*/Ω

*for values of Ω*

_{d}*≤ 10. Here, it must be commented that another approach in [39] shows that a single gamma-gamma distribution can be adopted as a very accurate approximation when all the variates of the sum are identically distributed, as found in the rate-adaptive transmission scheme here proposed, but wherein an adjustment parameter after applying non-linear regression methods has to be included. In this way, to evaluate the integral Eq. (19), we can use the Meijer’s G-function [31, Eq. (9.301)], available in standard scientific software packages such as Mathematica and Maple, in order to transform the integral expresion to the form in [40, Eq. (21)], expressing in Eq. (19)*

_{d}*K*(·) [40, Eq. (14)] in terms of Meijer’s G-function and using the fact that the Q-function is related to the complementary error function erfc(·) by $\text{erfc}(x)=2Q(\sqrt{2}x)$ and $\text{erfc}(\sqrt{z})=\frac{1}{\sqrt{\pi}}{G}_{1,2}^{2,0}\left(z|\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\right)$ [41, Eq. (06.27.26.0006.01)]. Finally, after a simple power transformation of the RV ${I}_{n}={I}_{T}^{2}$ in order to achieve a linear argument for the Meijer’s G-function related to erfc(·) and using [40, Eq. (21)], a closed-form solution for BER is derived as can be seen in

_{ν}*α,β*) = (4, 3) and (

*α,β*) = (4, 1) are assumed, corresponding to values of scintillation index of

*SI*= 0.66 and

*SI*= 1.5, respectively, and where rectangular pulse shapes with

*ξ*= 1 are used for values of TDO = {1, 2, 4}, rate reductions of

*RR*= {1, 2, 4, 8} and the three rate-adaptive transmission schemes presented in the previous section, based on: the only use of silence periods (Sil), the only use of repetition coding (Rep) and, finally, the joint use of repetition codes and variable silence periods (Rep&Sil).

BER simulation results are furthermore included as a reference. Due to the long simulation time involved, simulation results only up to BER=10^{−6} are included. Simulation results demonstrate an excellent agreement with the analytical results in Eq. (20) for the different rate-adaptive transmission schemes and levels of turbulence, as well as the relevant improvement in performance obtained when the potential time-diversity order available in the channel is fully exploited, especially, at high levels of turbulence. This improvement is even more significant when the PAOPR is increased by using variable silence periods once no more diversity gain is available. To the best of our knowledge, no closed-form expressions have been previously reported when the aproximation here considered, proposed in [4], is adopted, corroborating in this paper that the simulation results demonstrate an excellent agreement with the analytical results. With the purpose of analyzing the diversity order achieved for the rate-adaptive here proposed, we can use that the argument of the Meijer’s G-function in Eq. (20) tends to zero for high average electrical SNRs. Hence, following an asymptotic expansion of the Meijer’s G-function [41, Eq. (07.34.06.0006.01)] and considering the fact that the two parameters related to the atmospheric conditions are greater than each other, being *α* > *β* assumed in our case, the asymptotic BER can be expressed after some algebraic manipulations as

*RR*= 1}, scheme Sil-{

*RR*= 2}, scheme Rep-{

*RR*= 4, TDO = 2} and, finally, scheme Rep&Sil-{

*RR*= 8, TDO = 4}. In this way, it is straightforward to show that the average BER behaves asymptotically as ${({G}_{c}\sqrt{\gamma \xi})}^{-{G}_{d}}$, where

*G*and

_{d}*G*denote diversity order and coding gain, respectively [28], corroborating a diversity gain of Ω

_{c}*in relation to the absence of rate reduction or rate-adaptive transmission only based on variable silence periods, wherein the average BER varies as ${\left(\sqrt{\gamma \xi}\right)}^{-\beta}$. At high SNR, if asymptotically the error probability behaves as ${\left({G}_{c}\sqrt{\gamma \xi}\right)}^{-{G}_{d}}$, the diversity order*

_{d}*G*determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain

_{d}*G*(in decibels) determines the shift of the curve in SNR. This translates into a coding gain advantage for the rate-adaptive transmission scheme here proposed (Rep&Sil) relative to the scheme only based on the use of repetition coding (Rep) of

_{c}*RR*is greater than the time-diversity order available in the turbulent channel.

#### 4.2. Exponential atmospheric turbulence channel

In this subsection, considering a limiting case of strong turbulence conditions [4,15], a negative exponential model is adopted to describe turbulence-induced fading, leading to an easier mathematical treatment to evaluate error rate performance for different values of rate reduction and time-diversity order. Here, particularizing with the negative exponential distribution in Eq. (6) and knowing that this can be seen as a gamma distribution of parameters *μ* = 1 and *λ* = 1 [42], i.e. *I ∼ G*(*μ* = 1, *λ* = 1), the sum of Ω* _{d}* i.i.d. RVs represented by
${I}_{T}=\sum _{k=1}^{{\Omega}_{d}}{I}_{k}$ is also a RV following a gamma distribution with parameters

*μ*= Ω

*and*

_{d}*λ*= 1, i.e.

*I*∼

_{T}*G*(

*μ*= Ω

*,*

_{d}*λ*= 1) as

Next, substituting Eq. (23) in Eq. (19) and evaluating the integral by using [41, Eq. (06.27.21.0133.01)] after a simple transformation of the RV ${I}_{n}=\sqrt{\gamma \xi}{\Omega}_{c}{I}_{T}$, a closed-form solution for the average BER yields as

_{2}

*F*

_{2}(

*a*

_{1},

*a*

_{2};

*b*

_{1},

*b*

_{2};

*z*) is a generalized hypergeometric function [31, Eq. (9.14.1)]. The results corresponding to this FSO scenario are illustrated in Fig. 2, where rectangular pulse shapes with

*ξ*= 1 are used for values of TDO = {1, 2, 4} and rate reductions of

*RR*= {1,2, 4, 8}. BER simulation results are furthermore included as a reference. Due to the long simulation time involved, simulation results only up to BER=10

^{−6}are included. Simulation results demonstrate an excellent agreement with the analytical results in Eq. (24) for the different rate-adaptive transmission schemes here analyzed. As in previous subsection, with the purpose of analyzing the diversity order achieved for the rate-adaptive here proposed, making use in Eq. (24) of the fact that the series expansion corresponding to a generalized hypergeometric function can be simplified by

_{2}

*F*

_{2}(

*a*

_{1},

*a*

_{2};

*b*

_{1},

*b*

_{2};

*z*) ∝ 1+

*O*(

*z*), the asymptotic BER can be expressed after some algebraic manipulations as

*RR*= 1}, scheme Sil-{

*RR*= 2}, scheme Rep-{

*RR*= 4, TDO = 2} and, finally, scheme Rep&Sil-{

*RR*= 8, TDO = 4}. In this way, it is straightforward to show that the average BER behaves asymptotically as ${\left({G}_{c}\sqrt{\gamma \xi}\right)}^{-{G}_{d}}$, where

*G*and

_{d}*G*denote diversity order and coding gain, respectively, corroborating a diversity gain of Ω

_{c}*in relation to the absence of rate reduction or rate-adaptive transmission only based on variable silence periods, wherein the average BER varies as ${(\sqrt{\gamma \xi})}^{-1}$ [14, 15], and corroborating the same coding gain advantage for the rate-adaptive transmission scheme here proposed (Rep&Sil) relative to the scheme only based on the use of repetition coding (Rep) as in previous subsection.*

_{d}Additionally, a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10log_{10} *ξdB*. Recently, the use of pulses having a shortened duty cycle and high PAOPR has been proposed in [37], showing by simulation a relevant improvement in performance in terms of burst error rate. This superiority has been previously reported by the authors in terms of closed-form expressions corresponding to the average BER performance in several FSO scenarios [15, 18, 19], as well as from the point of view of information theory [43]. Following the approach presented in this paper, this improvement can be viewed as an additional coding gain, regardless of the rate-adaptive transmission scheme employed. So, for instance, when a rectangular pulse shape of duration *κT _{b}*, with 0

*< κ*≤ 1, is adopted, a value of

*ξ*= 1/

*κ*can be easily shown. Nonetheless, a significantly higher value of $\xi =4/\kappa \sqrt{\pi}$ is obtained when a Gaussian pulse of duration

*κT*as

_{b}*g*(

*t*) = exp (−

*t*

^{2}/2

*ρ*

^{2}) ∀|

*t*| <

*κT*/2 is adopted, where

_{b}*ρ*=

*κT*/8 and the reduction of duty cycle is also here controlled by the parameter

_{b}*κ*. In this fashion, 99.99% of the average optical power of a Gaussian pulse shape is being considered. Ultrashort pulses from mode-locked lasers often have a temporal shape which can be described with a Gaussian function. This improvement in performance due to the pulse shape can be even increased when a squared hyperbolic secant (sech

^{2}) function is employed which is another temporal shape proper to ultrashort pulses from mode-locked lasers [44]. In this way, a value of

*ξ*= 8/3

*κ*, greater than that of the Gaussian pulse, is obtained when a sech

^{2}pulse of duration

*κT*as

_{b}*g*(

*t*) = sech

^{2}(

*t*/

*ρ*) ∀|

*t*| <

*κT*/2 is adopted, where

_{b}*ρ*=

*κT*/8 and the reduction of duty cycle is also here controlled by the parameter

_{b}*κ*. As in previous pulse shape, 99.99% of the average optical power of a sech

^{2}pulse shape is being considered. Obtained results corresponding to the sech

^{2}pulse shape with

*κ*= 0.25 for the rate-adaptive transmission (Rep&Sil) here proposed with TDO = 2 and rate reductions of

*RR*= {1, 2, 4, 8} are displayed in Fig. 3(a) together with results previously displayed in Fig. 2 where rectangular pulse shapes with

*κ*= 1 are used. From this figure, it can be observed that a significant horizontal shift in the BER performance above 10 decibels is achieved in any case, regardless of the rate reduction assumed.

Finally, analytical results in Eq. (20) for the rate-adaptive transmission scheme (Rep&Sil) here proposed with TDO = 2, rectangular pulse shape with *κ* = 1 and rate reductions of *RR* = {1, 2, 4, 8}, when different levels of turbulence strength of (*α,β*) = (4, 1), (*α,β*) = (10, 1) and (*α,β*) = (20, 1) are assumed, corresponding to values of scintillation index of *SI* = 1.5, *SI* = 1.2 and *SI* = 1.1, respectively, are displayed in Fig. 3(b) together with the corresponding analytical results in Eq. (24). As previously commented in section 2, it can be deduced from this figure that the BER performance corresponding to the FSO scenario where the strength of atmospheric fading is represented by the gamma-gamma distributed turbulence model with channel parameters *β* = 1 and increasing *α* tends to be closer and closer to that corresponding to the negative exponential distributed turbulence model.

## 5. Achievable information rate performance analysis

In this section, the analysis of the achievable information rate corresponding to the three rate-adaptive transmission schemes here presented is considered. From the mode of operation based on reducing the initial rate, *R _{b}* = 1/

*T*, for a

_{b}*RR*parameter as

*R*/

_{b}*RR*in order to achieve a target BER requirement

*P*, the achievable information rate,

_{b}*R*, in bits/channel use, can be defined as

*R*= 1/

*RR*. While the required value of

*RR*to satisfy a target BER requirement can be numerically solved from analytical results in Eqs. (20) or (24), we can significantly simplify this analysis when target BER requirements and levels of turbulence imply a sufficiently tight performance for the corresponding asymptotic BER in Eqs. (21) or (25). From results displayed in Fig. 1 and Fig. 2, it can be deduced that the asymptotic BER expressions are closer and closer upper bounds for the BER performance as the level of turbulence is increased. This is an expected conclusion since a greater value of average SNR is required to satisfy the same predefined target BER requirement. In this sense, particularizing with the negative exponential distributed turbulence model and making use of the asymptotic BER in Eq. (25), the achievable information rate can be written as

*Y*by

*Y*/

*σ*and now considering

*X*∈ {0,1}. In this way, our channel model can be rewritten as

*I*(

*X*;

*Y*|

*i*) for this channel is derived as in [43, 45] as follows

*P*(

_{X}*x*= 0) =

*P*(

_{X}*x*= 1) = 1/2, ${f}_{Y}(y|x=1,i)=(1/\sqrt{2\pi})\text{exp}(-{(y-Ai)}^{2}/2)$, and ${f}_{Y}(y|x=0,i)={f}_{Y}(y|x=0)=(1/\sqrt{2\pi})\text{exp}(-{y}^{2}/2)$. In this way, the ergodic capacity in bits per channel use,

*C*(

*γ*), can be numerically obtained by averaging (30) over the PDF in (6) as follows This expression is computed using a symbolic mathematics package [46]. Figure 4 depict the mutual information in (31) for the exponential atmospheric turbulent optical channel together with the results corresponding to the achievable information rate in (26), (27) and (28) corresponding to the rate-adaptive transmission schemes Sil, Rep and Rep&Sil, respectively, for two different target BER requirements,

*P*= 10

_{b}^{−4}and

*P*= 10

_{b}^{−8}, and TDO={2, 4}. It can be noted that obtained results corroborate previous results presented in terms of BER performance in Fig. 2. For example, it can be observed that the required value of SNR to satisfy a target BER requirement of

*P*= 10

_{b}^{−8}is about 30 dB when using the rate-adaptive transmission scheme Rep&Sil with RR=8 and TDO=4; while, however, this value is about 70 dB when the rate-adaptive transmission scheme Rep with RR=4 and TDO=2 is assumed. These values can be contrasted in Fig. 4(b) from the point of achievable information rate, where capacity values of 1/8 bits/channel use and 1/4 bits/channel use are achieved, respectively.

From this figure, it can be deduced not only the superiority of the rate-adaptive transmission scheme based on the joint use of repetition coding and variable silence periods but also that an adaptive transmission design approach based on taking advantage out of the potential time-diversity order available in the turbulent channel is required, corroborating the fact that the rate-adaptive transmission scheme only based on variable silence periods implies a remarkably inefficient performance from the point of view of information theory. In this way, it can be observed that even when the available time-diversity order is low (TDO=2) a relevant improvement in achievable information rate is obtained, especially when a lower target BER is demanded. In spite of high values of TDO cannot be possible because of the latency introduced by the interleaver, to achieve a time diversity order available of TDO=2, perfect interleaving can be done by simply sending the same information delayed by the expected fade duration, as shown experimentally in [27] for a rate reduction of 2. It must be commented that the rate-adaptive transmission scheme proposed is not based on an adaptive signal constellation where more complicated modulation techniques can be defined, being assumed the use of OOK signaling due to its simplicity and low implementation cost, and, hence, an achievable information rate not higher than 1 bit/channel use can be achieved, since this is determined by the signal constellation and how the coding technique is able to take advantage of it. At the expense of a greater simplicity in hardware implementation, lower values of capacity are achieved if compared to rate-adaptive transmission schemes based on adaptive modulation or coding techniques more sophisticated than repetition coding and the inclusion of variable silence periods [21–23].

## 6. Conclusions

In this paper, a novel rate-adaptive transmission scheme for FSO communication systems with intensity modulation and direct detection over atmospheric turbulence channels is analyzed. This scheme is based on the joint use of repetition coding and variable silence periods, exploiting the potential time-diversity order available in the turbulent channel as well as allowing the increase of the PAOPR, which has shown to be a favorable characteristic in IM/DD FSO links [15, 18, 19]. Here, repetition coding is firstly used in order to accomodate the transmission rate to the channel conditions until the whole time diversity order available in the turbulent channel by interleaving is exploited. Then, once no more diversity gain is available, the rate reduction can be increased by using variable silence periods in order to increase the PAOPR. Novel closed-form expressions for the average BER as well as their corresponding asymptotic expressions are presented when the scintillation follows negative exponential and gamma-gamma distributions, covering a wide range of atmospheric turbulence conditions. Furthermore, we extend the concepts of diversity and coding gain, which are well known from the RF communication literature [28], to the rate-adaptive FSO systems under study, allowing us to provide simple, insightful, and accurate closed-form approximations for the BER performance at high SNR. Unlike rate-adaptive transmission only based on variable silence periods where diversity gain is not achieved [37], obtained results show a diversity gain as in the corresponding rate-adaptive transmission scheme only based on repetition codes, i.e. with the same slope of the BER versus average SNR, but providing a relevant improvement in coding gain, i.e. a significant horizontal shift in the BER performance in the limit of large SNR. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also OOK formats with any pulse shape, corroborating the advantage of using pulses with high PAOPR, such as Gaussian pulses or squared hyperbolic secant pulses, and concluding the fact that this improvement can be viewed as an additional coding gain, regardless of the rate-adaptive scheme employed or even in absence of rate-adaptive transmission.

In terms of pros and cons of using the adaptive transmission scheme Rep&Sil here analyzed and proposed, we can conclude that one of the pros of this adaptive transmission scheme is the greater simplicity, requiring a lower implementation complexity if compared to alternative rate-adaptive transmission schemes proposed for use in FSO systems [20–23]. In this sense, the use of OOK signaling with repetition coding implies a hardware implementation of a significant lower complexity if compared to that required by the variable rate turbo-coding scheme as proposed in [20], the adaptive LDPC-coded modulation and transmission scheme that varied both the power and the modulation order of a FSO system with M-ary pulse amplitude modulation as proposed in [21,22] and the adaptive transmission techniques employing subcarrier phase shift keying intensity modulation as proposed in [23]. Additionally, together with the use of repetition coding, the adaptive scheme Rep&Sil here proposed considers the inclusion of variable silence periods once no more diversity gain is available, providing a higher level of PAOPR and, hence, a better performance, without the need of increasing complexity in hardware implementation. On the contrary, one of the cons of the rate-adaptive transmission scheme here proposed in relation to previous rate-adaptive transmission schemes is the lower capacity achieved since no adaptive modulation is considered, proposing the use of OOK signaling due to its simplicity and low implementation cost and, hence, providing an achievable information rate not higher than 1 bit/channel use. Nonetheless, as revealed out by the results under no rate reduction or rate-adaptive transmission only based on variable silence periods, a slow change in the slope of performance curve can be observed in Fig. 1 and Fig. 2, especially at high levels of turbulence, justifying the adoption of rate-adaptive transmission schemes as here proposed since it is not practical for many applications to increase the power margin in the link budget to eliminate the deep fades observed under increasing levels of turbulence. For example, when no rate reduction is assumed, as can be seen in Fig. 4(b), to satisfy a target BER requirement of *P _{b}* = 10

^{−8}requires a considerably high value of SNR about 150 dB which can be remarkably reduced to a value of SNR about 74 dB at the expense of a reduction of capacity of 0.5 bits/channel use or a value of SNR about 68 dB at the expense of a lower capacity of 0.25 bits/channel use for a TDO=2. In this way, when a value of TDO=4 is available, this target BER requirement of

*P*= 10

_{b}^{−8}can be satisfied with a value of SNR about 37 dB at this same capacity value of 0.25 bits/channel use.

From the relevant results here obtained when an adaptive transmission scheme based on the joint use of repetition coding and variable silence periods is adopted, investigating in the atmospheric turbulent FSO case the design of trellis coding schemes, where not only the PAOPR is increased, as proposed in [47] for indoor optical wireless communications, but also the potential time-diversity order available in the turbulent channel is exploited, is an interesting topic for future research, emphasizing the fact that the analysis of the FSO scenario implies to take into account aspects different to those present in the study of indoor optical communications. In this sense, conclusions in [37] from the analysis by simulation results of the rate-adaptive transmission technique only based on variable silence periods, simply adapting to FSO scenario from the indoor optical communications [35], such as proposing the use of shortened pulses as the central core of this adaptive transmission scheme, and its superiority compared to schemes based on variable-rate repetition coding, must be revised in the light of the analytical results presented in this paper: firstly, in relation to the impact on performance of the pulse shape adopted, providing an additional coding gain with independence of the rate-adaptive scheme employed or even in absence of rate-adaptive transmission; and, secondly, in relation to the benefit of using the available time-diversity order, aspect ignored when the rate-adaptive scheme is only based on variable silence periods.

## Acknowledgments

The authors are grateful for financial support from the Junta de Andalucía (research group “Communications Engineering (TIC-0102)”).

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