Resource Allocation in Wireless Networks with Multiple Relays

Transcription

1 Resource Allocation in Wireless Networks with Multiple Relays Kağan Bakanoğlu, Stefano Toasin, Elza Erkip Departent of Electrical and Coputer Engineering, Polytechnic Institute of NYU, Brooklyn, NY, 0 Departent of Inforation Engineering University of Padova, via Gradenigo 6/B, 353 Padova, Italy Eail: kbakan0@students.poly.edu, toasin@dei.unipd.it, elza@poly.edu Abstract A cooperative network where transission between two nodes that have no direct link, but assisted by any relays is considered. We assue a broadband syste, such as OFDM, odeled by ultiple parallel Gaussian subchannels between the source and each relay, and also between each relay and the destination. We forulate the optiization proble for joint power and subchannel allocation under a short ter per-node power constraint to axiize the instantaneous total transission rate between the source and the destination. To solve this optiization proble, first we find the optial power allocation for a given subchannel allocation. Then we focus on a greedy algorith that jointly allocates subchannels and power. Siulation results reveal that our proposed algorith results in rates close to the optiu allocation. I. INTODUCTION Orthogonal Frequency Division Multiplexing OFDM is a proising technique for broadband wireless networks. OFDM can itigate the adverse effects of frequency selective ultipath fading by transitting signals over a nuber of narrowband channels which typically experience different fading levels. When the subchannel gains are known at the transitter, significant perforance iproveent can be achieved by dynaic power allocation []. In a wireless environent, the source and the destination nodes can be assisted by interediate nodes when the direct link is not available. The inforation fro the source to the destination is then transitted by ulti-hopping. Multihop transission is being considered to iprove coverage and increase throughput for the next generation of wireless networks, for instance 80.6j. In this paper, we study a broadband syste where the sourcedestination pair with no direct link is assisted by ultiple relays which utilize decode-and-forward. The transission is done in two hops where the source forwards the inforation to the relays and the relays forward the decoded inforation to the destination. The channels between nodes are assued to be frequency selective and OFDM is used in both hops. We study resource allocation, naely subchannel and power allocation to axiize the end-to-end rate. An overview of resource allocation techniques applied to relay networks is provided by []. In [3], [4] a cooperative single relay network is considered where nodes have partial channel state inforation CSIT about the ean channel attenuation. Optial power allocation for aplify and forward AF cooperation has been investigated in [5]. For a decode and forward DF strategy, tie and power are optiized in [6], under a This work is supported by WICAT and NSF grants No and No constraint on the average total syste power, with the goal of either axiizing the delay-liited capacity or iniizing the outage probability. Ergodic capacity for a DF relay syste has been also explored under various CSIT assuptions see [7], [6] and references therein. Soe of these papers do not consider OFDM. It would be classify the as so, that akes our contributions ore significant. In [8], a broadband relay channel where each link is coposed of any parallel, independent Rayleigh fading channels is analyzed. Under a long ter total average power constraint per node, power and transission tie are dynaically allocated either to iprove the delay liited capacity or to decrease the outage probability. However, [8] assues the availability of direct link. Resource allocation is even ore critical when ultiple relays are available, however only a few papers in literature consider this scenario. In [3], [4], ultiple relays are only considered when available in clusters around the source and the destination node. Opportunistic AF relaying has been investigated for ultiple-relay networks in [9] and [0]. Resource allocation for a wireless ultihop network with ultiple sources, ultiple relays and one destination is studied in [5]. A ultihop network with one source, one destination and ultiple relays where only one relay is utilized in each hop is investigated in detail. The proble we address can be thought of as a cobination of downlink and uplink resource allocation probles for ultiuser OFDM. On the other hand, rate atching in each relay, that is the constraint that outgoing rate at each relay is bounded by the incoing rate has to be aintained. This constraint akes the resource allocation proble ore challenging. In this context, we forulate an optiization proble over resource allocation strategies, naely subchannel and power allocation, over the source and relays to axiize the end-to-end rate. We first establish the optial power allocation for a given subchannel assignent. Then, we propose a greedy algorith for end-to-end rate axiization proble which jointly allocates subchannels and power. Downlink and uplink resource allocation for ultiuser OFDM are well investigated topics. In [3] the su rate is axiized in downlink. Su rate axiization with fairness and proportionally fairness aong users is considered in [] and [] respectively. Also in [4] uplink resource allocation proble is studied. These papers studied downlink and uplink probles separately, the challenging part, rate atching, is not considered. The reainder of this paper is organized as follows: In Section II syste odel is introduced and an optiization proble to axiize the instantaneous end-to-end rate is forulated. In Section III, the optiu power allocation is established for a given subchannel allocation. In Section IV, we propose a greedy

2 Fig.. S H, Syste odel H,N H M, H M,N RL RL M H,N H M, H, H M,N algorith which jointly allocates subchannels and power. Then nuerical results and conclusion follow. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a relay network with a source node S, a destination node D and M relay nodes RL, =,,..., M, eploying decode and forward. We assue that there is no direct link between S and D and all transissions are perfored in two hops with the assistance of relays. The links aong source/destination and each relay are coposed of N parallel subchannels corrupted by independent, unit variance coplex additive white Gaussian noise as shown in Fig.. For subchannel n, h,n denotes the coplex channel gain of the S RL link and h,n denotes the channel gain of the RL D link. The corresponding power gains are = h,n and = h,n. The source and relay nodes are subject to short ter power constraints, P S and P R, respectively. The relays are assued to be half-duplex, thus receiving and transitting in orthogonal tie slots. To this end, counication takes place in two phases coprising of equal tie slots: Phase and phase. In phase, the source transits to relays and in phase relays forward the inforation they decode to the destination. All subchannels can be used in both phases. In order to keep the ipleentation siple, we do not consider distributed space-tie coding and ipose a constraint on subchannel allocation where only one relay can be assigned to each subchannel in both phase and. Let the set E and Ē be the sets of subchannels which are assigned to RL in phase and phase, respectively. Moreover, source power allocated to subchannel n in phase and RL power allocated to subchannel n in phase are P n and P,n, respectively. Defining the total rate fro S to RL in phase as R and the total rate fro RL to D phase as R, we have R = + P n n E R = + P,n n Ē Since relays utilize decode-and-forward, the contribution of RL to the end-to-end rate is liited by the iniu of phase and phase rates and R, respectively. Our ai is to find optiu subchannel allocation, E and Ē and power allocations, P n and P,n, which axiize the instantaneous end-to-end achievable rate total. The resulting optiization proble can be forulated as ax = P n, P,n,E,Ē ax P n, P,n,E,Ē = D in R, R 3a subject to P,n P {,..., M} P n, P,n 0 for all, n E, E, E M are disjoint Ē, Ē, Ē M are disjoint E E E M {,, N} Ē Ē Ē M {,, N} 3b 3c 3d 3e 3f 3g 3h Note that, in this optiization proble, the su of the relay bottleneck rates is axiized. This results in a cobination of downlink and uplink resource allocation for ultiuser OFDM. With rate atching in each relay, the coupling of phase and phase leads to a challenging proble since an optial algorith has to solve phase, phase and rate atching probles jointly. Moreover, the optiization includes set selection, i.e. decision of E and Ē and the objective function involves iniu of two functions, and hence not a convex proble. Instead of solving this proble optially, we focus on providing a suboptial solution in two steps. We first find optial power allocation for given subchannel assignents in phase and. Then, based on this power allocation, a greedy algorith is proposed which jointly allocates subchannels and power. III. OPTIMAL POWER ALLOCATION FOR A GIVEN SUBCHANNEL ASSIGNMENT In this section, we investigate the optiu power allocation that achieves the axiu end-to-end rate for a given subchannel assignent. In other words, we focus on the optiization proble 3a together with power constraints, 3b, 3c and 3d, when the sets E and Ē are given for all {,, M}. Then, the optiization proble for power allocation can be written as ax P n, P,n = in R, R P,n P {,..., M} P n, P,n 0 for all, n 4a 4b 4c 4d where R and R are the rates of the RL in phase and phase, respectively and are given in and. Theore 3.: For a given subchannel assignent, E and Ē for all {,, M}, the optial power allocation that solves 4a - 4d for RL is found by waterfilling P,n = + for n Ē 5 where / is chosen to satisfy RL power budget, P R. The optial power allocation of the source is given by

3 / / +µ +µ 3 R < R R R R4 < R 3 = R = R 3 4 E E E 3 E 4 n Fig.. Optial source power allocation for a given phase subchannel assignent, E, and given optial phase rates, R, =,, M with M = 4 relays and N = 6 subchannels. + Pn = < R for n E ; + +µ = R 6 for n E. where R = n E + Pn and R = n Ē + H,n P,n are the rates corresponding to RL in phase and phase, respectively, with optial power allocation and, µ chosen to satisfy source power budget, P S. Proof: First, we consider optiu relay power allocation for given Ē. In 4a, each ter in the suation is subject to a separate relay power constraint 4c. Hence, relay power allocation P,n, n = {,..., N} and = {,..., M}, can be done only considering 4c. This results in M optiization probles =,, M ax R = ax P,n P,n n Ē + P,n P,n P R P,n 0 for all, n 7a 7b 7c The proble in 7 is equivalent to the standard proble of counication over parallel channels and the optiu solution is found by allocating RL power aong subchannels n Ē according to the waterfilling procedure [6]. The optiu power allocation for relays can be written as P,n = + n Ē 8 where / is chosen to satisfy power constraint, P R. The resulting optial phase rate of RL can be written as R = + H,n P,n 9 n Ē The optial source power allocation in phase can be now written as ax in R, R 0a P n = P n 0 for all n 0b 0c Note that 0a is the downlink power allocation proble constrained to eet relay phase rates R. This phase proble strictly depends on rates achieved in phase since the source has to allocate power according to phase optiu rates of each relay. The optiization in 0a can be written as ax P n = n E + P n n E + P n R 0 P n P S 0 P n 0 for all n To solve a-d, the Lagrangian is a b c d L = + P n = n E + µ + P n R = n E ν P n P S Using Karush-Kuhn-Tucker KKT conditions, the optial source power allocation is found as + Pn µ = + 3 where = ln ν and the copleentary slackness condition is µ n E + Pn R = 0 for all =,..., M. When n E + Pn < R, in order to satisfy the copleentary slackness condition, we choose µ = 0. Thus, optial source power allocation can be written as Pn = + < R ; + +µ = R. where R = n E + P n 4 As seen in the proof, the optial relay power allocation for RL is to do waterfilling on Ē, using power P R resulting in R, =,, M. The optial source power allocation is illustrated in Fig. for M = 4 relays and N = 6 subchannels when phase subchannel assignent, E and optial phase rates, R, are given for all =,, M. In the figure, subchannels allocated to the relays are ordered, the heights of the bars denote / and gray bars show the aount of power allocated to each subchannel. The source assigns power to all subchannels aintaining an equal water level, until the total phase rate R for RL eets the phase rate R. When R is atched with R, the optial source power allocation is to apply waterfilling

4 with water level + µ / to the subchannels allocated to RL. When the rate of phase can not be atched with phase rate for RL within the source power budget, P S, the source allocates power using waterfilling with water level / to the subchannels allocated to RL. Note that, the waterfilling level is the sae for all subchannels allocated to the relays whose phase rate is not atched with phase rate. If the source has ore than the power needed to atch all relay rates, the reaining power is not allocated although it can not be used in the next channel realizations. IV. GREEDY ALGORITHM FOR A JOINT SUBCHANNEL AND POWER ALLOCATION In the previous section, the optial power allocation for a given subchannel assignent is found. In this section, we propose a subchannel assignent algorith along with the corresponding optial power allocation in greedy fashion. Since the relays decode and forward each relay can only forward the iniu of phase and phase rates. Thus, a isatch of phase and phase rates for each relay is not favorable. A good algorith has to decrease the isatch by allocating ore resources to the bottleneck side of the transission. Motivated by this observation we propose a polynoial tie greedy algorith which jointly allocates subchannels and powers. The ain idea of this algorith is to increase the end-to-end rate at each step of the algorith by keeping the relay rate isatch sall. Let R opt E denote the end-to-end rate obtained by optial power allocation for a given subchannel allocation E = [E,..., E M ; Ē,..., Ē M ] as described in Section III. Greedy Algorith Initialization a Set A = {,..., N} and Ā = {,..., N} which are the available subchannels in the phase and phase, respectively. b Set E = and Ē = Until A = and Ā = a Set S = and or S = b For = to M i Find n = arg ax for n A and n = arg ax for n Ā ii Find R using R opt when n is tentatively allocated to RL in phase, that is E = E {n } iii Find R using R opt when n is tentatively allocated to RL in phase, that is Ē = Ē { n } iv Find R 3 using R opt when both n and n are tentatively allocated to RL in phase and phase, respectively, that is E = E {n } and Ē = Ē { n } v Find R = axr 3. The axiu suggests which phases to allocate an additional subchannel to RL vi Based on the axiu in step bv above, set S = {n } and or S = { n } c Find = arg axr d Update E = E S and or Ē = Ē S e Update A = A \ S and or Ā = Ā \ S In the greedy algorith we first initialize the sets A, Ā, E and Ē for all =,..., M where A and Ā are the available subchannel sets in the phase and phase and E, Ē denote Fig. 3. The odel for the source, the relay and the destination locations Opt, d =0.5 Greedy Algorith, d =0.5 Opt, d =0.3 Greedy Algorith, d =0.3 Opt, d =0.7 Greedy Algorith, d = P Fig. 4. Total rate vs node power where P S = P R = P for M = 3, N = 4 and d = 0.. the already allocated subchannels. At each step of the algorith at ost one subchannel for phase and phase are allocated. For each relay, the best subchannel aong available subchannels are tentatively chosen in phase and phase. In steps bii-bv, phases to allocate subchannels is decided. The corresponding rate axiizing subchannels for RL are stored in sets S and or S. After following sae procedure for all relays, the relayl, which has the largest end-to-end rate increase is chosen. The subchannels stored in the sets S and or S are allocated to RL and the available subchannel sets are updated. This algorith continues until all subchannels are allocated. Note that the algorith is greedy since at each step the R opt iproves the rates when additional subchannels are available. V. NUMERICAL RESULTS In order to evaluate the perforance of proposed greedy algorith, we consider the topoy shown in Fig. 3. The source, the center relay and the destination are located on a straight line and the distance between the source and the destination is noralized to. All relays are located on a line perpendicular to S - D axis. The source-center relay and the center relaydestination distances are d and d, respectively, where 0 < d <. The distance between the relays are d. The source-relay and relay-destination channels are odeled as N independent flat Rayleigh fading subchannels. Pathloss aong the nodes are taken into account by adjusting the eans of the channel coefficients and the pathloss exponent is assued to be 4. In our nuerical results, the end-to-end rate is optiized for each channel realization and the average end-to-end rate over all channel realizations are showed in Figures 4, 5 and 6.

5 To evaluate how well the greedy algorith does, we copare the perforance with the optiu resource allocation. The optiu resource allocation is found by full search where all possible subchannel allocations are tried with their corresponding optial power allocation of Sec. III. Hence coputing the optial solution using full search becoes highly intractable when a large nuber of subchannels are available. Thus, optial solution is given for only sall nuber of subchannels. Fig. 4 shows the total rate su as a function of the node power P = P S = P R for different d where we set d = 0., M = 3 and N = 4. The figure shows that the greedy algorith perfors very close to the optial resource allocation. The gap between the optiu solution and the greedy algorith diinishes when the relays are located close to the destination. On the other hand, for M = 3, highest end-to-end rate is achieved when the relays are close to the id-point of source destination pair, that is d = 0.5. In Fig. 5, relay power is fixed to P R = and d = 0.. We plot total rate su as a function of source power, P S, for different d and N = 4. In the region where source power is less than relay power the end-to-end rate is higher when the relays are close to the source. This is an expected results since the relays utilize decode-and-forward. Furtherore, the proposed greedy algorith perfors very close to the optial solution when the relays are close to the destination and the source power is sall. Fig. 6 shows the effect of nuber of relays M on total rate su, for three different source power values where relay power P R = is fixed and N = 4. When the source power is P S = 0., slightly increases as M increases. This is because phase is the bottleneck, thus increasing nuber of relays does not affect. However, in the case of P S = 0 total increases significantly since now phase becoes bottleneck. VI. CONCLUSION In this paper, we analyze resource allocation, naely subchannel and power allocation, in an OFDM syste eploying ultiple relays in a two-hop syste to facilitate counication aong a source and a destination. We forulate an optiization proble to axiize the instantaneous end-to-end rate. This proble is solved in two steps. First, we establish the optial power allocation given subchannel assignent. Then a greedy algorith is proposed which allocates subchannels and power jointly. Nuerical results show that proposed greedy algorith perfors very close to the optiu solution. Our future work includes investigation of the resource allocation proble for ultiple relays when the direct link between the source and the destination is present in the syste. REFERENCES [] B. S. Krongold, K. Rachandran, and D. L. Jones, Coputationally efficient optial power allocation algorith for ulticarrier counication systes, IEEE Trans. Coun., vol. 48, no., pp. 3 7, Jan [] Y.-W. Hong, W.-J. Huang, F.-H. 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