On Capacity Scaling in Arbitrary Wireless Networks


 Allan Newman
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1 On Capacity Scaing in Arbitrary Wireess Networks Urs Niesen, Piyush Gupta, and Devavrat Shah 1 Abstract arxiv: v3 [csit] 3 Aug 2009 In recent work, Özgür, Lévêque, and Tse 2007) obtained a compete scaing characterization of throughput scaing for random extended wireess networks ie, n nodes are paced uniformy at random in a square region of area n) They showed that for sma pathoss exponents α 2, 3] cooperative communication is order optima, and for arge pathoss exponents α > 3 mutihop communication is order optima However, their resuts both the communication scheme and the proof technique) are strongy dependent on the reguarity induced with high probabiity by the random node pacement In this paper, we consider the probem of characterizing the throughput scaing in extended wireess networks with arbitrary node pacement As a main resut, we propose a more genera nove cooperative communication scheme that works for arbitrariy paced nodes For sma pathoss exponents α 2, 3], we show that our scheme is order optima for a node pacements, and achieves exacty the same throughput scaing as in Özgür et a This shows that the reguarity of the node pacement does not affect the scaing of the achievabe rates for α 2, 3] The situation is, however, markedy different for arge pathoss exponents α > 3 We show that in this regime the scaing of the achievabe pernode rates depends cruciay on the reguarity of the node pacement We then present a famiy of schemes that smoothy interpoate between mutihop and cooperative communication, depending upon the eve of reguarity in the node pacement We estabish order optimaity of these schemes under adversaria node pacement for α > 3 Index Terms Arbitrary node pacement, capacity scaing, cooperative communication, hierarchica reaying, mutihop communication, wireess networks I INTRODUCTION Consider a wireess network with n nodes paced on [0, n] 2 usuay referred to as an extended network), with each node being the source for one of n sourcedestination pairs and the destination for another pair The performance of this network is captured by ρ n), the argest uniformy achievabe rate of communication between these sourcedestination pairs Whie the scaing behavior of ρ n) as the number of nodes n goes to infinity is by now we understood for random node pacement, itte is known for the case of arbitrary node pacements In this paper, we are interested in anayzing the impact of such arbitrary node pacement on the scaing of ρ n) A Reated Work The probem of determining the scaing of ρ n) was first anayzed by Gupta and Kumar in [1] They show that, under random pacement of nodes in the region, certain modes of communication motivated by current technoogy, and random sourcedestination pairing, the maximum achievabe pernode rate ρ n) can scae at most as On 1/2 ) Moreover, it was shown that mutihop communication can achieve essentiay the same order of scaing Since [1], the probem has received a considerabe amount of attention One stream of work [2] [8] has progressivey broadened the conditions on the channe mode and the communication mode, under which U Niesen and D Shah are with the Laboratory of Information and Decision Systems, Department of EECS at the Massachusetts Institute of Technoogy Emai: P Gupta is with the Mathematics of Networks and Communications Research Department, Be Labs, AcateLucent Emai: The work of U Niesen and D Shah was supported in parts by DARPA grant ITMANET) C and NSF grant CNS ; the work of P Gupta was supported in part by NSF Grants CCR and CNS
2 2 mutihop communication is order optima Specificay, with a power oss of r α for signas sent over distance r, it has been estabished that under high signa attenuation α > 3 and random node pacement, the best achievabe pernode rate ρ n) for random sourcedestination pairing scaes essentiay ike Θn 1/2 ) and that this scaing is achievabe with mutihop communication Another stream of work [8] [12] has proposed progressivey refined mutiuser cooperative schemes, which have been shown to significanty outperform mutihop communication in certain environments In an exciting recent work, Özgür et a [8] have shown that with nodes paced uniformy at random, and with ow signa attenuation α 2, 3], a cooperative communication scheme can perform significanty better than mutihop communication More precisey, they show that for α 2, 3], the best achievabe pernode rate for random sourcedestination pairing scaes as ρ n) = On 1 α/2+ε ) and cooperative communication achieves a pernode rate of Ωn 1 α/2 ε ) here, ε > 0 is an arbitrary but fixed constant) That is, cooperative communication is essentiay order optima in the attenuation regime α 2, 3] In summary, for random extended networks with random sourcedestination pairing, the optima communication scheme exhibits the foowing threshod behavior: for α 2, 3] the cooperative communication scheme is order optima, whie for α > 3 the mutihop communication scheme is order optima B Our Contributions The characterization of the scaing of ρ n) as a function of the pathoss exponent α mentioned in the ast paragraph depends criticay on the reguarity induced with high probabiity by pacing the nodes uniformy at random However, a wireess network encountered in practice might not exhibit this amount of reguarity Our interest is therefore in understanding the impact of the node pacement on the scaing of ρ n) To this end, we consider wireess networks with arbitrary ie, deterministic) node pacement with minimumseparation constraint) The impact of this arbitrary node pacement depends cruciay on the pathoss exponent α For sma pathoss exponents α 2, 3], we show that for random sourcedestination pairing, the rate of the best communication scheme is upper bounded as ρ n) = Oog 6 n)n 1 α/2 ) We then present a nove cooperative communication scheme that achieves for any pathoss exponent α > 2 a pernode rate of ρ HR n) n 1 α/2 o1) Thus, our cooperative communication scheme is essentiay order optima for any such arbitrary network with α 2, 3] In other words, in the sma pathoss regime, the scaing of ρ n) is the same irrespective of the reguarity of the node pacement The situation is, however, quite different for arge pathoss exponents α > 3 We show that in this regime the scaing of ρ n) depends cruciay on the reguarity of the node pacement, and mutihop communication may not be order optima for any vaue of α In fact, for ess reguar networks we need more compicated cooperative communication schemes to achieve optima network performance Towards that end, we present a famiy of communication schemes that smoothy interpoate between cooperative communication and mutihop communication, and in which nodes communicate at scaes that vary smoothy from oca to goba The amount of interpoation between the cooperative and mutihop schemes depends on the eve of reguarity of the underying node pacement We estabish the optimaity of this famiy of schemes for a α > 3 under adversaria node pacement In summary, for α 2, 3] the reguarity of the node pacement has no impact on the scaing of ρ n) Cooperative communication is order optima in this regime and achieves the same scaing as in the case of random node pacement For α > 3 the reguarity of the node pacement strongy impacts the scaing of ρ n), and a communication scheme interpoating between mutihop and cooperative communication depending on the reguarity of the node pacement is order optima under adversaria node pacement) In particuar, simpe mutihop communication may not be order optima for any α > 3 This contrasts with the case of random node pacement where mutihop communication is order optima for a α > 3 C Organization The remainder of this paper is organized as foows Section II describes in detai the communication mode Section III provides forma statements of our resuts Sections IV and V describe our new
3 3 cooperative communication scheme for the α 2, 3] regime) and interpoation scheme for the α > 3 regime) for arbitrary wireess networks Sections VI through XI contain proofs Finay, Sections XII and XIII contain discussions and concuding remarks II MODEL In this section, we introduce some notationa conventions and describe in detai the network and channe modes We use the foowing conventions: K i for different i denote stricty positive finite constants independent of n Vectors and matrices are denoted by bodface whenever the vector or matrix structure is of importance We denote by ) T and ) transpose and conjugate transpose, respectivey To simpify notation, we assume, when necessary, that fractions are integers and omit and operators Consider the square An) [0, n] 2 of area n, and et V n) An) be a set of V n) = n nodes on 1 An) We say that V n) has minimumseparation r min if r u,v r min for a u, v V n), where r u,v is the Eucidean distance between nodes u and v We use the same channe mode as in [8] Namey, the samped) received signa at node v is y v [t] = h u,v [t]x u [t] + z v [t] 1) u V n)\{v} for a v V n), and where {x u [t]} u,t are the samped) signas sent by the nodes in V n) Here {z v [t]} v,t are independent and identicay distributed iid) with distribution N C 0, 1) ie, circuary symmetric compex Gaussian with mean 0 and variance 1), and h u,v [t] = ru,v α/2 exp 1θ u,v [t]), for pathoss exponent α > 2 We assume that for each t N, the phases {θ u,v [t]} u,v are iid 2 with uniform distribution on [0, 2π) We either assume that for each u, v V n) the random process {θ u,v [t]} t is stationary ergodic in t, which is caed fast fading in the foowing, or that for each u, v V n) the random process {θ u,v [t]} t is constant in t, which is caed sow fading in the foowing In either case, we assume fu channe state information CSI) is avaiabe at a nodes, ie, each node knows a {θ u,v [t]} u,v at time t Whie the fu CSI assumption is quite strong, it can be shown that avaiabiity of a 2bit quantized version of {θ u,v [t]} u,v at a nodes is sufficient for the achievabe schemes presented here see Section XIIA for the detais) We aso impose an average power constraint of 1 on the signa {x u [t]} t for every node u V n) Each node u V n) wants to transmit information at uniform rate ρn) to some other node w V n) We ca u the source and w the destination node of this communication pair The set of a communication pairs can be described by a traffic matrix λn) {0, 1} n n, where the entry in λn) corresponding to u, w) is equa to 1 if node u is a source for node w We say that λn) is a permutation traffic matrix if it is a permutation matrix ie, every node is a source for exacty one communication pair and a destination for exacty one communication pair) For a traffic matrix λn), et ρ n) be the highest rate of communication that is uniformy achievabe for each sourcedestination pair For a permutation traffic matrix λn), ρ n) can aso be understood as the maxima achievabe pernode rate 1 The setting considered here with n nodes paced on a square of area n is caed an extended network If the n nodes are paced on a square of unit area, we speak of a dense network Whie dense networks are not treated in detai in this paper, we briefy discuss impications of the resuts for the dense setting in Section XIIC 2 It is worth pointing out that recent work [13] suggests that, under certain assumptions on scattering eements, for α 2,3), and for very arge vaues of n, the iid phase assumption as a function of u, v V n) used here is too optimistic However, subsequent work by the same authors [14] shows that under different assumptions on the scatterers, the channe mode used here is sti vaid even for α 2,3), and for very arge vaues of n This indicates that the question of channe modeing for very arge networks in the ow pathoss regime is somewhat deicate and requires further investigation We point out that for α 3 this issue does not arise
4 4 III MAIN RESULTS This section presents the forma statement of our resuts The resuts are divided into two parts In Section IIIA, we consider ow pathoss exponents, ie, α 2, 3] We present a cooperative communication scheme for arbitrary node pacement and for either fast or sow fading We show that this communication scheme is order optima for a node pacements when α 2, 3] In Section IIIB, we consider high pathoss exponents, ie, α > 3 We present a communication scheme that interpoates between the cooperative and the mutihop communication schemes, depending on the reguarity of the node pacement We show that this communication scheme is order optima under adversaria node pacement with reguarity constraint when α > 3 A Low Path Loss Regime α 2, 3] The first resut proposes a nove communication scheme, caed hierarchica reaying in the foowing, and bounds the pernode rate ρ HR n) that it achieves This provides a ower bound to ρ n), the argest achievabe pernode rate The hierarchica reaying scheme enabes cooperative communication on the scae of the network size In the random node pacement case, this cooperation coud be enabed in a custer around the source node cooperativey transmitting) and in a custer around its destination node cooperativey receiving) With arbitrary node pacement, such an approach does no onger work, as both the source as we as the destination nodes may be isoated The hierarchica reaying scheme circumvents this issue by reaying data between each sourcedestination pair over a densey popuated region in the network A detaied description of this scheme is provided in Section IV, the proof of Theorem 1 is contained in Section VII Theorem 1 Under fast fading, for any α > 2, r min 0, 1), and δ 0, 1/2), there exists ) b 1 n) n O og δ 1/2 n) such that for any n, node pacement V n) with minimum separation r min, and permutation traffic matrix λn), we have ρ n) ρ HR n) b 1 n)n 1 α/2 The same concusion hods for sow fading with probabiity at east )) 1 exp 2 Ω og 1/2+δ n) = 1 o1) as n Theorem 1 shows that the pernode rate ρ HR n) achievabe by the hierarchica reaying scheme is at east n 1 α/2 βn), where the oss term βn) converges to zero as n at a rate arbitrariy cose to O og 1/2 n) ) by choosing δ sma) The performance of the hierarchica reaying scheme can intuitivey be understood as foows As mentioned before, the scheme achieves cooperation on a goba scae This eads to a mutiantenna gain of order n On the other hand, communication is over a distance of order n 1/2, eading to a power oss of order n α/2 Combining these two factors resuts in a pernode rate of n 1 α/2 We note that Theorem 1 remains vaid under somewhat weaker conditions than having minimum separation r min 0, 1) Specificay, we show that the resut of Özgür et a [8] can be recovered through Theorem 1 as the random node pacement satisfies these weaker conditions We discuss this in more detai in Section XIID The next theorem estabishes optimaity of the hierarchica reaying scheme in the range of α 2, 3] for arbitrary node pacement The proof of the theorem is presented in Section VIII
5 5 Theorem 2 Under either fast or sow fading, for any α 2, 3], r min 0, 1), there exists b 2 n) = O og 6 n) ) such that for any n, node pacement V n) with minimum separation r min, and for λn) chosen uniformy at random from the set of a permutation traffic matrices, we have with probabiity 1 o1) as n ρ n) b 2 n)n 1 α/2 Note that Theorem 2 hods ony with probabiity 1 o1) for different reasons for the sow and fast fading case For fast fading, this is due to the randomness in the seection of the permutation traffic matrix In other words, for fast fading, with high probabiity we seect a traffic matrix for which the theorem hods For the sow fading case, there is additiona randomness due to the fading reaization Here, with high probabiity we seect a traffic matrix and we experience a fading for which the theorem hod Comparing Theorems 1 and 2, we see that for α 2, 3] the proposed hierarchica reaying scheme is order optima, in the sense that ogρ HR n)) im n ogn) ogρ n)) = im = 1 α/2 n ogn) Moreover, the rate it achieves is the same order as is achievabe in the case of randomy paced nodes Hence in the ow pathoss regime α 2, 3], the heterogeneity caused by the arbitrary node pacement has no effect on achievabe communication rates B High Path Loss Regime α > 3 We now turn to the high pathoss regime α > 3 In the case of randomy paced nodes, mutihop communication achieves a pernode rate of ρ MH n) = Ωn 1/2 ) with probabiity 1 o1) and is order optima for α > 3 For arbitrariy paced nodes, the situation is quite different as Theorem 3 shows The proof of Theorem 3 is contained in Section IX Theorem 3 Under either fast or sow fading, for any α > 3, for any n, there exists a node pacement V n) with minimum separation 1/2 such that for λn) chosen uniformy at random from the set of a permutation traffic matrices, we have as n with probabiity 1 o1) ρ n) 2 2+5α n 1 α/2, ρ MH n) 4 α n α/2, Comparing Theorem 3 with Theorem 1 shows that under adversaria node pacement with minimumseparation constraint the hierarchica reaying scheme is order optima even when α > 3 Moreover, Theorem 3 shows that there exist node pacements satisfying a minimum separation constraint for which hierarchica reaying achieves a rate of at east a factor of order n higher than mutihop communication for any α > 3 In other words, for those node pacements cooperative communication is necessary for order optimaity aso for any α > 3, in stark contrast to the situation with random node pacement, where mutihop communication is order optima for a α > 3 Theorem 3 suggests that it is the eve of reguarity of the node pacement that decides what scheme to choose for pathoss exponent α > 3 So far, we have seen two extreme cases: For random node pacement, resuting in very reguar node pacements with high probabiity, ony oca cooperation is necessary and mutihop is an orderoptima communication scheme For adversaria arbitrary node pacement, resuting in a very irreguar node pacement, goba cooperation is necessary and hierarchica reaying is an orderoptima communication scheme We now make this notion of reguarity precise, and show that, depending on the reguarity of the node pacement, an appropriate interpoation between mutihop and hierarchica reaying is required for α > 3 to achieve the optima performance We refer to this interpoation scheme as cooperative mutihop communication in the foowing
6 6 Before we state the resut, we need to introduce some notation Consider again a node pacement V n) An) with minimum separation r min 0, 1) Divide An) into squares of sideength dn) n, and fix a constant µ 0, 1] We say that V n) is µreguar at resoution dn) if every such square contains at east µd 2 n) nodes Note that every node pacement is triviay 1reguar at resoution n; a random node pacement can be shown to be µreguar at resoution ogn) with probabiity 1 o1) as n for any µ < 1; and nodes that are paced on each point in the integer attice inside An) are 1reguar at resoution 1 The cooperative mutihop scheme enabes cooperative communication on the scae of reguarity dn) Neighboring squares of sideength dn) cooperativey communicate with each other To transmit between a source and its destination, we use mutihop communication over those squares In other words, we use cooperative communication at sma scae dn), and mutihop communication at arge scae n For reguar node pacements, ie, dn) = 1, the cooperative mutihop scheme becomes the cassica mutihop scheme For very irreguar node pacement, ie, dn) = n 1/2, the cooperative mutihop scheme becomes the hierarchica reaying scheme discussed in the ast section The next theorem provides a ower bound on the pernode rate ρ CMH n) achievabe with the cooperative mutihop scheme The proof of the theorem can be found in Section X Theorem 4 Under fast fading, for any α > 2, r min 0, 1), µ 0, 1), and δ 0, 1/2) there exists ) b 3 n) n O og δ 1/2 n) such that for any n, node pacement V n) with minimum separation r min, and permutation traffic matrix λn), we have ρ n) ρ CMH n) b 3 n)d 3 α n)n 1/2, where d n) min{h : V n) is µ reguar at resoution h} The same concusion hods for sow fading with probabiity 1 o1) as n Theorem 4 shows that if V n) is reguar at resoution d n) then a pernode rate of at east ρ CMH n) d 3 α n)n 1/2 βn) is achievabe, where, as before, the oss term βn) converges to zero as n at a rate arbitrariy cose to O og 1/2 n) ) The performance of the cooperative mutihop scheme can intuitivey be understood as foows The scheme achieves cooperation on a scae of d 2 n) This eads to a mutiantenna gain of order d 2 n) On the other hand, communication is over a distance of order dn), eading to a power oss of order d α n) Moreover, each sourcedestination pair at a distance of order n 1/2 must transmit their data over order n 1/2 d 1 n) many hops, eading to a mutihop oss of n 1/2 dn) Combining these three factors resuts in a pernode rate of d 3 α n)n 1/2 The next theorem shows that Theorem 4 is tight under adversaria node pacement under a constraint on the reguarity The proof of the theorem is presented in Section XI Theorem 5 Under either fast or sow fading, for any α > 3, there exists b 4 n) = O og 6 n) ), such that for any n, and d n), there exists a node pacement V n) with minimum separation 1/2 and 1/2reguar at resoution d n) such that for λn) chosen uniformy at random from the set of a permutation traffic matrices, we have ρ n) b 4 n)d 3 α n)n 1/2, with probabiity 1 o1) as n As an exampe, assume that d n) = n η for some η 0 Then Theorem 4 shows that for any node pacement of reguarity d n) and α > 3, ρ CMH n) n 3 α)η 1/2 βn),
7 7 where βn) converges to zero as n at a rate arbitrariy cose to O og 1/2 n) ) In other words ogρ CMH n)) im 3 α)η 1/2 n ogn) Moreover, by Theorem 5 there exist node pacements with same reguarity such that for random permutation traffic with high probabiity ρ n) is essentiay) of the same order, in the sense that ogρ n)) im 3 α)η 1/2 n ogn) In particuar, for η = 0 ie, reguar node pacement), and for η = og ogn)/ ogn) ie, random node pacement), we obtain the order n 1/2 scaing as expected For η = 1/2 ie, competey irreguar node pacement), we obtain the order n 1 α/2 scaing as in Theorems 1 and 3 IV HIERARCHICAL RELAYING SCHEME This section describes the architecture of our hierarchica reaying scheme On a high eve, the construction of this scheme is as foows Consider n nodes V n) paced arbitrariy on the square region An) with a minimum separation r min Divide An) into squareets of equa size Ca a squareet dense, if it contains a number of nodes proportiona to its area For each sourcedestination pair, choose such a dense squareet as a reay, over which it wi transmit information see Figure 1) u 1 MAC u 2 BC u 3 w 1 w 2 w 3 Fig 1 Sketch of one eve of the hierarchica reaying scheme Here {u i, w i)} 3 i=1 are three sourcedestination pairs Groups of sourcedestination pairs reay their traffic over dense squareets, which contain a number of nodes proportiona to their area shaded) We time share between the different dense squareets used as reays Within a these reay squareets the scheme is used recursivey to enabe joint decoding and encoding at each reay Consider now one such reay squareet and the nodes that are transmitting information over it If we assume for the moment that a the nodes within the same reay squareet coud cooperate then we woud have a mutipe access channe MAC) between the source nodes and the reay squareet, where each of the source nodes has one transmit antenna, and the reay squareet acting as one node) has many receive antennas Between the reay squareet and the destination nodes, we woud have a broadcast channe BC), where each destination node has one receive antenna, and the reay squareet acting again as one node) has many transmit antennas The cooperation gain from using this kind of scheme arises from the use of mutipe antennas for these mutipe access and broadcast channes To actuay enabe this kind of cooperation at the reay squareet, oca communication within the reay squareets is necessary It can be shown that this oca communication probem is actuay the same as the origina probem, but at a smaer scae Hence we can use the same scheme recursivey to sove this
8 8 subprobem We terminate the recursion after severa iterations, at which point we use simpe TDMA to bootstrap the scheme The construction of the hierarchica reaying scheme is presented in detai in Section IVA A backoftheenveope cacuation of the pernode rate it achieves is presented in Section IVB A detaied anaysis of the hierarchica reaying scheme is presented in Sections VI and VII A Construction Reca that Ab) [0, b] 2 is the square region of area b The scheme described here assumes that n nodes are paced arbitrariy in An) with minimum separation r min 0, 1) We want to find some rate, say ρ 0, that can be supported for a n sourcedestination pairs of a given permutation traffic matrix λn) The scheme that is described beow is recursive and hence hierarchica) in the foowing sense In order to achieve rate ρ 0 for n nodes in An), it wi use as a buiding bock a scheme for supporting rate ρ 1 for a network of n 1 n 2γn) nodes over Aa 1 ) square of area a 1 ) with a 1 n γn) for any permutation traffic matrix λn 1 ) of n 1 nodes Here the branching factor γn) is a function such that γn) as n We wi optimize over the choice of γn) ater The same construction is used for the scheme over Aa 1 ), and so on In genera, our scheme does the foowing at eve 0 of the hierarchy or recursion) In order to achieve rate ρ for any permutation traffic matrix λn ) over n n 2 γ n) nodes in Aa ), with a n γ n), use a scheme achieving rate ρ +1 over n +1 nodes in Aa +1 ) for any permutation traffic matrix λn +1 ) The recursion is terminated at some eve Ln) to be chosen ater We now describe how the hierarchy is constructed between eves and + 1 for 0 < Ln) Each sourcedestination pair chooses some squareet as a reay over which it transmits its message This reaying of messages takes pace in two phases a mutipe access phase and a broadcast phase We first describe the seection of reay squareets, then the operation of the network during the mutipe access and broadcast phases, and finay the termination of the hierarchica construction 1) Setting up Reays: Given n nodes in Aa ), divide the square region Aa ) into γn) equa sized squareets Denote them by {A k a +1 )} γn) k=1 Ca a squareet dense if it contains at east n /2γn) = n +1 nodes In other words, a dense squareet contains a number of nodes of at east a 1/2 +1 fraction of its area We show that since the nodes in Aa ) have constant minimum separation r min, a squareet can contain at most Oa +1 ) ie Oa /γn))) nodes, and hence that there are at east Θ2 γn)) dense squareets Each sourcedestination pair chooses a dense squareet such that both the source and the destination are at a distance Ω a +1 ) from it We ca this dense squareet the reay of this sourcedestination pair We show that the reays can be chosen such that each reay squareet has at most n +1 communication pairs that use it as reay, and we assume this worst case in the foowing discussion
9 9 2) Mutipe Access Phase: Source nodes that are assigned to the same dense) reay squareet send their messages simutaneousy to that reay We time share between the Θ2 γn)) different reay squareets If the nodes in the reay squareet coud cooperate, we woud be deaing with a MAC with at most n +1 transmitters, each with one antenna, and one receiver with at east n +1 antennas In order to achieve this cooperation, we use a hierarchica ie, recursive) construction For this recursive construction, assume that we have access to a communication scheme to transmit data according to a permutation traffic matrix λn +1 ) between n +1 nodes ocated in a square of area a +1 We now show how this scheme at scae a +1 can be used to construct a scheme for scae a see Figure 2) u 1 x 11 x 1m y 11 y 1m ŷ 11 q 1 ŷ 1m ŷ 11 ŷ m1 ˆx 11 ˆx m1 v 1 P y x {λ k } m k=1 u m x m1 x mm y m1 y mm q m ŷ m1 ŷ mm ŷ 1m ŷ mm ˆx 1m ˆx mm v m Fig 2 Description of the mutipe access phase at eve in the hierarchy with m n +1 The first system bock represents the wireess channe, connecting source nodes {u i} n +1 i=1 with reay nodes {v i} n +1 i=1 The second system bock are quantizers {qi}n +1 i=1 used at the reay nodes The third system bock represents using n +1 times the communication scheme at eve + 1 organized as n +1 permutation traffic matrices {λ k n +1 )} n +1 k=1 ) to transpose the matrix of quantized observations {ŷij}n +1 i,j=1 In other words, before the third system bock, node v 1 has access to {ŷ 1j} n +1 j=1, and after the third system bock, node v1 has access to {ŷi1}n +1 i=1 The fourth system bock are matched fiters used at the reay nodes Suppose there are n +1 source nodes u 1,,u n+1 ocated anywhere in Aa )) that reay their message over the n +1 reay nodes v 1,,v n+1 ocated in the same dense squareet of area a +1 ) Each source node u i divides its message bits into n +1 parts of equa ength Denote by x ij the encoded part j of the message bits of node u i x ij is reay a arge sequence of channe symbos; to simpify the exposition, we sha, however, assume it is ony a singe symbo) The message parts corresponding to {x ij } n +1 i=1 wi be reayed over node v j, as wi become cear in the foowing Sources {u i } n +1 i=1, transmit {x ij} n +1 i=1 at time j for j {1, n +1 } Let y kj be the observed channe output at reay v k at time j Note that y kj depends ony on channe inputs {x ij } n +1 i=1 In order to decode the message parts corresponding to {x ij} n +1 i=1 at reay node v j, it needs to obtain the observations {y ij } n +1 i=1 from a other reay nodes In other words, a reays need to exchange information For this, each reay v k quantizes its observation {y kj } n +1 j=1 at an appropriate rate K independent of n to obtain {ŷ kj } n +1 j=1 Quantized observation ŷ kj is to be sent from reay v k to reay v j Thus, each of the n +1 reay nodes now has a message of size K for every other reay node This communication demand within the reay squareet can be organized as n +1 permutation traffic matrices {λ j n +1 )} n +1 j=1 between the n +1 reay nodes Note that these reay nodes are ocated in the same square of area a +1 In other words, we are now faced with the origina probem, but at smaer scae a +1 Therefore, using n +1 times the assumed scheme for transmitting according to a permutation traffic matrix for n +1 nodes in Aa +1 ), reay v j can obtain a quantized observations {ŷ ij } n +1 i=1 Now v j uses n +1 matched fiters on {ŷ ij } n +1 i=1 to obtain estimates {ˆx ij } n +1 i=1 of {x ij } n +1 i=1 In other words, each node v j computes 3 n +1 h u ˆxij = i,v k [j] k h u i,v k [j] 2ŷkj k=1 for every i {1,, n +1 } Using these estimates it then decodes the messages corresponding to {x ij } n +1 i=1 3 Note that, since we assume fu CSI, node v j has access to the channe gains {h ui,v k [j]} i,k at any time t j In particuar, this is the case at the time the matched fitering is performed
10 10 3) Broadcast Phase: Nodes in the same reay squareet then send their decoded messages simutaneousy to the destination nodes corresponding to this reay We time share between the different reay squareets If the nodes in the reay squareet coud cooperate, we woud be deaing with a BC with one transmitter with at east n +1 antennas and with at most n +1 receivers, each with one antenna In order to achieve this cooperation, a simiar hierarchica construction as for the MAC phase is used As in the MAC phase, assume that we have access to a scheme to transmit data according to a permutation traffic matrix λn +1 ) between n +1 nodes ocated in a square of area a +1 We again use this scheme at scae a +1 in the construction of the scheme for scae a see Figure 3) v 1 x 11 x m1 x 11 x m1 ˆx 11 ˆx m1 q 1 ˆx 11 ˆx 1m y 11 y 1m w 1 {λ k } m k=1 P y ˆx v m x 1m x mm x 1m x mm q m ˆx 1m ˆx mm ˆx m1 ˆx mm y m1 y mm w m Fig 3 Description of the broadcast phase at eve in the hierarchy with m n +1 The first system bock represents transmit beamforming at each of the reay nodes {v i} n +1 i=1 The second system bock are quantizers {qi}n +1 i=1 used at the reay nodes The third system bock represents using n +1 times the communication scheme at eve + 1 organized as n +1 permutation traffic matrices {λ k n +1 )} n +1 k=1 ) to transpose the matrix of quantized beamformed channe symbos {ˆx ij} n +1 i,j=1 In other words, before the third system bock, node v1 has access to {ˆx i1} n +1 i=1, and after the third system bock, node v1 has access to {ˆx1j}n +1 j=1 The fourth system bock is the wireess channe, connecting reay nodes {v i} n +1 i=1 with destination nodes {w i} n +1 i=1 Suppose there are n +1 reay nodes v 1,,v n+1 ocated in the same dense squareet of area a +1 ) that reay traffic for n +1 destination nodes w 1,,w n+1 ocated anywhere in Aa )) Reca that at the end of the MAC phase, each reay node v j has assuming decoding was successfu) access to parts j of the message bits of a source nodes {u i } n +1 i=1 Node v j reencodes these parts independenty; ca { x ij } n +1 i=1 the encoded channe symbos as before, we assume x ij is ony a singe symbo to simpify exposition) Reay node v j then performs transmit beamforming on { x ij } n +1 i=1 for the n +1 transmit antennas of {v k } n +1 k=1 to be sent at time T + j for some appropriatey chosen T > 0 not depending on j) Ca x kj the resuting channe symbo to be sent from reay node v k Then 4 x kj = i h v k,w i [T + j] k h v k,w i [T + j] 2 x ij In order to actuay send this channe symbo, reay node v k needs to obtain x kj from node v j Thus, again a reay nodes need to exchange information To enabe oca cooperation within the reay squareet, each reay node v j quantizes its beamformed channe symbos {x kj } n +1 k=1 at an appropriate rate K ogn) with K independent of n to obtain {ˆx kj} n +1 Now, quantized vaue ˆx kj is sent from reay v j to reay v k Thus, each of the n +1 reay nodes now has a message of size K ogn) for every other reay node This communication demand within the reay squareet can be organized as n +1 permutation traffic matrices {λ k n +1 )} n +1 k=1 between the n +1 reay nodes Note that these reay nodes are ocated in the same square of area a +1 Hence, we are again faced with the origina probem, but at smaer scae a +1 Using n +1 times the assumed scheme for transmitting according to a permutation traffic matrix for n +1 nodes in Aa +1 ), reay v k can obtain a quantized beamformed channe symbos {ˆx kj } n +1 j=1 Now each v k sends ˆx kj over the wireess channe at time instance T + j with T chosen to account for the preceding MAC phase and the oca cooperation in the BC phase) Ca y ij the received channe output at destination node 4 Note that, since we ony assume causa CSI, reay node v j does not actuay have access to {h vk,w i [T +j]} k,i at the time the beamforming is performed This probem can, however, be circumvented The detais are provided in the proofs see Lemma 10) k=1
11 11 w i at time instance T + j Using y ij, destination node w i can now decode part j of the message bits of its source node u i 4) Spatia ReUse and Termination of Recursion: The scheme does appropriatey weighted timedivision among different eves 0 Ln) Within any eve 1, mutipe regions of the origina square An) of area n are being operated in parae The detais reated to the effects of interference between different regions operating at the same eve of hierarchy are discussed in the proofs The recursive construction terminates at some arge enough eve L = Ln) to be chosen ater) At this scae, we have n L nodes in area Aa L ) A permutation traffic matrix at this eve comprises n L sourcedestination pairs These transmissions are performed using TDMA Again, mutipe regions in the origina square of area n at eve L are active simutaneousy B Achievabe Rates Here we present a backoftheenveope cacuation of the pernode rate ρ HR n) achievabe with the hierarchica reaying scheme described in the previous section The compete proof is stated in Section VII We assume throughout that ong bock codes and corresponding optima decoders are used for transmission Instead of computing the rate achieved by hierarchica reaying, it wi be convenient to instead anayze its inverse, ie, the time utiized for transmission of a singe message bit from each source to its destination under a permutation traffic matrix λn) Using the hierarchica reaying scheme, each message traves through L eves of the hierarchy Ca τ n) the amount of time spent for the transmission of one message bit between each of the n sourcedestination pairs at eve in the hierarchy We compute τ n) recursivey At any eve 1, there are mutipe regions of area a operating at the same time Due to the spatia reuse, each of these regions gets to transmit a constant fraction of time It can be shown that the addition of interference due to this spatia reuse eads ony to a constant oss in achievabe rate Hence the time required to send one message bit is ony a constant factor higher than the one needed if region Aa ) is considered separatey Consider now one such region Aa ) By the time sharing construction, ony one of its Θ2 γn)) dense reay squareets of area a +1 is active at any given moment Hence the time required to operate a reay squareets is a Θ2 γn)) factor higher than for just one reay squareet separatey Consider now one such reay squareet, and assume n +1 source nodes in Aa ) communicate each n +1 message bits to their respective destination nodes through a MAC phase and BC phase with the hep of the n +1 reay nodes in this reay squareet of area a +1 In the MAC phase, each of the n +1 sources simutaneousy sends one bit to each of the n +1 reay nodes The tota time for this transmission is composed of two terms i) Transmission of n +1 message bits from each of the n +1 source nodes to those many reay nodes Since we time share between Θ2 γn)) reay squareets, we can transmit with an average power constraint of Θ2 γn)) during the time a reay squareet is active, and sti satisfies the overa average power constraint of 1 With this bursty transmission strategy, we require a tota of a O n α/2 ) +1 = O n γ 1 α/2) n)n α/2 1) 2) γn)n +1 channe uses to transmit n +1 bits per source node The terms on the efthand side of 2) can be understood as foows: n +1 is the number of bits to be transmitted; a α/2 is the power oss since most nodes communicate over a distance of Θa 1/2 ); 2 γn) is the average transmit power; n +1 is the mutipeantenna gain, since we have that many transmit and receive antennas ii) We show that constant rate quantization of the received observations at the reays is sufficient Hence the n +1 bits for a sources generate On +1 ) transmissions at eve + 1 of the hierarchy Therefore, On +1 τ +1 n)) 3)
12 12 channe uses are needed to communicate a quantized observations to their respective reay nodes Combining 2) and 3), accounting for the factor 2 γn) oss due to time division between reay squareets, we obtain that the transmission time for one message bit from each source to the reay squareet in the MAC phase at eve is ) τ MAC n) = O 2 γ 1+1 α/2) n)n α/2 1 + τ +1 n) 4) Next, we compute the number of channe uses per message bit received by the destination nodes in the BC phase Simiar to the MAC phase, each of the n +1 reay nodes has n +1 message bits out of which one bit is to be transmitted to each of the n +1 destination nodes Since there are n +1 reay nodes, each destination node receives n +1 message bits As before the required transmission time has two components i) Transmission of the encoded and quantized message bits from each of the n +1 reay nodes to a other reay nodes at eve + 1 of the hierarchy We show that each message bit resuts in O + 1) og n ) quantized bits Therefore, O n ) og n ) bits need to be transmitted from each reay node This requires O n ) ogn)τ +1 n) ) 5) channe uses ii) Transmission of n +1 message bits from the reay nodes to each destination node As before, we use bursty transmission with an average power constraint of Θ2 γn)) during the fraction Θ2 γ 1 n)) of time each reay squareet is active this satisfies the overa average power constraint of 1) Using this bursty strategy requires a O n α/2 ) +1 = O n γ 1 α/2) n)n α/2 1) 6) γn)n +1 channe uses for transmission of n +1 bits per destination node As in the MAC phase, n +1 in the eft hand side of 6) can be understood as the number of bits to be transmitted, a α/2 as the power oss for communicating over distance Θa 1/2 ), 2 γn) as the average transmit power, and n +1 as the mutipeantenna gain Combining 5) and 6), accounting for a factor 2 γn) oss due to time division between reay squareets, the transmission time for one message bit from the reays to each destination node in the BC phase at eve is ) τ BC n) = O 2 γ 1+1 α/2) n)n α/ ) ogn)τ +1 n) 7) From 4) and 7), we obtain the foowing recursion τ n) = τ MAC n) + τ BC n) ) = O 2 γ 1 α/2)+1 n)n α/ ) ogn)τ +1 n) ) = O 2 L γn)n α/2 1 + L ogn)τ +1 n), 8) where we have used α > 2 This recursion hods for a 0 < L At eve L, we use TDMA among n L nodes in region Aa L ) with a permutation traffic matrix λn L ) Each of the n L sourcedestination pairs uses the wireess channe for 1/n L fraction of the time at power On L ), satisfying the average power constraint Assuming the received power is ess than 1 for a n so that we operate in the power imited regime), we can achieve a rate of at east Ωa α/2 L ) between any sourcedestination pair Equivaenty τ L n) = Oa α/2 L ) = O n α/2 γ Lα/2 n) ) = O n α/2 γ L n) ) 9)
13 13 Combining 8) and 9), we have τ 0 n) = O n α/2 1 2 L γn) + L ogn)τ 1 n) ) = = O n α/2 1 L ogn) )L 2 L γn) + L ogn) )L ) τ L n) = O n α/2 1 L ogn) ) L 2 L γn) + nγ L n) )) 10) The term ) L L ogn) 2 L γn) + nγ L n) ) is the oss factor over the desired order n α/2 1 scaing, and we now choose the branching factor γn) and the hierarchy depth L Ln) to make it sma Fix a δ 0, 1/2) and set With this Ln) og 1/2 δ n), γn) n 1/Ln)+1) Ln) ogn) ) Ln) n 2og 1/2 δ n)og ogn), 2 Ln) γn) n og 1/2 δ n)+og δ 1/2 n), nγ Ln) n) n ogδ 1/2 n) Since δ > 0, the n ogδ 1/2 n) term dominates in 10), and we obtain τ 0 n) bn)n α/2 1, where bn) n Oog δ 1/2 n)) Hence the pernode rate of the hierarchica reaying scheme is ower bounded as ρ HR n) = 1/τ 0 n) bn)n 1 α/2, with bn) n Oogδ 1/2 n)) Note that to minimize the oss term, we shoud choose δ > 0 to be sma V COOPERATIVE MULTIHOP SCHEME In this section, we provide a brief description of the cooperative mutihop scheme The detais of the construction and the anaysis of its performance can be found in Section X Reca that a node pacement V n) is µreguar at resoution dn) if every square [idn), i + 1)dn)] [jdn), j +1)dn)] for some i, j N contains at east µd 2 n) nodes Given such a node pacement V n), divide it into squares of sideength dn) Consider four adjacent squares, combined into a bigger square of sideength 2dn) By the reguarity assumption on V n), this bigger square contains at east 4µd 2 n) nodes Hence we can appy the hierarchica reaying scheme introduced in the ast section to support any permutation traffic within this bigger square at a pernode rate of bn)d 2 n)) 1 α/2 = bn)d 2 α n), where bn) is essentiay of order n og 1/2 n) By propery choosing the permutation traffic matrices within every possibe such bigger square of sideength 2dn), this creates a equivaent communication graph with n/d 2 n) nodes each corresponding to a square of sideength dn) in An), and with edges between nodes
14 14 corresponding to neighboring squares With the above communication procedure and appropriate spatia reuse, each such edge has a capacity of d 2 n)bn)d 2 α n) = bn)d 4 α n) The resuting communication graph is depicted in Figure 4 Fig 4 Communication graph in bod) resuting from the construction of the cooperative mutihop scheme The entire square has sideength n, and the dashed squares have sideength dn) Each bod) edge in the communication graph corresponds to using the hierarchica reaying scheme between the nodes in the adjacent squares of sideength dn) Now, to send a message from a source node in V n) to its destination node, we first ocate the squares of sideength dn) they are ocated in We then route the message over the edges of the communication graph constructed above in a mutihop fashion By the construction of the communication graph, each such edge is impemented using the hierarchica reaying scheme In other words, we perform mutihop communication over distance n with hop ength dn), and each such hop is impemented using hierarchica reaying over distance dn) Since each edge in the communication graph has a capacity of bn)d 4 α n) and has to support roughy n 1/2 dn) sourcedestination pairs, we obtain a pernode rate of per sourcedestination pair ρ CMH n) bn)d 4 α n)n 1/2 d 1 n) = bn)d 3 α n)n 1/2 VI ANALYSIS OF THE HIERARCHICAL RELAYING SCHEME In this section, we anayze in detai the hierarchica reaying scheme Throughout Sections VIA to VIC, we consider communication at eve, 0 < L = Ln), of the hierarchy A constants K i are independent of Reca that at eve, we have a square region Aa ) of area a n γ n) containing n n 2 γ n) nodes V n ) We divide Aa ) into γn) squareets of area a +1 Reca that a squareet of area a +1 in eve of the hierarchy is caed dense if it contains at east n +1 nodes We impose a power constraint of
15 15 P n) = Θ2 γn)) during the time any particuar reay squareet is active Since we time share between Θ2 γn)) reay squareets, this satisfies the overa average power constraint by choosing constants appropriatey) Since other regions of area a are active at the same time as the one under consideration, we have to dea with interference To this end, we consider a sighty more genera noise mode that incudes the experienced interference at the reay squareets More precisey, we assume that, for a u V n ), the additive noise term {z u [t]} t is independent of the signa {x u [t]} t and of the channe gains {h u,v [t]} v,t ; that the noise term is stationary and ergodic across time t, but with arbitrary dependence across nodes u; and that the noise has zero mean and bounded power N 0 independent of n Note that we do not require the additive noise term to be Gaussian In the above, N 0 accounts for both noise which has power 1 in the origina mode), as we as interference We show in Section VII that these assumptions are vaid Reca the foowing choice of γn) and Ln): Ln) og 1/2 δ n), γn) n 1/Ln)+1), with δ 0, 1/2) independent of n This choice satisfies γn) γñ) if n ñ, γ Ln) n) n for a n, 2 Ln) γn) as n, The first condition in 12) impies that the number of squareets γn) we divide An) into increases in n The second condition impies the squareet area a Ln) at the ast eve of the hierarchy is bigger than 1 As we sha see, the third condition impies that the number of dense squareets at the ast eve and hence at every eve) grows unbounded as n see Lemma 6 beow) Throughout Section VI, we consider the fast fading channe mode Sow fading is discussed in Section VIIB A Setting up Reays The first emma states that the minimumseparation requirement r min 0, 1) impies that a constant fraction of squareets must be dense We point out that this is the ony consequence of the minimumseparation requirement used to prove Theorem 1 Thus Theorem 1 remains vaid if we just assume that Lemma 6 beow hods directy See aso Section XIID for further detais Lemma 6 For any V n ) Aa ) with V n ) n and with minimum separation r min 0, 1), each of its squareets of area a +1 contains at most K 1 a /γn) nodes, and there are at east K 2 2 γn) dense squareets Proof Put a circe of radius r min /2 around each node By the minimumseparation requirement, these circes do not intersect Each node covers an area of πr 2 min/4 Increasing the sideength of each squareet by r min, this provides a tota area of a /γn) + r min ) 2 a γn) 1 + r min) 2 in which the circes around these nodes are packed Here we have used that γ +1 n) n by 12), and therefore γn) n/γ n) = a Hence there can be at most K 1 a /γn) nodes per squareet with K r min) 2 πrmin 2 11) 12)
16 16 Note that, since r min < 1, we have K 1 > 1 Let dn ) be the number of dense squareets in Aa ), and therefore γn) dn ) is the number of squareets that are not dense By the argument in the ast paragraph, each dense squareet contains at most K 1 a /γn) nodes, and those squareets that are not dense contain ess than n +1 nodes by the definition of dense squareets Hence dn ) must satisfy dn )K 1 a /γn) + γn) dn ) ) n +1 V n ) n Thus, using a = 2 n, n +1 = n /2γn), we have dn )K γn) dn ))/2 γn) As K 1 2 > 1, this yieds with dn ) 1 1/2 2 γn) γn) = K K γn), 1/2 2K 1 K 2 1 2K 1 Consider V n ) Aa ) with V n ), and choose arbitrary K 2 2 γn) dense squareets of area a +1 as guaranteed by Lemma 6) Ca those squareets {A k a +1 )} K 22 γn) k=1 For each sourdestination pair, we now seect one such dense squareet to reay traffic over To avoid bottenecks, this seection has to be done such that a reay squareets carry approximatey the same amount of traffic Moreover, for technica reasons, the distances from the source and the destination to the reay squareet cannot be too sma Formay, the seection of reay squareets can be described by the schedues S {0, 1} n K 2 2 γn) with s u,k = 1 if source node u reays traffic over dense squareet k, and S {0, 1} K 22 γn) n with s k,w = 1 if destination node w receives traffic from dense squareet k With sight abuse of notation, et r u,ak a +1 ) be the distance between node u V n ) and the cosest point in A k a +1 ), ie, Define the sets and Sn ) r u,ak a +1 ) { S {0, 1} n K 2 2 γn) : 0 n u=1 s u,k n +1 k, min r u,v 13) v A k a +1 ) 0 K 2 2 γn) k=1 s u,k 1 u, s u,k = 1 impies r u,ak a +1 ) } 2a +1 u, k Sn ) { S {0, 1} K 2 2 γn) n : S T Sn ) } 14) The sets Sn ) and Sn ) are the coection of schedues satisfying the conditions mentioned in the ast paragraph More precisey, the first condition in 14) ensures that at most n +1 sourcedestination pairs reay over the same dense squareet, the second condition ensures that each sourcedestination pair chooses at most one reay squareet, and the third condition ensures that sources and destinations are at east at distance 2a +1 from the chosen reay squareet Next, we prove that any node pacement that satisfies Lemma 6 aows for a decomposition of any permutation traffic matrix λn ) into a sma number of schedues beonging to Sn ) and Sn )
17 17 Lemma 7 There exist K 3 such that for a n arge enough independent of ), and every permutation traffic matrix λn ) {0, 1} n n we can find K 3 2 schedues {S i) n )} K 32 i=1 Sn ), { S i) n )} K 32 i=1 Sn ) satisfying λn ) = K 3 2 i=1 S i) n ) S i) n ) Proof Pick an arbitrary sourcedestination pair in λn ), and consider the squareets containing the source and the destination node Since each squareet has side ength a +1, there are at most 50 squareets at distance ess than 2a +1 from either of those two squareets As 2 Ln) γn) as n by 12), there exists K independent of ) such that for n K we have 50 K γn) Since there are at east K 2 2 γn) dense squareets by Lemma 6, there must exist at east K γn) dense squareets that are at distance at east 2a +1 from both the squareets containing the source and the destination node In order to construct a decomposition of λn ), we use the foowing procedure Sequentiay, each of the n sourcedestination pairs chooses one of the at east) K γn) dense squareets at distance at east 2a +1 that has not aready been chosen by n +1 other pairs If any sourcedestination pair can not seect such a squareet, then stop the procedure and use the sourcedestination pairs matched with dense squareets so far to define matrices S 1) n ) and S 1) n ) Now, remove a the matched sourcedestination pairs, forget that dense squareets were matched to any sourcedestination pair and redo the above procedure, going through the remaining sourcedestination pairs Let K 3 4/K 2 We caim that by repeating this process of generating matrices S i) n ) and S i) n ), we can match a sourcedestination pairs to some dense squareet with at most K 3 2 such matrices Indeed, a new pair of matrices is generated ony when a sourcedestination pair can not be matched to any of its avaiabe at east) K γn) dense squareets If this happens, a these dense squareets are matched by n +1 = n /2γn) pairs Hence at east K n sourcedestination pairs are matched in each round Since there are n tota pairs, we need at most n K n = K 3 2 matrices S i) n ) and S i) n ) For a permutation traffic matrix λn ), communication proceeds as foows Write λn ) = K 3 2 i=1 S i) n ) S i) n ) as in Lemma 7 Spit time into K 3 2 equa ength time sots In sot i, we use S i) n ) S i) n ) as our traffic matrix Consider without oss of generaity i = 1 in the foowing Write S 1) n ) S 1) n ) = K 2 2 γn) k=1 S 1,k) n +1 ) S 1,k) n +1 ), where S 1,k) n +1 ) S 1,k) n +1 ) is the traffic reayed over the dense squareet A k a +1 ) We time share between the schedues for k {1,, K 2 2 γn)} Consider now any such k In the worst case, there are exacty n +1 communication pairs to be reayed over A k a +1 ), and the reay squareet A k a +1 ) contains exacty n +1 nodes We sha assume this worst case in the foowing We focus on the transmission according to the traffic matrix S 1,1) n +1 ) S 1,1) n +1 ) Let V n +1 ) be the nodes in A 1 a +1 ), and et Un +1 ) and Wn +1 ) be the source and destination nodes of S 1,1) n +1 ) S 1,1) n +1 ), respectivey In other words, the source nodes Un +1 ) communicate to their respective destination nodes Wn +1 ) using the nodes V n +1 ) as reays
18 18 B Mutipe Access Phase Each source node in Un +1 ) spits its message into n +1 equa ength parts Part j at every node u Un +1 ) is to be reayed over the jth node in V n +1 ) Each part is separatey encoded at the source and separatey decoded at the destination After the source nodes are done transmitting their messages, the nodes in the reay squareet quantize their samped) observations corresponding to part j and communicate the quantized vaues to the jth node in the reay squareet This node then decodes the jth message parts of a source nodes Note that this induces a uniform traffic pattern between the nodes in the reay squareet, ie, every node needs to transmit quantized observations to every other node Whie this traffic pattern does not correspond to a permutation traffic matrix, it can be written as a sum of n +1 permutation traffic matrices A fraction 1/n +1 of the traffic within the reay squareet is transmitted according to each of these permutation traffic matrices This setup is depicted in Figure 2 in Section IVA Assuming for the moment that we have a scheme to send the quantized observations to the dedicated node in the reay squareet, the traffic matrix S 1,1) n +1 ) between Un +1 ) and V n +1 ) describes then a MAC with n +1 transmitters, each with one antenna, and one receiver with n +1 antennas We ca this the MAC induced by S 1,1) n +1 ) in the foowing Before we anayze the rate achievabe over this induced MAC, we need an auxiiary resut on quantized channes y 1 q 1 ŷ 1 w f x P y x y m q m ŷ m Pˆx ŷ ˆx ϕ ŵ Fig 5 Sketch of the quantized channe f and ϕ are the channe encoder and decoder, respectivey; {q k } m k=1 are quantizers; P y x and Pˆx ŷ represent stationary ergodic channes with the indicated margina distributions Consider the quantized channe in Figure 5 Here, f is the channe encoder, ϕ the channe decoder, {q k } m k=1 quantizers A these have to be chosen P y x and Pˆx ŷ, on the other hand, represent fixed stationary ergodic channes with the indicated margina distributions We ca R the rate of the channe code f, ϕ) and {R k } m k=1 the rates of quantizers {q k} m k=1 Lemma 8 If there exist distributions P x and {Pŷk y k } m k=1 such that R < Ix; ˆx) and R k > Iy k ; ŷ k ), k, then R, {R k } m k=1) is achievabe over the quantized channe Proof The proof foows from a simpe extension of Theorem 1 in Appendix II of [8] Lemma 9 Let the additive noise {z v } v V n+1 ) be uncorreated over v) For the MAC induced by S 1,1) n +1 ) with pernode average power constraint P n) n 1, a rate of +1 aα/2 ρ MAC n) K 4 P n)n +1 a α/2 per source node is achievabe, and the number of bits required at each reay node to quantize the observations is at most K 5 bits per n +1 tota message bits 5 sent by the source nodes Proof The source nodes send signas with a power of essentiay) n 1 +1 aα/2 for a fraction P n)n +1 a α/2 1 of time and are sient for the remaining time To ensure that interference is uniform, the time sots during which the nodes send signas are chosen randomy as foows Generate independenty for each region Aa ) a Bernoui process {B[t]} t N with parameter P n)n +1 a α/2 /1 + η) 1 for some sma η > 0 The nodes in Aa ) are active whenever B[t] = 1 and remain sient otherwise Since the bockength of the codes used is assumed to be arge, this satisfies the average power constraint of P n) with high probabiity for any η > 0 Since we are interested ony in the scaing of capacity, we ignore the additiona 5 Tota message bits refers to the sum of a message bits transmitted by the n +1 source nodes
19 19 1/1 + η) term in the foowing to simpify notation Ceary, we ony need to consider the fraction of time during which B[t] = 1 Let y be the received vector at the reay squareet, ŷ the componentwise) quantized observations We use a matched fiter at the reay squareet, ie, ˆx u = h u h u ŷ, where coumn vector h u = {h u,v } v V n+1 ) are the channe gains between node u Un +1 ) and the nodes in the reay squareet V n +1 ) The use of a matched fiter is possibe since we assume fu CSI is avaiabe at a the nodes We now use Lemma 8 to show that we can design quantizers {q v } v V n+1 ) of constant rate and achieve a pernode communication rate of at east K 4 P n)n +1 a α/2 The first channe in Lemma 8 see Figure 5) wi correspond to the wireess channe between a source node u and its reay squareet V n +1 ) The second channe in Lemma 8 wi correspond to the matched fiter used at the reay squareet To appy Lemma 8, we need to find a distribution for x u and for ŷ v y v Define r u r u,a1 a +1 )/ 2a 1 with r u,a1 a +1 ) as in 13), to be the normaized distance of the source node u Un +1 ) to the reay squareet A 1 a +1 ) For each u Un +1 ) et x u N C 0, r u αn 1 +1 aα/2 ) independent of xũ for u ũ, and et ŷ v = y v + z v for z v N C 0, 2 ) independent of y and for some 2 > 0 Note that the channe input x u has power that depends on the normaized distance r u ie, ony nodes u Un +1 ) that are at maxima distance 2a from the reay squareet transmit at fu avaiabe power) This is to ensure that a signas are received at roughy the same strength by the reays We proceed by computing the mutua informations Iy v ; ŷ v {hũ,ṽ }) and Ix u ; ˆx u {hũ,ṽ }) as required in Lemma 8 the conditioning on {hũ,ṽ } being due to the avaiabiity of fu CSI) Note first that by construction of S 1,1) n +1 ) see 14)), we have for u Un +1 ) and v V n +1 ) and hence From this, and since h u,v 2 = ru,v α, we obtain r u,a1 a +1 ) r u,v 2r u,a1 a +1 ), 2 3α/2 a α/2 2 3α/2 n +1 a α/ a r u r u,v 1 2a 15) h u,v 2 r α u 2 α/2 a α/2, h u 2 r α u 2 α/2 n +1 a α/2 We start by computing Iy v ; ŷ v {hũ,ṽ }) We have ŷ v = h u,v x u + z v + zṽ, and hence ŷ v has mean zero and variance E ŷ v 2 ) = u Un +1 ) u Un +1 ) h u,v 2 r α un 1 +1 aα/2 + N n +1 2 α/2 a α/2 n 1 +1 aα/2 + N = 2 α/2 + N 0 + 2, 16)
20 20 where we have used 16) Hence Iy v ; ŷ v {hũ,ṽ }) = hŷ v {hũ,ṽ }) hŷ v y v, {hũ,ṽ }) og 2πeE ŷ v 2 ) ) og2πe 2 ) og 2πe2 α/2 + N ) ) og2πe 2 ) = og α/2 + N ) 0 17) 2 We now compute Ix u ; ˆx u {hũ,ṽ }) We have Conditioned on {hũ}ũ Un+1 ), and E ũ Un +1 )\{u} ˆx u = h u x u + ũ Un +1 )\{u} h u h ũ h u x ũ + h u z + z) h u h u x u N C 0, hu 2 r ) un α 1 +1 aα/2, h u h ũ h u x ũ + h u h u z + z) 2 ) {hũ} = n 1 +1 aα/2 ũ Un +1 )\{u} r α ũ h u h ũ 2 h u 2 + N 0 + 2, where we have used the assumption that {z v } v V n+1 ) are uncorreated in the second ine Using 16), this is, in turn, upper bounded by r ũ α h u h ũ 2 + N α/2 r α u n 2 +1 aα ũ Un +1 )\{u} Simiary, we can ower bound the received signa power as E h u 2 x u 2) 2 3α/2 Since Gaussian noise is the worst additive noise under a power constraint [15], and appying Jensen s inequaity to the convex function og1 + 1/x), we obtain ) ) 2 3α/2 Ix u ; ˆx u {hũ,ṽ }) E og α/2 r un α 2 +1 aα ũ Un +1 )\{u} rα ũ h uhũ 2 + N ) 2 3α/2 og α/2 r u αn 2 +1 aα ũ Un +1 )\{u} rα ũ E h uhũ 2) 18) + N We have for u ũ, and hence using 15) E r u α ũ Un +1 )\{u} E h u h ũ 2) = Eh u h ũh ũ h u) = h u,v 2 hũ,v 2 r α ũ h uhũ 2 ) v V n +1 ) = v V n +1 ) = r α u,vr α ũ,v, 19) ũ Un +1 )\{u} v V n +1 ) 2 α n 2 +1 a α r α ur α u,v r α ũr α ũ,v
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