Improved Capacity Scaling in Wireless Networks With Infrastructure

UNDER REVISION FOR IEEE TRANSA CTIONS ON INFORMA TION THEOR Y 1 Impro v ed Capacity Scaling in W ireless Net works W ith Infrastructure W on-Y ong Shin, Member , IEEE , Sang-W oon Jeon, Stud ent Member , IEEE , Natasha De vroye, Mai H. V u, Member , IEEE , Sae- Y oung Chung, Senior Member , IEEE , Y ong H. Lee , Senior Member , IEEE , and V ahid T arokh, F ellow , IEEE The work of W .-Y . Shin and Y . H. Lee was supp orted by the Brain Ko rea 21 Project, The School of Information T echno logy , KAIST in 2008. The work of S.-W . Jeon and S.-Y . Chun g was supported by the MKE under the ITRC support program supervised by the IIT A (IIT A-2008-C1090-080 3-0002). The work o f M. H. V u was supported in part by ARO MURI grant number W911NF-07-1-0376. The material in this paper was presented in part at the IEEE Communication Theory W orkshop, St. Croix, US V irgin Islands, May 2008 and the IEEE International Symposium on Information Theory , T oronto, Canada, July 2008. W .-Y . Shin was with the Department of EE, KAIST , Daejeon 305-701 , Republic of K orea. He is no w with the School of Engineering and Applied Sciences, Harv ard Univ ersity , Cambridg e, MA 02138 USA (E-mail : wyshin@seas.harvard .edu). S.-W . Jeon, S.-Y . Chung, and Y . H. Lee are with the Department of EE, KAI ST , Daejeon 305-701, Republic of Korea (E-mail: swjeon@kaist.ac.kr; sychung@ee.kaist.ac.kr; yohlee@ee.kaist.ac.kr). N. Devro ye was with the School of Engineering and Applied Sciences, Harv ard Univ ersity , Cambridge, MA 02138 USA. S he is now with the Department of E lectrical and Computer Engineering, U ni versity of I llinois at Chicago, Chicago, Il linois 60607 US A (E-mail: de vroye@ece.uic.edu). M. H. V u was with the School of Engineering and Applied Sciences, Harv ard Univ ersity , Cambridge, MA 02138 USA. She is now with the Department of E lectrical and Computer Engineering, McGill Univ ersit y , Montreal, QC H3A 2A7, Canada (E-mail: mai.h.vu@mcgill.ca). V . T arokh is with th e School of Engineering and Applied Sciences, Harvard Univ ersity , Cambridge, MA 02138 USA (E-mail: v ahid@seas.harv ard.edu). UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 2 Abstract This pa per analy zes the impact and benefits of infra structure suppo rt in impr oving the th rough put scaling in networks of n r andom ly located wireless nodes. Th e infrastruc ture uses m ulti-antenn a base station s (BSs), in which the num ber of BSs an d the n umber of anten nas at each BS can scale at arbitrary rate s relativ e to n . Un der the mo del, capacity scaling laws ar e analyzed f or both den se and exten ded n etworks. T wo BS-ba sed routing schem es are first introdu ced in this stud y: an infrastructur e-suppor ted single-h op (ISH) routing protocol with multiple-access uplink and broad cast d ownlink a nd an infrastructu re-suppo rted m ulti-hop (IMH) routing protoc ol. Th en, their achiev able throug hput scalings are analyzed. These schemes ar e c ompared against two conventional schem es witho ut BSs: the multi-hop (MH) tran smission and hie rarchical cooperation (H C) schemes. I t is shown that a linear throug hput scaling is achiev ed in dense ne tworks, as in the case without help of BSs. In contrast, the pro posed BS-based routing schemes can, under realistic network condition s, improve the throug hput scaling sign ificantly in extended networks. The g ain comes fr om the fo llowing advantages of these BS-based proto cols. First, more nodes can transmit simultaneously in the p ropo sed scheme tha n in the MH scheme if the num ber of BSs and the number of antennas are large eno ugh. Seco nd, b y improving the lon g-distance sign al-to-no ise ratio (SNR), the received signal power can be larger than that of the HC, enabling a better through put scaling unde r exten ded networks. Furthermo re, b y deriving th e co rrespon ding in formation -theor etic cut-set upper bo unds, it is shown und er extended networks that a combin ation of fou r sche mes I MH, ISH, MH, a nd HC is or der-optimal in all operating r egimes. Index T erms Base station ( BS), infrastructu re, c ut-set upper boun d, hierarc hical cooper ation (HC), multi-an tenna, multi- hop (MH), single-hop, th roug hput scaling UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 3 I . I N T RO D U C T I O N In [1], Gu pta and Kumar introduced and st udied the t hroughput scaling in a large wireless ad hoc network. They s howed that, for a network of n source–desti nation (S–D) pairs rando mly d istributed in a unit area, the t otal throughput scales as Θ( p n/ log n ) . 1 This throug hput scaling is achie ved usin g a multi-hop (MH) com munication scheme. Recent results ha ve shown that an almost linear throughput i n the network, i.e., Θ( n 1 − ǫ ) for an arbitrarily small ǫ > 0 , is achie vable by usin g a hierarchical cooperation (HC) strategy [3], [4], [5], [6]. Besides the schemes in [3], [4], [5], [6], there has been a steady push to improve the th roughput of wi reless networks up t o a linear scaling in a variety of network scenarios by using novel techniques such as networks with node mobil ity [7], interference alignment schemes [8], and infrastructure support [9]. Although it would be good to h a ve such a linear scaling with onl y wireless connectivity , in practice there will be a price to pay in terms of hig her delay and higher cost o f channel estimation . For t hese reasons, it would stil l be go od to hav e infrastructure aiding wireless nodes. Such hybrid networks consistin g of bot h wireless ad hoc nodes and infrastructure nodes, or equivalently bas e s tations (BSs), have been introduced and analyzed in [10], [11], [9], [12], [13]. BSs are assumed to be interconnected by high capacity wired links. While it has been shown t hat BSs can be beneficial in wireless networks, the impact and benefits of infrastructure support are not fully understood yet. T his paper features analysis of the capacity scaling laws for a m ore general hybrid network where there are l antennas at each BS, all owing the exploitation of the spatial dimensi on at each BS. 2 By all owing the num ber m of BSs and th e n umber l of antennas to scale at arbit rary rates relative to the n umber n of wireless no des, achiev able rate scalin gs and in formation- theoretic upper boun ds are deriv ed as a function of th ese scalin g p arameters. T o show our achie vability results, two new routing protocol s utili zing BSs are proposed here. In the first protocol, mu ltiple sources (nodes) transmit simultaneous ly to each BS u sing a d irect si ngle-hop multip le-access in the uplink and a direct s ingle-hop broadcast from each BS in the downlink. In the second protocol, the high-speed BS links are combined with nearest-neighbor routin g vi a MH among the wireless nodes. Th e obtained results are also compared to two con ventional schemes without using BS s: the MH protocol [1] and HC protocol [3]. The proposed schemes are ev aluated in two different networks: dense networks [1], [14], [3] of unit area, and extended networks [15], [16], [17], [18], [3] o f unit node density . In dense networks, it is shown that an alm ost linear throughput scaling i s achieved as in [3], whi ch is rather obvious. On the contrary , i n extended n etworks, dependin g on the network configurations and the path-loss attenuation, the proposed BS-based p rotocols can improve the through put scali ng sign ificantly , com pared t o the case without help of BSs. Part of the im provement comes from t he following two adv antages over the con ventional s chemes: having more antennas enables m ore transmi t pairs that can be activ ated sim ultaneously (com pared to those of the MH schem e), i.e., enough degree-of-freedom (DoF) gain i s obtained, provided the num ber m of BSs and th e number l of antennas per BS are lar g e enough. In additio n, the BSs help to imp rove the long- distance sig nal-to-noise ratio (SNR) 3 , first term ed in [19], which leads to a larger recei ved s ignal power than that of the HC scheme, i .e., enough power g ain is ob tained, thus allowing for a better t hroughput scaling in extended networks. T o show the optimality of our proposed schemes, cut-set upper boun ds on the throughput scaling are deriv ed for a network wi th infrastructure. Not e th at t he pre vious upper bounds in [15], [16], [20], [17], [21], [3] assume pure ad h oc networks and those for BS-based networks are n ot rigorously characterized in both dens e and extended networks. In dense n etworks, it is shown that the obtained upper bou nd is the 1 W e use the follo wi ng no tations: i) f ( x ) = O ( g ( x )) means that there e xist constants C and c suc h that f ( x ) ≤ C g ( x ) for all x > c . ii) f ( x ) = o ( g ( x )) means that lim x →∞ f ( x ) g ( x ) = 0 . iii) f ( x ) = Ω( g ( x )) if g ( x ) = O ( f ( x )) . iv) f ( x ) = ω ( g ( x )) if g ( x ) = o ( f ( x )) . v) f ( x ) = Θ( g ( x )) if f ( x ) = O ( g ( x )) and g ( x ) = O ( f ( x )) [2]. 2 When the carrier frequenc y is very high, it is possible to deploy many antennas at each BS since the wav elength becom es very small. 3 In [19] , the long-distance SNR is defined as n times the recei ved SNR between two farthest nodes across the larg est scale in wireless networks. In our BS-based network, it can be interpreted as the total SNR transferred to any given node (or BS antenna) over a certain smaller scale reduced by infrastructure support. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 4 same as that of [3] assuming no BSs. Hence, it is seen t hat the BSs cannot improve the throughput scaling and the HC scheme i s order -op timal for all the operating regimes. In extended networks, the proposed approach is based in part on the characteristics at power -lim ited regimes shown in [3], [19]. It is shown that our upper bounds match the achiev able throughput scalings for all t he operating regimes wi thin a factor of n ǫ with an arbitrarily small exponent ǫ > 0 . T o achieve the order -optimal scaling, using one of the two BS-based routings, conv enti onal MH transmission , and HC strategy is needed depending on the operating regimes. The rest o f th is paper is organized as follows. Section II describes the p roposed network model with infrastructure support. The main results are briefly shown in Section III. The two proposed BS-based protocols are characterized in Section IV and their achie vable throughput scalings are analyzed in Section V. The corresponding information-th eoretic cut-set upper bound s ar e deri ved in Section VI. Finally , S ection VII summarizes this paper wi th some conclud ing remarks. Throughout this paper the superscripts T and † denote the transpose and conj ugate transpose, respec- tiv ely , of a matrix (or a vector). I n is the identity matrix of size n × n , [ · ] k i is the ( k , i ) -th element o f a matrix, tr( · ) is t he trace, and det( · ) is the determinant. C i s the field of compl ex numbers and E [ · ] is the expectation. Unless otherwise stated, all logarithms are ass umed to be to the base 2. I I . S Y S T E M A N D C H A N N E L M O D E L S Consider a two-dimensio nal wireless network that consist s of n S–D pairs uniformly and independentl y distributed on a square exce pt for th e area cover ed by BSs. Then, no nodes are physi cally located inside the BSs. The network is assumed to have an area of on e and n in dense and extended networks, respectiv ely . Suppose that the wh ole area is divided into m square cells , each of which is cov ered by one BS wi th l antennas at i ts center (see Fig. 1). It is assum ed that the tot al number of antennas in the network scales at most linearly with n , i.e., ml = O ( n ) . For analyti cal con venience, we would li ke to state that parameters n , m , and l are then related according to n = m 1 /β = l 1 /γ , where β , γ ∈ [0 , 1 ) sat isfying β + γ ≤ 1 . The nu mber of antennas is all owed to grow with the number of nodes and BSs in the network. The placement of these l antennas depends on h ow the number of antenn as scales as follows: 1) l antennas are regularly placed on the BS boundary if l = O ( p n/m ) , and 2) p n/m antennas are regularly placed on th e BS boundary and t he rest are uniformly pl aced insi de the boundary if l = ω ( p n/m ) and l = O ( n/ m ) . 4 Furthermore, w e assume that the BSs are connected to each other w ith su f ficiently large capacity such that the com munication between t he BSs is not the limit ing fa ctor t o overall throu ghput scaling. The required transmi ssion rate of each wired BS-to-BS link will be specified later (in Remark 4). It i s also assumed t hat these BSs are neither sources nor destination s. Suppose t hat the radius of each BS scales as ǫ 0 / √ m for dense networks and as ǫ 0 p n/m for extended networks, where ǫ 0 > 0 is an arbitrarily small constant i ndependent of n , w hich means that it is ind ependent of m and l as well. This radius scaling would ensure enough separatio n amo ng the antennas since the per-a ntenna distance scales at least as the a verage per-node di stance for any parameters n , m , and l . Let us first d escribe the uplink channel model. Let I ⊆ { 1 , · · · , n } denote the s et of simult aneously transmittin g wireless nodes. T hen, the l × 1 recei ved sign al vector y s of BS s ∈ { 1 , · · · , m } and the l × 1 uplink complex channel vector h u si between node i ∈ { 1 , · · · , n } and BS s are given by y s = X i ∈ I h u si x i + n s 4 Such an antenna deployment strategy guarantees both the nearest neighbor transmission from/to each B S antenna and enoug h space among the antennas, and thus enables our BS-based routing protocol via MH to work well, which will be discussed in Section IV -A. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 5 and h u si = " e j θ u si, 1 r u α/ 2 si, 1 e j θ u si, 2 r u α/ 2 si, 2 · · · e j θ u si,l r u α/ 2 si,l # T , (1) respectiv ely , where x i is the transmit signal of node i , and n s denotes the circularly symmetric com plex additive wh ite Gaussian noise (A WGN) vector whose element has zero-mean and variance N 0 . Here, θ u si,t represents th e random ph ases uniformly distributed over [0 , 2 π ) and independ ent for different i , s , t , and time (transmiss ion symbo l), i .e., fast fading. Not e that this random phase model i s based on a far-fi eld assumption , wh ich is va lid if the wav elength is s uffi ciently small . r u si,t and α > 2 denote the distance between node i and the t -th antenna of BS s , and th e path-loss exponent, respectiv ely . Similarly , the 1 × l downlink comp lex channel vector h d is between BS s and node i ( s ∈ { 1 , · · · , m } and i ∈ { 1 , · · · , n } ) and the complex channel h k i between nod es i and k ( i, k ∈ { 1 , · · · , n } ) are given by h d is = " e j θ d is, 1 r d α/ 2 is, 1 e j θ d is, 2 r d α/ 2 is, 2 · · · e j θ d is,l r d α/ 2 is,l # and h k i = e j θ ki r α/ 2 k i , (2) respectiv ely , where θ d is,t and θ k i hav e uniform distribution over [0 , 2 π ) , and are independent for different i , s , t , k , and tim e. r d is,t and r k i denote the dis tance between the t -th ant enna of BS s and nod e i , and the distance between n odes i and k , respectively . Suppose that each node and BS should satisfy an av erage transmit power constrain t P and nP /m , respectiv ely , during transm ission. 5 Then, the total transmi t powers allowe d for t he n wireless nodes and the m BSs are the same. Channel s tate information (CSI) is assumed t o be a vailable bot h at the recei vers and the trans mitters for downlink t ransmissions from BSs but onl y at the recei vers for transmis sions from wireless nodes. Let T n ( α, β , γ ) denote the to tal throughput of the network for the parameters α , β , and γ , and then its scaling exponent is defined by [3], [19] e ( α, β , γ ) = lim n →∞ log T n ( α, β , γ ) log n , (3) which captures the d ominant t erm in the exponent of the throug hput scaling. 6 It is assumed t hat each node transmits at a rate T n ( α, β , γ ) /n . I I I . M A I N R E S U LT S This section presents t he formal statement of our results, which are divided into two parts to show the capacity scaling laws: achiev able throug hput scalings and informati on-theoretic upper bounds. W e simply state these result s here and deriv e them in later sections. The following summ arizes our main results. In dense net works, the optimal scaling exponent is giv en by e ( α , β , γ ) = 1 , while t he opt imal scaling exponent e ( α , β , γ ) in extended networks is gi ven by e ( α, β , γ ) = max  1 + γ − (1 − β ) α 2 , min  β + γ , β + 1 2  , 1 2 , 2 − α 2  , (4) where the details are shown i n th e following two subsecti ons. 5 This assumption is reasonable since the balance between uplink and do wnlink would be maintained for the case where the transmit po wer of one BS in a cell increases proportionally to the total power consumed by all the users covered by the cell. In this case, note that although we allo w additional power for BS s, the total transmit po w er used by all wireless nodes and BSs still remains as Θ( n ) . 6 T o simplify notations, T n ( α, β , γ ) will be written as T n if droppin g α , β , and γ does not cause any confusion. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 6 A. Achievable Thr oughput Scaling In thi s s ubsection, th e throug hput scaling for both dense and extended networks under our rou ting protocols is shown. The following theorem first presents a lower bo und on the total capacity scaling T n in dense networks. Theor em 1: In a dense network, T n = Ω( n 1 − ǫ ) (5) is achie vable wit h h igh probability (whp ) for an arbit rarily s mall ǫ > 0 . Equation (5) is achiev able by simply usi ng the HC strategy [3]. 7 Although the HC provides an almost linear t hroughput scaling in dense networks, it may degrade t hroughput scalings in extended (or power - limited) networks. An achiev able throughput under extended networks is sp ecifically giv en as follows. Theor em 2: In an extended network, T n = Ω  max  ml  m n  α/ 2 − 1 , m min  l,  n m  1 / 2 − ǫ  , n 1 / 2 − ǫ , n 2 − α/ 2 − ǫ  (6) is achie vable whp for an arbitrarily small ǫ > 0 . The first to fourt h terms i n (6) corre spond to the achie vable rate scalings of the infrastructure-supported single-hop (ISH), infrastructure-supported multi-hop (IMH), MH, and HC prot ocols, respectively , where the t wo BS-based schemes wi ll be described in detail later (in Section IV). As a resul t, the best strategy among t he four s chemes ISH, IM H, MH, and HC depends on the path-loss exponent α , and t he parameters m and l under extended networks. Let us give an intuition for the achiev abil ity result above. For the first term in (6), ml represents the total number of si multaneously active sources in the ISH protocol while ( m/n ) α/ 2 − 1 comes from a performance degradation due to power lim itation. The s econd term in (6) represents the total number of sources that can s end their own packets s imultaneousl y using t he IMH protocol. From t he achie vable rates of each scheme, t he interesting result below is obtained under each network condit ion. Remark 1: The best achiev able one amo ng the four schemes and its scaling exponent e ( α , β , γ ) i n (3) are shown in T ABLE I according to the two-dimensional operatin g regimes on the achie vable throughput scaling with respect t o the scaling parameters β and γ (see Fig. 2). This result i s analyzed i n Appendix A. Operating regimes A–D are shown in Fig. 2. It is impo rtant to verify the b est proto col in each regime. In Regime A , w here β and γ are small , the infrastructure i s not helpful. In other regimes, we observe BS-based proto cols are dominant in some cases depending on the path-loss exponent α . For example, Regime D has the following charac teristics: the HC protocol has th e h ighest through put when t he path-los s attenuation is s mall, but as th e path-l oss e xponent α increases, the best scheme becom es t he ISH prot ocol. This i s because the penalty for long-range multiple-input multip le-output (MIMO) transm issions o f the HC increases. Finally , the IMH protocol becom es d ominant when α is large since the ISH protocol has a powe r li mitation at the hi gh p ath-loss attenuation regime. B. Cut-Set Up per Bound W e now turn our attention to presenti ng the cut-set upper bound of t he total through put T n . The upper bound [3] for p ure ad hoc networks of unit area is extended to our dense n etwork m odel. Theor em 3: The total throughput T n is upper-bounded by n log n whp in a dense network with i nfras- tructure. Note that the same upper boun d as that of [3] assum ing no BSs is found i n dense networks. This upper bound means that n S–D pairs can be active with genie-aided interference removal between simultaneously transmittin g nodes, w hile providing a power g ain of log n . In addition, i t is exa mined how the u pper b ound is close to th e achie vable t hroughput s caling. 7 Note that the HC always outperforms the proposed BS-based routing protocols in t erms of throughput performance under dense networks, e ven t hough the details are not shown in this paper . UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 7 Remark 2: Based on the above result, it is easy to see that the achie vable rate in (5) and th e up per bound are of the same order up to a factor log n in dense networks with the help of BSs, and th us the exponent of the capacity s caling is giv en by e ( α, β , γ ) = 1 . The HC is therefore order-optimal and we may con clude that infrastructure cannot im prove th e throughput s caling in dense networks. In extended networks, an upper bo und is established based on the charac teristics at power -lim ited regimes shown in [3], [19], and is p resented in t he following theorem. Theor em 4: In an extended network, t he t otal throughput T n is upper -bounded by T n = O  n ǫ max  ml  m n  α/ 2 − 1 , m min  l, r n m  , √ n, n 2 − α/ 2  (7) whp fo r an arbitrarily small ǫ > 0 . The relationship between the achiev able throughput and the cut-set upper bound is now examined as follows. Remark 3: The upper bound matches the achie vable t hroughput scaling wit hin n ǫ in e x tended networks with infrastructure, and th us the scaling exponent in (4) holds. In o ther words, it is shown that choos ing the best of the four schemes ISH, IM H, MH and HC is order-optimal for all the op erating regimes sho wn in Fig. 2 (see T ABLE I). I V . R O U T I N G P RO T O C O L S This section explains the two BS-based protocols. T wo con ventional schemes [1], [3] with no infras- tructure supp ort are also in troduced for com parison. Each rou ting protocol operates in di f ferent time slot s to av oid huge mutual interference. W e focus on the descripti on for extended networks since using the HC scheme [3] is eno ugh to provide a near -opt imal throughput in dense networks. A. Pr otocols W it h Infrastr uctur e Su pport W e generalize the con ventional BS-based transmi ssion scheme in [10], [11], [9], [12], [13]: a sou rce node transm its its packet to the clo sest BS, the BS ha ving the packet transmits it to the BS that is nearest to the dest ination of the s ource via wired BS-to-BS links, and the destination finally receiv es its data from the nearest BS. Sin ce there exist both access (to BSs) and exit (from BSs) routin gs, different time slots are used, e.g., e ven and od d time slots, respectively . W e start from the following l emma. Lemma 1: Suppose m = n β where β ∈ [0 , 1) . Then, t he number of nodes inside each cell i s between ((1 − δ 0 ) n 1 − β , (1 + δ 0 ) n 1 − β ) , i.e., Θ ( n/m ) , with probability lar g er than 1 − n β e − ∆( δ 0 ) n 1 − β , (8) where ∆( δ 0 ) = (1 + δ 0 ) ln(1 + δ 0 ) − δ 0 for 0 < δ 0 < 1 independent of n . The proof o f this lemma is given b y slig htly modifying the proof of Lemm a 4.1 in [3]. Not e that (8) tends to one as n goes to infinity . 1) Infrastructur e-supported single-hop (ISH) pr otocol: In cont rast with previous works, the s patial dimensions enabled by ha vi ng multip le antenn as at each BS are exploited here, and thus m ultiple trans- mission pairs can b e su pported using a single BS. Under e x tended networks, the ISH t ransmission protocol shown i n Fig. 3 is now propos ed as follows: • For the access routing, all source nodes in each cell, given by n/m nodes whp from Lemma 1 , transmit th eir independ ent packets sim ultaneously via single-hop multiple-access to the BS in the same cell. • E ach BS receiv es and jointly decodes packets from sou rce nodes in t he same cell. Signals receive d from t he other cells are treated as noise. • T he BS that completes decoding its packets transm its t hem to the BS closest to the corresponding destination by wired BS-to-BS links . UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 8 • For the exit routing, each BS t ransmits all packets receiv ed from other BSs, i.e., n/m packets, via single-hop b roadcast to t he destinations in t he cell. Since t he network is powe r-limited, the proposed ISH scheme is used with the full power , i.e., the transmit p owers at each node and BS are P and nP m , respectively . For the ISH protocol, m ore DoF gain is provided compared to transmissi ons via MH if m and l are lar ge enou gh. The power gain can also be obtai ned compared to t hat of the HC scheme in certain cases. 2) Infrastructur e-supported multi-ho p (IMH) pr otocol: The fact that the extended network is power - limited motiva tes the introduction of an IMH t ransmission protocol in which multiple s ource nodes in a cell transmi t their packets t o BS in the cell via MH, thereby ha ving m uch hig her receiv ed po wer , i.e., more power gain, than that of the direct one-hop t ransmission in extended n etworks. That is, bet ter long-di stance SNR is provided wi th the IMH protocol. Similarly , each BS deli vers its packets to the corresponding destinations by IMH transmi ssions. Und er extended networks, the IMH transmis sion protocol in Fig. 4 is proposed as follows: • Divide each cell into smaller sq uare cells of area 2 lo g n each, where these sm aller cells are called routing cells (wh ich i nclude at l east one node wh p [1], [14]). • For the access routing, min { l , p n/m } sou rce nodes in each cell transmit t heir i ndependent packets using MH rout ing to the correspondin g BS in t he cell as s hown in Fig . 5. It is assumed that each antenna placed only on the BS boundary recei ves its packet from one of t he no des in the nearest neighbor routing cell. It is easy to see that min { l , p n/m } MH paths can be su pported due to our antenna placement within BSs. Exi t routing is similar , where each antenna on the BS boundary uses power P that s atisfies t he power con straint. • T he BS-to-BS t ransmissions are t he sam e as the ISH case. • E ach routi ng cell operates based on 9 -t ime di vision mul tiple access (TDMA) t o avoid causing huge interference t o its neighbor cells. Note that the transm it power min { l , p n/m } at each BS, but not full p ower , i s eno ugh to perform the IMH protocol in the downlink. For the IMH protocol, more DoF gain i s poss ible compared to the M H scheme for l ar ge m and l . In addition, more po wer gain can also be obtained compared to the HC and ISH schemes in certain cases. B. Pr otocols W it hout Infrastru ctur e Supp ort The upper bound in Theorem 3 is only determined b y the number n of wireless nodes in dense networks. The upper bound in Theorem 4 also ind icates that either the number m of BSs or the number l of antennas per BS should be greater than a certain lev el in order to obtain im proved throughput scalings i n extended networks. This is because otherwise less DoF gain may be provided comp ared to that of the con ventional schemes with out help of BSs. Thus, transmissio ns o nly using wireless nodes may be enoug h to achie ve the capacity scalings in dense net works or in extended networks wi th small m and l . In this case, the MH and HC protocols, which were proposed in [1] and [3], respecti vely , are performed in our network with infrastructure. V . A C H I E V A B L E T H R O U G H P U T I N E X T E N D E D N E T W O R K S In thi s section, the achie vable throughput scaling i n Theorem 2 is rig orously analyzed in extended networks. It is dem onstrated that the th roughput scaling can be improved under some condi tions by applying two BS-based transm issions in extended networks. The transm ission rate o f the ISH p rotocol in extended networks will be shown first. Lemma 2: Suppose that an extended network uses the ISH protocol. Then, th e rate of Ω  l  m n  α/ 2 − 1  is achie vable at each cell for both access and exit routings . UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 9 Pr oof: In order to prove the result, we need t o quantify t he t otal amount of int erference when the ISH scheme i s used. W e first introduce t he following l emma and refer to Appendix B for th e detailed proof. Lemma 3: In an extended n etwork with the ISH protocol, the total interference power P u I in the uplink from n odes i n other cell s t o each BS antenna is upper -bounded by Θ(( m/n ) α/ 2 − 1 ) whp . Each node also has interference p ower P d I = Θ(( m/n ) α/ 2 − 1 ) whp i n the do wnlink from BSs in other cells. Note that when α > 2 th e term ( m/n ) α/ 2 − 1 tends to zero as n → ∞ . Now , the transm ission rate for the access routing is deriv ed as i n the following. The signal model from nod es in each cell to the BS with multiple ant ennas corresponds t o the sing le-input mult iple-output (SIMO) mu ltiple-access channel (MAC ). Since th e maximu m Euclidean distance among links of the above SIMO MA C scales as Θ( p n/m ) , i t is upper-bounded by as δ 1 p n/m , where δ 1 > 0 i s a certain cons tant. Let N I denote the sum of total interference power P u I recei ved from the other cells and noise va riance N 0 . Then, the worst case noise o f this channel has an uncorrelated Gaussian di stribution wit h zero-mean and variance N I [22], [23], [24], which lower -bo unds t he transmiss ion rate. By assumi ng full CSI at the recei ver (BS s ) and performing a minimum mean-square error (MMSE) estimati on [25 ], [26], [27] with successive i nterference cancellation (SIC) at BS s , the sum -rate of the SIMO MA C is given by [26], [27] I ( x s ; y s , H s ) ≥ E  log det  I l + P N I H s H † s  ≥ E  log det  I l + P δ α 1 ( n/m ) α/ 2 N I GG †  , (9) where x s denotes the n m × 1 transm it sig nal vector , whose elements are nodes in the cell covered by BS s , y s is th e l × 1 receiv ed signal vector at BS s , and H s = [ h u s 1 h u s 2 · · · h u s ( n/m ) ] ( h u si for i = 1 , · · · , n/m is given in (1)). G is the no rmalized matrix, whose element g ti is given by e j θ u si,t and represents the phase between node i and t he t -th antenna of BS s . Note t hat rotating the decoding order am ong n/m nodes in the cell leads to th e same rate of each node. Then, the above sum-rate is rewritten as I ( x s ; y s , H s ) ≥ E " l X i =1 log  1 + P δ α 1 ( n/m ) α/ 2 − 1 N I λ i  # ≥ l E  log  1 + P δ α 1 ( n/m ) α/ 2 − 1 N I λ 1  ≥ l log  1 + P δ α 1 ( n/m ) α/ 2 − 1 N I ¯ λ  Pr  λ 1 > ¯ λ  , (10) where { λ 1 , · · · , λ l } are the unordered eigen values of m n GG † [28] and ¯ λ is any nonnegativ e const ant. Due to the fac t that log (1 + x ) = (log e ) x + O ( x 2 ) for small x > 0 , (10) is given by I ( x s ; y s , H s ) ≥ c 0 l  m n  α/ 2 − 1 Pr  λ 1 > ¯ λ  (11) for some const ant c 0 > 0 ind ependent of n , s ince N I has a con stant scaling from Lemm a 3. By the Pale y -Zygmund inequality [29], i t is possi ble to lower -bo und th e sum-rate in the left-hand si de (LHS) of (11) by fol lowing the same l ine as Appendix I in [3], thu s y ielding I ( x s ; y s , H s ) ≥ c 1 l  m n  α/ 2 − 1 , where c 1 > 0 is some constant independent of n . For the exit routing, the sign al model from the BS with multiple antennas i n one cell t o nodes in the cell correspon ds to the multipl e-input single-output (MISO) broadcast channel (BC). From Lemma 3, it is seen that the total interference power recei ved from the other BSs is bounded. Hence, it is possible to deriv e the transmissio n rate for the exit routing by exploiting an upl ink-downlink duality [26], [27], [30], UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 10 [31]. In thi s case, the transmitters in the downlink are desi gned by an MMSE transmit precoding with dirty paper cod ing [32], [33], [34] at each BS, and the rate of the M ISO BC is then equal t o t hat o f the dual SIMO M A C wi th a sum p ower constraint . More precisely , with full CSI at the transmitter (BS) and the total transmit power nP m in the do wnlink, the sum -rate of the MISO BC is lower -bounded by [26] max Q x ≥ 0 E  log det  I l + 1 N ′ I H ′ s † Q x H ′ s  ≥ E  log det  I l + P N ′ I H ′ s † H ′ s  , ( 12) where H ′ s = [ h d T 1 s h d T 2 s · · · h d T ( n/m ) s ] T , N ′ I denotes the sum of t otal i nterference power P d I from BSs in the other cells and n oise variance N 0 , and Q x is the n m × n m positive semi -definite input cov ariance matrix which is diagonal and satis fies tr( Q x ) ≤ nP m . Here, the inequality holds since t he rate is reduced by simply applyin g t he same av erage powe r of each user . Due to the fact that (12) is equivalent to the right-hand side (RHS) of (9) (with a change of variables), Ω  l ( m/n ) α/ 2 − 1  is achiev able in the downlink of each cell by following the same approach as that for the access routing, which completes the proof of Lemma 2. Note that l corresponds to th e DoF at each cell pro vided by th e ISH prot ocol while ( m/n ) α/ 2 − 1 represents the t hroughput d egradation due to power loss . The transmi ssion rate for t he access and e xit routings of IMH protocol will now be analyzed in extended networks. The number of source nodes t hat can b e activ e simu ltaneously is examined under the IMH protocol, while maintaining a const ant t hroughput Θ(1) per S–D pair . Lemma 4: When an extended n etwork u ses t he IMH prot ocol, the rate of Ω  min  l,  n m  1 / 2 − ǫ  (13) is achie vable at each cell for both access and exit routings , where ǫ > 0 is an arbitrarily sm all constant. Pr oof: This result is obtained by m odifying the analysis in [1], [14], [35] on scaling laws under our BS-based network. W e mainly focus o n the aspects that are different from the conv enti onal schemes. From the 9-TDMA operation, the signal-to -interference-and-noise ratio (SINR) seen by any receiver is giv en by Ω(1) with a transmit power P . It can be interpreted that when the worst case n oise [22], [23], [24] is assum ed as in the ISH protocol, the achie vable throughput per S–D pair is lower -bounded by log(1 + SINR ) , thus providing a con stant scaling. First consider the case l = o ( p n/m ) where the number l o f antennas scales slower than t he number n/m of nodes in a cell. Then, it is p ossible to activ ate up to l source no des at each cell because there exist l routes for the last hop t o each BS antenna in the uplink. On the other hand, when l = Ω( p n/m ) , t he maximum num ber of simultaneou sly transmitting sources per BS is equal to the number of routing cells on the BS bound ary , which scales with ( n/m ) 1 / 2 − ǫ for an arbitrarily sm all ǫ > 0 . In t he downlink of each cell, the same numb er of S–D pairs as that in the uplink i s active simultaneou sly . Therefore, the transm ission rate per each BS is finally g iv en by (13), which completes t he proof of Lemm a 4. By using Lemmas 2 and 4, we are ready to sho w the achiev able throughput scaling in extended netw orks. The achie vable throughputs of the ISH and IMH protocols are give n by T n = Ω  ml  m n  α/ 2 − 1  (14) and T n = Ω  m min  l,  n m  1 / 2 − ǫ  , (15) respectiv ely , since there are m cell s in th e network. Throughput scalings of t wo con ventional protocols that do not u tilize the BSs are als o cons idered. From t he results of [1], [3], T n = Ω  n 1 / 2 − ǫ  (16) UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 11 and T n = Ω( n 2 − α/ 2 − ǫ ) (17) are yielded for the MH and HC schemes, respectively . Hence, the through put scali ng in e xtended networks is simply lower -bounded by t he maximum of (14)–(17), which com pletes the proof of Theorem 2. In addi tion, we would like to examine the required rate of each BS-to-BS trans mission. Remark 4: T o see ho w much data t raf fic flows on each BS-to-BS link, we first show t he following lemma. Lemma 5: Let X k i denote the number of destinations in the k -th cell wh ose source nodes are i n the i -th cell, where i, k ∈ { 1 , · · · , m } . Th en, for all i, k ∈ { 1 , · · · , m } , the following equatio n ho lds whp: X k i =  O  n m 2  if n = ω ( m 2 ) O (log n ) if n = O ( m 2 ) . (18) The proo f of this lemm a is present ed in Appendix C. Let C B S denote t he rate of each BS-to-BS link. Then, since each S–D pair transmi ts at a rate T n /n and the num ber of packets carried s imultaneousl y through each l ink is bo und by (18) from Lemm a 5, t he required rate C B S is gi ven by C B S =  Ω  T n m 2  if n = ω ( m 2 ) Ω  T n log n n  if n = O ( m 2 ) . V I . C U T - S E T U P P E R B O U N D T o see how closely the proposed schemes approach the fundamental l imits in a network with in- frastructure, new BS-based cut-set outer bounds on the t hroughput scaling are analyzed based on the information-theoretic approach [36]. A. Dense Networks Before showing the m ain proo f of Theorem 3, we start from the following lemma. Lemma 6: In our two-dimensional dense network where n n odes are un iformly distributed and there are m BSs with l regularly spaced antennas, the minim um di stance between any two nodes or between a node and an antenn a on th e BS boundary i s l ar ger than 1 /n 1+ ǫ 1 whp for an arbitrarily s mall ǫ 1 > 0 . The proof of this lemma is presented in Appendi x D. Now we present the cut-set upper bound o f the total throughpu t T n in dense networks. The p roof steps are si milar to those of [37], [3]. The through put per S–D pair i s simply upper- bounded by the capacity of the SIMO channel between a source node and the rest of no des i ncluding BSs. Hence, th e total throughput for n S–D pairs is bounded by T n ≤ n X i =1 log    1 + P N 0    n X k =1 k 6 = i | h k i | 2 + m X s =1 k h u si k 2       ≤ n log  1 + P N 0 n (1+ ǫ 1 ) α ( n − 1 + ml )  = c 2 n lo g n, where k · k denot es L 2 -norm of a vector and c 2 > 0 is some constant independent of n . The second inequality holds due to Lemm a 6. Thi s com pletes the proof of Theorem 3. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 12 B. Extended Networks Consider the cut L in Fig. 6 dividing the network area into two halves in an extended random network. Let S L and D L denote the sets of sources and destinati ons, respectively , for the cut in the network. More precisely , und er L , (wireless) source nodes S L are on the l eft half of t he network, while all nodes on the right h alf and all BS antennas are destinati ons D L . 8 In this case, we get the n × ( n + ml ) MIMO channel between the t wo set s of nod es and BSs separated by the cut. In extended networks, it i s necessary to narrow down the class of S–D pairs according to their Euclidean distance to ob tain a tight upper bound. In this subsection, t he up per bou nd based on the power transfer ar guments i n [3], [19] is shown, where an upper bound is proportio nal to the t otal receiv ed signal power from source nodes. The present problem is no t equiv alent t o the con ventional extended setup since a network wit h i nfrastructure support is taken into account. A new up per bou nd based on hybri d approaches that consider either th e sum of the capacities of t he multipl e-input si ngle-output (MISO) channel b etween transmitters and each receiv er or the amount of power transferred across the netw ork according to operating regimes, is thus deriv ed. W e start from the following lemma. Lemma 7: Assume a two-dimensional extended network. When the network area with the exclusion of BS area is di vided int o n squ ares of unit area, there are less than log n nodes in each square whp. This result can be obtained by applying our BS-based network and sligh tly modi fying the proof of Lemma 1 in [18]. For the cut L , the total throughput T n for sources on the left half is bounded by the capacity of the M IMO channel bet ween S L and D L , and th us T n ≤ max Q L ≥ 0 E h log det  I n + ml + H L Q L H † L i = max Q L ≥ 0 E h log det  I Θ( n ) + H L Q L H † L i , where the equality holds sin ce n = Ω( ml ) . 9 H L consists of h u si in (1) for i ∈ S L , s ∈ B , and h k i in (2) for i ∈ S L , k ∈ D r . Here, B and D r represent t he set of BSs i n the network and the set of (wireless) nodes on the right half, respectively . Q L is the positive semi-definite i nput cov ariance matrix whose k -th diagonal element satisfies [ Q L ] k k ≤ P for k ∈ S L . The set D L ( = B ∪ D r ) is partitioned into three groups according to t heir location, as s hown in Fig. 7. By generalized Hadamard’ s inequalit y [38] as in [16], [3], T n ≤ max Q L ≥ 0 E h log det  I √ n + H (1) L Q L H (1) L † i + max Q L ≥ 0 E h log det  I O ( √ ml ) + H (2) L Q L H (2) L † i + max Q L ≥ 0 E h log det  I Θ( n ) + H (3) L Q L H (3) L † i , (19) where H ( t ) L is the matrix with entries h H ( t ) L i k i for i ∈ S L , k ∈ D ( t ) L , and t = 1 , · · · , 3 . Here, D (1) L and D (2) L denote the sets of dest inations located o n the rectangular slab with width 1 immediately to the rig ht of the centerline (cut) and on the ring with width 1 imm ediately insi de each BS b oundary (cut) on the left half, respectiv ely . D (3) L is giv en by D L \ ( D (1) L ∪ D (2) L ) . Note that the sets ( D (1) L and D (2) L ) of destinations located very close to t he cut are considered separately since ot herwise their contribution to the to tal received power will be exce ssive, resulti ng i n a loo se bou nd. Each term in (19) will be analyzed belo w in detail. Be fore that, to get t he total p ower t ransfer of the set D (3) L , the same technique as th at in Section V of [3] i s used, whi ch is th e relaxatio n of the individual power constraints to a total weighted power constraint, where the weigh t assigned to each source corresponds 8 The other cut ˜ L can also be considered in the network. In this case, sources S ˜ L represent antennas at each BS as well as ad hoc nodes on the left half. The (wireless) destination nodes D ˜ L are on the ri ght half. Since the cut L provide s a tight uppe r bound compared to t he achie vable rate, the analysis for the cut ˜ L is not sho wn in this paper . 9 Here and in the sequel, the noise v ariance N 0 is assumed to be 1 to simplify the notation. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 13 to the total receiv ed power o n the ot her sid e of the cut. Specifically , each column i of the matrix H (3) L is normalized by the square root of t he total recei ved po wer on the other side of the cut from source i ∈ S L . The total weighted p ower P (3) L,i by s ource i i s then giv en by P (3) L,i = P d (3) L,i , (20) where d (3) L,i = X k ∈ ¯ D r \ D (1) L r − α k i + X s ∈ B l ,t ∈ [1 ,l ] r u − α si,t . (21) Here, ¯ D r is th e s et of destinati on nodes including BS antennas on the ri ght hal f and B l represents the set of BSs o n the l eft half. Then, th e third term in (19) is rewritten as max ˜ Q L ≥ 0 E h log det  I n + F (3) L ˜ Q L F (3) L † i , (22) where F (3) L is the matrix with entries h F (3) L i k i = 1 q d (3) L,i h H (3) L i k i , which are obtained from (21), for i ∈ S L , k ∈ D (3) L . Then, ˜ Q L is the matrix s atisfying h ˜ Q L i k i = q d (3) L,k d (3) L,i [ Q L ] k i , which means tr( ˜ Q L ) ≤ P i ∈ S L P (3) L,i (equal to the sum of th e total recei ved po wer from each source). W e next examine the behavior of the largest singular value for the normali zed channel m atrix F (3) L . From the fact that F (3) L is well-condit ioned w hp, this shows how much it essentiall y af fects an upper bound of (22), which wi ll be analy zed later in Lemm a 9. Lemma 8: Let F (3) L denote the normalized channel matrix whose element is given by h F (3) L i k i = 1 q d (3) L,i h H (3) L i k i . Then, E     F (3) L    2 2  ≤ c 3 (log n ) 3 , (23) where k · k 2 denotes the largest si ngular value of a m atrix and c 3 > 0 i s some constant independent of n . The proo f of this lemma i s present ed in Ap pendix E. Using Lemm a 8 yield s t he following result. Lemma 9: The term shown in (22) is upper-bounded by n ǫ X i ∈ S L P (3) L,i (24) whp w here ǫ > 0 is an arbitrarily small constant and P (3) L,i is give n by (20). Pr oof: Equation (22) is bounded by max ˜ Q L ≥ 0 E  log det  I n + F (3) L ˜ Q L F (3) L †  1 E F (3) L  + max ˜ Q L ≥ 0 E  log det  I n + F (3) L ˜ Q L F (3) L †  1 E c F (3) L  , (25) where the e vent E F (3) L is gi ven by E F (3) L =     F (3) L    2 2 > n ǫ  UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 14 for an arbitrarily small constant ǫ > 0 . Then, by using the result of Lemm a 8 and applyi ng the proof technique s imilar to that in Section V of [3], it is possible to prove that the first term in (25) decays polynomial ly to zero as n tends to infinity , and for the second term in (25), it follows that max ˜ Q L ≥ 0 E  log det  I n + F (3) L ˜ Q L F (3) L †  1 E c F (3) L  ≤ n ǫ X i ∈ S L P (3) L,i , which completes t he proof. Note that (24) represents the power transfer from the set S L of sources to the set D (3) L of the corre- sponding desti nations for a given cut L . For not ational con venience, let d (4) L,i and d (5) L,i denote the first and second terms in (21), respectiv ely . Th en, P d (4) L,i and P d (5) L,i correspond to the total recei ved power from source i to the destination sets ¯ D r \ D (1) L and D L \ ( D (2) L ∪ ¯ D r ) , respectiv ely . The comput ation of the total recei ved power of the set D (3) L will now be comp uted as fol lows: X i ∈ S L P (3) L,i = X i ∈ S L P d (4) L,i + X i ∈ S L P d (5) L,i , (26) which is e ventually used to compute (24). First, to get an upper bou nd on P i ∈ S L P d (4) L,i in (26), the network area is divided into n s quares of unit area. By Lemma 7 , since there are less than log n nodes inside each square whp, the powe r transfer can be upper-bounded by that under a regular network wit h at most log n nodes at each square (see [3] for the detailed description). Such a mo dification yields the following upper bound [3] for P i ∈ S L P d (4) L,i : X i ∈ S L P d (4) L,i ≤    c 4 n 2 − α/ 2 (log n ) 2 if 2 < α < 3 c 4 √ n (log n ) 3 if α = 3 c 4 √ n (log n ) 2 if α > 3 (27) whp for some constant c 4 > 0 independent of n . Next, the second term P i ∈ S L P d (5) L,i in (26) can be deri ved as in t he following lemma. Lemma 10: The term P i ∈ S L P d (5) L,i is gi ven by X i ∈ S L P d (5) L,i =            0 if l = o ( p n/m ) O  nl  m n  α/ 2 log n  if l = Ω( p n/m ) and 2 < α < 3 O  ml p m n (log n ) 2  if l = Ω( p n/m ) and α = 3 O  n √ l  ml n  α/ 2 log n  if l = Ω( p n/m ) and α > 3 (28) whp. The proo f of this lemma i s present ed in Ap pendix F. It is now possible to deriv e t he cut -set upper bound in Theorem 4 by using Lemm as 9 and 10. For notational con venience, let T ( i ) n denote t he i -th term in the RHS of (19) for i ∈ { 1 , 2 , 3 } . By generalized Hadamard’ s inequali ty [38] as in [16], [3], the first term T (1) n in (19) can be easily bo unded by T (1) n ≤ X k ∈ D (1) L log 1 + P N 0 X i ∈ S L | h k i | 2 ! ≤ ¯ c 1 √ n (log n ) 2 , (29) where ¯ c 1 > 0 is s ome constant independent o f n . Note that this upper b ound does no t depend on β and γ . The s econd inequality ho lds sin ce the m inimum distance b etween any source and destination is lar ger than 1 /n 1 / 2+ ǫ 1 whp for an arbit rarily small ǫ 1 > 0 , which is obtained by the deriv ation similar to that of Lemma 6, and th ere exist no more than √ n lo g n n odes in D (1) L whp by Lemma 7. The upper boun d UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 15 for T (2) n is now derived. Since s ome nodes in D (2) L are located very close to th e cut and th e information transfer to D (2) L is lim ited in DoF , t he second term T (2) n of (19) i s upper-bounded by the sum of the capacities of the M ISO channels. Mo re precisely , by generalized Hadamard’ s inequ ality , T (2) n ≤  ¯ c 2 ml log n if l = o ( p n/m ) ¯ c 2 √ nm log n if l = Ω( p n/m ) , ≤ ¯ c 2 m min  l, r n m  log n (30) where ¯ c 2 > 0 is som e constant independent of n . Next, the third term T (3) n of (19) will be sh own by using (24), (27) and Lem ma 10. If l = o ( p n/m ) , which corresponds to operating regimes A and B shown i n Fig. 2, t hen T (3) n is gi ven by T (3) n =  O ( n 2 − α/ 2+ ǫ ) if 2 < α < 3 O ( n 1 / 2+ ǫ ) if α ≥ 3 . Hence, under this network condition , T n ≤ c 5 n ǫ max  ml, √ n, n 2 − α/ 2  for so me con stant c 5 > 0 independent of n , which is upper-bounded by the RHS of (7). Now we focus on t he case for l = Ω( p n/m ) (regimes C and D in Fig. 2). In this case, T (3) n is upper -bounded by T (3) n ≤        ¯ c 3 n ǫ  n 2 − α/ 2 (log n ) 2 + nl  m n  α/ 2 log n  if 2 < α < 3 ¯ c 3 n ǫ  √ n (log n ) 3 + ml p m n (log n ) 2  if α = 3 ¯ c 3 n ǫ  √ n (log n ) 2 + n √ l  ml n  α/ 2 log n  if α > 3 ≤    ¯ c 3 n ǫ 2 max n n 2 − α/ 2 , nl  m n  α/ 2 o if 2 < α < 3 ¯ c 3 n ǫ 2 max n √ n, n √ l  ml n  α/ 2 o if α ≥ 3 (31) for s ome cons tant ¯ c 3 > 0 and an arbit rarily small constant ǫ 2 > ǫ > 0 . From (29), (30), and (31), we thus get the fol lowing result: T n ≤    ¯ c 4 n ǫ max n √ nm, n 2 − α/ 2 , nl  m n  α/ 2 o if 2 < α < 3 ¯ c 4 n ǫ max n √ nm, n √ l  ml n  α/ 2 o if α ≥ 3 ≤ ¯ c 4 n ǫ max  √ nm, n 2 − α/ 2 , ml  m n  α/ 2 − 1  , where the first and second inequaliti es hold since √ nm = Ω( √ n ) and √ nm = Ω  n √ l  ml n  α/ 2  , respec- tiv ely , whi ch results in (7). Thi s com pletes the proof of Theorem 4. Now we would like to examine in detail the amount of inform ation transfer by each separated destination set. Remark 5: The in formation transfer by the BS antennas on the left hal f, i .e., the desti nation set D L \ ¯ D r , becomes dominant under operating regimes B–D (especially at the high path-los s attenuat ion regimes) in Fig. 2. More specifically , compared to the pure network case w ith no BSs, as m and l increases (i.e., regimes B–D), enoug h DoF gain is obtain ed by exploiti ng multiple antennas at each BS, whi le the power gain is provided since all the BSs are connected by the wired BS-to-BS links. In additio n, note t hat the first to fourth terms in (7) represent the amount of in formation transferred to the desti nation sets D L \ ( D (2) L ∪ ¯ D r ) , D (2) L , D (1) L , and ¯ D r \ D (1) L , and can be achiev ed by the ISH, IMH, MH, HC schemes, respectiv ely . UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 16 V I I . C O N C L U S I O N The paper has analyzed the benefits of infrastructure support for generalized hybrid networks. Provided the number m of BSs and the number l of antennas at each BS scale at arbitrary rates relati ve to the number n of wireless nodes, the capacity scalin g laws were deriv ed as a function of these s caling parameters. Specifically , t wo routing protocols using BSs were propo sed, and th eir achiev able rate s calings were deriv ed and compared with t hose of th e two con ventional schemes MH and HC in bo th dense and extended networks. Furthermore, to show the opt imality of the achiev abil ity results, new information-theoreti c upper bounds were derived. In both dense and extended networks, it was shown that our achie vable schemes are order- optimal for al l the o perating regimes. A P P E N D I X A. Achievable Thr oughput W i th Respect to Operating Re gimes Let e ISH , e IMH , MH , and e HC denote the scalin g exponents for the achieva ble th roughput of the ISH, IMH, MH, and HC protocols, respectively . The scaling exponents amo ng the abo ve schemes are compared according t o operating re gimes A– D shown in Fig. 2 ( ǫ is om itted for notational con venience). From the result of Theorem 2, no te that e ISH , e MH , and e HC are giv en by 1 + γ − (1 − β ) α 2 , 1 2 , and 2 − α 2 , respective ly , regardless of operating regimes. 1) Regime A ( 0 ≤ β + γ < 1 2 ): e IMH = β + γ is obtained. Since e MH > e IMH > e ISH , pure ad hoc transmissio ns with no BSs out perform the ISH and IMH protocols. Hence, the results in Regime A of T ABLE I are obtained. 2) Regime B ( β + γ ≥ 1 2 and β + 2 γ < 1 ): e IMH is the same as that under Regime A. Since e IMH > e ISH and e IMH ≥ e MH , the IMH always outperforms t he ISH and th e M H. Hence, it is found that the HC scheme h as the largest scaling exponent und er 2 < α < 4 − 2 β − 2 γ , but if α ≥ 4 − 2 β − 2 γ t he IMH protocol becomes t he best. 3) Regime C ( β + 2 γ ≥ 1 and γ < 1 2 ( β 2 − 3 β + 2) ): Remark that e IMH = 1+ β 2 and e IMH ≥ e MH . Then, the following inequalities with respect to the path-los s exponent α are found : e ISH > e IMH for 2 < α < 1 + 2 γ 1 − β and e ISH ≤ e IMH for α ≥ 1 + 2 γ 1 − β ; e HC > e IMH for 2 < α < 3 − β and e HC ≤ e IMH for α ≥ 3 − β ; and e HC > e ISH for 2 < α < 2(1 − γ ) β and e HC ≤ e ISH for α ≥ 2(1 − γ ) β . The best schem e thus depends on the comparison amo ng 1 + 2 γ 1 − β , 3 − β , and 2(1 − γ ) β . Note that 3 − β < 2(1 − γ ) β and 3 − β > 1 + 2 γ 1 − β alwa ys hold under Regime C. Finally , th e b est achie vable schemes with respect to α are obtained and are shown in Fig. 8(a) . 4) Regime D ( β + γ < 1 and γ ≥ 1 2 ( β 2 − 3 β + 2) ): The s ame scaling exponents for our four p rotocols are the same as those under Regime C. The result is obtained by comparing 1 + 2 γ 1 − β , 3 − β , and 2(1 − γ ) β under Regime D. The following t wo inequalit ies 3 − β ≥ 2(1 − γ ) β and 3 − β ≤ 1 + 2 γ 1 − β are satisfied, and the best achiev able schem es with respect t o α are o btained and shown in Fig. 8(b). This coincides w ith the result s hown in T ABLE I. B. Pr oof of Lemma 3 First consider the uplink case. There are 8 k interfering cells, ea ch of which includes Θ( n/m ) nodes whp, in the k -th layer l k of the network as illustrated in Fig. 9. Let d k denote the E uclidean distance between a giv en BS antenna and any node in l k , whi ch is a random variable. Since d k scales as Θ( k p n/m ) , there exists c 7 > c 6 > 0 with con stants c 6 and c 7 independent of n , such t hat d k = c 8 k p n/m , where all c 8 lies in the interval [ c 6 , c 7 ] . Hence, the total interference power P u I at each BS ant enna from simul taneously UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 17 transmittin g nodes is upper -bounded by P u I ≤ ∞ X k =1 P ( m/n ) α/ 2 − 1 (8 k ) n m  m ( c 6 k ) 2 n  α/ 2 = 8 P c α 6 ∞ X k =1 1 k α − 1 ≤ c 9 , where c 9 > 0 i s some constant independent of n . Now let us consider the interference in th e downlink. The in terfering signal recei ved by node i , which is in t he cell cov ered by BS s , from the simult aneously operating BSs s ′ ∈ { 1 , · · · , m } \ { s } is given by X s ′ ∈{ 1 , ··· ,m }\{ s } h d is ′   n/m X j =1 u s ′ j x s ′ j   , where u s ′ j denotes the j -th transmit precoding vector at BS s ′ normalized so th at its L 2 -norm is uni ty and x s ′ j is the j -th transmit p acket at BS s ′ . Since u s ′ j is represented by a functio n o f the d ownlink channel coef ficients between BS s ′ and nodes com municating with BS s ′ , th e terms  h d is ′  k · h P j u s ′ j x s ′ j i k are independent for all k ∈ { 1 , · · · , n/m } and s ′ ∈ { 1 , · · · , m } \ { s } . Using the fact above and the l ayering technique sim ilar to th e uplink case, an upper bound of the av erage total interference p ower P d I at each node in the downlink is ob tained as the following: P d I ≤ ∞ X k =1 ( n/m ) P ( m/n ) α/ 2 − 1 (8 k )  m ( c 6 k ) 2 n  α/ 2 ≤ c ′ 9 , where c ′ 9 > 0 is some constant independent of n . C. Pr oof of Lemma 5 Let X i denote the number of sources in the i -th cell and E x denote the ev ent t hat X i is between ((1 − δ 0 ) n/m, (1 + δ 0 ) n/m ) for all i ∈ { 1 , · · · , m } , where 0 < δ 0 < 1 is s ome const ant independent o f n . Then, we ha ve Pr { X k i < a for all i, k ∈ { 1 , · · · , m }} ≥ Pr {E x } Pr { X k i < a for all i, k |E x } ≥ Pr {E x }   1 − m 2 Pr    (1+ δ 0 ) n/m X j =1 B j ≥ a      , (32) where P j B j is the sum of (1 + δ 0 ) n/m independent and identi cally dis tributed (i.i.d.) Bernoulli random var iables with probability Pr { B j = 1 } = 1 m . Here, the second inequality holds since the union bound is applied over all i, k ∈ { 1 , · · · , m } . W e first consider the case where n/ m = ω ( m ) , i.e., 0 ≤ β < 1 / 2 . By setting a = (1 + δ 0 ) 2 n/m 2 , we then get Pr    (1+ δ 0 ) n/m X j =1 B j ≥ (1 + δ 0 ) 2 n m 2    = Pr n e s P (1+ δ 0 ) n/m j =1 B j ≥ e s (1+ δ 0 ) 2 n/m 2 o ≤ e − (1+ δ 0 ) n/m 2 ( s (1+ δ 0 ) − e s +1) , (33) UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 18 which is deriv ed from the steps similar to the proof of Lemma 4.1 in [3], where the first inequalit y comes from an appl ication of Chebyshe v’ s inequality . Hence, us ing (8), (32), and (33) yields Pr n X k i < (1 + δ 0 ) 2 n m 2 for all i, k ∈ { 1 , · · · , m } o ≥  1 − n β e − ∆( δ 0 ) n 1 − β   1 − m 2 e − (1+ δ 0 )∆( δ 0 ) n/m 2  =  1 − n β e − ∆( δ 0 ) n 1 − β   1 − e 2 β ln n − (1+ δ 0 )∆( δ 0 ) n 1 − 2 β  , where ∆( δ 0 ) = (1 + δ 0 ) ln(1 + δ 0 ) − δ 0 , by choosing s = ln(1 + δ 0 ) , which con verges t o one as n goes to infinity . When n/m = O ( m ) , i.e., 1 / 2 ≤ β < 1 , s etting a = ln n and s = (2 + δ 0 ) β and following the approach similar to t he first case, we obt ain Pr { X k i < ln n for all i, k ∈ { 1 , · · · , m }} ≥  1 − n β e − ∆( δ 0 ) n 1 − β   1 − m 2 e − (1+ δ 0 ) ( 1 − e (2+ δ 0 ) β ) n/m 2 − (2+ δ 0 ) β ln n  =  1 − n β e − ∆( δ 0 ) n 1 − β   1 − e − δ 0 β ln n − (1+ δ 0 ) ( 1 − e (2+ δ 0 ) β ) n 1 − 2 β  , which con ver ges to one as n goes to infinity . This completes the proo f. D. Pr oof of Lemma 6 This result can be obtained by slightl y modi fying the asymptot ic analysis in [3], [14]. The mi nimum node-to-node di stance is easily deriv ed by foll owing the s ame approach as that in [3] and i s proved to scale at least as 1 /n 1+ ǫ 1 with probability 1 − Θ(1 /n 2 ǫ 1 ) . W e no w focus on how the distance between a node and an antenna on the BS boundary scales. Consider a circle of radius 1 /n 1+ ǫ 1 around one specific antenna on the BS boundary . Not e that there are no other antennas insi de the circle since the per- antenna distance is greater t han 1 /n 1+ ǫ 1 . Let E d denote the e vent th at n no des are located outside t he circle giv en by t he antenna. Then, w e ha ve P {E C d } ≤ 1 −  1 − c 10 π n 2+2 ǫ 1  n , where 0 < c 10 < 1 is some constant independent of n . Hence, by the union bound, the probability that the e vent E d is satisfied for all the BS antennas is lower -bou nded by 1 − ml P {E C d } ≥ 1 − ml  1 −  1 − c 10 π n 2+2 ǫ 1  n  ≥ 1 − n  1 −  1 − c 10 π n 2+2 ǫ 1  n  , where the s econd inequality holds since ml = O ( n ) , which tends to one as n goes to infinit y . This completes the proo f. E. Pr oof of Lemma 8 The size of matrix F (3) L is Θ( n ) × Θ ( n ) since ml = O ( n ) . Th us, the analys is essentially follows t he ar gument in [3] with a slight modi fication (see Appendix III in [3] for more precise description). Consider the network transformati on resulting i n a regular network wit h at m ost log n nodes at each square verte x except for the area cov ered by BSs. Then, the same nod e displ acement as shown i n [3] is performed, which will decrease the Eucli dean d istance between source and destin ation nod es. For con venience, the source node positions are in dexed in the resulting regular network. It is th us assu med that t he source nodes under the cut are located at p ositions ( − i x + 1 , i y ) where i x , i y = 1 , · · · √ n . In the following, P k ∈ D (3) L    h F (3) L i k i    2 and an upper bound for P i ∈ S L    h F (3) L i k i    2 are deriv ed: X k ∈ D (3) L    h F (3) L i k i    2 = 1 UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 19 and X i ∈ S L    h F (3) L i k i    2 = X i ∈ S L       1 q d (3) L,i h H (3) L i k i       2 =      P i ∈ S L r − α ki d (3) L,i if k ∈ ¯ D r \ D (1) L P i ∈ S L r u − α si,t d (3) L,i if k ∈ { t : t ∈ [1 , l ] for s ∈ B l } ≤            P i ∈ S L r − α ki P k ∈ ¯ D r \ D (1) L r − α ki if k ∈ ¯ D r \ D (1) L P i ∈ S L r u − α si,t P k ∈ ¯ D r \ D (1) L r − α ki if k ∈ { t : t ∈ [1 , l ] for s ∈ B l } ≤      c 11 log n P i ∈ S L x α − 2 i r − α k i if k ∈ ¯ D r \ D (1) L c 11 log n P i ∈ S L x α − 2 i r u − α si,t if k ∈ { t : t ∈ [1 , l ] for s ∈ B l } ≤      c 11 log n P i ∈ S L r − 2 k i if k ∈ ¯ D r \ D (1) L c 11 log n P i ∈ S L r u − 2 si,t if k ∈ { t : t ∈ [1 , l ] for s ∈ B l } ≤ c 11 (log n ) 2 √ n X i x ,i y =1 1 i 2 x + i 2 y ≤ c 12 (log n ) 3 , where ¯ D r is the set of nodes includ ing BS antennas on th e ri ght half, B l is th e s et of BSs i n the left half network, c 11 and c 12 are some positive constants independent of n , and x i denotes the x -coordinate of node i ∈ S L for o ur random network ( x i = 1 , · · · , √ n ). Here, the second and fifth inequalit ies hold since X k ∈ ¯ D r \ D (1) L r − α k i ≥ x 2 − α i c 11 log n and √ n X i x ,i y =1 1 i 2 x + i 2 y = O (lo g n ) , respectiv ely (see App endix III in [3] for th e detailed deriv ation). The fourth inequali ty comes from the result of Lemm a 7. Hence, i t is proved that b oth s caling result s are the same as the random network case shown i n [3]. Now it is possible to prove the inequality in (23) by following the same lin e as th at in Appendix III of [3], which resul ts in E h tr  F (3) L † F (3) L  q i ≤ C q n ( c 13 log n ) 3 q , where C q = (2 q )! q !( q +1)! is the Catalan nu mber for any q and c 13 > 0 is some constant independent of n . Then, from the property k F (3) L k 2 2 = lim q →∞ tr(( F (3) L † F (3) L ) q ) 1 /q (see [39]), the expectation of t he term k F (3) L k 2 2 is finally giv en by (23), which completes t he proof. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 20 F . Pr oof of Lemma 10 When l = o ( p n/m ) , there is no destin ation in D (5) L , and thus P i ∈ S L P d (5) L,i becomes zero. Hence, the case for l = Ω( p n/m ) is the focus from now o n. By the same argument as shown in the deri vation of P i ∈ S L P d (4) L,i , t he net work area is divided into n squares of unit area. Then, by Lemma 7, the power transfer und er our random network can be u pper- bounded b y that under a regular network with at most log n nod es at each square except for the area covered by BSs. As i llustrated in Fig. 10, the nodes in each square are moved together ont o one vertex o f th e correspondi ng square. The node displacement is performed in a sense of decreasing the Euclidean distance between node i ∈ S L and the antennas of the corresponding BS, thereby p roviding an upper bound for d (5) L,i . Layers of each cell are then introduced, as shown in Fig. 10, where t here exist 8( ǫ 0 p n/m + k ) vertices, each of which in cludes log n nodes, in the k -th layer l ′ k of each cell. The regular network described above can also be transformed i nto the other , which contains antennas regularly placed at s pacing ǫ 0 p nπ ml outside the shaded squ are for an arbi trarily small ǫ 0 > 0 . N ote that the s haded s quare of size 2 k × 2 k is drawn based on a s ource node i n l ′ k at its center (see Fig. 10). The modification yields an increase of the term d (5) L,i by source i . When d (5) L,i ( k ) is defined as d (5) L,i by node i that lies in l ′ k , the following up per bound for d (5) L,i ( k ) is obtained: d (5) L,i ( k ) ≤ ∞ X k x ,k y = ζ 1   ǫ 0 p nπ ml k x ) 2 + ( ǫ 0 p nπ ml k y  2  α/ 2 =  ml n  α/ 2 ∞ X k x ,k y = ζ η α ( k 2 x + k 2 y ) α/ 2 ≤  ml n  α/ 2 ∞ X k ′ = ζ 8 η α k ′ k ′ α ≤ c 14  ml n  α/ 2  1 ζ α − 1 + Z ∞ ζ 1 x α − 1 dx  = c 14  1 ζ + 1 α − 2   ml n  α/ 2 1 ζ α − 2 , where ζ = 1 + ⌊ k η ⌋ , η = 1 ǫ 0 q ml nπ , and c 14 is s ome constant independent of n . Here, ⌊ x ⌋ denotes the greatest integer less than or equal to x . Hence, d (5) L,i ( k ) is gi ven by d (5) L,i ( k ) = ( O   ml n  α/ 2  if k = O  p n ml  O  k 2 − α  ml n  if k = Ω  p n ml  , UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 21 finally yielding X i ∈ S L P d (5) L,i ≤ P m 2 log n √ n/m X k =1 8  ǫ 0 r n m + k  d (5) L,i ( k ) ≤ c 15 P √ nm log n    √ n/ml − 1 X k =1  ml n  α/ 2 + √ n/m X k = √ n/ml k 2 − α  ml n     ≤ c 15 P √ nm log n "  ml n  ( α − 1) / 2 +  ml n   ml n  α/ 2 − 1 + Z √ n/m √ n/ml 1 k α − 2 dx !# ≤      3 c 15 P 3 − α nl  m n  α/ 2 log n if 2 < α < 3 3 c 15 P 2 ml p m n (log n ) 2 if α = 3 3 c 15 P α − 3 n √ l  ml n  α/ 2 log n if α > 3 , (34) where c 15 is som e constant independent of n . Here, the first inequalit y holds si nce th ere exist 8( ǫ 0 p n/m + k ) vertices in l ′ k and at most log n nodes at each vertex. Equation (34) yi elds th e result in (28), which completes the proo f. R E F E R E N C E S [1] P . Gupta and P . R. 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K.: C ambridge Uni versity Press, 1999. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 23 Fig. 1. The wireless ad hoc network w ith infrastructure support. Fig. 2. Operating regimes on the ach iev able throughput scaling wi th respect t o β and γ . T ABLE I A C H I E V A B L E R A T E S F O R A N E X T E N D E D N E T W O R K W I T H I N F R A S T RU C T U R E . Regime Condition Scheme e ( α, β , γ ) 2 < α < 3 HC 2 − α 2 A α ≥ 3 MH 1 2 2 < α < 4 − 2 β − 2 γ HC 2 − α 2 B α ≥ 4 − 2 β − 2 γ IMH β + γ 2 < α < 3 − β HC 2 − α 2 C α ≥ 3 − β IMH 1+ β 2 2 < α < 2(1 − γ ) β HC 2 − α 2 D 2(1 − γ ) β ≤ α < 1 + 2 γ 1 − β ISH 1 + γ − α (1 − β ) 2 α ≥ 1 + 2 γ 1 − β IMH 1+ β 2 UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 24 Fig. 3. The infrastructure-suppo rted single-hop (IS H) protocol. Fig. 4. The infrastructure-suppo rted multi-hop (IMH) protocol. Fig. 5. The access routing in the IMH protocol. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 25 Fig. 6. The cut L in the two-dimensiona l rando m network. Fig. 7. The partit ion of destinations in the two-dimen sional random network. T o simplify the figure, one BS is sho wn in the left half network. (a) (b) Fig. 8. The best achie v able schemes with respect to α . (a) The Re gime C. (b) The Regime D. UNDER REVISION FOR IEEE TRANSACTIONS ON INFORMA TION THEO R Y 26 Fig. 9. Grouping of interfering cells. The first layer l 1 of the network represents the outer 8 shaded cells. Fig. 10. The displacement of the nodes to square vertices. The antennas are regularly placed at spacing p n ml outside the shaded square.