Learning Center
Plans & pricing Sign in
Sign Out

ubicc008 268


UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.

More Info
									                      A GAME THEORETIC POWER CONTROL
                       APPROACH FOR MIMO MC-DS/CDMA
                                    V.Nagarajan and P.Dananjayan †
                         Department of Electronics and Communication Engineering,
                        Pondicherry Engineering College, Pondicherry -605014, India
             , †
                                            † Corresponding author

               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        }
                                                                                                                                                    ( ( ))
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
                                                                                                                              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
                                                                                                       min Mt ,Mr N 1     }
                                                                                                                                                (          ( ))
         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
                                                            (      ( ))
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.
     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
The game G1 is given by,
                                                                                      f ( k ,i )
                                                                                      2                                                                                     2
                                                                               If                0 , it can be concluded that
                                                                                                                                u li
                              {       }
                            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
                                            Pi                                 which is derived form
                                                                                                                                 (    k ,i      )         , it can guarantee
                                                                                                                            (               )
                                                                       (6)                                                           k ,i
                                                                                                  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

                                                                                                  ( k ,i )           , the game G1 is supermodular.
                                                                                          (                  )
                               {       }
                            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
                                            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
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;
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
                                                                       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)


        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
 TX=4             260         10 -6         10 -7        [5] D. Goodman and N. Mandayam, “Power
 And                                                         control for wireless data”.lEEE Personal
                  330         10 -5.8      10 -6.8
 Rx=4                                                        Commun Mag vol. 7, pp. 454, Apr.2000.
                  450         10 -5.2      10 -6.2       [6] S. Catreux, P. F. Driessen, and L. J. Greenstein,
                                                             “Data throughputs using multiple-input
                           Iterations: 1,000                 multiple-output (MIMO) techniques in a noise
                                                             limited environment,” IEEE Trans. Wireless
6. Conclusion                                                Comm,vol. I, pp.226-234, Apr. 2002.
                                                         [7] E. Altman, 2. Altman, “S-Modular Games and
      In this paper a power control algorithm                Power Control in Wireless Networks”, IEEE
employing game theory approach is considered for a           Trans. Automat. Contr. vo1.48, pp. 839-842,
MIMO MC-DS/CDMA with a pricing scheme. The                   May. 2003.
pricing scheme is introduced to effectively control      [8] D.    M.     Topkis,     Supermodlarity      and
the power in the uplink. Simulation results show that        Complementarity. Princeton, NJ: Princeton
the utility in terms of equilibrium power is much less       Univ. Press, 1998.
in this approach compare to the traditional system. In   [9] H. Boleskei, D.Gesbert, A. J.Paulraj “On the
accession to the equilibrium power, equilibrium              Capacity      of   OFDM        Based     Spatial
utilities in terms of number of bits/s/Hz/W is               Multiplexing      Systems,”     IEEE      Trans.
considered for assaying the performance of the               Communication.,vol.50, pp.225-234, Feb.
propounded scheme with that of the traditional               2002.
system. It is discerned that the proposed scheme         [10]H.Ji and C.-Y. Huang, “Non-cooperative
achieves a 10% increase in equilibrium utilities at a        uplink power control in cellular radio
lesser power utilization. Also the pricing scheme            systems,” wireless Networks, vo1.7, pp.861-
proves to be an effective method in achieving a              874, Dec. 1998.
better performance in MIMO MC-DS/CDMA                    [11] W.Yu, W. Rhee, S.Boyd, and I. M. Cioffi,
system by mitigating the multiple access interference.       “Iterative Water-filling for Gaussian Vector
                                                             Mu1tiple Access Channels,” IEEE Trans.
REFERENCES                                                   Information.Theory, vol. 50, pp. 145--152, Jan.
[1] Chun-Hung       Liu,       “Low-complexity
    Performance Optimization for MIMO CDMA
    Systems”, IEEE publications WNCN, Mar.

To top