Hierarchical Beamforming for Large One-Dimensional Wireless Networks

Transcription

1 Hierarchical Beamformig for Large Oe-Dimesioal Wireless Networks Alla Merzakreeva, Olivier Lévêque Swiss Federal Istitute of Techology - Lausae, Switzerlad {alla.merzakreeva, olivier.leveque}@epfl.ch Ayfer Özgür Staford Uiversity, Staford CA aozgur@staford.edu Abstract We cosider a wireless etwork with a large umber of source-destiatio pairs distributed over a lie. Uder lie-ofsight propagatio, this etwork has oly oe degree of freedom for commuicatio. At high SNR, this oe degree of freedom ca be readily achieved by multi-hop relayig betwee odes. At low SNR, however, the performace is determied by the power trasfer i the etwork. We show that oe of the existig architectures ca achieve optimal capacity scalig. We develop a digital hierarchical beamformig architecture ad show that it is scalig optimal. This result reveals a ew regime for large wireless etworks, where beamformig techiques are eeded to achieve capacity. I. INTRODUCTION Multi-hop is the traditioal commuicatio architecture for wireless adhoc etworks. Iformatio is routed from source odes to destiatios via multiple poit-to-poit trasmissios betwee itermediate odes actig as relays. Ca we desig ew cooperatio architectures for wireless adhoc etworks that sigificatly outperform multi-hop? To shed some light o this questio i the large etwork regime, we follow the scalig law approach i [1]-[8]. This approach was iitiated i [1] ad cocetrates o the scalig of the capacity with the umber of odes whe other parameters of the etwork are coupled with i two specific ways, leadig to the so called dese ad exteded scaligs. We have show i [8],[9] that by extedig this approach to seekig a multi-parameter family of scalig laws, we ca use it to idetify the fudametal operatig regimes of large wireless etworks. Here may differet limits coupligs with respect to the parameters of the etwork are joitly cosidered. This allows to ucover a umber of qualitatively differet operatig regimes where the aswer to the above raised questio is differet. The curret uderstadig of the operatig regimes of large wireless etworks ca be roughly summarized as follows see [9] for a detailed overview: Whe SNR is high ad there are sufficiet spatial degrees of freedom i the etwork this is for example the case whe the pairwise chaels are subject to i.i.d. fadig, a hierachical cooperatio architecture based o distributed MIMO trasmissio ca provide sigificat capacity gai over multi-hop [5]. The situatio is more tricky for etworks at low SNR. Whe the SNR is low but there are still sufficiet spatial degrees of freedom i the etwork as with i.i.d. fadig, the optimal architecture depeds o the power path loss expoet of the eviromet α [5], [9]. α determies how fast sigal power decays with distace i a eviromet ad is typically betwee 2 ad 6 i a twodimesioal etwork. Whe 2 α < 3, a bursty versio of the hierarchical cooperatio architecture sigificatly outperforms multi-hop ad is scalig optimal. Whe α 3, multi-hoppig achieves the optimal scalig ad o strategy ca perform better. Iterestigly, oe of the two architectures, hierarchical cooperatio or multihop, uses beamformig, which is kow to be the right strategy for poit-to-poit MIMO chaels at low SNR [10]. This is because uder i.i.d. fadig, the distributed MIMO chaels available i the etwork are well-coditioed ad beamformig or waterfillig over the eigevalues of the chael provides little gai. Trasmittig idepedet streams from each ode durig the distributed MIMO trasmissios is scalig optimal. The recet work [6], however, reveals that degrees of freedom i a wireless etwork ca be limited by physical costraits i the spatial chael. This ca be thought of as the spatial chael itroducig correlatio betwee pairwise gais. I the regime whe the available degrees of freedom i the etwork are very few determied by the area of the etwork ad the carrier wavelegth [8], they ca be readily achieved by multi-hop. This makes multi-hop scalig optimal for such etworks i the high SNR regime [6]. However, at low SNR, the performace is determied by the power trasfer i the etwork ad it is ot clear whether ay of the existig architectures achieves the optimal scalig of the capacity. I this paper, we explore a ew regime where the etwork is both limited i power operatig at low SNR ad i spatial degrees of freedom operatig uder strogly correlated pairwise chaels. We show that a ew class of cooperative beamformig architectures outperform classical multi-hoppig whe α is small. To capture this regime i a simple setup, we focus o a oe-dimesioal wireless etwork i a lie-of-sight propagatio eviromet. This leads to the extremal case whe there is oly oe degree of freedom for commuicatig i the etwork. At high SNR, this sigle degree of freedom ca be readily achieved with multi-hop, or eve with simple timedivisio betwee the source-destiatio pairs i the etwork. At low SNR, we develop a hierarchical beamformig architecture ad show that it outperforms existig strategies whe 1 α 2. 1 Here, odes first broadcast their iformatio to 1 The assumptio α 1 i oe-dimesioal etworks replaces the traditioal assumptio α 2 made i two-dimesioal etworks.

2 Fig. 1: Oe-dimesioal etwork a small cluster aroud them. This allows to beamform this iformatio ad distribute it over a larger cluster i a secod step. Cotiuig i a hierarchical fashio, the iformatio of each source ode is broadcasted to the whole etwork, icludig the destiatio ode. We show that this broadcastig strategy achieves the optimal scalig whe α = 1. This is suprisig give that the origial requiremet is to commuicate the iformatio of each source ode oly to its correspodig destiatio ode. A future goal of this research program is to uderstad the power ad spatial degrees of freedom limited regimes of two-dimesioal wireless etworks. The multi-stage broadcastig architecture developed i the preset paper is remiiscet of other schemes existig i the literature, such as [11], [12], [13], [14]. The setups cosidered i these refereces are evertheless quite differet. The problem of broadcast capacity, i.e. the maximum data rate that ca be achieved from a sigle source ode to every other ode i the etwork, was cosidered i [11], [12], [13], while the reverse problem of the data gatherig chael was cosidered i [14]. Our itet here is to primarily study the multiple-uicast sceario. Aother differece is that the above refereces cosider primarily high or fixed SNR, while we are iterested i the low SNR regime. For oe S-D pair, the low SNR regime has bee studied i [15], where it was show that ampify-ad-forward based beamformig techiques achieve the optimal scalig i a diamod etwork. As opposed to this aalog strategy, the commuicatio architecture proposed for multiple S-D pairs i the preset paper is digital. II. MODEL There are odes uiformly ad idepedetly distributed alog a lie of legth L, as illustrated o Figure 1. Every ode is both a source ad a destiatio, ad the sources ad destiatios are radomly paired up oe-to-oe. All source odes wat to commuicate to their destiatio ode at the same rate R. The aggregate throughput of the etwork is defied as T = R. We assume that commuicatio takes place over a flat chael with badwidth W ad that the received sigal Y k [m] by ode k at time m is give by Y k [m] = j J h kj X j [m] + Z k [m] where J is the set of trasmittig odes, X j [m] is the sigal set at time m by ode j ad Z k [m] is additive white circularly symmetric Gaussia oise AWGN of power spectral desity N 0 /2 Watts/Hz. I a lie-of-sight eviromet, the complex basebad-equivalet chael gai h kj betwee trasmit ode j ad receive ode k is give by h kj = G exp2πir kj/λ r α/2 kj where G is give by the Friis formula, λ is the carrier wavelegth, r kj is the distace betwee ode k ad ode j ad α 1 is the power path loss expoet. I a oe-dimesioal etwork, this lie-of-sight model clearly departs from the classical i.i.d. phase assumptio: it implies i particular that the chael matrix betwee two distat clusters of odes is essetially rak 1 whe it is full-rak uder the i.i.d. model. Fially, we assume full chael state iformatio at both the trasmitters ad receivers which is a reasoable assumptio i a static lie-of-sight eviromet, as well as a commo average power budget per ode P. III. MAIN RESULT Let us deote by SNR s the sigal-to-oise ratio over the typical earest eighbor distace i the etwork. I a oedimesioal etwork, the typical earest eighbor distace is L, therefore, the short-distace SNR is SNR s = GP α. N 0 W L I this paper, we are iterested i the low SNR regime, where SNR s = γ for some γ > 0. We will use the otatio SNR s 0 db to deote this coditio. This models the sceario whe the pairwise chaels betwee earest eighbors are i the low SNR regime. Note that sice SNR s is joitly determied by a umber of system parameters P, W ad L, this ca be the case whe the available power per ode is small; whe the badwidth is large; or the distaces betwee the odes are large. A relatively straightforward aalysis reveals that i oedimesioal etworks, the multi-hop scheme described i [1] achieves with probability approachig 1 as icreases with high probability a aggregate throughput of order SNRs T = Ω log whe SNR s 0 db ad α 1. O the other had, the best kow iformatio theoretic upper boud o the throughput scalig of such etworks is give i [4]: for α 1, T = O log 3. This shows that for costat SNR s, multi-hop cooperatio is order optimal ad the aggregate throughput is costat, up to logarithmic factors. I the low SNR regime however that is, whe SNR s 0 db, the questio remais whether a more sophisticated strategy would ot allow to achieve higher throughput scalig tha multi-hop. We aswer this questio by the affirmative i the followig theorem, i the case where the path loss expoet α lies betwee 1 ad 2. Theorem 1: Let us assume that 1 α < 2 ad SNR s 0 db i.e. SNR s = γ for some γ > 0. The for ay ε > 0, there exists a commuicatio scheme referred to as 1

3 hierarchical beamformig i the sequel that achieves the followig aggregate throughput with high probability as gets large: T = Ω mi { SNR s 2 α ε, ε} 2 IV. HIERARCHICAL BEAMFORMING Let us start by cosiderig the situatio where the SNR i the etwork is very low. More precisely, let us assume that SNR s α 2 with 1 α < 2 3 The above aggregate throughput scalig is strictly higher tha that achieved by multi-hop. I particular, whe α = 1 ad SNR s 1/, T = ΩSNR s 1 ε, which is a order improvemet over multi-hop. The hierarchical beamformig architecture achievig this performace is described i detail i the ext sectio. Is this strategy optimal or ca we do better? Before aswerig this questio, let us itroduce the otio of a broadcastig scheme below. Defiitio 1: A commuicatio scheme achievig a perode throughput R for S-D pairs is said to be a broadcastig scheme if at this same rate R, all destiatios are able to decode the iformatio set ot oly by their correspodig source, but also by all the other sources. As we will see, hierarchical beamformig eters ito this category, ad so does classical multi-hop i oe-dimesioal etworks at the price of a small adaptatio of the origial scheme. The theorem below, together with Theorem 1 above, shows that: a hierarchical beamformig is scalig optimal whe α = 1; b amog all broadcastig schemes, hierarchical beamformig is scalig optimal whe 1 α < 2, ad multihoppig is whe α 2. Theorem 2: Cosider a oe-dimesioal etwork with α 1 ad SNR s 0 db. The: a The aggregate throughput scalig of ay commuicatio scheme is upper bouded with high probability by T = O mi { SNR s log 2, log 3 } b The aggregate throughput scalig of ay broadcastig scheme is upper bouded with high probability by T = { O mi { SNR s 2 α log 2, log 3 } if 1 α < 2 O SNR s log 2 if α 2 The proof of this result is omitted due to space limitatios, but let us idicate here briefly the mai ideas behid the proof. Part a ca be prove usig the classical SIMO boud showig that the commiucatio rate from oe source to the rest of the etwork is at most of order SNR s log 2. For part b, the followig observatio is useful: i the broadcastig sceario, each destiatio is waitig for messages from the other 1 odes i the etwork. So usig the MISO boud aroud a particular destiatio allows to upper boud the commo rate at which each source should trasmit towards this destiatio. A iterestig ope questio, that we do ot address i the preset paper, is whether ay scalig optimal scheme for the multiple-uicast problem i a oe-dimesioal etwork is also a broadcastig scheme or ot. I this regime, may trasmissios ca take place cocurretly i the etwork spatial reuse without creatig iterferece above the oise level. Uder this assumptio, the lower boud i Eq. 2 reads T = Ω SNR s 2 α ε 4 We first sketch the hierarchical beamformig strategy we propose ad the proceed to its performace aalysis which also provides a more detailed descriptio. Cosider oe particular source-destiatio pair s d i the etwork. For simplicity, assume that s has oe bit to commuicate to d. s ca commuicate this oe bit i two steps: it ca first broadcast this bit to a small cluster of M eighborig odes aroud itself. the ca the simultaeously trasmit this bit to the destiatio ode d by coheretly combiig their sigals. The beamformig gai due to the coheret combiig of the M sigals leads to a better performace tha simply trasmittig the bit from s to d. From the etwork poit of view, all source-destiatio pairs have to evetually accomplish these two steps. Step 2 is lograge commuicatio ad oly oe source-destiatio pair ca operate at a time. Step 1 ivolves local commuicatio ad ca be parallelized across source-destiatio pairs. This leads to the followig two phases i the operatio of the etwork: 1. The etwork is divided ito clusters of. Each source ode distributes oe bit to the i its cluster. There are M source odes i a cluster, which ca simply take turs to distribute their oe bit. Whe the total iterferece from the other clusters is below the oise level, this operatio ca be coducted i parallel amog all clusters. At the ed of this phase, each ode has therefore received ad decoded oe bit from every other ode i its cluster. 2. I the secod phase, the bits are beamformed to their actual destiatios oe at a time. Every cluster performs M successive trasmissios; i each trasmissio, the bit of oe particular source ode i the cluster is beamformed to its destiatio. There are a total of succesive beamformig trasmissios i this phase, oe for each source-destiatio pair i the etwork. A key observatio is that this two-phase scheme distributes the bits of every source ode to all other odes i the etwork, eve if this is ot what we set for. I the secod phase, the beamformig trasmissios doe oe at a time ca be decoded ot oly by the actual destiatio ode but simultaeoulsy by all the odes i the etwork. This is a cosequece of the fact that the etwork has oly oe degree of freedom. The trasmitted sigals from each cluster ca be arraged to coheretly combie simultaeously at all the remaiig odes

4 i the etwork. Therefore accordig to Defiitio 1, this twophase scheme is a broadcastig scheme. This brigs the idea of recursio. The broadcastig requiremet i the first phase ca be hadled by further dividig each cluster ito smaller clusters ad use the two-phase broadcast scheme we just described. The two-phase scheme is illustrated i Figure 2. The recursio is summarized i the followig lemma. Fig. 2: Two-phase beamformig Lemma 1: Cosider 1 α < 2 ad a oe-dimesioal etwork with odes ad SNR s 0 db, subject to a additive exteral iterferig source with bouded average power. If i this etwork, there exists a broadcastig scheme achievig with high probability a aggregate throughput T = Ω SNR s β for some β 2 α, the there exists aother broadcastig scheme achievig with high probability a aggregate throughput T = Ω SNR s fβ where fβ = 1 α1 β 2 β Notice that fβ > β for all 1 α < 2 ad β < 2 α, so the performace of the ew scheme is always strictly better tha that of the origial oe. Figure 3 below illustrates the behavior of fβ, for α = 1 ad α = 1.5. Fig. 3: Growth of the aggregate throughput expoet Proof of Lemma 1. Cosider a etwork of odes ad legth L, ad let us divide it ito clusters of legth LM/, with 1 M, so that each cluster cotais order M odes with high probability. Based o the assumptio made 5 i the lemma, the followig commuicatio scheme will be show to achieve the desired throughput scalig. Phase 1. Source odes broadcast iformatio to every other ode iside their cluster, usig the origial scheme with aggregate throughput T M = Ω SNR s M β. This step is parallelized across clusters 2. Phase 2. For each source ode iside a cluster of, all the odes iside the cluster simultaeously beamform the received bits to the rest of the etwork. Durig this secod phase, oly oe cluster operates at a time. Performace Aalysis. I the first phase, clusters work i parallel. I order to avoid collisios betwee eighborig clusters, a simple time-divisio scheme with two rouds is used, where half of the clusters are active at a time: this oly affects the throughput by a factor two ad allows clear receptio of the sigals iside each cluster. Oe ca ideed check that because of assumptio 3, the average power of the iterferece caused i oe cluster by simultaeous trasmissios i the other clusters is bouded. The broadcastig rate achieved by the scheme iside each cluster is RM = Ω SNR s M β 1, so the total time take by this first phase is upper bouded by t 1 = O 1 SNR s M β 1 I the secod phase, M broadcast trasmissios are performed sequetially from each cluster towards the rest of the etwork. As there are /M clusters, the total umber of trasmissios is therefore equal to that is, oe trasmissio takes place for each source ode. The SNR of each trasmissio is lower bouded by SNR s M M 2 α = SNR s M 1 α where the above factors are explaied as follows: - the factor /M is due to the fact that each cluster oly trasmits a fractio M/ of the time, so power ca be spared durig the rest of the time; - the factor M 2 is the beamformig gai otice that because of the lie-of-sight chael model 1 ad the assumptio of a oe-dimesioal etwork, it is ideed possible to beamform the sigal towards all destiatios simultaeously; - the factor α is a lower boud o the power atteuatio over distace. The total time take by this secod phase is therefore upper bouded by t 2 = O SNR s M 1 α 1 = O SNR s M α Optimal cluster size. I order to optimize the throughput of the ew scheme, the optimal cluster size M should be chose such that the duratios of the two phases are equal, i.e. t 1 = t 2, which leads to M β 1 = M α i.e. M = α/2 β 6 2 Notice that SNR s, that oly depeds o the distace betwee eighborig odes, remais uchaged for a cluster of or for the whole etwork.

5 Notice that α/2 β 1, as β 2 α by assumptio. Resultig aggregate throughput. With this cluster size, it is worthwhile oticig that the broadcastig rate of the ew scheme is the same as the oe achieved i each cluster with the origial scheme. However, as more odes participate i the trasmissio, the aggregate throughput icreases as follows: T = RM = Ω SNR s M β 1 = Ω SNR s fβ where fβ is give i 5. This completes the proof. Let us ow explai how applyig recursively Lemma 1 allows to obtai the lower boud 4 o the aggregate throughput scalig. Let us first use multi-hop for broadcastig iformatio at the lowest level of the hierarchy, that is, iside small clusters of M 1 odes. Note that multi-hop ca be easily trasformed ito a broadcastig scheme i the oe-dimesioal case without chagig its aggregate throughput scalig; sice iformatio is routed over a lie, each destiatio already observes the iformatio set by order odes o average. The aggregate throughput achieved iside each cluster is therefore SNRs T M 1 = Θ = Ω SNR s M β 1 β < 0 log M 1 Usig the the two-phase scheme described i the proof of Lemma 1, we reach for larger clusters of size M 2 to be specified below a aggregate throughput T M 2 = Ω SNR s M fβ 2 β < 0 Iteratig this procedure h 1 times, util the large cluster of size M h reaches the etwork size, we obtai the followig aggregate throughput T = Ω SNR s f h 1 β β < 0 As illustrated o Figure 3, the sequece f h 1 β coverges to the miimal solutio of the equatio β = fβ which is give by β = 2 α for 1 α < 2. For a fixed umber of hierarchical levels h, the achieved aggregate throughput scalig is therefore T = Ω SNR s 2 α ε, ad ε > 0 ca be made arbitrarily small by icreasig the umber h. I additio, let us describe how to compute the optimal cluster sizes M 1,..., M h i this process. From Eq. 6 i the proof of Lemma 1, we deduce that at level 1 k < h, M k = M k+1 α/2 βk where βk is the aggregate throughput expoet achieved at level k. This allows to compute recursively the cluster sizes, startig from M h =. From this aalysis, it turs out that as h gets large, the optimal cluster size M 1 at the lowest level of the hierarchy coverges to M 1 = α 1 So whe α = 1, the hierarchical beamformig scheme starts directly from tiy clusters, whereas whe 1 < α < 2, the optimal commuicatio strategy is first to perform multi-hop iside clusters of size α 1, ad the to use hierarchical beamformig. We therefore see that i the latter case, because of the higher value of the path loss expoet α, beamformig oly helps whe sufficietly may odes participate to the trasmissio. Fially, let us metio what happes at moderately low SNR, i.e. whe α 2 SNR s 1. I this case, the iterferece felt from the simultaeously trasmittig clusters might hurt the trasmissios iside a cluster. A simple solutio to this problem is to reduce the power used by each ode, so as to meet the equality SNR s = α 2. I this case, the aggregate throughput of the scheme is arbitrarily close to a costat, which proves the claim made i Theorem 1. ACKNOWLEDGMENT A. Merzakreeva s work was supported by Swiss NSF Grat Nr A. Özgür s work was supported by the ERC Startig Grat Project NOWIRE ERC-2009-StG , while the author was with the Swiss Federal Istitute of Techology - Lausae. REFERENCES [1] P. Gupta, P. R. Kumar, The Capacity of Wireless Networks, IEEE Tras. o Iformatio Theory 42 2, 2000, [2] L.-L. Xie ad P. R. Kumar, A Network Iformatio Theory for Wireless Commuicatios: Scalig Laws ad Optimal Operatio, IEEE Tras. o Iformatio Theory 50 5, 2004, [3] A. Jovicic, P. Viswaath ad S. R. Kulkari, Upper Bouds to Trasport Capacity of Wireless Networks, IEEE Tras. o Iformatio Theory 50 11, 2004, [4] A. Özgür, O. Lévêque, E. 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