Efficient Media Access Using Adaptive Array

Antennas

Thrasyvoulos Spyropoulos

INRIA,Sophia Antipolis, France

Caulgi S.Raghavendra

University of Southern California,Dept. of Electrical Engineering

Los Angeles, CA

Abstract

Wireless ad hoc networks have traditionally been assumed to use omni-directional anten-nas. However, it has been recently recognized that the use of advanced antenna technolo-gies could help alleviate the capacity limitations of multi-hop wireless networks, stemmingfrom interference generated in the shared medium [1]. Adaptive array antennas have theability to automatically respond to an unknown interference environment, in real time, bysteering nulls and reducing side lobe levels in the direction of interference, while retain-ing some desired signal beam characteristics. However, real-time beamforming usuallyrequires sophisticated DSP algorithms that use extra synchronization information, liketraining sequences, and might be quite “costly” for small wireless terminals. What ismore, simply equipping each node with an adaptive antenna is not merely sufficient toimprove performance.

In this paper, we present a cross-layer approach that enables nodes in an ad hoc networkto adapt the radiation pattern of a fully adaptive array antenna using information alreadyavailable or easily obtainable at the MAC layer. We propose a protocol, called ADAPT,that extends the basic mechanisms of 802.11, and adjusts the antenna pattern, on a per-packet basis, using only MAC layer information (semi-static beamforming). Additionally,we use analytical methods to assess the effect on performance due to adapting the antennapattern on a per-packet basis, instead of real-time, in the presence of node mobility andcontenting neighboring communications. Finally, we use ns-2 simulations to extensivelycompare the performance of our adaptive antenna configuration against typical omni-directional and directional configurations. Our protocol is shown to achieve up to a 3×improvement over the omni-directional case and up to a 50% improvement over thedirectional one, in most scenarios considered.

Key words: ad hoc networks, smart antennas, MAC protocol, cross-layer design

Preprint submitted to Elsevier 12 March 2007

1 Introduction

Wireless ad-hoc networks are multi-hop networks where all nodes cooperativelymaintain network connectivity without the need of any wired infrastructure (e.g.base stations, routers, etc.). It has commonly been assumed that nodes in ad-hoc networks are equipped with omni-directional antennas. However, it has beenrecently recognized that the use of directional or smart antennas could be verybeneficial in the context of ad-hoc networks [2]. Such antennas have the ability toconcentrate the radiated (received) power towards (from) the intended direction oftransmission (reception). As a result of this, they can help improve the performanceof ad-hoc networks in terms of energy efficiency [3,4], capacity [5–7], throughputand end-to-end delay [8–14].

However, simply equipping each node with a directional antenna is not merely suf-ficient to improve performance. In order to take advantage of the high potential ofadvanced antenna technologies, it is necessary to design appropriate communica-tion protocols that can synchronize the transmitting and receiving antennas. Theseprotocols will also depend on the type of antenna in hand. Two types of directionalantennas have attracted most researchers’ attention for use in ad-hoc networks.These are switched-beam antennas [9,12] and phased-array antennas [10,11,15,16].The former can only choose between a small set of fixed beams (more correctly, be-tween a set of fixed antenna patterns). The latter can steer the main beam towardsany direction but have no control over the accompanying sidelobes in the antennaradiation pattern. This lack of control on the whole antenna pattern results insignificant sidelobe interference, the cumulative effect of which can significantlyreduce the maximum number of simultaneous transmissions [5].

One way to overcome this difficulty is to use fully adaptive antennas (aka. “smartantennas”). Fully adaptive array antennas have the ability to automatically re-spond to an unknown interference environment, in real time, by steering nulls andreducing side lobe levels in the direction of the interference, while retaining somedesired signal beam characteristics. This process is often referred to as beamform-ing. These systems usually consist of an array of weighted antenna elements, whoseindividual weights are controlled, in real time, in order to produce the desired ra-diation pattern. A digital processing unit is usually responsible for controlling theelement weights towards some kind of optimization of output SINR (Signal to Inter-ference and Noise Ratio), in accordance with a control algorithm [17–19]. Finally,

Email addresses: Thrasyvoulos.Spyropoulos@sophia.inria.fr (ThrasyvoulosSpyropoulos), raghu@usc.edu (Caulgi S.Raghavendra).

2

in addition to minimizing interference, adaptive antennas can be used to estimatethe Direction-Of-Arrival (DOA) of one or more incoming signals using algorithmslike MUSIC, ESPRIT, etc. [17,19].

The use of adaptive array antennas has so far been limited to military applicationsand in base stations for cellular systems [19–21]. One of the reasons for this has beenthe high cost, complexity, and power consumption of their digital implementation.However, low cost, low power, analog adaptive array designs have been proposedand prototyped, specifically targeting the market of ad-hoc networks [22]. Anotherreason is that smart antennas usually require sophisticated algorithms that useextra synchronization information, like training sequences, to automatically beam-form. Examples of some recent efforts to apply such techniques in ad-hoc networkscan be found in [8,23–26]. In these works, each transmitter-receiver pair is assumedto share a unique training sequence or some other information in order to be ableto recognize the “signal-of-interest” (SOI) and iteratively adapt the antenna pat-tern. However, wireless ad hoc networks are often characterized by strict resourceconstrains (e.g. sensor networks) and/or no single adminstration (e.g. peer-to-peerwireless). These characteristics may render the application of schemes requiringsuch training sequence assignment extremely challenging and costly. Instead, anovel protocol is necessary that efficiently takes advantage of the increased capa-bilities of adaptive array antennas, yet is simple enough to be implemented byresource-limited ad hoc wireless terminals.

To this end, we propose a protocol called ADAPT, which uses a cross-layer ap-proach to take advantage of fully adaptive array capabilities. Rather than perform-ing real-time beamforming on the Physical layer: (i) it uses information alreadyavailable to the MAC layer through a modified version of 802.11 to control theantenna radiation pattern and cancel interferers, and (ii) the antenna pattern isadapted on a per-packet basis (instead of real-time). We shall use the term semi-static beamforming to refer to this type of beamforming, hereafter. We have usedthe ns-2 simulator [27] to compare our protocol to typical omni-directional anddirectional settings. We show that our protocol achieves superior performance, interms of network throughput and end-to-end delay, in most scenarios considered.

In the next section, we discuss some previous work related to ours. In Section 3 wepresent the antenna model, and describe the details of ADAPT. Then, in Section 4we analyze the effect that mobility and a phenomenon we call node deafness haveon the performance of ADAPT. Simulation results are given in Section 5, and weconclude this chapter with a discussion and some future directions in Section 6.

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2 Related Work and Contributions

One of the earliest efforts to modify existing MAC protocols to operate with di-rectional antennas can be found in [13]. In [9], the authors propose the use ofmultiple directional antennas, which can be switched on and off, to restrict thetransmission to only specific sectors, as well as to track and locate intended com-munication partners by noting the angle of arrival (AOA) of the incoming signal.They modify 802.11 to send DATA and ACK packets directionally, so as to reduceinterference. However, RTS and CTS packets are sent in every direction. Multi-ple directional antennas are also assumed in [14]. There, however, it is assumedthat the global positioning system (GPS) is used to locate communication peers.Furthermore, the authors explore the use of directional RTS packets, in order toallow for a higher number of simultaneous transmissions. A further increase in thenumber of simultaneous transmissions is accomplished in [10,11], by sending bothRTS and CTS packets directionally. This way, only nodes that can subsequentlyinterfere with the upcoming transmission are silenced, and any other node is freeto engage in communication towards a non-interfering direction. The 360o azimuthplane is divided into sectors and virtual carrier sensing [28] is performed separatelyfor each sector. We shall be referring to this protocol as directional virtual carriersensing (DVCS) or Directional MAC (DMAC). Finally, a thorough evaluation ofthe impact of sending different 802.11 packets directionally or omni-directionallyis attempted in [3].

A common characteristic of all the aforementioned protocols is their being randomaccess protocols in nature. Departing from this model, the authors in [12] propose adistributed scheduling scheme to arbitrate media access between competing nodes,so as to eliminate collisions. Their scheme requires knowledge of transmission inten-tion information in a 2-hop neighborhood, and it assumes that nodes are capableof simultaneously receiving multiple transmissions through the use of multi-beamadaptive arrays (MBAAs).

There has also been a large amount of research in the field of smart antennasfor the use in cellular, satellite, and military systems. A good presentation of thefield and different adaptive algorithms can be found in [17,19]. In the context ofad-hoc networks adaptive antennas have been used in [8,23–26]. In [8,24,25] eachnode is assumed to be using a training sequence, in order to allow the intendedreceiver to distinguish the signal-of-interest from interfering ones, and to adapt itsantenna pattern to maximize the Signal to Interference and Noise Ratio (SINR).An extensive, simulation-based evaluation of how the number of antenna elements,excitation pattern and training sequence length affect the capacity of the networkcan be found in [8]. DOA estimation algorithms like MUSIC [19] are used in [23], inconjunction with the emission of a tone by all nodes, in order to identify all potentialinterferers in the neighborhood of a node and beamform accordingly. Finally, thepromising technology of analog smart antennas for wireless terminals is described

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in [22], with appropriate MAC and routing protocols for the use of analog smartantennas in ad-hoc networks proposed in [26]. Nevertheless, all these proposalseither require the use of reference signals and thus would not scale well in large ad-hoc networks, or involve sophisticated real-time beamforming algorithms that maynot be applicable for small, energy-constrained terminals. Simpler protocols, likeADAPT, are necessary that utilize higher layer information to eliminate the needfor sophisticated physical layer beamforming algorithms. The work closest to oursis probably that of [29]. However, there only downlink semi-static beamforming isused. Furthermore, all nodes are assumed to be static and the effects of mobilityand deafness are ignored.

Summarizing, the contributions of this paper are the following: (i) a cross-layeralgorithm to adaptively perform interference prevention/cancelation using smartantennas, which is significantly more lightweight than commonly used physicallayer algorithms, (ii) the use of this algorithm in “listening” (uplink) mode, toeliminate the exposed terminal problem, and (iii) an analytical framework to eval-uate the effect of mobility and non-heard neighboring communication informationon the performance of such non-real-time beamforming schemes.

As a final note, a lot of attention has been given lately to MIMO systems due totheir potential to increase channel capacity by increasing the number of antenna el-ements [30,31]. These systems also use multiple antenna elements for transmissionand reception. However, a different data stream is transmitted over each antennaelement (“spatial multiplexing”). In the case of “beamforming”, the same datastream is transmitted (received) on each antenna element, but with possibly dif-ferent phase and amplitude. Although ad-hoc networks could possibly benefit fromspatial multiplexing as well, such a study is beyond the scope of this paper and isdeferred for future work.

3 ADAPT: MAC Layer Guided Interference Suppression

Before we proceed with the description of our antenna model, there are someimportant practical considerations that the designer of protocols for directionalantennas needs to be aware of. The power savings and interference suppressionof a directional antenna depend on how narrow the primary beam (lobe) of theantenna pattern is and also how suppressed the secondary lobes are compared tothe primary one [32]. Furthermore, the antenna efficiency, or how much of the powerfed into the antenna by the power amplifier is effectively transformed into radiatedpower, is another important issue. This depends: (i) on how well “matched” theantenna is to the transceiver, and (ii) on the physical size of the antenna comparedto the operating wavelength. It is a well-known fact that the antenna size must bein the same order of magnitude as the operating wavelength for efficiency close to100%. Further, if more than one antenna elements are used to increase gain, those

5

elements must be placed apart at distances of the same order of magnitude withthe wavelength. This creates some important limitations on the actual antennagain that one could achieve for a wireless terminal. The size of the terminal is themajor restricting factor. Depending on the size of the terminal (i.e. sensor, PDA,laptop, vehicle, etc.), one may not be able to use more than 3-4 elements for thefrequency bands currently used for ad-hoc networks (e.g. 2.4GHz). This might beless of a problem for vehicles or higher frequency bands foreseen for future use.

3.1 Antenna model

Our adaptive antenna model consists of a linear array of K + 1 omni-directionalelements, each of which is independently weighted at the output. Specifically, letus denote with si(t) the complex signal incident on element i ∈ {1, K + 1}, and wi

the complex weight of element i ∈ {1, K + 1}. Then, the combined signal z(t) atthe output (or input) of the antenna is given by

z(t) =∑

i

wi(t)si(t). (1)

There exist a large number of algorithms and techniques for adapting the antennaweights wi to achieve the desired radiation pattern, some of which can automati-cally respond to an unknown interference environment, in real time, with almostno information about the desired signal. However, the complexity of some of thealgorithms, and the digital hardware required, may render their use for resource-constrained wireless terminals prohibitive. It is out of the scope of this paper todescribe all existing algorithms or discuss the appropriateness of each one of themin the context of ad-hoc networks. A good presentation can be found in [1] [17][23]. In this work, in order to keep the implementation complexity low and our as-sumptions realistic, we use a much simpler method to adjust the antenna weights,which we call semi-static beamforming. For simplicity, we will assume that theadaptive array pattern is similar to a simple phased-array pattern, with a mainbeam and side lobes, except that it can additionally form K independent “nulls”in the pattern, areas of small width θnull where the antenna gain Gnull is close to0.

3.2 Semi-static beamforming

In the core of the ADAPT protocol is a mechanism called smart virtual carriersensing (SVCS). SVCS is an extension of the basic virtual carrier sensing mech-anism of 802.11, and uses similar ideas to the virtual carries sensing mechanism

6

of [10]. It is implemented using two tables, where overheard information regard-ing the direction-of-arrival (DOA Table) and duration of nearby communications(SVCS Table) is maintained as follows:

DOA table: The DOA table consists of entries of the form {node ID, angle}. It isa cache of the most recently encountered nodes along with the direction at whichthe node is expected to be found. Each node operates in promiscuous mode [28].Whenever a packet is overheard indicating the presence of a new node, its ID andDOA are logged in the table 1

SVCS table: This table consists of K entries of the form {angle, duration}, oneentry for each potential interferer that the antenna is capable of suppressing. As in802.11, each packet transmitted carries a value indicating the remaining commu-nication duration [28]. When a packet overheard from angle φ carries a durationvalue of t an entry {φ, t} is added in the table. If all K table entries are alreadypopulated, then the one with the shortest time to completion is evicted, and re-placed by the one just overheard. This is intuitively efficient, as there is a higherprobability to interfere with a longer communication. An SVCS table entry impliesthat, until time t:

(1) interference from direction φ may be experienced;(2) radiated energy from the node to which the table belongs, towards that direc-

tion, may harm the ongoing communication.

Semi-static beamforming uses the DOA and SVCS tables to adjust the node’stransmitting or receiving antenna pattern, in order to create nulls and minimizeinterference towards or from any other ongoing communication in its vicinity.

Semi-static downlink beamforming: Let’s assume a node S wants to send apacket to another node D. Node S acquires node D’s DOA with respect to itself,say φD, from its DOA table. Furthermore, let there exist up to K active entries inthe SVCS table of node S, whose DOAs are equal to φi, i ∈ [1, K]. “Active” tableentries are those whose predicted time of completion (t) indicates that they arestill ongoing. Finally, let Δx denote the distance between antenna elements [32],λ the signal’s center wavelength, and β be defined as β = 2π

λ. Then, downlink

beamforming consists of solving the following (K+1)×(K+1) system of equations,on a per packet basis, for the K + 1 element weights wn:

1 The issue of how DOA information is obtained is out of the scope of this work. In thesimplest case, a node could include its coordinates (e.g. available through GPS) in everypacket it sends.

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K+1∑n=1

wne−jβ(n−1)Δx cos φD = 1,

K+1∑n=1

wne−jβ(n−1)Δx cos φi = 0,∀i ∈ [1, K]. (2)

The solution vector w = [w1, w2, · · · , wK+1] forms a transmission beam, whichminimizes the interference towards known ongoing communications, and is usedto send node D an RTS packet. Note that, like any other cache, the DOA tableis there only to improve performance. A miss may occur during the DOA tablelookup, in which case an omni-directional RTS is sent instead. Finally, after nodeD receives the RTS from S it can use the same method to form a receiving beamtowards S, that maximizes the SINR.

Semi-static uplink beamforming: Eq.(2) can be used for uplink beamformingonly if the direction of the desired signal is known. This would be the case, forexample, when a node has already received successfully a packet from its peer (e.g.RTS packet), from which it could decode the respective angle-of-arrival (e.g. usingposition info contained in this RTS). However, a variant of the “exposed terminalproblem” [33] may prevent such knowledge, as shown in the following example.

Let’s assume some node A needs to transmit a packet to another node B, who isoverhearing a number of ongoing transmissions in its vicinity. If B was aware of A’sDOA, and A’s intention of sending B a packet, it could adapt its antenna weightsto create a semi-static receiving beam aimed at A, and cancel interference fromother nearby nodes. However, B cannot ever find out, because RTS packets fromA get garbled at B by interfering signals. Semi-static uplink beamforming tries toresolve this problem.

Once more, there exist up to K interfering signals whose DOA φi is known. How-ever, the node is in listening mode and there does not exist a signal-of-interest(SOI) at that moment. In this case, the node wishes to maximize the chance ofintercepting a future incoming signal from any direction other than the interferingones. Hence, it is desirable that nulls are formed towards all φi, while the receiv-ing gain remains close to 1 for all remaining directions. This can be achieved bysolving the following constrained optimization problem. Let superscript H denotethe complex conjugate transpose of a vector or matrix, w the vector of antennaweights, and RN the array correlation matrix for AWGN (Additive White GaussianNoise) [17]. Then, the optimal weight vector w is found by:

maximizew

wHRNw,

subject to∑K+1

n=1 wne−jβ(n−1)Δx cos φi = 0, ∀i.(3)

It is easy to calculate the optimal weights when the noise is AWGN. RN is a just a

8

Idle

CheckAOA table

packet to betransmitted to D

sendomni RTS

i) downlink beamformingusing SVCS tableii) send RTS

D not found D found

Perform SmartVirtual Carrier

Sensing

D lies inside a null

Backoff

No commtowards D

Update weightsand send DATA

received CTS received CTS

sendomni DATA

received ACKreceived ACK

TO countertimeout

< shortretry count

= shortretry count

TO counter

< longretry count

= longretry count

timeout

TO countertimeout

< TOD

= TOD

TO counter

timeout

= TOD

< TOD

Update SVCSand DOA

tables

Idle

packet received

destined forthis node

No

Yes

i) updateweightsii) receive packet

send back appropriatepacket (e.g. CTS, ACK)using the new weights

updateweights forreceivingpattern

uplink beamforming toavoid exposed terminal

Packet Transmission Packet Reception

Idle

CheckAOA table

packet to betransmitted to D

sendomni RTS

i) downlink beamformingusing SVCS tableii) send RTS

D not found D found

Perform SmartVirtual Carrier

Sensing

D lies inside a null

Backoff

No commtowards D

Update weightsand send DATA

received CTS received CTS

sendomni DATA

received ACKreceived ACK

TO countertimeout

< shortretry count

= shortretry count

TO counter

< longretry count

= longretry count

timeout

TO countertimeout

< TOD

= TOD

TO counter

timeout

= TOD

< TOD

Update SVCSand DOA

tables

Idle

packet received

destined forthis node

No

Yes

i) updateweightsii) receive packet

send back appropriatepacket (e.g. CTS, ACK)using the new weights

updateweights forreceivingpattern

uplink beamforming toavoid exposed terminal

Packet Transmission Packet Reception

Fig. 1. ADAPT flowchart for packet transmission and downlink beamforming (left), andpacket reception and uplink beamforming(right)

diagonal matrix with all entries equal to the average noise power received by eachomnidirectional element, and can easily be reversed [17,19].

In Figure 1 we depict a flowchart diagram of ADAPT for semi-static downlink anduplink beamforming. In order to clarify the functionality related to the adaptivearray antenna, we do not depict the detailed FSM of the protocol, but rather theparts that are changed from 802.11. Most other protocol mechanisms (e.g. backoff,deferring, etc,) remain more or less unchanged from the basic 802.11 implementa-tion. It is an advantage of ADAPT that it retains the general semantics of 802.11and requires only a few changes to be implemented.

4 The Effect of Mobility and Node “Deafness”

As we mentioned earlier, semi-static beamforming is performed on a per packetbasis. This implies that, while a single packet is being transmitted or received,no changes will occur in the antenna pattern. Semi-static beamforming sufficesfor situations where node mobility is low and traffic loads not high. However, theeffect of high mobility and a phenomenon we call “node deafness” can reducethe efficiency of semi-static beamforming. In this section, we will theoreticallyanalyze each of these two phenomena separately, in order to quantify their effecton performance.

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4.1 The effect of mobility

The DOA (direction-of-arrival) info used in semi-static beamforming to calculatethe optimal antenna weights are taken from the DOA and SVCS tables, describedin the previous section. Therefore, they me be outdated, as they correspond to therespective angles at some time in the past. If node mobility is low, then these valuescan serve as good estimates. However, consider the following scenario that providessome intuition of what can go wrong:

• time t1: A node Y at angle φY gets logged in the SVCS table of another nodeX, as currently receiving some packet until time T .

• time t2 (t2 < T ): node X starts transmitting a packet of its own. In order toavoid causing any interference at node Y , X creates a null in its antenna patterntowards φY .

• time t3 (t2 < t3 < T ): If node Y moves fast enough with respect to node X,then Y might move outside the sector corresponding to the null formed by nodeX towards Y , before it has finished receiving its packet (i.e. before T ). In thatcase, Y ’s reception will be ruined by X’s transmission.

Consequently, if nodes move with high speeds, many DOA and SVCS table entrieswill go stale quickly, before they can be of use in the beamforming process. It isimportant to note that the most challenging problem here is not that of trackingthe desired signal, as would be for example in the case of directional antennas.Rather, it is more important to make sure that all interfering signals are suppressedthroughout the duration of a packet reception. This is due to the fact that the angleof a null is much more narrow than that of a main beam. Theorem 4 calculates theprobability that a packet transmission will succeed despite other communicatingnodes in its neighborhood being mobile. This probability is as a function of nodes’speed, and average number of concurrently communicating nodes inside a node’sneighborhood.

Throughout our analysis we make the following assumptions:

• Assumption 1 : All nodes are moving independently according to the randomdirection model [34], with zero stopping time 2 . Specifically, a node chooses adirection uniformly distributed in [0, 2π), and moves towards that direction foran exponentially distributed amount of time, with a speed uniformly distributedin [vL, vH ] (reflection or a toroidal structural could be used at the boundaries).After a move has finished it repeats the above process.

2 In reality, this is a quite pessimistic assumption, as real nodes tend to pause for longperiods of time [35]. It is easy to see that such pauses have a beneficial effect for per-packet beamforming algorithms. Thus, our analysis actually gives a lower bound for thetransmission success probability.

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• Assumption 2 : Arrivals of packets to be transmitted at any node are Poissondistributed with parameter λ. Packet arrivals at different nodes are independent.

• Assumption 3 : The duration of any transmission is exponentially distributedwith rate μ and is independent from any other transmission.

• Assumption 4 : A node does not change its direction or speed of movementthroughout the duration of a single transmission.

• Assumption 5 : Any point-to-point packet exchange between two nodes A andB consists of the following packet transmissions: RTS(A → B), CTS(B → A),DATA(A → B), ACK(B → A). Each of these packets carries in its header avalue indicating the estimated time until completion of the full packet exchange(i.e. end of ACK’s transmission). Furthermore, a packet exchange is consideredsuccessful if and only if all four packets are received correctly. During these fourtransmissions (phases), both nodes A and B take turns being transmitters andreceivers. Therefore, we consider both nodes involved in a packet exchange simplyas active for the full duration of the exchange. An active node is considered asboth a source and subject of interference at any time until completion of thepacket exchange.

Although the assumptions of Poisson arrivals (Assumption 2) and exponentialtransmissions times (Assumption 3) may not hold in most real life situations, theyallow us to provide some analytical insight into how mobility affects the perfor-mance of our protocol.

The following definition expresses the velocity of an interferer in a coordinatessystem relative to the node of interest.

Definition 1 Let two nodes A and B, currently at positions→xA and

→xB, have

velocities→vA and

→v B, respectively. Further, let 〈→a,

→b 〉 denote the inner product of

two vectors→a and

→b . We define the unit vectors

→u⊥AB and

→u‖AB, such that (see also

Figure 2)

→u⊥AB : 〈→u⊥

AB,→xB − →

xA〉 = 0;→u‖AB : 〈→u‖

AB,→xB − →

xA〉 = ‖→xB − →xA‖.

Lemma 2 Let two nodes A and B move with speeds→v A and

→vB, respectively, and

let node A be pointing its main beam or null towards B. Let further v⊥AB denote

the rate by which node B drifts away from the sector corresponding to this beam

or null (i.e. v⊥AB = |〈→vB − →

vA,→u⊥AB〉|); Finally, let v

‖AB denote the rate by which

the distance between A and B changes along→u‖AB (i.e. v

‖AB = |〈→vB − →

v A,→u‖AB〉|);

Then, it holds that

E[v‖AB] = E[v⊥

AB] =vL + vH

π+

4(2v3H + 4v2

HvL + vHv2L + 3v3

L)

15π2. (4)

11

PROOF. See Appendix.

r

xA

xB

vA

vB

A

B

θ

⊥ABu

r

||ABu

r

r

xA

xB

vA

vB

A

B

θ

⊥ABu

r

||ABu

r

Fig. 2. Coordinates system defined by the relative movement of 2 nodes A and B.

Lemma 3 Let node A point a sector S of angle θ towards a node B at distance rfrom A (starting at time 0). S may correspond to A’s main beam or a null formedto that direction. Then, the average time Tc(r) it takes B to leave sector S, is givenby:

Tc(r) =1

2

r sin(0.5θ)

E[v⊥AB](1 − sin(0.5θ))

+1

2

r sin(0.5θ)

E[v⊥AB](1 + sin(0.5θ))

. (5)

PROOF. See Appendix.

Theorem 4 Let a node A initiate a packet transmission at time t, at which timethe are M ≤ K active entries in A’s SVCS table. Further, let θnull denote the angleof a null created by A’s antenna, R denote the maximum transmissions range, andlet α = μ sin(0.5θnull)

E[v⊥AB ]. Then the probability P m

succ that A’s transmission will not collide

due to mobility with any one of the M concurrent transmissions is lower boundedby

P msucc ≥

[1 −

(2e−αR(−1 + eαR − αR)

α2R

)]M

. (6)

PROOF. From the definition of ADAPT, A will form a null towards each of theM nodes corresponding to the M active entries in its SVCS table. Let AM denotethe set of all M active nodes and let all nodes j ∈ AM lie within range R from A.Then, the probability of a collision Pcoll = 1 − P m

succ is defined as the probabilityP (∃j ∈ AM : j leaves null sector before its transmission is finished ∧ A is stilltransmitting when this occurs). Now, let us define the following quantities:

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• rj = ‖→xj − →xA‖, j ∈ AM ;

• Xj: residual transmission time, j ∈ AM ∪ {A};• AM(t) = {j ∈ AM : Tc(rj) < t}.

Then, the probability that A will not collide with any of the M concurrent trans-missions is equal to

P msucc =

∫XA

P (Xi ≤ Tc(ri), ∀i ∈ AM(t))P (XA = t)dt.

However, all M active nodes move independently of each other and independentlyof A, and their average speed relative to node A is given by {E[v⊥

AB], E[v‖AB]}

(Lemma 2). Furthermore, all rj are also I.I.D. and distributed according to

P (rj = r) =

⎧⎪⎨⎪⎩

2rR2 , r ∈ (0, R]

0 , otherwise.

Finally, according to assumption (c) the residual transmission times Xj are alsoI.I.D., and exponentially distributed with parameter μ. Consequently, we can ex-press P m

succ as a product of independent probabilities. Now let us denote as P (n|t)the probability P (AM(t) = n). Since an ideal null in the antenna pattern has a very

small angle null we can approximate Tc(r) from Eq.(5) with Tc(r) = r sin(0.5θnull)

E[v⊥AB].

Furthermore, Tc(r) ≤ Tc(R), ∀r ∈ (0, R]. Therefore, we can calculate P (n|t) as

P (n|t ≥ Tc(R)) ∼=⎧⎪⎨⎪⎩

1 , n = M

0 , n �= M

⎫⎪⎬⎪⎭

P (n|t ≥ Tc(R)) ∼= B(n; M, f(t)),

where B(n; M, f(t)) =

⎛⎜⎝M

n

⎞⎟⎠ (f(t))n(1 − f(t))M−n denotes the binomial distribu-

tion with parameters M, f(t) =(

E[v⊥AB ]t

R sin(0.5θ)

)2

. We can now expand the integral in

the derivation of P msucc into

P msucc =

∫ ∞

Tc(R)

(∫ R

0P (X ≤ Tc(ρ))P (r = ρ)dρ

)M

P (XA = t)dt

+∫ Tc(R)

0

∑n=0

MP (n|t)⎛⎝∫ R

√f(t)

0P (X ≤ Tc(ρ))P (r = ρ|t)dρ

⎞⎠

n

P (XA = t)dt.

13

Noth that, for t ≤ Tc(R), r may only take values less than R√

f(t) in the secondinner integral and hence

P (r = ρ|t) =

⎧⎪⎪⎨⎪⎪⎩

2ρ(

sin(0.5θ)

E[v⊥AB]t

)2

, r ∈[0, sin(0.5θ)

E[v⊥ABt]

]

0 , otherwise.

Let us denote the first integral in the P msucc equation as I1 and the second as I2.

After performing some calculations we get

I1 = e−μTc(R)

[1 −

(2e−αR(−1 + eαR − αR)

α2R2

)]M

, α =μ sin(0.5θ)

E[v⊥AB]

,

I2 =M∑

n=0

∫ Tc(R)

0B(n; N, f(t))

e−μt(−2 + μ2t2 + e−μt(2 + 2μt))n

μ2n−2t2ndt.

Adding I1 and I2 gives the exact P msucc. Now, I1 is the probability that XA is long

enough, such that any node j ∈ AM that drifts out of its null, will collide with A,provided that j is still communicating. On the other hand, I2 captures the caseswhere some node j does drift away from its null, but A has already finished itstransmission/reception. If we ignore this latter case, we can define the approximateP m

succ as

P msucc =

[∫ R

0P (X ≤ Tc(ρ))P (r = ρ)dρ

]M

.

It is evident by the above discussion and preceding equations that P msucc is a more

pessimistic value than P msucc, since it assumes that any active node drifting out of

its null, while still active, will certainly interfere with A. Thus,

P msucc ≥ P m

succ =

[1 −

(2e−αR(−1 + eαR − αR)

α2R

)]M

.�

Note that the above bound becomes tighter when the traffic load is high. Forexample, in the extreme case where most nodes send packets back-to-back all thetime, Eq.(6) approaches equality. Eq.(6) implies that the probability of a successfultransmission, in the presence of node mobility, is a decreasing function of averagespeed, and number of concurrently active nodes. This is a result that agrees withour intuition. We will see how mobility affects performance using simulations also,in Sec. 5.

Finally, Theorem 5 also calculates the probability that a node’s main beam cantrack a desired signal for the duration of a packet transmission.

Theorem 5 Let a node A point its antenna main beam of angle θw towards somenode B, and start sending a packet to B. Then, the probability that A will be able

14

to track B throughout the duration of A’s packet transmission is given by

Ptrack = 1 − exp

(−μ

2R sin(0.5θw)

3E[v⊥AB] cos2(0.5θw)

). (7)

PROOF. This is straightforward to show that

Ptrack = 1 −∫ R

0

P (XA > Tc(r)|I = 1) + P (XA > Tc(r)|I = −1)

2dr,

where I is again the same indicator function as in the proof of Lemma 3. UsingEq.(5) and the fact that XA is exponentially distributed, we get Eq.(7) after somecalculations. �

4.2 The effect of node deafness

Node deafness occurs when an ongoing transmission in the vicinity of some nodedoes not get recorded in the SCVS table. Since a node performs beamforming onlybased on information recorded in the SVCS table it may potentially interfere witha transmission not logged there, if it decides to engage itself in communication.An example of how deafness can occur can be seen in the following sequence ofevents. We use the notation pkt(T ) to denote a packet of type pkt containing aremaining duration of time T in its header:

• time t1: node B overhears RTS(T1) from A; B forms a null towards A, to lastuntil t1 + T1.

• time t2 ∈ (t1, t1 + T1): B sends CTS(T2) with t2 + T2 > t1 + T1. A is busytransmitting and cannot receive B’s CTS.

• time t1 + T1: B removes the null formed towards A.• time t3 ∈ (t1 + T1, t2 + T2): A transmits a new packet; A is not aware of B’s

ongoing communication and does not form a null in its downlink beam towardsB; B’s reception gets garbled due to A’s transmission.

The following theorem measures the effect of deafness on the probability of asuccessful transmission, as a function of the average amount of activity in thesurrounding area of a node.

Theorem 6 Let a node A initiate a packet transmission at time t, at which timethere are M ≤ K active entries in A’s SVCS table. Then, the probability P d

succ

that A’s transmission will not collide, due to node deafness, with any one of the Mconcurrent transmissions, is at least

P dsucc ≥ 1 − 1

M + 1

M∑j=1

(jλ

μ + jλ

). (8)

15

PROOF. According to the specification of ADAPT, whether A is going to sendor receive a packet, it is going to form a null towards each of the M nodes corre-sponding to the M active entries in its SVCS table. Let AM denote again the set ofall such nodes. The probability of deafness is defined here as the probability thata forthcoming communication within a node’s vicinity does not get recorded in itsSVCS table (otherwise it would be able to create a null and avoid any interferencetowards or from the recorded direction). Let us denote this probability as Pdeaf .Furthermore, let us define the following quantities:

• Xj: residual transmission time, j ∈ AM ∪ {A};• Tj: time until next packet arrival, j ∈ AM ∪ {A};• AM = {j ∈ AM : Xj < XA}.

Now, if there are M active entries in A’s SVCS table at the time where A initiatesa packet exchange itself, Pdeaf is equal to

Pdeaf =K∑

j=1

P (‖AM‖ = j|M)P (∃i ∈ AM : Ti+1 ≤ XA − Xi|j).

In other words, it is the probability that j of the nodes in AM finish before A, andat least one of these j nodes starts again before A finishes, summed over all possiblej. In that case, the new activity of the node in question will not get recorded inA’s SVCS table, since A is busy. Furthermore, since any node in AM is busy whenA starts transmitting and does not record A’s activity in its SVCS table either, itwill not form a null towards A when (and if) it restarts, and therefore may interferewith it. Now, since all Xj are I.I.D and exponentially distributed with parameterμ it follows that

P (‖AM‖ = j|M) =

⎛⎜⎝M

j

⎞⎟⎠

⎛⎜⎝M + 1

j

⎞⎟⎠

1

M − j + 1=

1

M + 1.

This is reasonable since the above probability equals the probability that an expo-

nential clock competing with another M exact same clocks, will be the j + 1th toexpire. This probability is equal 1

M+1for any j. Furthermore, packet interarrival

times Tj are also I.I.D, exponentially distributed and independent from packetservice times. Therefore,

P (∃i ∈ AM : Ti+1 ≤ XA − Xi|j) = 1 − μ

μ + jλ=

jλ

μ + jλ.

16

Lower bound for P{success} under deafness

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3 4 5 6 7 8 9 10

M = number of concurrent transmissions

pro

bab

ility

of

succ

ess

Series2

Series3

λ=μ/4

λ=μ/10

Lower bound for P{success} under deafness

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3 4 5 6 7 8 9 10

M = number of concurrent transmissions

pro

bab

ility

of

succ

ess

Series2

Series3

λ=μ/4

λ=μ/10

Fig. 3. Probability of success, due to node deafness, as a function of the number ofconcurrent transmissions M .

Substituting the previous two equations in the equation for Pdeaf we get that

Pdeaf =1

M + 1

K∑j+1

jλ

μ + jλ.

This is the probability of deafness, as defined earlier. However, the fact that anongoing or incipient transmission does not get recorded in the node’s SVCS table,does not necessarily mean that there will be a collision. For example, even if somej ∈ AM restarts before A finishes, if A is still transmitting the same packet (e.g.not waiting say for an ACK packet), it won’t remove its null towards j, before thepacket transmission gets completed (remember that semi-static beamforming isperformed on a packet-per-packet basis). Hence if j finishes its new communication(e.g. transmission or reception of a short RTS or CTS packet) before A’s DATApacket is sent they still will not collide. It is evident therefore that Pcoll ≤ Pdeaf .Thus,

P dsucc = 1 − Pcoll ≥ 1 − 1

M + 1

M∑j=1

(jλ

μ + jλ

).�

In Figure 3 we depict how the lower bound of Eq.(8) decreases with increasing M ,for two different arrival to service rate ratios. We note that those are relativelyhigh rates for a shared media.

Real-time Uplink Beamforming: As Eq.(6),(8) and Fig. 3 imply, semi-staticbeamforming performs sufficiently well for a large range of mobility and trafficload values. One could possibly complement the basic algorithm with a “real-time”module, as well, to be able to cope with situations of high mobility or traffic, as wedescribe in the following. However, it is important to note that this scheme doesnot require training sequences or sophisticated blind beamforming algorithms. Whatis more, the real-time module needs only be used in situations of increased mobility

17

and/or high traffic loads. Thus, the overhead related to the physical layer algorithmwill only be incurred for a small fraction of the packets 3 .

Let’s assume that only the direction φS of the desired signal is known, while theinterference environment is unknown. (Note that, since the width of the antennamain beam is much larger than that of a null, we can safely use the φS entry fromthe DOA table in most cases.) Let further s(t) = [s1(t), s2(t), · · · , sK+1(t)] denotethe adaptive array received signal vector, R = E[ssH ] denote the array correlationmatrix, and svs = [1, ejβΔx cos φS , · · · , ejKβΔx cos φS ] denote the array steering vectorassociated with the direction of the desired signal. Then the beamforming algorithmsolves the following constrained optimization problem.

minimizew

wHRw,

subject to wHsvs = 1.

This algorithm seeks to minimize the mean output power of the array, while main-taining unity response towards φs. This is equivalent to maximizing the outputSINR. We are not concerned here with the specific algorithm to be used to es-timate matrix R online in order to calculate the optimal weights. The interestedreader in the available alternatives is referred to [17,19].

5 Simulation Results

We have used the ns-2 network simulator with the CMU wireless extensions forall our simulations. Our goal was to thoroughly evaluate the performance of ourADAPT protocol, running on an ad-hoc network of nodes with adaptive arrayantennas, by extensively comparing it to typical directional and omni-directionalconfigurations. In order to do so we have developed the following modules in ns-2:

• Directional antenna models: We have implemented models for a flat-toppedantenna [10], and a circular array of dipoles. We shall only present here resultsfor the former, but we have observed a similar behavior for the latter, also.

• Adaptive array antenna model: We have implemented an adaptive array an-tenna model with the capability of independently steering a main beam and Knulls to any direction in the 360o plane. We assume, throughout, that the num-ber of elements K in the adaptive array is 10, and we calculate the parametersfor the flat-topped model based on a linear end-fire array of 10 elements [32] 4 .

3 The MAC layer could, for example, turn this component on and off, based on theexperienced success of packet transmissions.4 We have also performed simulation runs for different numbers of antenna elements, and,thus, different resulting beamwidths and antenna gains. The basic conclusions drawn forthe configuration chosen was found to hold for other parameters, as well.

18

0

7

2

9

6

1

8

4

3

5

Fig. 4. Scenario A — 10 static nodes in an area of 400 × 400m and 5 CBR connectionsactive in parallel.

Each null is assumed to be of 2o width, and have a gain of −50dB [25]. We havealso implemented the real time beamforming algorithm described in Section 4.2.As mentioned earlier, it is not our focus here to delve into the details of differ-ent real-time adaptive algorithms. Therefore, we have not dealt with issues likeconvergence speed of the algorithm, and other physical layer details. Instead, wehave assumed that the antenna pattern adapts to the interference environmentautomatically (in one cycle), using only knowledge of the SOI (signal-of-interest)direction.

• Directional virtual carrier sensing (DVCS): We have implemented the ba-sic directional virtual carrier sensing mechanism, to our best understanding, asdescribed in [10], to be used in combination with flat-topped directional anten-nas. The reason for this choice is the simplicity and good performance of thisprotocol, as well as its being a random access MAC protocol and therefore moreappropriate to be compared to ADAPT.

• ADAPT protocol: We have implemented all the mechanisms of ADAPT, asdescribed in Section 3.

5.1 Single-hop network of static nodes (Scenario A)

We start our performance evaluation with a simple, proof-of-concept scenario,namely a single-hop network of 10 static nodes. Our motivation for this is toisolate, as much as possible, the effect of the routing and transport protocols, aswell as that of mobility, from the effect of the specific MAC protocol and antennatype chosen. We will look into multi-hop scenarios in the next sections. The 10nodes are distributed in an area of 400 × 400m. There exist 5 active connections,namely 3 → 5, 1 → 6, 4 → 8, 9 → 2, 0 → 7, as depicted in Figure 4. Although notall nodes are within range (transmission range is equal to 250m), they all interferewith each other and compete for the shared channel.

Each connection is sending CBR traffic, with packet sizes of 512 Bytes. We definenetwork throughput as the total amount of traffic that successfully gets through by

19

Single Hop Network (10 nodes - 5 connections)

0

5000

10000

15000

20000

25000

0.02 0.016 0.012 0.009 0.007 0.005

packet interrarival time (sec) ne

twor

k th

roug

hput omni

directional

ADAPT (MAC)

ADAPT (full)

Fig. 5. Network throughput comparison for Scenario A.

all connections in the network, throughout the simulation run. In some cases, we usethe terms total or aggregate throughput interchangeably with network throughput.In Figure 5, we compare the network throughput for traffic rates of 50 packets/sec(i.e. average packet inter-arrival time of 0.02 seconds) up to 200 packets/sec (i.e.average packet inter-arrival time of 0.005 seconds), for the following four configu-rations:

• omni = {omni-directional, 802.11},• directional = {directional, DVCS},• ADAPT(MAC) = {adaptive array, semi-static downlink beamforming},• ADAPT(full) = {adaptive array, semi-static uplink/downlink beamforming +

real-time algorithm}.

It can be seen in Figure 5 that the network is saturated in the omni-directionalcase for the range of traffic loads considered. All connections compete with eachother, resulting in a channel utilization that is merely as high as that of a singleone. The use of directional antennas improves the situation, achieving a maximumtotal throughput of around 14000 packets, which is about 2.2 times that of a singleconnection. However, we have noticed in our simulations that this comes at thecost of increased average packet delay and a large number of dropped packets.It is evident that the adaptive array configuration achieves the best performance.Even using only downlink semistatic beamforming, performance is considerablybetter, reaching a total throughput of around 3 times that of a single connection.Furthermore, using the full ADAPT stack, a high of 21000 packets is reached (forinput rate equal to 200 packets/sec).

Some insight into the performance of the four configurations is provided in Figure 6.There, it is depicted how the available bandwidth gets distributed among the fivecompeting connections by each of the four configurations. It is clear from this figure,that the bandwidth allocation achieved is not balanced among all connections.Specifically, due to the exposed terminal problem, some connections get almostshut down, being unable to capture incoming transmission requests due to excessiveinterference, and receive much less than their fair share of the bandwidth.

20

OM N ID IR EC T ION A L

0

500

1000

1500

2000

2500

3000

3500

0.02 0.016 0.012 0.008

D IR EC T ION A L

0

1000

2000

3000

4000

5000

6000

7000

0.02 0.016 0.012 0.008

A D A PT ( f ul l)

0

1000

2000

3000

4000

5000

6000

7000

0.02 0.016 0.012 0.008

3->5 1->6 4->8 9->2 0->7

A D A PT ( M A C o nly)

0

1000

2000

3000

4000

5000

6000

7000

0.02 0.016 0.012 0.008

Fig. 6. Distribution of available bandwidth among the 5 competing connections, for the4 configurations considered.

The omni-directional configuration is in the worse standing achieving a fair allo-cation for only 3 of the 5 connections, and that only for lower traffic loads. Onthe other hand, DVCS and ADAPT using only downlink semistatic beamforming,manage to distribute the bandwidth in a fair manner among 4 out of 5 of the con-nections, for lower traffic loads. Additionally, the first ADAPT configuration man-ages to allocate a considerable, albeit not equal, share to the fifth connection, aswell, for this same range of traffic loads. As is expected, the ADAPT configurationthat implements semi-static uplink beamforming, alleviates the exposed terminalproblem, and achieves the most fair allocation of bandwidth for the whole rangeof traffic loads considered. We should note here that the real-time part of ADAPT,although implemented in the latter configuration, does not play an important rolein the achieved performance (only in the few occurrences of node deafness), due toabsence of any mobility.

5.2 Multi-hop network of static nodes (Scenario B)

It has been noted repeatedly that, in the wireless context, there exists a stronginteraction between protocols operating in different layers [36]. Therefore, in orderto correctly account for the effect of the full protocol stack, we consider a multi-hop network, where each connection spans a number of nodes. Specifically, weconsider a scenario where 30 nodes are uniformly distributed on a plane of size1000 × 1000m. All nodes are still assumed to be static, in order to isolate theeffects of mobility. We consider the performance of the network, with an increasingnumber of simultaneous TCP connections with randomly chosen endpoints. Allnodes run the Ad-hoc On-Demand Distance Vector (AODV) routing protocol [37].

21

Multi Hop Network - No Mobility

10000

15000

20000

25000

30000

6 10 14 18

number of connectionsth

rou

ghpu

t (pk

ts)

omni directional fully adaptive

Fig. 7. Network throughput comparison for Scenario B.

Table 1End-to-End Delay for Scenario B.

Antenna Type omni directional adaptive

End-to-end Delay (sec) 0.8630 0.6643 0.5222

A comparison of the aggregate throughput achieved by all connections, for theomni-directional, directional, and adaptive array configurations, is presented inFigure 7 as a function of the number of connections.

As can be seen from the plots, ADAPT achieves the highest network throughput inall cases considered. The improvement becomes more evident with a higher numberof connections. Nevertheless, the attained improvement for both the directional andadaptive array configurations is not close to the actual SINR for a single reception(i.e. physical layer gain due to the antenna itself). The reason for this is that, inthe multi-hop setting, a significant number of routing and other broadcast packetsare exchanged, which do not take advantage of the additional antenna capabilities.These are route discovery packets, for example, which are generally broadcasted.

Delay also plays an important role in the context of ad-hoc networks, especially forreal-time/multimedia applications. In order to evaluate the effect of each configu-ration on end-to-end delay, we run simulations of 10 CBR connections each of rate100 packets/sec. Connection endpoints are randomly chosen among all 30 nodes.Average end-to-end delay results for the 3 configurations are given in Table 1. Asis evident from this table, in the case of a multi-hop network of non-mobile nodesadaptive array antennas offer the best performance in terms of end-to-end delay,as well.

22

Multi Hop Network - Speed: 1.5m/s

10000

15000

20000

25000

30000

6 10 14 18number of connections

thro

ug

hp

ut

(pkt

s)

omni directional adaptive array

Multi Hop Network - Speed: 10 m/s

4000

8000

12000

16000

20000

24000

6 10 14 18number of connections

thro

ug

hp

ut

(pkt

s)

omni directional ADAPT1 ADAPT2

Fig. 8. Network throughput for Scenario C: low mobility (left) and high mobility (right).ADAPT1 only implements semi-static beamforming, while ADAPT2 implements real–time adaptive beamforming, as well.

5.3 Multi-hop network of mobile nodes (Scenarios C)

In this last section we consider two multi-hop scenarios, where all nodes are mobile.30 nodes are constantly moving according to the ns-2 random waypoint mobilitymodel on a 1000 × 1000m (pause time is zero). In the first scenario nodes movewith an average speed of 1.5 m/s. This is a relatively low speed, better modelingwalking speeds. In the second scenario nodes move with average speed of 10 m/s.This scenario could model vehicles moving with moderate speed (e.g. cars in citytraffic, or heavy military vehicles).

We run simulations for an increasing number of TCP connections, randomly as-signed between all nodes. The attainable network throughput, for the omni, direc-tional, and fully adaptive configurations is depicted in Figure 8. When mobilityis low, it is evident that the adaptive antenna configuration attains superior per-formance than the other two configurations. This becomes especially pronouncedwhen the network is heavily loaded. For the high mobility scenario results for bothADAPT implementing only semistatic downlink beamforming and the full ADAPTare shown, in order to demonstrate the detailed effect of mobility. As was notedin Section 4, increased mobility has a detrimental effect on the performance ofsemi-static beamforming. Nevertheless, ADAPT using only semi-static beamform-ing still provides better performance in the majority of cases. Furthermore, whenthe full ADAPT stack is employed the adaptive antenna configuration retains aclear performance advantage.

Finally, we have evaluated the 3 configurations in terms of average packet end-to-end delay for the low mobility scenario. We have used 10 CBR connections,each of rate 100 pkts/sec, randomly assigned among all nodes. Results are givenin Table 2. It is evident from that table, that the adaptive configuration providesa clear advantage over the other two in terms of end-to-end delay, as well.

23

Table 2End-to-End Delay for Scenario C (low mobility)

Antenna Type omni directional adaptive

End-to-end Delay (sec) 0.5761 0.4453 0.2878

6 Conclusion and Future Directions

In this paper, we have proposed a cross-layer approach to adapt the radiationpattern of a fully adaptive array antenna (smart antenna) using MAC layer in-formation. We have shown using theory and simulations that our protocol, calledADAPT, can exploit the capabilities of adaptive array antennas, and improve theperformance of a mobile ad-hoc network, as compared to the cases where omni-directional or directional (phased array or switched beam) antennas are used. Al-though the use of an advanced antenna technology is partly responsible for theegains, it is important to note that simply equipping each node with an adaptiveantenna is not merely sufficient to improve performance. Our protocol successfullydeals with the issues of antenna synchronization and estimation of the surroundinginterference environment, a process necessary to realize the gains. However, it doesso using only information already available or easily obtainable at the MAC layer,rather than with costly real-time beamforming algorithms.

Finally, in this work, we have assumed relatively simple propagation models (2-ray)in our performance evaluation. Real-life propagation phenomena like fading andshadowing can have an adverse effect on the successful packet reception (yet, theyapply also to interfering signals and could sometimes help increase spatial reuse).Multi-path propagation, for example, introduces an angular spread in the directionof arrival for both the signal-of-interest and interfering signals. This makes it con-siderably more difficult to pinpoint and cancel interferers using only higher-layer,”semi-static” information, as in ADAPT. In future work, we intend to look intohow ADAPT could be modified to cope with multi-path propagation phenomena.Additionally, we intend to explore how similar cross-layer approaches could benefitthe potential application of MIMO systems for ad hoc networks.

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A Appendix

PROOF. [Proof of Lemma 2] Let→v i, i ∈ {A, B}, be a random speed vector,

chosen according to the Random Direction mobility model, namely:

Prob(‖→v i‖ = v)=

⎧⎪⎨⎪⎩

(vH − vL)−1 , v ∈ [vL, vH ]

0 , otherwise

Prob(∠→v i = θ)=

⎧⎪⎨⎪⎩

π−1 , θ ∈ [0, π]

0 , otherwise

27

The probability density function for v⊥i = 〈→v i,

→u⊥AB〉 can be calculated in the fol-

lowing way:

P (v⊥i = x) =

∫ vH

vL

P (∠→v i = cos−1 x

v)P (‖→v i‖ = v)dv.

Using a little bit of algebra we get that:

P (v⊥i = x) =

⎧⎪⎨⎪⎩

π−1 , |x| ≤ vL

π−1(

vH−xvH−vL

), vL ≤ |x| ≤ vH .

We would now like to calculate the ensemble average of the quantity v⊥AB =

〈→vB −→v A,

→uAB〉 = ‖→v⊥

B −→v⊥A‖. According to Assumption 1 of Section 4 the speed

vectors of different nodes are independently and identically distributed (I.I.D).Consequently, so will be v⊥

A and v⊥B . Hence,

E[v⊥AB] =

∫ vH

−vH

∫ vH

−vH

|xA − xB|P (v⊥A = xA)P (v⊥

B = xB)dxAdxB ⇒

E[v⊥AB] = 2

∫ vH

0

∫ 0

−vH

(xA − xB)P (v⊥A = xA)P (v⊥

B = xB)dxAdxB

+2∫ vH

0

∫ vH

0|xA − xB |2P (v⊥

A = xA)2P (v⊥B = xB)dxAdxB.

Using a bit of calculus we derive Eq.(4).

PROOF. [Proof of Lemma 3] We have assumed that, throughout the durationof a packet exchange, if A is receiving, no real-time DOA estimation is performed.Similarly, if A is transmitting, it is unable to receive any feedback on B’s updated

position. Therefore, S will remain pointed along→u‖AB and B will be drifting away

from S along→u⊥AB . Let x(t) denote the remaining distance along direction

→u⊥AB,

before node B steps outside S and let x(0) = r sin(0.5θ). That is, at time 0 B lies,on average, in the middle of sector S and at distance r from A. Finally, let’s definethe random indicator function I(·) such that I = 1, if A and B are moving away

from each other along→u‖AB, and I = −1, if they move towards each other. Note

that P (I = 1) = 12, P (I = −1) = 1

2. Then, the evolution of x(t) is given by

x(t) = r sin(0.5θ) − E[v⊥AB]t + IE[v

‖AB] sin(0.5θ)t.

The second term represents the distance covered along→u⊥AB and the third term

represents the change in total distance to be covered, due to B’s relative movement

moving away or towards A, along→u‖AB . The time Tc until B crosses either side of

sector S is found by solving the above equation for x(Tc) = 0.

28