Mitigating Turbulence-Induced Fading in Coherent FSO Links: An Adaptive Space-Time Code Approach

Abstract

Free space optical communication systems have witnessed a significant rise in attention over the last half a decade owing largely to their enormous bandwidth and relative ease of deployment. Generally, free space optical communication systems differ in their detection mechanism as various detection mechanisms are being reported, including intensity modulation/direct detection FSO, differential FSO and coherent FSO. In this chapter, we explore the prospect of obtaining an optimally performing FSO system by harnessing the cutting-edge features of coherent FSO systems and the coding gain and diversity advantage offered by a four-state space-time trellis code (STTC) in order to combat turbulence-induced fading which has thus far beleaguered the performance of FSO systems. The initial outcomes of this technique are promising as a model for various visible light communication applications.

1. Introduction

Telecommunication is one of the most important innovations in the history of mankind as it affords people the opportunity to communicate rapidly and reliably over long distances, often breaking physical and geographical barriers to make the world a global village as it is known today. One of the key ingredients in the heart of communication technologies over the last century is wireless communication. The advent of RF wireless communication techniques and protocols has been instrumental to the giant strides made in the communication domain as it eliminates the cumbrous requirement of lengthy wired connections, a requirement which has often been a great limitation for wired communication systems.

RF wireless communication systems enjoyed significant attention and penetration but soon became a victim of its success as more and more contents are demanded by users due to the proliferation of data, video, gaming and general broadband multimedia. These demands have prompted communication system engineers to explore more efficient, faster and reliable wireless communication techniques. To this end, free space optical (FSO) communication has been a viable solution [1].

Free space optical communication is a communication technology that employs light as carrier by modulating baseband information with optical carriers often from laser beams through free space to the receiver [2]. The path of connection between FSO transmitters and the receivers are known as FSO links. Even though the very first optical system dates back to the eighteenth century, modern FSO communication systems were first widely deployed by the National Aeronautics and Space Administration (NASA) and have since become a promising broadband wireless access technology. FSO communication systems are also being combined with standard RF systems in order to form hybrid communication systems that harness the unique features of RF and FSO communication systems to enhance performance, capacity and reliability.

Characteristically, FSO communication systems are highly secure as they have high immunity to interference with the use of secure point-to-point line-of-sight links, they require no licensing or regulatory permission, and they are fast and can be easily deployed and operated compared to other systems like the fiber optic systems [3]. These features make FSO communication the favored option for the provision of high-speed links for a variety of next generation optical applications including broadcast, security, wireless backhaul at a data rate as high as 40 Gbps [4], fiber backup and last mile communication [5]. Finally, the ease of setup and cost effectiveness of FSO systems have made them the preferred option for restoring connection in case of disaster.

However, the performance of FSO communication systems are greatly affected by turbulence-induced fading [6, 7, 8, 9, 10, 11, 12], and different investigations are currently being explored to address this challenge. The inhomogeneity of the temperature and pressure of the atmosphere causes local variations in the refractive index as light propagates from the transmitter to the photoreceptor; these variations degrade the performance of FSO links significantly.

Geared towards the improvement of the performance of coherent FSO communication systems in the presence of atmospheric turbulence, this work examines the error reduction schemes currently being employed for FSO links and presents an adaptive space-time trellis code (STTC) scheme for coherent FSO links.

2. Free space optical communication: types and variants

In terms of reception technique, however, the most commonly reported variants of FSO communication systems are the direct detection/intensity modulated (IM/DD) FSO system and the coherent FSO system. The IM/DD FSO communication systems convey the information to be transmitted only on the intensity of the emitted light, and the receivers simply decode the information as the light changes in intensity. In coherent FSO communication systems, however, other signal properties such as phase and frequency may be employed in conveying the information. At the receiving end, as against simply observing changes in light intensity as in the case of IM/DD FSO systems, coherent FSO communication systems, first, mix the received field optically with a local oscillator before the actual photodetection. So far, more works on the IM/DD FSO communication system are being reported in literature owing to its simplicity of detection as less complex receivers and algorithms are required. Coherent FSO systems, though more complex, however, offer superior performance in terms of improved receiver sensitivity and background noise rejection [13].

Finally, in terms of communication range, FSO communication systems can be characterized into short-range, medium-range, long-range and inter-terrestrial FSO systems depending on the application for which they are deployed, and these applications include inter-chip communication, inter-vehicular communication, metropolitan area communication as well as satellite and space exploration.

3. FSO system and channel models

Concerted efforts being expended by researchers in the quest of effectively modeling the FSO channel are geared towards understanding the channel and serving as the template upon which FSO modulators, demodulators, receivers and other devices can work. Accurate mathematical models for FSO communication system are the basis upon which the development of high-performing hardware is established, despite huge technical challenge of turbulence-induced fading. This challenge is however being addressed using various techniques as summarized in Figure 1.

Figure 1.

Some turbulence mitigation techniques in FSO systems.

In the wavelength diversity schemes, the source information is encoded into different wavelengths obtainable from different constituents of the infrared spectrum.

4. Turbulence models

One of the most crucial steps in the attempts to mitigate the degradation in the performance of optical communication systems is accurate modeling of the atmospheric turbulence under different conditions. Below are some of the irradiance functions presented in terms of probability distribution functions.

4.1 Lognormal distribution

The lognormal distribution is one of the most widely employed for weak atmospheric turbulence distribution. Here, the irradiance value received at the receiver follows the distribution [9]

$$f_A\left(a\right)=\frac{1}{2a{\left(2\pi {\sigma }_x^2\right)}^{\frac{1}{2}}}\mathit{exp}\left(-\frac{{\left(lna+2{\sigma }_x\right)}^2}{8{\sigma }_x^2}\right)$$E1

where ${\mu }_x$ is the mean value of fading and ${\sigma }_x^2$ is the fading covariance.

4.2 K-distribution

The K-distribution turbulence model is often used to describe strong atmospheric turbulence conditions (non-Rayleigh sea clutter). For K-distribution atmospheric turbulence model, the probability distribution function $p\left(I\right)$ is expressed as [14]

$$p\left(I\right)=\frac{2\alpha }{\mathrm{\Gamma }\left(\alpha \right)}{\left(\mathit{\alpha I}\right)}^{\frac{\alpha -1}{2}}{K}_{\alpha -1}\left(2\sqrt{\mathit{\alpha I}}\right),I>0,\alpha >0$$E2

$$K_m\left(\bullet \right)=\text{modified Bessel function of second kind and order}m.$$

4.3 Negative exponential distribution

Negative exponential turbulence model is employed for saturated turbulence cases where the probability distribution function of the received irradiance value is expressed as [15]

$$p\left(I\right)=\frac{1}{I_0}\mathit{exp}\left(\frac{-I}{I_0}\right),I_0>0$$E3

where $I_0$ denotes the mean irradiance.

4.4 Gamma-gamma distribution

The gamma-gamma model is very commonly used in FSO communication literatures because it is applicable for a wider range of turbulence conditions. In comparison with measured data, gamma-gamma distribution is effective in describing weak to strong atmospheric turbulence conditions. The PDF is expressed as [16]

$$f_{\stackrel{\sim }{H}}^{\mathit{GG}}\left(h\right)=\frac{2{\left(\mathit{\alpha \beta }\right)}^{\frac{\alpha +\beta }{2}}}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{h}^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\mathit{\alpha \beta h}}\right)$$E4

where $K_v\left(x\right)$ is the modified Bessel function of the second kind and $\alpha \text{and}\beta$ are the turbulence parameter.

5. Space-time trellis coded coherent FSO

The essence of space-time trellis encoder is to employ mapping functions which are representatives of their trellis diagrams to map binary data to modulation symbols. We design and evaluate the performance of space-time trellis code with two transmit antennas for FSO channel. In order to simplify the design and yet ensuring that there is no jeopardy to the intended MIMO configuration, we represent, for the two transmit antennas, the input bitstream $\mathit{c}$ as [17]

$$\mathit{c}=\left({\mathit{c}}_0,{\mathit{c}}_1,{\mathit{c}}_2,\dots ,{\mathit{c}}_{\mathit{t}},\dots \right)$$E5

where ${\mathit{c}}_{\mathit{t}}$, at any instant $\mathit{t}$ denotes a group of two information bits expressed as

$${\mathit{c}}_1=\left({\mathit{c}}_{\mathit{t}}^1,{\mathit{c}}_{\mathit{t}}^2\right)$$E6

As shown in Figure 2, the encoder, made up of feedforward shift registers, converts the input bit sequence into a sequence of modulated signals $x_0,x_1,x_2,\dots ,x_t,\dots$, where each element $x_t$ of this sequence is the space-time symbol at a given time $t$.

Figure 2.

STTC encoder [18].

The output $x_t^i$ of the encoder for the $i$th transmitter at time $t$ is expressed as [19]

$$x_t^i=\sum _{k=1}^m\sum _{j=0}^{v_k}{g}_{j,i}^k{c}_{t-j}^k\mathrm{Mod}2,i=1,2$$E7

Space-time code (STC) leverages on the features of both time diversity and space diversity to combat turbulence-induced fading in wireless communication systems. RF wireless systems in particular have witness an explosion of interest in the use of space-time coding to improve communication system performance in terms of error control and turbulence mitigation, and FSO communication systems are also witnessing a lot of interest in using this same tool for similar purpose.

In this chapter, we present an adaptive four-state space-time trellis coded coherent FSO system with two transmit lasers, as illustrated in Figure 3. Firstly, the error correction performance of the system is complemented by the interleaver, a mechanism put in place to distribute the burst errors—an effect of deep fade, onto different codeword lengths.

Figure 3.

Space-time trellis coded FSO communication system.

Denoting the average SNR as $\gamma$, we take the received signal matrix for each codeword $C$ as [20]

$$R=\sqrt{\gamma }\mathrm{CH}+Z$$E8

where $H$ and $Z$ are the channels and noise matrices, respectively, and $H$ is modeled in terms of the uniformly distributed channel gain phase ${\varphi }_{\mathrm{\mu v}}$ and the channel gain amplitude $a_{\mathrm{\mu v}}$ as [20]

$${\left[H\right]}_{\mathrm{\mu v}}=a_{\mathrm{\mu v}}{e}^{j{\varphi }_{\mathrm{\mu v}}}1\le \mu \le 2,1\le v\le N$$E9

We begin our analysis using the pairwise error probability (PEP), which is the probability that the decoder erroneously decodes a transmitted STTC codeword C as $C^{\prime }=\left[c_0^{\prime }\dots c_{T-1}^{\prime }\right]$. Then, assuming a gamma-gamma fading distribution as portrayed in Eq. (4), we represent the conditional PEP as [5]

$$P_e\left(E|H\right)=Q\left(\sqrt{\frac{\gamma d^2\left(E\right)}{2}}\right)$$E10

where

$$d^2\left(E\right)=\mathit{tr}\left\{H^H{E}^H\mathit{EH}\right\}$$E11

Now, writing a matrix B and its constituent elements as

$$B=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{12}^{\ast }& b_{22}\end{array}\right]$$E12

and by equating B with the positive semi-definite matrix $E^{H}E$ and comparing the elements thereof, where E represents the error matric in the decoding of the codewords and ${\left(\bullet \right)}^H$ denotes the Hermitian transpose function, we write the asymptotic pairwise error probability of the systems as [21]

$$\mathit{PEP}=\frac{{\left(\pi a_h^2{\mathrm{\Gamma }}^2\left(2\mu \right)F\left(\mu ,\mu ;1;{\xi }^2\right)\right)}^N\mathrm{\Gamma }\left(2\mathit{\mu N}+\frac{1}{2}\right){\gamma }^{-2\mathit{\mu N}}}{2\sqrt{\pi }\left(b_{11}{b}_{22}\right){\mu }^N\mathrm{\Gamma }\left(2\mathit{\mu N}+1\right){\mathrm{\Gamma }}^{2N}\left(\mu +\frac{1}{2}\right)}$$E13

where the Gaussian hypergeometric function $F\left(\bullet \right)$ is readily computed by using specialized computing functions from libraries of most engineering computing applications or by using fast-converging series [20] as

$$F\left(\frac{t}{2},\frac{t}{2};1;{\xi }^2\right)=\sum _{n=0}^{\infty }{\left[\left(\begin{array}{c}\frac{t}{2}+n-1\\ n\end{array}\right){\xi }^n\right]}^2$$E14

The function $\Gamma$ in Eq. (13) is a function of the channel parameters $\alpha$ and $\beta$; these parameters may be obtained through the Rytov variance, which in turn is a function of the refractive index, the transmission path length between the transmitter and the receiver and the optical wave number [22].

With proper modifications of the values of $\xi$, Eq. (13) and by extension, Eq. (14), could be modified for general case as well as specific non-orthogonal space-time codes for coherent free space optical communication system. We leverage onto this feature to introduce an adaptive orthogonality controller which adjusts its parameters to any STC supplied thereby not merely eliminating the orthogonality condition as presented in [21] but effectively introduces additional flexibility to the coding scheme.

Readers are to note, however, that several space-time code designs reported for IM/DD FSO communication systems cannot be simply employed for coherent FSO communication systems. This caveat is due to the peculiarities inherent in coherent FSO systems. In addition to this, it should also be noted that in this work, it is assumed that the transmit lasers simultaneously illuminate the receivers with the receivers far away enough from the transmit lasers to assume independent and identically distributed (iid) fading gain.

6. Results and discussion

In this section, the results of the space-time code technique for mitigating turbulence-induced fading in coherent FSO communication systems are presented. Free space optical systems often face the challenge of fading as well as pointing error, and the effect of the latter has been well addressed [23]. The performance of the link under gamma-gamma turbulence is investigated for two transmit lasers, first, with two receivers and then four and six receivers, respectively, as shown in Figure 4. Apart from the reduction of the average bit error rate with increase in SNR values, the result shows that at low average SNR, the average performance of the link under the turbulence condition for the different number of receivers are relatively close. However, the difference in performance becomes apparent at higher SNRs as evidenced from SNR 20 to SNR 38.

Figure 4.

Performance of coherent FSO link with different receivers.

Although gamma-gamma distribution have been well reported as suitable for modeling weak turbulence as well as strong turbulence scenarios, for the sake of analysis, we employ the values $\alpha =3.0$ and $\beta =2.7$. The choice of these values is informed by their popularity in literature as the performance of this work is compared with earlier works in this domain, many of which employed the gamma-gamma distribution parameter above.

FSO communication systems vary in their reception mechanisms as well as modulation techniques. Many works have employed OOK, PPM and QPSK or a combination of these modulation schemes all in a bid to mitigate turbulence-induced fading in FSO links. A few of these works and their corresponding features in comparison to this work are presented in Table 1.

Coding scheme	Modulation scheme	Detection type	SNR (dB)	References
STBC	OOK	IM/DD	—	[24]
STC variant—no additional constellation extension	PPM	IM/DD	$0\le \mathrm{SNR}\le 30$	[25]
STBC	OOK	IM/DD with maximum likelihood (ML)	—	[26]
Alamouti-type STC	OOK and PPM	Coherent detection and IM/DD	Additional 3 dB loss relative to BPSK	[27]
STTC	—	IM/DD	$0\le \mathit{SNR}\le 90$	[28]
Extended Alamouti STC with turbo coding	PPM	IM/DD	$0\le \mathrm{SNR}\mathrm{per}\mathrm{bit}\le 30$	[29]
STTC	QPSK	Coherent detection	$0\le \mathit{SNR}\le 35$	This work

Table 1.

Some coding schemes employed for FSO links.

Even though the efficiency of space-time codes for turbulence mitigation or error correction in intensity modulated/direct detection FSO systems remains inconclusive in literature, we establish that space-time coding—adaptive space-time trellis codes as in the case of this work, together with inherent potentials of coherent reception for FSO systems—remains a promising solution for free space optical communication systems.

7. Conclusion

For coherent free space optical communication links, we explore the space-time approach to mitigating turbulence-induced fading which thus far remains a serious performance degrading factor for FSO systems. Additionally, as an effort geared towards realizing the promising potentials of coherent free space optical communication systems, we propose an adaptive orthogonality controller for seamless deployment of space-time codes for coherent free space optical communication systems.