Resource Allocation for Co-Primary Spectrum Sharing in MIMO Networks
Tachporn Sanguanpuak, Sudarshan Guruacharya, Nandana Rajatheva, Matti Latva-aho
aa r X i v : . [ c s . N I] A ug Resource Allocation for Co-Primary SpectrumSharing in MIMO Networks
Tachporn Sanguanpuak ∗ , Sudarshan Guruacharya † , Nandana Rajatheva ∗ , Matti Latva-aho ∗∗ Centre for Wireless Communications (CWC), University of Oulu, Finland; † Dept. Elec. & Comp. Eng., University of Manitoba, CanadaEmail: { tachporn.sanguanpuak, nandana.rajatheva, matti.latva-aho } @oulu.fi; [email protected] Abstract —We study co-primary spectrum sharing concept intwo small cell multiuser network. Downlink transmission isexplored with Rayleigh fading in interfering broadcast channel.Both base stations and all the users are equipped with multipleantennas. Resource allocation with joint precoder and decoderdesign is proposed for weighted sum rate (WSR) maximizationproblem. The problem becomes mixed-integer and non-convex.We factor the main objective problem into two subproblems.First subproblem is multiuser with subcarrier allocation wherewe assume that each subcarrier can be allocated to multiple users.Gale-Shapley algorithm based on stable marriage problem andtransportation method are implemented for subcarrier allocationpart. For the second subproblem, a joint precoder and decoderdesign is proposed to obtain the optimal solution for WSRmaximization. Monte Carlo simulation is employed to obtain theresults.
I. I
NTRODUCTION
Future wireless networks will have to satisfy the require-ments and quality of service of a large amount of applicationssuch as various information flows like video streaming, dataapart from voice, and smart-phones etc. The number of devicesin wireless network is increasing exponentially, and it couldreach hundreds of billions within next decade. To satisfy amuch larger low-rate devices and also high-rate mobile users,network operators will need to increase network capacity.The networks operators will need to allocate spectrum moreefficiently to obtain higher network capacity.Multi-operator spectrum sharing in which the multiple op-erators agree on jointly using parts of their licensed spectrumbecomes a main contributor in this direction. This is mainlydue to the fact that using of dedicated spectrum by operatorshas been shown to be inefficient, spectrum is found to be idleat various times. Multi-operator spectrum sharing concept istherefore expected to be an important aspect to improve spec-tral efficiency. In [1], licensed shared access (LSA) concept in . GHz spectrum band is demonstrated for spectrum sharingbetween mobile network operator and incumbent users. Thesmart antennas technologies can be used to enhance LSAsystems [2]. Different types of incumbent users and factorsfor allowing spectrum sharing with LSA from their perspectiveare considered in [3].In [4], the authors consider co-primary spectrum sharingfor dense small cells in which each operator divides spectrumpool into dedicated and shared spectrum band. System levelsimulation is used to evaluate the throughput. The spectrum sharing aspect in cognitive radio system is studied in [5].The unlicensed secondary users try to maximize their overallcapacity by cooperating with primary users. Sharing spectrumbetween the co-located radio networks (RANs) supported bydifferent operators is studied in [6]. A heuristic search algo-rithm is proposed to maximize the inter-RAN sum rate basedon user-grouping, spectrum partitioning, and user scheduling.There are a number of technology features and capabili-ties, which should be considered to be greatly beneficial inoptimizing the utilization of available spectrum. For example,including multiple-input-multiple-output (MIMO), beamform-ing, and network densification. By employing MIMO capabil-ity, the diversity gain and the coding gain can be improvedthrough beamforming technique or space time coding com-pared to single-input-single-output (SISO) system [7]. Thejoint precoder and decoder (known as a joint transceiver)shoud be designed properly to achieve higher throughput.Most of the MIMO broadcast channel (MIMO BC) worksconsider the linear joint transceiver techniques according toeasily implementation [8], [9].In [10], multiple-BS transmit signal simultaneously to theirusers in their own cells and cause the interferences to users. Ajoint precoder and decoder design to maximize weighted sumrate (WSR) based on iterative minimization of weighted meansquare error (WMSE) is proposed.In 5th generation mobile communication (5G) system,multiple-input multiple-output (MIMO) and multi-operatorspectrum sharing in heterogeneous network will play crucialroles. In the earlier works mentioned above, the concept ofco-primary spectrum sharing in MIMO with multiple usersnetworks has not been studied. In our work, we propose a co-primary spectrum sharing in multiuser two small cells network.The small cell base stations and their own users employ mul-tiple antennas. Joint precoder-decoder design based on MIMOsystem with subcarriers allocation for downlink transmissionis proposed. We assume that both base stations allocate usersin dedicated subbands and shared subbands. Each base stationallocates its users to utilize the shared band when the numberof subcarriers in dedicated spectrum band is not enough toserve all users.The WSR maximization problem of both cells for multipleusers with multiple subcarriers allocation is studied. The prob-lem becomes non-convex and therefore we separate the mainproblem into two subproblems. In the first subproblem, wellocate subcarriers to users by using two methods. The firstmethod is Gale-Shapley algorithm which is based on stablemarriage problem [11]. The second method is implementedbased on transportation method [12], linear programming isused to solve this problem. The number of users whichis served by each subcarrier is less than or equal to thenumber of transmitting antennas for both methods. In thesecond subproblem, after a fixed subcarrier assignment, a jointprecoder-decoder design is employed. Bisection method isused to find the optimal precoder.II. S
YSTEM M ODEL
Both base stations allocate users in their dedicated bandsfirst and then in the shared bands. In each cell, the base stationallocates the same amount of bandwidth to each set of users.The base stations employ multiple number of antennas suchthat N T k and N T j for base station k and j , where j = k ,respectively. At both base stations, the number of total usersis more than the number of transmit antennas denoted as, I k ≥ N T k and I j ≥ N T j . I k , and I j are the set of users in the cell k and j where I k is the cardinality of I k , denoted as I k = |I k | .Each user in both cells utilize multiple number of antennas.Any user i k ∈ I k in the cell k uses N R k antennas and anyuser i j ∈ I j in the cell j uses N R j antennas. The system isdemonstrated in Fig. 1. Fig. 1. Spectrum Sharing Between Two Small Cells
The base station k and j allocate the set of subcarriers N , and M , respectively to their own users in the dedicatedspectrum band. We denote the cardinality set of N , and M as N = |N | and M = |M| , respectively. In the cell k , and j simultaneous transmission to multiple users on the samesubcarrier n , and m , respectively is allowed.Thus, in the dedicated band of the cell k , the received signalby user i k on subcarrier n is given by, y i k ,n = H i k ,n T i k ,n x i k ,n + I k X j k =1 ,j k = i k H i k ,n T j k ,n x j k ,n + z i k ,n , (1) where P I k j k =1 ,j k = i k H i k ,n T j k ,n x j k ,n is the intracell interfer-ence (ICI). The H i k ,n is N R k × N T k channel gain matrixbetween the base station k and any user i k on subcarrier n . T i k ,n is the N T k × a i k beamforming matrix allocated to user i k on subcarrier n and a i k denotes independent data streamsfrom the base station k to user i k . x i k ,n is the column matrixof length a i k × representing transmitted symbols to different I k users. The z i k ,n is additive white Gaussian noise with zeromean and covariance matrix σ i k ,n I N Rk .It is required to select t k ≤ N T k users out of the setof I k users in each subcarrier. Number of simultaneouslyserved users on each subcarrier is limited by the number oftransmit antennas. There are C k = P N Tk t k =1 (cid:0) I k t k (cid:1) combinationsof users who can utilize same subcarrier, each of them isdenoted as L t k , where L t k = { , ..., I k } , < |L t k | ≤ N T k .Assuming that the group of users L t k is assigned to subcarrier n , then received signal by any user ( i k ∈ L t k ) is as, y i k ,n = H i k ,n T i k ,n x i k ,n + P j k ∈L tk ,j k = i k H i k ,n T j k ,n x j k ,n + z i k ,n , where P j k ∈L tk ,j k = i k H i k ,n T j k ,n x j k ,n denotes the interfer-ence caused by users on the same subcarrier n .In the shared band part, both intracell and intercell interfer-ence affect all the users. The intracell interference is causedby the same subcarrier allocated to users and the intercellinterference is caused by the broadcast signal from the otherbase station in the same frequency for shared band.At the receiver, a decoding matrix U i k ,n with the size N R k × a i k is used to recover the original signal. We assumethat the receiver use MMSE decoder thus, the U i k ,n denotesMMSE decoding matrix from the base station k to user i k on subcarrier n . The output of the decoder can be writtenas r i k ,n = U Hi k ,n y i k ,n . In the cell j , we consider the samescenario as in the cell k . Thus, we can write the output of thedecoder in the cell j similar to cell k as r i j ,m = U Hi j ,m y i j ,m .The maximization of total weighted sum rate of two cellsin both dedicated band and shared band parts are studied innext section.III. W EIGHTED S UM R ATE M AXIMIZATION FOR C O -P RIMARY S PECTRUM S HARING
The optimization problem for maximization weighted sumrate of both cells can be written as,max P I k i k =1 P Nn =1 µ i k ,n R k + P I j i j =1 P Mm =1 µ i j ,m R j subject to P i k ∈ I k P n ∈N || T i k ,n || ≤ P maxk , P i j ∈I j P m ∈M || T i j ,m || ≤ P maxj P i k ∈I k ρ i k ,n,t k ≤ N T k ∀ n, t k P i j ∈I j ρ i j ,m,t j ≤ N T j ∀ m, t j ρ i k ,n,t k , ρ i j ,m,t j ∈ { , } ∀ n, m, t k , t j , (2)where R k = P C k t k =1 ρ i k ,n,t k R i k ,n , where R i k ,n is the rateof user i k on subcarrier n . And R j = P C j t j =1 ρ i j ,m,t j R i j ,m ,where R i j ,m is the rate of user i j on subcarrier m . ρ i k ,n,t k denotes whether subcarrier n is assigned to user i k ∈ L t k in the cell k , then ρ i k ,n,t k = 1 , otherwise ρ i k ,n,t k = 0 . And ρ i j ,m,t j denotes whether subcarrier m is assigned to user i j ∈ t j in the cell j , then ρ i j ,m,t j = 1 , otherwise ρ i j ,m,t j =0 . We assume that each particular subcarrier is allocated tomore than one user. In addition the number of users whichare served by each subcarrier is less than or equal to numberof transmitting antennas. µ i k ,n and µ i j ,m are weights whichare used to represent the priority of any user i k and i j in thecell k and j , respectively.This problem is mixed-integer and nonconvex, we proposea heuristic method to solve (2). To reduce the computationalcomplexity, first we find subcarrier assignment and in thesecond stage, we find the precoder for users in the particularsubcarrier by using fixed subcarrier assignment. The algorithmfor subcarrier allocation in the cell k is described as followsaccording to Gale-Shapley algorithm.[11] Essentially, Gale-Shapley algorithm solves a matching problem (also known asthe stable marriage problem) for a bipartite graph, where thetwo disjoint set of vertices are customarily referred to as theset of men and set of women. A man and a woman is said tohave a stable marriage if neither of them are better off beingwith a different partner. The basic idea behind the algorithm isto allow men to propose to women in the order of their prefer-ence, while women provisionally accepts the proposal until abetter proposal arrives. Unlike the assignment problems solvedby the Hungarian method for weighted bipartite graph, wherethe objective is to obtain a maximum weighted matching, thefocus of Gale-Shapley method is to obtain a stable matching.In our case, we consider a slight modification of the stablemarriage problem, in that a single woman can engage withmore than one man. We assume that the role of men isassumed by the users, while the role of women is assumed bythe subcarriers. However, our modification of the basic Gale-Shapley algorithm allows a subcarrier to accept more than oneuser. If the number of subcarriers is not enough to serve allof it’s users, the base station will assign the remaining usersto utilize the shared band. Note that in the cell j , we repeatthe same process as in the cell k . The algorithm for subcarrierallocation with joint precoder and decoder designs is describedin Table I. The worst case complexity of this algorithm is O (( |N | + |M| ) P k I k ) .IV. W EIGHTED S UM R ATE M AXIMIZATION VIA W EIGHTED SUM
MSE M
INIMIZATION FOR C O -P RIMARY S PECTRUM S HARING
In this section, we study a joint precoder-decoder design inMIMO system for the second stage subproblem. Due to theequivalence between WSR maximization problem and mini-mization of WMSE problem [10], we can obtain the optimalsolution of WSR maximization via WMSE minimization. Inthe dedicated band of the cell k , the MSE covariance matrixbetween the actual transmitted data vector and the receivedsignal at any user i k on the subcarrier n can be writtenas MSE ( T i k ,n , U i k ,n ) = E x ik,n , z ik,n [ || x i k ,n − U Hi k ,n y i k ,n || ] under the assumption that x i k ,n and z i k ,n are independent. It
1. Set R i k = 0 for i k ∈ I k
2. For each user i k ∈ I k , in cell k , make a preference list of dedicatedsubchannels B i k = [ α , ..., α N ] such that || H i k ,α l || ≥ || H i k ,α m || if l < m
3. For each dedicated subchannel n ∈ N , make a preference listof users Q n = [ β , ..., β N ] such that || H β l ,n || ≥ || H β m ,n || if l < m
4. For each subchannel n ∈ N , initialize the user acceptance list A n = ∅
5. Repeat6. For each user i k ∈ I k ,7. If ∄ n such that i k ∈ A n , (i.e. if there is no acceptance listto which user i k belongs to)8. Find the subchannel α t ∈ B i with the highest preference,9. If |A α t | < N T ,10. Put user i k in the acceptance list A α t
11. Else if ∃ γ ∈ A α t such that β − ( i ) < β − ( γ ) in thepreference list Q α t
12. Replace the user γ by user i in A α t
13. End If14. Remove α t from the preference list B i
15. End If16. End For17. Until {∃ n such that i k ∈ A n for every i k ∈ I k } OR {B i k = ∅ for any i k ∈ I k }
18. Find the subset of users I ′ k ⊂ I k which have not been assignedto any acceptance list19. Do steps 2 to 17 for users in I ′ k using shared subchannels.20. For each user i k ∈ I k ,21. Allocate power in that particular subchannel n based onoptimal joint transceiver design via WMMSEiterative algorithm22. End For23. For each user i k ∈ I k ,24. Compute the rate achieved R k
25. End For TABLE IS
UBCARRIERS ALLOCATION BY USING G ALE -S HAPLEY ALGORITHM WITHJOINT PRECODER - DECODER DESIGN leads to,MSE ( T i k ,n , U i k ,n )= E x ik,n , z ik,n [( x i k ,n − U Hi k ,n y i k ,n )( x i k ,n − U Hi k ,n y i k ,n ) H ]= ( I N Rk − U Hi k ,n H i k ,n T i k ,n )( I N Rk − U Hi k ,n H i k ,n T i k ,n ) H + X j k ∈L tk ,j k = i k U i k ,n H i k ,n T j k ,n T Hj k ,n H Hi k ,n U Hi k ,n + σ i k ,n U Hi k ,n U i k ,n (3)Additionally, in the dedicated band of the cell j , theMSE covariance matrix at any user i j on the subcarrier m denoted as MSE ( T i j ,m , U i j ,m ) can be written equivalent toMSE ( T i k ,n , U i k ,n ) . In the shared band of the cell k , the MSEcovariance matrix of any remaining user l k which has not beenassigned to the dedicated band can be given as,MSE ( T l k ,p , U l k ,p )= ( I N Rk − U Hl k ,p H l k ,p T l k ,p )( I N Rk − U Hl k ,p H l k ,p T l k ,p ) H + K X k =1 X b k ∈L tk share ,b k = l k U l k ,p H l k ,p T b k ,p T Hb k ,p H Hb k ,p U Hl k ,b + σ l k ,p U Hl k ,p U l k ,p (4)here p is the subcarrier in shared band which any remaininguser l k is assigned. The MSE covariance matrix of anyremaining user l j of the cell j which is assigned to sharedband can be written equivalent to MSE ( T l k ,p , U l k ,p ) . Withan extension of [10][Theorem 1], for W i k ,n ≥ be a weightmatrix at any user i k on the subcarrier n . The followingproblem has the same optimal solution as weighted sum ratemaximization problem and can be rewritten as,min P I k i k =1 P Nn =1 µ i k ,n F i k ,n + P I j i j =1 P Mm =1 µ i j ,m F i j ,m subject to P i k ∈ I k P n ∈N || T i k ,n || ≤ P maxk , P i j ∈I j P m ∈M || T i j ,m || ≤ P maxj (5)where F i k ,n = Tr ( W i k ,n × MSE ( T i k ,n )) − log det ( W i k ,n ) (6) F i j ,m = Tr ( W i j ,m × MSE ( T i j ,m )) − log det ( W i j ,m ) (7)For fixed receivers which are given by the MMSE solution, itleads to, U MMSE i k ,n = J − i k ,n H i k ,n T i k ,n , (8) J i k ,n = P I k j k =1 ,j k = i k P Nn =1 H i k ,n T j k ,n T Hj k ,n H Hi k ,n + σ i k ,n I .With the MMSE receiver U MMSE i k ,n , the following MSE matrixis given byMSE ( T i k ,n , U MMSE i k ,n ) = I − T i k ,n H i k ,n J − i k ,n H i k ,n T i k ,n (9)The coordinate descent algorithm based on [10] is used tosolve (5). The coordinated descent algorithm is performed bythree sets of variables which are precoder, decoder, and weightmatrices. Each set of these three variables is solved seperatelyin a sequential manner, while assuming the other two sets arefixed. Thus, we update the transmit beamformer T i k ,n andweight W i k ,n by fixing the decoder matrix.The Karush-Kuhn-Tucker (KKT) condition can be used tofind the optimal solution of the problem (5). With the firstorder optimality condition of Lagragian function respect to T i k ,n , this yields T opt i k ,n = µ i k ,n (cid:18) I k X j k =1 ,j k = i k N X n =1 H Hj k ,n U MMSE j k ,n W j k ,n × ( U MMSE j k ,n ) H H j k ,n + λ ∗ i k ,n I (cid:19) − H Hi k ,n U MMSE i k ,n W i k ,n (10)where the optimum λ ∗ i k ,n must be positive and it can beobtained by using the bisection method. To obtain the jointprecoder and decoder design for WMSE minimization aftersubcarrier assignment, an iterative algorithm is proposed inTable II. Note that the data rate of each user i k on thesubcarrier n in the dedicated band of cell k can be obtainas R i k ,n = log det (( MSE ( T i k ,n , U MMSE i k ,n )) − ) . All the userssdata rate are calculated similar as R i k ,n .
1. Initialize the precoding matrix T i k ,n such that Tr ( T i k ,n T Hi k ,n ) = P t I K
2. Repeat3. U i k ,n ← U MMSE i k ,n = J − i k ,n H i k ,n T i k ,n in (8)4. W i k ,n ← ( I − U Hi k ,n H i k ,n T i k ,n ) −
5. Find the optimum Lagrange multiplier value λ ∗ i k ,n by usingbisection method6. Substitute the optimum λ ∗ i k ,n in T opt i k ,n in (10)7. Put T i k ,n ← T opt i k ,n in (10)8. Untill | ( log det ( W i k ,n )) b +1 − ( log det ( W i k ,n )) b | ≤ ǫ where b denotes the iteration number and ǫ is a tolerence value( < ǫ << ) TABLE IIA N ITERATIVE ALGORITHM FOR A JOINT PRECODER AND DECODERDESIGN VIA
WMMSE
AFTER SUBCARRIER ALLOCATION BASED ON G ALE -S HAPLEY ALGORITHM IN THE CELL k In the cell j , an iterative algorithm to obtain the optimalprecoder and decoder design can be proposed similar to as inthe cell k . V. N UMERICAL R ESULTS
The numerical results illustrate performance of maximiza-tion of WSR via WMSE minimization problem for MIMOwith co-primary spectrum sharing. Rayleigh fading channelsare assumed in MIMO interfering broadcast channels. Theuncorrelated fading channels are generated for each userequipment independently in each time slot.Monte Carlo simulation is used by assuming samplesto obtain the results. After both base stations finish allocatingsubcarriers to their users in the dedicated band there may stillbe some remaining users. Both base stations will assign thoseusers to employ shared band. In all the simulation, we assumethat the number of antennas at both base stations N T k , N T j =4 and at each user, N R k , N R j = 2 To compare the performance of Gale-Shapely method, wemodel the subchannel allocation problem as a transportationproblem, which is a special case of maximum weight flowproblem, which is then solved by using linear programming.The transportation problem itself is a generalization of theassignment problem which can be solved using the Hungarianmethod. The Hungarian method is not applicable in our casesince a subchannel can be assigned to more than one user. In anutshell, we have a bipartite graph where the subchannels aresource nodes with outflow constraint equal to the maximumnumber of users they can support, which is equal to the numberof transmit antennas. Similarly, the destination nodes are theusers with inflow constraint equal to unity, that is, a user canbe assigned to a single subchannel. Lastly, the weight matrixis given by the channel gain matrix. .Fig.2. demonstrates the overall WSR of both small cells. Weshow the overall WSR in the dedicated band and shared bandof both cells versus iterations. We assume that the number ofusers in both cells ( I k and I j ) is equal ( I k = I j = 10 , ) foreach case. The number of subcarriers for dedicated band inthe cell k and j is N = M = 3 , . The number of subcarriersfor shared band N share = M share = 0 , , in both cells. Itcan be observed that when I k = 18 , N = 3 , N share = 2 , the Iterations W e i gh t ed S u m R a t e Gale−ShapelyTransportation Method I k =10, N=4, N share =0I k =18, N=3, N share =2I k =18, N=4, N share =1I k =18, N=4, N share =2I k =18, N=4, N share =2 Total weighted sum rate in the dedicated band I k =22, N=4, N share =2Total weighted sum rate in the shared band Fig. 2. Weighted sum rate versus iterations for MIMO co-primary spectrumsharing
Gale-Shapley converge faster than the transportation method.When I k = 10 , N = 4 , N share = 0 , the transportationmethod converges little faster. In most cases, the Gale-Shapleyand transportation method give similar WSR values versusiterations and converge the same time. SNR W e i gh t ed S u m R a t e Gale−ShapelyTransportation Method N=3, N share =2N=3, N share =3N=4, N share =1N=3, N share =1N=4, N share =2 Fig. 3. Weighted sum rate versus SNR for MIMO co-primary spectrumsharing
Fig.3. demonstrates the the overall WSR in both cells versusSNR. We assume that the number of users in both cells ( I k and I j ) is equal ( I k = I j = 18 ) for each case. We assume that thenumber of subcarriers for dedicated band in the cell k and j N = M = 3 , and the number of subcarriers for shared band N share = M share = 1 , , in both cells. When the number ofsubcarriers in shared band is increased, the weighted sum rateimproved for both Gale Shapley and transportation methods. For N = 3 , N share = 1 , , the transportation method giveshigher WSR throughput. But for N = 4 , N share = 1 , bothGale-Shapley and transportation methods give the same values.The WSR is enhanced with increasing number of subcarriersin the shared band according to the number of users ismore than number of transmit antennas multiplied by numberof dedicated subcarriers thus, both base stations assign theremaining users to utilize their shared band. Although, intercellinterference is available at remaining users allocated in theshared band. Number of Users W e i gh t ed S u m R a t e Gale−ShapelyTransportation Method N=4, N share =0, SNR=15 dBm N=4, N share =2, SNR=15 dBm N=4, N share =0, SNR=10 dBm N=5, N share =2, SNR=15 dBm
Fig. 4. Weighted sum rate versus number of users for MIMO co-primaryspectrum sharing
Fig.4. demonstrates the the overall WSR in both cells versusnumber of users with the fixed SNR values. We illustrate thetotal WSR of both cells by varying with number of users withSNR = 10 , dBm. The number of subcarriers for dedicatedband in the cell k and j N = M = 4 , . The number ofsubcarriers for shared band N share = M share = 0 , . We canobserve that with increasing the values of SNR, the WSR isimproved. When the number of users is from to , theGale-Shapley and transportation method give similar values.For higher number of users such that from to users, theGale-Shapley provides higher throughput for all cases.VI. C ONCLUSION
We propose co-primary spectrum sharing concept with theMIMO multiuser for two small cell network in downlinktransmission. The overall weighted sum rate maximization ofboth small cells is studied with subcarrier allocation and ajoint precoder and decoder design. The weighted sum ratemaximization problem becomes mixed-integer and non-convexproblem. We separate the main optimization into two sub-problems. In the first subproblem, we assume that both basestations allocate the subcarriers to each user by employingtwo methods. The first method is Gale-Shapley based onstable marriage problem and the second one is based ontransportation problem. The linear programming is used toolve the transportation problem. In both methods, each sub-carrier can be allocated to multiple users. In the second sub-problem, a joint precoder-decoder design for MIMO multiuseris proposed. The maximization of weighted sum rate is solvedvia weighted mean square error minimization problem for ajoint transceiver design. The optimal precoder is obtained bythe bisection method.Numerical results illustrate the over all weighted sum ratefor both base stations with uncorrelated antennas. We showthe convergence of the Gale-Shapley and the transportationproblem. The overall weighted sum rate throughput is demon-strated for different number of subcarriers served in dedicatedand shared bands for both cells. Then, with different numberof users and fixed the value of SNR, the overall weightedsum rate is explored. In terms of the convergence, these twomethods provide similar behavior. For the WSR versus numberof users, the Gale-Shapley gives higher WSR throughput thanthe transportation problem for high number of users. Whenthe number of subcarriers in the dedicated band is not enoughto serve all users, both base stations will assign the remainingusers to utilize the shared band. The overall average weightedsum rate is improved significantly in this instance. In termsof the complexity, the transportation method has leads morecomplexity because the linear programming is used to solvedthe problem. Thus, the Gale-Shapley is a good method formultiple subcarriers with multiple users assignment problem.R
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