Multihop Hierarchical MIMO A Multicast Structure in wireless ad hoc networks

Transcription

1 Multihop Hierarchical MIMO A Multicast Structure in wireless ad hoc networks January 11, 2008 Abstract In this paper, we study multicast in large-scale wireless ad hoc networks. Consider N nodes that are randomly deployed in an area of size N. Each is a multicast source with n d destination nodes randomly chosen. The fundamental question is as follows: what is the multicast capacity from an information-theoretic viewpoint and how can one achieve it? Our preliminary work presents an optimistic view: node cooperation has the potential to improve capacity significantly in certain regime. In particular, when 2 α < 3, the aggregated achievable multicast source rate scales as Ω((N/n d ) 2 α/2 ɛ), where ɛ > 0 can be arbitrarily small. This is a significant improvement from (N/n d ) 1/2 obtained in previous work assuming a simple protocol model [1, 2]. This is achieved through a MIMO-based multihop architecture. The key ingredient is to choose an appropriate cluster size for MIMO cooperation (and thus the corresponding distance for each hop) that optimally integrate the multicast gain and the power transfer gain (loss). 1 Introduction Multicast data transfer has many applications. In military networks it is often stated that multicast traffic dominates due to the need for group communications. In the civilian context, an emerging application that has already been tested in universities is the use of wireless ad hoc networks to broadcast replays during football games. A situation like a football game would have a large number of spectators, each having a mobile device and a desire for a replay of an important moment in the game. There is almost no infrastructure available from which they could obtain such data, and there is a strong incentive to form an ad hoc network for this purpose. Some of the users might be close to data sources (perhaps if they were close to an Internet access point), and they would act as sources for the multicast traffic. Other nodes would act as relays and sinks for the data. The questions arise as to how many multicast sessions can be supported by such a network, what the total capacity of the network would be, and how to achieve the capacity in a simple manner. Earlier work has studied multicast scaling using the protocol-model first introduced in [3]. Consider a randomly deployed network of N nodes, each is a multicast source with n d receivers randomly chosen, where n d = N γ, 0 < γ < 1. It has been shown that the aggregated multicast source rate scales as Θ( N/ n d log N) [1, 2]. In addition, in [2], a simple comb structure was proposed to achieve the capacity upperbound. The results in [1, 2] assume the simple protocol model, where a transmission is successful if and only if all other transmissions are outside a certain range from the receiver. This model captures the current widely-used physical layer technology: signals from other transmitters are interference that degrades the communication. A nature question is whether more sophisticated physical-layer technology can improve the scaling law. 1

2 The question has received a lot of research attention in the context of unicast. Xie and Kumar first show that nearest neighbor multihop is order-optimal when the pass loss exponent α is larger than 6 [4]. Later on, researchers gradually tighten the regime where multihop is order-optimal, α > 5 in [5], α > 4.5 in [6], and α > 4 in [7]. Recently, in [8], the authors identify the scaling laws of the information-theoretic capacity of the network in the previous open range of 2 < α < 4. They show that the capacity scales as n 2 α/2 for 2 α < 3 and n for α > 3 in extended networks. In addition, almost linear scaling is achievable in dense networks or when α = 2. When the path attenuation is low, i.e., 2 α < 3, the capacity is achieved within an arbitrarily small exponent by intelligent node cooperation and distributed multiple-input multiple-output (MIMO) communication. To realize the cooperation, nodes exchange information through a hierarchical and digital architecture. When the path attenuation is high, i.e., α 3, the capacity is achieved through nearest neighbor multihop. The optimistic result using hierarchical MIMO motivates us to revisit the multicast problem. The fundamental question is as follows: what is the multicast capacity from an information-theoretic viewpoint and how can one achieve it? Our preliminary work presents an optimistic view: cooperation has the potential to improve capacity significantly in certain regime, as we have constructed such a multicast structure. 2 System Model We consider an extended network where N wireless nodes are located randomly in a square of area N. Each node is a multicast source, each is associated with n d destination nodes chosen randomly. Thus, the total number source-destination pairs is N d, where N d = N n d. Each multicast source has the same traffic rate r s to send to all its destination nodes. The aggregated multicast source rate is R s = N s r s. Each node has a common average transmit power constraint of P. We assume the same physical model as in [8]. Let α be the power path loss exponent, where α = 2 in free space. We assume a narrow-band communication channel where the carrier frequency is much larger than the channel bandwidth. The complex based-band channel gain between node i and node k at time m is given by H ik [m] = Gr α/2 ik exp(jθ ik [m]) where r ik is the distance between the nodes, θ ik [m] is the random phase at time m, uniformly distributed in [0, 2π] and {θ ik [m], 1 i n, 1 k n} is a collection of independent and identically distributed (i.i.d.) random processes. The θ ik [m] and r ik are also assumed to be independent. The parameters G and α are assumed to be constant. We assume that the channel is random, depending on the node locations and the phases. Node locations are fixed. The phase changes are stationary and ergodic. We assume that the channel gains are known at all the nodes. Because we consider an extended network, the average distance between neighboring nodes remains a constant as the network size increases. In this case, we assume the far-field path-loss model, which renders a random phase model reasonable. As in [8], multipath effects are ignored. We summarize the main notations used in the paper in Table 2 for easy reference. 2.1 Hierarchical MIMO In [8], the authors propose a hierarchical MIMO structure to achieve asymptotically tight unicast scaling law (within an arbitrarily small exponent) when 2 α < 3. The scheme is used as a basic component in our proposed multicast scheme, and thus is summarized as follows. 2

3 α : path loss exponent N : total number of nodes in the network N d : total number of multicast destinations. n d : # of destinations per source, n d = N γ, 0 < γ < 1 r s : per multicast source rate R s : aggregated multicast source rate, R s = N r s M : cluster size We restate Theorem 5.1 from [8] The interference constraints stated in the following theorem come from Lemma 3.1 and Lemma 3.2 in [8]. Theorem 2.1 Consider an extended network on a N N square. There are two cases. 2 α < 3: For any ɛ > 0, w.h.p., an aggregated throughput of KN 2 α/2 ɛ is achievable in the network for all possible pairings between sources and destinations. K > 0 is a constant independent of N and the source-destination pairing. The capacity is achievable when the network experiences external interference. In particular, when α > 2, the capacity is achievable if the external interference signal received by nodes is a collection of uncorrelated zero-mean stationary and ergodic random process with its power upperbounded by K I. When α = 2, the capacity is achievable if the external interference signal received by nodes is a collection of uncorrelated zero-mean stationary and ergodic random process with its power upperbounded by K I log N. In both cases, K I is a constant independent of N. α 3: w.h.p., an aggregated throughput K N is achievable in the network for all possible parings between sources and destinations. K > 0 is a constant independent of N and the source-destination pairing. The following scheme achieves the capacity within an arbitrarily small exponent. It operates in an iterative manner in three phases through clustering and long-range MIMO transmissions between clusters. One can start with a simple scheme that achieves an aggregated capacity of N b. Then divide the network into clusters of size M = N 1/(2 b). Source nodes send bits to destinations in three steps. Phase I. Setting Up Transmit Cooperation Clusters work in parallel. Within a cluster, each source node has to distribute M bits to the other nodes, 1 bit for each node, such that at the end of the phase, each node has 1 bit from each of the source nodes in the same cluster using the scheme with aggregated capacity of M b. Phase II. MIMO Transmission One performs long-distance MIMO transmissions (of size M M) between source-destination pairs, one at a time. In particular, nodes in the source cluster together can form a distributed transmit antenna array, sending the bits simultaneously to the corresponding destination cluster. 3

4 Phase III. Cooperate to Decode Clusters work in parallel. Each node in the destination cluster obtained one observation from the MIMO transmission for each source, and it quantizes and ships the observation back to the destination node, which can then do joint MIMO processing of all the observations and decode the transmitted bits. After the three steps, one obtains a scheme that achieves a capacity of N 1/(2 b), which is higher than the original scheme with capacity N b. The key ingredient of the capacity achieving scheme is iteration. One can start with a simple scheme that achieves an aggregated capacity of 1 using simple TDMA (i.e., b = 0). Applying the above steps h times, one gets a scheme achieving an aggregated throughput of Θ(N h/(h+1) ). Given any ɛ > 0, we can now choose h such that and we get a scheme that achieves aggregate throughput scaling with high probability. 3 Main Results In this section, we present the MHM scheme that can achieve the upper bound arbitrarily close. We note that the theorems are presented in their accurate form. However, for easy illustration, we outline the proof and the construction of the MHM scheme in a hand-waving fashion. In particular, we consider a special network where destination nodes of a multicast source is evenly distributed. In other words, if we divide the network into clusters of size N/n d, then each cluster has one and only one destination node from each source. We will present the the rigorous proofs in Section 4 where destinations are randomly distributed in the network. 3.1 Achievability Theorem 3.1 Consider an extended network on a N N square. There are two cases. 2 α < 3: For any ɛ > 0, w.h.p., an aggregated source rate of KN (2 α/2 ɛ)(1 γ) is achievable, where K > 0 is a constant independent of N. An optimal operation scheme is MHM. α 3: w.h.p., an aggregated throughput N K nd log N is achievable, where K > 0 is a constant independent of N. nearest-neighbor multihop. An optimal operation is the We construct a multihop hierarchical MIMO (MHM) structure to achieve the upper bound within an arbitrarily small exponent. The construction is inspired by the derivation of the upperbound. To achieve the upperbound, we notice that the cutset size is M = N 1 γ when 2 α < 3 and is 1 when α 3. We construct a similar structure. Let s first consider the case where 2 α < 3. We divide the network into clusters of size M = N 1 γ. On average, each cluster has one destination node from each source. For simplicity, we assume that there is exactly one destination node in each cluster from each source. The operation is long range MIMO enabled multihop. The length of each hop is M, which is achieved through M M long-range MIMO transmission, as shown in Figure 1. 4

5 S2: MxM MIMO S S1&3: intra-cluster Hierarchical MIMO Figure 1: MHM Routing is extremely simple in the scheme. Consider a multicast flow. Its source node locates in an arbitrary cluster. The information bit is first spread out vertically and then horizontally, cluster-by-cluster, as shown in Figure 1. Inter-cluster communication is through M M MIMO transmission between neighboring clusters. Step I: In each cluster of size M, each node distributes a bit to each of the other nodes in the cluster. This distribution uses hierarchical MIMO unicast within the cluster. There are total c 1 N/M = c 1 n d simultaneous transmissions in the network. This step takes M 2 /(c 1 M 2 α/2 ɛ ) unit of time. Step II: Multihop transmission. Each cluster of size M constructs a M M MIMO transmission to its neighboring cluster. There are total c 2 N/M = c 2 n d simultaneous transmissions in the network. This step takes M 2 /(c 2 M 2 α/2 ) unit of time. Step III: In each cluster of size M, each node has an observation for each other node in the cluster. Quantize each observation into Q bits. Use hierarchical MIMO unicast within the cluster to distribute the quantized symbol to the node that needs it. This step takes Q M 2 /(c 1 M 2 α/2 ɛ ) unit of time. Note that a bit for each multicast source is transmitted from the source to each of the clusters of size M hop-by-hop following the previous three steps. Each transmission reaches a constant number (one to four) of neighboring clusters. Therefore, there are totally N/M such transmissions. On the other hand, there are N/M simultaneous transmissions in each step. The aggregated source rate 5

6 can be calculated as follows: R s = 1 N M M 2 N M M 2 KM 2 α 2 ɛ + M 2 M M/M α 2 + Q M 2 KM 2 α 2 ɛ (1) = K M 2 α 2 ɛ (2) = K N (2 α 2 ɛ)(1 γ) (3) Because each source node s sending rate is limited by O(log N). R s = min(n log N, N (2 α 2 ɛ)(1 γ) ) = N (2 α 2 ɛ)(1 γ). (4) For comparison, we also consider using Unicast MIMO to deliver multicast traffic. In particular, each source sends its data to each of its destinations independently as if they are NOT multicast. Using unicast MIMO, there are N d s-d pairs. Following the capacity results in Theorem 2.1, each source node can achieve a throughput of KN 2 α 2 ɛ N d = KN 1 α 2 ɛ γ. There are N sources. So the aggregated source rate using unicast MIMO is 3.2 Path Attenuation Low Attenuation R UMIMO = KN 2 α 2 γ ɛ. (5) We have α = 2. In this case, we have two schemes that significantly improve the capacity compared to nearest-neighbor multihop: MHM and unicast MIMO. The performance of MHM, based on Eq. 3, is R s = N 1 γ ɛ. The performance using unicast MIMO is: R s = N N N d = N 1 γ ɛ. 3.3 Medium Attenuation We consider the case where 2 α < 3. Compare the achievability result obtained at Eq. 3. We note that MHM is strictly better than unicast MIMO and nearest-neighbor multihop [2, 1] High Attenuation When α 2, we use the standard multihop structure where information is relayed to the nearest neighbor toward the destination. In Figure 2, we compare MHM, Unicast MIMO, and the nearest-neighbor multicast. The x-axis is α and y-axis is exponent e(α) defined as log R n (α) e(α) := lim n log n, (6) where R n (α) is the aggregated source rate. The scaling exponent enables us to ignore less important factors (e.g., log n factor) and focus on the most salient component. For example, n and n/ log n both have scaling exponent of 1/2. 6

7 e(α) ( 1 γ ) / 2 multihop 1 γ Unicast MIMO MHM ( 2 α / 2 γ ) ( 2 α / 2)(1 γ ) 1/ 2 γ 2 3 α Figure 2: Compare MHM, Unicast MIMO, and the nearest-neighbor multicast. 3.4 Multicast Gain In this section, we provide an intuitive explanation on multicast gain using a back-of-the-envelop calculation. The metric is the aggregated multicast transport capacity, and the unit is bit-meter/sec. If a source node sends a bit to all its destinations, which has an average distance of d, then it contributes to dn d bit-meter. Normalized over the time to send this information, T, we obtain the multicast transport capacity of dn d /T bit-meter/sec. We define the multicast gain in the following sense. Consider an optimal multicast scheme with aggregated transport capacity of C m. Consider a structure that uses the optimal unicast scheme to deliver multicast traffic. In particular, it assumes that each multicast flow is n d independent unicast flows and results in an aggregated transport capacity of C u. We call the ratio between C m and C u the multicast gain. In this section, we answer the question how much the multicast gain is. The answer is somewhat counter-intuitive at the first glance. The multicast gain depends on the path loss exponent. If the path loss exponent is small (i.e., α = 2), there is NO multicast gain. On the other hand, if the pass loss exponent is large (i.e., α > 2), then there is a multicast gain. The intuition is explained as follows using a back-of-the-envelop calculation. It has been shown that the hierarchical MIMO structure achieves the optimal unicast capacity within an arbitrarily small exponent. Therefore, it is the scheme used. We call it the unicast MIMO. Its aggregated multicast transport capacity is ) ( ) C u = Θ (N 2 N α 2 N = Θ N 3 2 N 1 α 2. (7) The first term N 2 is the number of sender-receiver pairs when we use a N N MIMO structure; the second term N α/2 is the path loss; the third term N is the average distance between the multicast sender and a receiver (i.e., sending a bit to a destination contributes to 1 N bit-meter). Consider a MHM structure. Its multicast capacity is C M = Θ ( M 2 M α 2 N M N M M ) ( ) = Θ N 3 2 M 1 α 2. (8) The first term M 2 is the average number of sender-receiver pairs when we use a M M MIMO structure; the second term M α/2 is the pass loss for the M M MIMO transmission; the third 7

8 term N/M is the number of spatial reuse; and the last term M is the distance of this hop. The fourth term N/M is the number of destinations this bit contributes to. To elaborate, the average distance in terms of the number of M M long-rang transmissions is N/M, there are total N/M receiver clusters, and it takes N/M to reach all clusters. Therefore, each transmission contributes to an average of N M Θ N ( ) M N = Θ N M M receivers. Compare Eqs. 7 and 8, it is clear that when α = 2, the two schemes are equivalent. On the other hand, if 2 < α < 3, MHM is superior. In MHM, we observe two types of gains. The first type of gain is M 2 M α 2, which we refer to as the power gain. When the path attenuation is large, power transfer is more efficient among small clusters. The second type of gain is N/M N/M M, due to the benefit of exploring multicast where one bit is potentially used by multiple destinations. When 2 < α < 3, the two types of gains contribute to the optimal performance. Compare Eqs. 7 and 8, we see that the multicast gain, G m, is G m = 1, α = 2 n α 2 1 d, 2 α < 3 nd, α 3. (9) 4 Detailed Analysis In Section 3, to simplify the discussion, we assume that destination nodes of a multicast source is evenly distributed. In other words, if we divide the network into clusters of size N/n d, then each cluster has one and only one destination node from each source. Of course, this is not realistic. In this section, we consider the case where destinations are randomly distributed in the network. We first need an appropriate cluster size so that each cluster has at least one destination node (which is necessary to prove the upper bound). Let M be the cluster size. Lemma 1 Let the cluster size be M = 5N ln N n d, (10) Then each cluster is guaranteed to have at least one destination node from each source w.h.p., and each cluster is bounded to have at most 10 ln N = 10γ ln n d destination nodes w.h.p. In addition, the number of nodes in each cluster is between ((1 δ)m, (1 + δ)m) with probability larger than 1 N/Me Λ(δ)M where Λ(δ) is a positive constant independent of N and δ > 0. The first part of the lemma can be proved using Chernoff bound and union bound. The second half of the lemma on the number of nodes in each cluster is a restatement of Lemma 4.1 in [8], and thus the proof is omitted here. Assume M takes the value in Eq. (10) for the rest of the session. As in [8], we assume that there are exactly M nodes in each cluster to simplify the discussion. 8

9 4.1 Achievability We select one delegate for each multicast source in each cluster. If the source node is in the cluster, the source node is the delegate. Otherwise, the delegate is arbitrarily chosen, where it may or may not be a destination node of the multicast flow. There are totally N sources and each cluster has M nodes. Therefore, we can evenly distribute the load of delegates among all M nodes and let each node be delegates for N/M sources (including itself). Thus, all nodes have the same amount of load. The role of the delegate is as follows: in Steps I and III, the delegate serves as the source and destination, respectively. In Step III, through cooperation to decode, the delegate recovers the original multicast message. We then introduce a new step, Step IV. Step IV: The delegates sends the message to all other destination nodes in the cluster through unicast using hierarchical MIMO. The routing is modified accordingly. Consider a multicast flow. Its source node locates in an arbitrary cluster. Its multicast message is first spread out vertically and then horizontally, clusterby-cluster, as shown in Figure 1. Inter-cluster communication is through M M MIMO transmission between neighboring clusters. In each cluster, there are O(log N) destinations. Within each cluster, the delegate recovers the message in Step III through cooperative decoding. Then it relays the message to all other destination nodes in the cluster through unicast using hierarchical MIMO (which causes a capacity loss of O(log N)). If the destination nodes are within the same cluster as the source node, the source node uses unicast to deliver the information to each of the destinations, again with a capacity loss factor of O(log N). For a cluster, it receives the message through M M MIMO transmission at most once. It sends out information using M M MIMO transmission to at most four neighboring clusters. The multicast gain comes from that fact that a message delivered to a neighboring cluster can potentially benefit many receivers. We count the aggregated source rate as follows: each source generates M L bits-long messages. These messages are sent to all delegates hop-by-hop (A hop is a hop of a M M MIMO transmission to a neighboring cluster) and then be delivered to all destinations. To calculate the aggregated source rate, we divide the total amount of source bits (N M L) by the total time to deliver them. The system operates in a pipelining fashion. After a long time of operation, each delegate holds M message of L bits for each source it represents. In other words, each node holds N (N = M N/M) L bits-long messages. Next, we calculate the time to deliver these messages in detail. To establish the achievability result, we need to prove the following Lemmas. First, for Steps I and III, given the cluster size of M, Θ(N/M) clusters can transmit simultaneously while achieving a capacity of M 2 α/2 ɛ. Second, for Step II, there are Θ(N/M) simultaneously M M long range transmissions between neighboring clusters. Step I. Intra-Cluster Distribution Each node has M messages for each multicast source. In the worst case, it needs to deliver the messages to one delegate in each of the four neighboring clusters. Consider one delegate in one cluster. Following a similar approach in [8], we divide each cluster into two half-clusters of equal size, each with M/2 nodes (within a constant factor). One half-cluster is farther away from the destination cluster and one is close to the border. The node delivers two messages to each node in the half-cluster that is farther away from the destination cluster. We use the hierarchical MIMO scheme developed in [8] for these deliveries. Theorem 2.1 states the achievable rate of the scheme. This step may need to repeat at most four times for four neighboring clusters. We need to apply Theorem 2.1 to Step I, where each cluster is of size M. The average power constrain of each user is P. The difference is that the clusters work in parallel and thus we need to 9

10 Figure 3: 9-TDMA scheduling scheme. consider the interference among simultaneously transmitting clusters. Lemma 2 Consider clusters of size M. The clusters follow a 9-TDMA scheduling scheme, as shown in Figure 3. For α > 2, the interference power received by a node from the simultaneously operating clusters transmissions is upperbounded by a constant K I, which is independent of N. For α = 2, the interference-power is bounded by K I2 log N. Moreover, the interference signals received by different nodes in the cluster are zero-mean and un-correlated. Proof: In the 9-TDMA schedule, the clusters with the same number operate simultaneously while other clusters remain inactive. There are at least two inactive clusters between any two clusters that are active. The proof of the lemma is similar to that of Lemma 4.2 in [8]. The modifications needed are as follows. In Lemma 4.2 in [8], the network is a dense network of size 1 and each node is constrained to an average power of P/M (note that M is the largest cluster in the hierarchical MIMO transmission in our context). Because we consider an extended network, the distance is scaled up by a factor of M, and the power is scaled up by a factor of M α/2 to compensate the path attenuation. To meet the average power constraint of P, the scheme only runs a fraction of time, where the fraction is 1/M α/2 1. The rest of the proof follows that of Lemma 4.2 [8] while having P j P M α/2 1 and A c = M. The above interference constraint satisfies the interference constraint required in Theorem 2.1. Therefore, applying Theorem 2.1, each cluster can finish its intra-cluster traffic within a time T (Step I) = 4 9 M N k 1 M 2 α/2 ɛ = 36 k 1N M α/2+ɛ 1 where k 1 > 0 is a constant and ɛ > 0. Step II: neighboring cluster transmission. We construct M/2 M/2 MIMO transmissions that transmit from the farther away half-cluster to the destination half-cluster. The distance between a node in the sending half-cluster and a node in the receiving cluster is lower bounded by M/2 and upper bounded by 2 M. The transmit power is P M α/2 1. To maintain the average power constraint of P, nodes operate only a fraction of time, and the fraction is M 1 α/2. This guarantees that the power received by each node in the destination bluster is bounded below and above by constants that are independent of M when transmitting. 10

11 Figure 4: 25-TDMA scheduling scheme. Lemma 3 Consider clusters of size M. Following a 25-TDMA scheme, as shown in Figure 4, the interference power received by a node from the simultaneously operating inter-cluster transmissions is upperbounded by a constant K I, which is independent of N, when α > 2. For α = 2, the interference-power is bounded by K I2 log N. The lemma holds following the same proof as in Lemma 2 using the fact that the closest interfering cluster is at least two clusters away from the receiving cluster. The lemma will be useful in determining the mutual information achieved by the simultaneous M M transmissions. In total, there are N/M/25 simultaneously transmitting clusters. Each transmission consists of an L-bits long packet, encoded into C symbols using a randomly generated Gaussian code. The M/2 nodes transmit their encoded sequences of length C symbols simultaneously to M/2 nodes in the destination half-cluster. The nodes in the destination cluster quantize the signals they observe during the C transmissions and store there quantized signals without trying to decode the transmitted symbols. Each cluster may need to transmit to four neighboring clusters. Since there are N/M/25 simultaneously transmitting clusters, the total time to complete Step II is N T (Step II) = C M 1 α/2 Step III: Cooperate to decode. In this phase, we aim to provide each delegate, the observations of the symbols that were originally intended for it, so that a delegate can decode the original message. These observations are accumulated as all nodes in the cluster in Step II. Each observation is quantized into Q bits and sent to the delegate using hierarchical MIMO unicast. Again, we use a 9-TMDA scheme as in Step I. Therefore, the total time to complete Step III is We restate Lemma 4.4 in [8]. T (Step III) = 9 Q M N k 1 M 2 α/2 ɛ = 9 k 1 Q N M α/2+ɛ 1 Lemma 4 There exists a strategy to encode the observations at a fixed rate Q bits per observation and get a linear growth of the mutual information for the resultant M M quantized MIMO channel when the network experiences interference from the exterior. 11

12 Using this lemma, and Lemma 3, we conclude that the three phases can effectively realize a virtual MIMO channels achieving spatial multiplexing gain between two delegates in neighboring clusters. Step IV: Distribution from the delegate to destinations. Each delegate has O(log N) destinations in its cluster. It uses unicast to deliver the information to the destination through the hierarchical MIMO scheme. Following the same 9-TMDA scheme as in Step I, the time to deliver the message is T (Step IV) = 9 M N log N k 1 M 2 α/2 ɛ = 9 k 1 N log N M α/2+ɛ 1 Therefore, in total, the total time to deliver M L bits-long message for each source to all its destination is T t = T (Step I) + T (Step II) + T (Step III) + T (Step IV) c 2 N log N M α/2+ɛ 1 Therefore, the aggregated source rate is R s (N) = NML T t K 2 N (2 α/2 ɛ )(1 γ). We note that all the steps take roughly the same amount of time. (The time for Step I and III is higher than that of Step II by a factor of M ɛ where ɛ is positive but arbitrarily small. The time for Step IV is higher than Step II by a factor of M ɛ log N because each cluster has O(log N) destination nodes.) This implies that they are evenly loaded. The cluster size is an important design factor. For α > 2, a larger cluster size is less efficient in power transfer, and a smaller cluster size is less efficient in exploring the multicast gain. 5 Conclusion In this report, we study multicast in large-scale wireless ad hoc networks in an extended network. When 2 α < 3, MIMO is indeed more efficient in terms of power transfer. The MIMO-based multihop scheme can achieve the capacity within an arbitrarily small exponent. The key ingredient is to choose an appropriate cluster size (and thus the corresponding distance for each hop) that optimally integrate the multicast gain and the power transfer gain (loss). References [1] X.-Y. Li, S.-J. Tang, and O. Frieder, Multicast capacity for large scale wireless ad hoc networks, in MobiCom 07: Proceedings of the 13th annual ACM international conference on Mobile computing and networking. New York, NY, USA: ACM, 2007, pp [2] S. Shakkottai, X. Liu, and R. Srikant, The multicast capacity of large multihop wireless networks, in MobiHoc 07: Proceedings of the 8th ACM international symposium on Mobile ad hoc networking and computing. New York, NY, USA: ACM, 2007, pp [3] P. Gupta and P. R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory, vol. IT-46, no. 2, pp , March

13 [4] L. L. Xie and P. R. Kumar, A network information theory for wireless communication: Scaling laws and optimal operation, IEEE Transactions on Information Theory., vol. 50, no. 5, [5] A. Jovicic, P. Viswanath, and S. Kulkarni, Upper bounds to transport capacity of wireless networks, IEEE TRANSACTIONS ON INFORMATION THEORY, vol. 50, no. 11, [6] S. H. A. Ahmad, A. Jovicic, and P. Viswanath, On outer bounds to the capacity region of wireless networks, IEEE/ACM Trans. Netw., vol. 14, no. SI, pp , [7] L.-L. Xie and P. R. Kumar, On the path-loss attenuation regime for positive cost and linear scaling of transport capacity in wireless networks, IEEE/ACM Trans. Netw., vol. 14, no. SI, pp , [8] A. Ozgur, O. Leveque, and D. Tse, Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks, IEEE Transactions on Information Theory, vol. 53, no. 10, pp ,