Marco Di Renzo
email: marco.direnzo@lss.supelec.fr

Jesus Alonso Zarate

Alonso Luis
email: luisg@tsc.upc.edu

Christos Verikoukis

J Alonso-Zarate

L Alonso

On the Impact of Shadowing on the Performance of Cooperative Medium Access Control Protocols

Accurate representation of the physical layer is instrumental for a sound design and optimization of Medium Access Control (MAC) protocols for cooperative wireless networks. However, the vast majority of MAC protocols are designed and analyzed by considering simplified physical layer and channel models, which often lead to too optimistic performance predictions. In particular, even though many experimental activities have showcased the important role played by shadow-fading, most protocols are designed and evaluated by taking into account only the transmission distance (circular coverage model) or only the fast-fading. Motivated by the proved unsuitability of these models, the contribution of this paper is threefold: i) we provide important considerations on how to adequately include the effect of shadowing into the design of MAC protocols for cooperative networks; ii) we provide an analytical framework to determine the subset of active relays in order to meet a given Quality-of-Service (QoS) requirement; and iii) we study, through analysis and simulation, the performance of a promising MAC protocol for cooperative networks, which is called Persistent Relay Carrier Sensing Multiple Access (PRCSMA), by explicitly taking into account the effect of shadowing. Our study shows that shadowing can dramatically change system and protocol performance.

I. INTRODUCTION

Automatic Repeat reQuest (ARQ) mechanisms are extensively exploited by many link-layer protocols to reduce the packet error probability in communication networks. However, when applied to wireless networks, the effectiveness of ARQ schemes highly depends on the coherence time of the wireless channel: the longer the coherence time is, the less effective the ARQ mechanism is [START_REF] Zorzi | ARQ error control for fading mobile radio channels[END_REF]. In fact, a longer coherence time implies that subsequent retransmitted packets might experience almost the same channel conditions, thus resulting in consecutive failures. To overcome this limitation, in [START_REF] Dianati | A node-cooperative ARQ scheme for wireless ad hoc networks[END_REF] the authors have recently proposed an improved ARQ mechanism, which exploits distributed cooperation among nearby network nodes to improve the performance of link-layer protocols. The proposal in [START_REF] Dianati | A node-cooperative ARQ scheme for wireless ad hoc networks[END_REF] is today known as Cooperative ARQ (C-ARQ) scheme, and it foresees that whenever a packet is received with insufficient quality, a retransmission can be requested not only to the original source node, but also to network nodes that are nearby the destination, which might have overheard the original transmission (due to the broadcast nature of the wireless channel) and might be willing to relay independent received versions of the same packet and provide an inherent form of spatial diversity, which is not affected by temporal correlation of fading. Recent results have shown the benefits of C-ARQ as opposed to non-cooperative schemes [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF].

Research leading to these results has received funding from the research projects, CO2GREEN (TEC2010-20823), CENTENO (TEC2008-06817-C02-02), and GREENET (PITN-GA-2010-264759).

When designing and optimizing the performance of C-ARQ protocols, a fundamental issue is to efficiently coordinate the access to the wireless medium of the many potential network nodes (i.e., relays) that can be involved in the cooperative phase. In fact, the set of active cooperative nodes compete and contend to retransmit the packet towards the destination. In this context, an adequate Medium Access Control (MAC) protocol is crucial to enable an effective exploitation of cooperation by reducing latency and collisions. In [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF], we have recently proposed Persistent Relay Carrier Sensing Multiple Access (PRCSMA), which is a new CSMA-based MAC protocol specifically designed to coordinate the retransmissions from the relays in a C-ARQ scheme. In [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF] and [START_REF] Alonso-Zárate | Performance analysis of a persistent relay carrier sensing multiple access protocol[END_REF], we have shown that PRCSMA can significantly outperform non-cooperative ARQ protocols. However, the assessment of the performance of this protocol has been made by considering simplified physical layer and channel models, which, on the other hand, play a crucial role to understand the actual performance of any protocol. In fact, many recent papers have clearly shown that oversimplifying the wireless channel for protocol design might eventually provide solutions that simply do not work in practical operating conditions (see, e.g., [START_REF] Newport | Experimental evaluation of wireless simulation assumptions[END_REF]- [START_REF] Hanzo | QoS-aware routing and admission control in shadow-fading environments for multi-rate MANETs[END_REF]).

More specifically, recent simulations [START_REF] Hanzo | The effects of shadow-fading on QoS-aware routing and admission control protocols designed for multi-hop MANETs[END_REF], [START_REF] Hanzo | QoS-aware routing and admission control in shadow-fading environments for multi-rate MANETs[END_REF] and experimental [START_REF] Agrawal | Correlated link shadow fading in multihop wireless networks[END_REF] studies have clearly pointed out that shadow-fading is the crucial impairment to be considered for the analysis and design of protocols for cooperative wireless networks. In particular, in this context, including shadowing requires to carefully consider three fundamental issues:

• Non-ergodicity -Unlike fast-fading, shadowing is, in general, a non-ergodic process for the duration of a communication that is composed by the transmission of many packets [START_REF] Mary | Mary symbol error outage over Nakagami-m fading channels in shadowing environments[END_REF], [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]. In other words, shadowing might change during the duration of a communication, but the fluctuations are not fast enough to go through all the states of the distribution. This implies that the most suitable metric for design, optimization, and analysis of communication protocols in the presence of shadowing is the Outage Packet Error Probability (OPEP), as opposed to the Average Packet Error Probability (APEP) [START_REF] Mary | Mary symbol error outage over Nakagami-m fading channels in shadowing environments[END_REF].

• Mobility -Unlike conventional cellular systems, in cooperative networks all the nodes might be in motion. This yields fairly different propagation channel models. Recent results have shown that shadowing time-correlation plays an important role to assess the performance of multi-hop mobile networks [START_REF] Wang | Joint shadowing process in urban peer-to-peer radio channels[END_REF], as several consecutive packets might experience the same attenuation and, therefore, might be in outage for a long period of time. In this context, it is crucial to characterize second-order statistics for a sound system design [START_REF] Di Renzo | Secondorder statistics of amplify-and-forward multi-hop wireless networks: A framework for computing the end-to-end SNR auto-correlation function over log-normal shadowing channels[END_REF].

• Spatial topology -Unlike non-cooperative protocols, when spatially-close neighbor nodes are willing to collaborate it is important to consider that radio links that are geographically close to one another often experience similar environmental shadowing effects, and, thus, might have correlated shadow-fading [START_REF] Agrawal | Correlated link shadow fading in multihop wireless networks[END_REF]. In these scenarios, neglecting the correlation among the cooperative links might overestimate the end-to-end performance [START_REF] Newport | Experimental evaluation of wireless simulation assumptions[END_REF].

S D dSD R1 R2 R3 R4 Rn d R 2 D d R 1D d R3 D d R 4D d R n D dSD >> dRiD
This paper is a first attempt to include all the above issues into the analysis, design, and optimization of cooperative MAC protocols. Due to space constraints, we focus our attention on only the first problem, i.e., replacing averages with outages, and postpone the analysis of time-and spatial-correlation of shadow-fading to future research. More specifically, our contribution is the following: i) by exploiting the frameworks proposed in [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]Sec. 2.4], we provide a closed-form analytical framework to choose the subset of active relays of a C-ARQ protocol for a given Quality-of-Service (QoS) requirement imposed at the application layer (i.e., the APEP in this paper); and ii) by using this physical layer driven relay selection mechanism, we provide an analytical framework to compute the throughput of the PRCSMA protocol, as well as study the potential benefits of cooperation and the role played by random fluctuations of the received signal caused by shadowing.

The reminder of this paper is organized as follows. In Section II, system model, channel model, and PRCSMA protocol are introduced. In Section III, the problems related to introducing shadowing in our system model are described, and an adequate relay selection mechanism is proposed. In Section IV, the performance of the PRCSMA protocol is studied in closed-form. In Section V, some numerical results are shown to validate our analytical derivation and claims. Finally, Section VI, concludes the paper.

II. SYSTEM MODEL

In this paper, we consider the system model in Fig. 1, which can be encountered in many real applications, such as a base station communicating with subscribers, a sink node connected to a sensor network, or an access point in a wireless local area network. The scenario is composed by a source (S) node which wishes to establish a communication with a destination (D) node. The message that S wants to send to D is composed by many packets, which are transmitted in different time slots. The communication ends when all the packets are delivered to D. Also, we assume a cooperative networking scenario in which n relay nodes (R i for i = 1, 2, . . . , n) are willing to help S to deliver, with high reliability, the packets to D. In particular, we consider a cooperative networking setup where the source-to-destination distance (d SD ) is much larger than the relay-to-destination distance (d R i D ). In this scenario, if all the nodes use the same transmission power, the relay-todestination links will be, on average, of better quality than the source-to-destination link because of the shorter distance.

A. Channel Model

Let l be the generic l-th wireless link of the network in Fig. 1. In particular, l can be the source-to-destination link (l=SD) or a relay-to-destination link (l=R i D for i = 1, 2, . . . , n). The receive-power on this link is [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]Eq. (1.1)] P

(l) r = (P t /d α l ) h (l) fast 2 h (l)
shad , where: i) P t is the transmit-power that is common to all the nodes; ii) α is the path-loss coefficient; iii) h (l) fast is the envelope of the fast-fading; iv) and h (l) shad is the shadow-fading. Thus, the channel model is quite general and accounts for all relevant terms that are usually estimated from measurements [START_REF] Newport | Experimental evaluation of wireless simulation assumptions[END_REF]. In particular, we assume that h (l) fast is a Nakagami-m random variable with fading parameters m l ≥ 0.5 and, without loss of generality1 , E h

(l) fast 2 = 1
for all the wireless links [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]. Also,

h(l) shad = (P t /d α l ) h (l) shad
denotes the random variable including path-loss and shadowing. As substantiated by many experimental measurements [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF], it is well-modeled by a log-normal random variable with parameters, in dB, equal to µ l = E 10 log 10 h(l) shad and

σ 2 l = E 10 log 10 h(l) shad -µ l 2
. Given the system model in Fig. 1, it is reasonable to assume: i) m l = m and σ 2 l = σ 2 for each wireless link; ii) µ SD = µ SRi for i = 1, 2, . . . , n; and iii) µ RiD = µ RD for i = 1, 2, . . . , n. We emphasize that, in general, µ SD = µ RD because source-to-destination and relay-to-destination distances are different. Finally, since, as mentioned in Section I, in this paper we are interested neither in time-nor in spatial-correlation of shadowing but only on the importance of considering it as a non-ergodic process, it is reasonable to assume that fading and shadowing are both timeand spatial-uncorrelated during the overall communication.

B. PRCSMA Protocol

In this paper, we are interested in studying the performance of the PRCSMA protocol in [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF], which has been specifically designed to coordinate the channel access of the relay nodes in C-ARQ schemes. The interested reader can find the details of this protocol in [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF]. In this section, we summarize only the most important aspects useful to highlight the impact of shadowing on its performance. Furthermore, we emphasize that, with very few changes, the considerations in this section are applicable to any C-ARQ based MAC protocol.

PRCSMA is based on the Distributed Coordination Function (DCF) of the IEEE 802.11 standard for wireless local area networks [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF]. In particular, it is a CSMA-based protocol, where the relays sense the channel for a Distributed Inter Frame Space (DIFS) before attempting to seize it. A slotted-time reference is considered, and the collision resolution algorithm relies on a binary exponential backoff with a Contention Window (CW) of size CW i = CW min •2 i for i = 1, 2, . . . , m b , where m b is the maximum backoff period.

PRCSMA works as follows. For each packet sent by the source to the destination, two events can take place: 1) if the received packet is of sufficient quality to meet the QoS requirement set by the application2 , then it is kept and considered as a correct packet that can be further processed by the upper layers. In that case, an ACK (acknowledgment) packet is broadcast so that relays and source can flush their cooperative and data buffers, respectively, and the communication ends. According to the rules of the DCF, the transmission of the ACK is performed after a Short Inter Frame Space (SIFS), which is needed to process the data and to compensate different propagation delays; 2) on the other hand, if the packet is of not sufficient quality, then the destination broadcasts a Call for Cooperation (CFC) packet to the nearby relay nodes to invite them to the cooperation phase. In that case, all the relay nodes that have overheard (with sufficient quality to meet the QoS requirement set by the application) the packet transmitted from the source become active relays and attempt to forward the packet to the destination. When this happens, the active relays will contend the access to the channel for relaying. More specifically, when an active relay seizes the channel to retransmit, if its packet is received by the destination with sufficient quality, then the same operations as in 1) are executed. On the other hand, if the packet is received by the destination with insufficient quality, then the destination waits for another active relay. The procedure is iterated until either a packet of sufficient quality is received and an ACK can be broadcast or a time out period has elapsed. In this latter case, the packet is discarded for the benefit of backlogged data.

III. IMPACT OF SHADOWING ON PROTOCOL DESIGN

In this section, we aim at providing some guidelines to optimize the performance of the PRCSMA protocol to explicitly meet the QoS requirement of the application. More specifically, we are interested in providing a formal definition for "packet of sufficient quality", which determines the set of active relays along with the correct reception of a packet by the destination. As foreseen by many communication systems, we consider a QoS requirement defined in terms of packet error probability, and denote the target value by P * p . Therefore, a packet will be accepted by the MAC layer as long as the APEP is below P * p .

A. Averages vs. Outages

To better understand the role played by log-normal shadowing, let us give a closer look at the operation of the system without and with shadowing. For a more comprehensive description, the interested reader might consult [START_REF] Mary | Mary symbol error outage over Nakagami-m fading channels in shadowing environments[END_REF] and [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF].

• Without Shadowing. From the channel model in Section II-A, the receive-power on the l-th link reduces to P

(l) r = (P t /d α l ) h (l) fast 2
. In this case, it is reasonable and common practice to assume that fast-fading is fast enough to be modeled as an ergodic process for the duration of a packet, such that the performance can be characterized through the APEP (where the average is computed over the distribution of the fast-fading). Thus, for a given physical layer setup, the APEP is a monotonic increasing function of only the distance d l , and we can write 3 APEP l = f (d l ) [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]. Accordingly, for a given distance the APEP is fixed, and a QoS requirement on the APEP, i.e., APEP l ≤ P * p , turns out to be equivalent to a maximum communication range, i.e., APEP l ≤ P * p ⇒ d l ≤ d * l . This reduces to the well-known circular coverage model, which has extensively been shown to be inadequate for the analysis and design of communication protocols [START_REF] Newport | Experimental evaluation of wireless simulation assumptions[END_REF].

• With Shadowing. In the presence of shadowing, things change substantially. From Section II-A, the receive-power on the l-th link depends on h (l) shad as well, which is a slowlyvarying random variable that can be reasonably assumed to be almost constant for the duration of a packet, such that it can experience only a few channel states for the overall duration of the communication [14, see figure in slide 6]. In other words, unlike h (l) fast , the process h (l) shad cannot be assumed to be ergodic for the duration of the communication. In that case, we are allowed to compute the average of the packet error probability over fast-fading, but we are not allowed to do the same over shadowing. Thus, for a given physical layer setup, the APEP is a function of both the distance d l and the shadowing. With the notation in Section II-A, we have

APEP l = f d l , h (l) shad = f h(l)
shad . Thus, a QoS requirement on the APEP, i.e., APEP l ≤ P * p , turns out to be equivalent to [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF], where γ * l is a threshold value that depends on the physical layer setup. Further details about γ * l are given in Section III-B. However, even though shadowfading is non-ergodic and we cannot compute the average over its distribution, h(l) shad is still a random variable with log-normal distribution. So, the quality of the link cannot be characterized only by the proximity (distance) between the nodes, but needs to be characterized statistically. In this case, the most suitable metric is the OPEP [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF], i.e., the probability of receiving a packet of insufficient quality, which is defined as 4

APEP l ≤ P * p ⇒ h(l) shad ≥ γ * l
OPEP l = Pr APEP l ≥ P * p = Pr f h(l) shad ≥ P * p = Pr h(l) shad ≤ γ * l = Q ([µ l -10 log 10 (γ * l )]/σ),
where the last identity follows from the assumption that h(l) shad is lognormal distributed.

From the considerations above, we are now ready to define in a formal way the concept "packet of sufficient quality" when shadowing is taken into account. In particular, a packet is accepted by the MAC layer if and only if the mean receivepower is above the threshold γ * l . In fact, both relays and destination can measure the mean receive-power, compare it with the pre-assigned threshold γ * l , and decide whether the relay can become active and whether the destination should request a retransmission. With this policy, and from the definition of γ * l above, i.e., γ * l = f -1 P * p , we can guarantee that if the mean receive-power is above γ * l , then the packet is decoded with APEP l ≤ P * p , which allows us to meet the QoS requirement. Furthermore, since h(l) shad is a random variable on its own, we can characterize statistically the quality of a link by exploiting the OPEP, which yields the probability that a packet is of not sufficient quality by taking into account the random nature of the shadowing. It is usual to say that a link is in outage when h(l) shad ≤ γ * l . Thus, the set of active relays introduced in Section II-A is given by only those relays which are not in outage, and the simple formula

P active l = 1 -OPEP l = 1 -Q ([µ l -10 log 10 (γ * l )]
/σ) provides a closed-form expression of the probability that a link is active by taking into account fast-fading, shadowing, and a realistic physical layer setup.

B. Computation of γ *

l for a Realistic Physical Layer The crucial point in the relay selection procedure described above in the presence of shadowing is the closed-form computation of γ * l = f -1 P * p , by explicitly taking into account the most important parameters of the physical layer. The computation of this threshold for a generic physical layer is not a trivial task, especially when the QoS requirement of interest is the APEP, i.e., APEP l ≤ P * p . This is an important contribution of this paper, as this threshold is often set to arbitrary values that do not account for the actual physical layer implemented by the communication system, such as modulation, coding, packet length, etc. To this end, we exploit the methodology recently proposed in [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]. As a realistic illustrative example, at the physical layer we assume: i) single-input-single-output transceivers for all the nodes of the network in order to comply the with lowcomplexity requirement of the nodes, which exploit distributed rather than co-located multiple-antenna diversity; ii) a general Multilevel Phase Shift Keying (M-PSK) modulation scheme; iii) a channel block-code; iv) hard-decision decoding; and v) ideal interleaving. Furthermore, we define a packet error as the probability that at least one bit of the packet is wrong. With these assumptions, γ * l = f -1 P * p can be computed in closed-form as follows [START_REF] Mary | Etude analytique des performances des systèmes radio-mobiles en présence d'évanouissements et d'effet de masque[END_REF]Sec. 2.4]:

γ * l = m g mod 8 > > < > > : 2 6 4 P * b k mod • v u u t 1 -t • m s P * b k mod ! 3 7 5 -(1/m) -1 9 > > = > > ; (1)
where we have defined:

P * b = 8 > < > : P * c • (t + 1) • B (t + 1, J -t) n 1 -ỹ [P * c • (t + 1) • B (t + 1, J -t)] (t+1) -1 o J-t-1 9 > = > ; (t+1) -1 (2)
and: i) ξ) dξ is the Gamma function; viii) M is the modulation order of M-PSK; ix) N is the number of bits in a packet; x) k is the number of information bits in a codeword; xi) J is the number of bits (information and redundancy) in a codeword; and xii) t is the number of errors the channel code can correct.

P * c = 1 -1 -P * p k/N ; ii) g mod = sin 2 (π/M ); iii) t = m/(m + 1); iv) k mod = Γ (m + 0.5)/[ √ π max {log 2 (M ) , 2} Γ (m + 1)]; v) ỹ = (t + 1)/(t + 2); vi) B (x, y) = [Γ (x) Γ (y)]/Γ (x + y); vii) Γ (z) = +∞ 0 ξ z-1 exp (-
It is worthwhile mentioning that, in spite of the generality of the physical layer we consider, the final expression in ( 1) is very simple to compute, and it only depends on the fading parameters rather than on instantaneous channel state information. This means that it can be computed during the initial setup and used for the whole duration of the communication.

Also, we emphasize that in our system model we have m l = m for all the wireless links, which implies that γ * = γ * l is the same for all the wireless links. Finally, we note that the QoS requirement, P * p , is explicitly given in (1).

IV. CLOSED-FORM PERFORMANCE OF PRCSMA

Having defined that most appropriate performance metric to take into account the effect of shadowing, and provided a relay selection mechanism that accounts for the QoS requirement requested by the application for a realistic physical layer and channel model, we are now ready to analyze the performance of the PRCSMA protocol. In particular, we focus our attention on the analysis of the throughput with the aim of understanding the improvement introduced by cooperative networking and the number, n, of possible active relays.

The average throughput of the system, expressed in bits per second (bps), can be computed as the ratio between the average size of the data packets, i.e., E[L], and the average total time required to transmit a packet (by including transmission time, MAC overhead, and possible retransmissions). Accordingly, the throughput, S, of the system can be expressed as:

S [bps] = E {L} (1 -OPEP SD • p nR ) (1 -OPEP SD ) T S + OPEP SD • E ˘T S|COOP ¯ (3)
where: i) in our case E {L} = N , as we consider the transmission of packets with a constant number of bits; ii) OPEP SD is the outage packet error probability that can be obtained from Section III-A; iii) p nR is the probability that there are no active relays during the cooperation phase or, in other words, that all the relays are in outage. Accordingly, we have p nR = OPEP n SD ; iv) T S is the duration of the transmission of a single packet; and v) E T S|COOP is the average duration of a packet transmission when cooperation is requested by the destination. Furthermore, it is worthwhile mentioning that the substraction in the numerator takes into account the fraction of packets that are discarded after cooperation when all the relays are in outage, and, thus, there are no active relays that can help. Also, the denominator is a weighted sum of the probability that the source-to-destination link is in outage.

Finally, an important consideration is worth being made about (3) to emphasize, once again, the difference of the analysis in the presence of shadowing. By carefully looking at previous analysis neglecting shadowing [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF], [START_REF] Alonso-Zárate | Performance analysis of a persistent relay carrier sensing multiple access protocol[END_REF], we notice that in (3) the APEP over the source-to-destination link is actually replaced by the OPEP over the same link. This confirms that in the presence of shadowing the metric that should be used to characterize the performance of the protocol is the outage probability rather than the average error probability. More specifically, it would have been erroneous to replace in [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF], [START_REF] Alonso-Zárate | Performance analysis of a persistent relay carrier sensing multiple access protocol[END_REF] the PEP averaged over both fast-fading and shadowing because, as explained in Section III-A, shadowing is a nonergodic process for the duration of the communication, and, thus, it does not experience all the fading states (see, e.g., [14, figure in slide 6] and [START_REF] Mary | Mary symbol error outage over Nakagami-m fading channels in shadowing environments[END_REF] for further comments).

To compute (3), we need a closed-form expression of E T S|COOP . This parameter can be obtained by using considerations similar to [START_REF] Alonso-Zárate | Persistent RCSMA: A MAC protocol for a distributed cooperative ARQ scheme in wireless networks[END_REF], [START_REF] Alonso-Zárate | Performance analysis of a persistent relay carrier sensing multiple access protocol[END_REF], but updating the analytical development to incorporate the notion of outage instead of average. Due to space constraints, we are unable to report all the details of the analytical derivation and only summarize the final result. In particular, E T S|COOP can be computed as:

 E ˘T S|COOP ¯= T CFC + T ACK + 2T SIFS + T COOP T COOP = p nR T TO-COOP + (1 -p nR ) E {T CONT } (4)
where: i) T CFC is the transmission time of the CFC packet sent by the destination to initiate the cooperation phase; ii) T ACK is the transmission time of the ACK related to the CFC packet; iii) T SIFS is the duration of the SIFS that is used to compensate for the propagation delays; iv) T TO-COOP is the time-out after which the cooperation phase ends if no retransmissions from the relays are received; and v) E {T CONT } is the average contention time due to the PRCSMA protocol. It can be computed by resorting to the procedure described in [START_REF] Alonso-Zárate | Performance analysis of a persistent relay carrier sensing multiple access protocol[END_REF] and by considering the actual behavior of the new protocol based on outages. More specifically, only two parameters need to be replaced in the framework in [START_REF] Alonso-Zárate | Performance analysis of a persistent relay carrier sensing multiple access protocol[END_REF], i.e., the average number of active relays, i.e., E {R}, and the average number of transmissions from all relays to deliver a packet to the destination, i.e., E {K}. By taking into account the considerations in Section III-A, they can be computed in closed-form as follows:

E {R} = n X r=0 n r h" n r " (1 -OPEP r SD ) OPEP n-r SD io = nOPEP SD (5) E {K} = +∞ P k=0 ˆ(k + 1) OPEP k RD (1 -OPEP RD ) = (1 -OPEP RD ) -1 (6)
where • • is the binomial coefficient, and the last identities are obtained with simple algebraic manipulations.

V. NUMERICAL RESULTS

The aim of this section is twofold: i) to validate the accuracy of the analytical frameworks developed in Section III for the physical layer and in Section IV for the MAC protocol; and ii) to analyze the impact of shadowing and cooperation on the performance of the overall PRCSMA protocol. The analytical results are compared to Monte Carlo simulations by applying all the rules of the protocol without any simplifications. The following set of parameters is used to obtain the presented figures. The physical layer parameters are: i) M = 8; ii) m = 3; iii) Hamming block-code with J = 7, k = 4, and t = 1; iv) N = 128 bytes; and v) µ RD = 40dB. Furthermore, the QoS requirement is set to P * p = 0.1, which yields a threshold equal to γ * = 41.12 = 16.14dB. Note that these parameters are common to all links, and thus we are considering that the relays use all the same transmission rate. The MAC layer parameters are: i) SlotT ime = 10µs; ii) the minimum and maximum values of CW are CW min = 16 and CW MAX = 32, respectively; iii) the overall duration of MAC header and physical-layer preamble is 232 µs; iv) DIFS = 50µs; v) SIFS = 10µs; vi) the length of RTS (Request-to-Send) packets is 20 bytes; and vii) the lengths of CTS (Clear-to-Send), CFC, and ACK packets are 14 bytes each.

In Fig. 2 and Fig. 3, we show the normalized throughput, i.e., S norm = S/(T s N ) for n = 1 and n = 10, respectively. We notice a very good agreement between Monte Carlo simulations and the analytical models developed in Section III and in Section IV. As far as the system behavior is concerned, the following conclusions can be drawn: i) S norm increases when the quality of the source-to-destination link improves (i.e., µ SD increases), as expected; ii) if µ SD ≥ 18dB, then S norm decreases when the severity of the shadowfading increases (i.e., σ increases). On the other hand, S norm increases if µ SD = 10dB. This behavior can be explained by taking into account the working principle of the cooperative network under analysis and the system setup. In fact, if µ SD < γ * = 16.14dB the transmit-power is too low to even offset the deterministic path-loss caused by the transmission distance. In this case, the system is almost always in outage and the throughput is very low. However, if σ increases, there are occasional situations in which shadowing results in a constructive gain, which yields a higher throughput than the scenario with no shadowing (i.e., σ = 0dB). In this case, shadowing is beneficial. However, we emphasize that a system in good operation should, at least, be designed to have a receive-power that guarantees a good communication in the absence of shadowing, i.e., µ SD ≥ γ * in our case. These two trends highlight the important role played by shadowing to assess the performance of MAC protocols, as well as that severe shadowing conditions (σ = 12dB) might reduce the throughput by up to 37% when µ SD > γ * ; and iii) the throughput increases with the number of potential relays n.

In Fig. 4 and Fig. 5, we study the gain introduced by cooperation for different shadowing conditions, when µ SD = 18dB and µ SD = 24dB, respectively. The main aim is to understand if there is an optimal value of n to achieve the best throughput gain by exploiting the cooperative networking paradigm. To this end, we plot the normalized throughput gain with respect to the setup with no cooperative nodes, i.e., n = 0. In formulas, we have S gain [%] = 100 × {[S norm (n) -S norm (n = 0)]/S norm (n = 0)}. The results are very interesting, and three main conclusions can be drawn: i) there is an optimal value of n to achieve the best throughput. In particular, too many potential relays result in a higher overhead, which reduces the net throughput; ii) the gain of cooperation is higher for more severe shadowing conditions (i.e., higher values of σ). In fact, cooperation has primarily been introduced to counteract severe shadowing through distributed diversity. Thus, the gain of cooperation in the absence of shadowing is close to zero, since we can always deterministically set the transmit-power for a fixed distance and avoid outages. This highlights, once again, the important role of shadowing for the analyzed cooperative scenario; and iii) the throughput gain introduced by cooperation increases for worse source-to-destination links (i.e., µ SD decreases). This is an expected result, and shows that cooperation should be used only when actually needed, i.e., when the direct link is of not sufficient quality. The PRCSMA protocol is actually designed to account for that, as the cooperation phase is initiated only if this link fails to deliver the packets, thus controlling the overhead when cooperation is not needed.

VI. CONCLUSION

In this paper, we have proposed an analytical methodology for the analysis and design of cooperative MAC protocols with a realistic physical layer and wireless channel model. Notably, we have carefully addressed the issues related to accounting for slowly-varying shadow-fading in the analysis and design of C-ARQ based MAC protocols. We have proved that the appropriate performance metric to be considered is the outage error probability, as opposed to usual practice of considering either just the propagation distance or the error probability averaged over fast-fading. Also, we have studied the PRCSMA protocol, and have highlighted that shadowing can significantly affect its performance, as well as that cooperation can substantially improve the end-to-end performance. Ongoing research is concerned with the extension of the proposed framework to take into account time-and, especially, spatial-correlation of shadow-fading in the analysis and design of C-ARQ based MAC protocols.

Fig. 1 .

 1 Fig. 1. System model.

Fig. 2 .

 2 Fig. 2. Normalized throughput against the quality of the source-to-destination link (µ SD ) and the severity of shadowing (σ). Solid lines show the analytical framework and markers show Monte Carlo simulations (n = 1).

Fig. 3 .

 3 Fig. 3. Normalized throughput against the quality of the source-to-destination link (µ SD ) and the severity of shadowing (σ). Solid lines show the analytical framework and markers show Monte Carlo simulations (n = 10).

Fig. 4 .

 4 Fig. 4. Throughput gain against the number of potential relays (n) and the severity of shadowing (σ) when µ SD = 18dB.

Fig. 5 .

 5 Fig. 5. Throughput gain against the number of potential relays (n) and the severity of shadowing (σ) when µ SD = 24dB.

E {•} is the expectation operator.

In Section III, we provide a formal definition of "packet of sufficient quality" by taking into account the channel model introduced in Section II-A.

f (•) denotes "function of".

Q (x) = " 1 . √ 2π " R +∞ x exp `-t 2 ‹ 2 ´dt is the Q-function,and Pr {•} denotes probability.


Thilo Fath
email: thilo.fath@eads.net

Harald Haas
email: h.haas@ed.ac.uk

Marco Di Renzo
email: marco.direnzo@lss.supelec.fr

Raed Mesleh
email: r.mesleh@jacobs-university.de

Spatial Modulation applied to Optical Wireless Communications in Indoor LOS Environments

Keywords: optical wireless communications, modulation, diversity, SM, repetition coding

In this paper, we study the performance of Spatial Modulation (SM) applied to optical wireless communications (OWC) in indoor environments with line-of-sight (LOS) characteristics. To this end, we consider setup scenarios with different numbers of optical transmitters and receivers which are arranged within a room. SM is compared to repetition coding (RC). Because RC is known to achieve very good performance in OWC systems due to the use of intensity modulation and the resulting constructive superposition of the power signals. The results show that SM can outperform RC when high spectral efficiencies are desirable, e.g. 4 bit/s/Hz and greater, since it can operate with reduced signal modulation orders by conveying additional data bits in the spatial domain. We also demonstrate that SM benefits from receive-diversity to a larger extent while at the same time requiring less computational complexity. Furthermore, we give a general framework to numerically approximate the average bit error probability of both SM and RC.

I. INTRODUCTION

The first pioneering studies in optical free-space transmissions for indoor environments [START_REF] Gfeller | Wireless In-House Data Communication Via Diffuse Infrared Radiation[END_REF] have shown the potential of optical wireless communications (OWC) to provide flexible and efficient indoor data transmission. With the advent of cheap and powerful light emitting diodes (LEDs), appropriate optical transmitters are available which can be used e.g. in home and office scenarios. As OWC transmission does not interfere with delicate electronic systems, it can even be applied to sensitive environments like hospitals and aircrafts. However, like all wireless communication systems, OWC also has to cope with the ever increasing demand for higher data rates. Therefore, it is important to provide high spectral efficiencies at low error ratios.

Commonly, OWC transmission schemes employ intensity modulation of the optical carrier and direct detection at the receiver side [START_REF] Hranilovic | Wireless Optical Communication Systems[END_REF], [START_REF] Kahn | Wireless Infrared Communications[END_REF]. To this end, up-and down-conversion of the signals can be done by low-cost diodes without the need for sophisticated high-frequency circuit designs. Most optical wireless links use simple modulation techniques like on-offkeying (OOK) or pulse amplitude modulation (PAM) because they offer easy implementation. PAM provides an enhanced spectral efficiency by using several intensity levels of the optical signal in contrast to OOK which encodes information by simply switching the device on and off. However, the provision of enhanced spectral efficiencies by using larger constellation sizes leads to worse bit error ratio (BER) performance. Thus, the main drawback of simple single-input-single-output (SISO) systems is that the achievable data rate at reasonable BER performance is low since a high order modulation scheme is needed. A known solution to improve the error performance of wireless communication systems is by exploitation of diversity. The reliability can be enhanced by using several receiving devices, e.g. photo diodes, which enables receive-diversity. These so called single-input-multiple-output (SIMO) methods receive several replicas of the same transmitted signal. Hence, an enlarged portion of the emitted power can be collected and the multiple signal receptions can be combined to improve the quality of the wireless link. Besides, it is possible to use several transmit devices as well. In order to provide transmitdiversity, repetition coding (RC) can be used which works by the principle that the same information is sent from multiple transmitters simultaneously. RC is known to achieve very good performance in free-space optical communications with lineof-sight (LOS) as the intensities coming from the multiple transmitters constructively add up at the receiver side [START_REF] Safari | Do we really need OSTBCs for free-space optical communication with direct detection?[END_REF], [START_REF] Navidpour | BER Performance of Free-Space Optical Transmission with Spatial Diversity[END_REF]. Because of this, RC can outperform orthogonal space-time block codes (OSTBCs) and even SIMO transmission [START_REF] Safari | Do we really need OSTBCs for free-space optical communication with direct detection?[END_REF].

In [START_REF] Chau | Space Modulation on Wireless Fading Channels[END_REF], [START_REF] Mesleh | Spatial Modulation -A New Low Complexity Spectral Efficiency Enhancing Technique[END_REF] a new and promising modulation technique called Spatial Modulation (SM) has been proposed. It has been shown that SM can achieve high data rates while providing good BER performance and low system complexity. Like RC, SM employs several emitters for transmitting data. As opposed to common modulation techniques where information is conveyed by modulating the signal, SM additionally conveys data bits in the spatial domain. In order to accomplish this, SM considers the transmitter array as an additional (spatial) constellation diagram. Unlike RC, SM works by the principle that only one transmitter is active at any time instance. In addition to modulating the signal, e.g. modulation of the signal amplitude, information is also encoded in the index of the transmitter which emits the signal. Therefore, high spectral efficiencies can be achieved which depend not only on the signal constellation size, but also on the number of transmitters. This leads to a degree of freedom as both parameters can be traded off against each other. Furthermore, SM can deal with high channel correlation and power imbalances of wireless links [START_REF] Di Renzo | On the Performance of Space Shift Keying MIMO Systems Over Correlated Rician Fading Channels[END_REF]. As these are characteristics of optical wireless links [START_REF] Fath | On the Performance of Space Shift Keying for Optical Wireless Communications[END_REF], [START_REF] Mesleh | Optical Spatial Modulation[END_REF], SM seems to be especially suitable for OWC.

Hence, in this paper, we apply SM to OWC in indoor LOS environments. We compare SM and RC transmission with regard to their BER performance for different spectral efficiencies. Setup scenarios with different numbers of optical transmitters and receivers are considered. Furthermore, we give a numerical framework to calculate the respective average bit error probability (ABEP) of both schemes.

The remainder of the paper is organized as follows: In Section II we introduce the system model and the notations. Section III presents the optical wireless setup from which we derive the channel coefficients for the considered indoor scenarios. The BER performance of SM and RC is studied in Section IV, where numerical and simulation results are shown. Finally, Section V concludes the paper.

II. SYSTEM MODEL

The following notations are used throughout the paper: lower case bold symbols denote vectors and upper case bold symbols denote matrices. We use [•]

T for the transpose operator, |•| for the absolute value and • F for the Frobenius norm. The signal constellation size is given by M and d H (•, •) denotes the Hamming distance of two bit assignments.

Q(x) = 1 √ 2π +∞ x exp( -t 2 2
) dt is the Q-function. N t is the number of transmitters and N r is the number of receivers, where n t is the transmitter index and n r is the receiver index. We assume perfect knowledge of the channel and ideal time synchronisation at the receiver side. Channel coding is not taken into account.

The received signal vector is given by:

y = H x + n, (1)
where

x = [x 1 . . . x Nt ]
T is the signal vector to be transmitted. H is the N r ×N t channel matrix. Its single elements h nrnt represent the respective channel coefficient of the wireless link between transmitter n t and receiver n r . Furthermore, n is the noise, which we assume as zero mean additive white Gaussian noise (AWGN) of power E n . The detection at the receiver is based on the maximum-likelihood (ML) principle. The ML detector decides for the signal vector x which minimises the Euclidean distance between the actual received signal y and all potential received signals leading to

x = arg max x p y (y|x, H) = arg min x y -H x 2 F , (2)
where p y is the probability density function of y conditioned on x and H.

The intensities

I PAM m
which can be used for signal modulation applying M -PAM are given by

I PAM m = 2 I M -1 m for m = 0, 1, . . . , (M -1), (3)
with I being the mean optical intensity emitted. Hence, M -PAM provides a spectral efficiency of log 2 (M ) bit/s/Hz. As RC works by the principle that all transmitters emit the same signal, x 1 = x 2 = . . . = x Nt In order to ensure comparability, the mean optical power transmitted has to be fixed, irrespective of the number of transmitters. Thus, for RC transmission, the optical power is equally distributed across all N t emitters and the intensities given in (3) have to be divided by factor N t . In (4) we denote the ABEP of RC employing M -PAM for an arbitrary N t × N r scenario. It is a generalization of the common M -PAM ABEP, which is e.g. given in [START_REF] Hranilovic | Wireless Optical Communication Systems[END_REF]. At this end, E s = (r I) 2 is the mean electrical energy of the intensity modulated optical signal, with r being the optical-to-electrical conversion coefficient.

Because in intensity modulated optical communications the electrical energy is proportional to the square of the optical power [START_REF] Kahn | Wireless Infrared Communications[END_REF]. By using SM, the bit sequence to be transmitted is passed to the SM encoder. As illustrated in Fig. 1, the encoder maps the bits to the signal vector x. At any given time instance, only one transmitter radiates optical power. This means that only one element of x is non-zero. The index of this respective element states the index of the transmitter (e.g. LED) which is activated. Therefore, the emitter index depends on the bit sequence at the encoder input. In this way, one part of the data is transmitted. The other part of the data is conveyed in a conventional way via PAM. This means that the nonzero element of x is a specific signal constellation point with intensity I m representing additional information bits. b m nt is the bit assignment of the signal when intensity I m is emitted from transmitter n t . However, classical PAM must be modified because a signal with I m = 0 cannot be used for conveying information as in this case all elements of x would be zero and the spatial information would be lost. As a consequence, the intensities I SM m which can be used for SM are

I SM m = 2 I M +1 m for m = 1, 2, . . . , M. (5)
In the example displayed in Fig. 1, the data bits are arranged in blocks of 4 bits leading to a spectral efficiency of 4 bit/s/Hz. The last two bits define the transmitter index and the first two ones the signal intensity assuming M = 4. As shown, for instance the bit sequence "1 0 1 1" corresponds to LED

ABEP RC ≈ 2 (M -1) M • log 2 (M ) • Q    1 M -1 • E s E n N 2 t • Nr nr=1 Nt nt=1 h nrnt 2    (4) ABEP SM ≤ 1 MN t • log 2 (MN t ) • M m (1) =1 Nt n (1) t =1 M m (2) =1 Nt n (2) t =1 d H b m (1) n (1) t , b m (2) n (2) t • Q   r 2 4 E n • Nr nr=1 I SM m (2) • h nrn (2) t -I SM m (1) • h nrn (1) t 2   (6)
number 4 emitting an optical pulse with intensity I 3 . At the receiver side, the detector has to perform two detection tasks.

First, it has to estimate the index of the respective LED which is active and second it has to decode the information encoded in the signal from the received intensity level. Only if both the index and the signal constellation point are detected correctly, the bit sequence can be decoded error free. As ML detection is assumed, both estimation tasks are jointly done by the decoding algorithm given in (2). According to [START_REF] Navidpour | BER Performance of Free-Space Optical Transmission with Spatial Diversity[END_REF], the minimum distance between two possible SM intensities is 2 I M +1 , whereas the minimum distance for PAM is 2 I M -1 . The lower signal distance of SM leads to a worse BER performance as the error probability depends on the Euclidean distance of the transmitted signals. But as SM conveys additional information bits in the spatial domain, it provides a higher spectral efficiency which is log 2 (M ) + log 2 (N t ) = log 2 (MN t ) bit/s/Hz. In other words: SM can achieve the same efficiency as M -PAM, but with a reduced signal constellation size of M = M Nt , hence effectively enlarging the distance of the signal points. In [START_REF] Di Renzo | On the Performance of Space Shift Keying MIMO Systems Over Correlated Rician Fading Channels[END_REF] and [START_REF] Di Renzo | Performance Analysis of Spatial Modulation[END_REF] it is shown that the error performance of SM depends on the differences between the channels. Moreover, it is demonstrated that the ABEP of SM can be approximated by union bound methods. Due to space constraints, we omit the detailed calculation and report only the final result for the ABEP of SM which is given in [START_REF] Chau | Space Modulation on Wireless Fading Channels[END_REF].

If we consider the computational complexity at the receiver side, it can be seen that the detection of SM transmission requires fewer mathematical operations compared to RC. For RC, a total of M N r (2 N t + 1) multiplications, additions and subtractions are needed. In contrast, SM merely requires 3 M N t N r = 3 M N r operations and, therefore, is less computationally expensive. For instance, if M = 16, N t = 4 and N r = 4, RC requires 576 operations, whereas SM requires only 192 operations.

III. OPTICAL WIRELESS SETUP SCENARIO

In the following, we consider an LOS environment with different N t × N r setup scenarios. We assume that the transmitters are placed at a height of z = 2.5 m and are oriented downwards to point straight down from the ceiling. The receivers are located at a height of z = 0.75 m (e.g. height On the basis of this setup scenario, we derive the channel coefficients of the optical wireless links. Fig. 3 illustrates the geometries used to calculate the coefficients h nrnt . As shown, φ is the angle of emergence with respect to the transmitter axis ψ is the angle of incidence with respect to the receiver axis. Furthermore, d depicts the distance between transmitter and receiver. According to [START_REF] Kahn | Wireless Infrared Communications[END_REF], the DC channel gain is the most distinctive parameter describing an optical wireless link. Therefore, the channel coefficient of a directed LOS link can Fig. 3. Geometric scenario used for calculation of channel coefficient hn r n t .

TABLE I VALUES OF PARAMETERS USED IN SIMULATIONS TO CALCULATE

CHANNEL COEFFICIENT hn r n t .

Parameter

Value used in simulations φ according to specific TX and RX position within setup scenario, given in degree ψ according to specific TX and RX position within setup scenario, given in degree

Φ 1 2 15 • Ψ 1 2 30 • A 1 cm 2 r 1 A/W d
according to specific TX and RX position within setup scenario, given in m be calculated as follows:

h nrnt = (k+1) A 2 π d 2 nr n t cos k φ nrnt cos ψ nrnt 0 ≤ ψ nrnt ≤ Ψ 1 2 0 ψ nrnt > Ψ 1 2 (7) with the order k = -ln(2) ln(cos Φ 1 2
) and the transmitter semiangle Φ 1 2 . Furthermore, Ψ 1 2 denotes the FOV semiangle of the receiver and A is the detector area of the receiver. Hence, the channel coefficients depend on the specific position of each transmitter and receiver within the setup scenario. Table I gives the respective values considered in the following simulations to calculate h nrnt .

IV. RESULTS ON BIT ERROR RATIO PERFORMANCE

In this section, we study the BER performance of SM and RC using the scenario introduced in section III. We consider setups with different numbers of optical transmitters and receivers. Furthermore, we analyse the error performance for varying spectral efficiencies. In order to ensure comparability, the mean emitted power is the same in each scenario as well as for both transmission techniques. First of all, we consider the 4×4 setup scenario illustrated in Fig. 2. For this scenario, Fig. 4 depicts the BER performance of SM and RC assuming a spectral efficiency of 2, 4 and 6 bit/s/Hz. It can be seen that the numerical ABEPs (markers) given in ( 4) and ( 6) very closely match the simulation results (lines). In general, the given ABEPs provide a good means to evaluate the BER performance of SM and RC because they do not depend on specific channel statistics, but only on the transfer factors. As seen in Fig. 4, for a spectral efficiency of 2 and 4 bit/s/Hz, RC applying 4-PAM, respectively 16-PAM, achieves a better performance than SM. But, if we consider improved spectral efficiency of 6 bit/s/Hz, SM outperforms RC up to a signal to noise ratio (SNR) of about 36 dB. This is because SM operates with a reduced signal constellation size of M = 16 compared to 64-PAM RC transmission. At a SNR above 36 dB RC gets superior because of its transmit-diversity gain.

As SM only uses one transmitter at any time instance, it cannot provide transmit-diversity. However, it can utilize receive-diversity to a larger extent than RC by offering higher SNR gains with increasing N r . This finding is shown in Fig. 5, where we study the performance of SM and RC for 5 bit/s/Hz for a scenario with N t = 8 and a varying number of optical receivers. It can be seen that when consecutively doubling N r from 2 up to 16, RC achieves a performance gain of about 3 dB in each step. In contrast, SM achieves larger improvements as it provides a performance gain of about 5 dB by moving from N r = 4 to N r = 8 and of about 9 dB by moving from N r = 8 to N r = 16. Consequently, in the 8 × 16 scenario SM even outperforms RC by about 2 dB.

Besides these improvements, SM has another essential advantage over RC transmission. If more bits are to be transmitted per channel use, RC needs a higher increase in SNR to be able to provide the same BER performance. This observation is taken from Fig. 6, which shows the error ratios for a 16 × 16 transmission system providing different spectral efficiencies (4, 5 and 6 bit/s/Hz). It can be seen that RC needs a SNR betterment of about 6 dB to achieve the same BER when providing 5 instead of 4 bit/s/Hz, whereas SM requires an increase of only 3 dB. Consequently, SM with M = 2 outperforms 32-PAM RC by about 8 dB and even 16-PAM by about 2 dB. If the spectral efficiency is increased by 1 bit to 6 bit/s/Hz, RC requires additional 6 dB, in contrast to SM which needs only an increase of about 4 dB. In summary, for a spectral efficiency of 6 bit/s/Hz SM outperforms RC by about 10 dB and even outperforms the less efficient 32-PAM RC transmission by about 4 dB. Hence, the benefits of SM over RC largely increase with greater spectral efficiencies.

V. SUMMARY & CONCLUSION

We have studied the performance of Spatial Modulation applied to OWC in indoor environments and compared it to RC. The simulation results were substantiated by numerical ABEP calculations. We have demonstrated that SM achieves significant gains if several receivers are employed because it provides large SNR improvements with increasing number of receivers. Especially for higher spectral efficiencies, SM provides a better error performance as it can operate with a reduced signal constellation size by conveying data bits in the spatial domain. Furthermore, SM can achieve better error ratios particularly at low SNR regions because the transmitdiversity gain of RC prevails only at higher SNRs. Moreover, SM even provides less computational complexity. Thus, SM is a suitable modulation technique for OWC to provide high data rates at good BER performance. Future work will deal with the adaptation of SM in order to achieve benefits by transmitdiversity as this will enhance its performance, especially for low spectral efficiencies and at high SNR regions.

4 Fig. 1 .

 41 Fig. 1. Illustration of SM operation providing 4 bit/s/Hz with Nt = 4.

Fig. 2 .

 2 Fig. 2. Positioning of a 4 × 4 setup within room.

Fig. 4 .

 4 Fig. 4. Comparison of SM and RC for spectral efficiency of 2, 4 and 6 bit/s/Hz in 4 × 4 setup scenario (lines show simulation results and markers numerical ABEP results).

4 Fig. 5 .

 45 Fig. 5. Comparison of SM and RC for 5 bit/s/Hz with Nt = 8 and varying number of optical receivers Nr (lines show simulation results and markers numerical ABEP results).

Fig. 6 .

 6 Fig. 6. Comparison of SM and RC for spectral efficiency of 4, 5 and 6 bit/s/Hz in 16×16 setup scenario (lines show simulation results and markers numerical ABEP results).

ACKNOWLEDGEMENT

Professor Haas acknowledges the Scottish Funding Council support of his position within the Edinburgh Research Partnership in Engineering and Mathematics between the University of Edinburgh and Heriot Watt University.