WIRELESS relays are commonly used to extend the coverage

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1 On Capacity of Relay-Assisted Communication Ashutosh Sabharwal Department of Electrical and Computer Engineering Rice University, Houston TX 77. Abstract In this paper, we study communication in wireless networks where there is no central control available, like wireless ad hoc and sensor networks. We derive the outage probability of multi-hop communication and study it in detail for a simple linear topology. The results indicate that an appropriate choice of route can lead to large performance gains over alternate routes. In the second part of the paper, we consider the problem of estimating outage-minimizing routes using channel measurements. We show that finding the optimal routes reliably requires many independent channel measurements, which is equivalent to sending multiple route discovery requests. The large overhead in route discovery can prove to be counter-productive in networks which transmit information at the same time-scale as the mobility of nodes. I. INTRODUCTION WIRELESS relays are commonly used to extend the coverage of infrastructure-based wireless networks into regions with low signal power [ ], reduce infrastructure cost [6 8], or for load balancing across cells [9]. In wireless ad hoc networks, nodes rely on packet forwarding by other mobile nodes in the system to reach their destination [], thereby ensuring network connectivity. Appropriate network and packet level coding in relay networks can lead to significant gain in achievable communication rates [ 4]. In this paper, we focus on relay-assisted communication in wireless networks where network topology (number of nodes, their locations and the channels between them) is unknown to all the nodes. Thus, there are no central controllers (like basestations or access points) which can coordinate communication among multiple competing nodes. We first show that perfect topology information at the source node can allow the source to optimally choose its multi-hop forwarding strategy, which can result in large improvements in end-to-end performance over other possible routes. Though the results are studied for a simple linear topology, but they convey an already well-appreciated message in ad hoc networking literature: only good routes lead to good performance [, ]. In the second part of the paper, we show that the network resources required to learn the topology and then derive gains from that knowledge, can be significant. Thus, leading to a tradeoff between consuming network resources to learn the unknown topology and the improvement in network capacity when equipped with this topology information. The tradeoff discussed in this paper points to the possibility that route discovery as a precursor to actual data communi- Ricochet TM network is an example of low-cost infrastructure using wireless relays (see for more information). cation may not be the best solution for all wireless networks. We conjecture that the time-scale of mobility and the packet arrivals in the network should determine whether routing based transmission is a suitable solution or not. We contend that routing based communication is useful only if node mobility is at a much longer time-scale compared to the packet arrival rate. The rest of the paper is organized as follows. In Section II, we describe the channel model and review the concept of outage probability. In Section III, we derive the outage of multihop communication and study the linear topology in detail. The measurement-communication efficiency tradeoff associated with routing based transmission is discussed in Section IV. We conclude in Section V. II. SYSTEM MODEL In this section, we outline the channel model used to analyze relay-assisted systems. We also review the performance metric, outage probability, used for performance analysis. A. Channel Model In our subsequent analysis, we will use a block flat fading channel [6], where the fading is assumed to be constant for consecutive symbols. The length represents the coherence interval of the channel, which is assumed to be long enough to permit reliable channel tracking at the receiver. Let the transmitted signal be represented by, where is the node number. The signal can either be the signal transmitted by a user or the relay. The received signal at the relay or destination is given by () where is the power transmitted by node, is the distance between the transmitter and receiver, and is the path loss exponent (for simplicity, assumed to be same for all senderreceiver pairs). The channel coefficients and are assumed to be independent of each other and identically distributed as complex circularly symmetric Gaussian variables with zero mean and unit variance. The second term represents the interference caused by other simultaneous transmissions in the network. The additive noise is assumed to be complex circularly symmetric Gaussian with zero mean and variance. Finally, each user is assumed to have the available average power of.

2 - Q B. Outage Probability In block fading channels with long coherence intervals, outage probability is a good indicator of packet loss in coded systems [6]. An outage occurs when the instantaneous mutual information is less than the requested data rate. For the channel model in () and assuming that the interfering nodes use Gaussian codes, a transmission is in outage if where is the rate of transmission in bits/s/hz. When there are no interfering users (the second term in () is zero), then the probability of outage is given by The factor $#& where. $#& () & is the average received SNR at the receiving node. Prob +*,(- $#& ( / ) III. OUTAGE OF MULTI-HOP COMMUNICATIONS In this section, we present the outage probability analysis of relay-assisted multihop communications, where the data traverses multiple relays to reach the destination. Consider multihop communication with hops, which implies that there are nodes in the communication link. The zero-hop link is equivalent to direct communication between the source and the destination. For the sake of simplicity, let nodes and be the source and destination nodes, respectively. Further assume that data is forwarded from node to node in the link. Let distance between node and be represented by. For communication between nodes and, the outage probability for an end-to-end transmission rate of bits/s/hz is given by (see Section II-B) Prob 4 6& 7 7 () The factors and depend on the number of spatially concurrent communinations, and ensures that the total transmission rate is. For example, if only one node can be active in the whole link at any time, then and 98. On the other hand, if every link can be simultaneously active, then and 7. Note that with more simultaneously active links, smaller rate per transmission suffices to achieve the end-to-end target rate of bits/s/hz but each transmission encounters more interference (from other active links). In the rest of the paper, we will focus on the case when only one node Fig.. Outage Probability 4 hop hop hop hop Direct SNR per link (in db) Outage performance of direct,,, and 4-hop communication with equi-spaced relay nodes for bits/s/hz. transmits at any given time; thus and 7 A8. In this case, the outage is given by (). For a packet to successfully reach the destination, it should not be in outage at all links. Thus, the probability of outage for hop communication is CBED F +*,JILKNMPO Q4R O Note that A8 corresponds to direct communication between the source and destination. To understand the outage performance of multi-hop links, SBED F, we consider the case where all nodes are located on the straight line joining the source and destination. In this case,, where is the distance between the source and the destination. We further assume that only one link can be active at any given time, which implies. Though each link transmits at times the rate of the single-hop link, the power used per link is equal to. Further, total average network power (sum of average transmit powers of all the nodes) is also equal to, since each node is active only out slots. Though the physical layer resources (bandwidth and power) are same for all, the average packet delay increases with. Figures and show the outage performance for -hop communication, where VU EW. The results are for the #X for -hop commu- T8 straight line topology, with nication. For A8 Y (4) bits/s/hz, the example in Figure shows that the 4-hop performance is better than all other cases, with gains of.,, and db over -hop, -hop, -hop and direct communication, respectively. For all practical purposes, the performance of 4-hop and -hop communication is same, and hence either of the routes can be chosen when the objective is to minimize end-to-end outage probability. In fact, it is very hard to distinguish between the two routes (4 and -hop) in practice (see the next section for relevant results). As the data rate requirements increase, routes with fewer hops are favored. Figure shows the case for bits/s/hz,

3 F, 4 hop hop hop hop Direct 9 8 Outage Probability Optimal Number of Hops (K) SNR per link (in db)... Rate (bits/s/hz) Fig.. Outage performance of direct,,, and 4-hop communication with equi-spaced relay nodes for :; bits/s/hz. Fig.. Optimal number of hops for the straight line relay topology. in which direct communication outperforms all other routes. The incremental loss of adding an additional hop over direct route is.,.,.8 and. db. The effect of relay location for the straight line topology can also be easily analyzed using () and (4). Minimizing BVD F is equivalent to minimizing the following quantity,. The outage CBVD F is minimum when #& for all. Thus, the optimal locations for the relay nodes to achieve minimal outage is when they are equi-spaced. The above locations for the relay nodes ensures that all hops have identical outage performance, since the received SNR is identical for each node. The straight line topology can be further analyzed to compare the performance of -hop communication with single-hop (direct) communication. For simplicity, we assume that relay nodes are placed in their outage-minimizing locations along # the straight line joining source and destination, i.e.,, which leads to. ) The optimum number of hops can be obtained for each by choosing a configuration which minimizes (), and is shown in Figure. From Figure, it is clear that for higher rates, fewer hops are preferred. Finally, the hop communication is better than the single hop if.., where CBVD *, -. The critical rate can thus be found by solving () (6) beyond which direct communication The critical rate is better than the -hop communication with equally spaced relays is shown in Figure 4. Critical Rate (bits/s/hz) Number of Hops (K) Fig. 4. Critical rate beyond which single hop communication is better than -hop communication. IV. IMPACT ON OPTIMAL ROUTE DISCOVERY The main purpose of analyzing the simplified straight line topology (in Section III) was to gain insight into how the routes should be chosen in wireless (ad hoc and sensor) networks. Typically, wireless networks without any infrastructure (central coordinating nodes like base-stations or access points) perform a route discovery and route selection before actual transmission of data []. From the analysis in Section III, it is evident that the choice of routes can have significant impact on the achievable end-to-end reliability of multi-hop communication. In practice, the choice of optimal (minimal outage) routes has to be based on the channel measurements. In this section, we discuss the tradeoffs involved in choosing optimal routes. To make the following discussion concrete, we will restrict our attention to two commonly used metrics of route selection from the set of discovered routes: minimum hop count route and the best quality (minimum outage probability) route. To find the route with minimum hop count requires no channel quality measurements, while ensuring network connectivity. From Figure, it is clear that choosing a route with the minimum number of hops is not always outage optimal. To

4 4 find the route with minimal outage, the source has to rely on channel measurements. The outage optimal route is estimated by comparing the estimated outage probabilities, CBVD F +* KNMPO, ILKNM O Q R O O Q (7) where is the estimated average SNR for the link, using multiple independent measurements of the channel conditions. In Figures and 7, we consider the probability of correctly detecting the optimal route using multiple channel measurements for 8 and bits/s/hz. As the number of channel measurements increase, Y the outage optimal route is found with higher success rate. Figures 6 and 8 show the histograms with and independent measurements for rates 8 and bits/s/hz, respectively. The seemingly poor performance Y for 8 can be attributed to the fact that the performance difference Y between the 4-hop (optimal) and -hop is only. db (see Figure ), thus the two paths are virtually indistinguishable. The combined probability of detection for 4 and -hop links is close to one for independent measurements. The main reason for poor performance for small number of measurements is that outage probability depends on average SNR, estimation of which requires measuring the channel multiple times. Most route discovery requests traverse the network once, and hence cannot hope to estimate average SNR with a single measurement. In some cases, each node can maintain the performance of its links with other neighbouring nodes, which is equivalent to having large number of channel measurements []. But these cases require all links to constantly active, a situation unlikely to happen in practice for dense networks, because spatial concurrency of transmissions does not allow all links to be active all the time. The above simulation experiments clearly show the important tradeoff in communicating over wireless networks with unknown topology. To achieve optimal performance resulting in large system capacity, it is important to correctly discover the best paths to the destination. But discovery of the best paths requires repeated measurements which can tax the total network resources. Thus reliable network topology knowledge is critical to achieve the highest resource utilization, but reliability of topology information comes at a significant resource cost. The tradeoff between accurately knowing network topology and using it to increase net throughput is reminiscent of a well understood tradeoff in wireless communication [7]. Power control leads to significant reduction in outage probability, but requires using valuable network resources to learn the channel at both the receiver and transmitter. Replacing single link with a time-varying multi-hop channel and power control with combined coding and forwarding, we arrive at a similar tradeoff in routing accuracy and resultant performance. We conclude the section by conjecturing that routing based communication is best suited when the mobility of nodes is at much longer time-scale than the packet arrivals. If each route is used for a number of packets before it is no longer optimal Probability of Finding the Outage Optimal Path Number of Independent Route Measurements Fig.. Accuracy of estimating outage optimal routes using channel measurements. In this example, 4-hop route is optimal. The source is presented with routes. Frequency Number of Hops Fig. 6. Histogram of selected routes for : ; bits/s/hz, for and independent measurements. Note that due to small difference in outage performance of 4-hop and -hop connection, most of the probability mass for measurements is shared by the two routes. For all practical purposes, either of the two routes have identical performance. or available, then the resources used in the route discovery constitute only a small fraction of the total network resources. A more concrete statement of the above conjecture is the focus of our current research. V. CONCLUSIONS In this paper, we highlighted a fundamental tradeoff in the resource utilization of commonly used routing-based communication methods in ad hoc networks. Using elementary information theoretic methods, we quantified the gains in using some routes over the other for a linear topology. The results show that proper selection of routes is critical for routing methods to result in efficient network resource utilization. We then showed that channel estimation based route selection can prove to be very resource inefficient. REFERENCES [] M. R. Bavafa and H. H. Xia, Repeaters for CDMA systems, in Proc. Vehicular Technology Conference, vol., pp. 6 6, 998.

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