Average Channel Capacity of Amplify-and-forward MIMO/FSO Systems Over Atmospheric Turbulence Channels

Received Oct 6, 2017 Revised Jul 18, 2018 Accepted Aug 1, 2018 In amplify-and-forward (AF) relay channel, when the direct link between source and destination terminals is deeply faded, the signal from the source terminal to the destination terminal propagates through the relay terminals, each of which relays a signal received from the previous terminal to the next terminal in series. This paper, we theoretically analyze the performance of multiple-input multiple-output (MIMO) AF free-space optical (FSO) systems. The AF-MIMO/FSO average channel capacity (ACC), which is expressed in terms of average spectral efficiency (ASE) is derived taking into account the atmospheric turbulence effects on the MIMO/FSO channel. They are modeled by log-normal and the gamma-gamma distributions for the cases of weak-to-strong turbulence conditions. We extract closed form mathematical expression for the evaluation of the ACC and we quantitatively discuss the influence of turbulence strength, link distance, different number of relay stations and different MIMO configurations on it. Keyword:


INTRODUCTION
The necessity of a cost-effective, license-free, high security and high bandwidth access technique has lead to a continuous research and commercial interest in optical wireless communication systems [1]- [3]. One of major degradations to the performance of FSO communications is the influence of atmospheric turbulence caused by variations in the refractive index, pressure fluctuations in the air along the propagation path of the laser beam [4]. . The electrical signal at the output of the PD corresponding to the nth receive aperture output can be expressed as follows [9]     21

ATMOSPHERIC TURBULENCE MODELS
There exist a few published papers concerning the probability density function (pdf) of multi-hop transmission over amplify-and-forward relay fading channels [9], [10]. With amplify-and-forward free-space optical systems using SC-QAM modulation, for weak atmospheric turbulence condition, the turbulence included fading is assumed to be a random process that follows that log-normal distribution [9], [10], whereas for strong turbulence conditions, a gamma-gamma distribution is used [11].

The Log-Normal Turbulence Model of AF-MIMO/FSO Systems
In log-normal fading channel, the probability density function (pdf) for an normalized irradiance with log-normal, 0 mn X  , is described as [9], [ In equation (6), is the optical wave number, L is the link distance in meters,  is the optical wavelength, and D is the receiver aperture diameter of the PD. The parameter 2 2  is the Rytov variance and in this case, is expressed by [

The Gamma-gamma Turbulence Model of AF-MIMO/FSO Systems
The pdf of gamma-gamma fading channel, 0, mn X  the pdf for an normalized irradiance with gamma-gamma, 0 mn X  , is described as [11]  where ()   is the Gamma function and () K    is the modified Bessel function of the second kind of order     , while the parameters  and  are directly related to atmospheric conditions through the following expressions:

AVERAGE CHANNEL CAPACITY
An optical wireless channel is a randomly time-variant channel and the received instantaneous electrical signal-to-noise ratio (SNR) is a random variable. Thus, the channel capacity must be considered as a random variable, and its average value, known as average channel capacity, C . The ASE in bits/s/Hz if the frequency response of the channel is known. The average channel capacity is given by [12].
where B is the channel's bandwidth and is the total channel SNR and can be reduced to a product of the first-order pdf of each element nm  . The pdfs of  are respectively described in log-normal [9], [10] and gamma-gamma [11] distributions as follows   .

NUMERICAL RESULTS
Using the above closed mathematical forms as shown in Equation 18 and 20, we can estimate the average capacity of AF -MIMO/FSO system over atmospheric turbulence channel. We use (18) for the case of weak turbulence with log-normal distribution model, while (20) is used for strong case with the gammagamma distribution. In the analysis below, we evaluate the average capacity for three different values of turbulence strength 2 AF P  Assumes that, the relay stations are placed equidistant, turbulence conditions between relay of stations is the same. Relevant parameters considered in our analysis are provided in Table 1. It can be also seen that the ASE strongly depends on MIMO configurations, at achieve the SNR=10 dB for MIMO configurations M=N=4, C=0, ASE=7 b/s/Hz with L=2000 m but L=4000 m ASE=6 b/s/Hz. Similar, for M=N=2 the corresponding result is ASE=5 dB with and ASE=4 dB with L=4 000 m. For the case C=1 and C=2 the results are similar, ASE decreases 1 dB. The ASE strongly depends on the number relay stations, with L=2000 m, M=N=4 when number of relay stations increases from C=0 to C=1 and from C=1 to C=2 ASE of systems decreases 3 dB and 1.5 dB, respectively to achieve the SNR=10 dB. The ASE strongly depends on the link distance gets longer, with C=0, M=N=4, when increasing distance from L=2000 m to L=4000 m ASE of systems decreases 1 dB. The impact of SNR on the ASE of systems is more significant in low regions than in high regions. The reason of this is that as optical link distance gets longer, the signal propagates in the atmospheric with longer distances. It has been that observed that with increase in the value of C, capacity performance of the system deteriorates. ASE of systems decreases 0.3 dB and 0.4 dB, respectively to achieve the SNR=20 dB. The ASE strongly depends on number relay stations, and as the link distance gets longer, the ASE under the weak turbulence conditions is higher than in the cases of moderate and strong turbulence, especially with longer link distance L. The reason of this is that as optical link distance gets longer, the signal propagates in the atmospheric with longer distances; the influence of the turbulence therefore becomes stronger. On the other hand, as expected, the ASE could be improved by approximately 2 (b/s/Hz) when the system is upgraded from SISO/FSO to 2×2 MIMO/FSO or from 2×2 MIMO/FSO to 4×4 MIMO/FSO for number of relay stations 0, C  approximately 1 (b/s/Hz) for 1, C  and approximately 0.7 (b/s/Hz) for 2.  AF P  It is clearly shown that the ASE performance is improved significantly with the increase of number of lasers and receivers and the ASE performance is decreases significantly with the increase of number of relay stations. In addition, as expected, ASE increases as number of lasers M and receiver N increase from SISO to 2×2 MIMO and 4×4 MIMO. It can be also seen that the ASE strongly depends on the atmospheric turbulence strength, MIMO configurations, and as the link distance gets longer. Increasing the number of relay stations, capacity performance of the system deteriorates but increases the gain of power, this problem is very important in the FSO system.

CONCLUSION
The purpose of this paper is to present the performance analysis on the average channel capacity of AF -MIMO/FSO systems using SC-QAM signals over atmospheric turbulence channels. The log-normal and gamma-gamma distribution models were used to describe the fluctuation of the optical propagating over atmospheric turbulence channels. The ASE is calculated based on the probability density function of the FSO system combining the relay stations, ASE considering different link conditions, number of relay stations and MIMO configurations. The numerical results showed that, with the similar link distance and turbulence strength, regardless the link distance and turbulence condition, the ASE of the FSO link could be improved by approximately 2 (b/s/Hz) when the system is upgraded from from SISO/FSO to 2×2 MIMO/FSO or from 2×2 MIMO/FSO to 4×4 MIMO/FSO for number of relay stations 0, C  approximately 1 (b/s/Hz) for 1, C  and approximately 0.7 (b/s/Hz) for 2.

C 
The ASE strongly depends on the atmospheric turbulence strength, MIMO configurations, number of relay stations and as the link distance. Increasing the number of relay stations, capacity performance of the system deteriorates but increases the gain of power, this problem is very important in the FSO system.