## Abstract

Atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance. In this paper, a scheme combining transmit laser selection (TLS) and space-time trellis code (STTC) for multiple-input-single-output (MISO) free-space optical (FSO) communication systems with intensity modulation and direct detection (IM/DD) over strong atmospheric turbulence channels is analyzed. Assuming channel state information at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available *L* lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas. Based on a pairwise error probability (PEP) analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2*L*-1 when *L* transmit lasers are available and a simple two-state STTC with rate 1 *bit*/(*s∙Hz*) is used. Simulation results are further demonstrated to confirm the analytical results.

© 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*, above all in densely populated urban areas, as well as a supplement to radio-frequency (RF) links [2] and the recent development of radio on free-space optical links (RoFSOLs) [3,4]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance [5,6]. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [7–13]. In particular, heuristic space-time code (STC) designs such as repetition codes (RCs) [14–16] and orthogonal space-time block codes (OSTBCs) [17–19] have been proposed for FSO systems with IM/DD.

It must be emphasized that a simple translation of the analysis of STCs from RF systems is not plausible due to peculiarities proper to FSO scenario. It is well known from the vast literature on wireless RF systems that simply sending the same signal from different antennas (i.e., repetition coding) does not realize any transmit diversity advantage. However, in [20] it was shown that simple RCs not only are able to extract full diversity but also always outperform OSTBCs, because of the fact that the transmitted signal is an intensity and, hence, it is subject to a nonnegativity constraint. In this way, unlike in the RF case [21], performance bounds and systematic design guidelines for general FSO STCs are not available. In [22], 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.

Selection transmit diversity technique is well known for RF systems, presenting a vast amount literature on RF multiple-input-multiple-output (MIMO) schemes that pay attention to transmission using such scheme, being a promising approach for reducing complexity since one can employ a reduced number of RF chains [23–25]. This idea is proposed in [26, 27] 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 the STC designs previously commented, such as OSTBCs and RCs, implementing transmit diversity but not using channel state information (CSI) at the transmitter (CSIT). The knowledge of CSIT is feasible for FSO channels given what scintillation is a slow time varying process relative to the large symbol rate. This has been recently considered for FSO links from the point of view of information theory [28, 29].

In this paper, a scheme combining transmit laser selection (TLS) and space-time trellis code (STTC) for multiple-input-single-output (MISO) FSO communication systems with IM/DD over strong atmospheric turbulence channels is analyzed, where the turbulence-induced fading is described by the negative exponential and K distributions and the channel fade level is tracked by both the transmitter and receiver. Assuming CSI at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available *L* lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas, providing a better performance compared to a similar selection transmit diversity scheme investigated previously by the authors for uncoded links [26, 27]. Here, STTC is considered in order to assume a fast fading channel, where the channel coefficients fade independently from one symbol to the next. Such a channel is a suitable model for a fully interleaved flat fading channel where an interleaver of length longer than the coherence time of the channel is employed. This may be done for example to make sure that the consecutive symbols transmitted see almost independent fades in an attempt to improve the diversity order, taking advantage not only of the number of transmit lasers but also of the shortest error event length in the trellis code [30]. In relation to difficulty of STTC design and decoding complexity and the requirement of a large number of transmit antennas in order to achieve a full diversity order [21], the combination of transmit laser selection and existing STTCs designed for two transmit antennas is a good approach to achieving a high diversity order while maintaining low decoding complexity at the receiver. Based on a pairwise error probability analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2*L*-1 when *L* transmit lasers are available and a simple two-state STTC with rate 1 *bit*/(*s ∙ Hz*) is used. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also on-off keying (OOK) formats with any pulse shape, corroborating the advantage of using pulses with high peak-to-average optical power ratio (PAOPR), such as Gaussian pulses with reduced duty cycle.

## 2. Atmospheric turbulence channel model

The use of infrared technologies based on IM/DD links is considered; in this way, having a single-input-single-output (SISO) system as a reference, the instantaneous current in the receiving photodetector, *y*(*t*), can be written as

where *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 *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 ∀*tx*(*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 [31]. 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. Considering strong turbulence conditions [5,16], negative exponential and K distribution for the i.i.d. channel irradiances can be assumed. The probability density function (PDF) corresponding to the K turbulence model is given by

where *α* is a channel parameter related to the effective number of discrete scatterers, Γ(∙) is the well-known Gamma function, and *K _{v}*(∙) is the

*v*th-order modified Bessel function of the second kind [32]. Since the mean value of this turbulence model is

*E*[

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

*E*[

*I*

^{2}] = (2 + 2/α), 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 [5, 18, 28], 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 any diversity order. This distribution can be seen as the K-distributed turbulence model in (2) when the channel parameter *α* → ∞.

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [18, 26]. A new basis function ϕ(*t*) is defined as ϕ(*t*) = *g*(*t*)/√*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}^{∞}

_{-∞}

*g*

^{2}(

*t*)dt is the electrical energy. In this way, an expression for the optical intensity can be written as

where *G*(*f* = 0) represents the Fourier transform of *g*(*t*) evaluated at frequency *f* = 0, i.e. the area of the employed pulse shape. The random variable (RV) *a _{k}* follows a Bernoulli distribution with parameter

*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

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. The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fast fading statistics. In spite of 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, this approach is valid because temporal correlation can in practice 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 [8,9,11,12].

## 3. Proposed transmit diversity scheme

The use of optical arrays, similar to the use of antenna-array technology for microwave systems, is considered as a means of combatting fading. In particular, we adopt a MISO array based on *L* laser sources, assumed to be intensity-modulated only and all pointed towards a distant photodetector, assumed to be ideal noncoherent (direct-detection) receiver. The sources and the detector are physically situated so that all transmitters are simultaneously observed by the receiver, and the separation distance between the lasers is large enough so that the fading experienced between source-detector pairs *I _{j}*(

*t*) is assumed to be statistically independent. Assuming channel state information at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available

*L*lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas [21]. To illustrate the proposed scheme, we adopt in this paper the example shown in Fig. 1, where a two-state STTC with rate 1

*bit*/(

*s∙Hz*) using OOK is displayed [33, Fig. 6.5.]. The incoming symbol stream is first encoded using the trellis structure and the encoded stream is then distributed among the two sources out of the available

*L*lasers corresponding to greater values of scintillation,

*I*

_{(L)}(

*t*) and

*I*

_{(L-1)}(

*t*), where

*I*

_{(1)}(

*t*),

*I*

_{(2)}(

*t*),…,

*I*

_{(L)}(

*t*) is a new sequence of

*L*auxiliary random variables obtained by arranging the random sequence

*I*

_{1}(

*t*),

*I*

_{2}(

*t*),…,

*I*(

_{L}*t*) in an increasing order of magnitude. The labeling

*i/kk*along each branch of the trellis refers to the input bit (

*i*) and the corresponding pair of output symbols (

*kk*) that result from the transition between the states at the beginning and end of the branch.

## 4. Performance analysis

In this section, an optical array based on *L* = {1,2,4,8} laser sources, all pointed towards a distant photodetector, is considered. We present aproximate closed-form expressions for the bit error rate (BER) using a pairwise error probability analysis when the scintillation follows negative exponential and **K** distributions, which cover a wide range of strong atmospheric turbulence conditions. The PEP represents the probability of choosing the space-time sequence **X**̂ when in fact the sequence **X** was transmitted [30, Chapter 16]. Assuming that the correct path is the all-zeros sequence, then for the shortest error event path of length *N* = 2 illustrated by shading in Fig. 1, we have

where each column of **X** and **X**̂ is associated with the two symbols transmitted from the two lasers in a given symbol interval (time slot) and *d* is the Euclidean distance in (6) corresponding to the OOK signaling (i.e., the OOK symbols are the elements of the **X** and **X**̂ matrices associated with the trellis). In the proposed scheme, for example, we associate the first and second rows with the (L-1)th and Lth order statistics corresponding to the scintillation. Under the assumption of perfect CSI, the conditional PEP with respect to scintillation coefficients of greater value, *I*
_{(L)} and *I*
_{(L-1)}, is given as [30, Chapter 16]

where *Q*(∙) is the Gaussian-*Q* function. Similar expressions to evaluate the pairwise error probability of coded FSO IM/DD links using OOK signaling can be found in [7,9]. Here, the division of *d* by 2 is considered so as to maintain the average optical power in the air at a constant level of *P*, being transmitted by each laser an average optical power of *P*/2. Substituting the value of *d* obtained in (6) gives

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 [6], knowing that PDF in (2) or (4) is normalized. Under the assumption of perfect interleaving, we can exploit independency among fading coefficients to obtain the average PEP,

*P*(

**X**→

**X**̂), by averaging (9) as follows

According to order statistics [34], for i.i.d. RVs of {*I _{j}*}

_{j=1,2,⋯L}, the PDF corresponding to

*I*

_{(L)}and

*I*

_{(L-1)}can be written as

being *F _{I}*(

*i*) the cumulative density function (CDF) corresponding to the turbulence model. An union bound on the average BER can be found as [30, eq. (13.44)]

where *P*(**X**) is the probability that the coded sequence **X** is transmitted, *n*(**X**,**X**̂) is the number of information bit errors in choosing another coded sequence **X**̂ instead of **X** and *n _{c}* is the number of information bits per transmission. Next, if we were to choose to approximate the average BER by considering only error event paths of minimum length (i.e.,

*N*= 2) [30, Section 14.6.4], we can use (10) to obtain

*P*(

_{b}*E*) ≃

*P*(

**X**→

**X**̂). To simplify the expression in (10), we use the approximation for the

*Q*-function presented in [35, Eq. (14)] (i.e.,

*Q*(

*x*) ≃ (1/12)exp(-

*x*

^{2}/2)+(1/4)exp(-2

*x*

^{2}/3)), finally obtaining

$$+\frac{1}{4}{\int}_{0}^{\infty}\mathrm{exp}\left(\frac{-\gamma \xi {i}_{1}^{2}}{3}\right){f}_{{I}_{\left(L\right)}}\left({i}_{1}\right)d{i}_{1}{\int}_{0}^{\infty}\mathrm{exp}\left(\frac{-\gamma \xi {i}_{2}^{2}}{3}\right){f}_{{I}_{\left(L-1\right)}}\left({i}_{2}\right)d{i}_{2}$$

To evaluate the improvement in performance, we compare the proposed scheme, which is referred to as the TLS/STTC scheme, with the transmit diversity technique TLS presented in [26] for uncoded FSO links, based on the selection of the optical path with a greater value of scintillation, where the average BER is given by

In the same way, we also include the performance corresponding to the STTC scheme when no transmit laser selection is used, where, following an approach as in (13), the average BER can be written as

#### 4.1. K atmospheric turbulence channel

Particularizing with the K distribution in (2) and using [36, Eq. (03.04.21.0013.01)] together with the fact that *K _{v}*(∙) is a even function with respect to its parameter, the derived integral for the CDF of the K channel can be written as

The results corresponding to this FSO scenario are illustrated in the Fig. 2, when different levels of turbulence strength of *α* = 1 and *α* = 4 are assumed, corresponding to values of scintillation index of *SI* = 3 and *SI* = 1.5, respectively, and where rectangular pulse shapes with ξ = 1 are used. Additionally, a relevant improvement in performance must be noted as a consequence of pulse shape used, providing an increment in the average SNR of 10log_{10} ξ *dB*. 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 ξ = 4/

*κ*√π is obtained when a Gaussian pulse of duration

*κT*as

_{b}*g*(

*t*) = exp (-

*t*

^{2}/2σ

^{2}) ∀|

*t*| <

*κT*/2 is adopted, where σ =

_{b}*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. Then, a Gaussian pulse shape with

*κ*= 0.25 is also adopted when

*L*= 2 in order to show the improvement in performance obtained with pulse shapes having a high PAOPR. Numerical results for TLS/STTC in (14), TLS in (15) and STTC without laser selection in (16) are computed using a symbolic mathematics package [37]. 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 for

*L*= {2,4,8}, as well as the greater diversity order for the transmit diversity technique here proposed if compared with TLS and STTC, being superior to the number of available transmit lasers

*L*.

#### 4.2. Exponential atmospheric turbulence channel

In this subsection, considering a limiting case of strong turbulence conditions [5,18,28], a negative exponential model is adopted to describe turbulence-induced fading, leading to an easier mathematical treatment to evaluate error rate performance for any number of transmit lasers. Here, particularizing with the negative exponential distribution in (4), the derived integral for the CDF of the exponential turbulent channel can be written as *F _{I}*(

*i*) = 1 - exp (-

*i*). Using the binomial theorem in (11) and (12), we obtain

Next, substituting (18) and (19) in (14) and evaluating the integrals by using [38, eqn. (7.4.32)], a closed-form solution for the aproximate average BER yields as

$$\left(4\mathrm{\pi erfc}\left(\frac{m}{\sqrt{\gamma \xi}}\right)\mathrm{erfc}\left(\frac{n}{\sqrt{\gamma \xi}}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\phantom{\rule{.2em}{0ex}}\left(\frac{{m}^{2}+{n}^{2}}{\gamma \xi}\right)+9\mathrm{\pi erfc}\left(\frac{\sqrt{3}m}{2\sqrt{\gamma \xi}}\right)\phantom{\rule{.2em}{0ex}}\mathrm{erfc}\phantom{\rule{.2em}{0ex}}\left(\frac{\sqrt{3}n}{2\sqrt{\gamma \xi}}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\phantom{\rule{.2em}{0ex}}\left(\frac{3\left({m}^{2}+{n}^{2}\right)}{4\gamma \xi}\right)\right)$$

where erfc(∙) is the complementary error function. The results corresponding to this FSO scenario are illustrated in the Fig. 3, where rectangular pulse shapes with *κ* = 1 are used.

As before in Fig. 2, a Gaussian pulse shape with *κ* = 0.25 is also adopted when *L* = 2 in order to show the improvement in performance obtained with pulse shapes having a high PAOPR. Here, results for TLS/STTC from evaluating the expression in (20) are displayed together with numerical results for the TLS and STTC schemes computed as in previous subsection. BER simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical results for *L* = {2,4,8}, as well as a better performance in terms of diversity gain, as previously concluded for the K channel. With the purpose of analyzing the diversity order achieved for the TLS/STTC scheme here proposed when *L* transmit lasers are available, we can use in (20) the series expansions corresponding to the exponential function [38, eqn. (4.2.1)] (i.e., exp (*x*) = ∑^{∞}
_{k=0}
*x ^{k}*/

*k*!) and the error function [38, eqn. (7.1.5)] (i.e., erfc(

*x*) = 1 - (2/√π)∑

^{∞}

_{k=0}(-1)

^{k}

*x*

^{2k+1}/((2

*k*+ 1)

*k*!)). In this way, it is straightforward to show that the average BER behaves asymptotically as 1/γ

^{(2L-1)/2}, corroborating a diversity gain of 2

*L*- 1 in relation to the absence of space-time trellis coding with laser selection, wherein the average BER varies as 1/γ

^{1/2}[17,18]. For example, in comparison to the selection transmit diversity scheme in [27], where a diversity order of

*L*can be demonstrated in a similar way as before, the additional use of the simple two-state STTC in Fig. 1 provides a performance improvement of 20 dB at a target BER rate of 10

^{-9}with only two transmit lasers.

## 5. Conclusions

In this paper, a scheme combining transmit laser selection and space-time trellis code for MISO FSO communication systems with IM/DD over strong atmospheric turbulence channels is analyzed, where the turbulence-induced fading is described by the negative exponential and K distributions and the channel fade level is tracked by both the transmitter and receiver. Assuming CSI at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available *L* lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas, providing a better performance compared to a similar selection transmit diversity scheme investigated previously by the authors for uncoded links [26, 27]. Based on a pairwise error probability analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2*L*-1 when *L* transmit lasers are available and a simple two-state STTC with rate 1 *bit*/(*s∙Hz*) is used. As revealed out by the results under no diversity assumption, a slow change in the slope of performance curve can be observed. This justifies the adoption of diversity techniques 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 strong turbulence. Additionally, the use of pulse shapes having a high PAOPR has shown to be a key factor to achieve a significant improvement in performance. From the relevant results here obtained when a simple two-state STTC is used, investigating in the FSO case the impact on the diversity order of STTCs designs of greater minimum symbol-wise Hamming distance, which is directly related to the shortest error event length in the trellis code, is an interesting topic for future research.

## Acknowledgments

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

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