Transmit Optimization for Relay-based Cellular OFDMA Systems

Transcription

1 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the ICC 007 proceedings. Transit Optiization for Relay-based Cellular OFDMA Systes Wooseok Na, Woohyuk Chang, Sae-Young Chung, Yong H. Lee Departent of Electrical Engineering Coputer Science Korea Advanced Institute of Science Technology 373- Guseong-dong, Yuseong-gu, Daejeon, , Republic of Korea E-ail: wsna, sychung, Abstract This paper considers a broadb cellular orthogonal frequency-division ultiple-access OFDMA syste with relay nodes operating in decode--forward half-duplex ode. Two transit resource allocation probles for su-rate axiization are forulated for such a syste. The first one is the optiization of subcarrier allocation with predeterined power assignent for each subcarrier, the other is the joint optiization of power subcarrier allocation. Since these probles can t be solved easily in direct fors, we ake continuous relaxation solve the dual probles using a subgradient ethod. Nuerical results show that a rearkable increent in su-rate is achieved, with the aid of relay nodes sophisticated resource allocation, copared to a syste without relay nodes. I. INTRODUCTION Currently, due to its suitability for efficient high datarate transission, orthogonal frequency-division ultipleaccess OFDMA is considered as a proising technology for packet-based cellular counications [3] next generation cellular systes. In designing such cellular systes, a foridable perforance issue is the channel ipairent at the edge of a cell induced by a bad propagation environent, such as path-loss shadowing, co-channel interference fro adjacent cells. In any previous works []-[4], it is shown that these channel ipairents can be itigated through the use of cooperative transission by wireless relay nodes. That is, by an efficient design of relaying strategy resource allocation, the reliability transission rate of a wireless network can be iproved. In this paper, we consider the use of wireless relays for the su-rate axiization of a broadb cellular OFDMA syste. We assue the decode--forward strategy the half-duplex operation of relays, which is considered to be practical [4]. We further assue that the channel is fixed or slowly-varying, thus the channel can be estiated fed-back fro obile stations MSs relay stations RSs to the base station BS for centralized processing. Then, we investigate the downlink resource allocation for such a syste. This research was supported in part by MIC Ministry of Inforation Counication & IITA Institute for Inforation Technology Advanceent, Korea, through TI-KAIST international joint progra conducted by MMPC Mobile Media Platfor Center of KAIST. MS MS Fig.. RS RS Even index frae Odd index frae Syste odel As an optiization proble, we first forulate the optial subcarrier allocation proble with predeterined power assignent for each subcarrier. After that, we also forulate the optial joint power subcarrier allocation proble, which is ore coplex than the previous one. Then, it is observed that these probles belong to the class of integer prograing probles, thus, it is hard to find the optial solution in a direct approach. Instead of solving these probles directly, we use the continuous relaxation, as in [5] [6], to recast the as linear convex optiization probles, respectively, solve the dual probles using the subgradient ethod [7]- [9]. As the result of optiization, we show nuerically that the su-rate of a cellular OFDMA syste can be arkedly increased with the aid of RSs, copared to a syste without RSs. II. SYSTEM DESCRIPTION We consider a cellular syste, as shown in Fig. ; this syste is coposed of a single BS, M RSs, K MSs sharing the sae frequency b. We assue that the wireless channel between each pair of transitting receiving nodes is frequency selective, OFDMA is eployed for data counication to covert the channel into a set of N orthogonal subcarriers with flat channel responses additive white Gaussian noises AWGN. It is also assued that the channel BS /07/$ IEEE 574

2 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the ICC 007 proceedings. reains unchanged for at least two fraes, all channel inforation is fed-back to BS for centralized processing. Many previous works on this kind of relay-based network assue that the RS can transit receive siultaneously in the sae frequency b []-[3], i.e., full-duplex. However, since any liitations in radio ipleentation preclude the terinals fro full-duplex operation, we assue the halfduplex operation of RSs. Due to the half-duplex characteristics, each RS operates in two phases; it receives data fro BS at, say, an even index frae, then relays the data to MSs at the following odd index frae. We further assue the decode--forward operation of RSs. Fro the next section, we will focus on the downlink resource allocation for the syste, which axiizes the surate. III. OPTIMIZATION WITH FIXED POWER In this section, we forulate the su-rate axiizing downlink subcarrier allocation proble while assuing that the average power for each subcarrier is predeterined. A. Fixed Power Subcarrier Allocation Forulation For the ease of exposition, we assue that the average transit power, denoted as P, for each subcarrier of RSs BS is equal. For each subcarrier of index n, we define variables ρ,n indicating whether the subcarrier is allocated to the BS to th RS =,...,MorMS = M +,...,M+ K link, which has signal-to-interference-plus-noise ratio SINR α,n, in an even index frae. We also define σ,k,n indicating whether the subcarrier is allocated to the BS =0orth RS =,...,Mtokth MS link, which has SINR β,k,n, in an odd index frae. If the channel reains unchanged, α M+j,n = β 0,j,n j =,...,K holds. Then, using these variables, the subcarrier allocation proble is forulated as ax ρ,σ in A,B } + = B A M =0 M+K M+K =M+ A + B 0 = ρ,n =, n a K k= σ,k,n =, n b ρ,n 0, },, n σ,k,n 0, },, k, n c d ρ,n log + α,n P 3 n= n= k= K σ,k,n log + β,k,n P. 4 The OFDMA constraint guarantees that each subcarrier is occupied by only one transitter-receiver pair, that is, there is no subcarrier reuse in the syste. In, the first ter represents the rate of data flow fro BS to MSs via RSs operating in decode--forward strategy, the second ter represents the data rate of direct channels fro BS to MSs. Unfortunately, due to the OFDMA constraint, the above proble belongs to the class of integer prograing probles [], for which an exact solution usually involves an exhaustive search, which is coputationally extensive when the nuber of variables is large. Therefore, for an easier forulation, we relax the integer constraints c d as ρ,n 0 σ,k,n 0 respectively. This continuous relaxation corresponds to peritting tie sharing of each subcarrier was shown to be optial in the liiting case N in [5] [6]. Accordingly, the following lea holds. Lea : The optial solution to - with continuous relaxation satisfies A = B for =,...,M. Proof: Assue that the optiu is such that A B for soe,...,m}. We first consider the case that A >B. Then, for soe ρ,n 0, we can decrease ρ,n increase ρ,n for soe M +,...,M+K} while aintaining A >B satisfying a. Then, we obtain a new feasible solution with a larger objective value, which is a contradiction. Siilarly, we can show the contradiction for the case that A <B. Physically, the above lea iplies that when the subcarriers are allocated optially, the rates of receiving transitting data of each RS are equal. Using Lea, the subcarrier allocation proble with continuous relaxation can be rewritten as ax ρ,σ 4 A + B + = M+K =M+ A + B 0 5 A = B, for =,...,M 6a M+K = ρ,n =, n 6b K k= σ,k,n =, n 6c M =0 ρ,n 0,, n σ,k,n 0,, k, n 6d 6e This is a linear optiization proble which can be efficiently solved using the siplex [] or interior-point ethod [0]. B. Dual Proble for Subcarrier Allocation Although the stard optiization tools, such as siplex or interior-point ethod, can solve 5-6 efficiently, they still have large coplexity when the nuber of variables is large. Furtherore, unless the nuber of subcarriers is sufficiently large, the continuous relaxation generally yields a fractionalvalued solution, we should round it to 0 or to get a integer-valued solution. As an indirect approach, we can solve the dual proble of 5-6. Since 5-6 is a linear optiization proble, the duality gap is zero, thus the solution of the dual proble is equal to that of the prial proble. For the forulation of 575

3 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the ICC 007 proceedings. the dual proble, we first define the Lagrangian as follows: L ρ, σ, λ = 4 + = M+K =M+ A + B A + B 0 λ A B, 7 = λ [λ,...,λ M ] T is the vector of Lagrangian dual variables for the equality constraint 6a. Then, the dual objective function proble is coputed as ax L ρ, σ, λ gλ = ρ,σ 8 6b-6e the dual proble is given as in λ gλ. 9 Since the dual proble is always convex, it is guaranteed that the gradient-type algoriths converge to the global optiu. For 9, we can use a subgradient ethod [7]-[9], since it is hard to differentiate with λ. As the first step of this ethod, we initialize λ 0, given λ i, i is the iteration index, copute the dual objective function 8. Fro 7 8, we observe that the dual objective function can be decoupled into two independent subprobles as gλ =g λ+g λ, M ax g λ = ρ = 4 λ A + M+K =M+ A 6b 6d 0 g λ = ax σ B 0 + M = 4 + λ B 6c 6e. g λ can be further decoupled into N independent persubcarrier probles as g λ = N n= g,nλ, M ax ρ = 4 λ ρ,n log + α,n P g,n λ = + M+K =M+ ρ,n log + α,n P M+K = ρ,n = ρ,n 0,. Then, is a knapsack proble [], given λ i,the solution is given for each n as γ i,n ρ i,n = i 4 λi, = i 0, otherwise. 3 arg ax γ,n i 4 =,...,M+K log + α,n P,=,...,M log + α,n P,= M +,...,M + K. 5 Siilarly, g λ also decouples into N subprobles as g λ = N n= g,nλ. One clear fact is that if the solution of g,n λ is σ, k,n =for a, k pair σ,k,n =0for, k, k, then k is given as k = arg ax β,k,n. 6 k=,...,k This iplies that, if the th RS uses the nth subcarrier, then it always transits to the MS with the best channel for su-rate axiization. Using this property, for each n, the solution of g,n λ i is given as, =,k,n = i,k= 0, otherwise. σ i k i 7 i arg ax δ,n, i =0,...,M 8 k i arg ax β,k,n, i 9 δ i,n k=,...,k log + β,n P,=0 4 + λi log + β,n P,=,...,M 0 β,n = ax β,k,n. Note that ρ,n σ,k,n k=,...,k obtained in coputing the dual objective function are always integer-valued, that is, there is no tie-sharing. They can have fractional values only when there are ore than one equal axia of γ,n or δ,n. However, due to the nuerical error induced fro the finite-precision coputation, such a case seldo happens in practice. After coputing gλ i as above, λ i is updated using the subgradient ethod as λ i+ = λ i + s i A i B i, =,...,M s i is a sequence of step size. Since the subgradient ethod is guaranteed to converge to the optiu provided that s i is designed properly [9], we can obtain a nearoptial solution for 5-6 with a sufficiently large nuber of iterations. Then, since the solution is integer-valued, it is also a solution of -. IV. OPTIMAL POWER AND SUBCARRIER ALLOCATION Although the fixed-power subcarrier allocation proble considered in the previous section is siple, the available transission power is inflexibly inefficiently exploitted in that proble. Therefore, in this section, we consider the joint power subcarrier allocation proble axiizing the surate of a given relay-based cellular syste under the OFDMA total power constraint. A. Joint Power Subcarrier Allocation Forulation For the forulation of the joint power subcarrier allocation proble, instead of indication variables, such as ρ σ, we define positive variables p,n q,k,n, which represent the aount of power allocated to the nth subcarrier of BS to th RS =,...,MorMS = M +,...,M + K link BS =0orth RS =,...Mtokth MS 576

4 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the ICC 007 proceedings. link, respectively. Then, using these variables, the joint power subcarrier allocation proble is forulated as ax in Ã, p,q B } + M+K à + B 0 = =M+ N M+K n= = p,n P 0 3a K k= q,k,n P, 3b N n= p,n p,n =0,,n q,k,n q,k,n =0,, k,k,n à B p,n 0,, n q,k,n 0,, k, n 3c 3d 3e 3f log + α,n p,n, 4 n= n= k= K log + β,k,n q,k,n, 5 P P 0 is the total transit power of th RS BS. In this proble, the OFDMA constraint is represented by 3c 3d. As in, the first ter of represents the data rate of BS-RS-MS channels, the second ter represents the data rate of direct channels fro BS to MSs. Siilarly with the fixed-power case, -3 is a nonlinear integer prograing proble [], it is hard to find the optial solution. For this case also, we can consider a continuous relaxation by allowing the tie sharing of each subcarrier. For this purpose, we define additional variables v,n w,k,n, which represent the fraction of tie that the nth subcarrier is occupied by BS to th RS =,...,M or MS = M +,...,M + K link BS =0or th RS =,...Mtokth MS link, respectively. Lea also holds in this case, thus the relaxed probles are written as ax p,q,v,w 4 =  + ˆB + M+K  + ˆB 0 =M+ 6  = ˆB, for =,...,M 7a N M+K n= = p,n P 0 7b K k= q,k,n P, 7c  N n= M+k M =0 n= = v,n =, n 7d K k= w,k,n =, n 7e v,n 0,, n w,k,n 0,, k, n p,n v,n log +α,n v,n 7f 7g 8 ˆB K n= k= q,k,n w,k,n log +β,k,n. 9 w,k,n Then, this is a convex optiization proble can be solved efficiently using stard optiization tools [0]. B. Dual Proble for Power Subcarrier Allocation As we did in Section III, we try to tackle -3 by solving its dual proble. Since it is a convex proble, the duality gap is zero. The Lagrangian is fored as follows: L p, q, v, w,λ,µ,ξ= 4  + ˆB = + M+K  + ˆB 0 =M+ M+K + µ P 0 p,n = n= = n= k= λ A B = K + ξ P q,k,n. 30 λ, µ, ξ are the vectors of dual variables. Then, the dual objective function is coputed as ax L p, q, v, w,λ,µ,ξ hλ, µ, ξ = p,q,v,w 3 7d-7g the dual proble is given as in hλ, µ, ξ λ,µ,ξ µ 0,ξ 0 3 represents the eleent-wise inequality. To solve the dual proble with a subgradient ethod, we first initialize λ 0, µ 0, ξ 0 copute the dual objective function given λ i, µ i, ξ i. We observe that the dual objective function can be decoupled into two subprobles as hλ, µ, ξ =h λ, µ+h λ, ξ, h λ, µ = ax p,v M = 4 λ  + M+K +µ P 0 N M+K n= = p,n 7d 7f =M+  33 ax ˆB q,w 0 + M = 4 + λ ˆB h λ, ξ = + M = ξ P N K n= k= q,k,n 6c 6e. 34 Then, are equal to the dual objective functions of ulti-carrier broadcast ultiple-access channel optiization probles considered in [5] [6], respectively. Given 577

5 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the ICC 007 proceedings. λ i, µ i, ξ i, the solution of 33 is given for each n as v,n i, =ˆ = i X i ˆ i,n > , otherwise, p i,n = v i,n [ c i µ i α,n 36 MS RS BS c i ˆ i arg ax =,...,M+K [ X,n i c i c ln +α i,n µ i µ i [ c i X,n, i 37 α,n µ i α,n 38, 39 4 λi, =,...,M, = M +,...,M + K. 40 Siilarly, for each n, the solution of 34 is given as w i,k,n =, =ˇ i,k= ǩi, Y i ˇ i,n > 0 0, otherwise, q i,k,n = wi,k,n [ d i ξ i β,k,n 4 4 ˇ i arg ax Y,n, i 43 =0,...,M ǩ i arg ax β ˇ,k,n, 44 i k=,...,k [ ] Y,n i d i d ln +β i +,n 45 ξ i β,n [ ] ξ i d i +, 46 ξ i β,n d i, = λi, =,...,M, β,n ax β,k,n. k=,...,k Note that the tie sharing paraeters v,n w,k,n obtained in coputing the dual objective function are always integer-valued, that is, there is no tie-sharing. Tie-sharing is possible only when there are ore than one equal axia of X,n or Y,n. However, as entioned previously, this case seldo happens due to the nuerical error. After coputing hλ i,µ i,ξ i as above, it is iniized using a subgradient ethod [9]. That is, λ i, µ i, ξ i Fig.. are updated as follows: λ i+ Topology of a relay-based cellular syste. = λ i + s i  i µ i+ = µ i + t i P 0 ξ i+ = ξ i + u i P i ˆB,=,...,M, 48 M+K n= = K n= k= p i,n q i,k,n, 49,=0,...,M. 50 s i, t i, u i are sequences of step size designed properly [9]. Then, with this subgradient ethod, we can obtain a near-optial solution of 6-7 with a sufficiently large nuber of iterations. Then, since the solution is integervalued, it is also a solution of -3. V. SIMULATIONS This section copares the perforance of two optiization schees for the relay-based network treated in previous sections through coputer siulation. We consider a single cell odel with a ring-shaped boundary region. The radius of the inner outer bound of this region is , respectively. MSs are uniforly distributed in the boundary region, RSs are evenly located on the inner bound of this region. We also assue that all MSs reain stationary the TABLE I SIMULATION PARAMETERS. Paraeter Assuption Cell radius 000 Propagation odel BS-MS, RS-MS: log 0 R path-loss, in db BS-RS: log 0 R R in kiloeters RS antenna gain 0 db MS antenna gain 0dB MS noise figure 9dB Theral noise density -74 db/hz BS RS Tx power 4 db FFT size 8 # of used subcarriers 96 Channel bwidth.5 MHz Sall-scale fading odel Exponential power delay profile with decaying rate 0 µs delay spread #ofmss 8 578

6 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the ICC 007 proceedings. Su rate bit/sec/hz Fixed power optial subcarrier allocation Optial power subcarrier allocation # of RSs Fig. 3. Su rate of relay-based cellular syste. channel does not change. A siplified view of this topology is shown in Fig.. The siulation paraeters are listed in Table I, which refers ainly to [3] [4]. For perforance coparison, we evaluate the su-rate of each optiization schee using a different nuber of RSs, the results are shown in Fig. 3. We can observe that, as the nuber of RSs increases, the su-rate also increases. When the nuber of RSs is zero, i.e., when there is no RS, the fixed power subcarrier allocation proble is reduced to a knapsack proble [], the solution is siply to choose the best receiver MS for each subcarrier. Without RSs, the joint power subcarrier allocation proble is reduced to the su-rate axiization proble for broadcast channels [5], the solution is to choose the best receiver for each subcarrier to perfor waterfilling over all subcarriers. For the considered syste, the power level is sufficiently high, thus there is a little difference between the fixed-power allocation waterfilling []. When there are ore than one RS, the su-rate is evaluated using the subgradient ethod with 000 iterations. The result shows that the joint power subcarrier allocation schee has a uch higher su rate than the fixed power subcarrier allocation schee, which is because the forer exploits the available power ore efficiently than the latter. VI. CONCLUSION In this paper, we showed that the su-rate of a broadb cellular OFDMA syste can be rearkably increased by using relay nodes sophisticated resource allocations. In particular, we forulated two different resource allocation probles for su-rate axiization. The first one is the optiization of subcarrier allocation with a predeterined power assignent for each subcarrier, the other is the joint optiization of power subcarrier allocation. Due to the OFDMA constraint which forces the exclusive use of each subcarrier by transitter-receiver pairs, these probles belong to the class of integer prograing probles, solving the is coputationally prohibitive when the nuber of variables is large. Therefore, for easier forulations, we relaxed the probles as linear convex optiization probles, respectively, by allowing tie-sharing usage of each subcarrier by ultiple transitter-receiver pairs. Then, we forulated the dual proble of each relaxed proble solved it using a subgradient ethod. Fro the result of the subgradient ethod, we always obtained integer-valued solutions, which are the near-optial solutions to the original unrelaxed probles. Throughout this paper, we have assued that all channel inforation is perfectly known to the BS, which is alost ipossible in practice because of the iperfectness of channel estiation liited bwidth of the feedback channel. Therefore, our future work will include a study on the effects of nonideal liited channel inforation on the perforance of the relay-based syste. Another topic of future work is solving the uplink resource allocation proble for the relaybased syste. Since there is a larger nuber of constraints on the transission power of each MS, this proble is ore coplex than its downlink counterpart. However, we believe that it also can be easily tackled with approaches siilar to those treated in this paper. REFERENCES [] T. M. Cover A. A. El Gaal,, Capacity theores for the relay channel, IEEE Trans. Infor. Theory, vol. IT-5, no. 5, pp , Sep [] G. Kraer, M. Gastpar, P. Gupta, Cooperative strategies capacity theores for relay networks, IEEE Trans. Infor. 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