26

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

27

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.

28

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)

29

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)

30

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.

31

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

32

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

33

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

34

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.

35

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

36

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)

37

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

38

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,

39

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

40

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.

41

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

42

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.

43

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.

44

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

45

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.