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CHAPTER 2
GAME THEORY BASED CHAOTIC CODE SPREADING FOR
MIMO MC-DS/CDMA SYSTEM
2.1 INTRODUCTION
The enormous growth of wireless services during the last decade gives
rise to the need for a bandwidth efficient modulation technique that can transmit
high data rates in a reliable way. Due to the wide bandwidth requirement for the
wireless communication system the combination of MC-DS/CDMA with chaotic
code spreading has recently attracted a lot of interest in wireless communication and
provides an efficient approach to reduce the chip rate and the spreading code length.
The utility improvement is achieved when more users are active, since
the performance degradation due to the MAI becomes more obvious with large
system capacity. The ability of the receiver to detect the desired signal in the
presence of interference relies to a great extent on the correlation properties of the
spreading codes (sequences). As the number of interferers or their power increases,
MAI becomes substantial and can seriously degrade the BER performance of the
system, as a whole. This work aims at employing the NPGP with chaotic sequences
such that the MAI is effectively reduced in a MC-DS/CDMA environment
comparing with Walsh spreading sequences. Further the system performance for
MIMO MC-DS/CDMA is studied with chaotic spreading sequences.
2.2 NON COOPERATIVE POWER CONTROL GAME
Game theory is an appropriate tool in the setting of communications networks and deals primarily with distributed optimization [107-112]. Game theory
typically assumes that all players seek to maximize their utility functions in a
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manner which is perfectly rational. A game has three components such as, a set of players, a set of possible actions for each player and a set of utility functions mapping action profiles into the real numbers. The most important equilibrium
concept in game theory is the concept of Nash Equilibrium. Nash equilibrium is an action profile at which no user may gain by unilaterally deviating. So Nash equilibrium is a stable operating point because no user has any incentive to change strategy.
In CDMA systems where each mobile interacts with others by affecting the signal to interference ratio (SIR) through interference, game theory provides a natural framework for analyzing and developing power control mechanisms. For a mobile in such a network, obtaining individual information on the power level of
each of the other users is practically impossible due to the excessive communication and processing overhead required. Therefore, in a distributed power control setting, each user attempts to minimize its own cost (or maximize its utility) in response to the aggregate information on the actions of the other users. This makes the use of non-cooperative game theory for uplink power control most appropriate. Shah et al [93] proposed a mechanism by which each terminal adjusts its transmit power
in a single-cell data network, in order to maximize its individual satisfaction of the use of network resources in a distributed fashion. The interaction between self-optimizing terminals is called a non-cooperative power control game. At the outcome of the power control game, it has shown that there exists an equilibrium vector of transmit powers where no terminal can improve its utility individually any further. However, through a mechanism called pricing, improvement in the utilities for all terminals is noted.
For data communication, information is sent in the form of packets. It is
assumed that all errors in the received signal can be detected and that the incorrect
data has to be retransmitted. The achieved throughput T can then be expressed by,
T R f where R is the information transmit rate and f is a measure of the
efficiency of the transmission protocol. The efficiency of the protocol should depend
on the SIR achieved over the channel, and its value varies from zero to
one . . 0,1i e f . The efficiency of the protocol is better, when the SIR is higher.
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Further, it is assumed that the user i has a battery with energy content E Joules and
transmit power is Pi. The utility is defined as ,U Pi i i , which a user derives from the
network as the total number of bits that can be transmitted correctly in the lifetime of
the battery. Formally, the utility function is defined as
,i i i
i
EU P R f
P
(2.1)
The unit of the above utility function is bits.
The utility function of user i, for single carrier transmission is expressed by
( )i
i ii
fLsU R
D P
(2.2)
where
L is the number of information bits.
D is number of bits in a packet
Ri is the transmission rate for user i
Pi is the transmit power of user i
f(γi) is the efficiency function for the transmission of user i
( ) 1 2M
fi
Pi (2.3)
where M is the number of sub-carriers
Similarly, the utility function of user i for the MC-DS/CDMA is defined
as the ratio of the total throughput over the total transmits power among all sub-
carriers:
1
( )
,
M CM
m
ii i
i m
fLU R
D P
(2.4)
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The Nash equilibrium is widely involved in game theoretic problems. In
NPG, for i, PiN is Nash equilibrium if and only if
ˆ ˆ (P ,P ) (P ,P ), for all P and 1, 2.., M N M
i i i i i iU U i Ii (2.5)
where 1 ,2 ,P { , , ......, },i i i i MP P P represents any other transmit power vectors different
from PNi , P̂ i contains the transmit power vectors of all users except user i and
maximum number of user I, ( )
, max[0, ]m
i mP P , and ( )
maxm
P is the maximum transmit
power assigned to the mth sub-carrier. That is, the Nash equilibrium is a constrained
maximum solution.
Then, the unconstrained maximum solution of transmit power ,ˆi mP and
related SNR, k are evaluated, and then the maximum power limitation is applied.
As for an unconstrained maximum solution,,
ˆ(P ,P )0
Mi i i
i m
UP
, for any i and m. If
,
,
i mi
i mP
then,
1
' 12,
, ''
2' '( ) ( )ˆ , ' , '(P ,P )
=
MM i imi i i
ii m M
Pi mm
f P fi m i m iU LR
P D
i,m'
i ii
P' 1
' 'L f (γ)γ -f (γ )= R 2D M
m
(2.6)
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That is i satisfies, ' '( ) ( )f fi i i and the corresponding
1
2,
, 22, '
'
i
Mi
i mPi m
i mm
(2.7)
The Nash equilibrium is ( ), max, min ,N m
i mi mP P P . If ( )
, max ,,mi m i mP P P
belongs to the power strategy space and , ,N
i m i mP P , since ,i mP is global maximum
for the utility function. When ( )
, maxm
i mP P , the efficiency function is given by
22 / 2( ) 1M
i
itf e dt
(2.8)
122 2'
( ) ( ) /1M
i
i i i kf f te dt
(2.9)
12 / 22 / 2 221
'( ) ( )
Mi
itMi i i ie dtf f e
(2.10)
From equations (2.8) and (2.9),
2 222 2 / 21/ i
i tM ie e dt
(2.11)
0k for i < i%, which is equivalent to 0i and the resulting
,
ˆ(P ,P )0
Mi i i
i m
UP
when , ,P Pi m i m , the utility function monotonically increasing in
,Pi m for max, 0, m
i mP P , and ( )
, maxN mP Pi m to maximize the utility function value.
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Although the Nash equilibrium provides a self-optimizing power control
solution for individual user, it is not necessarily the best operating point for the
whole system [95]. Therefore, there exist the other power solutions to make the
utilities of all the users greater than those at the Nash equilibrium. In order to make
efficient use of system resource, non-cooperative power control game with pricing is
introduced to force each user to efficiently share rather than aggressively
approaches the system resource. Utility (and pricing) based network control
algorithms have extensively been studied in the literature [90, 96]. These are not
new concepts and have been studied in economics. The utility represents the degree
of a user’s satisfaction when it acquires certain amount of the resource and the price
is the cost per unit resource which the user must pay for this resource. The basic idea
of these algorithms is to control a user’s behavior through the price of resources to
obtain the desired results, e.g., high utilization for the overall system and fairness
among users.
2.3 MC–DS/CDMA SYSTEM MODEL
The transmitter model of the MC-DS/CDMA system for the ith user is
shown in Figure 2.1. At the transmitter side, the binary data stream bi(t) is first
divided into m parallel branches and each branch signal is multiplied by a
corresponding chip value of the spreading sequence
C i= [Ci[1], Ci[2], Ci[3] .............. Ci[M]]T . Following this each of the M branch
signals modulates a sub-carrier frequency using binary phase shift keying (BPSK).
Then, the M numbers of sub-carrier modulated sub-streams are added in order to
form the transmitted signal. Hence, the transmitted signal of user i can be expressed
by
2, 1, 2,3....m
P MiS (t)= b (t)c [m]cos( t) i Ii i iM m = 1 (2.12)
where Pi is transmit power of each user
ωmt is the sub-carrier frequency set
bi(t) is binary data stream ci(m) is spreading sequence
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The binary data stream’s bi(t) waveform consists of a sequence of
mutually independent rectangular pulses of duration Tb and of amplitude +1 or -1
with equal probability. The spreading code ci(m) denotes the spreading sequence of
the user. Denoting the spreading factor as N where N=Tb/TC. The sub-carrier signals
are assumed to be orthogonal and the spectral main-lobes of the sub-carrier signals
are not overlapping with each other. The received signal may be expressed by
)I M2Pir(t)= b (t)c [m]g ×cos( t + )+n(ti i m,i m m,iMi=1 m=1
(2.13)
where n(t) represents the AWGN having zero mean and double-sided power spectral
density of N0/2. MC-DS/CDMA signal is identified with the aid of the spreading
sequences as shown in Figure 2.2. Low pass filter(LPF) is type of filter which allow
frequency below cutoff frequency.
Figure 2.1 Transmitter model of MC-DS/CDMA
[1]ic
[2]ic
[ ]ic M
1co s ( )t
2cos( )t
cos( )mt
( )is t
( )ib t
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Figure 2.2 Receiver model of MC-DS/CDMA
2.3.1 Chaotic Code Spreading
Chaos based communication systems qualify as broadband systems in
which the natural spectrum of the information signal is spread over a very large
bandwidth. This class of systems, is called spread spectrum communication systems,
since they make use of a much higher bandwidth than that of the data bandwidth to
transmit the information. Nowadays, pseudo-noise sequences such as Gold
sequences and Walsh Hadamard sequences are by far the most popular spreading
sequences and have good correlation properties, limited security and it can be
reconstructed by linear regression attack for their short linear complexity.
A chaotic sequence generator can visit an infinite number of states in a
deterministic manner and therefore produce a sequence which never repeats itself.
The designer gets the flexibility in choosing the spreading gain as the sequences can
LPF
LPF
LPF
1 ,12 cos( )it
2 ,22 cos( )it
,2 cos( )M i Mt
( )r t( )iZ t
,1ic
,2ic
,i Mc
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be truncated to any length. Many authors have shown [52, 55, 56] that chaotic
spreading sequences can be used as an inexpensive alternative to the linear feedback
shift register (LFSR). However, the search for the best set of codes contributing
reduced MAI is still one of the severe requirements of future MC-DS/CDMA
systems. Generation of good set of sequences demands for large set dimension,
period and limited privacy. To overcome these limitations, new chaotic spreading
codes, is being used in this work. Instead of using other spreading codes in
MC-DS/CDMA, this chaotic code has produced good result in utility and reducing
MAI. A single system described by its discrete chaotic map can generate a very
large number of distinct chaotic sequences, each sequence being uniquely specified
by its initial value. This dependency on the initial state and the non-linear
characteristic of the discrete map make the MC-DS/CDMA system highly secure.
A chaotic map is a dynamic discrete-time continuous-value equation that
describes the relation between the present and next value of chaotic system.
Let Xn+1 and Xn be successive iterations of the output X and M is the forward
transformation mapping function. The general form of multidimensional chaotic
map is Xn+1 = M (Xn, Xn-1… Xn-m).
A simple logistic map is given in equation
Xn+1 =μ Xn (1-Xn) , 0 < Xn <1 and 1 ≤μ ≤4 (2.14)
where μ is the bifurcation parameter and the system exhibits a great variety of
dynamics depending on the value of μ, (3.6 ≤μ ≤ 6). Using logistic map the chaotic
spreading sequences for the MC-DS/CDMA/BPSK system is generated. After
assigning different initial condition to each user, the chaotic map is started with the
initial condition of the intended receiver and iterated repeatedly to generate multiple
codes. It is assumed that the transmitter and the intended receiver have agreed upon
a starting point, x00 and two chaotic maps, C1(x, r1) and C2(x, r2) with their
corresponding bifurcation parameters, r1 and r2. The chaotic maps and their
bifurcation parameters may or may not be the same and their uniqueness among the
different pairs of transmitters and receivers is no necessary.
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Figure 2.3 Generation of chaotic sequences
In Figure 2.3, x00 initiates a chaotic sequences X0={xn0: n = 0, 1, 2..N}
through the chaotic map C1(x, r1). The elements of this sequence are then used to
generate the sequences Sn = {xni: i = 0, 1, 2..N.}, n = 0, 1, 2... through the chaotic
map C2(x, r2). The sequences Sn, so obtained are the spreading sequences to be used
for each data bit. Note that the spreading sequence changes from one bit to another.
The receiver regenerates the sequence Sn is exactly the same manner as the
transmitter does. Every receiver will be assigned distinct x00, C1(x, r1), C2(x, r2), r1
and/or r2, and therefore, the resulting spreading sequences for each receiver in a
multiple-access communication system will be completely independent and
uncorrelated.
2.3.2 Pricing Strategy to Increase Entire System Utility
In the NPG, each terminal aims to maximize its own utility by adjusting
its own power, but it ignores the cost (or harm) it imposes on other terminals by the
interference it generates. The self-optimizing behaviour of an individual terminal is
said to create an externality when it degrades the quality for every other terminal in
the system. Among the many ways to deal with externalities, pricing (or taxation)
( , )1 1 1X C x rn n
2( , )1 1X C x rn n
0n 1n
( mod ) 0i N
( mod ) 0i N
0 0x
1 /ch ip b it
nix
/Nchips bit
0nx
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has been used as an effective tool both by economists and researchers in the field of
computer networks. Pricing is motivated by different objectives, such as generate
revenue for the system and encourage players to use system resources more
efficiently. It does not refer to monetary incentives, rather refers to a control signal
to motivate users to adopt a social behavior. An efficient pricing mechanism makes
decentralized decisions compatible with overall system efficiency by encouraging
efficient sharing of resources rather than the aggressive competition of the purely
non cooperative game. A pricing policy is called incentive compatible if pricing
enforces a Nash equilibrium that improves social welfare. A social welfare is
defined as the sum of utilities.
In order to improve the equilibrium utilities of NPG in the Pareto sense,
the usage-based pricing schemes has been introduced in [93, 95, 96]. Through
pricing, system performance can be increased by implicitly inducing cooperation
and the non-cooperative nature of the resulting power control solution had been
maintained. An efficient pricing scheme should be tailored for the problem at hand.
Within the context of a resource allocation problem for a wireless system, the
resource being shared is the radio environment and the resource usage is determined
by terminal’s transmit power. Although the Nash equilibrium provides a self-
optimizing power control solution for individual user, it is not necessarily the best
operating point for the whole system. Consequently, the other power solutions to
make the utilities of all the users greater than those at the Nash equilibrium is
established [113-115]. In order to make efficient use of system resource, NPGP is
introduced to force each user to efficiently share rather than aggressively
approaches the system resource.
The utility function of user i for the MC-DS/CDMA is defined as the
ratio of the total throughput over the total transmit power among all sub-carrier is
given by
( )
1 ,
fLMC iU Ri i MD Pm i m
(2.15)
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The utility function by means of pricing is re-organized by
( ) ( ) ( ) ( )MP Mu t u t g t p ti i i i i i i i (2.16)
where
uiMP is utility function with pricing
uiM is traditional utility function
gi is set of updated instances for all the users
P i λ is transmit power
In NPGP each terminal maximizes its net utility given by the difference
between the utility function and pricing function. The class of pricing functions
studied is linear in transmit power, where the pricing function is simply the product of
a pricing factor and the transmit power. Such a pricing function allows easy
implementation of the power control algorithm and is realized by the base station
announcing the pricing factor to all the users. This in turn is used for choosing the
transmit power from its strategy space that maximizes its net utility
2.3.3 Simulation Results
For comparison of chaotic and Walsh spreading codes, it is assumed that
the number of information bits per frame L =64, while the total number of bits per
frame D = 80; the transmission rate for each user Ri is assumed to be 105 bits/s; the
total transmit power is limited by Pmax = 6 Watts and Pmin = 0.1 Watts.
The total utility for fifty users is compared in Figure 2.4 for Walsh
Hadamard and chaotic codes. From this figure, it is observed that, by employing
Walsh code of length 64 bits yielded around 9×106 bits/joule, utility whereas chaotic
code yielded 10×106 bits/joule. This shows ten percent ballpark amelioration in the
total utility has been achieved by the use of chaotic code. NPGP sets up certain
cooperation among terminals via the pricing strategy, and each terminal tries to
increase its own utility and reduces the interference to other users as well. Although
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Nash equilibriums are the best operating points to increase the traditional utility
function Ui, each player in NPG cares about its own utility, and may generate
significant interference to other users. Therefore, considerable improvement is
achieved by NPGP with chaotic sequence when many users are active, circumventing
MAI.
Figure 2.4 Total utility of MC-DS/CDMA with number of users
The total utility for chaotic and Walsh codes is compared in
Figure 2.5 by varying the noise power from 10-6 Watts to 10-2 Watts and observed
that, when noise power is around 10-3 Watts, the utility start to drastically decrease
due to increase in noise power. But there is a remarkable improvement in utility
factor by using chaotic code when the noise power levels increases. Hence chaotic
code outperforms by 9% when compared with Walsh Hadmard code in term of
utility factor. Thus, the chaotic code is an excellent candidate for mitigating MAI,
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which in turn contribute for the amelioration of the system capacity. Furthermore,
when the noise power at the receiver side increases, the SNR(γi) and the efficiency
function f(γi) decreases. Thus, fewer utility is obtained with increase in σ2N. When
the noise power is limited to 10-4 Watts, the utility factor is appreciable. On the other
hand, when noise power increases, the efficiency function decreases significantly.
When σ2N>10-3 Watts, total utility drops drastically for conventional NPGP, more
than that of proposed NPGP with chaotic sequences. This is due to the fact that
the terminals in the modified NPGP carefully manage the transmit power to
increase individual utility and combat MAI as well, since modified NPGP can
tolerate higher noise power than conventional NPGP.
Figure 2.5 Total utility of MC-DS/CDMA with noise power
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2.4 MIMO MC-DS/CDMA SYSTEM MODEL
Figures 2.6 and 2.7 illustrate the transmitter and receiver model of the
MIMO MC-DS/CDMA system of the ith user. At the transmitter, the binary data
stream bi(t) is divided into number of parallel branches, where each branch-signal is
multiplied by a corresponding chip value spreading sequence Ci= [Ci[1], Ci[2],
Ci[3] ….. Ci[M]]T and each of the branch signals modulates a sub-carrier frequency
using binary phase shift keying (BPSK). Then, the numbers of sub-carrier modulated
sub streams are added in order to form the transmitted signal. Hence, the transmitted
MIMO MC-DS/CDMA signal of S(t) for user i can be expressed
,min
1
2, 1, 2,3....
t rM M
i
P MiS (t)= b (t)c [m]cos( t) i Ii i i mM m = 1
(2.17)
Mt and Mr are the number of transmit and receive antennas respectively. The binary
data stream bi(t) consists of sequence of mutually independent rectangular pulses of
duration Tb and amplitude +1 or -1 with equal probability. The spreading sequence,
ci(m) denotes the spreading waveform of the ith user with spreading factor N=Tb/TC,
represents the number of chips per bit-duration. It is assumed that the sub-carrier
signals are orthogonal and their spectral main lobes are not overlapping with each
other. The received signal can be expressed by
,min
1
)t rM M
i
I M2Pir(t)= b (t)c [m]cos( t+ )+n(ti i m m,iMi=1 m=1
(2.18)
where n(t) represents the AWGN having zero mean and double-sided power spectral
density of N0/2. MIMO MC-DS/CDMA signal is identified with the aid of the
spreading sequences as shown in Figure 2.7.
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Figure 2.6 Transmitter model of MIMO MC-DS/CDMA
Figure 2.7 Receiver model of MIMO MC-DS/CDMA
[1]ic
[2]ic
[ ]ic M
1co s( )t
2cos( )t
cos( )mt
( )is t
( )ib t
1( )is t
( )mi
s t
LPF
LPF
LPF
1 ,12cos( )it
2 ,22cos( )it
,2cos( )M i Mt
( )r t( )iZ t
,1ic
,2ic
,i Mc
( )is t
1( )is t
( )mi
s t
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2.4.1 Pricing Strategy to Increase Entire System Utility
The utility function of ith user for a MIMO MC-DS/CDMA is given as
,m in
1
( )
1 ,
t rM M
i
fLM C iU Ri i MD Pm i m
(2.19)
The utility function of equation (2.19) is reformulated as
,min
1( ) ( ) ( ) ( )
t rM M
i
MP Mu t u t g t p ti i i i i i i i
(2.20)
In a MIMO MC-DS/CDMA system for each terminal that employs NPGP,
the utility is maximized and given by the difference between utility function. Also the
pricing function considered here has a linear transmit power, similar to that of
MC-DS/CDMA system. Moreover it allows easy implementation of NPGP and is
realized by the base station by announcing the pricing function to all user, whereby
each terminal can choose its transmit power from its strategy space.
2.4.2 Simulation Results
The total utility for fifty users is compared in Figure 2.8 for Walsh
Hadamard and chaotic codes. From this figure, it is discerned that, employing Walsh
code of length 64 bits yielded around 8.2×107 bits/joule , utility where as chaotic
code yielded 9.8×107 bits/joule. This shows sixteen percent improvement in the total
utility has been achieved by the use of chaotic sequence. NPGP sets up certain
cooperation among terminals via the pricing strategy, and each terminal tries to
increase its own utility and reduces the interference to other users as well. Although
Nash equilibriums are the best operating points to increase the traditional utility
function Ui each player in NPG cares about their own utility, and may generate
significant interference to other users. Therefore, considerable improvement is
achieved by NPGP with chaotic sequence when many users are active, mitigating
MAI.
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Figure 2.8 Total utility of MIMO MC-DS/CDMA with number
of users
The total utility for chaotic and Walsh codes are compared with different users in Figures 2.9 and 2.10 by varying the noise power from 10-6 Watts to 10-2 Watts and observed that, when noise power is around 10-3 Watts, the utility start to decrease drastically due to increase in noise power. But there is a remarkable improvement in utility factor by using chaotic code when the noise power level increases. It is noted that chaotic code outperforms by 9% compared to Walsh Hadmard code in terms of utility factor and is good in mitigating MAI, which in turn enhance the system capacity. Furthermore, when the noise power at the receiver side increases, the SNR and efficiency function decreases. Thus, less utility is obtained with increase in σ2
N. When the noise power is limited to 10-4 Watts, the utility factor is appreciable. On the other hand, when noise power increases, the efficiency function decreases significantly. When σ2
N>10-3 Watts, the total utility of conventional NPGP is less compared to that of proposed NPGP scheme. This is due to the fact that the terminals in the modified NPGP carefully manage the transmit power to increase individual utility and combat MAI as well, since modified NPGP can tolerate higher noise power than conventional NPGP.
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Figure 2.9 Total utility of MIMO MC-DS/CDMA with noise power for 12 users
Figure 2.10 Total utility of MIMO MC-DS/CDMA with noise power for 30 users
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2.5 SUMMARY
The proposed NPGP with chaotic spreading codes performs effectively
by reducing the MAI in MC-DS/CDMA system. The application of chaotic codes
performs better in the system than the classical codes in terms of exterminating
MAI. Care has been taken to drastically minimize the interference factor in
MC-DS/CDMA systems through the chaotic codes spreading. To further enhance
the capacity of MC-DS/CDMA system, in this work MIMO MC-DS/CDMA has
been considered for the analysis. A NPG algorithm based on pricing performs
effectively by reducing the MAI in MIMO MC-DS/CDMA system. The multi user
scenario with large number of users, is discerned with the help of numerical results
that by initializing chaotic spreading code, the utility performance is improved
compared to traditional NPGP. The simulation result shows that the proposed
chaotic code spreading approach achieves a significant improvement in the system
utility of about twenty percent by effectively combating MAI.