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A GAME THEORETIC POWER CONTROL APPROACH FOR MIMO MC-DS/CDMA SYSTEMS V.Nagarajan and P.Dananjayan † Department of Electronics and Communication Engineering, Pondicherry Engineering College, Pondicherry -605014, India nagarajanece31@rediffmail.com,pdananjayan@rediffmail.com † † Corresponding author ABSTRACT A major challenge to enhance the performance of multiuser multiple-input multiple-output (MIMO) multi-carrier direct sequence code division multiple access (MC -DS/ CDMA) system relies on the effective multiple access interference suppression. In this work a novel distributed non cooperative power control game with pricing (NPGP) is considered for utilizing the system resource more efficiently. The ratio of throughput versus power is referred to as the utility function which should be maximized by combating the multiple access interference (MAI). Simulation results show that the propounded scheme achieves significant performance improvement, compared with the conventional system without NPGP. Keywords: Game theory, power control, pricing, MIMO, MC-DS/CDMA. 1 INTRODUCTION economic model has been proposed [3]. In [3] game theoretic approach is employed to study the power The enormous growth of wireless services control in the multi user scenario for the proposed during the last decade gives rise to the need for a model. It is a powerful tool in modeling interactions bandwidth efficient modulation technique that can between self-interested users and predicting their reliably transmit high data rates. As multi carrier choice of strategies. Each player in the game technique combine good bandwidth efficiency with maximizes some function of utility in a distributed an immunity to channel dispersion, these techniques fashion [3, 4]. The game settles at Nash equilibrium have received considerable attention. To able to if one exists. Since users act selfishly, the support multiple users, the multicarrier transmission equilibrium point is not necessarily the best technique can be combined with a CDMA scheme. operating point from a social point of view. To In tandem the demand for wireless services increases, circumvent this, pricing the system resources appears efficient use of resources has gained a significant to be a powerful tool for achieving a more socially importance. Ever increasing need for wireless desirable result [2,3]. In the MC-DS/CDMA, raising systems to provide high data transmission rates need one’s power not only increase their signal-to a system which performs well under severe fading interference-and–noise ratio (SINR), but also conditions. Though MIMO MC-DS /CDMA seem to increases the interference observed by other users, be an excellent candidate for high data rate thereby declining their SINR, each tend to increase communication, its performance is limited by their own power levels, thereby reaching the Nash multiple access interference (MAI) and near-far equilibrium. To overcome this situation a distributed effect. The power control algorithm plays a game theoretic power control algorithm to provide significant role in combating this effect. Compared efficient use of the radio resources in CDMA system with single antenna MC-DS /CDMA, MIMO MC- has been established [4,5]. The power control DS /CDMA exhibits better performance, but it has problem in multi-user MIMO CDMA system, using the traditional impairment as the single carrier game theory framework has been proposed in [2,6] is system [1,2]. Hence the performance of a MIMO considered in this work. A new utility functions for MC-DS /CDMA consequently lies in the area of the NPG by using singular value decomposition interference suppression and power control in multi (SVD) is proposed to solve the problem. The new user scenario. utility functions, which are based on MIMO MC-DS Recently, an alternative approach to the power /CDMA system for wireless data, refer to the control problem in wireless systems based on an spectral efficiency and power efficiency is considered. The utility functions also reflect to the convenience. Since each antenna can quality of service (QoS) that the data users get, accommodates sub carriers, the total throughput where utility is defined as the ratio of throughput to will be the summation of the throughput of transmit power. Then Nash equilibrium and the individual carrier. In order to solve the power performance of the power control games in a single control problem in the MIMO MC –DS/ CDMA cell MIMO MC- DS/ CDMA system is considered system, a marginal utility function which is which seems to be an ideal solution to use the system expressed in Eq (3) is established. resource more efficiently. The paper is organized as follows. Section 2 um = T / P i i i explains MIMO MC –DS/ CDMA system and the utility function of the power control game. Section 3 { min Mt ,Mr N 1 } ( ( )) L shows the two NMPCGs for the MIMO MC– log M 1 BER 2 k ,i k ,i DS/CDMA system. Section 4 discusses the existence k=1 S=1 and uniqueness of the games and the algorithm to = reach the Nash equilibrium. Simulation results are { min Mt ,Mr N 1}P ,i (3) k=1 k given and discussed in section 5. Finally, Section 6 S=1 draws the conclusion. The power control utility function is given in Eq (4) 2 MIMO MC –DS/CDMA SYSTEM AND UTILITY FUNCTIONS { min Mt ,Mr N 1 } ( ( )) L The uplink of a single cell N-users MIMO log Mk ,i 1 2BER 2 k ,i MC- DS/ CDMA system with feedback is k=1 S=1 u = considered for our analysis. Each user is assumed to have Mt transmit antennas and the base station is i { min Mt ,Mr N 1 } P ,i k=1 k equipped with Mt x Mr antennas. Each antenna is S=1 capable of transmitting 1x Mr Subchannel. { min Mt ,Mr N 1 } ( ) subcarriers and processing gain G are considered. In this system, the user's bit stream is demultiplexed log Mk ,i f 2 k ,i k=1 N=1 (4) among several transmitting antennas, each of which = transmits an independently modulated signal, { min Mt ,Mr N 1 P ,i } simultaneously and in the same frequency band. The k=1 k base station receives these signal components by an S=1 antenna array whose sensor outputs are processed such that the original data stream can be recovered. where, f( k,i) =(1-2BER(( k,i))L is called efficiency Assume that the channel state information (CSI) is function. The frame successive rate (FSR) is perfectly known to receiver, and the transmitter can approximated by, f( ,i), which closely follows the get the CSI through feedback. Assume H, which is behaviour of the probability of correct reception the channel matrix of user i can be decomposed while producing FSR equals zero at Pi =0. using SVD is given in Eq. (1). The pricing mechanism was introduced into { m in M t ,M r } the CDMA non-cooperative power control game H i =U i iV i = U i ( k ) i ( k )V i ( k ) (1) [4]. By using the pricing mechanism, the power k =1 control game was more efficient. A new utility where U i( k ) and Vi( k )are M r×I and function with pricing in MIMO MC- DS/ CDMA M t×I unitary vectors, respectively, power control game is developed. It is expressed in Eq. (5) and i ( k ) are the singular values of Hi. The per- user attainable normalized throughput, in bit per { min Mt ,Mr N 1 } second Hertz, of MIMO MC- DS /CDMA system is the sum of the normalized throughputs of the c k =1 S =1 log Mk ,i f 2 ( k ,i ) min (Mt, Mr) decoupled sub channels. Then the u = tP i Pi i normalized throughput of ith user is given in Eq (2). { min Mt ,Mr } { } min Mt ,Mr N 1 ( ( )) L T = Tk ,i = log Mk ,i 1 BER ,i (2) (5) i k=1 k=1 S=1 2 k P = k =1 { min Mt ,Mr N 1 P ,i } i k where k,i is to represent the SINR of ith user in kth S =1 where Pi is the total transmitting power of the ith sub channel, which is using sth sub carrier for user, and t is a positive scalar. This proposed utility no user may gain by unilaterally deviating Nash function, which gives attention to both spectral equilibrium. Hence, Nash equilibrium is a stable efficiency and power efficiency, are based on operating point because no user has any incentive MlMO MC- DS/ CDMA system. to change strategy [3]. The Nash equilibrium of . proposed NMCPGs are given in sec 4.1 and 4.2. 3. NON COOPERATIVE MIMO POWER CONTROL GAME 4.1. The NMCPG, GI, G2 are supermodular games with appropriate strategy space Ai = P i , Pi Let G = N ,{ Ai},{Ui (.)} denote the non respectively [8,9]. cooperative MlMO power control game (NMCPG) where N = {l, 2... N} is the index set for the mobile Consider the game G1 first. users currently in the cell. The ith user select a total transmit power strategy Pi, such that Pi Ai where Ai, denotes the strategy space of ith user. Let the in t uli 1 m M ,Mr N 1 { } f k,i ( ) vector P =( P1,........, PN ) denote the outcome of the = 2 log2Mk,i k,i f k,i ( ) ( ) game in terms of the selected power levels of all P P i i k=1 S=1 k,i ( ) (8) users, and P-i, denotes the vector consisting of elements of P other than the ith element. The strategy space of all the users excluding the ith user is denoted A-i. According to the analysis, two NMCPGs are established. These games have the 2u min Mt,Mr N=1 { } 2f ( k,i ) ( k,i ) li 1 k,i = 2 log2 Mk,i same player space and strategy space, but different utility functions P Pj P i i k=1 S=1 ( ) 2 k,i Pj (9) The game G1 is given by, f ( k ,i ) 2 2 If 0 , it can be concluded that u li for { } min Mt ,Mr N 1 ( 2 k ,i ) Pi P j k=1 S=1 log M f 2 k ,i k ,i ( ) all jKi. Assume there exists a P-i such that 0<P-i < Pi G1 = max U1i( P ,P i ) = P Ai i i Pi which is derived form 2f ( k ,i ) , it can guarantee ( ) 0 2 (6) k ,i 2u li 0 for all jKi.So, it can be concluded that with Pi P j The game G2 is given by, the strategy space Ai = P i , Pi where P-i is derived from 2f ( k ,i ) , the game G1 is supermodular. ( ) 0 { } min Mt ,Mr N 1 2 k ,i log2 Mk ,i f k=1 S=1 k ,i The following theorems, proven in [9, 10], G2 = max U2i( P ,P i ) = tPi P A i i i Pi (7) guarantees the existence and the uniqueness of a nash equilibrium of supermodular game, and give the algorithm that can converge to the equilibrium. for all i N 4.2. The set of Nash equilibrium of a supermodular In outdoor, macro cell with the typical game is nonempty. parameters of outdoor channel, the maximum The best response is singular value i ( k ) and U i ( k ) , V i ( k ) can BR ( P i ) ={ pi Ai :ui ( pi , P i ) ui ( p 'i , P i ) ! P ' Ai (10) successfully approximate Hi . In the NMCPGs, that each user is assumed rational and selfish. Assume that for all i=1, 2,…N, Ai are compact, Users always maximize their own utilities by convex, lower semi continuous in its argument, and selecting the best transmit power strategy, which hold scalability property. Further assume that for depends on the transmit power strategies of all the each i=1,2,…..N, BR(P-i)>0 for all Pj Aj, j " i. other users in the system. In the games, a set of Then the Nash equilibrium is unique and general powers can be found where the users are satisfied. updating algorithm converges monotonically to an equilibrium whose convergence holds for any initial 4. NASH EQUILIBRIUM policy in the strategy space. It can be concluded that each of our NMCPGs has unique Nash equilibrium Nash equilibrium is the most widely used point and then the asynchronous power control solution in NPG [4]. It is an action profile in which algorithm, we considered in this work, converges to a unique Nash equilibrium point. In this algorithm users update their transmission powers in the same for k =1 to K. manner as in [2].Assume user i updates its { transmission power at time instances in the set Ti ={ti1 ti2 …..}, with tik< tik+1 and ti0 for all i € N. Let for S =1 to N-1(IFFT size) T={t1,T2,…} where T=T1 # T2 # …… # TN with { tk<tk+1. The NMCPG generates a sequence of power eff. function of k th subchannel of vector following the iterative procedure as follows. user1 =(1-BER ( k,i))L/** L=frame size. } The power vector P(0)=P is set at time t=0. For all i N. Calculating ri ( tk ) = argmax p p ui ( pi, p t (tk t ) ) . Given end for i } that pi(tk)=min(ri(tk),pmax).If p(tk) equivalent to p(tk- 1),the iterative procedure ends and Nash equilibrium end for power vector is divided to be p(tk).If it is not the calculate throughput of user ‘i’at transmit_ case the iterative procedure is repeated the power ‘t’. predetermined number of times until p(tk)=p(tk-1). calculate utility of user ‘i’ without pricing at transmit_ power ‘t’. 4.3. Proposed game theoretic power control if utility1(t)=utility max ‘t’. algorithm for MC- DS/CDMA { Assuming ‘N’ users in a single cell, the SINR power for ith user power(i)= t. is estimated for all the ‘N’ users participating in power for ith user utility(i) = utility(t). the game. Suppose if a particular user increases the } power level beyond the required threshold, then end if access to that particular user will be denied so as to keep the interference level well within control. } This procedure is followed for all the users end for whoever tend to increase the power level thereby Power_subchannel =Power(1/K) . contributing to the MAI.This scheme is called power_ iteration =Power(iteration-1). pricing whereby allowing all the users. Simulation results have shown that by employing this pricing } scheme, the overall utility of a particular user end while. achieves significant performance amelioration, by Results: Power without pricing (power), Utility mitigating the MAI. without pricing (Utility) Iterative algorithm 1: Iterative algorithm 2: Initiliation () Initiliation () Distance d; Mr -Transmitting antenna;Mt- Distance d; Mr -Transmitting antenna; Mt - Receiving antenna; S-IFFT size; Receiving antenna; S-IFFT size; Generate Channel Matrix H; Generate Channel Matrix H; iteration iteration while(Power " Power iteration) /**Initially while(Power " Power iteration) /**Initially Power Power iteration is a random matrix. iteration is a random matrix. iteration =iteration+1. iteration =iteration+1. for k =1 to K. /**k=min (Mr Mt) for k =1 to K. /**k=min(Mr Mt) { { for txt power(t)= Min_power to Max_power for txt power(t)= Min_power to Max_power { { Power_subchannel =Power/K /**k=min (Mr Mt) Power_subchannel =Power/K /**k=min(Mr Mt) for k =1 to K. for k =1 to K. { { Calculate SNR of Kth subchannel of user i( k,i ). Calculate SNR of Kth subchannel of user i( k,i ). } } end for end for scatter components and is a zero-mean unit- variance complex Gaussian random variable [11]. for k =1 to K. The following parameters are considered for { simulation. for s=1 to N-1(IFFT size) eff. function of k th subchannel of Table 1 Simulation Parameter user1 =(1-BER ( k,i))L/** L=frame size. Parameters value } Distance in meter 260,330,450, end for (d) 560,660,800, } 900, 950, 1000 end for Block size(L) 80 bits calculate throughput of user ‘i’at transmit_ power Maximum total 2watts for each transmit power user ‘t’. constraint Pi calculate utility of user ‘i’ without pricing at Path loss exponent T 3.6 transmit_ power ‘t’. Median of the mean 0.097 if utility1(t)=utility max ‘t’. path gain c { AWGN power at 5 × 10-5(watts) power for ith user power(i) = t. receiver U2 Spread gain G 100 power for ith user utility(i) = utility(t). Users 9 } IFFT size 512 end if } end for Power_subchannel =Power(1/K) . power_ iteration =Power(iteration-1). } end while. Results: Power with pricing (power), Utility with pricing (Utility) 5. NUMERICAL RESULTS Consider a single cell wireless data MIMO MC– DS/CDMA system with stationary multi-user, fixed frame size, no forward error correction, with Mt=Mr=2 and Mt=Mr=4 The channel matrix of the MlMO system is given by Fig.1.Performance of MIMO MC-DS/CDMA with and without Pricing Distance vs. equilibrium power H i= (i ) hm n ;1 m M r ,1 n M t (11) Fig.1 and Fig.2 elucidates the equilibrium utility where hmn is the complex signal path gain from for a function of distance between a user and the transmitter n to receiver m. This gain is modeled base station. It is discerned that, as the number of by antennas increases, the equilibrium utility for a (i) particular user at some distance away from the base h m n = c / d i$ s Z m n (12) station decreases. By introducing the concept of where di, is the base-mobile distance in pricing, the equilibrium utility increases. Thus the kilometer of ith user is the path loss exponent, c is equilibrium utility without pricing, for a user with the median of the mean path gain at a reference two antennas is lesser than that with pricing. With distance d = 1 km, s is a log-normal shadow fading pricing, a user with four antennas has higher utility, variable, where 10log(s) is a zero-mean Gaussian when compared to the user with two antennas. Thus random variable with standard deviation % and users with four antennas have more utility than that Zmn represents the phasor sum of the multi path of a user with two antennas. the overall system to keep the interference level as low as possible to achieve better overall performance. Here, in MIMO the number of transmitting and receiving antenna is assumed to be two and four. For a particular user at some distance away from the base station, the equilibrium power increases. By introducing the concept of pricing, the equilibrium power decreases. Thus the equilibrium power with pricing, for four antennas is lower than that without pricing and it is also comparatively lower with the equilibrium power of the user with two antennas. Comparison of Distance Vs equilibrium utilities performance and Distance Vs equilibrium power performance of MC-DS/CDMA system is given in table 1 and table 2. Fig.2.Performance of MIMO MC-DS/CDMA with and without Pricing Distance vs. equilibrium power Fig.4. Performance of MIMO MC-DS/CDMA with and without Pricing Distance vs. equilibrium Utility Table 1 Comparison of Distance Vs equilibrium utilities performance of MC-DS/CDMA system. Number of Distance Without With antenna in meter pricing pricing (bits/s/ (bits/s/ Fig.3.Performance of MIMO MC-DS/CDMA Hz/W) Hz/W) with and without Pricing Distance vs. 260 105.2 106 equilibrium Utility TX=2 and RX=2 330 105 105.7 Fig.3 and Fig.4 elucidates the performance of MIMO 450 104.2 105 MC-DS/CDMA with and without pricing to the overall utilization. It can be discerned that utilization 260 106.2 107.4 of power with pricing is less compared to the scheme TX=4 and without pricing. This represents the performance Rx=4 330 106 107 bound for power allocation in a MC-DS/CDMA 450 105.4 106.4 system which is mainly controlled by the pricing scheme with the aid of Nash equilibrium. Thus the performance bound derived can be generally used for Iterations: 1,000 Table 2 Comparison of Distance Vs equilibrium [2] Wei zhong“Distributed game theoretic power power performance of MC-DS/CDMA system. control for wireless data over MIMO CDMA system” IEEE Trans.Commun., vol. 50, pp: Number Distance Without With 237-241, Feb. 2005. of in meter pricing pricing [3] A.B.Mackenzie,S. E. Wicker, “Game Theory antenna (power (power in Communications.Motivation, Explanation, in in and Application to Power Control”, in watts) watts) Proc.IEEE GLOBECOM, pp.25-29, Nov. 2001. TX=2 260 10 -6 10 -7 [4] C. Saraydar, N. B. Mandayam, and D. J. and Goodman, “Efficient power control via pricing RX=2 330 10 -5.8 10 -6.8 in wireless data networks”, IEEE Trans.Commun., vol. 50, pp: 291-303, Feb. 450 10 -5.2 10 -6.2 2002. TX=4 260 10 -6 10 -7 [5] D. Goodman and N. 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