Gamma-gamma Distribution Research Articles

Performance of quadrature amplitude modulation orthogonal frequency division multiplexing‐based free space optical links with non‐linear clipping effect over gamma–gamma modelled turbulence channels

The free space optical (FSO) communication systems have attracted significant research and commercial interest in the last few years because of their low installation and operational cost along with their very high performance characteristics. However, for terrestrial FSO links, the optical signal propagates through the atmosphere which exhibits time-varying behaviour that implies variations in the links' performance. In this study, the authors estimate the performance metrics for terrestrial FSO links which are using the orthogonal frequency division multiplexing (OFDM) technique with a quadrature amplitude modulation scheme over turbulence channels. More specifically, the authors investigate the influence of the non-linear clipping effect of the OFDM scheme, along with the atmospheric turbulence modelled using the gamma-gamma distribution. Both effects significantly influence the performance of the link and here the authors derive closed form mathematical expressions for the estimation of the average signal to noise ratio, the outage probability and the average bit error rate that are vital for FSO system performance characterisation. Finally, using these expressions, the authors present the corresponding numerical results for common parameter values of the FSO links and investigate the accuracy of the expressions for marginal cases with nearly negligible turbulence effect.

Novel space-time trellis codes for free-space optical communications using transmit laser selection.

In this paper, the deployment of novel space-time trellis codes (STTCs) with transmit laser selection (TLS) for free-space optical (FSO) communication systems using intensity modulation and direct detection (IM/DD) over atmospheric turbulence and misalignment fading channels is presented. Combining TLS and STTC with rate 1 bit/(s · Hz), a new code design criterion based on the use of the largest order statistics is here proposed for multiple-input/single-output (MISO) FSO systems in order to improve the diversity order gain by properly chosing the transmit lasers out of the available L lasers. Based on a pairwise error probability (PEP) analysis, closed-form asymptotic bit error-rate (BER) expressions in the range from low to high signal-to-noise ratio (SNR) are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results show diversity orders of 2L and 3L when simple two-state and four-state STTCs are considered, respectively. Simulation results are further demonstrated to confirm the analytical results.

On the Ergodic Capacity of MIMO Free-Space Optical Systems Over Turbulence Channels

The free-space optical (FSO) communications can achieve high capacity with huge unlicensed optical spectrum and low operational costs. The corresponding performance analysis of FSO systems over turbulence channels is very limited, especially when using multiple apertures at both transmitter and receiver sides. This paper aim to provide the ergodic capacity characterization of multiple-input multiple-output (MIMO) FSO systems over atmospheric turbulence-induced fading channels. The fluctuations of the irradiance of optical channels distorted by atmospheric conditions is usually described by a gamma-gamma ($\Gamma \Gamma$) distribution, and the distribution of the sum of $\Gamma \Gamma$ random variables (RVs) is required to model the MIMO optical links. We use an $\alpha$-$\mu$ distribution to efficiently approximate the probability density function (PDF) of the sum of independent and identical distributed $\Gamma\Gamma$ RVs through moment-based estimators. Furthermore, the PDF of the sum of independent, but not necessarily identically distributed $\Gamma \Gamma$ RVs can be efficiently approximated by a finite weighted sum of PDFs of $\Gamma \Gamma$ distributions. Based on these reliable approximations, novel and precise analytical expressions for the ergodic capacity of MIMO FSO systems are derived. Additionally, we deduce the asymptotic simple expressions in high signal-to-noise ratio regimes, which provide useful insights into the impact of the system parameters on the ergodic capacity. Finally, our proposed results are validated via Monte-Carlo simulations.

Performance Analysis of FSO Communications With Diversity Methods: Add More Relays or More Apertures?

This paper targets the performance analysis of multiple-input-multiple-output (MIMO) point-to-point single-aperture relay-assisted and MIMO relay-assisted free-space optical (FSO) communications. The objective of the paper is to compare the different FSO diversity methods via an outage probability analysis over gamma-gamma turbulence channels in the case of independent fading among the different apertures of the communicating nodes. Through a well-tailored calculation method, we approximate the gamma-gamma distribution and the gamma-gamma sum-distribution by appropriate gamma distributions. This method resulted in simple closed-form outage probability expressions that are accurate for average-to-large values of the signal-to-noise ratio (SNR) and in asymptotic expressions that are useful for evaluating the diversity gains and coding gains. In particular, we derived the diversity gains of MIMO repetition coding (RC), MIMO transmit laser selection (TLS), MIMO all-active relaying (AR) and MIMO selective-relaying (SR). We prove that MIMO-RC and MIMO-TLS, on one hand, and MIMO-AR and MIMO-SR, on the second hand, achieve the same diversity gain and we evaluate the coding gain of TLS with respect to RC and of SR with respect to AR. By comparing the MIMO and cooperative methods with point receivers, we reach the central result that it is always better, from a diversity gain point of view, to add more apertures to the source and destination rather than adding more relays in their vicinity despite the fact that the fading variance along FSO links decreases with the distance. For receivers that implement aperture averaging, we derive conditions under which MIMO techniques are unconditionally better than cooperative techniques with any network topology.

Ergodic capacity analysis for DF strategies in cooperative FSO systems.

This paper focuses on the ergodic capacity analysis in the context of cooperative free-space optical (FSO) systems when the line of sight is available. Novel asymptotic closed-form expressions for the ergodic capacity corresponding to two different decode-and-forward (DF) strategies are obtained for a cooperative FSO communication system. Here, the atmospheric turbulence is modeled by a gamma-gamma distribution of parameters α and β which allows to study a wide range of turbulence conditions (moderate-to-strong) as well as the effect of the misalignment with zero boresight. It is demonstrated that cooperative communications are able to achieve not only a better performance in terms of the error rate performance as well as outage probability than direct transmission, but also in terms of the channel capacity in the context of FSO systems without much increase in hardware. In this way, a 3-way FSO communication setup is considered, in which the cooperative protocol can be applied to achieve a greater ergodic capacity compared to a direct transmission. It can be concluded that a greater and robust capacity strongly dependent on the relay location is achieved compared to a direct transmission without cooperative communication when line of sight is available. Here, the line of sight is taken into account in order to achieve a significant robustness under different turbulence conditions and more severe pointing errors regardless of the relay location. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

Asymptotic Analysis of TWR Assisted FSO Links With Partial Dual-Relay Selection

In this letter, a partial-dual relay selection (P-DRS) scheme is proposed for amplify-and-forward protocol based two-way relay (TWR) assisted free space optical (FSO) communication systems. Closed form asymptotic expression is developed for the moment generating function (mgf) of the equivalent signal-to-noise ratio of the system, considering an optical channel that is modeled by the Gamma-Gamma distribution and affected by path losses and pointing errors. Expressions for the outage probability and average error probability are calculated based on the mgf. Finally, the mathematical analysis is accompanied with simulations to demonstrate the effectiveness of the proposed P-DRS scheme.

Unified Performance Analysis of Mixed Radio Frequency/Free-Space Optical Dual-Hop Transmission Systems

The mixed radio frequency (RF)/free-space optical (FSO) relaying is a promising technology for coverage improvement, while there lacks unified expressions to describe its performance. In this paper, a unified performance analysis framework of a dual-hop relay system over asymmetric RF/FSO links is presented. More specifically, we consider the RF link follows generalized $\kappa$-$\mu$ or $\eta$-$\mu$ distributions, while the FSO link experiences the gamma-gamma distribution, respectively. Novel analytical expressions of the probability density function and cumulative distribution function are derived. We then capitalize on these results to provide new exact analytical expressions of the outage probability and bit error rate (BER). Furthermore, the outage probability for high signal-to-noise ratios and the BER for different modulation schemes are deduced to provide useful insights into the impact of system and channel parameters of the overall system performance. These accurate expressions are general, since they correspond to generalized fading in the RF link and account for pointing errors, atmospheric turbulence and different modulation schemes in the FSO link. The links between derived results and previous results are presented. Finally, numerical and Monte-Carlo simulation results are provided to demonstrate the validity of the proposed unified expressions.

Performance Analysis of Polarization Modulated DirectDetection Optical CDMA Systems over Turbulent FSO LinksModeled by the Gamma-Gamma Distribution

This paper proposes a theoretical study to characterize the transmission of optical code division multiple access (CDMA) systems deploying polarization shift keying (PolSK) over a free space optical (FSO) link under the impact of atmospheric turbulence. In our analysis, a novel transceiver architecture for atmospheric OCDMA FSO systems based on polarization modulation with direct detection is proposed and discussed. A detailed analytical model for PolSK-OCDMA systems over a turbulent FSO link is provided. Further, we derive a closed-form bit error ratio (BER) and outage probability expressions, taking into account the multiple-access interference (MAI), optical noise and the atmospheric turbulence effect on the FSO link modeled by the Gamma-Gamma distribution. Finally, the results of this study show the most significant parameters that degrade the transmission performance of the PolSK-OCDMA signal over FSO links and indicate that the proposed approach offers improved bit error ratio (BER) performances compared to the on-off-keying (OOK) modulation scheme in the presence of turbulence.

Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications.

The impact of relay placement on diversity order in adaptive selective decode-and-forward (DF) cooperative strategies is here investigated in the context of free-space optical (FSO) communications over atmospheric turbulence channels with pointing errors when line of sight is available. The irradiance of the transmitted optical beam here considered is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution together with a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Novel closed-form approximate bit error-rate (BER) expressions are obtained for a cooperative FSO communication setup with N relays, assuming that these relays are located in an area similar to an annulus around source or destination node. An analytical expression is here found that determines the best selection criterion based on the knowledge of the channel state information (CSI) of source-relay or relay-destination links in order to significantly increase the diversity order corresponding to the cooperative strategy under study. It is concluded that the highest diversity order is achieved when the relation β(SR(min)) > β(SD) + β(R(min)D) is satisfied, wherein β(SR(min)), β(R(min)D) and β(SD) are parameters corresponding to the atmospheric turbulence conditions of source-relay and relay-destination link with the greatest scintillation index, and source-destination link, respectively.

Mixed RF/FSO Relaying with Outdated Channel State Information

We investigate the outage probability and the average bit error rate (BER) performance of a dual-hop amplify-and-forward (AF) relaying system, composed of a mixed radio frequency (RF)/free-space optical (FSO) link, when simultaneously outdated channel state information (CSI) is assumed at the relay and there is a misalignment between transmitter and receiver apertures in FSO link. In contrast to the majority of works on CSI-assisted AF relays, in this paper, we assume that the estimated CSI is outdated, when the relay amplifies the transmitted signal. The RF link experiences Rayleigh fading, while the FSO link is under the influence of atmospheric turbulence, modeled by the Gamma-Gamma distribution. Novel analytical expressions for the outage probability and the average BER are derived in a power series form, which in some special cases are simplified to offer engineering insight into the effects of important transceiver and channel parameters on the system performance. Numerical and simulation results show that there is an optimal value of the transmitter beam waist, which minimizes the overall outage probability. This optimal value strongly depends on the pointing errors standard deviation. Furthermore, the outage probability varies for several orders of magnitude depending on the transmitter beam waist.

Outage performance of DF relay-assisted FSO communications using time-diversity

Outage probability as a performance measure for a decode-and-forward (DF) relaying scheme based on repetition coding with the relay is presented in the context of free-space optical (FSO) communication systems with intensity modulation and direct detection over atmospheric turbulence and misalignment fading channels. Novel closed-form asymptotic expressions for the outage probability are derived for a three-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size, and jitter variance is considered. Fully exploiting the potential time-diversity available in the relay turbulent channel, a greater robustness to the relay location and pointing errors is achieved, maintaining a high diversity gain regardless of the source-destination link distance and remarkably superior to that of the two-transmitter case as well as the conventional DF scheme. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

AVERAGE CHANNEL CAPACITY OF FREE-SPACE OPTICAL MIMO SYSTEMS OVER ATMOSPHERIC TURBULENCE CHANNELS

In this paper, we theoretically analyze the performance of multiple-input multiple-output (MIMO) free-space optical (FSO) systems. The 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 quantitatively discuss the influence of turbulence strength, link distance, and different MIMO configurations on the system ASE. Numerical and computer simulation results are presented in order to verify the validity of the mathematical analysis.

Modelling and Characterization of Subcarrier Intensity Modulation Based Free Space Optical Communication

This paper is an investigation to model and characterize subcarrier intensity modulation employing M-ary phase shift keying and rectangular quadrature amplitude modulation for optical wireless communication over different turbulence channels. Atmospheric turbulence results in many effects causing fluctuation in the received optical power .The popular models for describing atmospheric turbulence are the lognormal distribution (for weak turbulence induced irradiance fluctuation), the K- distribution (for strong turbulence induced irradiance fluctuation), the negative exponential distribution (for irradiance fluctuation in saturated turbulence conditions), and the Gamma-Gamma distribution (for describing irradiance fluctuation over a wide range of turbulence regimes from weak to strong turbulence). The adaptive schemes offer efficient utilization of optical wireless communication channel capacity by adapting the modulation order according to the received signal-to-noise ratio and a pre-defined target bit-error rate requirement. The improved spectral efficiency can be achieved by the adaptive systems without increasing the transmitter power.

The Gamma-Gamma Signal Fading Model: A Survey [Wireless Corner

In a wireless communication medium, the received signal envelope fluctuates randomly due to the scattering nature of these channels. This paper is intended to present a comprehensive literature review on the gamma-gamma signal fading model and its use in the performance analysis of different communication systems, with more elaboration on wireless radio-frequency (RF) communication networks, including cellular and relay networks, and wireless optical networks. First, the different communication channels where the gamma-gamma distribution is used to model the signal-fading phenomena are described. Second, the reported results on the statistics for the gamma-gamma distribution, including the product, the quotient, the sum, and the order statistics are presented. Finally, the performance analyses carried out in the open literature for wireless RF systems, relay networks, and wireless optical systems are summarized, and some possible directions for future research are highlighted.

Transmit alternate laser selection with time diversity for FSO communications.

In this paper, a new transmit alternate laser selection (TALS) scheme for FSO communication systems using intensity modulation and direct detection (IM/DD) over atmospheric turbulence and misalignment fading channels is presented when limited time diversity is available in the turbulent channel. Assuming channel state information (CSI) at the transmitter and receiver and a time diversity order (TDO) limited, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation. Implementing repetition coding with blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. Closed-form asymptotic bit error-rate (BER) expressions are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Fully exploiting the potential time-diversity TDO available in the turbulent channel, a significant diversity gain is achieved, providing a diversity order of (2L + 1 - TDO)TDO/2.

Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels.

The average bit error rate (BER) for binary phase-shift keying (BPSK) modulation in free-space optical (FSO) links over turbulence atmosphere modeled by the exponentiated Weibull (EW) distribution is investigated in detail. The effects of aperture averaging on the average BERs for BPSK modulation under weak-to-strong turbulence conditions are studied. The average BERs of EW distribution are compared with Lognormal (LN) and Gamma-Gamma (GG) distributions in weak and strong turbulence atmosphere, respectively. The outage probability is also obtained for different turbulence strengths and receiver aperture sizes. The analytical results deduced by the generalized Gauss-Laguerre quadrature rule are verified by the Monte Carlo simulation. This work is helpful for the design of receivers for FSO communication systems.

Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels

In this paper, a novel adaptive cooperative protocol with multiple relays using detect-and-forward (DF) over atmospheric turbulence channels with pointing errors is proposed. The adaptive DF cooperative protocol here analyzed is based on the selection of the optical path, source-destination or different source-relay links, with a greater value of fading gain or irradiance, maintaining a high diversity order. Closed-form asymptotic bit error-rate (BER) expressions are obtained for a cooperative free-space optical (FSO) communication system with Nr relays, when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions, following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. A greater robustness for different link distances and pointing errors is corroborated by the obtained results if compared with similar cooperative schemes or equivalent multiple-input multiple-output (MIMO) systems. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

A finite mixture model of vehicle-to-vehicle and day-to-day variability of traffic network travel times

This study proposes an approach to modeling the effects of daily roadway conditions on travel time variability using a finite mixture model based on the Gamma–Gamma (GG) distribution. The GG distribution is a compound distribution derived from the product of two Gamma random variates, which represent vehicle-to-vehicle and day-to-day variability, respectively. It provides a systematic way of investigating different variability dimensions reflected in travel time data. To identify the underlying distribution of each type of variability, this study first decomposes a mixture of Gamma–Gamma models into two separate Gamma mixture modeling problems and estimates the respective parameters using the Expectation–Maximization (EM) algorithm. The proposed methodology is demonstrated using simulated vehicle trajectories produced under daily scenarios constructed from historical weather and accident data. The parameter estimation results suggest that day-to-day variability exhibits clear heterogeneity under different weather conditions: clear versus rainy or snowy days, whereas the same weather conditions have little impact on vehicle-to-vehicle variability. Next, a two-component Gamma–Gamma mixture model is specified. The results of the distribution fitting show that the mixture model provides better fits to travel delay observations than the standard (one-component) Gamma–Gamma model. The proposed method, the application of the compound Gamma distribution combined with a mixture modeling approach, provides a powerful and flexible tool to capture not only different types of variability—vehicle-to-vehicle and day-to-day variability—but also the unobserved heterogeneity within these variability types, thereby allowing the modeling of the underlying distributions of individual travel delays across different days with varying roadway disruption levels in a more effective and systematic way.

Performance Analysis of FSO Array Receivers in Presence of Atmospheric Turbulence

We evaluate the performance of a free space optical link using an array of direct detection receivers in terms of the bit error rate (BER) under the influence of various atmospheric conditions and turbulence strengths. The expression for BER is derived by modeling the turbulence as a gamma-gamma distribution. The effect of weather conditions is incorporated using Beer-Lambert's law. Results show that increasing the number of detectors reduces the BER as expected. In addition, the diversity gain increases with the increasing turbulence strength but remains almost constant for a fixed value of the turbulence strength and the number of detectors over various atmospheric conditions such as very clear air, drizzle, haze, fog, and so forth.

Performance Comparison of OFDM based FSO Communication System under Log-normal and Gamma-Gamma Distribution

Recently, researches on Orthogonal Frequency Division Multiplexing (OFDM) based Free Space Optics (FSO) communication system gain interest since it combines the advantages of both techniques by mitigating the multipath propagation effects to improve the transmission data rate. The performance of such systems depends on base band modulation used, number of subcarriers, turbulence modeling, nature of laser beam and so on. In this paper, a performance comparison is carried out considering three different levels (weak, medium and strong) of atmospheric turbulence with log-normal and Gamma- Gamma scintillation model for plane, spherical and Gaussian beams. The Gamma-Gamma distribution analysis shows that for all kinds of laser beams, Bit Error Rate (BER) is high for weak turbulence which decreases further for medium and strong. The only difference is that the BER variation for Gamma-Gamma distribution with Gaussian beam is more prominent than for plane beam. The decrease in BER is clearly observed at higher SNR value. This kind of improvement in BER is observed with log-normal PDF at low SNR with weak turbulence. Thus we can say that weak atmospheric turbulence can be modeled by log-normal PDF and medium to strong condition can be modeled by Gamma-Gamma PDF.