6. Results and discussion

½ � H <sup>μ</sup><sup>v</sup> ¼ aμve

portrayed in Eq. (4), we represent the conditional PEP as [5]

d2

Now, writing a matrix B and its constituent elements as

asymptotic pairwise error probability of the systems as [21]

2 ffiffiffi

applications or by using fast-converging series [20] as

F t 2 ; t 2 ; <sup>1</sup>; <sup>ξ</sup><sup>2</sup> � �

the receiver and the optical wave number [22].

identically distributed (iid) fading gain.

the coding scheme.

110

PEP <sup>¼</sup> <sup>π</sup>a<sup>2</sup>

Peð Þ¼ EjH Q

as C<sup>0</sup> ¼ c<sup>0</sup>

where

0…c<sup>0</sup> T�1

We begin our analysis using the pairwise error probability (PEP), which is the probability that the decoder erroneously decodes a transmitted STTC codeword C

> 0s @

ffiffiffiffiffiffiffiffiffiffiffiffiffi γd<sup>2</sup> ð Þ E 2

1

ð Þ¼ <sup>E</sup> tr HHEHEH � � (11)

2 � �γ�2μ<sup>N</sup>

ξn

2

� � (13)

� �. Then, assuming a gamma-gamma fading distribution as

Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies

<sup>B</sup> <sup>¼</sup> <sup>b</sup><sup>11</sup> <sup>b</sup><sup>12</sup> b∗ <sup>12</sup> b<sup>22</sup> � �

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

and by equating B with the positive semi-definite matrix EHE and comparing the

<sup>h</sup>Γ<sup>2</sup>ð Þ <sup>2</sup><sup>μ</sup> <sup>F</sup> <sup>μ</sup>; <sup>μ</sup>; <sup>1</sup>; <sup>ξ</sup><sup>2</sup> � � � � <sup>N</sup><sup>Γ</sup> <sup>2</sup>μ<sup>N</sup> <sup>þ</sup> <sup>1</sup>

where the Gaussian hypergeometric function Fð Þ∙ is readily computed by using specialized computing functions from libraries of most engineering computing

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

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

¼ ∑ ∞ n¼0

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

<sup>π</sup> <sup>p</sup> ð Þ <sup>b</sup>11b<sup>22</sup> <sup>μ</sup><sup>N</sup>Γð Þ <sup>2</sup>μ<sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>Γ</sup><sup>2</sup><sup>N</sup> <sup>μ</sup> <sup>þ</sup> <sup>1</sup>

t <sup>2</sup> þ n � 1 n � �

� �<sup>2</sup>

<sup>j</sup>ϕμ<sup>v</sup> 1≤μ≤2, 1≤v≤ N (9)

A (10)

(12)

(14)

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.

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

#### Figure 4.

Performance of coherent FSO link with different receivers.


#### Table 1.

Some coding schemes employed for FSO links.

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 gammagamma distribution parameter above.

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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 turbulenceinduced fading in FSO links. A few of these works and their corresponding features in comparison to this work are presented in Table 1.

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.
