IEEE ICC 2014 - Wireless Communications Symposium

Coordinated Resource Allocation with Vertical Beamforming in 3D MIMO-OFDMA Networks Weidong Zhang, Ying Wang, Peilong Li State Key Laboratory of Networking and Switching Technology Beijing University of Posts and Telecommunications Email: [email protected]

Abstract— This paper investigates coordinated resource allocation for 3-dimension (3D) antenna array systems in multicell multiple-input multiple-output (MIMO) and orthogonal frequency division multiple-access (OFDMA) wireless networks. Cell-center user and cell-edge user speciﬁc downtilts are accordingly partitioned through dynamic vertical beamforming in the 3D MIMO-OFDM communication systems. Taking these user speciﬁc downtilts into consideration, the objective of our proposed coordinated resource allocation scheme is to maximize both the cell-edge users’ and cell-center users’ throughput, subject to per base-station (BS) power, cell-center user and cell-edge user speciﬁc downtilt constraints. To solve the coordinated resource allocation problem, resource blocks (RBs) are accordingly partitioned for cell-center users and cell-edge users, by referring to the fractional frequency reuse (FFR) scheme. Based on such RB partitioning, FFR-based dual decomposition method (FDDM) are proposed, where RB assignment, power allocation (RAPA) and downtilts adjustment are jointly optimized. Simulation results demonstrate the eﬃcacy of our proposed coordinated resource allocation scheme.

I. Introduction Multiple input multiple output (MIMO) is an attractive physical layer technology that can support high date rate communications and increase reliability through the use of multiple transmit and receive antennas [1]. The combination of multiple input multiple output (MIMO) and orthogonal frequency division multipleaccess (OFDM) technologies have been developed for the next generation wireless networks, to facilitate the spatial multiplexing on the time-frequency resource blocks (RBs) [2]. Since the adjacent cells reuse the frequency spectrum in MIMO-OFDM systems, co-channel interference caused by transmission in neighboring cells remains a major impairment that limits throughput. Therefore, interference coordination has been studied and investigated to guarantee user experience in [3] and references therein. Dynamic coordinated beam steering is regarded as an eﬀective interference coordination technique by forming the dedicated beams for particular users [4]. Conventional 2-dimension (2D) antenna installations, adopting dynamic horizontal beam steering schemes, have been widely researched and developed [4]. However, the well known 2D antenna array systems only adapts the shape of the horizontal antenna pattern through coordinated beamforming and precoding to users, whereas the vertical array pattern is ﬁxed.

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We focus on 3D MIMO-OFDM systems with the additional capability to dynamically adapt the shape of the vertical antenna pattern to the location of users, thus realizing cellcenter user and cell-edge user speciﬁc downtilt. Recently, there are some works studying the coordinated vertical beamforming with fractional frequency reuse (FFR) in 3D MIMO-OFDM systems. [5] evaluated the performance of the dynamic vertical beamforming for cell users, and have not considered the RB assignment and power allocation problem. [6] studied coordinated resource allocation scheme with FFR in 3D antenna array systems, without guaranteeing the cell-center users’ and cell-edge users’ performance. To the best of our knowledge, there are few researches that study an eﬀective coordinated resource allocation algorithm, where joint RB assignment and power allocation (RAPA) are performed for cell-center users and cell-edge users with dynamic downtilt adaption. This paper jointly optimizes the downtilts and resource allocation scheme. The objective of our proposed coordinated resource allocation scheme is to maximize both of the cellcenter and cell-edge users’ throughput, subject to per basestation power constraints in multiple cell scenario. Taking the cell-center user and cell-edge user speciﬁc downtilts into consideration, such resource allocation problem is formulated into a mixed-integer nonlinear optimization problem [7]. Note that the optimization problem is formulated with three sets of variables, consisting of the cell-center user and cell-edge user speciﬁc downtilt adjustment, and RAPA. To solve the optimization problem, we ﬁrstly partition the available RBs for cell-center users and cell-edge users with FFR scheme. Based on such RB partitioning, a FFR-based dual decomposition method (FDDM) is proposed to solve the above-mentioned resource allocation problem, by adopting sub-gradient methods. On the basis of the proposed RB partitioning, the local optimal downtilt adjustment and RAPA solutions can be achieved by FDDM, to maximize system throughputs. Simulation results show that our coordinated resource allocation scheme outperforms other classical coordinated RAPA schemes with ﬁxed downtilts. The rest of the paper is organized as follows. Section II introduces the system model and formulates the coordinated optimization problem. The RB partitioning based FDDM are then proposed in Section III. In section IV, simulation results are presented. Finally, section V concludes this paper.

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II. System Model and Problem Formulation Cell-center band Cell-edge band Other two cell-edge bands

Otherwise, cell-center users reuse the partitioned resources for all M cells as shown in Fig.1. Consequently, in case u is a cell-center user and served by the m-th BS on RB n, the corresponding SINR can be written as

Available Frequency Band

S INR[n] m,u (θm,1 ) =

Vertical azimuth

De sir ed

sig

where σ2 denotes the thermal noise. For u, the corresponding achievable data rate (in bits per channel use) is given by the following Shannons formula:

na l

Cell-edge user

Fig. 1.

[n] R[n] m,u (θm ) = log2 [1 + S INRm,u (θm )].

Network layout using FFR scheme under investigation.

We consider a coordination cluster of M ≥ 2 BSs in a downlink OFDMA based cellular network, in which multiple clusters are deployed. Denote Um as the set of users assigned to BS m. Deﬁne k(m, n) as the selected users served by m-th BS on RB n, and K U1 ∪ ... ∪ U M . Let k (k[1] , ..., k[N] ), where k[n] (k(1, n), ..., k(M, n)), k(m, n) ∈ Um and k ⊆ K. In this paper, the status of users (cell-edge or not) is distinguished by their signal-to-interference-plus-noise ratio (SINRs) on the common RB [8]. The user’s status is determined based on a chosen threshold T R . Each user u computes its SINR on the selected common RB, and if it is less than the threshold T R , u is classiﬁed as the cell-edge user. Otherwise, u is the cell-center user [8]. The overall frequency resources is divided into a set of N orthogonal RBs, where n represents the subcarrier used by cell-edge users. Using FFR scheme, both cell-center users and cell-edge users are served by only one BS. Referring to [9], RBs are partitioned into two regions for cell-center users and cell-edge users, with reuse factor of 1 and α accordingly. In case M = 3 BSs are deployed in each coordination cluster, α is set to 3 as shown in Fig.1. Let Nm,c and Nm,e represent the set of RBs allocated for cell-center users and cell-edge users in the m-th cell respectively, Nm Nm,c ∪ Nm,e . There are two downtilts, cell-center user and cell-edge user speciﬁc downtilts, employed in each cell as illustrated in Fig.1. Denote θm,1 and θm,2 as the cell-center user and cell-edge user speciﬁc downtilts for the m-th BS, where θm (θm,1 , θm,2 ) and θ (θ1 , ..., θ M ). In Fig.1, the vertical angle between the serving beamforming and desired signal radiated towards a cell-edge user u is θm,2 − θm,u . Consequently, the desired signal strength should be [n] [n] calculated by p[n] m G m,u cos(θm,2 − θm,u ), where G m,u represents the channel gain of u in the m-th cell on subcarrier n, and p[n] m denotes the allocated power for subcarrier n in cell m. [n] Let p = (p[1] , ..., p[N] ), where p[n] (p[n] 1 , ..., p M ). When u is a cell-edge user with FFR scheme, the SINR of u served by m on RB n can be written as S INR[n] m,u (θm,2 ) =

[n] p[n] m G m,u cos(θm,1 − θm,u ) , (2) [n] σ2 + j∈M, jm p[n] j G j,u cos(θ j,1 − θ j,u )

[n] p[n] m G m,u cos(θm,2 − θm,u ) . σ2 m∈M

(1)

(3)

From Eq.(1) - Eq.(3), we formulate following optimization problem, and aims to maximize the total transmission throughput subject to per-BS power, cell-center user and cell-edge user speciﬁc downtilt constraints. max p,k,θ

wk(m,n) R[n] m,k(m,n) (θm,1 ) +

m∈M

m∈M n∈Nm,c

s.t. C1 :

n ∈N

[n ] wk(m,n ) Rm,k(m,n ) (θm,2 ),

m,e

[n] p[n] m ≤ Pm , pm ≥ 0, ∀m ∈ M,

n∈Nm

C2 : 0 ≤ θm,1 ≤ θc , 0 ≤ θm,2 ≤ θe , ∀m ∈ M,

(4)

where wk(m,n) > 0 is a weight factor that accounts for the priority of selecting user k(m, n) to use an RB n in the m-th cell and Pm is the maximum transmit power of the m-th BS for overall cell users. Inequations C1 and C2 indicate the perBS power, cell-center user and cell-edge user speciﬁc downtilt constraints, respectively. To solve this optimization problem, we should ﬁnd solutions with respect to Nm , k, p, and θm . III. Coordinated Resource Allocation in 3D MIMO Systems Since (4) is a constrained non-convex optimization problem, computing its global optimal solution may not be feasible in practice [10]. Thus, a coordinated resource allocation scheme is proposed to solve this optimization problem suboptimally in this section. Firstly, partition available RBs for the cell-center users and cell-edge users respectively, and obtain Nm for the m-th cell as described in Section A. Secondly, using FDDM proposed in Section B, jointly optimize the downtilt adjustment and RAPA. Finally, the convergence and computation complexity are veriﬁed and analyzed in Section C. A. RB Partitioning for Each Cell Assume that each cell performs equal power allocation (EPA). We calculate the data rate gain of the n-th RB for all cell-edge users in the m-th cell, according to Rm,n = R[n] (5) m,u (θm,2 ), ∀n ∈ N. u∈Um,e

For cell-edge users in the m-th cell , the best RB has the maximum m∈M Rm,n value among all available RBs as follows.

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IEEE ICC 2014 - Wireless Communications Symposium

nˆ = arg max n∈N

Then, the dual optimization problem can be given by Rm,n .

(6)

Algorithm I RB Partitioning Step 1: Initialization • For m-th BS, N = {1, · · · , N} and M = {1, · · · , M}. Nm,c = Nm,e = φ. Step 2: Iteration • Calculate the data rate gain of N RBs for both cell-center and cell-edge users in each cell according to Eq.(5). • for m = 1 : M 1) Determine nˆ m for the m-th cell according to Eq.(6), Nm = N. 2) In case card(Nm,e ) < γN, Nm,e = Nm,e ∪ nˆ m , N = N \ nˆ m , and go to step 1 in this loop. 3) Otherwise, m = m + 1, and go to step 1 in this loop. end for Step 3: Finalization • Obtain Ne = N1,e ∪ . . . ∪ N M,e and Nc = N − Ne . • The available RBs of the m-th cell are Nm,e and Nm,c = Nc for cell-center users and cell-edge users, respectively.

+

m∈M

[n] wk(m,n) R[n] m,k(m,n) (p ) +

m∈M n ∈Nm,e

=

[n] [n] wk(m,n) R[n] m,k(m,n) (pm ) − λm pm ,

(11)

=

[n ] [n ] [n ] wk(m,n ) Rm,k(m,n ) (pm ) − λm pm . (12)

Fixed downtilts are initialized for cell-center users and celledge users, respectively. Eq.(10) can be decomposed into N independent optimization problems as follows. ⎧ [n] ⎪ ⎪ ⎨maxp[n] ≥0,k[n] ∈K gn (λ, p[n] , k , θm,1 ), ∀n ∈ Nm,c , ⎪ ⎪ [n ] ⎩maxp[n ] ≥0,k[n ] ∈K gn (λ, p , k[n ] , θm,2 ), ∀n ∈ Nm,e .

(13)

EPA is initialized for p[n] and p[n ] in M BSs. The multivariate maximization (13) indicates a rule to ﬁnd the optimal [n] [n ] user kˆ and kˆ for the speciﬁc RBs n and n . [n] [n ] After kˆ and kˆ have been ﬁxed, the optimal power allocation solution can be derived by ⎧ [n] [n] ⎪ ⎪ gn (λ, p[n] , kˆ ), ∀n ∈ Nm,e , ⎨ pˆ m = arg max p[n] m ≥0 ⎪ ⎪ ] [n ] ˆ [n ] ⎩ pˆ [n ] (λ, p g , k ), ∀n ∈ Nm,e . n m = arg max p[n m ≥0

(8)

(14)

Applying Karush-Kuhn-Tucker (KKT) condition [7], by taking the derivation of Eq.(14) with respect to p[n] and p[n ] , we have ⎤† ⎡ [n] ⎥⎥ ⎢⎢⎢ wk(m,n) σ2 + jm p[n] j G j,k(m,n) cos[θ j,1 − θ j,k(m,n) ] ⎥ ⎥⎥⎥ , ⎢⎢⎢ pˆ [n] = − m ⎦ ⎣ λ ln 2 + t[n] [n] Gm,k(m,n) cos[θm,1 − θm,k(m,n) ] m m (15) ⎤† ⎡ ⎥⎥⎥ ⎢⎢⎢ wk(m,n ) σ2 ⎥⎥⎥ , pˆ m[n ] = ⎢⎢⎣⎢ − [n ] (16) λm ln 2 Gm,k(m,n ) cos[θm,1 − θm,k(m,n ) ] ⎦

βm,e (θm,e − θm,2 ), (7)

where βc (β1,c , ..., β M,c ), βe (β1,e , ..., β M,e ) and λ (λ1 , ..., λ M )T denote the vector of nonnegative Lagrangian multipliers. Note that p satisﬁes the two constraints: 1) p[n] m ≥ 0, ∀m ∈ M, ∀n ∈ Nm . 2) Only one p[n] m is positive for each n occupied by m-th BS. The Lagrange dual objective function, with Λ(·) given by Eq.(7) can be formulated as p≥0,k∈K

(10)

m∈M

m∈M

g(λ, βc , βe ) = max Λ(λ, βc , βe , p, k).

[βm,c (θm,c − θm,1 ) + βm,e (θm,e − θm,2 ) + λm Pm ]

m∈M [gn (λ, p[n] , k[n] ) + gn (λ, p[n ] , k[n ] ),

gn (λ, p[n ] , k[n ] )

βm,c (θm,c − θm,1 )

m∈M

m∈M [n ] [n ] wk(m,n ) Rm,k(m,n )+ ) (p

max

gn (λ, p[n] , k[n] )

p[n] m )

=

where

n∈Nm

m∈M n∈Nm,c

+

g(λ, βc , βe )

n∈Nm,c

B. FFR-based Dual Decomposition Method We consider the Lagrangian of the optimization problem (4) dualized with respect to the system constraints and cell-center users’ throughput requirement λm (Pm −

Note that the Lagrangian Λ(λ, βc , βe , p, k) is linear with respect to λ, βc and βe for ﬁxed p, k. And g(λ, βc , βe ) is the maximum of these linear functions. Thus, the dual problem (9) is convex. The duality gap between g(λ∗ , β∗c , β∗e ) and the solution to the primal problem (5) is tend to zero [11]. Moreover, observe that Eq.(8) can be recast as

+

Note: In Algorithm I-Iteration step, the priority of selecting cell-edge RBs among cells is based on the round robin scheduling algorithm. On the basis of the derived RB partitioning for each cell, FDDM is proposed to jointly optimize the downtilt adjustment and RAPA in the following.

(9)

λ,β

The proportion between the available RBs of cell-edge users N and all system RBs in the m-th cell is denoted by γm = Nm,e , where Nm,e = card(Nm,e ), and card(·) represents the cardinality of a certain set. In this paper, such proportion factor is set to be equal and denoted by γ = γm . Then, the available RBs need to be chosen for cell-center and cell-edge users in each cell respectively. And such RB partitioning among BSs can be carried out in the following procedure.

Λ(λ, βc , βe , p, k)

{λ∗ , β∗c , β∗e } = min g(λ, βc , βe ),

m∈M

where ∀n ∈ Nm,c , ∀n ∈ Nm,e , ∀m ∈ M, tm[n]

=

j∈M, jm

wk( j,n)

[n] [n] G[n] m,k( j,n) cos[θm,1 − θm,k( j,n) ]S INR j,k( j,n) (p ) . (17) [n] σ2 + l∈M p[n] l G l,k( j,n) cos[θl,1 − θl,k( j,n) ]

Here, [x]† max(0, x) and the taxation term tm[n] ’s physical meaning is the summation of the interference that the m-th BS causes to cell-center users in other BSs on the RB n. We T denote t (t[1] , ..., t[N] ), where t[n] (t1[n] , ..., t[n] M) .

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IEEE ICC 2014 - Wireless Communications Symposium

] In Eq.(15) and Eq.(16), pˆ [n] ˆ [n m and p m have similar form in equation but with diﬀerent meaning. Remark 1: The optimal power allocation for cell-center user has the same form as improved iterative water-ﬁlling algorithm (IIWF) [10]. On RBs partitioned to be utilized by cell-center users in Algorithm I, BS not only considers its own transmission but also intend to reduce the interference caused for other BSs. More power would be allocated to the RB which has better SINR with less interference for others according to Eq.(15). Remark 2: On the other hand, The optimal power allocation for cell-edge user has the same form as the traditional WF intuitively [10]. More power would be allocated to the RB [n ] which has better SNR. Substituting Eq.(15), Eq.(16), kˆ and [n] kˆ into Eq.(11) and Eq.(12), we have

where l is the iteration index, and lλm lβm,c and lβm,e are the appropriate positive step-size sequences. Eq.(26) can be iterated until convergence [7]. A summary of our proposed FDDM is given in Algorithm III. A summary of the proposed FDDM is described in the Algorithm II.

Algorithm II FFR Based Dual Decomposition Method Step 1: Initialization • Initialize p based on equal power allocation. • Initialize λ, β, θ, and set l = 0. Step 2: Iteration • repeat for m ∈ M for n ∈ Nm update k(m,n) according to the maximization (13) and calculate pˆ [n] m according to Eq.(15), Eq.(16). [n] ˆ [n] [n ] ˆ [n ] g( λ, βc , βe ) = gm (λ, βc , pˆ m , k , θm,1 ) + gm (λ, βe , pˆ m , k , θm,2 ) end for m∈M m∈M end for for m ∈ M [βm,c θm,c + βm,e θm,e + λm Pm ], (18) + calculate θˆm,1 and θˆm,2 according to Eq.(22), (23). m∈M end for where Update λ, βc and βe according to Eq.(26), set l = l + 1. [n] [n] [n] ≤ ε, and until Convergence, λl − λ(l−1) ≤ ε, βlc − β(l−1) ˆ c , k , θ ) = −β θ + w R (θ )−λ p ˆ , gm (λ, βc , pˆ [n] m,1 m,c m,1 k(m,n) m,1 m m m m,k(m,n) (l−1) l − β ≤ ε. β [n] e e k(m,n)∈kˆ n∈Nm,c Step 3: Finalization (19) • p, k, and θ are derived for users. [n ] [n ] ˆ [n ] [n ] wk(m,n ) Rm,k(m,n ) (θm,2 )−λm pˆ m . gm (λ, βe , pˆ m , k , θm,2 ) = −βm,e θm,2 + [n ]

k(m,n)∈kˆ n∈Nm,c

(20)

Note: In Algorithm II, ε represents the convergence tolerance, and we set ε = 10−2 in this paper.

C. Convergence and Computation Complexity Analysis After the above-mentioned procedure, the optimal downlilt solutions for cell-center users and cell-edge users can be derived by ⎧ [n] [n] ⎪ ⎪ ⎨θˆm,1 = arg maxθˆm,1 ≥0 gm (λ, βc , pˆ m , kˆ , θm,1 ), ∀n ∈ Nm,c , ⎪ [n ] ⎪ ⎩θˆm,2 = arg max ˆ gm (λ, βe , pˆ m[n ] , kˆ , θm,2 ), ∀n ∈ Nm,e . θm,2 ≥0

(21)

Applying Karush-Kuhn-Tucker (KKT) condition [7], by taking the derivation of Eq.(21) with respect to θm,1 and θm,2 , we have −βm,c ln 2 + φm,1 }† , (22) θˆm,1 = {arcsin A2m + B2m θˆm,2 = {φm,2 + arcsin −σ2 βm,e ln 2 }† . [ n∈Nm,e cnm cos θm,k(m,n ) ]2 + [ n∈Nm,e cnm sin θm,k(m,n ) ]2 (23)

The detailed proof of Eq.(22), Eq.(23) and deﬁnitions of φm,1 , φm,2 , Am , Bm , cnm are given in Appendix A. g(λ, β) may not be diﬀerentiable, sub-gradient methods can guarantee to converge to the global optimal solution [7]. Therefore, the Lagrangian dual multipliers can be updated as ⎧ † (l+1) ⎪ ⎪ = βlm,c − lβm,c θm,c − θˆm,1 , βm,c ⎪ ⎪ ⎪ † ⎪ ⎨ (l+1) (24) βm,e = βlm,e − lβm,e θm,e − θˆm,2 , ⎪ ⎪ ⎪ ⎪ † ⎪ ⎪ [n] l l ⎩λ(l+1) = λm + λm ( n∈Nm pˆ m − Pm ) , m

Using sub-gradient methods, FDDM can guarantee convergence to a local optimal solution for Eq.(8) according to [11]. For FDDM, the number of iterations required to get optimal Lagrangian dual multipliers λ, (βc , βe ) and achieve ε-optimality, i.e. g∗ − g < ε, is O( ε12 ) [11]. In Algorithm I, the computational complexities for RB partitioning is O(N + (N − γ1 N)+, ..., +(N − γ M MN))|K|, which is much lower than 2N K by using exhaustive method. In RAPA and downtilt adjustment procedures of FDDM, p[n] m [n ] and pm , ∀n ∈ Nm,c , ∀n ∈ Nm,e , have the same form as IIWF and the traditional WF as mentioned in Remark 1 and Remark 2. Solving WF and IIWF has a computational burden O(N M log2 N) [10], while updating k and θ has the complexities O(N M|K|) and O(M|K|) accordingly. In summary, the total computation complexity of Algorithm M(N|K|+N log2 N+|K|) I and FDDM is much lower than O(2N K ). ε2 Therefore, the proposed coordinated resource allocation scheme can eﬀectively optimize downtilt adjustment and RAPA in 3D MIMO systems, especially when ε is small. IV. Simulation Result and Discussion We evaluate the performance of our coordinated resource allocation scheme by using a LTE system level simulator. Urban macro-cell (UMa) scenarios is considered in which 7 coordination clusters are deployed [12]. The overall BSs are arranged in grid fashion with appropriate cell-wrap to ensure

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IEEE ICC 2014 - Wireless Communications Symposium

•

•

WF/IIWF-Fixed Downtilt: Use traditional WF/IIWF to implement RAPA for overall cell users. And both cellcenter user and cell-edge user speciﬁc downtilts are set to be 15 degree according to [12]. FDDM-AlgorithmI: Based on the RB partitioning in Algorithm I, implement FDDM by utilizing Algorithm II, respectively.

Proportional fairness (PF) scheduling algorithm and full buﬀer traﬃc model are utilized for simulation. Monte Carlo simulation is performed for each subframe and iterated over a total of 1000 subframes. To verify the overall cell users’ performance of diﬀerent interference coordinations, the SINR cumulative density function (CDF) curves of cell users are illustrated in Fig.2 and Fig.3. 1 0.9

WF IIWF FDDM

0.8 0.7

CDF

0.6 0.5

coordinated resource allocation scheme is improved by 713dB. Moreover, the overall cell users’ SINR performance of FDDM-AlgorithmI is much higher than traditional WF-Fixed Downtilt. 1 0.9

WF IIWF FDDM

0.8 0.7

CDF

0.6 0.5 0.4 0.3 0.2 0.1 0 -5

0

5

10

15

SINR(dB)

20

25

30

Fig. 3. CDF curves of cell-edge users’ SINR for diﬀerent coordinated resource allocation schemes in 3D MIMO systems.

Fig.3 further shows the cell-edge users’ SINR performance for ﬁve coordination schemes. It is clear that FDDMAlgorithmII is still better than classic IIWF-Fixed Downtilt. Compared with traditional WF-Fixed Downtilt, the SINR performance of our proposed coordinated resource allocation scheme is improved by 10-20dB. This is because our proposed coordinated resource allocation scheme is implemented by guaranteeing the balanced experience of cell-center users and cell-edge users with dynamic downtilt adaption. 2.5

Downlink Spectral Efficiency(bps/Hz/cell)

elimination of edge eﬀects. The inter-cell distance between two BSs is 500m, and maximum transmit power is 46dBm. Center frequency and bandwidth are 2.3GHz and 10MHz, respectively, and thermal noise density is −174dBm/Hz. There are 50 RBs and 10 users in each cell. The 3D MIMO channel modeling parameters in the horizontal dimension, including fast fading, shadow fading and path loss models, is set based on ITU channel model [12], [13]. Other parameters including antenna conﬁguration, receiver method, and minimum separation between UE to BS etc, are set according to [12]. Users in each cell are generated uniformly based on random distribution, and T R = 10dB is set to divide cell-center and cell-edge users. Using FFR scheme, cell-edge users’ reuse factor α = 3, and universal reuse is utilized for cell-center users. Therefore, card(Ne ) = 3γN, card(Nc ) = (1 − 3γ)N, and set γ = 0.2 according to [14]. We introduce three coordinated resource allocation schemes compared in the simulation.

Cell-average Spectral Efficiency Cell-edge Spectral Efficiency

2

1.5

1

0.5

0.4

0

0.3 0.2 0.1 0 -5

0

5

10

SINR(dB)

15

20

25

WF

IIWF

FDDM

Fig. 4. Cell-average and cell-edge spectral eﬃciency for diﬀerent coordinated resource allocation schemes in 3D MIMO systems.

30

Fig. 2. CDF curves of overall cell users’ SINR for diﬀerent coordinated resource allocation schemes in 3D MIMO systems.

In Fig.2, it is observed that the overall cell users’ performance of FDDM-AlgorithmI outperforms other coordinated resource allocation schemes, especially in low SINR region. This is because cell-edge users’ performance can be further improved by utilizing FFR scheme. Compared with classic IIWF-Fixed Downtilt, the SINR performance of our proposed

To further verify the overall cell throughput performance of diﬀerent coordinated resource allocation schemes, the downlink cell-average and cell-edge spectral eﬃciency deﬁned in [12], is evaluated and illustrated in Fig.4. It is observed that the average spectral eﬃciency of FDDM - AlgorithmI is about 30% and 15% higher than WF-Fixed Downtilt and IIWF-Fixed Downtilt, respectively. Furthermore, FDDM yields up the best cell-edge users’ performance through vertical beamforming and allocating more RBs for cell-edge users in 3D MIMO systems.

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V. Conclusion This paper investigates coordinated resource allocation scheme in 3D MIMO systems, where cell-center user and cell-edge user speciﬁc downtilts are accordingly partitioned through dynamic vertical beamforming. The FFR scheme is introduced in this paper, where an RB partitioning scheme is presented accordingly. Based on such RB partitioning, FDDM is proposed where downtilts adjustment and RAPA are jointly optimized. Simulation results have shown that based on the proposed RB partitioning algorithms, FDDM outperforms the classical coordinated algorithms signiﬁcantly with lower computation complexity. Appendix A: The calculation procedure of downtilts Applying Karush-Kuhn-Tucker (KKT) condition [7], by taking the derivation of Eq.(21) with respect to θm,1 , we have

Let

n n ∈Nm,e cm cos θm,k(m,n ) cos φm,2 = , [ n ∈Nm,e cnm cos θm,k(m,n ) ]2 + [ n ∈Nm,e cnm sin θm,k(m,n ) ]2 (33)

the optimal downtilt for cell-edge users in M BSs can be obtained as follows: θˆm,2 = {φm,2 + arcsin −σ2 βm,e ln 2 }† . n 2 n 2 [ n∈Nm,e cm cos θm,k(m,n ) ] + [ n∈Nm,e cm sin θm,k(m,n ) ] (34)

Note that θˆm,2 must satisfy cell-edge user speciﬁc downtilt π ⎞ constraints C2 , θˆm,2 < 2 .

⎛ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜an sin[θm,1 − θm,k(m,n) ] − −βm,c ln 2 = bnm sin[θm,1 − θm,k( j,n) ]⎟⎟⎟⎠ , ⎝ m n∈Nm,c

j∈M, jm

(25)

where anm =

σ2 +

wk(m,n) pnmG[n] m,k(m,n)

[n] [n] l∈M pl G l,k(m,n)

cos[θl,1 − θl,k(m,n) ]

.

(26)

[n] [n] pnmG[n] m,k( j,n) S INR j,k( j,n) (p ) = wk( j,n) . [n] σ2 + l∈M p[n] j∈M, jm l G l,k( j,n) cos[θl,1 − θl,k( j,n) ] (27) The expansion of Eq.(25) can be written as:

bnm

sin θm,1 Am − cos θm,1 Bm = −βm,c ln 2,

(28)

where Am = n∈Nm,c anm cos θm,k(m,n) − j∈M, jm bnm cos θm,k( j,n) , Bm = n∈Nm,c anm sin θm,k(m,n) − j∈M, jm bnm sin θm,k( j,n) . Let cos φm,1 =

Am A2m + B2m

,

(29)

the optimal downtilt for cell-center users in M BSs can be derived as follows: −βm,c ln 2 θˆm,1 = {arcsin + φm,1 }† . 2 2 Am + Bm

(30)

Note that θˆm,1 must satisfy cell-center user speciﬁc downlilt constraints C2 , θˆm,1 < π2 . Similarly, taking the derivation of Eq.(21) with respect to θm,2 , we have −σ2 βm,e ln 2 = sin θm,2 cnm cos θm,k(m,n ) − cos θm,2 cnm sin θm,k(m,n ) . n ∈Nm,e

n ∈Nm,e

(31)

where

cnm

=

This work is supported by Key Project (2013ZX03001025002), National 863 Project (2014AA01A701), National Nature Science Foundation of China (61121001). References [1] J. Zyren and W. McCoy, “Overview of the 3GPP long term evolution physical layer,” Freescale Semiconductor, Inc., white paper, 2007. [2] V. Jungnickel, M. Schellmann, L. Thiele, T. Wirth, T. Haustein, O. Koch, W. Zirwas, and E. Schulz, “Interference-aware scheduling in the multiuser mimo-ofdm downlink,” Communications Magazine, IEEE, vol. 47, no. 6, pp. 56–66, 2009. [3] G. Boudreau, J. Panicker, N. Guo, R. Chang, N. Wang, and S. Vrzic, “Interference coordination and cancellation for 4g networks,” Communications Magazine, IEEE, vol. 47, no. 4, pp. 74–81, 2009. [4] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multi-antenna wireless system,” Wireless Communications, IEEE Transactions on, vol. 9, no. 5, pp. 1748–1759, 2010. [5] K. Safjan, V. D’Amico, D. Bultmann, D. Martin-Sacristan, A. Saadani, and H. Schoneich, “Assessing 3gpp lte-advanced as imt-advanced technology: the winner+ evaluation group approach,” Communications Magazine, IEEE, vol. 49, no. 2, pp. 92–100, 2011. [6] V. D’Amico, Botella, et al., “Advanced interference management in artist4g: Interference avoidance,” in European Wireless Technology Conference, Paris, France, September 2010, 2010, pp. 21–24. [7] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004. [8] T. Novlan, R. Ganti, A. Ghosh, and J. Andrews, “Analytical evaluation of fractional frequency reuse for ofdma cellular networks,” Wireless Communications, IEEE Transactions on, no. 99, pp. 1–12, 2011. [9] S. Ali and V. Leung, “Dynamic frequency allocation in fractional frequency reused ofdma networks,” Wireless Communications, IEEE Transactions on, vol. 8, no. 8, pp. 4286–4295, 2009. [10] L. Venturino, N. Prasad, and X. Wang, “Coordinated scheduling and power allocation in downlink multicell ofdma networks,” Vehicular Technology, IEEE Transactions on, vol. 58, no. 6, pp. 2835–2848, 2009. [11] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” Communications, IEEE Transactions on, vol. 54, no. 7, pp. 1310–1322, 2006. [12] ITU-R Rep. M.2135, “Guidelines for evaluation of radio interface technologies for IMT-Advanced,” Jul.2009. [13] E. Dahlman, 3G evolution: HSPA and LTE for mobile broadband. Academic Press, 2008. [14] M. Rahman and H. Yanikomeroglu, “A downlink dynamic interference avoidance scheme with inter-cell coordination,” Wireless Communications, IEEE Transactions on, vol. 9, no. 4, pp. 1414–1425, 2010.

[n ] wk(m,n ) pnm Gm,k(m,n )

Acknowledgment

[n ] [n ] 1 + S NRm,k(m,n ) (pm )

.

(32)

4770

Coordinated Resource Allocation with Vertical Beamforming in 3D MIMO-OFDMA Networks Weidong Zhang, Ying Wang, Peilong Li State Key Laboratory of Networking and Switching Technology Beijing University of Posts and Telecommunications Email: [email protected]

Abstract— This paper investigates coordinated resource allocation for 3-dimension (3D) antenna array systems in multicell multiple-input multiple-output (MIMO) and orthogonal frequency division multiple-access (OFDMA) wireless networks. Cell-center user and cell-edge user speciﬁc downtilts are accordingly partitioned through dynamic vertical beamforming in the 3D MIMO-OFDM communication systems. Taking these user speciﬁc downtilts into consideration, the objective of our proposed coordinated resource allocation scheme is to maximize both the cell-edge users’ and cell-center users’ throughput, subject to per base-station (BS) power, cell-center user and cell-edge user speciﬁc downtilt constraints. To solve the coordinated resource allocation problem, resource blocks (RBs) are accordingly partitioned for cell-center users and cell-edge users, by referring to the fractional frequency reuse (FFR) scheme. Based on such RB partitioning, FFR-based dual decomposition method (FDDM) are proposed, where RB assignment, power allocation (RAPA) and downtilts adjustment are jointly optimized. Simulation results demonstrate the eﬃcacy of our proposed coordinated resource allocation scheme.

I. Introduction Multiple input multiple output (MIMO) is an attractive physical layer technology that can support high date rate communications and increase reliability through the use of multiple transmit and receive antennas [1]. The combination of multiple input multiple output (MIMO) and orthogonal frequency division multipleaccess (OFDM) technologies have been developed for the next generation wireless networks, to facilitate the spatial multiplexing on the time-frequency resource blocks (RBs) [2]. Since the adjacent cells reuse the frequency spectrum in MIMO-OFDM systems, co-channel interference caused by transmission in neighboring cells remains a major impairment that limits throughput. Therefore, interference coordination has been studied and investigated to guarantee user experience in [3] and references therein. Dynamic coordinated beam steering is regarded as an eﬀective interference coordination technique by forming the dedicated beams for particular users [4]. Conventional 2-dimension (2D) antenna installations, adopting dynamic horizontal beam steering schemes, have been widely researched and developed [4]. However, the well known 2D antenna array systems only adapts the shape of the horizontal antenna pattern through coordinated beamforming and precoding to users, whereas the vertical array pattern is ﬁxed.

978-1-4799-2003-7/14/$31.00 ©2014 IEEE

We focus on 3D MIMO-OFDM systems with the additional capability to dynamically adapt the shape of the vertical antenna pattern to the location of users, thus realizing cellcenter user and cell-edge user speciﬁc downtilt. Recently, there are some works studying the coordinated vertical beamforming with fractional frequency reuse (FFR) in 3D MIMO-OFDM systems. [5] evaluated the performance of the dynamic vertical beamforming for cell users, and have not considered the RB assignment and power allocation problem. [6] studied coordinated resource allocation scheme with FFR in 3D antenna array systems, without guaranteeing the cell-center users’ and cell-edge users’ performance. To the best of our knowledge, there are few researches that study an eﬀective coordinated resource allocation algorithm, where joint RB assignment and power allocation (RAPA) are performed for cell-center users and cell-edge users with dynamic downtilt adaption. This paper jointly optimizes the downtilts and resource allocation scheme. The objective of our proposed coordinated resource allocation scheme is to maximize both of the cellcenter and cell-edge users’ throughput, subject to per basestation power constraints in multiple cell scenario. Taking the cell-center user and cell-edge user speciﬁc downtilts into consideration, such resource allocation problem is formulated into a mixed-integer nonlinear optimization problem [7]. Note that the optimization problem is formulated with three sets of variables, consisting of the cell-center user and cell-edge user speciﬁc downtilt adjustment, and RAPA. To solve the optimization problem, we ﬁrstly partition the available RBs for cell-center users and cell-edge users with FFR scheme. Based on such RB partitioning, a FFR-based dual decomposition method (FDDM) is proposed to solve the above-mentioned resource allocation problem, by adopting sub-gradient methods. On the basis of the proposed RB partitioning, the local optimal downtilt adjustment and RAPA solutions can be achieved by FDDM, to maximize system throughputs. Simulation results show that our coordinated resource allocation scheme outperforms other classical coordinated RAPA schemes with ﬁxed downtilts. The rest of the paper is organized as follows. Section II introduces the system model and formulates the coordinated optimization problem. The RB partitioning based FDDM are then proposed in Section III. In section IV, simulation results are presented. Finally, section V concludes this paper.

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IEEE ICC 2014 - Wireless Communications Symposium

II. System Model and Problem Formulation Cell-center band Cell-edge band Other two cell-edge bands

Otherwise, cell-center users reuse the partitioned resources for all M cells as shown in Fig.1. Consequently, in case u is a cell-center user and served by the m-th BS on RB n, the corresponding SINR can be written as

Available Frequency Band

S INR[n] m,u (θm,1 ) =

Vertical azimuth

De sir ed

sig

where σ2 denotes the thermal noise. For u, the corresponding achievable data rate (in bits per channel use) is given by the following Shannons formula:

na l

Cell-edge user

Fig. 1.

[n] R[n] m,u (θm ) = log2 [1 + S INRm,u (θm )].

Network layout using FFR scheme under investigation.

We consider a coordination cluster of M ≥ 2 BSs in a downlink OFDMA based cellular network, in which multiple clusters are deployed. Denote Um as the set of users assigned to BS m. Deﬁne k(m, n) as the selected users served by m-th BS on RB n, and K U1 ∪ ... ∪ U M . Let k (k[1] , ..., k[N] ), where k[n] (k(1, n), ..., k(M, n)), k(m, n) ∈ Um and k ⊆ K. In this paper, the status of users (cell-edge or not) is distinguished by their signal-to-interference-plus-noise ratio (SINRs) on the common RB [8]. The user’s status is determined based on a chosen threshold T R . Each user u computes its SINR on the selected common RB, and if it is less than the threshold T R , u is classiﬁed as the cell-edge user. Otherwise, u is the cell-center user [8]. The overall frequency resources is divided into a set of N orthogonal RBs, where n represents the subcarrier used by cell-edge users. Using FFR scheme, both cell-center users and cell-edge users are served by only one BS. Referring to [9], RBs are partitioned into two regions for cell-center users and cell-edge users, with reuse factor of 1 and α accordingly. In case M = 3 BSs are deployed in each coordination cluster, α is set to 3 as shown in Fig.1. Let Nm,c and Nm,e represent the set of RBs allocated for cell-center users and cell-edge users in the m-th cell respectively, Nm Nm,c ∪ Nm,e . There are two downtilts, cell-center user and cell-edge user speciﬁc downtilts, employed in each cell as illustrated in Fig.1. Denote θm,1 and θm,2 as the cell-center user and cell-edge user speciﬁc downtilts for the m-th BS, where θm (θm,1 , θm,2 ) and θ (θ1 , ..., θ M ). In Fig.1, the vertical angle between the serving beamforming and desired signal radiated towards a cell-edge user u is θm,2 − θm,u . Consequently, the desired signal strength should be [n] [n] calculated by p[n] m G m,u cos(θm,2 − θm,u ), where G m,u represents the channel gain of u in the m-th cell on subcarrier n, and p[n] m denotes the allocated power for subcarrier n in cell m. [n] Let p = (p[1] , ..., p[N] ), where p[n] (p[n] 1 , ..., p M ). When u is a cell-edge user with FFR scheme, the SINR of u served by m on RB n can be written as S INR[n] m,u (θm,2 ) =

[n] p[n] m G m,u cos(θm,1 − θm,u ) , (2) [n] σ2 + j∈M, jm p[n] j G j,u cos(θ j,1 − θ j,u )

[n] p[n] m G m,u cos(θm,2 − θm,u ) . σ2 m∈M

(1)

(3)

From Eq.(1) - Eq.(3), we formulate following optimization problem, and aims to maximize the total transmission throughput subject to per-BS power, cell-center user and cell-edge user speciﬁc downtilt constraints. max p,k,θ

wk(m,n) R[n] m,k(m,n) (θm,1 ) +

m∈M

m∈M n∈Nm,c

s.t. C1 :

n ∈N

[n ] wk(m,n ) Rm,k(m,n ) (θm,2 ),

m,e

[n] p[n] m ≤ Pm , pm ≥ 0, ∀m ∈ M,

n∈Nm

C2 : 0 ≤ θm,1 ≤ θc , 0 ≤ θm,2 ≤ θe , ∀m ∈ M,

(4)

where wk(m,n) > 0 is a weight factor that accounts for the priority of selecting user k(m, n) to use an RB n in the m-th cell and Pm is the maximum transmit power of the m-th BS for overall cell users. Inequations C1 and C2 indicate the perBS power, cell-center user and cell-edge user speciﬁc downtilt constraints, respectively. To solve this optimization problem, we should ﬁnd solutions with respect to Nm , k, p, and θm . III. Coordinated Resource Allocation in 3D MIMO Systems Since (4) is a constrained non-convex optimization problem, computing its global optimal solution may not be feasible in practice [10]. Thus, a coordinated resource allocation scheme is proposed to solve this optimization problem suboptimally in this section. Firstly, partition available RBs for the cell-center users and cell-edge users respectively, and obtain Nm for the m-th cell as described in Section A. Secondly, using FDDM proposed in Section B, jointly optimize the downtilt adjustment and RAPA. Finally, the convergence and computation complexity are veriﬁed and analyzed in Section C. A. RB Partitioning for Each Cell Assume that each cell performs equal power allocation (EPA). We calculate the data rate gain of the n-th RB for all cell-edge users in the m-th cell, according to Rm,n = R[n] (5) m,u (θm,2 ), ∀n ∈ N. u∈Um,e

For cell-edge users in the m-th cell , the best RB has the maximum m∈M Rm,n value among all available RBs as follows.

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IEEE ICC 2014 - Wireless Communications Symposium

nˆ = arg max n∈N

Then, the dual optimization problem can be given by Rm,n .

(6)

Algorithm I RB Partitioning Step 1: Initialization • For m-th BS, N = {1, · · · , N} and M = {1, · · · , M}. Nm,c = Nm,e = φ. Step 2: Iteration • Calculate the data rate gain of N RBs for both cell-center and cell-edge users in each cell according to Eq.(5). • for m = 1 : M 1) Determine nˆ m for the m-th cell according to Eq.(6), Nm = N. 2) In case card(Nm,e ) < γN, Nm,e = Nm,e ∪ nˆ m , N = N \ nˆ m , and go to step 1 in this loop. 3) Otherwise, m = m + 1, and go to step 1 in this loop. end for Step 3: Finalization • Obtain Ne = N1,e ∪ . . . ∪ N M,e and Nc = N − Ne . • The available RBs of the m-th cell are Nm,e and Nm,c = Nc for cell-center users and cell-edge users, respectively.

+

m∈M

[n] wk(m,n) R[n] m,k(m,n) (p ) +

m∈M n ∈Nm,e

=

[n] [n] wk(m,n) R[n] m,k(m,n) (pm ) − λm pm ,

(11)

=

[n ] [n ] [n ] wk(m,n ) Rm,k(m,n ) (pm ) − λm pm . (12)

Fixed downtilts are initialized for cell-center users and celledge users, respectively. Eq.(10) can be decomposed into N independent optimization problems as follows. ⎧ [n] ⎪ ⎪ ⎨maxp[n] ≥0,k[n] ∈K gn (λ, p[n] , k , θm,1 ), ∀n ∈ Nm,c , ⎪ ⎪ [n ] ⎩maxp[n ] ≥0,k[n ] ∈K gn (λ, p , k[n ] , θm,2 ), ∀n ∈ Nm,e .

(13)

EPA is initialized for p[n] and p[n ] in M BSs. The multivariate maximization (13) indicates a rule to ﬁnd the optimal [n] [n ] user kˆ and kˆ for the speciﬁc RBs n and n . [n] [n ] After kˆ and kˆ have been ﬁxed, the optimal power allocation solution can be derived by ⎧ [n] [n] ⎪ ⎪ gn (λ, p[n] , kˆ ), ∀n ∈ Nm,e , ⎨ pˆ m = arg max p[n] m ≥0 ⎪ ⎪ ] [n ] ˆ [n ] ⎩ pˆ [n ] (λ, p g , k ), ∀n ∈ Nm,e . n m = arg max p[n m ≥0

(8)

(14)

Applying Karush-Kuhn-Tucker (KKT) condition [7], by taking the derivation of Eq.(14) with respect to p[n] and p[n ] , we have ⎤† ⎡ [n] ⎥⎥ ⎢⎢⎢ wk(m,n) σ2 + jm p[n] j G j,k(m,n) cos[θ j,1 − θ j,k(m,n) ] ⎥ ⎥⎥⎥ , ⎢⎢⎢ pˆ [n] = − m ⎦ ⎣ λ ln 2 + t[n] [n] Gm,k(m,n) cos[θm,1 − θm,k(m,n) ] m m (15) ⎤† ⎡ ⎥⎥⎥ ⎢⎢⎢ wk(m,n ) σ2 ⎥⎥⎥ , pˆ m[n ] = ⎢⎢⎣⎢ − [n ] (16) λm ln 2 Gm,k(m,n ) cos[θm,1 − θm,k(m,n ) ] ⎦

βm,e (θm,e − θm,2 ), (7)

where βc (β1,c , ..., β M,c ), βe (β1,e , ..., β M,e ) and λ (λ1 , ..., λ M )T denote the vector of nonnegative Lagrangian multipliers. Note that p satisﬁes the two constraints: 1) p[n] m ≥ 0, ∀m ∈ M, ∀n ∈ Nm . 2) Only one p[n] m is positive for each n occupied by m-th BS. The Lagrange dual objective function, with Λ(·) given by Eq.(7) can be formulated as p≥0,k∈K

(10)

m∈M

m∈M

g(λ, βc , βe ) = max Λ(λ, βc , βe , p, k).

[βm,c (θm,c − θm,1 ) + βm,e (θm,e − θm,2 ) + λm Pm ]

m∈M [gn (λ, p[n] , k[n] ) + gn (λ, p[n ] , k[n ] ),

gn (λ, p[n ] , k[n ] )

βm,c (θm,c − θm,1 )

m∈M

m∈M [n ] [n ] wk(m,n ) Rm,k(m,n )+ ) (p

max

gn (λ, p[n] , k[n] )

p[n] m )

=

where

n∈Nm

m∈M n∈Nm,c

+

g(λ, βc , βe )

n∈Nm,c

B. FFR-based Dual Decomposition Method We consider the Lagrangian of the optimization problem (4) dualized with respect to the system constraints and cell-center users’ throughput requirement λm (Pm −

Note that the Lagrangian Λ(λ, βc , βe , p, k) is linear with respect to λ, βc and βe for ﬁxed p, k. And g(λ, βc , βe ) is the maximum of these linear functions. Thus, the dual problem (9) is convex. The duality gap between g(λ∗ , β∗c , β∗e ) and the solution to the primal problem (5) is tend to zero [11]. Moreover, observe that Eq.(8) can be recast as

+

Note: In Algorithm I-Iteration step, the priority of selecting cell-edge RBs among cells is based on the round robin scheduling algorithm. On the basis of the derived RB partitioning for each cell, FDDM is proposed to jointly optimize the downtilt adjustment and RAPA in the following.

(9)

λ,β

The proportion between the available RBs of cell-edge users N and all system RBs in the m-th cell is denoted by γm = Nm,e , where Nm,e = card(Nm,e ), and card(·) represents the cardinality of a certain set. In this paper, such proportion factor is set to be equal and denoted by γ = γm . Then, the available RBs need to be chosen for cell-center and cell-edge users in each cell respectively. And such RB partitioning among BSs can be carried out in the following procedure.

Λ(λ, βc , βe , p, k)

{λ∗ , β∗c , β∗e } = min g(λ, βc , βe ),

m∈M

where ∀n ∈ Nm,c , ∀n ∈ Nm,e , ∀m ∈ M, tm[n]

=

j∈M, jm

wk( j,n)

[n] [n] G[n] m,k( j,n) cos[θm,1 − θm,k( j,n) ]S INR j,k( j,n) (p ) . (17) [n] σ2 + l∈M p[n] l G l,k( j,n) cos[θl,1 − θl,k( j,n) ]

Here, [x]† max(0, x) and the taxation term tm[n] ’s physical meaning is the summation of the interference that the m-th BS causes to cell-center users in other BSs on the RB n. We T denote t (t[1] , ..., t[N] ), where t[n] (t1[n] , ..., t[n] M) .

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IEEE ICC 2014 - Wireless Communications Symposium

] In Eq.(15) and Eq.(16), pˆ [n] ˆ [n m and p m have similar form in equation but with diﬀerent meaning. Remark 1: The optimal power allocation for cell-center user has the same form as improved iterative water-ﬁlling algorithm (IIWF) [10]. On RBs partitioned to be utilized by cell-center users in Algorithm I, BS not only considers its own transmission but also intend to reduce the interference caused for other BSs. More power would be allocated to the RB which has better SINR with less interference for others according to Eq.(15). Remark 2: On the other hand, The optimal power allocation for cell-edge user has the same form as the traditional WF intuitively [10]. More power would be allocated to the RB [n ] which has better SNR. Substituting Eq.(15), Eq.(16), kˆ and [n] kˆ into Eq.(11) and Eq.(12), we have

where l is the iteration index, and lλm lβm,c and lβm,e are the appropriate positive step-size sequences. Eq.(26) can be iterated until convergence [7]. A summary of our proposed FDDM is given in Algorithm III. A summary of the proposed FDDM is described in the Algorithm II.

Algorithm II FFR Based Dual Decomposition Method Step 1: Initialization • Initialize p based on equal power allocation. • Initialize λ, β, θ, and set l = 0. Step 2: Iteration • repeat for m ∈ M for n ∈ Nm update k(m,n) according to the maximization (13) and calculate pˆ [n] m according to Eq.(15), Eq.(16). [n] ˆ [n] [n ] ˆ [n ] g( λ, βc , βe ) = gm (λ, βc , pˆ m , k , θm,1 ) + gm (λ, βe , pˆ m , k , θm,2 ) end for m∈M m∈M end for for m ∈ M [βm,c θm,c + βm,e θm,e + λm Pm ], (18) + calculate θˆm,1 and θˆm,2 according to Eq.(22), (23). m∈M end for where Update λ, βc and βe according to Eq.(26), set l = l + 1. [n] [n] [n] ≤ ε, and until Convergence, λl − λ(l−1) ≤ ε, βlc − β(l−1) ˆ c , k , θ ) = −β θ + w R (θ )−λ p ˆ , gm (λ, βc , pˆ [n] m,1 m,c m,1 k(m,n) m,1 m m m m,k(m,n) (l−1) l − β ≤ ε. β [n] e e k(m,n)∈kˆ n∈Nm,c Step 3: Finalization (19) • p, k, and θ are derived for users. [n ] [n ] ˆ [n ] [n ] wk(m,n ) Rm,k(m,n ) (θm,2 )−λm pˆ m . gm (λ, βe , pˆ m , k , θm,2 ) = −βm,e θm,2 + [n ]

k(m,n)∈kˆ n∈Nm,c

(20)

Note: In Algorithm II, ε represents the convergence tolerance, and we set ε = 10−2 in this paper.

C. Convergence and Computation Complexity Analysis After the above-mentioned procedure, the optimal downlilt solutions for cell-center users and cell-edge users can be derived by ⎧ [n] [n] ⎪ ⎪ ⎨θˆm,1 = arg maxθˆm,1 ≥0 gm (λ, βc , pˆ m , kˆ , θm,1 ), ∀n ∈ Nm,c , ⎪ [n ] ⎪ ⎩θˆm,2 = arg max ˆ gm (λ, βe , pˆ m[n ] , kˆ , θm,2 ), ∀n ∈ Nm,e . θm,2 ≥0

(21)

Applying Karush-Kuhn-Tucker (KKT) condition [7], by taking the derivation of Eq.(21) with respect to θm,1 and θm,2 , we have −βm,c ln 2 + φm,1 }† , (22) θˆm,1 = {arcsin A2m + B2m θˆm,2 = {φm,2 + arcsin −σ2 βm,e ln 2 }† . [ n∈Nm,e cnm cos θm,k(m,n ) ]2 + [ n∈Nm,e cnm sin θm,k(m,n ) ]2 (23)

The detailed proof of Eq.(22), Eq.(23) and deﬁnitions of φm,1 , φm,2 , Am , Bm , cnm are given in Appendix A. g(λ, β) may not be diﬀerentiable, sub-gradient methods can guarantee to converge to the global optimal solution [7]. Therefore, the Lagrangian dual multipliers can be updated as ⎧ † (l+1) ⎪ ⎪ = βlm,c − lβm,c θm,c − θˆm,1 , βm,c ⎪ ⎪ ⎪ † ⎪ ⎨ (l+1) (24) βm,e = βlm,e − lβm,e θm,e − θˆm,2 , ⎪ ⎪ ⎪ ⎪ † ⎪ ⎪ [n] l l ⎩λ(l+1) = λm + λm ( n∈Nm pˆ m − Pm ) , m

Using sub-gradient methods, FDDM can guarantee convergence to a local optimal solution for Eq.(8) according to [11]. For FDDM, the number of iterations required to get optimal Lagrangian dual multipliers λ, (βc , βe ) and achieve ε-optimality, i.e. g∗ − g < ε, is O( ε12 ) [11]. In Algorithm I, the computational complexities for RB partitioning is O(N + (N − γ1 N)+, ..., +(N − γ M MN))|K|, which is much lower than 2N K by using exhaustive method. In RAPA and downtilt adjustment procedures of FDDM, p[n] m [n ] and pm , ∀n ∈ Nm,c , ∀n ∈ Nm,e , have the same form as IIWF and the traditional WF as mentioned in Remark 1 and Remark 2. Solving WF and IIWF has a computational burden O(N M log2 N) [10], while updating k and θ has the complexities O(N M|K|) and O(M|K|) accordingly. In summary, the total computation complexity of Algorithm M(N|K|+N log2 N+|K|) I and FDDM is much lower than O(2N K ). ε2 Therefore, the proposed coordinated resource allocation scheme can eﬀectively optimize downtilt adjustment and RAPA in 3D MIMO systems, especially when ε is small. IV. Simulation Result and Discussion We evaluate the performance of our coordinated resource allocation scheme by using a LTE system level simulator. Urban macro-cell (UMa) scenarios is considered in which 7 coordination clusters are deployed [12]. The overall BSs are arranged in grid fashion with appropriate cell-wrap to ensure

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•

•

WF/IIWF-Fixed Downtilt: Use traditional WF/IIWF to implement RAPA for overall cell users. And both cellcenter user and cell-edge user speciﬁc downtilts are set to be 15 degree according to [12]. FDDM-AlgorithmI: Based on the RB partitioning in Algorithm I, implement FDDM by utilizing Algorithm II, respectively.

Proportional fairness (PF) scheduling algorithm and full buﬀer traﬃc model are utilized for simulation. Monte Carlo simulation is performed for each subframe and iterated over a total of 1000 subframes. To verify the overall cell users’ performance of diﬀerent interference coordinations, the SINR cumulative density function (CDF) curves of cell users are illustrated in Fig.2 and Fig.3. 1 0.9

WF IIWF FDDM

0.8 0.7

CDF

0.6 0.5

coordinated resource allocation scheme is improved by 713dB. Moreover, the overall cell users’ SINR performance of FDDM-AlgorithmI is much higher than traditional WF-Fixed Downtilt. 1 0.9

WF IIWF FDDM

0.8 0.7

CDF

0.6 0.5 0.4 0.3 0.2 0.1 0 -5

0

5

10

15

SINR(dB)

20

25

30

Fig. 3. CDF curves of cell-edge users’ SINR for diﬀerent coordinated resource allocation schemes in 3D MIMO systems.

Fig.3 further shows the cell-edge users’ SINR performance for ﬁve coordination schemes. It is clear that FDDMAlgorithmII is still better than classic IIWF-Fixed Downtilt. Compared with traditional WF-Fixed Downtilt, the SINR performance of our proposed coordinated resource allocation scheme is improved by 10-20dB. This is because our proposed coordinated resource allocation scheme is implemented by guaranteeing the balanced experience of cell-center users and cell-edge users with dynamic downtilt adaption. 2.5

Downlink Spectral Efficiency(bps/Hz/cell)

elimination of edge eﬀects. The inter-cell distance between two BSs is 500m, and maximum transmit power is 46dBm. Center frequency and bandwidth are 2.3GHz and 10MHz, respectively, and thermal noise density is −174dBm/Hz. There are 50 RBs and 10 users in each cell. The 3D MIMO channel modeling parameters in the horizontal dimension, including fast fading, shadow fading and path loss models, is set based on ITU channel model [12], [13]. Other parameters including antenna conﬁguration, receiver method, and minimum separation between UE to BS etc, are set according to [12]. Users in each cell are generated uniformly based on random distribution, and T R = 10dB is set to divide cell-center and cell-edge users. Using FFR scheme, cell-edge users’ reuse factor α = 3, and universal reuse is utilized for cell-center users. Therefore, card(Ne ) = 3γN, card(Nc ) = (1 − 3γ)N, and set γ = 0.2 according to [14]. We introduce three coordinated resource allocation schemes compared in the simulation.

Cell-average Spectral Efficiency Cell-edge Spectral Efficiency

2

1.5

1

0.5

0.4

0

0.3 0.2 0.1 0 -5

0

5

10

SINR(dB)

15

20

25

WF

IIWF

FDDM

Fig. 4. Cell-average and cell-edge spectral eﬃciency for diﬀerent coordinated resource allocation schemes in 3D MIMO systems.

30

Fig. 2. CDF curves of overall cell users’ SINR for diﬀerent coordinated resource allocation schemes in 3D MIMO systems.

In Fig.2, it is observed that the overall cell users’ performance of FDDM-AlgorithmI outperforms other coordinated resource allocation schemes, especially in low SINR region. This is because cell-edge users’ performance can be further improved by utilizing FFR scheme. Compared with classic IIWF-Fixed Downtilt, the SINR performance of our proposed

To further verify the overall cell throughput performance of diﬀerent coordinated resource allocation schemes, the downlink cell-average and cell-edge spectral eﬃciency deﬁned in [12], is evaluated and illustrated in Fig.4. It is observed that the average spectral eﬃciency of FDDM - AlgorithmI is about 30% and 15% higher than WF-Fixed Downtilt and IIWF-Fixed Downtilt, respectively. Furthermore, FDDM yields up the best cell-edge users’ performance through vertical beamforming and allocating more RBs for cell-edge users in 3D MIMO systems.

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V. Conclusion This paper investigates coordinated resource allocation scheme in 3D MIMO systems, where cell-center user and cell-edge user speciﬁc downtilts are accordingly partitioned through dynamic vertical beamforming. The FFR scheme is introduced in this paper, where an RB partitioning scheme is presented accordingly. Based on such RB partitioning, FDDM is proposed where downtilts adjustment and RAPA are jointly optimized. Simulation results have shown that based on the proposed RB partitioning algorithms, FDDM outperforms the classical coordinated algorithms signiﬁcantly with lower computation complexity. Appendix A: The calculation procedure of downtilts Applying Karush-Kuhn-Tucker (KKT) condition [7], by taking the derivation of Eq.(21) with respect to θm,1 , we have

Let

n n ∈Nm,e cm cos θm,k(m,n ) cos φm,2 = , [ n ∈Nm,e cnm cos θm,k(m,n ) ]2 + [ n ∈Nm,e cnm sin θm,k(m,n ) ]2 (33)

the optimal downtilt for cell-edge users in M BSs can be obtained as follows: θˆm,2 = {φm,2 + arcsin −σ2 βm,e ln 2 }† . n 2 n 2 [ n∈Nm,e cm cos θm,k(m,n ) ] + [ n∈Nm,e cm sin θm,k(m,n ) ] (34)

Note that θˆm,2 must satisfy cell-edge user speciﬁc downtilt π ⎞ constraints C2 , θˆm,2 < 2 .

⎛ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜an sin[θm,1 − θm,k(m,n) ] − −βm,c ln 2 = bnm sin[θm,1 − θm,k( j,n) ]⎟⎟⎟⎠ , ⎝ m n∈Nm,c

j∈M, jm

(25)

where anm =

σ2 +

wk(m,n) pnmG[n] m,k(m,n)

[n] [n] l∈M pl G l,k(m,n)

cos[θl,1 − θl,k(m,n) ]

.

(26)

[n] [n] pnmG[n] m,k( j,n) S INR j,k( j,n) (p ) = wk( j,n) . [n] σ2 + l∈M p[n] j∈M, jm l G l,k( j,n) cos[θl,1 − θl,k( j,n) ] (27) The expansion of Eq.(25) can be written as:

bnm

sin θm,1 Am − cos θm,1 Bm = −βm,c ln 2,

(28)

where Am = n∈Nm,c anm cos θm,k(m,n) − j∈M, jm bnm cos θm,k( j,n) , Bm = n∈Nm,c anm sin θm,k(m,n) − j∈M, jm bnm sin θm,k( j,n) . Let cos φm,1 =

Am A2m + B2m

,

(29)

the optimal downtilt for cell-center users in M BSs can be derived as follows: −βm,c ln 2 θˆm,1 = {arcsin + φm,1 }† . 2 2 Am + Bm

(30)

Note that θˆm,1 must satisfy cell-center user speciﬁc downlilt constraints C2 , θˆm,1 < π2 . Similarly, taking the derivation of Eq.(21) with respect to θm,2 , we have −σ2 βm,e ln 2 = sin θm,2 cnm cos θm,k(m,n ) − cos θm,2 cnm sin θm,k(m,n ) . n ∈Nm,e

n ∈Nm,e

(31)

where

cnm

=

This work is supported by Key Project (2013ZX03001025002), National 863 Project (2014AA01A701), National Nature Science Foundation of China (61121001). References [1] J. Zyren and W. McCoy, “Overview of the 3GPP long term evolution physical layer,” Freescale Semiconductor, Inc., white paper, 2007. [2] V. Jungnickel, M. Schellmann, L. Thiele, T. Wirth, T. Haustein, O. Koch, W. Zirwas, and E. Schulz, “Interference-aware scheduling in the multiuser mimo-ofdm downlink,” Communications Magazine, IEEE, vol. 47, no. 6, pp. 56–66, 2009. [3] G. Boudreau, J. Panicker, N. Guo, R. Chang, N. Wang, and S. Vrzic, “Interference coordination and cancellation for 4g networks,” Communications Magazine, IEEE, vol. 47, no. 4, pp. 74–81, 2009. [4] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multi-antenna wireless system,” Wireless Communications, IEEE Transactions on, vol. 9, no. 5, pp. 1748–1759, 2010. [5] K. Safjan, V. D’Amico, D. Bultmann, D. Martin-Sacristan, A. Saadani, and H. Schoneich, “Assessing 3gpp lte-advanced as imt-advanced technology: the winner+ evaluation group approach,” Communications Magazine, IEEE, vol. 49, no. 2, pp. 92–100, 2011. [6] V. D’Amico, Botella, et al., “Advanced interference management in artist4g: Interference avoidance,” in European Wireless Technology Conference, Paris, France, September 2010, 2010, pp. 21–24. [7] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004. [8] T. Novlan, R. Ganti, A. Ghosh, and J. Andrews, “Analytical evaluation of fractional frequency reuse for ofdma cellular networks,” Wireless Communications, IEEE Transactions on, no. 99, pp. 1–12, 2011. [9] S. Ali and V. Leung, “Dynamic frequency allocation in fractional frequency reused ofdma networks,” Wireless Communications, IEEE Transactions on, vol. 8, no. 8, pp. 4286–4295, 2009. [10] L. Venturino, N. Prasad, and X. Wang, “Coordinated scheduling and power allocation in downlink multicell ofdma networks,” Vehicular Technology, IEEE Transactions on, vol. 58, no. 6, pp. 2835–2848, 2009. [11] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” Communications, IEEE Transactions on, vol. 54, no. 7, pp. 1310–1322, 2006. [12] ITU-R Rep. M.2135, “Guidelines for evaluation of radio interface technologies for IMT-Advanced,” Jul.2009. [13] E. Dahlman, 3G evolution: HSPA and LTE for mobile broadband. Academic Press, 2008. [14] M. Rahman and H. Yanikomeroglu, “A downlink dynamic interference avoidance scheme with inter-cell coordination,” Wireless Communications, IEEE Transactions on, vol. 9, no. 4, pp. 1414–1425, 2010.

[n ] wk(m,n ) pnm Gm,k(m,n )

Acknowledgment

[n ] [n ] 1 + S NRm,k(m,n ) (pm )

.

(32)

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