The new concepts are used to study the ... we focus on routing in multi-hop wireless networks.1 ... 1Note that routing is used in both wired and wireless networks,.
Network Channel Estimation in Cooperative Wireless Networks Ahmad Khoshnevis, and Ashutosh Sabharwal Dept. of Electrical and Computer Eng. Rice University Houston, TX 77005 E-mail: {farbod,ashu}@rice.edu Abstract — In distributed wireless networks, where nodes actively participate in helping communication for other nodes, they are typically unaware of their neighourhood and hence have to “estimate” it before sending any useful data. In this paper, we formalize the concept of node neighbourhood by introducing the notion of network channel, in which all nodes become part of a large channel. The notion of network channel is then used to study routing in decode and forward networks. To do the same, we introduce the concept of network coherence time, which denotes the time for which the network topology remains approximately constant. The new concepts are used to study the tradeoffs between encoding rate of route discovery packets, number of discovered routes and accuracy of subsequent network channel estimation. Finally, we propose a simple adaptive algorithm for route selection using outage capacity as the metric for route selection, and show that our algorithm outperforms the existing route selection based on the minimum hop count.
I. Introduction Information theoretic analysis of wireless networks has highlighted the utility of node collaboration (e.g., [1, 2, 3]). Though the role of channel coding as a fundamental construct is clear in the above cited analysis, the role of network protocols like routing and medium access remains unclear. In fact, it is not even clear if these protocols are fundamental constructs or not. If they are fundamental to the operation of wireless networks (and communication networks in general), then it is important to understand how they should be optimally designed. Motivated by these fundamental questions, in this paper we focus on routing in multi-hop wireless networks.1 Our contributions in this paper are two-fold. First, we “reverse engineer” the reason why routing protocols are used in practical wireless networks. The main reason is that nodes are unaware of the presence of their destination and their neighbourhood (and hence the channel probability law between different nodes). The above realization naturally leads to the concept of a network channel between source and destination nodes. By treating the whole network of nodes as a channel, we show that routing protocols are simply network channel estimators, where the estimation is performed to find the channel probability law. Thus the objective of network channel estimation is different from conventional channel estimation, where the current channel realization is estimated to 1 Note
that routing is used in both wired and wireless networks, but our work is directly applicable to only wireless networks.
aid in reception. Finally, note that if there is no mobility in the system and the channel probability law between nodes does not change with time, network channel estimation is not an interesting problem. Since, without mobility, nodes can estimate the probability laws once and then use them forever at a vanishingly small asymptotic rate loss. Second, to demonstrate the utility of the network channel formalism, we consider conventional minimum-hop routing (for decode and forward coding) and show it can be improved upon by no additional overhead. The main idea of the proposed improvement is simple, measure the loss probabilities of the routes before committing to one. The loss probabilities can be calculated by simply measuring the number of unacknowledged packets sent at rate R. Even with our elementary setup, several important tradeoffs and results appear. First, the rate at which route discovery packets are encoded impacts the number of discovered routes; lower encoding rates implies more routes can be potentially discovered. Second, if more routes are discovered, then the likelihood of both good and bad routes getting discovered increases, making the process of selecting the optimum routes harder. For our proposed network estimation based approach, it implies that longer estimation phase is needed for the case with more routes. Lastly, minimum hop-count route are seldom good and the transmitter is mostly better off selecting a route at random. Throughout the paper, we study systems which first estimate the network channel probability law and then adapt their transmissions accordingly. Though this technique is not shown to be capacity achieving in any sense, it should be noted that none of the known informationtheoretic results apply to the problem of unknown network channels. Lack of knowledge about the exact channel probability law is typically studied as the problem of compound channels, but that analysis is typically limited to systems where there is no feedback available to learn the probability law. Furthermore, as will become clear from the next section, network channel is markedly different from the channels studied in compound channel capacity theorems. The network channel is not passive, i.e., it consists of other nodes which are equipped with both power and computational capabilities. The rest of the paper is organized as follows. In Section II, we introduce the concept of network channel via several examples. The main results are presented in Section III and we conclude in Section IV.
II. Network is the Channel Consider a network N which has N nodes. Generically, the nodes are considered to belong to one of the two classes. The first class, labelled as C, consists of
Communicating nodes which can take three roles: source, destination or a relay for other peer nodes. The class represented by C consists of typical mobile nodes in a wireless network. The other class, denoted by G, are Gateway nodes, and act as gateways to the backbone network for all other nodes in class C. The number of nodes in class C is denoted by nC and in class G by nG , such that nC + nG = N or equivalently, C ∪ G = N . The time-varying channel between any two nodes i and j is denoted by hij . The channel hij could represent a multipath channel or a multiple antenna channel, and in general, belongs to the set of multidimensional time-varying impulse responses. The first order probability distribution of hij is denoted by pt (hij ). In typical wireless channels, there are two time-scales of variations. The probability distribution, pt (·) varies according to the slow time-scale, due to large-scale mobility. The fast time-scale changes in phase and amplitude of the channel, known as fading, cause short time-scale variations. The set of all inter-node channels is denoted by H = {hij : i, j ∈ N , i 6= j}. Depending on the allowable cooperation and coding, only a subset of channels in H may be of interest. Following three examples clarify the above definitions.
Figure 2: Ad hoc Network. cooperate in relaying or forwarding other users information, He = {hij : i ∈ C & j ∈ N \ {i} OR i ∈ G & j ∈ C}. That is all channels between different mobiles are of interest, but the channels between different base-stations are not of interest.
Figure 3: Ad hoc extension to cellular network. Example 1 (Cellular Network) A typical cellular network is shown in Figure 1(a). In this case, all mobile nodes m1 , m2 , . . . belong to the set C and the base-stations belong to the set G. If all communication happens directly between base-stations and mobiles, then the set of channels of interest is Hc = {hij : i ∈ G & j ∈ C OR i ∈ C & j ∈ G} ⊂ H; the set of channels of interest are depicted in Figure 1(b).
III. Main Results In this section, we review the concept of outage capacity, introduce network channel estimation and then propose a simple method of estimating outage capacity with no additional overhead to current protocols.
A
Outage Capacity
The quality of any route is measured using its outage capacity defined as Ck (R) = (1 − αk (R))R,
Figure 1: Cellular Network.
Example 2 (Ad hoc Network) An ad hoc network is a collection of nodes with no central infrastructure, shown in Figure 2(a). In this case, all nodes in the network belong to the set C and set G is empty. Variations of ad hoc networks exist which may have asymmetric node roles (some nodes act as only relays) and/or some nodes act as gateways to outside world. In ad hoc networks, Ha = H, i.e., all channels between nodes are of interest, as shown in Figure 2(b).
Example 3 (Cellular with ad hoc Extension) Consider the network in Figure 3(a), where nodes in a cellular network also act as relays for other peer nodes [4, 5, 6, 7]. In this case, the set of C and G are defined similar to Example 1, but the network channel is richer than that in a cellular network. Since nodes can
(1)
where R is the rate of packet transmission and αk is the probability of outage of route k under consideration. Let route k constitute of g hops and the outage probability of each hop be given by αk,i , for i = 1, . . . , g. Then αk can be calculated as αk (R) = 1 −
g Y
(1 − αk,i (R))
(2)
i=1
The outage probability for each hop, αk,i is defined as the probability that the instantaneous mutual information is less than the desired rate R, αk,i (R) = Probθ (I(X; Y |θ) < R) .
(3)
The parameter θ denotes the state of system and changes in each transmission. For example, if there is no other concurrent transmission in the system which interferes the current transmission, then θ = h, the current channel state. If there is an interfering transmission, then θ refers to a multidimensional parameter which includes additional information about the interference.
When there are no interfering users (θ = h), then for a Rayleigh block faded channel, the probability of outage is given by 2R − 1 2 αk,i (R) = Prob |hi | < SNRi 2R − 1 = 1 − e−γ , where γ = . (4) SNRi The factor SNRi is the average received SNR at the receiving node of the ith hop. αk (R)
= =
B
1−
g Y
(1 − αk,i (R))
i=1 Pg − i=1 γi
1−e
.
(5)
Network Channel Estimation
In practice, the source node is not unaware of the network channel, i.e., the network topology and the probability law governing the channel between any two nodes is unknown. Almost all of information-theoretic analysis which proposes to use advanced collaborative coding to achieve higher end-to-end throughput assumes that these probability laws and the network topology are known apriori [1, 2, 8]. Since the network channel is unknown, either it has to be estimated before the source can determine at what rates it can reliably send the data, or it can use an estimation-free method, much like in compound channel problems [9]. Note that compound channel capacity analysis consists of methods which do not aim at estimating the channel distribution and using feedback at the transmitter. Thus, with the availability of feedback, it is not immediately clear which transmission strategies are optimal over the network channel. Routing algorithms perform two tasks. First, they determine if the destination node is available or not.2 Second, they provide an estimate of the network topology and the inter-node distributions. Below we present three results, which show the tradeoff involved in probability of discovering a route, number of routes and the probability of finding the capacity optimal route. Route Discovery: Consider that the route discovery is performed by using packets encoded at rate R0 . Then route k is discovered with probability 1 − αk (R0 ). Now, it is straightforward to show the following result. Fact 1 For any two routes x and y (with possibly unequal number of hops), αx (R) < αy (R) ⇐⇒ αx (R0 ) < αy (R0 ), for any value of R and R0 if αx and αy are given by Equation (5). The above statement implies that routes with higher outage capacity are more likely to be discovered, independent of the rate used to encode the route discovery packets. Note that αk (R) decreases monotonically as R is decreased. Thus smaller is R0 (the rate for route discovery 2 In
packets), the higher is the chance that the route(s) with maximum outage capacity Ck will be discovered. But reducing R0 means that more routes will be discovered and more network time will be consumed since smaller R0 implies longer packets. Number of discovered routes: The choice of rate R0 determines the number of discovered routes. As R0 reduces the outage probability for all routes reduces, thus increasing their chances of being discovered. Discovering routes at smaller R0 not only increases the chances of finding the outage optimal route but also poor routes. Thus, with more discovered routes, the choice of finding the optimal route becomes more challenging, as will become clear from the following results.
a usual information theoretic setup, destination node is assumed to be ready to receive the data. Destination discovery is in essence a binary hypothesis test performed by the whole network to determine the presence of the destination node.
C
Route Selection
By the end of route discovery procedure, the transmitter is provided with a set U = {ri }i∈A , where A is a finite indexing set of routes ri , i ∈ A, from source node to destination node. In the next step, source node selects a route, r ∈ U, for sending its data to the destination node. In current systems there are variety of criteria, such as minimum hop-count, minimum end-to-end delay, or highest end-to-end throughput, used for route selection. However since the route discovery is performed by a single transmission, the transmitter has only a single sample of the network channel. Hence network channel estimation based on one measurement is by design very inaccurate. For example, in [10], it is shown that minimum hop count route is not always throughput optimal or outage capacity optimal, and finding the optimal routes requires many network channel measurements. Note that to estimate outage of any route requires knowing the cummulative exponent in Equation (5), which can only be obtained by repeated measuremens of instantaneous SNRs on each of the links. But such a procedure is not feasible in a real network. If the instantaneous SNR on any hop is low for a transmission, then that packet is lost and then the receiver possibly gets no feedback on the actual channel during that transmission. Thus, instead of directly measuring the average SNR in Equation (5), we propose to measure the packet loss or equivalently outages αk (R) directly by measuring the number of missing ACK packets. Definition: Network coherence time, Tc , is defined as the time during which the topology of the network is not affected significantly by the mobility of nodes. Network coherence time is measured in packets and not in symbols. Let Tc be the network coherence time, Ttr Tc be the channel estimation time, and U be the set of discovered routes. We divide Tc into two portions, i.e., channel estimation phase, lasting Ttr packet durations, and transmission period, lasting Ttx packet durations. Therefore Tc = Ttr + Ttx . During the estimation phase, Ttr , source node multiplexes through all the available routes for sending the data. Receipt of acknowledgment of a transmission using a route, r ∈ U , can be seen as a one bit of feedback that gives the source node information about the reliability of that route. At the end of the estimation phase, the source node calculates the outage associated with each route by simply dividing the num-
ber of losses by the total number of packets sent on each route. Network channel estimation phase provides more samples of network channel for the source node, allowing it to estimate outage capacities of each route. As one can notice, the decision made by the source node is highly dependent on the number of samples of network channel, ns . Let |U | be the number of discovered routes. Then ns = Ttr /|U |. If length of the estimation phase is fixed, then ns depends on the number of discovered routes. Higher ns means fewer measurements per route, which in turn leads to poorer accuracy estimate for each of the routes. Hence there is a trade-off between the number of discovered routes (which depends on the rate of discovery R0 ) and the accuracy in finding outage optimal route.
IV. Conclusions In this paper we introduced the notion of Network Channel, in which the intermediate nodes between the source and destination nodes along with the communication links between them are considered as the communication channel. We showed that by estimating the network channel, route selection in mobile ad-hoc networks can be improved so that the overall throughput of the network increases significantly comparing to a network with route selection based on minimum hop-count.
References [1] T. M. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Transactions on Information Theory, vol. 25, pp. 572–584, September 1979. [2] L.-L. Xie and P. R. Kumar, “A network information theory for wireless communication: Scaling laws and optimal operation,” submitted to IEEE Transactions on Information Theory, April 2002. [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behaviour,” submitted to IEEE Transactions on Information Theory, January 2002.
Figure 4: Probability of outage versus number of discovered routes for a network with 6 nodes. Figure 4 shows the probability of outage versus number of discovered routes for a network with 6 nodes, estimation phase is 100 packet transmissions, and data transmission period constitutes of 1000 packet transmissions. As can be seen in Figure 4, a system with route selection based on network channel estimation outperforms system with route selected based on the minimum hop count at all times. Also one can see in Figure 4 that as the number of discovered routes increases the performance improves, because the probability of having the optimum route in the set of discovered routes increases. However as the number of discovered routes exceeds a certain threshold (in this case 10 routes) the performance degrades. Because, although the probability of having the optimum route among the set of discovered routes is increased, but due to the lack of sufficient number of samples, source node is not capable of detecting the optimum route. Note that the minimum hop count uniformly underperforms the proposed method, which for large number of routes selects a route randomly. We close this section by stating a practically useful observation. Though there may be only outage optimal route, there are many routes with approximately the same performance in the network. The main purpose of route selection procedure is to eliminate poor routes, which is simpler than finding the optimal route.
[4] C. Qiao and H. Wu, “iCAR: an integrated cellular and ad-hoc relay system,” in Int. Conf. on Computer Communications and Network (IC3N), October 2000. [5] C. Qiao, H. Wu, and O. Tonguz, “Load balancing via relay in next generation wireless systems,” in IEEE Workshop on Mobile Ad Hoc Networking and Computing (MobiHOC), August 2000. [6] Y. Lin and Y. Hsu, “Multihop cellular: a new architecture for wireless communications,” in Proc. IEEE INFOCOM, March 2000. [7] A. Zadeh and B. Jabbari, “On the capacity modeling of multi-hop cellular CDMA networks,” in Proceedings of IEEE MILCOM, October 2001. [8] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “On the capacity of ’cheap’ relay networks,” in Proc. of CISS, (Baltimore, MD), March 2003. [9] A. Lapidoth and P. Narayan, “Reliable communication under channel uncertainity,” IEEE Transactions on Information Theory, vol. 44, pp. 2148–2177, Oct. 1998. [10] A. Sabharwal, “On capacity of relay-assisted communications,” in Proc. GLOBECOM 2002, (Taipei, Taiwan R.O.C.), November 2002.
