On Capacity Scaling in Arbitrary Wireless Networks

On Capacity Scaling in Arbitrary Wireless Networks

Urs Niesen, Piyush Gupta, and Devavrat Shah U. Niesen and D. Shah are with the Laboratory of Information and Decision Systems, Department of EECS at the Massachusetts Institute of Technology. Email: {uniesen,devavrat}@mit.eduP. Gupta is with the Mathematics of Networks and Communications Research Department, Bell Labs, Alcatel-Lucent. Email: pgupta@research.bell-labs.comThe work of U. Niesen and D. Shah was supported in parts by DARPA grant (ITMANET) 18870740-37362-C and NSF grant CNS-0546590; the work of P. Gupta was supported in part by NSF Grants CCR-0325673 and CNS-0519535.
Abstract

In recent work, Özgür, Lévêque, and Tse (2007) obtained a complete scaling characterization of throughput scaling for random extended wireless networks (i.e., nodes are placed uniformly at random in a square region of area ). They showed that for small path-loss exponents cooperative communication is order optimal, and for large path-loss exponents multi-hop communication is order optimal. However, their results (both the communication scheme and the proof technique) are strongly dependent on the regularity induced with high probability by the random node placement.

In this paper, we consider the problem of characterizing the throughput scaling in extended wireless networks with arbitrary node placement. As a main result, we propose a more general novel cooperative communication scheme that works for arbitrarily placed nodes. For small path-loss exponents , we show that our scheme is order optimal for all node placements, and achieves exactly the same throughput scaling as in Özgür et al. This shows that the regularity of the node placement does not affect the scaling of the achievable rates for . The situation is, however, markedly different for large path-loss exponents . We show that in this regime the scaling of the achievable per-node rates depends crucially on the regularity of the node placement. We then present a family of schemes that smoothly “interpolate” between multi-hop and cooperative communication, depending upon the level of regularity in the node placement. We establish order optimality of these schemes under adversarial node placement for .

{keywords}

Arbitrary node placement, capacity scaling, cooperative communication, hierarchical relaying, multi-hop communication, wireless networks.

I Introduction

Consider a wireless network with nodes placed on (usually referred to as an extended network), with each node being the source for one of source-destination pairs and the destination for another pair. The performance of this network is captured by , the largest uniformly achievable rate of communication between these source-destination pairs. While the scaling behavior of as the number of nodes goes to infinity is by now well understood for random node placement, little is known for the case of arbitrary node placements. In this paper, we are interested in analyzing the impact of such arbitrary node placement on the scaling of .

I-a Related Work

The problem of determining the scaling of was first analyzed by Gupta and Kumar in [1]. They show that, under random placement of nodes in the region, certain models of communication motivated by current technology, and random source-destination pairing, the maximum achievable per-node rate can scale at most as . Moreover, it was shown that multi-hop communication can achieve essentially the same order of scaling.

Since [1], the problem has received a considerable amount of attention. One stream of work [2, 3, 4, 5, 6, 7, 8] has progressively broadened the conditions on the channel model and the communication model, under which multi-hop communication is order optimal. Specifically, with a power loss of for signals sent over distance , it has been established that under high signal attenuation and random node placement, the best achievable per-node rate for random source-destination pairing scales essentially like and that this scaling is achievable with multi-hop communication.

Another stream of work [9, 10, 11, 12, 8] has proposed progressively refined multi-user cooperative schemes, which have been shown to significantly out-perform multi-hop communication in certain environments. In an exciting recent work, Özgür et al. [8] have shown that with nodes placed uniformly at random, and with low signal attenuation , a cooperative communication scheme can perform significantly better than multi-hop communication. More precisely, they show that for , the best achievable per-node rate for random source-destination pairing scales as and cooperative communication achieves a per-node rate of (here, is an arbitrary but fixed constant). That is, cooperative communication is essentially order optimal in the attenuation regime .

In summary, for random extended networks with random source-destination pairing, the optimal communication scheme exhibits the following threshold behavior: for the cooperative communication scheme is order optimal, while for the multi-hop communication scheme is order optimal.

I-B Our Contributions

The characterization of the scaling of as a function of the path-loss exponent mentioned in the last paragraph depends critically on the regularity induced with high probability by placing the nodes uniformly at random. However, a wireless network encountered in practice might not exhibit this amount of regularity. Our interest is therefore in understanding the impact of the node placement on the scaling of . To this end, we consider wireless networks with arbitrary (i.e., deterministic) node placement (with minimum-separation constraint).

The impact of this arbitrary node placement depends crucially on the path-loss exponent . For small path-loss exponents , we show that for random source-destination pairing, the rate of the best communication scheme is upper bounded as . We then present a novel cooperative communication scheme that achieves for any path-loss exponent a per-node rate of . Thus, our cooperative communication scheme is essentially order optimal for any such arbitrary network with . In other words, in the small path-loss regime, the scaling of is the same irrespective of the regularity of the node placement.

The situation is, however, quite different for large path-loss exponents . We show that in this regime the scaling of depends crucially on the regularity of the node placement, and multi-hop communication may not be order optimal for any value of . In fact, for less regular networks we need more complicated cooperative communication schemes to achieve optimal network performance. Towards that end, we present a family of communication schemes that smoothly “interpolate” between cooperative communication and multi-hop communication, and in which nodes communicate at scales that vary smoothly from local to global. The amount of “interpolation” between the cooperative and multi-hop schemes depends on the level of regularity of the underlying node placement. We establish the optimality of this family of schemes for all under adversarial node placement.

In summary, for the regularity of the node placement has no impact on the scaling of . Cooperative communication is order optimal in this regime and achieves the same scaling as in the case of random node placement. For the regularity of the node placement strongly impacts the scaling of , and a communication scheme “interpolating” between multi-hop and cooperative communication depending on the regularity of the node placement is order optimal (under adversarial node placement). In particular, simple multi-hop communication may not be order optimal for any . This contrasts with the case of random node placement where multi-hop communication is order optimal for all .

I-C Organization

The remainder of this paper is organized as follows. Section II describes in detail the communication model. Section III provides formal statements of our results. Sections IV and V describe our new cooperative communication scheme (for the regime) and “interpolation” scheme (for the regime) for arbitrary wireless networks. Sections VI through XI contain proofs. Finally, Sections XII and XIII contain discussions and concluding remarks.

Ii Model

In this section, we introduce some notational conventions and describe in detail the network and channel models.

We use the following conventions: for different denote strictly positive finite constants independent of . Vectors and matrices are denoted by boldface whenever the vector or matrix structure is of importance. We denote by and transpose and conjugate transpose, respectively. To simplify notation, we assume, when necessary, that fractions are integers and omit and operators.

Consider the square

of area , and let be a set of nodes on111The setting considered here with nodes placed on a square of area is called an extended network. If the nodes are placed on a square of unit area, we speak of a dense network. While dense networks are not treated in detail in this paper, we briefly discuss implications of the results for the dense setting in Section XII-C. . We say that has minimum-separation if for all , where is the Euclidean distance between nodes and . We use the same channel model as in [8]. Namely, the (sampled) received signal at node is

(1)

for all , and where are the (sampled) signals sent by the nodes in . Here are independent and identically distributed (i.i.d.) with distribution (i.e., circularly symmetric complex Gaussian with mean and variance ), and

for path-loss exponent . We assume that for each , the phases are i.i.d.222It is worth pointing out that recent work [13] suggests that, under certain assumptions on scattering elements, for , and for very large values of , the i.i.d. phase assumption as a function of used here is too optimistic. However, subsequent work by the same authors [14] shows that under different assumptions on the scatterers, the channel model used here is still valid even for , and for very large values of . This indicates that the question of channel modeling for very large networks in the low path-loss regime is somewhat delicate and requires further investigation. We point out that for this issue does not arise. with uniform distribution on . We either assume that for each the random process is stationary ergodic in , which is called fast fading in the following, or that for each the random process is constant in , which is called slow fading in the following. In either case, we assume full channel state information (CSI) is available at all nodes, i.e., each node knows all at time . While the full CSI assumption is quite strong, it can be shown that availability of a -bit quantized version of at all nodes is sufficient for the achievable schemes presented here (see Section XII-A for the details). We also impose an average power constraint of on the signal for every node .

Each node wants to transmit information at uniform rate to some other node . We call the source and the destination node of this communication pair. The set of all communication pairs can be described by a traffic matrix , where the entry in corresponding to is equal to if node is a source for node . We say that is a permutation traffic matrix if it is a permutation matrix (i.e., every node is a source for exactly one communication pair and a destination for exactly one communication pair). For a traffic matrix , let be the highest rate of communication that is uniformly achievable for each source-destination pair. For a permutation traffic matrix , can also be understood as the maximal achievable per-node rate.

Iii Main Results

This section presents the formal statement of our results. The results are divided into two parts. In Section III-A, we consider low path-loss exponents, i.e., . We present a cooperative communication scheme for arbitrary node placement and for either fast or slow fading. We show that this communication scheme is order optimal for all node placements when . In Section III-B, we consider high path-loss exponents, i.e., . We present a communication scheme that “interpolates” between the cooperative and the multi-hop communication schemes, depending on the regularity of the node placement. We show that this communication scheme is order optimal under adversarial node placement with regularity constraint when .

Iii-a Low Path Loss Regime

The first result proposes a novel communication scheme, called hierarchical relaying in the following, and bounds the per-node rate that it achieves. This provides a lower bound to , the largest achievable per-node rate. The hierarchical relaying scheme enables cooperative communication on the scale of the network size. In the random node placement case, this cooperation could be enabled in a cluster around the source node (cooperatively transmitting) and in a cluster around its destination node (cooperatively receiving). With arbitrary node placement, such an approach does no longer work, as both the source as well as the destination nodes may be isolated. The hierarchical relaying scheme circumvents this issue by relaying data between each source-destination pair over a densely populated region in the network. A detailed 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 , , and , there exists

such that for any , node placement with minimum separation , and permutation traffic matrix , we have

The same conclusion holds for slow fading with probability at least

as .

Theorem 1 shows that the per-node rate achievable by the hierarchical relaying scheme is at least , where the “loss” term converges to zero as at a rate arbitrarily close to (by choosing small). The performance of the hierarchical relaying scheme can intuitively be understood as follows. As mentioned before, the scheme achieves cooperation on a global scale. This leads to a multi-antenna gain of order . On the other hand, communication is over a distance of order , leading to a power loss of order . Combining these two factors results in a per-node rate of .

We note that Theorem 1 remains valid under somewhat weaker conditions than having minimum separation . Specifically, we show that the result of Özgür et al. [8] can be recovered through Theorem 1 as the random node placement satisfies these weaker conditions. We discuss this in more detail in Section XII-D.

The next theorem establishes optimality of the hierarchical relaying scheme in the range of for arbitrary node placement. The proof of the theorem is presented in Section VIII.

Theorem 2.

Under either fast or slow fading, for any , , there exists such that for any , node placement with minimum separation , and for chosen uniformly at random from the set of all permutation traffic matrices, we have

with probability as .

Note that Theorem 2 holds only with probability for different reasons for the slow and fast fading case. For fast fading, this is due to the randomness in the selection of the permutation traffic matrix. In other words, for fast fading, with high probability we select a traffic matrix for which the theorem holds. For the slow fading case, there is additional randomness due to the fading realization. Here, with high probability we select a traffic matrix and we experience a fading for which the theorem hold.

Comparing Theorems 1 and 2, we see that for the proposed hierarchical relaying scheme is order optimal, in the sense that

Moreover, the rate it achieves is the same order as is achievable in the case of randomly placed nodes. Hence in the low path-loss regime , the heterogeneity caused by the arbitrary node placement has no effect on achievable communication rates.

Iii-B High Path Loss Regime

We now turn to the high path-loss regime . In the case of randomly placed nodes, multi-hop communication achieves a per-node rate of with probability and is order optimal for . For arbitrarily placed 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 slow fading, for any , for any , there exists a node placement with minimum separation such that for chosen uniformly at random from the set of all permutation traffic matrices, we have

as with probability .

Comparing Theorem 3 with Theorem 1 shows that under adversarial node placement with minimum-separation constraint the hierarchical relaying scheme is order optimal even when . Moreover, Theorem 3 shows that there exist node placements satisfying a minimum separation constraint for which hierarchical relaying achieves a rate of at least a factor of order higher than multi-hop communication for any . In other words, for those node placements cooperative communication is necessary for order optimality also for any , in stark contrast to the situation with random node placement, where multi-hop communication is order optimal for all .

Theorem 3 suggests that it is the level of regularity of the node placement that decides what scheme to choose for path-loss exponent . So far, we have seen two extreme cases: For random node placement, resulting in very regular node placements with high probability, only local cooperation is necessary and multi-hop is an order-optimal communication scheme. For adversarial arbitrary node placement, resulting in a very irregular node placement, global cooperation is necessary and hierarchical relaying is an order-optimal communication scheme. We now make this notion of regularity precise, and show that, depending on the regularity of the node placement, an appropriate “interpolation” between multi-hop and hierarchical relaying is required for to achieve the optimal performance. We refer to this “interpolation” scheme as cooperative multi-hop communication in the following.

Before we state the result, we need to introduce some notation. Consider again a node placement with minimum separation . Divide into squares of sidelength , and fix a constant . We say that is -regular at resolution if every such square contains at least nodes. Note that every node placement is trivially -regular at resolution ; a random node placement can be shown to be -regular at resolution with probability as for any ; and nodes that are placed on each point in the integer lattice inside are -regular at resolution .

The cooperative multi-hop scheme enables cooperative communication on the scale of regularity . Neighboring squares of sidelength cooperatively communicate with each other. To transmit between a source and its destination, we use multi-hop communication over those squares. In other words, we use cooperative communication at small scale , and multi-hop communication at large scale . For regular node placements, i.e., , the cooperative multi-hop scheme becomes the classical multi-hop scheme. For very irregular node placement, i.e., , the cooperative multi-hop scheme becomes the hierarchical relaying scheme discussed in the last section.

The next theorem provides a lower bound on the per-node rate achievable with the cooperative multi-hop scheme. The proof of the theorem can be found in Section X.

Theorem 4.

Under fast fading, for any , , , and there exists

such that for any , node placement with minimum separation , and permutation traffic matrix , we have

where

The same conclusion holds for slow fading with probability as .

Theorem 4 shows that if is regular at resolution then a per-node rate of at least is achievable, where, as before, the “loss” term converges to zero as at a rate arbitrarily close to . The performance of the cooperative multi-hop scheme can intuitively be understood as follows. The scheme achieves cooperation on a scale of . This leads to a multi-antenna gain of order . On the other hand, communication is over a distance of order , leading to a power loss of order . Moreover, each source-destination pair at a distance of order must transmit their data over order many hops, leading to a multi-hop loss of . Combining these three factors results in a per-node rate of .

The next theorem shows that Theorem 4 is tight under adversarial node placement under a constraint on the regularity. The proof of the theorem is presented in Section XI.

Theorem 5.

Under either fast or slow fading, for any , there exists , such that for any , and , there exists a node placement with minimum separation and -regular at resolution such that for chosen uniformly at random from the set of all permutation traffic matrices, we have

with probability as .

As an example, assume that

for some . Then Theorem 4 shows that for any node placement of regularity and ,

where converges to zero as at a rate arbitrarily close to . In other words

Moreover, by Theorem 5 there exist node placements with same regularity such that for random permutation traffic with high probability is (essentially) of the same order, in the sense that

In particular, for (i.e., regular node placement), and for (i.e., random node placement), we obtain the order scaling as expected. For (i.e., completely irregular node placement), we obtain the order scaling as in Theorems 1 and 3.

Iv Hierarchical Relaying Scheme

This section describes the architecture of our hierarchical relaying scheme. On a high level, the construction of this scheme is as follows. Consider nodes placed arbitrarily on the square region with a minimum separation . Divide into squarelets of equal size. Call a squarelet dense, if it contains a number of nodes proportional to its area. For each source-destination pair, choose such a dense squarelet as a relay, over which it will transmit information (see Figure 1).

BCMAC
Fig. 1: Sketch of one level of the hierarchical relaying scheme. Here are three source-destination pairs. Groups of source-destination pairs relay their traffic over dense squarelets, which contain a number of nodes proportional to their area (shaded). We time share between the different dense squarelets used as relays. Within all these relay squarelets the scheme is used recursively to enable joint decoding and encoding at each relay.

Consider now one such relay squarelet and the nodes that are transmitting information over it. If we assume for the moment that all the nodes within the same relay squarelet could cooperate then we would have a multiple access channel (MAC) between the source nodes and the relay squarelet, where each of the source nodes has one transmit antenna, and the relay squarelet (acting as one node) has many receive antennas. Between the relay squarelet and the destination nodes, we would have a broadcast channel (BC), where each destination node has one receive antenna, and the relay squarelet (acting again as one node) has many transmit antennas. The cooperation gain from using this kind of scheme arises from the use of multiple antennas for these multiple access and broadcast channels.

To actually enable this kind of cooperation at the relay squarelet, local communication within the relay squarelets is necessary. It can be shown that this local communication problem is actually the same as the original problem, but at a smaller scale. Hence we can use the same scheme recursively to solve this subproblem. We terminate the recursion after several iterations, at which point we use simple TDMA to bootstrap the scheme.

The construction of the hierarchical relaying scheme is presented in detail in Section IV-A. A back-of-the-envelope calculation of the per-node rate it achieves is presented in Section IV-B. A detailed analysis of the hierarchical relaying scheme is presented in Sections VI and VII.

Iv-a Construction

Recall that

is the square region of area . The scheme described here assumes that nodes are placed arbitrarily in with minimum separation . We want to find some rate, say , that can be supported for all source-destination pairs of a given permutation traffic matrix . The scheme that is described below is “recursive” (and hence hierarchical) in the following sense. In order to achieve rate for nodes in , it will use as a building block a scheme for supporting rate for a network of

nodes over (square of area ) with

for any permutation traffic matrix of nodes. Here the branching factor is a function such that as . We will optimize over the choice of later. The same construction is used for the scheme over , and so on. In general, our scheme does the following at level of the hierarchy (or recursion). In order to achieve rate for any permutation traffic matrix over

nodes in , with

use a scheme achieving rate over nodes in for any permutation traffic matrix . The recursion is terminated at some level to be chosen later.

We now describe how the hierarchy is constructed between levels and for . Each source-destination pair chooses some squarelet as a relay over which it transmits its message. This relaying of messages takes place in two phases – a multiple access phase and a broadcast phase. We first describe the selection of relay squarelets, then the operation of the network during the multiple access and broadcast phases, and finally the termination of the hierarchical construction.

Iv-A1 Setting up Relays

Given nodes in , divide the square region into equal sized squarelets. Denote them by . Call a squarelet dense if it contains at least nodes. In other words, a dense squarelet contains a number of nodes of at least a fraction of its area. We show that since the nodes in have constant minimum separation , a squarelet can contain at most (i.e. ) nodes, and hence that there are at least dense squarelets. Each source-destination pair chooses a dense squarelet such that both the source and the destination are at a distance from it. We call this dense squarelet the relay of this source-destination pair. We show that the relays can be chosen such that each relay squarelet has at most communication pairs that use it as relay, and we assume this worst case in the following discussion.

Iv-A2 Multiple Access Phase

Source nodes that are assigned to the same (dense) relay squarelet send their messages simultaneously to that relay. We time share between the different relay squarelets. If the nodes in the relay squarelet could cooperate, we would be dealing with a MAC with at most transmitters, each with one antenna, and one receiver with at least antennas. In order to achieve this cooperation, we use a hierarchical (i.e., recursive) construction. For this recursive construction, assume that we have access to a communication scheme to transmit data according to a permutation traffic matrix between nodes located in a square of area . We now show how this scheme at scale can be used to construct a scheme for scale (see Figure 2).

Fig. 2: Description of the multiple access phase at level in the hierarchy with . The first system block represents the wireless channel, connecting source nodes with relay nodes . The second system block are quantizers used at the relay nodes. The third system block represents using times the communication scheme at level (organized as permutation traffic matrices ) to “transpose” the matrix of quantized observations . In other words, before the third system block, node has access to , and after the third system block, node has access to . The fourth system block are matched filters used at the relay nodes.

Suppose there are source nodes (located anywhere in ) that relay their message over the relay nodes (located in the same dense squarelet of area ). Each source node divides its message bits into parts of equal length. Denote by the encoded part of the message bits of node ( is really a large sequence of channel symbols; to simplify the exposition, we shall, however, assume it is only a single symbol). The message parts corresponding to will be relayed over node , as will become clear in the following. Sources , transmit at time for .

Let be the observed channel output at relay at time . Note that depends only on channel inputs . In order to decode the message parts corresponding to at relay node , it needs to obtain the observations from all other relay nodes. In other words, all relays need to exchange information. For this, each relay quantizes its observation at an appropriate rate independent of to obtain . Quantized observation is to be sent from relay to relay . Thus, each of the relay nodes now has a message of size for every other relay node.

This communication demand within the relay squarelet can be organized as permutation traffic matrices between the relay nodes. Note that these relay nodes are located in the same square of area . In other words, we are now faced with the original problem, but at smaller scale . Therefore, using times the assumed scheme for transmitting according to a permutation traffic matrix for nodes in , relay can obtain all quantized observations . Now uses matched filters on to obtain estimates of . In other words, each node computes333Note that, since we assume full CSI, node has access to the channel gains at any time . In particular, this is the case at the time the matched filtering is performed.

for every . Using these estimates it then decodes the messages corresponding to .

Iv-A3 Broadcast Phase

Nodes in the same relay squarelet then send their decoded messages simultaneously to the destination nodes corresponding to this relay. We time share between the different relay squarelets. If the nodes in the relay squarelet could cooperate, we would be dealing with a BC with one transmitter with at least antennas and with at most receivers, each with one antenna. In order to achieve this cooperation, a similar hierarchical 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 between nodes located in a square of area . We again use this scheme at scale in the construction of the scheme for scale (see Figure 3).

Fig. 3: Description of the broadcast phase at level in the hierarchy with . The first system block represents transmit beamforming at each of the relay nodes . The second system block are quantizers used at the relay nodes. The third system block represents using times the communication scheme at level (organized as permutation traffic matrices ) to “transpose” the matrix of quantized beamformed channel symbols . In other words, before the third system block, node has access to , and after the third system block, node has access to . The fourth system block is the wireless channel, connecting relay nodes with destination nodes .

Suppose there are relay nodes (located in the same dense squarelet of area ) that relay traffic for destination nodes (located anywhere in ). Recall that at the end of the MAC phase, each relay node has (assuming decoding was successful) access to parts of the message bits of all source nodes . Node re-encodes these parts independently; call the encoded channel symbols (as before, we assume is only a single symbol to simplify exposition). Relay node then performs transmit beamforming on for the transmit antennas of to be sent at time (for some appropriately chosen not depending on ). Call the resulting channel symbol to be sent from relay node . Then444Note that, since we only assume causal CSI, relay node does not actually have access to at the time the beamforming is performed. This problem can, however, be circumvented. The details are provided in the proofs (see Lemma 10).

In order to actually send this channel symbol, relay node needs to obtain from node . Thus, again all relay nodes need to exchange information.

To enable local cooperation within the relay squarelet, each relay node quantizes its beamformed channel symbols at an appropriate rate with independent of to obtain . Now, quantized value is sent from relay to relay . Thus, each of the relay nodes now has a message of size for every other relay node.

This communication demand within the relay squarelet can be organized as permutation traffic matrices between the relay nodes. Note that these relay nodes are located in the same square of area . Hence, we are again faced with the original problem, but at smaller scale . Using times the assumed scheme for transmitting according to a permutation traffic matrix for nodes in , relay can obtain all quantized beamformed channel symbols . Now each sends over the wireless channel at time instance (with chosen to account for the preceding MAC phase and the local cooperation in the BC phase). Call the received channel output at destination node at time instance . Using , destination node can now decode part of the message bits of its source node .

Iv-A4 Spatial Re-Use and Termination of Recursion

The scheme does appropriately weighted time-division among different levels . Within any level , multiple regions of the original square of area are being operated in parallel. The details related to the effects of interference between different regions operating at the same level of hierarchy are discussed in the proofs.

The recursive construction terminates at some large enough level (to be chosen later). At this scale, we have nodes in area . A permutation traffic matrix at this level comprises source-destination pairs. These transmissions are performed using TDMA. Again, multiple regions in the original square of area at level are active simultaneously.

Iv-B Achievable Rates

Here we present a back-of-the-envelope calculation of the per-node rate achievable with the hierarchical relaying scheme described in the previous section. The complete proof is stated in Section VII. We assume throughout that long block codes and corresponding optimal decoders are used for transmission.

Instead of computing the rate achieved by hierarchical relaying, it will be convenient to instead analyze its inverse, i.e., the time utilized for transmission of a single message bit from each source to its destination under a permutation traffic matrix . Using the hierarchical relaying scheme, each message travels through levels of the hierarchy. Call the amount of time spent for the transmission of one message bit between each of the source-destination pairs at level in the hierarchy. We compute recursively.

At any level , there are multiple regions of area operating at the same time. Due to the spatial re-use, each of these regions gets to transmit a constant fraction of time. It can be shown that the addition of interference due to this spatial re-use leads only to a constant loss in achievable rate. Hence the time required to send one message bit is only a constant factor higher than the one needed if region is considered separately. Consider now one such region . By the time sharing construction, only one of its dense relay squarelets of area is active at any given moment. Hence the time required to operate all relay squarelets is a factor higher than for just one relay squarelet separately. Consider now one such relay squarelet, and assume source nodes in communicate each message bits to their respective destination nodes through a MAC phase and BC phase with the help of the relay nodes in this relay squarelet of area .

In the MAC phase, each of the sources simultaneously sends one bit to each of the relay nodes. The total time for this transmission is composed of two terms.

  1. Transmission of message bits from each of the source nodes to those many relay nodes. Since we time share between relay squarelets, we can transmit with an average power constraint of during the time a relay squarelet is active, and still satisfies the overall average power constraint of . With this “bursty” transmission strategy, we require a total of

    (2)

    channel uses to transmit bits per source node. The terms on the left-hand side of (2) can be understood as follows: is the number of bits to be transmitted; is the power loss since most nodes communicate over a distance of ; is the average transmit power; is the multiple-antenna gain, since we have that many transmit and receive antennas.

  2. We show that constant rate quantization of the received observations at the relays is sufficient. Hence the bits for all sources generate transmissions at level of the hierarchy. Therefore,

    (3)

    channel uses are needed to communicate all quantized observations to their respective relay nodes.

Combining (2) and (3), accounting for the factor loss due to time division between relay squarelets, we obtain that the transmission time for one message bit from each source to the relay squarelet in the MAC phase at level is

(4)

Next, we compute the number of channel uses per message bit received by the destination nodes in the BC phase. Similar to the MAC phase, each of the relay nodes has message bits out of which one bit is to be transmitted to each of the destination nodes. Since there are relay nodes, each destination node receives message bits. As before the required transmission time has two components.

  1. Transmission of the encoded and quantized message bits from each of the relay nodes to all other relay nodes at level of the hierarchy. We show that each message bit results in quantized bits. Therefore, bits need to be transmitted from each relay node. This requires

    (5)

    channel uses.

  2. Transmission of message bits from the relay nodes to each destination node. As before, we use bursty transmission with an average power constraint of during the fraction of time each relay squarelet is active (this satisfies the overall average power constraint of ). Using this bursty strategy requires

    (6)

    channel uses for transmission of bits per destination node. As in the MAC phase, in the left hand side of (6) can be understood as the number of bits to be transmitted, as the power loss for communicating over distance , as the average transmit power, and as the multiple-antenna gain.

Combining (5) and (6), accounting for a factor loss due to time division between relay squarelets, the transmission time for one message bit from the relays to each destination node in the BC phase at level is

(7)

From (4) and (7), we obtain the following recursion

(8)

where we have used . This recursion holds for all . At level , we use TDMA among nodes in region with a permutation traffic matrix . Each of the source-destination pairs uses the wireless channel for fraction of the time at power , satisfying the average power constraint. Assuming the received power is less than for all (so that we operate in the power limited regime), we can achieve a rate of at least between any source-destination pair. Equivalently

(9)

Combining (IV-B) and (IV-B), we have

(10)

The term

is the “loss” factor over the desired order scaling, and we now choose the branching factor and the hierarchy depth to make it small. Fix a and set

With this