Cooperative Energy Efficient Power Allocation Algorithm for downlink massive MIMO

Cooperati v e Ener gy Ef ficient Po wer Allocation Algorithm for Do wnlink Massiv e MIMO Saeed Sadeghi V ilni Abstract Massiv e multiple input multiple output (MIMO) is proposed to increase network capacity and reduce power consumption in next generation of wireless networks. In this paper , po wer allocation in the massiv e multiple input multiple output (MIMO) cellular networks is studied. Considering circuit power consumption, energy efficienc y is utilized to ev aluate the network performance, since circuit po wer consumption increases by increasing both the number of antenna in transmitter and the number of network users. T o optimize transmit power of the network, an energy efficient optimization problem is considered under both maximum transmit power and quality of service (QoS) constraints. Where the users cooperate with the base station (BS) to minimize network power usage. T o solve the problem, an energy efficient power allocation algorithm is proposed. Con ver gence of the proposed algorithm is sho wn numerically , in addition, the appropriate number of transmit antenna and users are obtained. Finally , pilot contamination effect is ev aluated, where it is shown energy efficienc y decreases dramatically . Index T erms Massiv e MIMO, power allocation, energy efficienc y , cooperation, pilot contamination. I . I N T RO D U C T I O N Cellular network traf fic is increased greatly todays, due to rapid gro wth of smart terminal usage and high data rate requirements services [1]- [2]. In order to increase the spectral and energy ef ficiency up to around ten times massiv e multiple input multiple output (MIMO) is proposed in the next generation of wireless networks. Massive MIMO means using large number of transmit antenna (about one hundred antenna) and can enhance spectral ef ficiency and reliability [3]- [4]. W ith a large number of transmit antenna, massive MIMO can focus ener gy at desired point and increase spectral ef ficiency but circuit power consumption, which connected to the number of antenna, also increased. T o achiev e maximum energy effi ciency , unlimited number of transmit antenna is optimal The author is with the Department of Electrical and Computer Engineering, T arbiat Modares University , P . O. Box 14115-194, T ehran, Iran. Corresponding author is S. S. V ilni (email: saeed.sadeghi@modares.ac.ir). 2 if circuit po wer consumption is not considered [5]. Hence to ev aluate network ener gy ef ficiency , which is throughput to po wer consumption ratio, circuit power consumption should be taken into account. Energy ef ficient resource allocation is one of open problems in wireless system design, wich is considered in [6]- [16]. In [6]- [8] authors try to maximize the energy ef ficiency but circuit po wer consumption is ignored; Also in [9] authors assume a constant circuit power consumption. From another point of view , QoS plays an important role in wireless system resource allocation; for example, satisfying QoS constraint in addition to providing a roughly fair network leads a practical network. No QoS constraint is considered in [8]- [13]. In [6] and [12]- [15], authors assume that the number of transmit antenna is very greater than users, which is impractical assumption. In this paper , we try to make this assumption practical, we consider any number of transmit antenna and users. In [16] authors consider imperfect channel state information (CSI), but in system design, they utilize estimated channel and real channel together; which appears they hav e a design error . T o obtain channel state information (CSI), The base stations (BSs) use time division duplexing (TDD) protocol as shown in Fig. 1. As the protocol shows, users send a pilot sequence after their data to the BS. The BS estimates the users’ channel using the receiv ed pilot and sends the channel data back to users. Note that in spite of the pilots are sent in the same bandwidth, but they must be orthogonal to data; hence this is not the perfect CSI. Due to resource limitation, it is not possible for all users in all cells to have orthogonal pilots, therefore each pilot is reused by users of the neighbor cells, this creates pilot contamination. Pilot contamination leads to imprecisely channel estimation which reduces system performance [17]. Pilot contamination in [9] and [12]- [15] is not considered too. In this paper , we in vestigate energy ef ficient power allocation in a cellular network with large number of transmit antenna (massiv e MIMO), that users cooperate with their BS. W e consider energy efficienc y as users’ utility where ov erall transmit power in a cell is limited. In addition, a minimum data rate should be provided for each user . W e utilize the fractional programming technique to solve the problem, since the energy ef ficiency objecti ve function is a fractional function. W e assume cooperation between users and their BS, and found a closed form for optimal transmit po wer of users. The algorithm of cooperativ e energy efficient power allocation is gi ven in Algorithm 1. W e e xtend the problem to multi cell case under pilot contamination. Finally , the con ver gence of the proposed algorithm is in vestigated by simulation. In addition, we explore pilot contamination effect on energy efficiency . The remainder of this paper is or ganized as follo ws. In Section II the system model is presented. In Section III the energy ef ficient optimization problem is formulated. The solution for the problem is presented in Section IV . In section V our scenario extended to multi cell case. Numerical results are described in Section VI follo wed by conclusion in Section VII. 3 Notation: V ectors and matrices are denoted by boldface small and big letters, respectively . The super- script H stands for conjugate transpose. Euclidean norm with k . k is denoted and I K sho ws an Identity K × K matrix. N (0 , σ 2 ) denotes a Gaussian probability density function with zero mean and variance σ 2 . Fig. 1. TDD protocol to acquire CSI I I . S Y S T E M M O D E L Consider a do wnlink single cell network including a BS equipped with M antennas and K single antenna users which are distributed in the cell uniformly . Assume flat fading channels between BS and users with perfect channel state information (CSI) and all users get service on a single frequency . Channel v ector between BS and user i is modeled by g i = √ β i × h i ∈ C M × 1 where h i sho ws fast fading and β i is large scale fading that consists of path loss and shado w fading. Fast fading has a distribution as C N (0 , 1) and β i = ψ ( d 0 d i ) v where ψ shows shadow fading with the distribution 10 log ψ 10 ∼ N (0 , σ 2 sh ) , d 0 is minimum distance from users to the BS, and d i is distance from user i to the BS and v is the path loss exponent. The recei ved signal by user i can be expressed as y i = K X k =1 √ p k g i w k x k + n i (1) Where p k is transmit power of user k which is an element of transmit power vector p = [ p 1 ...p k ...p K ] , x k is transmission symbol for user k with k x k k 2 = 1 which is assumed that transmission symbols are uncorrelated among different users, n i is additiv e noise with C N (0 , σ 2 ) distribution and w l ∈ C M × 1 is beamforming vector for l th user , by maximum ratio transmission (MR T) beamforming we hav e w k = g H k k g k k (2) The recei ved signal by user i can be written as y i = √ p i g i w i x i + K X k =1 ,k 6 = i √ p k g i w k x k + n i (3) Equation (3) is the summation of three parts: desired signal, interference and noise. According to (3), 4 SINR of user i can be expressed as following. sinr i ( p i ) = p i k g i w i k 2 P K k =1 ,k 6 = i p k k g i w k k 2 + σ 2 (4) Where σ 2 is noise power on the channel of user i . The instantaneous data rate for user i can be written as follo wing r i ( p i ) = B log(1 + Γ i sinr i ( p i )) (5) Where B is bandwidth and Γ i is SINR gap between Shannon channel capacity and a practical modulation and coding scheme. This SINR gap is equal to − 2 3 ln(5 e i ) that e k is target bit error rate [18]. I I I . P RO B L E M S TA T E M E N T A. Energy Efficiency Po wer consumption in the network includes transmit power and power consumed in the circuits. Circuit power consumption in BS has two parts: constant part P fix in volv ed site cooling, transmitting filter , conv erter and local oscillator; and v ariable part, P a , is the po wer to run an antenna. If circuit power consumption of each user is displayed with P ue , the po wer consumption P can be defined as P ( p ) = K X k =1 p l + ( M × P a ) + P fix + ( K × P ue ) (6) According to [19] and [20] the energy efficienc y η in communication system calculated in bit/Joule and is the ratio of total rate to po wer consumption which can be formulated as the following η ( p ) = P K k =1 r l ( p ) P ( p ) (7) B. Optimization Pr oblem According to abov e discussion, the goal of the network is to maximize the energy efficienc y (7) under maximum po wer and minimum data rate constraints, i.e. following problem max p η ( p ) s.t. C 1 : P K k =1 p k ≤ P max C 2 : r k ( p ) ≥ R min ∀ k (8) Therefore, the netw ork should allocate transmit power to each users such that maximize the energy ef ficiency . In (8), constraint C 1 is the power constraint where P max is total maximum transmit power , and constraint C 2 shows minimum data rate R min that must be provided for each user . 5 I V . S O L V I N G T H E P RO B L E M There are two main challenges in (8): the problem has fractional form and (7) is not concav e. T o solve the problem (8), we transform the the objecti ve function of (8) to a subtractiv e form by the following Theorem [21]. Theorem 1. The maximum ener gy efficiency η ∗ is achieved in (8) if and only if max p P K k =1 r k ( p ) − η ∗ P ( p ) = P K k =1 r k ( p ∗ ) − η ∗ P ( p ∗ ) = 0 for P K k =1 r k ( p ) ≥ 0 and P ( p ) > 0 . According to Theorem 1, the problem (8) is transformed to following problem. max p P K k =1 r k ( p ) − η P ( p ) s.t. C 1 : P K k =1 p k ≤ P max C 2 : r k ( p ) ≥ R min ∀ k (9) T o solve (8), we solve (9) iteratively . W e propose to solve problem (9) with a specific v alue of energy ef ficiency η and obtain users transmit power p t 1 for iteration t 1 ; and then compute A = P K k =1 r k ( p t 1 ) − η P ( p t 1 ) . If A is zero, η is optimal energy efficienc y and p t 1 is optimal transmit po wer; otherwise energy ef ficiency with p t 1 is computed and then p t 1 is used in (9) and solve the problem again until A goes to zero. T o address non-conca vity of the objectiv e function of (8), we propose to relax the objectiv e function by the follo wing lower bound [22]. log (1 + Γ i sinr i ( p i )) ≥ a i log (Γ i sinr i ( p i )) + b i (10) Where a i and b i are auxiliary multiplicati ve and additi ve variables for user i . Equation (10) means utilizing ˆ r i = a i log (Γ i sinr i ( p i )) + b i instead of r i = log (1 + Γ i sinr i ( p i )) as the objecti ve function. The v ariable of problem p is changed to ˆ p = ln p . Therefore we hav e the following problem. max ˆ p P K l =1 ˆ r l ( ˆ p ) − η P ( ˆ p ) s.t. C 1 : P K k =1 e ˆ p k ≤ P max C 2 : ˆ r k ( ˆ p ) ≥ R min ∀ k (11) Since log-sum-exp is concav e [23], The problem (11) is a con ve x optimization problem. In order to solve (9), we solve (11) iterati vely until transmit power conv erged; also a i and b i are updated in each iteration by obtained transmit po wer from current iteration with following equation. a i = sinr i ( ˆ p i ) 1 + sinr i ( ˆ p i ) (12a) 6 b i = log(1 + sinr i ( ˆ p i )) − a i log( sinr i ( ˆ p i )) (12b) Since the problem (11) is a con ve x optimization problem, dual Lagrangian function is utilized to solve that. Let λ and φ Lagrange multiplier corresponding to maximum transmit power and minimum data rate constraint, the dual Lagrangian function of (11) can be written as following [23]. L ( ˆ p , λ, φ ) = P K k =1 ˆ r k ( ˆ p ) − η P ( ˆ p ) − φ ( P K k =1 e ˆ p k − P max ) + P K k =1 λ l ( R min − r k ) (13) Optimal transmit power can be obtained by the Karush Kuhn T ucker (KKT) conditions as following for user i . ∂ L ( ˆ p , λ, φ ) ∂ p i = 0 (14) According to (14), optimal transmit po wer for user i can be obtained as following. e ˆ p i = p i = [ ( λ i + 1) B a i ln 2 B ln 2 P K k =1 ,k 6 = i ( λ k +1) a k ( k w i g k k 2 ) I k + ( η + φ ) ] + (15) Where I k is interference plus noise suffered by user k . In this paper cooperating of users with BS is considered such as they measure interference I k and feed back to BS. I k can be calculated as I k = K X u =1 ,u 6 = k e ˆ p u ( k w u h k k 2 ) + σ 2 (16) By the subgradient method the Lagrange multipliers can be updated as following. φ ( t 3 + 1) = [ φ ( t 3 ) + γ φ ( K X k =1 e ˆ p k − P max )] + (17) λ k ( t 3 + 1) = [ λ l ( t 3 ) + γ λ ( R min − r k )] + (18) Where t 3 is iteration index for Lagrange multipliers update also γ φ and γ λ are step size for φ and λ respecti vely . Algorithm for cooperativ e energy efficient po wer allocation is presented in Algorithm 1. V . E X T E N S I O N T O M U LT I C E L L C A S E In this section, the single cell scenario in Section II is extended to multi cell case, where each cell has K users and all users operate in a single frequency . Pilot contamination and estimate channel with 7 least square (LS) method is considered in multi cell case as in [24]. As previous section optimal transmit po wer for user m in cell j can be obtained as following p j m = [ ( λ j m + 1) B α j m ln 2 B ln 2 P L l =1 P K k =1 ,k 6 = m z lk I lk − ( η + φ j ) ] + (19) Where z lk = k w j m ˆ g j l k k 2 α lk ( λ lk + 1) , ˆ g j l k sho ws estimated channel between user k in cell l and j th BS, I lk is interference plus noise that measured by users and send for corresponding BSs, α lk is SCA multiplicati ve variable, λ lk is Lagrange multiplier corresponding to minimum data rate constraint and w lk is beamforming vector for user k in cell l , respectiv ely . Also, L shows the number of cells, φ j is the Lagrange multiplier corresponds to maximum transmit power in cell j and η is network energy efficienc y . The method to solve the optimization problem is giv en in the appendix A. Algorithm 1 Energy Ef ficient Cooperativ e Power Allocation 1: Initialize con ver gence tolerance  and initialize arbitrary η ( t 1 ) = 0 , con = 0 and Dink elbach algorithm iteration index t 1 = 1 2: repeat 3: initialize with a feasible ˆ p t 2 , Set a l ( t 2 ) = 1 , 4: b l ( t 2 ) = 0 and SCA iteration index t 2 = 1 5: repeat 6: initialize arbitrary λ l ( t 3 ) , φ ( t 3 ) , γ λ , γ φ and Lagrangian multipliers iteration index t 3 = 1 7: repeat 8: Compute I l by user and feed back to BS 9: Compute the optimal po wer transmit ˆ p t 2 +1 according to (15) 10: Update φ and λ l according to (17) and (18) 11: t 3 = t 3 + 1 12: until Con ver gence of λ l and φ 13: Compute sinr l ( ˆ p t 2 +1 ) and update a l ( t 2 + 1) and b l ( t 2 + 1) 14: t 2 = t 2 + 1 15: until con ver gence of ˆ p 16: if P K l =1 ˆ r l ( ˆ P ( t 2 + 1)) − η ( t 1 ) P ( ˆ p t 2 +1 ) ≤  then 17: Set con = 1 18: Return η ( t 1 ) as optimal energy efficienc y and ˆ p 19: as optimal transmit po wer 20: else 21: Set con = 0 22: t 1 = t 1 + 1 23: Return η ( t 1 + 1) = P K l =1 r l ( ˆ p t 2 ) P ( ˆ p t 2 ) 24: until con = 1 8 T ABLE I S I M U L AT I O N P A R A M E T E R S Parameter Description V alue σ 2 Noise po wer -174dBm σ 2 sh Shado w fading variance 8dB P max T otal maximum transmit po wer 1W P a Po wer consumption per antenna 1W P ue Po wer consumption per antenna 0.1W P fix Fixed power consumption in BS 20W R min Minimum requirement rate for users 14Kbps B Bandwidth 10KHz e k T arget bit error rate 10e-3 - Cell radius 500m d 0 Minimum distance 50 α Path loss exponent 3 V I . A N A LY S I S A N D S I M U L A T I O N R E S U LT S In this section, we ev aluate cooperativ e energy efficient po wer allocation in algorithm 1. Simulation parameters are given in T able I. First, the conv ergence of the Algorithm 1 is sho wn. Fig. 2 shows the con ver gence of algorithm 1 when number of antennas M = 100 and number of users K = 5 , as it can be seen, the algorithm con ver ges in three iterations. In fig. 3 energy efficienc y versus number of transmit antenna M is shown for K = 5 ; which is a concav e curve. When the number of transmit antennas increase, the performance of beamforming gets better; therefor SINR and data rate gro w . On the other hand, circuit power consumption increased when M increased. In figure 2, first, increasing of data rate overcome circuit power consumption but increasing number of antenna leads circuit po wer consumption overcome data rate increases. T o consider the effect the number of users, energy efficienc y in terms of the number of users is depicted in fig. 4. It can be seen that firstly , energy efficienc y increased b ut by increasing the number of users near to number of antenna energy efficienc y start decreasing. this is due to inter user interference increasing. Fig. 5 shows average transmit power for one user versus M . It shows that transmit po wer decreases by increasing the number of transmit antennas which is expected. Fig. 6 shows energy ef ficiency in multi cell versus transmit antenna. Transmit po wer is obtained by Algorithm 1 when the estimated channel is in access, by utilizing designed transmit po wer , a bit with BPSK modulation in a real channel is transmitted. Energy efficienc y for any number of transmit antenna in receiv er is computed, this process is implemented 10000 times for any number of antenna whenev er the channel changes, and the av erage of energy efficienc y over 10000 is plotted. Ener gy ef ficiency in 9 Fig. 2. Con vergence of the proposed algorithm for energy efficiency at M = 100 Fig. 3. Energy efficienc y versus number of transmit antenna for K = 5 multi cell related to single cell, due to pilot contamination and inter cell interference, decreased to about 0.75 of single cell. In channel estimation with pilot training, estimated channel is summation of desired user channel and the other users with same pilot sequence in neighbor cells, so MR T beam-forming , which uses channel vector directly , in addition to focus on desire user, also focus an amount of power to all users with the same pilot. By increasing number of transmit antenna interference on users with same pilot in the other cells gets higher . Therefore, in large number of transmit antenna when we use MR T beam-forming, inter-cell interference increases and the energy ef ficiency is placed in less than 0.75 of 10 Fig. 4. Energy efficienc y versus number of users for M = 82 Fig. 5. A verage transmit power per one user over number of transmit antenna single cell. V I I . C O N C L U S I O N In this paper , a cooperative energy efficient power allocation algorithm for do wnlink massi ve MIMO system is proposed. Based on cooperation between users and BS, a closed form for optimal transmit power is found and an algorithm is proposed. Simulation results show con ver gence of the proposed algorithm. 11 Fig. 6. Energy efficienc y versus number of transmit antenna for perfect and imperfect CSI The simulations show that there is an optimal number of antenna and optima number of users in each cell due to circuit power consumption and inter user interference. In addition, the simulations shown that the more number of transmit antenna the less transmit po wer . Finally the scenario is extended to multi cell case, in which pilot contamination is considered. The network total energy efficienc y decreases in comparison to single cell case. Also, results show that considering pilot contamination for large number of transmit antenna with MR T beamforming increases inter cell interference. A P P E N D I X A T R A N S M I T P O W E R F O R M U LT I C E L L C A S E Each user sends a pilot with po wer p u to its corresponding BS. The received signal at j th BS from users, Y j , can be written as follo ws. Y j = L X l =1 √ p u G j l Φ H l + Z j (20) where Y j is a M × τ matrix, G j l = [ g j l 1 ... g j l K ] is a M × K matrix which shows channel gains between users in cell l and j th BS, Φ l = [ φ j l ... φ lK ] is K × τ pilot matrix of users in cell l that Φ l × Φ H l = I K 12 and Φ l 1 × Φ l 2 6 = 0 ∀ l 1 6 = l 2 due to pilot reuse. By utilizing LS estimation method ˆ g j l k can be obtained as follo wing ˆ g j l k = 1 √ p u Y j φ lk (21) Recei ved signal to interference plus noise ratio (SINR) of user m in cell j can be obtained as following. sinr j m = p j m k ˆ g j j m w j m k 2 P L l =1 P K k =1 , 6 = m p lk k ˆ g lj m w lk k 2 + σ 2 (22) Therefore the transmission rate for the same user can be formulated as follows. r j m = B log 2 (1 + Γ sinr j m ) (23) Finally , the energy ef ficiency of the network can be expressed as η = P L l =1 P K k =1 r lk P L l =1 P K k =1 p lk + P L l =1 P c l = A ( P ) B ( P ) (24) where P c l is circuit power consumption in cell l and P is L × K transmit power matrix, the energy ef ficiency maximization problem can be expressed as max P η = A ( P ) B ( P ) s.t. C 1 : P K k =1 p lk ≤ P max ∀ l C 2 : r lk ≥ R min (25) Using Dinkelbach and SCA algorithms the problem (25) can be written max ˆ P A ( ˆ P ) − η B ( ˆ P ) s.t. C 1 : P K k =1 e ˆ p lk ≤ P max ∀ l C 2 : r lk ≥ R min (26) The problem (26) is a con ve x optimization problem and utilizing Lagrange method, transmit power and Lagrange multipliers are updated as follo ws p j m = [ ( λ j m + 1) B α j m ln 2 B ln 2 P L l =1 P K k =1 ,k 6 = m z lk I lk − ( η + φ j ) ] + (27) λ lk ( t 3 + 1) = ( λ lk ( t 3 ) − γ λ ( r min lk − R min )) + (28) φ l ( t 3 + 1) = ( φ l ( t 3 ) − γ φ ( P max − K X k =1 p lk )) + (29) where γ λ and γ φ are step size and t 3 is Lagrange iteration index. 13 R E F E R E N C E S [1] M. U. Farooq, M. W aseem, M. T . Qadri and M. W aqar, “Understanding 5G wireless cellular network: challenges, emerging research directions and enabling technologies, ” W ireless P ersonal Communications , 95, no. 2, pp. 261-285, Jul. 2017. [2] C. Sexton, N. Kaminski, J. M. Marquez-Barja, N. Marchetti and L. A. 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