Relay Selectio ad Max-Mi Resource Allocatio for Multi-Source OFDM-Based Mesh Networks Kiaoush Hosseii ad Raviraj Adve The Edward S. Rogers Sr. Departmet of Electrical ad Computer Egieerig, Uiversity of Toroto 10 Kig s College Road, Toroto, ON M5S 3G4, Caada Email: {kiaoush, rsadve}@comm.utoroto.ca Abstract We cosider a multi-source mesh etwork of static access poits wherei sources use decode-ad-forward to cooperate with each other. All trasmissios use orthogoal frequecy divisio multiplexig (OFDM). Our objective is to maximize the miimum achievable rate across all flows. We fid a tight upper boud o the performace of the subcarrier-based cooperatio ad show that selectig a sigle relay for each subcarrier is optimal for almost all subcarriers. The solutio to the related optimizatio problem simultaeously solves the relay, power, ad subcarrier assigmet problems. Secod, ulike previous works, we also cosider relay selectio for the etire OFDM block. This addresses the fact that, i additio to the sychroizatio problems caused, it is likely impractical for a relay to oly decode a subset of subcarriers. We propose three selectio-based cooperatio schemes to relay the etire OFDM block with varyig complexity. Simulatio results show that uder the COST-231 chael model, the performace of the simplest scheme almost exactly tracks that of a exhaustive search. I. INTRODUCTION Cooperative diversity is a class of spatial diversity techiques made possible through relayig. The odes of a distributed commuicatio etwork share resources to achieve the beefits of multiple-iput multiple-output systems with oly a sigle atea at each trasmitter/receiver. Cooperatio ca also help address large-scale fadig. Two popular relayig schemes are decode-ad-forward (DF - the relay decodes ad re-ecodes the source data) ad amplify-ad-forward (AF - the relay amplifies ad retrasmits its received sigal). The iitial work [1] [3] has led to much research activity i this area. The most relevat literature here is the work o selectio which showed that pairig a source with a sigle best relay miimizes the overhead due to relayig ad avoids the eed for sychroizatio across relays [4], [5]. I multisource etworks, selectio also makes better use of limited power tha distributed space time codes [5]. Selectio has sice bee exteded to several relayig scearios [6], [7]. O the separate track, orthogoal frequecy divisio multiplexig (OFDM) has bee show to be a promisig, ad icreasigly popular, techique to mitigate the impact of multipath fadig. I OFDM, data is trasmitted i parallel over multiple (N) frequecy subcarriers. The key here is the ability to create the trasmit sigal usig a N-poit iverse Fast Fourier Trasform (IFFT) of the data symbols. Furthermore, because each subcarrier experieces a differet chael realizatio, resource allocatio ca sigificatly ehace performace [8] [11]. Recetly, the combiatio of OFDM ad cooperatio diversity has attracted itese iterest. Specifically, selectio of the right relayig ode ad dyamic allocatio of subcarrier ad power are cosidered critical. Li ad Liu studied the capacity of OFDM-based relay etworks for both AF ad DF strategies [12] ad the problem of maximizig the sum rate with fairess costraits i a multiple-source multiple-relay etwork usig a graph theoretical approach [13]. Fairess is imposed by limitig the umber of sources a sigle relay ca help. Ng ad Yu [14] used a utility maximizatio framework to choose the optimal relay strategy ad resource allocatio i cellular etworks. Dai et al. [15] provided outage aalysis of two differet relayig strategies i OFDM-based multihop etworks ad show that potetially choosig a differet optimal parter for each subcarrier achieves full diversity order while there is o diversity gai if selectig the relay which has the highest combied sigal-to-oise ratio (SNR). The work i [16] allows for subcarrier permutatio i a cooperative etwork i cotext of max-mi fairess. Weg et al. [17] proposed a resource allocatio scheme i which a group of odes ca share their redudat resources with others. All these works deal with the OFDM trasmissio o a persubcarrier basis, i.e., as if each were a idepedet trasmissio. This is, ufortuately, ot true. Give the importace of time ad frequecy sychroizatio i OFDM, it is urealistic to expect differet relays to relay idividual subcarriers idepedetly. Furthermore, i OFDM, the raw data is chael ecoded before data modulatio ad the IFFT; DF requires decodig all subcarriers. Most of the subcarrier-based selectio is, therefore, theoretically optimal, but impractical. I this paper, we cosider max-mi optimizatio for a multisource multi-destiatio cooperative OFDM-based static mesh etwork of access poits (APs). Each ode is a source as well as a potetial relay for other odes. We begi with per-subcarrier selectio. It has bee claimed that selectio is the exact power allocatio solutio [18]. Buildig o the work i [19], we show that this is true for most, ot all, the subcarriers. Usig the Karush-Kuh-Tucker (KKT) coditios, we characterize a upper boud to the origial problem which leads to the joit relay ad power allocatio for each subcarrier. This solutio also leads to a simple, heuristic that is also a lower boud. Simulatios show that the two bouds are idistiguishable whe usig the COST-231 [20] chael model. We the deal with selectio for a etire OFDM block.
Fig. 1. Cooperative multi-source multi-destiatio etwork We propose a simple selectio scheme but with performace close to exhaustive search ad ot much differet from persubcarrier selectio. The rest of the paper is structured as follows. Sectio II, develops the system model for the multiple-source OFDMbased etwork uder cosideratio. I Sectio III the issue of resource allocatio o a per-subcarrier basis is ivestigated i some detail. Sectio IV deals with selectio based o a etire OFDM block. Sectio V presets the results of simulatios that illustrate the workigs of the theory preseted. Sectio VI cocludes this paper. II. SYSTEM MODEL The system uder cosideratio is a outdoor static mesh etwork of APs. Some of the odes are physically coected to the iteret backboe ad are a gateway for other APs. The odes are istalled at some height ad have a Ricia chael, with a lie-of-sight compoet, to eighborig odes (potetial relays), but a Rayleigh chael to the destiatio. Each ode i the etwork acts as both a source as well as a potetial relay for other sources. I additio, each ode has its ow destiatio which is ot withi the set of K source odes. The system model is illustrated i Fig. 1. All trasmissios use OFDM withi their ow frequecy bad, i.e., simultaeous trasmissios do ot iterfere ad, eve with a half-duplex costrait, a ode ca receive the trasmissios from other sources while trasmittig. Because each user experieces a differet chael realizatio o each subcarrier, power ad subcarrier allocatio ehaces system performace. For each subcarrier, the chael betwee a ode i ad destiatio j is modeled as a flat ad slowly-fadig Rayleigh chael. We assume all iter-ode chaels vary slowly eough for chael state iformatio to be fed back with limited overhead, makig resource allocatio possible. All APs are attached to a power supply ad trasmit with costat ad maximum total power of P Joules/symbol. We cosider the DF protocol wherei each relay receives, decodes ad re-ecodes the iformatio with the same codebook as the trasmitter, ad forwards it to the destiatio. We place a half duplex costrait ad commuicatio happes i two phases. I Fig. 1, the solid arrows idicate the first, data sharig, stage. The dashed arrows represet the secod phase wherei the sources relay for oe aother. Each source trasmits its data usig N subcarriers ad receives iformatio from other sources. Durig the secod time slot, oly oe ode relays each subcarrier of a source. Fially, the destiatio ode combies messages received i the two phases to decode the origial iformatio. Cosider for ow a system where a relay is chose persubcarrier. O subcarrier, with relay l helpig, the rate at which source k ca trasmit is: d k = max l s k r l = 1 2 log 2 = 1 2 log 2 { } mi I s () k r l,, (1) ( ) 1+sr (), (2) kl ( 1+sr () 0k +sr() ), (3) where, sr () kl = P h () kl 2 /(N N 0 ) is the received sigalto-oise ratio (SNR) of the th subcarrier at relay l from source k while sr () 0k = P h() 0k 2 /(N N 0 ) ad sr () = h() 2 /N 0 are the received SNR of the th subcarrier of source k through the direct ad relayig paths, respectively. I these expressios, N 0 is the power spectral desity of the white receiver oise, h () kl is the chael gai betwee ode k ad relay l o the th subcarrier; h () 0k ad h() are the chael gais of source-destiatio ad relay-destiatio o the th subcarrier, respectively. The factor of (1/2) accouts for the fact that commuicatio happes i two phases. s k r l is the rate at which source k ca commuicate with relay l, while is the rate at which it trasmits iformatio to its destiatio ode with the help of relay l o the th subcarrier. The overall trasmissio rate of user k is the sum over all N subcarriers: R k = N =1 d k. (4) Note that the source distributes its power equally while is the power that the l th relay allocates to the th subcarrier of ode k. Therefore, (1) states that the maximum trasmissio rate is the rate at which both relay ad destiatio ca decode iformatio. Like most other works i this area, i order to make the problem tractable, we assume that all iter-source chaels are strog eough that s k r l d k = k,. (5) This is a crucial assumptio justified by the modelig of source-relay chaels as Ricia while the source-destiatio ad relay-destiatio chaels are Rayleigh. III. SUBCARRIER-BASED RESOURCE ALLOCATION As described above, all source odes ca decode each others data ad, from (5), the rate limitig factor is the compoud source-relay-destiatio chael. This sectio develops the optimal relay selectio ad power allocatio scheme to achieve max-mi fairess, i.e., to maximize the miimum rate across all sources. As far as possible, this metric leads to a equal rate for all source odes. I keepig with its may beefits,
we impose selectio i the secod, relayig, phase. Therefore, the optimizatio problem we wish to solve is: {max}mi R k, (6) k s.t. C 1 : l 1k p() l = 0, k, ad = l 2, (7) C 2 : 0, l,k,, C 3 : K P, l. (8) Costrait C 1 eforces selectio by allowig oly oe ode to devote power to each subcarrier. Costraits C 2 ad C 3 state that power must be o-egative ad that the total available power of the l th relay is limited to P Joules/symbol. Due to the selectio costrait, (6)-(8) is a, essetially itractable, mixed-iteger programmig optimizatio problem. Oe proposed solutio [9], [11] separates the power allocatio ad selectio problems. First, subcarriers are selected assumig equal power allocatio; the, power is distributed based o this selectio. However, with K sources ad N subcarriers, there are K 2 N relay assigmets to be checked. Therefore, eve this scheme is ifeasible for realistic values of N ad K. We build o a alterative approach developed i [19] to form a approximate solutio that is also a upper boud 1. A. Approximate Solutio ad Upper Boud Other tha the iteger costrait i (7), the costraits i the origial problem of (6)-(8) are affie. To fid a approximate solutio we igore the costrait i (7). Hece, the solutio to the ew problem is a upper boud to the subcarrier based (UBSB) resource allocatio problem of (6)-(8). The revised formulatio, stated here i the epigraph form, is a cocave maximizatio problem with efficiet solvers available [21]: {max}t (9) t, C 1 : N K 1 2 log 2 1+sr () 0k + =1 C 2 : 0, l,k,, C 3 : l =k K k =l sr () t 0, k, (10) = P, l. (11) The solutio to this optimizatio problem is characterized by the KKT coditios [21]; the Lagragia is give by: ( ) L,α k,μ l,λ = t K N K 1 + α k 2 log 2 1+sr () 0k + sr () t =1 l =k 1 It is worth emphasizig that while the solutio methodology here is similar to that of [19], both our problem formulatio ad solutio are sigificatly differet. The developmet here, usig the epigraph form, leads to effective solutios to OFDM-based relayig ad allows us to show that selectio is a sub-optimal solutio to the resource allocatio problem. + K μ l K P k =l K K + k =l λ, (12) where the α k, μ l, ad λ are the Lagrage multipliers associated with rate, total power, ad positive power costraits, respectively. For the sake of clarity, assume that K = 3. Ay solutio for the power that odes l 1 ad l 2 allocate to the j th subcarrier of source k 1 satisfies KKT coditios, which are: ) ) L (,α k,μ l,λ (p = L (),α k,μ l,λ p (j) l 1 k 1 p (j) l 2 k 1 = 0, (13) 0 α k 0, μ l 0, λ 0, l,k,, α k (R k ξ) = 0, λ = 0, l,k,. Now suppose that both relays, l 1 ad l 2, allocate some power to the j th subcarrier of the specified source. Usig the KKT coditios ad the fact that λ l1k 1j = λ l2k 1j = 0: = h(j) μ 2 h (j) 2 2. (14) Similarly, if the power that these two relays allocate to the i th subcarrier of the same source are ot zero, usig the same KKT coditios, oe ca coclude that: = h(i) μ 2 h (i) 2 2. (15) Equatios (14) ad (15) caot be simultaeously satisfied sice chael gais are cotiuous radom variables. Thus, ulike i [18], at most oe subcarrier of each source ca be helped by more tha oe ode i the system. Now, let us evaluate all the possible relay selectios i the etwork with four source odes, where the j th subcarrier of source k 1 is relayed by all other odes i the system. The KKT coditios state that: h (j) = μ 2 2 h (j) = μ 3 (16) 2 h (j) l 3k 1 2. Now suppose that the i th subcarrier of the same source is relayed through both odes l 1 ad l 2. The, h (i) = μ 2 (17) 2 h (i) 2. As a result, alog with (16), we have h (j) 2 / h (j) 2 = h (i) 2 / h (i) 2, which is a zero-probability evet. Now assume that oe of the subcarriers ca be helped with all three relays. As a example, cosider the case i which the j th subcarrier is relayed via ode l 1 ad l 2 ad the i th subcarrier ca be helped by ode l 1 ad l 3 i the system. Applyig the same KKT coditios, it follows that: h (j) = μ 2 ad 2 h (j) 2 h (i) = μ 3 (18) 2 h (i) l 3k 1 2. Now, the th subcarrier ca be helped by ode l 2 ad l 3 oly if h (j) 2 l 2 k 1 = h (j) l 1 k 2 h () 1 l 2 k 2 1 (i) h 2 l 3 k 1 h (i) l 1 k 2 h () 1 l 3 k 1 2, which happes with zero
probability. Therefore, whe two subcarriers are relayed with two odes, all others ca be helped by at most oe ode. Geeralizig this to the etwork with K source odes, oe cocludes that at most K 2 subcarriers of each source ca be helped by more tha oe relay ad selectio is imposed o (N K +2) subcarriers. I practice, N K which meas that a large fractio of subcarriers meet the selectio criterio, i.e., selectio is the approximate, though ot optimal solutio, to the relaxed optimizatio problem i (9)-(11). B. A Heuristic Algorithm ad a Lower Boud By eglectig the selectio costrait, the solutio to the problem i (9)-(11) provides a upper boud to that of the origial optimizatio problem i (6)-(8). Here, we use this to develop a heuristic solutio to the origial problem. We force the (maximum of K 2) subcarriers that do ot meet the costrait to receive power oly from the sigle relay that achieves a higher data rate. Sice this is a solutio that meets all the costraits of the origial problem, this is also a lower boud o the subcarrier based (LBSB) optimizatio problem. I Sectio V, we will show that the performace gap betwee the upper ad lower bouds is idistiguishable. As a result, this heuristic approach provides almost the exact solutio to the origial mixed-iteger optimizatio problem with sigificatly reduced solutio complexity. C. Optimal Power Allocatio Usig (13), the power that relay l 1 allocates to the j th subcarrier of the source k 1 ca be characterized as: 1+sr (j) 0k 1 + K (j) + l =l 1 1 p (j) = α k1 2l2μ l1 h (j) l 1 k 1 2 N 0 l =k 1 sr. (19) Equatio (19) shows that subcarriers which suffer more oise or ca receive more power from other relays will be allocated with less power. Cosequetly, the solutio to the power allocatio problem i the multi-source etwork follows multi-level waterfillig. Practical algorithms to solve differet waterfillig problems are provided i [22]. IV. BLOCK-BASED RELAY SELECTION The optimizatio problem ad solutio detailed so far is i keepig with existig literature. It allows differet subcarriers withi a OFDM block to be helped by differet relays. This is problematic for two reasos. Oe, while ot explicitly stated, most of the previous work assumes a relay ca treat each subcarrier as a idepedet trasmissio. I DF-based relayig, the decodig costrait is at the level of a subcarrier, e.g., (1). However, i OFDM, the data is first protected by a chael code, modulated ad the a block of N subcarriers is formed. It is ot possible to decode iformatio without receivig ad decodig a etire OFDM block. Secod, practical OFDM systems deped heavily o accurate time ad frequecy sychroizatio. This would be extremely difficult i a distributed mesh etwork. To the best of our kowledge, there has bee o cosideratio i the existig literature about selectio at the level of a etire OFDM block. I a multi-source etwork, as log as each relay has to divide its available power amogst all allocated sources, the solutio to the relay assigmet problem is ot immediate. Ufortuately, both the optimizatio formulatio ad solutio of joit selectio ad power allocatio are extremely complicated. Here we separate the problems ito selectio followed by power allocatio (via waterfillig) across subcarriers. As i [5], we propose three relay selectio schemes with differet levels of complexity ad compare the results i terms of the max-mi rate. A. Optimal Relay Selectio I a etwork with K sources where each source ca act as a relay for other odes, there are (K 1) K differet possible relay assigmets. The optimal scheme is exhaustive search over all possible relay selectios ad pick the oe which provides the maximum miimum rate i the system. This is clearly impossible for ay reasoable K. B. Sequetial Relay Selectio I this subsectio, we propose sequetial relay selectio to approximate the results of the optimal relay assigmet with less complexity. Based o this scheme, the first ode evaluates its achievable rate through selectig its best relay, r s1. The, s 2 picks r j ad r i odes with the best ad secod best relaydestiatio chaels. If its best relay has ot bee assiged to the first ode, i.e. r(s 1 ) = r j, it will be allocated to s 2. Otherwise, cosiderig the fact that its best relay distributes its available power amogst both sources, it evaluates the rate of commuicatio over both compoud source-relay-destiatio chaels ad selects the oe with higher rate. This process repeats sice oe relay has bee assiged to each source. I this scheme, K(K 1) waterfillig algorithms have to be solved. Although sequetial relay selectio is simpler tha the optimal relay selectio scheme, it is still too complex to be implemeted i practice. C. Decetralized Relay Selectio The decetralized or simple relay selectio scheme igores all other sources. Each source selects its best relay with the assumptio that the correspodig relay distributes its power equally over all subcarriers of oly that source. I particular: r k = r m if m = argmax j N P h() log 2 1+ N N 0 =1 r jk 2, (20) where r k is the relay assiged to ode k, j {1,...,K}, ad j = k. With each source havig selected the relays, the relays allocate power, via waterfillig, to the assiged sources. Note that sice each source picks its best relay idepedetly of all other source odes, this scheme ca be implemeted i the decetralized maer. I a etwork withk source odes, oly K water-fillig problems eed be solved.
Miimum rate across all users (bits/s/hz) 18 16 14 12 10 8 6 4 2 UBSB LBSB Direct Trasmissio Optimal Relay Selectio Sequetial Relay Selectio Simple Relay Selectio Miimum rate across all source odes (bits/s/hz) 50 45 40 35 30 25 20 UBSB LBSB Optimal Relay Selectio Sequetial Relay Selectio Simple Relay Selectio Direct Trasmissio 0 0 2 4 6 8 10 12 14 16 18 20 SNR (db) 15 26 27 28 29 30 31 32 33 34 Trasmitted Power (dbm) Fig. 2. Miimum rate across all potetial sources of differet cooperatio strategies i Equal Average Chael sceario with K=3 ad N=16. V. SIMULATION RESULTS I this sectio, we preset simulatio results to evaluate ad compare three resource allocatio algorithms i differet scearios. They are subcarrier-based relay selectio, blockbased relay selectio, ad direct trasmissio (o cooperatio). All iter-ode wireless chaels are modeled as frequecyselective chaels cosistig of four resolvable paths. Also, 16 subcarriers are used. We cosider two differet geometries: i the first sceario, all iter-ode chaels are idepedet ad have equal average power. The secod case is more realistic, where odes are distributed i the space radomly ad chaels are characterized by ode positios i the etwork. Therefore, iter-ode chaels have ueve power. To solve the relaxed optimizatio problem, we used CVX, a package for specifyig ad solvig covex problems [23], [24]. A. Simulatio Results for Equal Average Chaels Our first example uses three APs with all iter-ode chaels havig the same average power. Fig. 2 plots the miimum rate across all users for differet values of SNR. As see i the figure, the upper ad lower bouds (the heuristic) are idistiguishable. Furthermore, give the additioal flexibility of subcarrier-based cooperatio, both outperform block-based resource allocatio schemes. Note that at higher values of SNR, direct trasmissio outperforms all cooperatio based protocols. This result validates the fact that cooperatio is meaigful oly whe the relay-destiatio chael ca compesate for the factor of (1/2) due to relayig over two time-slots. Fially, for the block-based selectio scheme, the optimal ad sequetial schemes perform sigificatly better tha the simple selectio scheme. B. Simulatio Results for U-Equal Average Chaels I this subsectio, we provide simulatio results i the more realistic sceario where odes are radomly distributed i the etwork. Thus, iter-ode chaels have differet average Fig. 3. Miimum rate across all potetial sources of differet cooperatio strategies i Uequal Average Chael sceario with K=3 ad N=16. TABLE I PARAMETER VALUES IN COST-231 Parameter Value Parameter Value AP Height 15m Frequecy 3.5 GHz Buildig Spacig 50m Rooftop Height 30m Destiatio Height 15m Road Orietatio 90 deg. Street Width 12m Noise PSD -174 dbm power. The wireless chaels are simulated usig the COST- 231 chael model recommeded by IEEE 802.16j workig group [20]. This approach models both large ad small scale fadig. Parameters chose for this model are summerized i Table I. The variace of the log-ormal fadig is set to 10.6dB. We geerate radom ode locatios over a square area of 0.04 square kilometers. We fix the trasmitted power of each potetial ode to [26, 28, 30, 32, 34] dbm. Results are averaged over both source locatios ad chael realizatios. Fig. 3 plots the max-mi achievable rate across all APs ad compares the performace of various resource allocatio schemes. From the figure, the performace gap betwee LBSB ad UBSB is, agai, egligible. This proves that the heuristic method to fid the solutio of the origial covex optimizatio problem is almost exact. Furthermore, we compare the performace of block-based schemes. Not surprisigly, the optimal relay selectio method outperforms the other two schemes. Simple relay selectio closely tracks the sequetial relay selectio method, but with sigificatly less complexity. This result idicate that simple relay selectio scheme ca be implemeted i decetralized maer without sigificat performace loss. Moreover, direct trasmissio has the worst performace which validates the fact that relayig ca ehace the miimum rate of the system i this realistic sceario. Fig. 4 illustrates the importace of ode locatios o the performace of differet trasmissio/resource allocatio schemes. This example simulates a sigle source-destiatio
Source trasmissio rate (bits/s/hz) 40 35 30 25 20 15 10 5 UBSB Simple Relay Selectio Direct Trasmissio 0 0.1 0.15 0.2 0.25 0.3 0.35 Relay Destiatio distace (Km) Fig. 4. Source trasmissio rate i a sigle source-destiatio pair etwork with two odes act as relays ad N=16. pair with two relay odes i the system. The source-destiatio distace is fixed to0.2 2 km. Relays are located o both sides of source-destiatio path. Results are averaged over differet chael realizatios. Clearly oe wats the relay close to the destiatio; however, ote that this may impact o the assumptio that the relay ca always decode. Simulatio results show that relayig schemes outperform direct trasmissio wheever relays are located betwee the source ad destiatio odes. While the upper boud o subcarrier-based selectio outperforms block-based selectio, the performace loss for this more practical approach is surprisigly small. VI. CONCLUSION This paper ivestigates resource allocatio algorithms for cooperatio i a multi-source OFDM-based etwork. As has bee show earlier, selectio cooperatio has may advatages i distributed etworks, especially miimizig overhead ad avoidig issues of sychroizatio. We set up the uderlyig problem with a selectio costrait o each subcarrier to esure max-mi fairess across all sources. Sice this mixediteger programmig problem is computatioally complex, we relaxed the selectio costrait ad formulated a covex optimizatio problem that provides a tight upper boud. We showed that selectio is violated i oly K 2 of N subcarriers. This i tur leads to a heuristic solutio to the origial problem ad a tight lower boud. A secod cotributio i this paper is to formulate blockbased selectio for a multi-source etwork. Block-based selectio avoids issues of sychroizatio i OFDM-based etworks. We proposed three cooperatio schemes with varyig complexity. 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