Transmission Capacity of Carrier Sensing Ad Hoc Networks with Multiple Antennas

Transcription

1 1 Transmsson Capacty of Carrer ensng Ad Hoc Networs wth ultple Antennas Andrew. Hunter, Radha Krshna Gant, and Jeffrey G. Andrews Abstract ultple antennas have become a common component of wreless networs, mprovng range, throughput, and spatal reuse, both at the ln and networ levels. At the same tme, carrer sensng s a wdely used method of mprovng spatal reuse n dstrbuted wreless networs, especally when there s lmted coordnaton among non-communcatng nodes. Whle the combnaton of carrer sensng and multple antennas has been consdered n the lterature, physcal layer spatal models and the attendant consequences have not been ncluded. The prmary reason for ths has been the dffculty of analyzng functonals of nteractng pont processes. Havng developed new methods of quantfyng physcal layer performance wth robust spatal networ models, we use these technques to address the followng questons: What multple antenna technques produce the best networ performance, and what s the performance gan? And, how should multple antennas nteract wth carrer sensng parameters? Overall, the analyss confrms the sgnfcant beneft of multple antennas n dstrbuted wreless networs. I. INTRODUCTION As the number of wrelessly connected devces grows, so wll the need for these devces to create flexble, decentralzed networs. At the same tme multple antennas and carrer sensng-based medum access are two common performance enhancng features of modern wreless systems, even n energy- and cost-constraned systems. Theoretcal studes of decentralzed networs wth carrer sensng have struggled wth the analytcal expressons for the nterference of correlated pont processes, dffcultes whch are only compounded by more complcated physcal layers. In ths paper we develop a model to ln ey carrer sensng parameters to envronmental and networ performance parameters n wreless ad hoc networs of nodes equpped wth adaptve mult-antenna rados. A few studes have approached the dffcult problem of medum access control AC analyss n random spatal feld of nterference ncludng [1], [2], [3], and [4], whle at the same tme [5], [6], and [7] have consdered multantenna IO rados, though they have focused manly on the Aloha case. As a pont of comparson, the study [5] developed a partcular coordnated protocol as a comparson pont aganst Aloha. The present wor develops a new model for the combnaton of a multple-antenna physcal layer wth a tunable CA model whch ncludes Aloha as a specal case. Ths allows the nteracton of IO and CA to be studed and provdes a framewor for determnng whch The authors are wth the Wreless Networng and Communcatons Group WNCG of the Electrcal and Computer Engneerng Department, The Unversty of Texas at Austn, Austn, TX, UA emal: hunter, rgant, jandrews@ece.utexas.edu. IO confguratons yeld the hghest gans for carrersensng networs as a whole, as well as enablng comparsons aganst Aloha. II. YTE ODEL For the underlyng physcal layer model, we consder a set of nodes, each havng smlar rados, randomly dstrbuted on R 2, communcatng n a slotted system wthout any central controller. The nodes are assumed to be dstrbuted as a spatal Posson pont process pror to the operaton of any medum access mechansm. A path loss exponent of α characterzes large scale fadng, whle small scale fadng s Raylegh between any par of antennas. The analyss wll focus on a typcal recever located at the orgn wth ts ntended transmtter a dstance R away. At the physcal layer, a transmsson s assumed to be successful f the INR s greater than some target threshold θ, where the INR s gven by the expresson: INR o R α j Φ t j d α j + 1, 1 where o s the fadng sgnal level at the detecton pont, and j and d j are the fadng nterference level and dstance to the jth nterferer, respectvely, and Φ t s the pont process of actvely nterferng nodes around a typcal recever. Gven that θ s met over a slot, the achevable rate wll be log θ. A. Performance etrcs Ultmately, the prncpal desgn goal s to maxmze networ sum throughput, or equvalently, area spectral effcency. However, recognzng that physcal layer mplementaton and user experence cannot abde degenerate cases, two types of qualty requrements are ntroduced. Frst, as mentoned above, a target INR θ s requred on any partcular attempted transmsson, whch results n an outage f not met. econd, the outage probablty ϵ can also be taen as a constrant on any gven ln or transmsson. Transmsson capacty s defned as the mean densty of successful transmssons that can occur wth gven qualty constrants.e., θ, and ϵ are satsfed. Gven the success probablty functon P λ t, the transmsson capacty s TC arg max λ t P λ t. 2 λ t Lastly, area spectral effcency s the transmsson capacty scaled by the data rate acheved n each of those transmssons: t log θ λ t P λ t, where t s the number of data

2 2 streams transmtted on each ln. For smplcty of presentaton, when multple antennas are used at the transmtter for spatal multplexng we assume two thngs: frst, each spatal mode wll treated as a separate transmsson, carryng a separate pacet and decoded separately. econd, the same outage constrant wll be appled across all modes of all lns ndependently. ore elaborate arrangements of modes and outages can be devsed and analyzed, but these necessarly lead to more cumbersome expressons best consdered outsde the space here. B. The IO Channel Each transmtter has N t and each recever has N r antennas to decode t mnn t, N r ndependent transmtted data streams. In Raylegh fadng, the channel between the recever of nterest and ts ntended transmtter s R α 2 H whch s an N t N r matrx of..d. complex Gaussan entres wth unt varance scaled by a power-law path loss factor. Ths channel can be decomposed nto spatal modes by means of ts VD when channel state nformaton CI s avalable at the transmtter, whle n the absence of transmtter CI, the recever can decode equal power and rate streams by varous technques ncludng zero-forcng ZF or maxmalrato-combnng [5]. The ZF beamformng soluton for a recever wth CI s the pseudo-nverse: H H H 1 H H, where H denotes the Hermtan transpose. The results here wll be restrcted to the case of openloop zero-forcng multplexng methods. A number of other technques wth and wthout CI at the transmtter follow very smlar expressons, but ths provdes a rch enough set of optons to demonstrate the relatonshps between CA and IO technques. C. oft Carrer ensng We consder a soft CA model whch taes nto account two effects: 1 Large-scale densty λ t of transmtters. 2 hot range nhbton. Large-scale densty : We use a atern model smlar to [1] for obtanng the fnal densty of transmtters. In ths model each node s assocated wth a set ˆN consstng of transmtters whch ndvdually cause an nterference greater than θ. A node s fnally selected to transmt f ts tmer a unform random varable n [0, 1] s the smallest. Gven an ntal densty λ of nodes attemptng to access the channel, after carrer sensng the resultng large-scale densty λ t s [1] λ t λ 1 exp N, 3 N where N s the average number of nodes n a typcal contenton set ˆN and equals N λ e θ α x dx, R 2 πγ1 + 2/αθ 2/α λ. 4 R Rx Tx oft offender Inhbted Fg. 1. Contenton around the typcal recever. Fadng and channel sensng errors lead to a soft CA model unle the correspondng hard care boundary mared here. hot range nhbton: We mae a further approxmaton that from the perspectve of a typcal recever, the set of nterferers s dstrbuted around t as a radally symmetrc nhomogeneous Posson process wth densty 1 e θ X α λ t, where λ t s the average densty of the set of currently transmttng nodes. e θ X α s the probablty that an exponentally dstrbuted power fade wll exceed a threshold θ at that dstance. The average densty s all that s necessary to model the nterference contrbuton from nodes at long dstances, whle at close ranges the presence of the communcatng node has the domnant effect on the dstrbuton of nterferers. Furthermore, the model treats the channel through whch carrer sensng decsons were made as ndependent of the Posson shot nose nterference process of data transmsson. In ths sense the carrer sensng s soft, beng nether an exact geographc ds, nor exactly representatve of mutually nterferng nodes durng data communcaton. Fnal model: From the perspectve of a typcal recever the nterferers form a non-homogeneous Posson pont process of densty 1 e θ X α λ t. Ths model has several advantages over the hard-core model: Frst, t brngs fadng nto play when modelng carrer sensng. econd, t allows for modelng mperfect carrer sensng. Thrd, t enables modelng of a behavor specfc to mult-antenna systems n whch networ nterference dffers between the control/carrer sensng modes and hgh data rate modes. Lastly, t provdes more tractable results that gve smpler functonal relatonshps between the carrer sensng parameters and other envronmental and system parameters. In addton, a model n whch the probablty of nhbton rses at least as fast as the power-law path loss functon results n the nterference power from the nearest nterferer havng fnte mean. Ths removes problems related to the behavor of the path loss model near the orgn a feature shared by the atern hard-core model. D. CA ensng wth ultple Antennas Another way a CA ad hoc networ can tae advantage of multple antennas s n the carrer sensng tself. When the carrer sensng mechansm s subject to fadng, more conservatve thresholds must be met to guarantee outage requrements. One approach whch s readly tractable n the current model s to use selecton or combnng dversty on

3 3 Fg. 2. Comparson of throughput and outage for CA and Aloha, for IO system. Densty s ntal densty,.e., before thnng va carrer sensng. Outage s counted among actve transmtters.e., those not thnned by CA. Parameters were R 1, α 4, θ 0dB. multple receve antennas for carrer sensng and to mpose a threshold on the selected or combned output the former approach beng approprate when CI over the control channel s dffcult to obtan. As an example, suppose a node wth c multple receve antennas performs carrer sensng on each antenna, but averages the results across antennas before applyng the threshold. In ths case N R2 2π 2 α θ α λ c 1 λ e θ c x α c 1 θ c x α dx! Γ 2 α α c Γ + 1 Note that n the lmt of a large number of antennas, the carrer sensng behavor approaches that of a hard-core process as the nfluence of small scale fadng on sensng decsons dsappears. Whle channel hardenng technques mae the carrer sensng mechansm more effcent, prmarly by reducng errors due to deep fades, the net beneft turns out to be small n partcular small compared to the beneft of CA over Aloha. Therefore, for the man analyss of multplexng systems, a IO carrer sensng channel s assumed. III. OPEN-LOOP PATIAL ULTIPLEXING WITH CA We now state the central result of the paper whch deals wth the probablty of success or equvalently the INR dstrbuton. Theorem 1: Gven a networ of nodes performng CA, open-loop spatal multplexng, and wth per-stream decodng success defned n 1, P θ N r 1 [ s! d ds L I Φ se 1 s ]sn t θ R α, where L IΦ s exp CN α t T s 2π α s/n t 2 α λt [CN α t T s] 1 1 BN t α, α ΓN t α U N t α, 1 α, θ s N t wth α 2/α for notatonal convenence and Bx, y ΓxΓy Γx+y beng the beta functon, and where Ua, b, x 1 Γa 0 e xt t a t b a+1 dt s Kummer s confluent hypergeometrc U-functon. Proof: ee the appendx, whch develops these expressons usng technques smlar to those of [1] and [6]. In addton the appendx derves an effcent method of evaluatng the dervatves both symbolcally and numercally. Agan we note that as s, Ua, b, s 0 and thus T 0 whch corresponds to the Aloha case. As s 0, ΓaUa, b, s Ba, b, whch mples that T CN α t as the threshold θ 0. In ths case the probablty of success approaches 1 among actve transmtters, at the expense of λ t vanshng. A. Optmal CA threshold pecalzng Theorem 1 to the case N r N t 1, we obtan the success probablty n a IO system as P θ exp θ 2/α R 2 λ t [Cα T ] e θrα /, where Cα πγ1 + 2/αΓ1 2/α s the famlar Aloha constant and T 2π α eθ θ R α Γ 1 2α, θ θ R α 6

4 4 the two-parameter gamma functon beng the upper ncomplete gamma functon. Note that T s a factor related to the carrer sensng threshold, but also that λ t s dependent on ths threshold. Ths expresson holds for any outage level or transmtter densty. As the nhbton threshold rses so that very few nodes are nhbted, λ t λ and T 0 recoverng the Aloha result [1]. Note also that wth a fxed threshold 1 lm λ t, πγ1 + 2/αθ 2/α, 7 λ whch ndcates that regardless of the number of users requestng access, the carrer sensng process lmts the set of actve transmtters to a lmtng densty, as t should. A natural queston then s to fnd the optmal carrer sensng threshold. The optmum threshold s then the unque maxmum to: θ arg max λ t exp θ θ 2/α R 2 λ t [Cα T ]. 8 Ths expresson s log-concave n θ and hence s relatvely easy to optmze numercally. At ths pont there are a few observatons that can be made to assst further analyss: Frst, as λ the optmum threshold approaches a lmt whch s the unque soluton to 8 at λ t,. econd, as λ becomes small, for a fxed threshold, very lttle nhbton taes place and the sum throughput performance of Aloha and CA are nearly dentcal 1 so that there s very lttle dfference n an optmum versus a non-optmum threshold. Hence, a useful approxmaton s to consder the optmum threshold for any ntal densty to be roughly equal to that for the lmtng densty λ t,. Lastly, as λ 0, the networ s sparse enough that very lttle nhbton taes place and λ t λ, and the spatal densty of transmssons can be related to the outage probablty: λ t ϵ θ 2/α R 1 2 [Cα T ] + oϵ2. 9 The factor T s always less than or equal to Cα, and hence carrer sensng wth an approprate threshold represents a strct mprovement over Aloha. For the multple antenna case, the optmum θ can be found numercally. As n the sngle-antenna case, the optmum threshold reaches a lmt as λ grows large, whch s a relatvely good selecton at all. IV. EXAPLE AND DICUION Fg. 2 compares throughput and outage of CA and Aloha for the base IO case. Here the CA s not fully optmzed, but rather has a fxed threshold for all ntal contenton denstes. Note that the fgures plot throughput and outage versus the ntal contenton densty, pror to thnnng va carrer sensng, n order for the comparson wth Aloha to be far, and ths conventon was used for all plots n the paper. 1 Ths s very dfferent from sayng that the transmsson capacty at a fxed outage s nearly the same for both. At a fxed λ, f Aloha experences 1% outage, then CA can ncrease throughput at the same λ by no more than 1% at best. On the other hand, for a fxed 1% outage constrant, CA can mantan a substantally greater λ t than Aloha can. Fg. 3. Comparson of throughput for CA at a hgh base. Note that no outage constrants are mposed. Parameters were R 1, α 4, θ 0dB. Fg. 4. Throughput vs. ntal densty for 4-antenna systems wth optmzed CA and an outage constrant of 10% Parameters were R 1, α 4, θ 0dB, 10dB. Fg. 3 compares 4-antenna IO confguratons at a hgh base wth fully optmzed CA. Fg. 4 compares 4- antenna IO confguratons wth a relatvely small outage constrant appled. These curves demonstrate several dstnctons between the performance of CA networs versus Aloha networs. The frst s smply that as the ntal contenton densty ncreases, CA throughput overtaes Aloha substantally and reaches a non-zero plateau. Furthermore, comparng throughput levels for Fg. 2 and Fg. 4, t becomes clear that CA IO systems acheve a hgher throughput at substantally lower outage among transmttng nodes. Ths can substantally reduce the burden on PHY layer decodng mplementatons and conserve energy. In partcular, note that a 1 3 antenna confguraton wth an optmzed threshold can

5 5 acheve a networ throughput equvalent to the best Aloha confguraton but wth a fourfold reducton n the outage probablty. V. CONCLUION Ths paper developed a model for CA and ts nteracton wth multple antennas n ad hoc wreless networs. The results confrm the beneft of CA and IO technques n ad hoc networs, alone or n tandem. Frst, the throughput at any contenton densty s superor for optmzed CA systems over ther Aloha counterparts. econd, the carrer sensng mechansm results n a throughput plateau whch s mantaned wth fxed opertng parameters, unle Aloha, even as networ access requests ncrease. Thrd, CA systems acheve a hgher throughput at substantally lower outage among transmttng nodes. And lastly, CA mproves the envronment for spatal multplexng, n certan cases mang a hgher number of data streams more attractve at any spatal traffc densty. APPENDIX For a gven stream at the recever usng the ZF soluton o GammaN r, 1/N t wth F c o x e N tx N r 1 N t x! so that o R α P θ P Φ d α > θ 0 N r 1 N r F c o θ R α tf IΦ + 1 tdt 1! 0 [ s! st e s t f IΦ + 1 tdt d ] ds L I Φ sl 1 s, sn tθ R α where the Laplace transform of the sum of the random varables s the product of the transforms. Now L 1 s e s/ and f n addton GammaN t, 1/N t as n open-loop multplexng across the networ, the Laplace transform of the Posson shot nose s [ L IΦ s exp E ] e x α s 1 dφ R 2 1 exp 2πλ t u s/n t u α 1 e θuα du exp 2π α θ 2 α R 2 λ t [BN t α, α 1 ΓN t α UN t α, 1 α, θ θ R α ] exp 2π α θ 2 α R 2 λ t [CN α t T ]. Our last tas s to fnd an explct and effcent method of calculatng dervatves of these Laplace transforms. For ths we turn to Faà d Bruno s formula for dervatves of d composte functons fgs, whch can be represented ds as a determnant [8]: d fgs det fgs ds for the matrx g 1 D g 2 D g 1 D g 3 D 2g 2 D g 1 D 1..., g 4 D 3g 3 D 3g 2 D g 1 D where D fgs f gs, and where the coeffcents appled to each row are rows of Pascal s trangle. To apply Faà d Bruno s formula to the dervatves of L IΦ sl 1 s, we can wrte L IΦ sl 1 s fgs, where fs es and gs log L IΦ sl 1 s whose dervatves are for > 1: g s 2π 1 α λ ts/n t 2 2 α C N α t α j 1 j0 n0 K j s j U j0 [ ] α + j, θ s, 10 1 α + j N t where K j θ /N t j ΓN t α N t α + j j j j 1 2 α n. To accomplsh ths we needed the relaton for the nth dervatve of Ua, b, x whch s easly expressble as d n dx n Ua, b, x 1 n a n Ua + n, b + n, x where n s the Pochhammer symbol for the rsng factoral. REFERENCE [1] F. Baccell, B. Blaszczyszyn, and P. uhlethaler, An ALOHA protocol for multhop moble wreless networs, IEEE Trans. on Inf. Theory, vol. 52, no. 2, pp , Feb [2] A. Hasan and J. G. Andrews, The guard zone n wreless ad hoc networs, IEEE Trans. on Wreless Comm., vol. 6, no. 3, pp , ar [3]. Kayna, G. Oen, and N. Jndal, Impact of fadng on the performance of ALOHA and CA, n Proc., IEEE gnal Proc. Adv. n Wreless Comm. PAWC, Italy, Jun [4] R. K. Gant and J. G. Andrews, A new method for computng the transmsson capacty of non-posson wreless networs, n Proc., IEEE Intl. ymposum on Informaton Theory, Austn, TX, [5] R. H. Y. Loue,. R. ckay, and I. B. Collngs, Open-loop spatal multplexng and dversty communcaton n ad hoc networs, avalable at arxv: v2, ep [6] A.. Hunter, J. G. Andrews, and. Weber, Transmsson capacty of wreless ad hoc networs wth spatal dversty, IEEE Trans. on Wreless Comm., vol. 7, no. 12, Dec [7] A.. Hunter and J. G. Andrews, Adaptve rate control over multple spatal channels n ad hoc networs, n Proc., Worshop on patal toch. odels for Wreless Net. pawn, Berln, Germany, Apr [8] W. P. Johnson, The curous hstory of Faà d Bruno s formula, n Amer. ath. onthly, vol. 109, no. 3, 2002.