A Game theoretic Power Control algorithm with Pricing for Spectral Efficient Communication in MIMO MC-DS/ CDMA System V.Nagarajan and P.Dananjayan† Department of Electronics and Communication Engineering, Pondicherry Engineering College, Pondicherry -605014, India [email protected], [email protected] †Corresponding author A distributed non cooperative power control game with pricing (NPGP) for multiple-input multiple-output (MIMO) multi-carrier direct sequence code division multiple access (MC -DS/ CDMA) system for different modulation is considered in this work. The utility functions for assaying the performance of MIMO MC-DS /CDMA for spectral efficient communication of the system carrying wireless data are envisaged. The spectral efficiency and power efficiency are referred as utility which divulges the level of satisfaction a user may get. According to the utility functions, two NPGP’s are propounded, which determines the existence and individuality of the Nash equilibria. A novel power control algorithm that delves the performance of the anticipated power control games to achieve the Nash equilibria is presented in this paper. The simulated results elucidate that a significant amelioration in terms of utilities specifically spectral efficiency for all users can be achieved using this approach. Also the propounded scheme exhibits better performance in the MIMO MC-DS /CDMA in terms of spectral efficiency as compared to the traditional system. Index Terms—MC-DS CDMA, MIMO, Pricing

I. 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 technique have received considerable attention. In tandem the demand for wireless services increases, efficient use of resources has gained a significant importance. Also there is need for obtaining very high data rate which is the prime aim of future communication. Hence there has always been a need for spectral efficient communication. The elemental component of radio resource management is transmitter power control. It is well know that mitigating interference using power control algorithm increases the capacity and also extends battery life. Ever increasing need for wireless systems to provide high data transmission rates needs a system which performs well under severe fading conditions. To meet these demands MIMO MCDS /CDMA is an excellent candidate. In spite of the fact that, the performance of MC-DS/CDMA is limited by multiple access interference (MAI) in addition to the near-far effect, power control algorithm plays a vital role in combating these effects. Though MIMO MC-DS/CDMA system has much better performance compared with single antenna MC-DS /CDMA, it still comes into the CDMA traditional impairments [1]. The challenge to augment the performance of a MIMO MC-DS /CDMA consequently lies in the techniques of interference suppression and power control for a MIMO multi user system. Recently, an alternative approach to the power control problem in wireless systems based on an economic model has been offered. In this model, service preferences for each user

are represented by a utility function. As the name implies, the utility function quantifies the level of satisfaction a user gets from using the system resources. Game theoretic methods are applied to study power control under this new model [2]. Game theory 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]. 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. 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 level not only increases their signal-to interference-and–noise ratio (SINR), but also increases the interference observed by other users. This drives the SINR of other users to decrease which in turn will force other users to increase their own power levels to reach 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 MC-DS-CDMA system, using game theory framework for various modulation techniques has been proposed in this work. A new utility functions for the NPG (non cooperative power control game) 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, power efficiency. Then Nash equilibria and the performance of the power control games in a single cell MIMO MC- DS/ CDMA system is considered. In this paper the power control for MlMO MC- DS /CDMA systems is considered for various modulation techniques. The game theory approach is implemented to use the system resource more efficiently

The paper is organized as follows. Section II explains MIMO MC–DS/CDMA system and the utility function of the power control game. Section III shows the two NMPCGs (non cooperative MIMO power control game with pricing) for the MIMO MC–DS/ CDMA system. Section IV discusses the existence and uniqueness of the games and the algorithm to reach the Nash equilibrium with proposed game theoretic power control algorithm for MIMO MC-DS/CDMA system for different modulation. Simulation results are given and discussed in section V. Finally, Section VI draws the conclusion. II. MIMO MC FUNCTIONS:

–DS/

CDMA SYSTEM

AND

.

H i =U i

where

iV i =

U i( k )

{

k =1

Vi( k )

and

}

U i ( k ) i ( k )V i ( k )

are

M r×I

and

(1) M t×I

unitary

( k )

are the singular values of Hi. vectors, respectively, and i The per-user 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

{

}

min Mt ,Mr N 1 L Tk ,i = log Mk ,i 1 BER ,i k 2 k=1 S=1

( ))

(

th

um = T / P i i i

{

=

th

k,i is to represent the SINR of i user in k sub channel, which

is using sub carrier for 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/

(

{

( ))

}

(3)

min Mt ,Mr N 1 Pk ,i k=1 S=1

The power control utility function is given in Eq (4)

{

}

min Mt ,Mr N 1 L log Mk ,i 1 2 BER k ,i 2 k=1 S=1

u = i

(

{

( ))

}

min Mt ,Mr N 1 Pk ,i k=1 S=1

{

=

where,

}

min Mt ,Mr N 1 log Mk ,i f k ,i 2 k=1 N=1

( )

{

(4)

}

min Mt ,Mr N 1 Pk ,i k=1 S=1 f

k ,i

=

(1 2 BER ( 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. 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 M t , M r N 1 u

c i

k =1

=

S =1

log

Pi

{

(2) where

}

min Mt ,Mr N 1 L log M 1 BER 2 k ,i k ,i k=1 S=1

UTILITY

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 capable of transmitting 1x Mr subchannel. 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). m in M t ,M r

CDMA system, a marginal utility function which is expressed in Eq (3) is established.

}

min M t , M r N 1 Pi = k =1 Pk ,i S =1

2

M k ,i f

( k ,i ) tP i

(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.

III. NON COOPERATIVE MIMO POWER CONTROL GAME Let G = N ,{ Ai},{Ui (.)}

denote the non cooperative

A. The NMCPG, GI, G2 are supermodular games with appropriate strategy space Ai = P i , Pi

MlMO power control game (NMCPG) where N = {l, 2... N} is the index set for the mobile users currently in the cell. The ith

Consider the game G1 first.

user select a total transmit power strategy Pi , such that

P i A i 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. All of these games have the same player space and strategy space, but different utility functions The game G1 is given by, min{M t , M r } N G1 =

max Pi Ai

S =1

k =1

U1 i ( Pi , P i ) =

log

2

M

k ,i

f

=

1 Pi 2

{

k =1

2u 1 li = 2 Pi P j Pi

(

(

}

min M t , M r

k ,i

)

2

)

2f

( k ,i )

N 1 S =1

{

log 2 M k , i

min M t , M r k =1

}

N =1 S =1

( k ,i )

f

(

log 2 M k , i

( k ,i ) f ( k ,i ) k ,i )

( k ,i )

2f

(

(6)

}

Pi

form

(

can

be

2 k ,i

Ai = P i , Pi tPi

(7) for all i

( k ,i ) 2

k ,i

k ,i Pj

)

(9)

2u li

0

)

0,

it can guarantee

concluded

that

2u li Pi Pj

with

the 2f

min Mt ,Mr N 1 log2 Mk,i f k ,i k=1 S=1

(8)

, it can be concluded that P P for all jLi. k ,i i j Assume there exists a P-i such that 0
The game G2 is given by,

G2 = max U2i( Pi ,P i ) = Pi Ai

Pi

If

( k ,i )

Pi

{

uli

2f

1

respectively [6, 7].

where P-i is derived from

(

0 for

all jLi.So, it

strategy

( k,i ) 2 k,i

)

space

0 , the game

G1 is supermodular. The following theorems, proven in [8, 9], guarantees the existerice and the uniqueness of a nash equilibrium of supermodular game, and give the algorithm that can converge to the equilibrium.

N

In outdoor, macro cell with the typical parameters of outdoor channel, the maximum singular (k )

can successfully value i ( k ) and U i ( k ) , V i Hi . In the NMCPGs, that each user is assumed approximate 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. IV. NASH EQUILIBRIUM Nash equilibrium is the most widely used solution in NPG [4]. It is an action profile in which 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. Nash equilibrium is the most widely used solution in NPG [4]. It is an action profile in which 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 A and B.

B.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; Mt-Receiving antenna; S-IFFT size; Initialize m: m=1 for BPSK,2 for QPSK and 4 for QAM and 8 for 8-PSK: Generate Channel Matrix H; iteration

while(Power ! Power iteration) /**Initially 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.

Power

matrix

of

{ }

( i) H = hmn i

the ; 1

MlMO m

M ,1 r

system n

M

is t

given

by

(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 lognormal shadow fading variable, where 10-logs is a zero-mean Gaussian random variable with standard deviation # and zmn represents he phasor sum of the multi path scatter components and is a zero-mean unit-variance complex Gaussian random variable [6]. The parameters considered for simulation is given in Table .1 Table. 1 Input parameters consider for simulation Parameters value

Distance in meter 260,330,450,560,660,800, 900, } (d) 950, 1000 end for Block size(L) 80 bits } Maximum total 2watts for each user end for transmit power constraint Pi calculate throughput of user ‘i’at transmit_ power‘t’.calculate utility of user.‘i’(a). without pricing t transmit_ power t’. and Path loss exponent 3.6 " ( b). with pricing at transmit_ power ‘t’. Median of the 0.097 if utility1(t)=utility max ‘t’. mean path gain c { AWGN power at 5 × 10-5 power for ith user power(i)= t. 2 receiver $ power for ith user utility(i) = utility(t). Spread gain G 100 } Users 9 end if Modulation BPSK,QPSK,QAM,8PSK } Technique end for Power_subchannel =Power(1/K) . Fig. 1 elucidates the equilibrium utility as a function of distance for a multi-user scenario under Rayleigh conditions. Also the power_ iteration =Power(iteration-1). modulation techniques taken into considerations are BPSK and } QPSK .Channel model were built on the classical understanding end while. by taking into consideration delay spread, Doppler spread and a. Results: Power without pricing (power), Utility without fading. To obtain a reasonably statistical estimate Monte Carlo pricing (Utility) simulation were carried out. Investigation revealed that the b. Results: Power with pricing (power), Utility with pricing proposed scheme achieved a significant performance (Utility) improvement compared to the traditional system which does not employ game theory without pricing. Also the implication here IV. SIMULATION RESULTS is that by employing pricing scheme in MIMO MC-/DS CDMA For Consider a single cell wireless data MIMO MC– system a considerable spectral efficiency could be achieved. 6.2 6.4 , 10 instance the traditional scheme exhibits a utility of 10 DS/CDMA system with stationary multi-user, fixed frame size, no forward error correction, with Mt=Mr=4 .The channel for BPSK and QPSK whereas for the same modulation techniques the propounded scheme gives a utility of 107.2,107.4 for BPSK and QPSK

Fig.3.Performance of MIMO MC-DS/CDMA with and without Pricing (Mt=Mr=4) Fig.1.Performance of MIMO MC-DS/CDMA with and without Pricing (Mt=Mr=4)

Fig.3. analysis reveals that the propounded scheme achieves a spectral efficient communication at a reasonably less power compared to the scheme without pricing. The modulation techniques that are considered are BPSK and QPSK. Simulation results show that the proposed scheme utilizes a power of 10-7 for 9 users, whereas the traditional scheme requires a power of 10-6. This clearly shows that about 10% less power is required to carry same amount of data in this scheme.

Fig.2.Performance of MIMO MC- DS/CDMA with and without Pricing (Mt=Mr=4) Fig. 2 shows the equilibrium utility as a function of distance for a multi-user scenario under Rayleigh conditions. Also the modulation techniques taken into considerations are QAM and 8-psk. Investigation revealed that the proposed scheme achieved a significant performance improvement compared to the traditional system which does not employ game theory with without pricing. Also the implication here is that by employing pricing scheme in MIMO MC CDMA system a considerable spectral efficiency could be achieved. For instance the traditional scheme exhibits a utility of 1011.2,1018 for QAM and 8-PSK whereas for the same modulation techniques the propounded scheme gives a utility of 1012,1019 for QAM and 8-PSK.

Fig.4.Performance of MIMO MC-DS/CDMA with out Pricing (Mt=Mr=4)

and

Fig.4 shows that the propounded scheme achieves a spectral efficient communication at a reasonably less power compared to the scheme without pricing. The modulation techniques that are considered are QAM and 8PSK. Simulation results show that .The proposed scheme utilizes a power of 10-7 for 9 users, whereas the traditional scheme requires a power of 10-6. This clearly shows that about 10 % less power is required to carry same amount of data in this scheme .

VI. CONCLUSION In this paper a novel power control algorithm employing game theory approach for spectral efficient communication is considered for a MIMO MC-DS/CDMA with a pricing scheme. This scheme is introduced to effectively control the power in the uplink. The analysis is carried out for various modulation techniques. Simulation results show that the utility in terms of equilibrium power is much less in this approach compare to the conventional 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 and this scheme establishes spectral efficient communication whereby by carrying more no of bits per symbol duration at a lesser power. Also the pricing scheme proves to be an effective method in achieving a better performance in MIMO MCDS/CDMA system by extenuating 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 on.Commun., vol. 50, pp: 237241, 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,”Supermodularity 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. Commrm.,vol.50, pp.225234, Feb. 2002. [10] H.Ji and C.-Y. Huang, “Non-cooperative uplink power control in cellular radio systems,” wireless Networks, vo1.7, pp.861-874, Dec. 1998.

V.Nagarajan received the Bachelors Degree in Electronics and Communication Engineering from Madras University in 1999. He completed his Masters degree in Communication system from Pondicherry University in 2002. He is pursuing research in the area of wireless communication. He has published two international journals and presented 10 paperes in international and national conferences in the same area. His areas of interest include signal processing and mobile communication.

P. Dananjayan received Bachelor of Science from University of Madras in 1979, Bachelor of Technology in 1982 and Master of Engineering in 1984 from the Madras Institute of Technology, Chennai and Ph.D. degree from Anna University, Chennai in 1998. He is working as a Professor and Head of the Department of Electronics and Communication Engineering, Pondicherry Engineering College, Pondicherry, India. He has more than 60 publications in National and International Journals. He has presented more than 130 papers in National and International conferences. He has produced 6 Ph.D candidates and is currently guiding eight Ph.D students. His areas of interest include power electronics, Spread spectrum Techniques and Wireless Communication.