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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 The enormous growth of wireless services during the last decade gives rise to the need for a bandwidth efficient modulation technique that can reliably transmit high data rates. As multi carrier technique combine good bandwidth efficiency with an immunity to channel dispersion, these techniques have received considerable attention. To able to support multiple users, the multicarrier transmission technique can be combined with a CDMA scheme. In tandem the demand for wireless services increases, efficient use of resources has gained a significant importance. Ever increasing need for wireless systems to provide high data transmission rates need a system which performs well under severe fading conditions. Though MIMO MC-DS /CDMA seem to be an excellent candidate for high data rate communication, its performance is limited by multiple access interference (MAI) and near-far effect. The power control algorithm plays a significant role in combating this effect. Compared with single antenna MC-DS /CDMA, MIMO MCDS /CDMA exhibits better performance, but it has the traditional impairment as the single carrier system [1,2]. Hence the performance of a MIMO MC-DS /CDMA consequently lies in the area of interference suppression and power control in multi user scenario. Recently, an alternative approach to the power control problem in wireless systems based on an economic model has been proposed [3]. In [3] game theoretic approach is employed to study the power control in the multi user scenario for the proposed model. It is a powerful tool in modeling interactions between self-interested users and predicting their choice of strategies. Each player in the game maximizes some function of utility in a distributed fashion [3, 4]. The game settles at Nash equilibrium if one exists. Since users act selfishly, the equilibrium point is not necessarily the best operating point from a social point of view. To circumvent this, pricing the system resources appears to be a powerful tool for achieving a more socially desirable result [2,3]. In the MC-DS/CDMA, raising one’s power not only increase their signal-to interference-and–noise ratio (SINR), but also increases the interference observed by other users, thereby declining their SINR, each tend to increase their own power levels, thereby reaching the Nash equilibrium. To overcome this situation a distributed game theoretic power control algorithm to provide efficient use of the radio resources in CDMA system has been established [4,5]. The power control problem in multi-user MIMO CDMA system, using game theory framework has been proposed in [2,6] is considered in this work. A new utility functions for the NPG by using singular value decomposition (SVD) is proposed to solve the problem. The new utility functions, which are based on MIMO MC-DS /CDMA system for wireless data, refer to the spectral efficiency and power efficiency is considered. The utility functions also reflect to the quality of service (QoS) that the data users get, where utility is defined as the ratio of throughput to transmit power. Then Nash equilibrium and the performance of the power control games in a single cell MIMO MC- DS/ CDMA system is considered which seems to be an ideal solution to use the system resource more efficiently. The paper is organized as follows. Section 2 explains MIMO MC –DS/ CDMA system and the utility function of the power control game. Section 3 shows the two NMPCGs for the MIMO MC– DS/CDMA system. Section 4 discusses the existence and uniqueness of the games and the algorithm to reach the Nash equilibrium. Simulation results are given and discussed in section 5. Finally, Section 6 draws the conclusion. 2 MIMO MC –DS/CDMA SYSTEM AND UTILITY FUNCTIONS The uplink of a single cell N-users MIMO MC- DS/ CDMA system with feedback is considered for our analysis. Each user is assumed to have Mt transmit antennas and the base station is equipped with Mt x Mr antennas. Each antenna is Subchannel. capable of transmitting 1x Mr subcarriers and processing gain G are considered. In this system, the user's bit stream is demultiplexed among several transmitting antennas, each of which transmits an independently modulated signal, simultaneously and in the same frequency band. The base station receives these signal components by an antenna array whose sensor outputs are processed such that the original data stream can be recovered. Assume that the channel state information (CSI) is perfectly known to receiver, and the transmitter can get the CSI through feedback. Assume H, which is the channel matrix of user i can be decomposed using SVD is given in Eq. (1). H i =U i iV i = m in M t ,M r k =1 convenience. Since each antenna can accommodates sub carriers, the total throughput will be the summation of the throughput of individual carrier. In order to solve the power control problem in the MIMO MC –DS/ CDMA system, a marginal utility function which is expressed in Eq (3) is established. um = T / P i i i min Mt ,Mr N 1 L log M 1 BER 2 k ,i k ,i k=1 S=1 { } ( ( )) = min Mt ,Mr N 1 P ,i k k=1 S=1 { } (3) The power control utility function is given in Eq (4) min Mt ,Mr N 1 L log Mk ,i 1 2BER k ,i 2 k=1 S=1 { } ( ( )) u = i min Mt ,Mr N 1 P ,i k k=1 S=1 min Mt ,Mr N 1 log Mk ,i f k ,i 2 k=1 N=1 min Mt ,Mr N 1 P ,i k k=1 S=1 { } { } ( ) = (4) { } where, f( k,i) =(1-2BER(( k,i))L is called efficiency function. The frame successive rate (FSR) is approximated by, f( ,i), which closely follows the behaviour of the probability of correct reception while producing FSR equals zero at Pi =0. The pricing mechanism was introduced into the CDMA non-cooperative power control game [4]. By using the pricing mechanism, the power control game was more efficient. A new utility function with pricing in MIMO MC- DS/ CDMA power control game is developed. It is expressed in Eq. (5) min Mt ,Mr N 1 u c i = k =1 S =1 { } U i ( k ) i ( k )V i ( k ) M r×I (1) where M t×I U i( k ) and unitary Vi( k ) are vectors, and respectively, and i ( k ) are the singular values of Hi. The peruser attainable normalized throughput, in bit per second Hertz, of MIMO MC- DS /CDMA system is the sum of the normalized throughputs of the min (Mt, Mr) decoupled sub channels. Then the normalized throughput of ith user is given in Eq (2). min Mt ,Mr T = i k=1 { } log Mk ,i f 2 ( k ,i ) tP i P i { } min Mt ,Mr N 1 L log Mk ,i 1 BER ,i Tk ,i = 2 k k=1 S=1 { } ( ( )) (2) where k,i is to represent the SINR of ith user in kth sub channel, which is using sth sub carrier for min Mt ,Mr N 1 P = k =1 P ,i i k S =1 { } (5) where Pi is the total transmitting power of the ith user, and t is a positive scalar. This proposed utility function, which gives attention to both spectral efficiency and power efficiency, are based on MlMO MC- DS/ CDMA system. . 3. NON COOPERATIVE MIMO POWER CONTROL GAME Let G = N ,{ Ai},{Ui (.)} no user may gain by unilaterally deviating Nash equilibrium. Hence, Nash equilibrium is a stable operating point because no user has any incentive to change strategy [3]. The Nash equilibrium of proposed NMCPGs are given in sec 4.1 and 4.2. 4.1. The NMCPG, GI, G2 are supermodular games with appropriate strategy space Ai = P i , Pi denote the non respectively [8,9]. Consider the game G1 first. cooperative MlMO power control game (NMCPG) where N = {l, 2... N} is the index set for the mobile 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 vector P =( P1,........, PN ) denote the outcome of the game in terms of the selected power levels of all 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 same player space and strategy space, but different utility functions The game G1 is given by, min Mt ,Mr N 1 log M f 2 k ,i k ,i k=1 S=1 P i in t f k,i uli 1 m M ,Mr N 1 = 2 f k,i log2Mk,i k,i P P i k= S= 1 1 i k,i { } ( ) ( ) ( ) ( ) 2f (8) 2u min Mt,Mr N=1 1 li = 2 log2 Mk,i P Pj P i k=1 S=1 i { } ( k,i ) ( ) 2 k,i ( k,i ) k,i Pj (9) 2 { } G1 = max U1i( P ,P i ) = i P Ai i ( ) (6) If f ( k ,i ) u li for 0 , it can be concluded that Pi P j ( 2 k ,i ) all jKi. Assume there exists a P-i such that 0<P-i < Pi 2 which is derived form 2u li Pi P j 0 2f ( ( k ,i k ,i 2 ) ) 0 , it can guarantee for all jKi.So, it can be concluded that with The game G2 is given by, min Mt ,Mr N 1 log2 Mk ,i f k=1 S=1 P i the strategy space Ai = P i , Pi where P-i is derived { } from k ,i tPi 2f ( ( k ,i ) 2 k ,i ) 0 , the game G1 is supermodular. G2 = max U2i( P ,P i ) = i P A i i (7) The following theorems, proven in [9, 10], guarantees the existence and the uniqueness of a nash equilibrium of supermodular game, and give the algorithm that can converge to the equilibrium. 4.2. The set of Nash equilibrium of a supermodular game is nonempty. The best response is BR ( P i ) ={ pi Ai :ui ( pi , P i ) ui ( p 'i , P i ) ! P ' Ai (10) for all i N In outdoor, macro cell with the typical parameters of outdoor channel, the maximum singular value i ( k ) and U i ( k ) , V i ( k ) can successfully approximate Hi . In the NMCPGs, that each user is assumed rational and selfish. Users always maximize their own utilities by selecting the best transmit power strategy, which depends on the transmit power strategies of all the other users in the system. In the games, a set of powers can be found where the users are satisfied. 4. NASH EQUILIBRIUM Nash equilibrium is the most widely used solution in NPG [4]. It is an action profile in which Assume that for all i=1, 2,…N, Ai are compact, convex, lower semi continuous in its argument, and hold scalability property. Further assume that for each i=1,2,…..N, BR(P-i)>0 for all Pj Aj, j " i. Then the Nash equilibrium is unique and general updating algorithm converges monotonically to an equilibrium whose convergence holds for any initial policy in the strategy space. It can be concluded that each of our NMCPGs has unique Nash equilibrium point and then the asynchronous power control algorithm, we considered in this work, converges to a unique Nash equilibrium point. In this algorithm users update their transmission powers in the same 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 T={t1,T2,…} where T=T1 # T2 # …… # TN with tk<tk+1. The NMCPG generates a sequence of power vector following the iterative procedure as follows. 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 i for k =1 to K. { for S =1 to N-1(IFFT size) { eff. function of k th subchannel of user1 =(1-BER ( k,i))L/** L=frame size. } 1) that pi(tk)=min(ri(tk),pmax).If p(tk) equivalent to p(tk,the iterative procedure ends and Nash equilibrium power vector is divided to be p(tk).If it is not the case the iterative procedure is repeated the predetermined number of times until p(tk)=p(tk-1). 4.3. Proposed game theoretic power control algorithm for MC- DS/CDMA Assuming ‘N’ users in a single cell, the SINR is estimated for all the ‘N’ users participating in the game. Suppose if a particular user increases the power level beyond the required threshold, then 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 whoever tend to increase the power level thereby contributing to the MAI.This scheme is called pricing whereby allowing all the users. Simulation results have shown that by employing this pricing scheme, the overall utility of a particular user achieves significant performance amelioration, by mitigating the MAI. Iterative algorithm 1: Initiliation () Distance d; Mr -Transmitting antenna;MtReceiving antenna; S-IFFT size; Generate Channel Matrix H; iteration while(Power " Power iteration) /**Initially Power iteration is a random matrix. iteration =iteration+1. for k =1 to K. /**k=min (Mr Mt) { for txt power(t)= Min_power to Max_power { Power_subchannel =Power/K /**k=min (Mr Mt) for k =1 to K. { Calculate SNR of Kth subchannel of user i( k,i ). } end for end for } end for calculate throughput of user ‘i’at transmit_ power ‘t’. calculate utility of user ‘i’ without pricing at transmit_ power ‘t’. if utility1(t)=utility max ‘t’. { power for ith user power(i)= t. power for ith user utility(i) = utility(t). } end if } end for Power_subchannel =Power(1/K) . power_ iteration =Power(iteration-1). } end while. Results: Power without pricing (power), Utility without pricing (Utility) Iterative algorithm 2: Initiliation () Distance d; Mr -Transmitting antenna; Mt Receiving antenna; S-IFFT size; Generate Channel Matrix H; iteration while(Power " Power iteration) /**Initially Power iteration is a random matrix. iteration =iteration+1. for k =1 to K. /**k=min(Mr Mt) { for txt power(t)= Min_power to Max_power { Power_subchannel =Power/K /**k=min(Mr Mt) for k =1 to K. { Calculate SNR of Kth subchannel of user i( k,i ). } end for for k =1 to K. { for s=1 to N-1(IFFT size) eff. function of k th subchannel of user1 =(1-BER ( k,i))L/** L=frame size. } end for } end for calculate throughput of user ‘i’at transmit_ power ‘t’. calculate utility of user ‘i’ without pricing at transmit_ power ‘t’. if utility1(t)=utility max ‘t’. { power for ith user power(i) = t. power for ith user utility(i) = utility(t). } 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 H i= scatter components and is a zero-mean unitvariance complex Gaussian random variable [11]. The following parameters are considered for simulation. Table 1 Simulation Parameter Parameters Distance in meter (d) Block size(L) Maximum total transmit power constraint Pi value 260,330,450, 560,660,800, 900, 950, 1000 80 bits 2watts for each user Path loss exponent T Median of the mean path gain c AWGN power at receiver U2 Spread gain G Users IFFT size 3.6 0.097 5 × 10-5(watts) 100 9 512 (i ) hm n Fig.1.Performance of MIMO MC-DS/CDMA with and without Pricing Distance vs. equilibrium power Fig.1 and Fig.2 elucidates the equilibrium utility for a function of distance between a user and the base station. It is discerned that, as the number of antennas increases, the equilibrium utility for a particular user at some distance away from the base station decreases. By introducing the concept of pricing, the equilibrium utility increases. Thus the equilibrium utility without pricing, for a user with two antennas is lesser than that with pricing. With pricing, a user with four antennas has higher utility, when compared to the user with two antennas. Thus users with four antennas have more utility than that of a user with two antennas. ;1 m M r ,1 n M t (11) where hmn is the complex signal path gain from transmitter n to receiver m. This gain is modeled by h m n (i) = c / d i$ s Z m n (12) where di, is the base-mobile distance in kilometer of ith user is the path loss exponent, c is the median of the mean path gain at a reference distance d = 1 km, s is a log-normal shadow fading variable, where 10log(s) is a zero-mean Gaussian random variable with standard deviation % and Zmn represents the phasor sum of the multi path 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 antenna Fig.3.Performance of MIMO MC-DS/CDMA with and without Pricing Distance vs. equilibrium Utility Fig.3 and Fig.4 elucidates the performance of MIMO MC-DS/CDMA with and without pricing to the overall utilization. It can be discerned that utilization of power with pricing is less compared to the scheme without pricing. This represents the performance bound for power allocation in a MC-DS/CDMA 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 Distance in meter 260 TX=2 and RX=2 330 450 260 TX=4 and Rx=4 330 450 Without pricing (bits/s/ Hz/W) 105.2 105 104.2 106.2 106 105.4 With pricing (bits/s/ Hz/W) 106 105.7 105 107.4 107 106.4 Iterations: 1,000 Table 2 Comparison of Distance Vs equilibrium power performance of MC-DS/CDMA system. Number of antenna TX=2 and RX=2 TX=4 And Rx=4 Distance in meter Without pricing (power in watts) 10 -6 10 -5.8 10 -5.2 10 -6 10 -5.8 10 -5.2 With pricing (power in watts) 10 -7 10 -6.8 10 -6.2 10 -7 10 -6.8 10 -6.2 260 330 450 260 330 450 Iterations: 1,000 6. Conclusion In this paper a power control algorithm employing game theory approach is considered for a MIMO MC-DS/CDMA with a pricing scheme. The pricing scheme is introduced to effectively control the power in the uplink. Simulation results show that the utility in terms of equilibrium power is much less in this approach compare to the traditional system. In accession to the equilibrium power, equilibrium utilities in terms of number of bits/s/Hz/W is considered for assaying the performance of the propounded scheme with that of the traditional system. It is discerned that the proposed scheme achieves a 10% increase in equilibrium utilities at a lesser power utilization. Also the pricing scheme proves to be an effective method in achieving a better performance in MIMO MC-DS/CDMA system by mitigating the multiple access interference. REFERENCES [1] Chun-Hung Liu, “Low-complexity Performance Optimization for MIMO CDMA Systems”, IEEE publications WNCN, Mar. 2005. [2] Wei zhong“Distributed game theoretic power control for wireless data over MIMO CDMA system” IEEE Trans.Commun., vol. 50, pp: 237-241, Feb. 2005. [3] A.B.Mackenzie,S. E. Wicker, “Game Theory in Communications.Motivation, Explanation, and Application to Power Control”, in Proc.IEEE GLOBECOM, pp.25-29, Nov. 2001. [4] C. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efficient power control via pricing in wireless data networks”, IEEE Trans.Commun., vol. 50, pp: 291-303, Feb. 2002. [5] D. Goodman and N. Mandayam, “Power control for wireless data”.lEEE Personal Commun Mag vol. 7, pp. 454, Apr.2000. [6] S. Catreux, P. F. Driessen, and L. J. Greenstein, “Data throughputs using multiple-input multiple-output (MIMO) techniques in a noise limited environment,” IEEE Trans. Wireless Comm,vol. I, pp.226-234, Apr. 2002. [7] E. Altman, 2. Altman, “S-Modular Games and Power Control in Wireless Networks”, IEEE Trans. Automat. Contr. vo1.48, pp. 839-842, May. 2003. [8] D. M. Topkis, Supermodlarity and Complementarity. Princeton, NJ: Princeton Univ. Press, 1998. [9] H. Boleskei, D.Gesbert, A. J.Paulraj “On the Capacity of OFDM Based Spatial Multiplexing Systems,” IEEE Trans. Communication.,vol.50, pp.225-234, Feb. 2002. [10]H.Ji and C.-Y. Huang, “Non-cooperative uplink power control in cellular radio systems,” wireless Networks, vo1.7, pp.861874, Dec. 1998. [11] W.Yu, W. Rhee, S.Boyd, and I. M. Cioffi, “Iterative Water-filling for Gaussian Vector Mu1tiple Access Channels,” IEEE Trans. Information.Theory, vol. 50, pp. 145--152, Jan. 2004.

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