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 c as [17]

$$\mathcal{L} = (\mathcal{c}\_0, \mathcal{c}\_1, \mathcal{c}\_2, \dots, \mathcal{c}\_l, \dots) \tag{5}$$

where ct, at any instant t denotes a group of two information bits expressed as

$$\mathfrak{c}\_1 = \left(\mathfrak{c}\_t^1, \mathfrak{c}\_t^2\right) \tag{6}$$

As shown in Figure 2, the encoder, made up of feedforward shift registers, converts the input bit sequence into a sequence of modulated signals

Mitigating Turbulence-Induced Fading in Coherent FSO Links: An Adaptive Space-Time Code… DOI: http://dx.doi.org/10.5772/intechopen.84911

x0, x1, x2, …, xt, …, where each element xt of this sequence is the space-time symbol at a given time t.

The output xi <sup>t</sup> of the encoder for the ith transmitter at time t is expressed as [19]

$$\mathbf{x}\_{t}^{i} = \sum\_{k=1}^{m} \sum\_{j=0}^{v\_{k}} \mathbf{g}\_{j,i}^{k} \boldsymbol{\varepsilon}\_{t-j}^{k} \mathbf{Mod} \ \mathbf{2}, i = \mathbf{1}, \mathbf{2} \tag{7}$$

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.

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

$$R = \sqrt{\gamma} \mathbf{C} \mathbf{H} + \mathbf{Z} \tag{8}$$

where H and Z are the channels and noise matrices, respectively, and H is modeled in terms of the uniformly distributed channel gain phase ϕμ<sup>v</sup> and the channel gain amplitude aμ<sup>v</sup> as [20]

Figure 2. STTC encoder [18].

4.2 K-distribution

expressed as [14]

expressed as [15]

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 Ið Þ is

Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies

<sup>2</sup> <sup>K</sup><sup>α</sup>�<sup>1</sup> <sup>2</sup> ffiffiffiffiffi

KmðÞ¼ ∙ modified Bessel function of second kind and order m:

Negative exponential turbulence model is employed for saturated turbulence cases where the probability distribution function of the received irradiance value is

> exp �<sup>I</sup> I0 � �

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]

> αþβ <sup>2</sup> �1

where Kvð Þ x is the modified Bessel function of the second kind and α and β are

The essence of space-time trellis encoder is to employ mapping functions which

where ct, at any instant t denotes a group of two information bits expressed as

c<sup>1</sup> ¼ c 1 t ;c 2 t

converts the input bit sequence into a sequence of modulated signals

As shown in Figure 2, the encoder, made up of feedforward shift registers,

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 repre-

sent, for the two transmit antennas, the input bitstream c as [17]

<sup>K</sup>α�<sup>β</sup> <sup>2</sup> ffiffiffiffiffiffiffiffi

c ¼ c0;c1;c2; …;ct ð Þ ; … (5)

� � (6)

αβh

� � <sup>p</sup> (4)

2 <sup>Γ</sup>ð Þ <sup>α</sup> <sup>Γ</sup>ð Þ <sup>β</sup> <sup>h</sup> αI

� � <sup>p</sup> , I > 0, <sup>α</sup> > 0 (2)

, I<sup>0</sup> > 0 (3)

p IðÞ¼ <sup>2</sup><sup>α</sup>

4.3 Negative exponential distribution

where I<sup>0</sup> denotes the mean irradiance.

5. Space-time trellis coded coherent FSO

4.4 Gamma-gamma distribution

the turbulence parameter.

108

<sup>Γ</sup>ð Þ <sup>α</sup> ð Þ <sup>α</sup><sup>I</sup> <sup>α</sup>�<sup>1</sup>

p IðÞ¼ <sup>1</sup> I0

fH<sup>e</sup> GGð Þ¼ <sup>h</sup> <sup>2</sup>ð Þ αβ <sup>α</sup>þ<sup>β</sup>

Figure 3. Space-time trellis coded FSO communication system.

$$[H]\_{\mu\nu} = \mathfrak{a}\_{\mu\nu} \mathfrak{e}^{i\phi\_{\mu\nu}} \mathbf{1} \le \mu \le \mathbf{2}, \mathbf{1} \le \nu \le N \tag{9}$$

6. Results and discussion

DOI: http://dx.doi.org/10.5772/intechopen.84911

Figure 4.

higher SNRs as evidenced from SNR 20 to SNR 38.

Performance of coherent FSO link with different receivers.

scheme

PPM

STBC OOK IM/DD with

STBC OOK IM/DD — [24]

STTC — IM/DD 0≤ SNR≤90 [28]

STTC QPSK Coherent detection 0≤ SNR≤35 This work

maximum likelihood (ML)

Coherent detection and IM/DD

Detection type SNR (dB) References

Additional 3 dB loss relative to BPSK

— [26]

[27]

PPM IM/DD 0≤ SNR ≤30 [25]

PPM IM/DD 0≤SNR per bit ≤30 [29]

Coding scheme Modulation

Alamouti-type STC OOK and

STC variant—no additional constellation extension

Extended Alamouti STC with

Some coding schemes employed for FSO links.

turbo coding

Table 1.

111

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

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

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 α ¼ 3:0 and β ¼ 2:7. The choice of these values is

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<sup>0</sup> ¼ c<sup>0</sup> 0…c<sup>0</sup> T�1 � �. Then, assuming a gamma-gamma fading distribution as portrayed in Eq. (4), we represent the conditional PEP as [5]

$$P\_e(E|H) = Q\left(\sqrt{\frac{\gamma d^2(E)}{2}}\right) \tag{10}$$

where

$$d^2(E) = \text{tr}\{H^H E^H E H\}\tag{11}$$

Now, writing a matrix B and its constituent elements as

$$B = \begin{bmatrix} b\_{11} & b\_{12} \\ b\_{12}^\* & b\_{22} \end{bmatrix} \tag{12}$$

and by equating B with the positive semi-definite matrix EHE and comparing the elements thereof, where E represents the error matric in the decoding of the codewords and ð Þ<sup>∙</sup> <sup>H</sup> denotes the Hermitian transpose function, we write the asymptotic pairwise error probability of the systems as [21]

$$\text{PEP} = \frac{\left(\pi a\_h^2 \Gamma^2(2\mu) F(\mu, \mu; 1; \xi^2)\right)^N \Gamma\left(2\mu N + \frac{1}{2}\right) \chi^{-2\mu N}}{2\sqrt{\pi} (b\_{11} b\_{22}) \mu^N \Gamma(2\mu N + 1) \Gamma^{2N} \left(\mu + \frac{1}{2}\right)} \tag{13}$$

where the Gaussian hypergeometric function Fð Þ∙ is readily computed by using specialized computing functions from libraries of most engineering computing applications or by using fast-converging series [20] as

$$F(\frac{t}{2}, \frac{t}{2}; 1; \xi^2) = \sum\_{n=0}^{\infty} \left[ \binom{\frac{t}{2} + n - 1}{n} \xi^n \right]^2 \tag{14}$$

The function Γ in Eq. (13) is a function of the channel parameters α and β; 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 ξ, 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.

Mitigating Turbulence-Induced Fading in Coherent FSO Links: An Adaptive Space-Time Code… DOI: http://dx.doi.org/10.5772/intechopen.84911
