Siplified Analysis and Design of MIMO Ad Hoc Networks Sunil Srinivasa and Martin Haenggi Departent of Electrical Engineering University of Notre Dae Notre Dae, IN 46556, USA Eail: {ssriniv, haenggi}@nd.edu Abstract The siple, yet powerful concept of an erristor and its erristance has been recently introduced for ad hoc networks and applied to scenarios such as retransission tie diversity, path diversity, or a cobination thereof. We extend this foralis to the case of spatial diversity, realized by eploying ultiple antennas at each node. Based on this fraework, one can efficiently analyze and design Rayleigh-faded MIMO ad hoc networks that eploy selection cobining. The atheatically tractable definition of the erristor ter greatly siplifies the study of a ultiple-antenna network and helps solve probles based on end-to-end reliability or resource allocation easily, which we illustrate in an exaple. Moreover, this technique deonstrates the superiority in perforance of MIMO over single-antenna routing schees, particularly at high SNR. I. INTRODUCTION In an ad hoc network, nodes are free to ove randoly and organize theselves arbitrarily, and thus the network topology ight change rapidly and in an unpredictable anner. Energy constraints often entail ultihop routing between nodes far apart, where relays assist in the delivery of packets to their destinations []. While these decentralized systes are easily deployable and reconfigurable, their perforance is severely susceptible to fading and interference [], [2]. A well-known procedure to gain fro fading is to eploy ultiple antennas at each node of the ad hoc network [3]-[6]. However, the design of ultiple-input ultiple-output MIMO ad hoc networks is a challenging task that can easily becoe intractable as the nuber of nodes in the network increases. The design proble involves allocating transit power to each active link, to achieve at least the desired end-to-end reliability under the specified syste energy constraints. In this paper, we extend the erristor foralis, originally developed in [7] for single-antenna ad hoc links, to ultihop MIMO systes. The erristor fraework is very useful in characterizing transissions in an ad hoc link. It can be used to greatly siplify analysis and design probles for Rayleighfaded MIMO ad hoc networks. For systes that eploy selection cobining, we show that there exists a logarithic apping fro link reliabilities to erristor values. This relationship can be exploited to effortlessly solve probles of resource reallocation given the network end-to-end packet delivery probability. We illustrate this with an exaple. The concepts developed also provide useful insights into the benefits of spatial diversity at high signal-to-noise ratios SNRs. II. SYSTEM AND CHANNEL MODEL We consider a general ultihop MIMO relay network, with a single transitting node, a single receiving node and n relay nodes Fig.. Thus, the signal fro the transitter reaches the receiver after n hops. Each node has antennas. The channel between any two adjacent nodes is odeled as a flat narrow-band Rayleigh block fading channel, with an additive white Gaussian noise AWGN process Z of variance N. The input/output relationship at tie instant t for the k th link can be described as Y k t = H k t X k t + Z k for k =... n, where X k t and Y k t are the input and output vectors respectively and H k t, the channel atrix, whose entries are the large-scale path loss ultiplied by the corresponding i.i.d. fading coefficients. Relay Relay i d ij Relay Relay j n n Fig.. The general MIMO relay odel, with n links and antennas at each node. d ij is the physical separation between node i and j. In the following discussion, we will only consider the MIMO link between node i and node j Fig.. Analytical results derived for this link can be extended to other links without loss of generality. Let the transitter at relay i have a transit power budget of P. Assuing that it has no knowledge of the channel state inforation, each of the antennas transits with a power level P/ in order to axiize the throughput [8]. The MIMO transission strategy ais at diversity axiization, eaning that copies of the sae signal are sent through the antennas. Selection cobining is eployed at the receiver, i.e., the received signal with the axiu SNR is picked for decoding. We consider a large-scale path loss propagation With the coplete knowledge of the channel obtained usually via feedback, waterfilling would be eployed to optiize the throughput.
odel in which the transitted power falls off with distance as d α [9], where α is the path loss exponent. Let the physical separation between nodes i and j be d ij. This is assued to be uch greater than the antenna separation, so that the path loss is the sae for all signals eanating fro a node. We do not consider interference, i.e., we assue a perfect MAC schee or light traffic for deriving our results. III. EXTENSION OF THE ERRISTOR FORMALISM In this section, we define the erristor characterizing the transission in the MIMO link i j. For our analysis, we eploy a Rayleigh-faded odel which relates transit power, path loss and the reliability of the link 2 [7]. We also study the asyptotic behavior of the link reliability which leads us to a critical value of SNR in selection cobining. A. Erristor Modeling for the MIMO Link We begin by noting that the power Q at each receive antenna at node j is a su of i.i.d. exponentials. Consequently, the pdf of Q follows the central chi-square distribution with 2 degrees of freedo, and has a ean Q = P d α ij = P d α ij, where P is proportional to the transit power. We can write the cdf of the chi-square r.v. Q [] as follows: q/ Q F Q q = e k! q Q k= k, q. With Q i being the signal power at the i th receive antenna, the selection cobining strategy picks the signal with power S = ax Q, Q 2,..., Q for decoding. The cdf of S is given by F S s = F Q s. The transission fro node i to node j is successful if the axiu SNR at node j, γj ax, is greater than a certain threshold, Θ, which depends on the detector structure and the odulation and coding schee [2]. Therefore, the reception probability is given by p r = Pr[γj ax Θ] = Pr[S ΘN ]. Thus, we get p r = e ΘN / Q k= k! k ΘN. 2 Let R denote the noralized average NSR at the receiver, i.e., R := ΘN / Q. Then, for, we have p r = e R k! Rk = e R k= k= k! Rk Q. 3 Fig. 2 is a plot of the reception probability 3 as a function of the nuber of antennas. We reark that under good channel conditions, i.e., low values of R, the MIMO syste has a higher reception probability copared to the singleantenna syste, while at very low SNR values, using fewer 2 This odel is preferred not only for its siplicity, but also because it overcoes soe of the liitations in the disk odel [], [2] often used in the literature on ad hoc networks. Reception probability p r.9.8.7.6.5.4.3.2. R =.25 R =.95 R =.65 R = 2.35 R = 3.5 2 4 6 8 2 4 6 8 2 No. of antennas at each node Fig. 2. Reception probabilities as a function of the nuber of antennas eployed, for various values of the NSR R. Notice the contrast in asyptotic behavior as one set of curves approach, while the others tend to. antennas akes the link ore reliable. Siilar results on the sub-optiality of MIMO at low SNR have also been observed in ters of the behavior of capacity versus SNR in [3], [4] where even the receiver does not have any channel state inforation, and in an interference-liited environent [5]. B. Asyptotic Behavior of the Reception Probability We now discuss the asyptotic behavior of the reception probability as the nuber of antennas increases. To investigate this, first note that a Poisson distribution with paraeter λ can be approxiated by a Gaussian distribution with ean and variance λ, for large λ, with equality in the liiting case when λ [6]. Also note that the ter inside the parentheses in 3 is an infinite su over the pf of a Poisson-distributed variable with λ = R. Using the Gaussian approxiation for large and nonzero R, we can approxiate 3 as p r 2πR e k R2 2R dk 4 where we have replaced the discrete su by a continuous integral. Writing in ters of the Q-function area under the tail of the standard Gaussian probability density function [9], we have R p r Q. 5 R A well known tight upper bound for the Q-function [9] is Qx x 2 2π e x /2, x >. Using this, for R <, we can bound 5 as R p r R e 2 R 2 /2R. 6 2π
As, the approxiation becoes an equality, and thus p r as the nuber of antennas increases. For R >, R <. Using the property that Q x = Qx for x >, we can bound 5 as p r R R 2 2π e R /2R, 7 which yields p r as. Therefore, we conclude that there exists a phase transition, i.e., a critical value of R, R c = corresponding to an average SNR of Θ, above which the reception probability is, and below which, success is always guaranteed, assuing infinite antennas are eployed. This is, to the best of our knowledge, the first analytical derivation of this critical SNR level in selection cobining. C. Markov Approxiation To siplify the expression for p r 3, we apply Markov s inequality [7]. The Markov bound is rather a frequently loose bound, but nevertheless provides valuable insights on tail probabilities. Applying it to the tail probability ter gives e R k= k! Rk R = R. Using this in 3, we have p r R. The deviation of the Markov bound R fro the actual value 3 is plotted in Fig. 3 for different values of the paraeters R and. The threshold value of R for which the difference is less than. is calculated to be.3 for = 2,.25 for = 3 and.37 for = 4. For R >, R >, and the approxiation does not ake sense. Actual reception probability inus bound.9.8.7.6.5.4.3.2. = 2..2.3.4.5.6.7.8.9 Noralized average NSR R = 6 Fig. 3. The Markov approxiation for p r is tight at low values of R. However, it becoes increasingly inaccurate as R increases. It is seen that the Markov bound is very tight only for good SNR values R and/or when eploying any antennas. Under these conditions, first-order approxiations hold, and we have p r = R p r = e R. 8 Following the erristor definition fro [7, Eqn. 6], the erristor of the MIMO link i j is denoted by R = R, and its value is known as its erristance. Henceforth, we shall operate in the high SNR regie R, so that this approxiated representation is accurate. Thus, we can characterize transissions in any MIMO link by a network eleent, the erristor, whose value depends on the average noralized NSR at the receiving node and the nuber of antennas eployed. IV. DESIGN OF MIMO AD HOC NETWORKS A powerful application of the erristor representation is the efficient design of MIMO ad hoc networks. The design proble is to set the transit power levels at each node or, equivalently, choose erristances such that the end-to-end reliability, denoted by p EE, is at least at the desired level p D. In order to be able to do so, we need to find a relationship between the erristances involved with each transission link and the equivalent erristance of the network as a whole. We discuss the two ost fundaental link topologies in this regard: the ultihop series connection and links eploying tie/path diversity parallel connection. A. Multihop Connection Over an n-hop MIMO serial link, the end-to-end reliability is given by the product of the reception probabilities for each link. Equivalently, with R i denoting the erristance of the ith link, we get p EE = e P n i= R i. 9 The equivalent erristance R tot is given by n n R tot = ln p EE = R i = Ri. B. Parallel Connection For a ore coplicated network with retransission, the transission is successful if any one copy of the signal is successfully decoded by the receiver. Extending the result fro [7, Th. 2] to a MIMO syste, we conclude that erristors connected in parallel have to be ultiplied, i.e., n n R tot = R i = Ri. i= i= i= i= With the knowledge of the series and parallel erristor equivalents, we can siplify ost networks, since they can be put down as a cobination of these two fundaental connections. The following exaple describes how the erristor fraework is able to reduce coplex design probles to siple polynoial equations, that are analytically tractable. Exaple: Consider the three-hop MIMO network in Fig. 4, where each node has antennas and the spacing between any two adjacent nodes is equal to d. Node transits its packet twice, once to node 2 and again over the link 3. However, node 2 also listens when the packet is being sent to node 3. This iplicit transission is odeled by adding an erristor in parallel to link 2, whose value is denoted by R 2,i. Since d 3 = 2d 2, it is easy to see that R 2,i = 2 α R 3. The design
a b 2 3 d d d R 2,i 2 3 R 2 R R23 R 3 Fig. 4. An exaple: a A three-hop network, with node having to transit twice. b The corresponding erristor odel, which is uch easier to analyze. The dashed box R2,i denotes the iplicit erristor. proble is to choose the erristances to guarantee atleast the desired end-to-end reliability p D, which is specified as 9% say. p EE p D iplies R tot ln p EE.5. Let us assue the power required per transission to be the sae. Since the distances between adjacent nodes are the sae, we require R = R 2 = R 23 = R. Also, since d 3 = 2d 2, we need to set R 3 = R 2 α. Thus, R tot = R +R 2 + R R 2 α. Notice that node 2 needs to transit twice and thereby uses up twice the power copared to the other two nodes. At R tot =.5, a solution for = and α = 3.5 is R =.6. For = 2, R =.57 satisfies the equation. Just by increasing the nuber of antennas at each node by one, a assive 6% reduction in transit power is observed. Consider now a ore realistic scenario where each node expends the sae net transission power. Clearly, R = R 23 = R, while node having to transit with the sae net transit power requires d α R = dα R 2 + 2dα R 3. One possible setting to achieve this is by letting R 3 = 2 α R 2, which results in R 2,i = R 2 and R 2 = 2R. To eet the desired reliability, we require R +2R 2 +R 2 α 2R.5. For a single-antenna syste and assuing α = 3.5, the above equation is satisfied with equality at R =.48. Likewise, for = 2, R =.5, which confors to less than /2 the original power. With = 3, the solution is given by R =.43, resulting in a further 2% reduction in power consuption. This siplified procedure also lets us solve probles based on resource reallocation effortlessly. To see this, just iagine the scenario when node 2 exhausts all its energy. This would effectively ake the link 2 useless. Our erristor network would then consist of just R in series with R 3. Depending on the syste constraints, we can suitably design these two erristors to eet the desired requireents. V. COMPARISON OF MIMO WITH SINGLE-ANTENNA ROUTING SCHEMES In this section, we apply the erristor foralis to copare the perforance of the ultihop MIMO schee with two conventional single-antenna routing schees ultihop routing and connections with tie diversity retransissions, to study the relative benefits of spatial diversity. Priarily, we focus on two iportant aspects: total network energy consuption per packet and transission delay. To ake the coparison fair, we take the sae nuber of total transissions and the sae end-to-end link distance d for each of the three schees. With n transissions and outgoing paths fro each node, the total noralized energy consuption per packet is easily seen to be = n i= j= d α ij R ij. 2 Consider a MIMO network, with antennas at each node and n hops, each of length d/n. Then, the total nuber of transissions is n. With R denoting the NSR at each receive antenna, R tot = nr, and the total energy consued α d n = n. 3 n R tot For the single-antenna ultihop schee with n hops, each of length d/n, R tot = nr, where R is the NSR at each receiving node. Thus, the total energy consuption is α d n = n. 4 n R tot For the syste with n retransissions, R tot = R n, where R denotes the NSR at the receiver, and = nd α n. 5 R tot a b c Fig. 5. The three schees considered, with each having to transit four ties. a The MIMO network, eploying two antennas at each node. b The single-antenna ultihop schee c Connections with retransission involved. The end-to-end distance is d in each case. We now copare the three routing schees for the particular case of n = 2 and = 2 Fig. 5. The ratio between the total consued energies of MIMO and the single-antenna ultihop schees can be siplified to E tot d = 2 α 3 2 R 2 tot. 6 Fro this, we can deduce that the spatial diversity schee is ore energy-efficient if R tot < 2 3 2α p D > e 23 2α. 7
The area in the α, p D plane over which MIMO networks consue less energy than ultihop networks and the energy gain for various path loss exponents are plotted in Fig. 6. For practical scenarios such as high p D and oderate α, the MIMO schee clearly outperfors the serial link. Substantial energy gains are observed as p D oves closer to unity..95.9.85.8.75.7.65 MIMO schee is ore efficient 2 3 4 5 Path loss exponent α Energy Gain [db] 5 5 5 α=2 α=5.8.85.9.95 Link reliability p D Fig. 6. Left MIMO is ore energy-efficient than the ultihop schee in the region above the curve. Right The corresponding energy gain involved for different path loss exponents. The ratio between the energies of the MIMO and the tie diversity schees is given by = 2 α+ 2 R 4 tot. 8 Fro this, we can deduce that MIMO consues less energy than the network with retransission if R tot > 2 2 4α p D < e 22 4α. 9 This curve is plotted in Fig. 7. For the retransission schee to outperfor MIMO, p D ust be extreely sall 3. This is very hard to realize, and for all practical purposes, MIMO is ore energy-efficient. The plots in Figs. 6, 7 establish that huge energy gains are possible by using MIMO transission strategies..998.996.994.992 MIMO schee is ore efficient.99 2 3 4 5 Path loss exponent α Energy Gain [db] 5 5 α=5 α=2 5.8.85.9.95 Fig. 7. Left The region in the α, p D plane where MIMO consues less energy than the retransission schee. Right The corresponding energy gain as a function of p D for different path loss exponents. Another advantage of MIMO over the single-antenna systes is in the saller end-to-end transission delay. Considering the schees in Fig. 5, we see that the MIMO syste can transit in half the tie required for the other two. On using antennas, the transission delay reduces by a factor of /. Equivalently, the MIMO syste would see ore independent realizations of the channel than the single-antenna systes in a given tie interval, resulting in huge diversity benefits. VI. CONCLUSIONS The erristor foralis is developed for MIMO ad hoc networks eploying selection cobining, which greatly siplifies probles related to their analysis and design. With the knowledge of series and parallel erristor equivalents, coplex resource reallocation probles reduce to siple polynoial equations, which are easier to handle, as we have deonstrated in an exaple. Based on the erristor fraework, the MIMO ultihop network is shown to be ore energy-efficient and to have lower transission delays copared to the single-antenna schees with or without retransission, establishing the fact that spatial diversity can greatly benefit at high SNR values. Further, the asyptotic behavior of the MIMO link reliability as the nuber of antennas goes to infinity is studied, and as a useful side result, the critical value of the SNR, above which the reception probability is always one and below which it is always zero, is calculated. REFERENCES [] A. J. Goldsith and S. B. 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