## Abstract

In free space optical (FSO) communication, atmospheric turbulence causes fluctuation in both intensity and phase of the received light signal what may seriously impair the link performance. Additionally, turbulent inhomogeneities may produce optical pulse spreading. In this paper, a simple rate adaptive transmission technique based on the use of variable silence periods and on-off keying (OOK) formats with memory is presented. This technique was previously proposed in indoor unguided optical links by the authors with very good performance. Such transmission scheme is now extensively analyzed in terms of burst error rate, and shown in this paper as an excellent alternative compared with the classical scheme based on repetition coding and pulse-position modulation (PPM), presenting a greater robustness to adverse conditions of turbulence.

© 2010 Optical Society of America

## 1. Introduction

Atmospheric optical communication is receiving considerable attention recently for use in high data rate wireless links [1–6]. Considering their narrow beamwidths and lack of licensing requirements as compared to microwave systems, atmospheric optical systems are appropriate candidates for secure, high data rate, cost-effective, wide bandwidth communications. Furthermore, atmospheric free space optical (FSO) communications are less susceptible to the radio interference than radio-wireless communications. Thus, FSO communication systems represent a promising alternative to solve the last mile problem, above all in densely populated urban areas.

However, signal fading due to atmospheric turbulence can degrade seriously the performance of atmospheric optical links. In this respect, inhomogeneities in the temperature and pressure of the atmosphere lead to variations of the refractive index along the transmission path. As a consequence of this phenomenon, the quality of the received signal is deteriorated, undergoing fluctuations in both the intensity and the phase of an optical wave propagating through this medium. Such fluctuations can lead to an increase in the link error probability limiting the performance of communication systems. In this specific scenario, the turbulence-induced fading is called scintillation. Certainly, weather-induced attenuation caused by rain, snow and fog can also degrade the performance of atmospheric optical communication systems in the way shown in [7], but are not considered in this paper.

Nevertheless turbulence-induced intensity fluctuations may be quicker or slower according to a parameter called the correlation time of intensity fluctuations, *τ*
_{0}. This *τ*
_{0} influences on the atmospheric coherence and thus in the burst error rate, being an important specification in the quality of a system with a burst-mode transmission. To study this effect, a computationally efficient model for the atmosphere as an optical communication channel was presented in [8] and extended in [9, 10]. Taking into account the proposed channel model, the propagation of pulses in turbulent atmosphere is reviewed and a discussion about pulse distortion and broadening due to eddy motion is depicted. From this analysis, temporal spreading may only be considered at high data rate in the way shown in [10], particularly when operating in special scenarios where dust particles are likely present. This fact more than justifies the promotion of a simple rate-adaptive transmission scheme based on the use of a variable silence periods and on-off keying (OOK) formats with Gaussian pulses (OOK-GS) and reduced duty cycle [11] in most high-performance links. This technique was previously proposed in indoor unguided optical links by the authors and now it is adapted to atmospheric optical links, corroborating a very good performance in terms of burst error rate. A memory-version of such rate-adaptive transmission scheme is also provided and shown to be an efficient alternative for atmospheric optical wireless communications, corroborating a great robustness to adverse channel conditions.

## 2. Optical communication through turbulent atmosphere

Atmospheric turbulence can be physically described by Kolmogorov cascade theory [12–14]. Turbulent air motion represents a set of eddies of various scales sizes. Large eddies become unstable due to very high Reynolds number and break apart, so their energy is redistributed without loss to eddies of decreasing size until the kinetic energy of the flow is finally dissipated into heat by viscosity. The scale sizes of these eddies extend from a large scale size, *L*
_{0}, called the outer scale of turbulence, to a small scale size, *l*
_{0}, denoted the inner scale of turbulence, the scale where the energy is dissipated into heat. It is assumed that each eddy is homogeneous, although with a different index of refraction. These atmospheric index-of-refraction variations produce fluctuations in the irradiance of the transmitted optical beam, what is known as *atmospheric scintillation*. A widely used model with good accuracy to describe the spatial power spectrum of refractive index, Φ_{n}(*κ*) was proposed by Kolmogorov, which assumes the wavenumber spectrum to be:

as indicated in [12]. In Eq. (1), *κ* is the spatial wave number and *C*
^{2}
_{n} is the refractive-index structure parameter. Under the so-called Rytov approximation [14], the optical field, *u*(**r**), of an optical wave propagating at distance *L* from the source can be expanded as [6]:

where **r** is the observation point in the transverse plane at propagation distance *L*, *A*(**r**,*L*) is the amplitude of the electric vector of the optical wave and *u*
_{0}(**r**,*L*) is the optical field amplitude without air turbulence expressed, from [6], as *u*
_{0}(**r**,*L*) = *A*
_{0}(**r**,*L*) · exp [*jϕ*
_{0}(**r**,*L*)]. Finally, the exponent of the perturbation factor is:

Thus, in this latter equation, *χ* is the log-amplitude fluctuation and *S* is the phase fluctuation. The resulting expression for Φ_{1}, described in [14], is directly related to the continuum scale sizes of the turbulent eddies and so, by the Central Limit Theorem, Φ_{1}(**r**) is governed by Gaussian statistics and, therefore also, the log-amplitude, *χ*(**r**), and the phase, *S*(**r**), of the optical wave. Assuming that the receiving aperture is smaller than the correlation length of the irradiance fluctuations, i.e., the aperture behaves essentially as a point detector [12]; and since a system based on an intensity modulation and direct detection (IM/DD) scheme is considered in this paper, then the irradiance fluctuations are primarily concerned. Thus, if we apply the definition *I*(**r**,*L*) = *U*(**r**,*L*)*U*
^{*}(**r**,*L*), then the light intensity, *I*(*t*), is related to *χ*(*t*) by

as shown in [12]. In Eq. (4), *I*
_{0} = ∣*A*
_{0}∣^{2} is the level of irradiance fluctuation in the absence of air turbulence that ensures that the fading does not attenuate or amplify the average power, i.e., *E*[*I*] = ∣*A*
_{0}∣^{2}. This may be thought of as a conservation of energy consideration and requires the choice of *E*[*χ*] = *−σ*
^{2}
_{χ}, as was explained in [15, 16], where *E*[*χ*] is the ensemble average of log-amplitude, whereas *σ*
^{2}
_{χ} is its variance. ence, from the Jacobian statistical transformation, the probability density function of the intensity can be identified to have a log-normal distribution typical of weak turbulence regime [6]. With these considerations taken into account, an efficient channel model for FSO communications using intensity modulation and direct detection was presented in [8] under the assumption of weak turbulence regime. For these systems, the received optical power, *Y*(*t*), can be written as *Y*(*t*) = *α _{sc}*(

*t*)

*X*(

*t*) +

*N*(

*t*), with

*X*(

*t*) being the received optical power without scintillation, whereas

*α*(

_{sc}*t*) = exp[2

*χ*(

*t*)] is the temporal behavior of the scintillation sequence and represents the effect of the intensity fluctuations on the transmitted signal. To generate

*α*(

_{sc}*t*), a scheme based on a lowpass filtering of a random Gaussian signal is implemented as in [8].

*χ*(

*t*) is, as was explained above, the log-amplitude of the optical wave governed by Gaussian statistics. Finally, the additive white Gaussian noise,

*N*(

*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.

As was indicated in [8], to obtain the expression of the lowpass filter, *H _{sc}*(

*f*), we started with the covariance function for irradiance fluctuations,

*B*(

_{I}*r*), that for a plane wave and homogeneous and isotropic turbulence leads to [17]:

where *k* is the wave number, *L* is the propagation path length, *κ* is the scalar spatial wave number, Φ_{n}(*κ*) is the spatial power spectrum of refractive index and *J*
_{0}(·) is the Bessel function of the first kind and order zero. Based on the Kolmogorov spectrum given in Eq. (1) and after some mathematical manipulations indicated in [8], it is possible to obtain

being *τ*
_{0} the turbulence correlation time, whereas *u*
_{⊥} is the component of the wind velocity vector perpendicular to the propagation direction. Hence, the low-pass filter can be obtained from

This will be the channel model considered in this paper in order to include the atmospheric coherence effects in numerical simulations presented in Section 3. As a final remark, the simplified Gaussian channel model differs from the theoretical one obtained from Eq. (5) when the frequency spread is calculated, as can be deduced by inspecting the tails of the different spectrums displayed in [8]; but there is no difference between Gaussian and Kolmogorov bells at lower frequencies, i.e. there is no shift of the relative frequency content of the irradiance power spectrum, so the temporal cycles of the scintillation sequences obtained from both the Gaussian and the Kolmogorov model have the same temporal variability. In addition, both models are in total agreement when the average frequency shift is obtained, as it is developed in Appendix I.

#### 2.1. Propagation of Pulses Through Atmospheric Turbulence

Furthermore, in atmospheric optical communications, propagating pulses may be influenced by pulse spreading owing to the scattering caused by turbulent inhomogeneities and hydrometeors in the troposphere, and especially sand and dust particles, which are likely present in extreme scenarios such as the one detailed in [18], and causing a distortion of the optical pulse shape during propagation [19]. This problem was studied in [20], requiring a solution for the statistical moments of the wave field at different frequencies and at different positions. It is useful to know the two-frequency mutual coherence function (MCF), studied by many researchers [21,22]. The MCF is an important quantity by itself as it provides a measure of the coherence bandwidth [13]. From the MCF, the pulse broadening can be calculated [21] for a Gaussian beam. Thus, the estimate of the received pulse half-width is given by:

with *T*
_{0} being the input pulse half-width at the 1/*e* point [23] whereas the remaining parameter is defined by:

where *L*
_{0} is the outer scale of turbulence, *C*
^{2}
_{n} is the index of refraction structure parameter and *c* is the speed of light. Some specific examples for pulse broadening were calculated, from Eq. (8), in [23], finding that at optical frequencies, only pulses with width less than 1 ps (10^{−12} s) duration have considerable broadening in horizontal paths. Therefore, taking such conclusion into account, we can promote a rate-adaptive transmission technique based on shortening the width of the transmitted pulse shape in order to accommodate the transmission rate to the channel conditions as an efficient technique for atmospheric optical links because pulse broadening does not limit its performance, at least, not if the binary rate is under various hundreds of Gbps. If it were necessary, this temporal spreading may be included in the proposed channel model [8, 9] in the way detailed in [10], particularly when operating in special scenarios where dust particles are likely present. In such work, the long-term temporal broadening of a space-time Gaussian pulse propagating along a horizontal path through weak optical turbulence was modeled by a Bessel filter, where its cutoff frequency is related to the physical parameters of the link. So, if necessary, we can include this effect in a direct way to our numerical simulation model. The effect of broadening implies the appearance of a pulse-pulse interference that may seriously degrade the performance of the system. Nevertheless, a signalling technique with a higher peak-to-average optical power ratio (PAOPR) parameter, as the schemes presented in this paper, can compensate for the higher distortion induced on transmitted pulses by the effect of broadening.

## 3. Rate-Adaptive Links Using Variable Silence Periods as a Promising Alternative

#### 3.1. General Comments

To study the performance of both Kolmogorov and the proposed Gaussian spectra, IM/DD links are assumed operating through a 250 m horizontal path at a bit rate of 50 Mbps and transmitting pulses with either pulse-position modulation (PPM) and OOK formats under the assumption of equivalent bandwidth of 50 MHz. The criterion of constant average optical power is adopted being one of the most important features of IM/DD channels, as we show through this section. In relation to the detection procedure, a maximum likelihood (ML) detection and a soft-decision decoding are considered respectively for the PPM and for the OOK format unless the contrary is indicated. A 830-nm laser wavelength is employed. All these features are included in the system model [8], where its remarkable elements are: first, the channel model depicted in this paper corresponding to a turbulent atmospheric environment, where the component of the wind velocity transverse to the propagation direction, *u*
_{⊥} is taken to be 8 m/s, a typical magnitude, at least in southern Europe. This is the main reason to employ this concrete magnitude. On the other hand, the values of turbulence strength structure parameter, *C*
^{2}
_{n} were set to 1.23 × 10^{−14} and 1.23 × 10^{−13} m^{−2/3} for *σ*
^{2}
_{χ} = 0.01 and 0.1, respectively and for plane waves; second, a three-pole Bessel high-pass filter with a −1 dB cut-off frequency of 500 kHz for natural (solar) light suppression; and last, a five-pole Bessel low-pass filter employed as a matched filter. The receivers employed here are point receivers whereas the weather-induced attenuation is neglected so that we concentrate our attention on turbulence effects. Furthermore, the atmospheric-induced beam spreading that causes a power reduction at the receiver is also neglected because we are considering a terrestrial link where beam divergence is typically on the order of 10 *µ*Rad.

As a remarkable comment, with the inclusion of a wind speed, concretely 8 m/s as was said before, we can study the effect of the channel coherence in terms of burst error rate [8] so that we obtain highly reliable link performance predictions. In addition, in urban atmospheres, especially near or among roughness elements, strong wind shear is expected to create high turbulent kinetic energy, as was detailed in [24,25]. In such assumptions, we could have employed a higher magnitude for the wind speed without loss of generality. This fact even avoids a higher numerical complexity when we generate the lognormal scintillation sequence. Finally, and for simplicity, we assume that the wind direction is entirely transverse to the path of propagation. For special scenarios where Taylor’s hypothesis may not be fully satisfied (scenarios affected by strong wind shear, urban environments or tropical areas), the procedure needed to generate the scintillation pattern may be modified as detailed in [9]. In such cases, scintillation sequences registered by a receiver will not be identical to the patterns seen by another receiver except for a small shift in time, but the entrance of new structures into the optical propagation path may introduce new fluctuations into the received irradiance. Although Taylor’s hypothesis is a good estimate for many cases, and for mathematical convenience this Taylor’s hypothesis is assumed to be fully satisfied in this paper, however, the corrections proposed in [9] may be very useful to obtain more realistic results in particular environments.

Then, with all those values detailed at the begining of this section, and from Eqs. (8), (9), the received pulse half-width defined by the 1/*e* point [10,23] is on the order of *T*
_{2} ≈ *T*
_{0} = 4.73 ns, being the broadening negligible, as we had anticipated above. To obtain an appreciable broadening on the order of 10% of *T*
_{0}, we could use transmission rates up to 87.8 Gbps and 278 Gbps for *C*
^{2}
_{n} = 1.23 × 10^{−13} and 1.23 × 10^{−14} m^{−2/3} respectively with a 100% of duty-cycle (d.c.) and *L*
_{0} = 30.34 m (this last value coincident with the height of our emplacement). These values let us utilize very high binary rates or ultrashort transmitted pulses and still neglecting the temporal broadening of the transmitted pulses. Thus, in this paper, we take advantage of this latter fact by adopting pulses with reduced duty cycle. As the criterion of limited average optical power is adopted, the signal amplitude can be increased as the duty cycle is decreased in order to maintain constant the average optical power, and then, the peak-to-average optical power ratio (PAOPR) can be higher. As shown through this section, a technique that increases the PAOPR parameter is preferred as it provides better performance in atmospheric optical links, overcoming the imposed distortion when a system bandwidth constraint is required. The obtained performance for all analyzed signaling techniques are in terms of burst error rate average. Hence, the impact of the atmospheric channel coherence on the behavior of the different signalling schemes can be taking into account, as was indicated in [8], due to the burst error rate average represents a second order of statistics and so, the temporal variability of the received irradiance fluctuations can influence on such metric of performance; however, this fact is not considered simply by doing a bit error rate analysis since bit error rate does not change with the variable wind speed, i.e., bit error rate is the first order of statistics and, consequently, it is just a function of the lognormal channel variance. Thus, we followed Deutsch and Miller’s [26] definition of a burst error with lengths of 192 and 64 bits respectively for each figure, not containing more than 4 consecutive correct bits (*L _{b}* = 5 as explained in [26]) any sequence of burst error.

#### 3.2. Peak-to-average optical power ratio (PAOPR)

Let *x*(*t*) the instantaneous optical power defined by

where *a _{k}* is a random variable with values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse),

*P*is the pulse peak power,

_{peak}*p*(

_{n}*t*) is the pulse shape with normalized peak power, and

*T*is the bit period. Then, the average transmitted optical power can be expressed as

_{b}*P*=

*P*·

_{peak}*P*·

_{n}*pr*(

*a*= 1), where

_{k}*P*is the optical power averaged over

_{n}*T*given by ${P}_{n}=\frac{1}{{T}_{b}}{\int}_{0}^{{T}_{b}}{P}_{n}\left(t\right)dt$ , and $pr({a}_{k}=1)=\frac{1}{2}$ as we assume equiprobability of binary symbols. Thus, we can write the peak-to-average optical power ratio as

_{b}From Eq. (11), it is deduced that we must reduce *P _{n}*, depending on the adopted pulse shape, to improve the PAOPR while maintaining the average transmitted optical power at a constant level. Hereafter, to compare the different pulses performance, we define the increase factor in PAOPR as follows:

where PAOPR_{ref} is the PAOPR obtained with non-return to zero (NRZ) signaling and rectangular pulse shape, i.e. *P _{nref}* = 1. Note that Γ

_{PAOPR}only depends on the pulse shape characteristics.

Now, we obtain the Γ_{PAOPR} for a rectangular pulse with a duty cycle of *ξ*, where 0 < *ξ* ≤ 1. In this case, from Eq. (12), it is easily derived that

Next, the expression for a normalized Gaussian pulse centered in *t*
_{0} = *ξT _{b}*/2 is given by

where *σ _{p}* =

*ξ · σ*

_{0}depends on the duty cycle,

*ξ*, and on

*σ*

_{0}=

*T*/

_{b}*n*, which characterizes the pulse width in relation to the bit period. In particular,

*n*defines the amount of optical energy contained within

*T*, so that for

_{b}*n*≥ 6 at least the 99.8% of the transmitted energy is within

*T*. In our analysis, we assume

_{b}*n*= 6 and, thus, from Eq. (12) and Eq. (14), the Γ

_{PAOPR}for the Gaussian pulse is given by

where erf(·) is the error function. Its pulse power spectral density can be easily calculated and, of course, it can be checked that the occupied bandwidth of Gaussian pulse shape is quite larger than rectangular one. However, the higher *P _{peak}* transmitted more than compensates for the higher distortion induced on Gaussian pulse by a limited channel bandwidth.

Table 1 summarizes the increase in PAOPR for the analyzed pulse shapes and several values of the duty cycle, *ξ*, and, as expected, the Γ_{PAOPR} offered by the Gaussian pulse is substantially higher than the one obtained by the rectangular pulse shape.

In this respect, OOK-format with Gaussian pulse shapes (OOK-GS) and reduced duty cycle are employed in a first scenario shown in Fig. 1. Their performance are compared with a classic NRZ format, both formats with two different values of duty cycle and for two different intensities of turbulence: *σ*
^{2}
_{χ} = 0.1 and 0.01. In addition, a classic signalling technique based on variable-rate repetition coding is implemented, where *RR* is the rate reduction factor. As we can observe, Gaussian pulse shapes obtain better performance than rectangular pulse shapes (for a same duty cycle and a same rate reduction factor). Moreover, when the duty cycle decreases a better performance is achieved, obtaining that an OOK-GS format and a 25%-duty cycle achieves a remarkable better performance than an identical OOK-GS format but with a higher duty cycle. In addition, a Gaussian pulse shape presents better performance than another one with a NRZ format. These two aspects confirms that an increase in the PAOPR parameter provides better performance in the atmospheric optical link as it is depicted in Fig. 1.

Furthermore, results obtained for the OOK-GS format are illustrated in Fig. 1 for the proposed Gaussian channel developed in [8,9] together with the results obtained from the theoretical covariance function shown in Eq. (5) including a Kolmogorov spectrum in Φ_{n}(*κ*). It should be further noted that our Gaussian channel approach works perfectly not only for OOK-GS format but also for all the remaining cases in this paper so results obtained from the use of the more precise expression in Eq. (5) are omitted here for the sake of brevity.

Figure 2 shows a comparative between the OOK-GS pulse shapes displayed in Fig. 1 and PPM scheme for *σ*
^{2}
_{χ}=0.1 and 0.01. Then, an OOK-GS format and a 25%-duty cycle already achieves similar performance than the 4PPM scheme, but with a lower complexity. Again, a classic signalling technique based on variable-rate repetition coding is implemented. The fact that broadening distortion does not affect to transmitted pulses until very reduced widths, and the better performance obtained in Fig. 2 for schemes with a reduced duty cycle are an interesting starting point to apply our rate-adaptive transmission technique as we explain below.

#### 3.3. Rate Adaptive Links Using Variable Silence Periods

In this paper, a rate-adaptive transmission technique that was developed in [11] for indoor optical links with promising results is now implemented in turbulent atmospheric optical communications and fully evaluated in terms of burst error rate. The scheme is based on the insertion of *RR _{s}* − 1 silence bit periods after an information bit so as to obtain an effective code rate with a rate reduction factor of

*RR*depending on the channel conditions. The average optical power is always maintained at a same constant level. So, in this scenario, $pr({a}_{k}=1)=\frac{1}{2R{R}_{s}}$ so, from Eq. (12) and Eq. (14), the Γ

_{s}_{PAOPR}for the Gaussian pulse is now given by

As we said above, the distortion of transmitted pulses may be neglected until ultrashort pulse widths of 1 ps or even less, so that we can take advantage of this feature to reduce the width of transmitted pulses in order to implement our rate-adaptive transmission technique. The idea is applied to accommodate the transmission rate to the channel conditions using variable silence periods. In this respect, the worse channel conditions we have, the higher the rate reduction *RR _{s}* we need to provide a significant improvement in terms of burst error rate. Thus, the increase in the PAOPR parameter is then used being the average optical power maintained at a constant level, and for a same system bandwidth constraint.

Figure 3 shows a comparison between this rate adaptive technique using variable silence periods and the classic one proposed and specified in the Advanced Infrared (AIr) standard based on variable-rate repetition coding in a turbulent atmosphere and for *σ*
^{2}
_{χ} = 0.1 and *σ*
^{2}
_{χ} = 0.01. In such figure, all results are obtained for OOK-GS formats and a 25%-duty cycle. From Fig. 3, a remarkable improvement in performance can be noted when the rate adaptive technique based on inserting variable silence periods is adopted in relation to the classic variable-rate repetition coding, because there is no alteration in the statistics of occurrence of the transmitted pulses in this latter option and, consequently, Γ_{PAOPR} is not increased. This superiority is even more relevant when the rate reduction factor is increased, showing a greater robustness to the more severe atmospheric turbulence conditions. For instance, the technique based on variable silence periods can achieve an improvement in average optical power requirements above 1 and 2 optical dB at a burst error rate of 10^{−6} for a rate reduction of 2 and 4 respectively and *σ*
^{2}
_{χ} = 0.1 in regard to classic variable-rate repetition coding.

#### 3.4. Rate Adaptive Links Using Variable Silence Periods and Memory

Now, this variable silence time periods are used for modifying the statistics of the amplitude sequence regarding the statistics of the message sequence with the purpose of increasing the PAOPR parameter. Thus, a signaling technique based on giving memory by using the pulse position is included on the rate-adaptive technique based on inserting silence periods. The resulting signalling scheme was named OOK-GSc format as was described in [11]. OOK-GSc format is designed to avoid the appearance of more than one pulse in sets of two consecutive symbol periods, increasing the peak optical power while maintaining the average optical power at the same constant level, as before. Figure 4 shows a generic scheme where this OOK-GSc format is combined with the rate-adaptive technique based on inserting silence periods, where *RR _{b}* represents the initial bit rate of the system,

*RR*is the rate reduction factor inherent to the OOK-GSc scheme,

_{c}*RR*is the rate reduction factor introduced by inserting

_{s}*RR*− 1 silence bit periods after an information bit, as explained above, and

_{s}*RR*=

*RR*·

_{c}*RR*is the total effective rate reduction factor of the system.

_{s}OOK-GSc is based on a Markov chain of three states, providing two coded bits each information bit and thus, representing a rate reduction factor of *RR _{c}*=2. Then, and when

*RR*=1, $pr({a}_{k}=1)=\frac{1}{3RR}$ , so from Eq.(12) and Eq.(14), the Γ

_{s}_{PAOPR}for a Gaussian pulse shape is now given by

Nonetheless, the adopted approach to configure the OOK-GSc format can be generalized to avoid the appearance of more than one pulse in sets of more than two consecutive symbol per-iods. For instance, following [11], we can adopt the OOK-GScc scheme, where the appearance of more than one pulse in sets of three consecutive symbol periods is avoided. In this case, OOK-GScc format is based on a Markov chain of nine states, providing four coded bits each information bit and thus, representing a rate reduction factor of *RR _{c}* = 4. Hence,
$pr({a}_{k}=1)=\frac{1}{4RR}$
if

*RR*= 1, so from Eq.(12) and Eq.(14), the Γ

_{s}_{PAOPR}for a Gaussian pulse is now given by

We can employ the same generic scheme shown in Figure 4, but substituting the OOK-GSc stage for an OOK-GScc technique. Then, this OOK-GScc format is combined with the already explained rate-adaptive technique based on inserting silence periods; where, again, *RR _{b}* represents the initial bit rate of the system,

*RR*= 4 is the rate reduction factor inherent to the OOK-GScc scheme,

_{c}*RR*is the rate reduction factor introduced by inserting

_{s}*RR*− 1 silence bit periods after an information bit and

_{s}*RR*=

*RR*·

_{c}*RR*is the total effective rate reduction factor.

_{s}Figure 5 shows obtained results for the OOK-GS format using variable silence times with and without memory in a turbulent atmospheric environment for *σ*
^{2}
_{χ} = 0.1 and *σ*
^{2}
_{χ} = 0.01. As it is shown, a remarkable improvement in performance can be observed when OOK-GS format with memory is adopted if compared with the OOK format without memory, where excellent agreement can be noted by using different rate reductions. Nevertheless, OOK-GSc and OOK-GScc formats present a very similar behavior in terms of burst error rate for an identical rate reduction factor. This fact may not comply with the Γ_{PAOPR} heuristic law in the sense that the higher magnitude in Γ_{PAOPR} we have the better performance we should expect from a signalling technique. However, the answer to this apparent incongruence raises from the definition adopted in this paper for a burst error in addition to the particular distribution of error bits in the different signalling techniques with memory analyzed in this paper in conjunction with the Viterbi algorithm employed in reception. In this sense, Fig. 6 shows the number of occurrences for different sets of consecutive correctly received bits for both the OOK-GSc scheme with a rate reduction factor of *RR*=4 (where *RR*=*RR _{c}* ·

*RR*, with

_{s}*RR*=2 and

_{c}*RR*=2); and the OOK-GScc format with its inherent rate reduction factor of 4, for different values of additive white Gaussian noise power.We can check that the occurrence of sequences consisting of an association of ten or more consecutive correct bits is higher when using an OOK-GScc scheme than employing an OOK-Gsc format. This fact is effectively confirmed when we change the definition of a burst error. Now, if any burst error can contain until 9 consecutive correct bits (

_{s}*L*= 10 as explained in [26]), then we can obtain the expected results in terms of burst error rate, as displayed in Fig. 7, where the superiority of OOK-GScc format versus OOK-GSc one is again established, as expected from the heuristic analysis of the Γ

_{b}_{PAOPR}parameter.

Moreover, a bit-error study is derived to show the lower total number of error bits obtained from the OOK-GScc format with regard to the OOK-GSc scheme. Since bit error rate (BER) does not change with the variable wind speed, i.e., BER is the first order of statistics, then the BER performance of the link is just a function of the lognormal channel variance. Thus, Fig. 8 includes the performance of both OOK formats with memory studied in this paper in terms of bit error rate. In this respect, a significant improvement in performance is achieved when OOK-GScc format is adopted. For instance, at a bit error rate of 10^{−6} for a rate reduction of 4 and 8 and *σ*
^{2}
_{χ}=0.1, OOK-GScc obtains an improvement in average optical power requirements above 1.2 and 1.4 optical dB respectively with respect to OOK-GSc format.

Finally, to complete this comparative analysis, we use a Gaussian pulse shape for configuring the PPM scheme in order to obtain the 4GPPM format, as was carried out in [27]. This scheme shows a remarkable improvement with regard to the classic PPM as it is deduced when comparing their performance shown respectively in Fig. 2 and Fig. 9 (both figures, again, with *L _{b}* = 5). In this respect, the same conclusion that was obtained before for the OOK scheme is deduced now for the 4GPPM format: hence, when a Gaussian pulse shape is employed instead of a rectangular pulse for all values of

*RR*, then an improvement in performance is obtained, confirming that an increase in the PAOPR parameter provides better performance and a great robustness in the atmospheric optical link for a same system bandwidth constraint. The validity of this conclusion is extended not only for a ML detection procedure but also for a threshold (TD) detector, although with an important degradation in performance as a direct consequence of a more rudimentary detection algorithm. All in all, as was shown in Fig. 9, the inclusion of memory with OOK-GS formats provides a very attractive improvement in performance being, furthermore, the studied scheme with a higher PAOPR. Hence, this latter ratio can be used as a reliable figure of merit in IM/DD atmospheric optical links, where an increase in such ratio involves an increase in performance, as was effectively demonstrated through this paper.

## 4. Concluding Remarks

In this paper, an alternative rate-adaptive transmission technique successfully proposed in indoor optical communications is adapted here to atmospheric optical links. Such rate-adaptive technique is based on the fact that an increase in the peak-to-average optical power ratio involves higher performance. This feature is intensively studied in this paper providing an extremely complete numerical analysis. To increase such figure of merit (PAOPR), pulses having a shortened duty cycle must be employed taking into advantage the fact that it is possible to neglect temporal broadening, unless until various hundred of Gbps if atmospheric conditions are weaker, as was indicated in Section 3. This fact is the central core of the alternative rate adaptive transmission scheme successfully proved in this paper for turbulent environments. Pulses with reduced duty cycle in addition to the possibility of diminishing the presence of transmitted pulses by employing memory on OOK formats with Gaussian pulse shapes increase significantly the PAOPR parameter. In this sense, better performance are obtained for such signalling schemes as it is shown in the different comparatives included in this work as, for instance, versus 4PPM, 4GPPM or versus classic variable rate repetition coding scheme. Thus, a greater robustness is provided with the proposed signalling scheme with a low required complexity. Finally, we are currently working with pulse shapes based on the power pulse shape of the widely known optical soliton, corresponding to the hyperbolic secant square function, whose PAOPR parameter is higher than the one obtained for Gaussian and rectangular pulses [28]. Qualitatively, an optical power pulse shape for different normalized pulses, *pN*(*t*), with a reduced duty cycle, *ξ*, is presented in Fig. 10, showing that the solitonic pulse shape is narrower than rectangular and Gaussian pulses, so that we can increase the peak optical power to a higher lever maintaining the average optical power at the same constant level for all the pulse shapes under study. Then, as concluded in this paper, an increase in the peak-to-average optical power ratio involves higher performance. In this case, the higher pulse peak power transmitted, *P _{peak}*, compensates for the higher distortion induced on solitonic pulse by the system limited bandwidth, so a better performance is obtained when increasing the PAOPR parameter. In addition, we are currently researching about mode locking solitons [29, 30] and the possibility to properly employ them in an atmospheric optical communication environment.

## Appendix I

In this Appendix, it is shown that the average frequency shift is coincident for both the theoretical covariance function shown in Eq. (5); and the Gaussian spectrum obtained from Eq. (6) and employed in this paper. This average frequency shift is defined, from [31], as:

For the proposed Gaussian spectrum given in Eq. (7) the following relation holds:

where, for *τ* = 0, *B _{I}*(0) =

*σ*

^{2}

_{1}and $\frac{d}{d\tau}\left({B}_{I}\left(0\right)\right)=0$ , so ${B}_{{\mu}_{i}{\mu}_{i}}^{\left(1\right)}=0$ . On the other hand, for the theoretical spectrum obtained from Eq. (5), solving the integral for a Kolmogorov spectrum, we obtain

where the convergence of Eq. (21) is ensured due to the principle of analytic continuation. From Fried [32],

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.3em}{0ex}}-\mathrm{cos}\left(\frac{\pi \alpha}{2}\right)\mathrm{Im}\left[{\phantom{\rule{.2em}{0ex}}}_{1}{F}_{1}(\alpha ;1;j\frac{{\beta}^{2}}{4})\right]\}.$$

Thus, using Eqs. (22), (23) in Eq. (21), and knowing that *r* = *u*
_{⊥}
*τ*,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+\Gamma \left(-\frac{11}{6}\right)\mathrm{Im}\left\{\mathrm{exp}\left(j\frac{11\pi}{12}\right){\phantom{\rule{.2em}{0ex}}}_{1}{F}_{1}\left(-\frac{11}{6},1,j\frac{k{u}_{\perp}^{2}{\tau}^{2}}{4L}\right)\right\}].$$

where, obviously, for *τ* = 0, the temporal covariance function written in Eq. (21) reduces to *B _{I}*(0) = 1.23

*C*

^{2}

_{n}

*L*

^{11/6}

*k*

^{7/6}=

*σ*

_{2}

_{1}, i.e., the scintillation index. To obtain $\frac{d}{d\tau}\left({B}_{I}\left(0\right)\right)$ , we use the following property associated with the confluent hypergeometric function of the first kind:

being (*a*)_{k} the Pochhammer symbol. Thus, to obtain
$\frac{d}{d\tau}\left({B}_{I}\left(0\right)\right)$
, *k*=1 in Eq. (25). If
$z=j\frac{k{r}^{2}}{4L}$
and *r*=*u*
_{⊥}
*τ*, and applying the chain rule that in the Leibniz notation is stated in the form

then:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}-\frac{11}{6}\Gamma \left(-\frac{11}{6}\right)\mathrm{Im}\left\{\mathrm{exp}\left(j\frac{11\pi}{12}\right)j\frac{k{u}_{\perp}^{2}\tau}{2L}{\phantom{\rule{.2em}{0ex}}}_{1}{F}_{1}(-\frac{5}{6},2,j\frac{k{u}_{\perp}^{2}{\tau}^{2}}{4L})\right\}].$$

Therefore, it is possible to obtain that $\frac{d}{d\tau}\left({B}_{I}\left(0\right)\right)=0$ and, thus again, ${B}_{{\mu}_{i}{\mu}_{i}}^{\left(1\right)}=0$ .

## Acknowledgment

This work was fully supported by the Spanish Ministerio de Educación y Ciencia (MEC), Project TEC2008-06598.

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