Protocols for Relay-Assisted Free-Space Optical Systems

Protocols for Relay-Assisted Free-Space Optical Systems

Nestor D. Chatzidiamantis,   Diomidis S. Michalopoulos,  Emmanouil E. Kriezis,  George K. Karagiannidis,  and Robert Schober,  This paper has been submitted in part to the IEEE Global Communications Conference (GLOBECOM’11).N. D. Chatzidiamantis, E. E. Kriezis, and G. K. Karagiannidis are with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (E-mail: {nestoras, mkriezis, geokarag}@auth.gr).D. S. Michalopoulos and R. Schober are with the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada (E-mail: {dio, rschober}@ece.ubc.ca).
Abstract

We investigate transmission protocols for relay-assisted free-space optical (FSO) systems, when multiple parallel relays are employed and there is no direct link between the source and the destination. As alternatives to all-active FSO relaying, where all the available relays transmit concurrently, we propose schemes that select only a single relay to participate in the communication between the source and the destination in each transmission slot. This selection is based on the channel state information (CSI) obtained either from all or from some of the FSO links. Thus, the need for synchronizing the relays’ transmissions is avoided and the slowly varying nature of the atmospheric channel is exploited. For both relay selection and all-active relaying, novel closed-form expressions for their outage performance are derived, assuming the versatile Gamma-Gamma channel model. Furthermore, based on the derived analytical results, the problem of allocating the optical power resources to the FSO links is addressed, and optimum and suboptimum solutions are proposed. Numerical results are provided for equal and non-equal length FSO links, which illustrate the outage behavior of the considered relaying protocols and demonstrate the significant performance gains offered by the proposed power allocation schemes.

\pubid\specialpapernotice{keywords}

Atmospheric turbulence, cooperative diversity, distributed switch and stay relaying, free-space optical communications, relay-assisted communications, relay selection, power allocation.

I Introduction

The constant need for higher data rates in support of high-speed applications has led to the development of the Free Space Optical (FSO) communication technology. Operating at unlicensed optical frequencies, FSO systems offer the potential of broadband capacity at low cost [1], and therefore, they present an attractive remedy for the ”last-mile” problem. However, despite their major advantages, the widespread deployment of FSO systems is hampered by major impairments, which have their origin in the propagation of optical signals through the atmosphere. Rain, fog, and atmospheric turbulence are some of the major atmospheric phenomena that cause attenuation and rapid fluctuations in the received optical power in FSO systems, thereby increasing the error rate and severely degrading the overall performance [2].

In the past, several techniques have been applied in FSO systems for mitigating the degrading effects of the atmospheric channel, including error control coding in conjunction with interleaving [3], multiple-symbol detection [4], and spatial diversity [5, 6, 7]. Among these techniques, spatial diversity, which is realized by deploying multiple transmit and/or receive apertures, has been particularly attractive, since it offers significant performance gains by introducing additional degrees of freedom in the spatial dimension. Thus, numerous FSO systems with multiple co-located transmit and/or receive apertures, referred as Multiple-Input Multiple-Output (MIMO) FSO systems, have been proposed in the technical literature [5, 6, 7]. However, in practice, MIMO FSO systems may not always be able to offer the gains promised by theory. This happens in cases where the assumption that all the links of the MIMO FSO system are affected by independent channel fading becomes invalid [7]. Furthermore, since both the path loss and the fading statistics of the channel are distance-dependent, a large number of transmit and/or receive apertures is required in long-range links in order to achieve the desired performance gains, thus increasing the complexity of MIMO FSO systems.

In order to overcome such limitations, relay-assisted communication has been recently introduced in FSO systems as an alternative approach to achieve spatial diversity [8, 9, 10, 11]. The main idea lies in the fact that, by employing multiple relay nodes with line-of-sight (LOS) to both the source and the destination, a virtual multiple-aperture system is created, often referred as cooperative diversity system, even if there is no LOS between the source and the destination. In [8], various relaying configurations (cooperative diversity and multihop) have been investigated under the assumption of a lognormal channel model. Subsequently, several coding schemes for 3-way cooperative diversity FSO systems with a single relay and a direct link between the source and the destination have been proposed in [9], while the performance of such systems has been investigated in [10] and [11] assuming amplify-and-forward and decode-and-forward relaying strategies, respectively. It is emphasized that in all these previous works, all the available relays participated in the communication between the source and the destination, requiring perfect synchronization between the relays such that the FSO signals can arrive simultaneously at the destination, while the optical power resources are equally divided between all FSO links.

In view of the above, in this paper we present alternative transmission protocols which can be applied to relay-assisted FSO systems with no LOS between the source and the destination. For the signaling rates of interest, the atmospheric channel does not vary within one packet. Thus, channel state information (CSI) can be easily obtained for all or for some of the involved links. Capitalizing on this fact, the presented protocols select only a single relay to take part in the communication in every transmission slot, thus avoiding the need for synchronization between the relays. It should be noted that similar relay selection protocols have been also proposed in the context of radio-frequency relaying systems [12, 13, 14]. In particular, two types of relay selection protocols are presented: the select-max protocol that selects the relay that maximizes an appropriately defined metric and requires CSI from all the available FSO links, and the distributed switch and stay (DSSC) protocol which switches between two relays and requires CSI only from the FSO links used in the previous transmission slot. Furthermore, assuming the versatile Gamma-Gamma channel model [2] and decode-and-forward relay nodes, we derive novel closed-form analytical expressions for the outage performance of the proposed transmission schemes, as well as the scheme where all the available relays transmit simultaneously; thus, extending the analysis presented in [8] to the case of the Gamma-Gamma channel model. Finally, based on the derived outage results, we address the problem of optimizing the allocation of the optical power resources to the FSO links for minimization of the probability of outage; hence, rendering the relay-assisted FSO system under consideration more power efficient.

The remainder of the paper is organized as follows. In Section II, the system model and the considered relaying protocols are discussed. The outage performance of the relaying protocols under investigation is analyzed in Section III, while the problem of optimizing the allocation of the optical power resources to the FSO links is addressed in Section IV. Numerical results for various relay-assisted FSO architectures are presented in Section V, and, finally, concluding remarks are provided in Section VI.

Ii System Model

The system model under consideration is depicted in Fig. 1. In particular, we consider an intensity-modulation direct detection (IM/DD) FSO system without LOS between the source, , and the destination, , and the communication between these two terminals is achieved with the aid of multiple relays, denoted by , . The source node is equipped with a multiple-aperture transmitter, with each of the apertures pointing in the direction of the corresponding relay, and an optical switch111Optical switches can be implemented with either spatial light modulators (SLM) [15, Ch. (27)] or optical MEMS devices [16]., which either allows the simultaneous transmission from all the transmit apertures or selects the direction of transmission by switching between the transmit apertures.

The presence of a large field-of-view (FOV) detector at the destination is assumed allowing for the simultaneous detection of the optical signals transmitted from each relay. Moreover, all optical transmitters are equipped with optical amplifiers that adjust the optical power transmitted in each link. The relaying terminals use a threshold-based decode-and-forward (DF) protocol; that is, they fully decode the received signal and retransmit it to the destination only if the signal-to-noise ratio (SNR) of the receiving FSO link exceeds a given decoding threshold. Finally, throughout this paper, we assume that binary pulse position modulation (BPPM) is employed.

Ii-a Signal and Channel Model

For an FSO link connecting two terminals and , the received optical signal at the photodetector of is given by

(1)

where and represent the signal and the non-signal slots of the BPPM symbol, respectively, while and denote the average optical signal power transmitted from and the background radiation incident on the photodetector of , respectively. Furthermore, represents the percentage of the total optical power allocated to the FSO link between terminals and , is the channel gain of the link, is the photodetector’s responsivity, is the duration of the signal and non-signal slots, and and are the additive noise samples in the signal and non-signal slots, respectively. Since background-noise limited receivers are assumed, where background noise is dominant compared to other noise components (such as thermal, signal dependent, and dark noise) [5, 17], the noise terms can be modeled as additive white Gaussian, with zero mean and variance . After removing the constant bias from both slots, the instantaneous SNR of the link can be defined as [8]

(2)

Due to atmospheric effects, the channel gain of the FSO link under consideration can be modeled as

(3)

where accounts for path loss due to weather effects and geometric spread loss and represents irradiance fluctuations caused by atmospheric turbulence. Both and are time-variant, yet at very different time scales. The path loss coefficient varies on the order of hours while turbulence induced fading varies on the order of 1–100 ms [5]. Thus, taking into consideration the signaling rates of interest, which range from hundreds to thousands of Mbps, the channel gain can be considered as constant over a given transmission slot, which consists of hundreds of thousands (or even millions) of consecutive symbols.

The path loss coefficient can be calculated by combining the Beer Lambert’s law [2] with the geometric loss formula [1, pp. 44], yielding

(4)

where and are the receiver and transmitter aperture diameters, respectively; is the optical beam’s divergence angle (in rad), is the link’s distance (in km), and is the weather dependent attenuation coefficient (in 1/km).

Under a wide range of atmospheric conditions, turbulence induced fading can be statistically characterized by the well-known Gamma-Gamma distribution [2]. The probability density function (pdf) for this model is given by

(5)

where is the Gamma function [18, Eq. (8.310)] and is the th order modified Bessel function of the second kind [18, Eq. (8.432/9)]. Furthermore, and are parameters related to the effective atmospheric conditions via and [2], where denotes the Rytov variance222The Rytov variance is indicative of the strength of turbulence-induced fading. More specifically, values correspond to weak turbulence conditions, while values correspond to the moderate-strong turbulence regime [19]., is the weather dependent index of refraction structure parameter, and represents the wavelength of the optical carrier.

Ii-B Mode of Operation

Throughout this work, three different cooperative relaying protocols are considered: the all-active protocol, originally presented in [8], where all the available relays are activated, and the select-max and the distributed switch and stay combining (DSSC) protocols, which are both based on the concept of selecting a single relay.

Ii-B1 All-active

In this relaying scheme, the source activates all relays and the total power is divided between all available FSO links. Since the relay nodes operate in the DF mode only the relays that successfully decode the received optical signal remodulate the intensity of the optical carrier and forward the information to the destination. At the destination, owing to the presence of a large FOV aperture, aperture averaging occurs [6] and all the received optical signals are added. Hence, assuming perfect synchronization, the output of the combiner can be expressed as

(6)

where denotes the decoding set formed by the relays that have succesfully decoded the signal. Since the total power is divided between all available links, it follows that

The advantage of this scheme is that CSI is not required neither at the transmitter nor the receiver side, since the source transmits to all available relays, regardless of their channel gain. However, since it is assumed that all the signals arrive at the destination at the same time, this scheme requires accurate timing synchronization in order to account for the different propagation delays of the different paths, resulting in high complexity.

Ii-B2 Select-Max

This relaying protocol selects a single relay out of the set of available relays in each transmission slot. In particular, the relay which maximizes an appropriately defined metric is selected. This metric accounts for both the - and - links and reflects the quality of the th end-to-end path. Here, we adopt the minimum value of the intermediate link SNRs,

(7)

as the quality measure of the th end-to-end path, which will be referred as the ”min equivalent SNR” throughout the paper. Note that (7) represents an outage-based definition of the selection metric, in the sense that an outage on the th end-to-end link occurs if falls below the outage threshold SNR. Hence, the single relay that is activated in the select-max relaying protocol, is selected according to the rule

(8)

Since a single relay is activated in the select-max protocol, the total available optical power is divided between the - and - links, i.e., and in the case that has successfully decoded the received optical signal, i.e., , the signal at the destination can be expressed as

(9)

This relaying scheme requires the CSI of all the available - and - FSO links in order to perform the selection process. This can be achieved by some signalling process that takes advantage of the slowly-varying nature of the FSO channel . Here, each receiver estimates the correspponding link CSI and feeds it back to the source through a reliable low-rate RF feedback link. It is emphasized that since only one end-to-end path is activated in each transmission slot, only one signal arrives at the destination and thus synchronization between the relays is not needed.

Ii-B3 Dssc

Requiring less CSI than select-max, the DSSC protocol applies to the case where there are only two relays available and one of them is selected to take part in the communication between the source and the destination, in a switch-and-stay fashion [13]. More specifically, in each transmission slot the destination compares the equivalent SNR of the active end-to-end path with a switching threshold, denoted by . If this SNR is smaller than , the destination notifies the source and the other available relay is selected for taking part in the communication, regardless of  its end-to-end performance metric.

Mathematically speaking, denoting the two available relays by and and the equivalent SNR of the th end-to-end path during the th transmission period by , the active relay in the th transmission period, , is determined as follows:

(10)

and

(11)

Hence, in the case that has successfully decoded the received signal, the optical signal at the destination is given by

(12)

Since in this protocol only a single relay assists in the communication between the source and the destination, the power allocation rule of the select-max protocol also holds for DSSC relaying.

When there are more than two available relays in the system, i.e., , a modified version of DSSC protocol could initially sort all the available paths based on their end-to-end distance, defined as

(13)

with , and, then, use as and the two relays that correspond to the paths with the minimum end-to-end distance. It should be noted that end-to-end distance is an indicative of the path’s end-to-end performance, taking into consideration that both path loss and Rytov variance are monotonically increasing with respect to the link distance.

The simplicity of this scheme compared to the select-max protocol lies in the fact that only the CSI of the active end-to-end path is required for the selection process, resulting in less implementation complexity. Furthermore, as in the select-max scheme, no synchronization among the relays is needed, since only one end-to-end path is activated in each transmission slot.

Iii Outage Analysis

At a given transmission rate, the outage probability is defined as

(14)

where is the instantaneous capacity, which is a function of the instantaneous SNR. Since is monotonically increasing with respect to , (14) can be equivalently rewritten as

(15)

where denotes the threshold SNR. If the SNR, , drops below , an outage occurs, implying that the signal cannot be decoded with arbitrarily low error probability at the receiver. Henceforth, it is assumed that the threshold SNR, , is identical for all links of the relaying system.

Iii-a Outage Probability of the Intermediate Links

Since DF relaying is considered, an outage event in any of the intermediate links may lead to an outage of the overall relaying scheme. Therefore, the calculation of the outage probability of each intermediate link is considered as a building block for the outage probability of the relaying schemes under investigation.

By combining (2) with (15), the outage probability of the FSO link between nodes and is defined as

(16)

which can be equivalently rewritten as

(17)

where is the power margin given by . Using the cumulative density function (cdf) of the Gamma-Gamma distribution [20, Eq. (7)], the outage probability of the FSO link between nodes and can be analytically evaluated for any and , yielding

(18)

where is the Meijer’s -function [18, Eq. (9.301)].

To gain more physical insights from (18), it is meaningful to explore the outage probability in the high power margin regime.

Theorem 1

For high values of power margin and when , the outage probability of the FSO link between nodes and can be approximated by

(19)

where and .

{proof}

A detailed proof is provided in Appendix I.

It should be noted that in the analysis that follows it is assumed that holds for every possible FSO link. Although this condition may seem restrictive, it can be relaxed in practical applications by inserting an infinitely small perturbation term , so that , when [21].

Iii-B Outage Probability of All-Active Relaying

In this scheme an outage occurs when either the decoding set is empty or the SNR of the multiple-input single-output link between the decoding relays and the destination falls below the outage threshold. Hence, the outage probability of this scheme can be expressed as [8, Eq. (30)]

(20)

where denotes the th possible decoding set, is the total number of decoding sets, and is the probability of event given by

(21)

Iii-B1 Exact Analysis

In order to evaluate (20), the cdf of the sum of weighted non-identical Gamma-Gamma variates, , needs to be derived first. However, to the best of the authors’ knowledge, there are no closed-form analytical expressions for the exact distribution of the sum of non-identical Gamma-Gamma variates. Therefore, the numerical method of [22, Eq. (9.186)], which is based on the moment generating function (MGF) approach, is applied and thus the cdf of , denoted as , is evaluated via

(22)

where is the MGF of the channel gain of the FSO link given by [23, Eq. (4)], while the parameters , , are calculated based on the numerical error term obtained by [22, Eq. (9.187)].

Theorem 2

The outage probability of the all-active relaying protocol in Gamma-Gamma fading is given by

(23)
{proof}

The proof follows straightforwardly by combining (20) with (18) and (22).

Iii-B2 Asymptotic Analysis

In order to gain more physical insights into the performance of the relaying protocol under consideration, we further consider the high power margin regime, i.e., when . In order to perform this analysis, an asymptotic expression for needs to be derived first.

Lemma 3

For high values of power margin, the cdf for the weighted sum of non-identical Gamma-Gamma variates that corresponds to decoding set , , can be approximated as

(24)
{proof}

A detailed proof is provided in Appendix II.

The asymptotic expression for the outage probability of the all-active relaying scheme is given by the following theorem.

Theorem 4

For high values of power margin, the outage probability of the all-active relaying scheme in Gamma-Gamma fading can be approximated by

(25)
{proof}

We first observe that in the high power margin regime, i.e., , (21) can be approximated by

(26)

Hence, by combining (20) with (19), (24), and (26), the asymptotic expression in (25) is obtained. This concludes the proof.

An important result derived from the asymptotic expression of the previous theorem is the diversity gain of the transmission protocol under consideration which is summarized in the ensuing corollary.

Corollary 1

For the all-active relaying protocol, the diversity gain of a relay-assisted FSO system with relays is given as

(27)
{proof}

We define the diversity order as

(28)

Hence, (27) follows straightforwardly by observing that the term which corresponds to the power of with the minimum of dominates in the sum of (25), when .

Iii-C Outage Probability of Select-Max Relaying

In the select-max protocol a single relay out of the available relays is selected according to the selection rule in (8). Hence, the outage probability of the relaying scheme under consideration is given by

(29)

where denotes the probability of outage when only relay is active. Given that is active, an outage occurs when either or have not decoded the information successfully, i.e.,

(30)

Hence, the probability of outage of the select-max relaying scheme is obtained by combining (29) with (30).

Iii-C1 Exact Analysis

The following theorem provides an accurate analytical expression for the performance evaluation of the select-max relaying scheme.

Theorem 5

The probability of outage of a relay-assisted FSO system that employs the select-max relaying protocol in Gamma-Gamma turbulence-induced fading is given by

(31)
{proof}

The proof follows straightforwardly by combining (29) with (30) and (18).

Iii-C2 Asymptotic Analysis

In order to gain more physical insights into the performance of the relaying protocol under consideration, we investigate its asymptotic behavior when , in the ensuing theorem and corollary.

Theorem 6

For high values of power margin, the outage probability of the select-max relaying scheme can be approximated as

(32)
{proof}

The proof starts by observing that as the probability of outage given relay is active, can be approximated by

(33)

Hence, by combining (29) with (19) and (33), the asymptotic expression in (32) is obtained. This concludes the proof.

Corollary 2

The diversity gain of a relay-assisted FSO system employing the select-max relaying protocol and relays is given as

(34)
{proof}

The proof follows straightforwardly from (32). When , the term that corresponds to the power of with the minimum of dominates in the sum inside the product. Hence, after taking the product of the dominating terms, the diversity order is obtained.

Iii-D Outage Probability of DSSC Relaying

In the DSSC protocol, the selection of the single relay which takes part in the communication is based on (10) and (11). Hence, an outage occurs when there is an outage either in the end-to-end link of the first relay, given that the first relay is selected in the th transmission slot, or in the end-to-end link of the second relay, given that the second relay is selected in the th transmission slot, i.e.,

(35)

Following the analysis of [22, Sec. (9.9.1.2)], the above equation can be rewritten as

(36)

Iii-D1 Exact Analysis

The following theorem provides an accurate analytical expression for the performance of the DSSC relaying scheme.

Theorem 7

The probability of outage of a relay-assisted FSO system that employs the DSSC relaying protocol is given by

(37)

where

(38)

and .

{proof}

We first note that the cdf of the equivalent SNR defined in (7) can be expressed according to [24, pp. 141] as

(39)

which is equivalent, after a variable transformation, to

(40)

Using (16) the cdf of the equivalent channel gain is derived as (38) and, thus, according to (36), (37) is obtained. This concludes the proof.

Corollary 3

The performance of the DSSC relaying scheme is minimized when and in that case it becomes equal to that of the select-max scheme for two relays, and .

{proof}

Following the analysis in [22, (Ch. 9.9.1.1)], the performance of DSSC relaying is minimized when is set. In that case, (37) yields

(41)

which is equivalent to (31) when . This concludes the proof.

Iii-D2 Asymptotic Analysis

In order to gain more physical insights of the DSSC protocol with optimized threshold, we investigate the asymptotic behavior of its performance when in the ensuing corrolary.

Corollary 4

The minimum outage probability of the DSSC relaying protocol can be approximated at the high power margin regime, by (32) with , and the maximum achieved diversity gain is given by

(42)
{proof}

The proof follows straightforwardly by combining (41) with (32) and (34).

Iv Optimal Power Allocation

In this section, we are interested in optimizing the optical power resources in both the - and - links, in order to minimize the outage probability of the relay-assisted FSO system for a given total optical power. Hence, in the following, we optimize the parameters and for each of the relaying protocols under consideration.

Iv-a Power Allocation in All-Active Protocol

Since in the all-active scheme the power is divided among all the underlying links, the minimization of its outage probability is subject to two constraints; the total power budget of all links is equal to and the optical power emitted from each transmitter is less than . Consequently, the optimum power allocation can be found by solving the following optimization problem

(43)

where is given by (23). It should be noted that that the above optimization problem is convex problem. This can be explained as follows. Since the objective function is an outage probability, it is convex according to [25]. Furthermore, since all the constraints are linear, they form a convex set [26], which leads to a convex optimization problem and, thus, a unique optimal solution.

Using the exact outage expression in (23), it is difficult to find the optimum solution for the problem in (43), even with numerical methods, due to the involvement of the Meijer’s G-functions. Therefore, the asymptotic expression of (25) is used as objective function instead and hence the optimization problem is reformulated as

(44)

which is a geometric program that can be numerically solved using numerical optimization techniques, such as the interior point method [26, Sec. 14].

Since the derivation of the exact solution is cumbersome and motivated by the dependence of the outage probability on the link distance, the following suboptimal power allocation scheme for all-active relaying is proposed.

Proposition 1

For all-active relaying, the fraction of the total optical power which is allocated to each link is given by

(45)

for the - and - links, respectively, with .

Iv-B Power Allocation in Select-Max Protocol

Similar to the all-active scheme, the outage probability of the select-max protocol can also be minimized by optimizing the optical power resources which are allocated to each of the links. However, in this scheme, both the objective function and the constraints are different from the problem in (43).

Since the total optical power is divided only between the - and - links of the active relay, the problem can be formulated as

(46)

where is the the probability of outage when is active, given by (30). Based on the same reasoning as in the previous relaying scheme, we conclude that the above optimization problem is convex, thus, leading to a unique optimal solution.

Due to the involvement of the Meijer’s G-functions, it is again difficult to find the optimum solution if the exact expression in (30) is used as objective function, even with numerical methods. Therefore, the asymptotic expression in (33) is employed and hence the power allocation optimization problem is reformulated as