Robust synchronization for asynchronous multi-user chaos-based DS-CDMA
Georges Kaddoum a,b,?,Daniel Roviras a ,Pascal Charge
′b ,Danie `le Fournier-Prunaret b a IRIT Laboratory,University of Toulouse,2rue charles camichel,31071Toulouse Cedex,France
b
LATTIS Laboratory,University of Toulouse,135Avenue de Rangueil,31077Toulouse Cedex 4,France
a r t i c l e i n f o
Article history:
Received 18July 2008Received in revised form 10October 2008
Accepted 22October 2008
Available online 31October 2008Keywords:
Chaos-based DS-CDMA Asynchronous multi-user Synchronization Code acquisition
Probability of detection Probability of false alarm
a b s t r a c t
In this paper we propose two systems for achieving synchronization in asynchronous multi-user chaos-based DS-CDMA.For the ?rst system,synchronization process is realized thanks to a binary code used as an additive pilot sequence to the spreaded signal.Gold sequences are used as pilot signals for the different users to accomplish the synchronization.For the second synchronization system,the synchronization is made through a binary code used as a multiplicative pilot signal for the spreaded data sequence.These synchronization processes are evaluated under the assumption of an additive white Gaussian noise channel together with multi-user interferences.In this paper we will focus on the initial synchronization phase (code acquisition)and we assume that the system can achieve correctly the code tracking after this ?rst synchronization phase.The code acquisition for the two systems is evaluated in terms of the probability of detection and probability of false alarm.
&2008Elsevier B.V.All rights reserved.
1.Introduction
Within the past decade,several research efforts have addressed the use of chaotic signals in digital communica-tions.Chaotic signals can offer very attractive properties such as the security of transmission and low probability of interception [1].In addition,the improvement of the system performance when chaotic sequences are applied instead of conventional binary codes for spreading spec-trum motivates the developments of chaos-based DS-CDMA transmission techniques [1,2].The major problem of chaos based DS-CDMA systems remains the synchroni-zation of the received chaotic signal with the local chaotic signal generated in the receiver despread information.The intensive work of Pecora and Carroll in synchronization ?eld [3]has opened the way for chaotic transmission
systems implementation [4–7].For classical DS-CDMA using binary pseudo-noise (PN)codes,an other synchroni-zation method has been proposed in the literature [8–11].The synchronization problem is solved via a two-step approach:An acquisition search is ?rst activated in order to align the local sequence to the received sequence within an uncertainty of a half time chip duration [8].The time uncertainty,which is basically determined by the transmis-sion time of the transmitter and the propagation delay,can be much longer than a chip duration.As initial acquisition is usually achieved by a search through all possible phases (delays)of the sequence,a larger timing uncertainty means a larger search area.Moreover,in many cases,initial code acquisition must be accomplished in low signal-to-noise-ratio (SNR)environments and in presence of jammers.The acquisition procedure is possible when the spreading sequence exhibits some kind of periodicity.Given the initial acquisition,code tracking takes place and is usually accomplished by a delay lock loop (DLL).The tracking loop keeps on operating during whole communication period.If the channel changes abruptly,the DLL lose track of the
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0165-1684/$-see front matter &2008Elsevier B.V.All rights reserved.doi:10.1016/j.sigpro.2008.10.023
Corresponding author at:IRIT Laboratory,University of Toulouse,2
rue charles camichel,31071Toulouse Cedex,France.
E-mail address:georges.kaddoum@enseeiht.fr (G.Kaddoum).Signal Processing 89(2009)807–818
correct timing and initial acquisition will be re-performed [8].Sometimes,we perform initial code acquisition periodically no matter whether the tracking loop loses track or not.
This classical synchronization technique has been applied for chaos-based DS-CDMA systems in[12–22].In [13,14]the authors have studied the performance of the acquisition process when a Markov chaotic sequence is used as a spreading code for DS-CDMA.They have shown in[13]that the Markov code outperforms the independent and identically distributed code in acquisition and bit error rate(BER)frameworks.The authors have shown in [12]that the Bernoulli and tailed shift map give a better performance in the acquisition phase than Gold se-quences.Noise perturbations have not been included in [12,16,18,22]in order to study the effects of the multi-user interferences on the acquisition process.However, the presence of noise is inevitable for any real commu-nication system and we have included it in the study of the system performance.In most cases,when evaluating the sequence synchronization of chaos-based DS-CDMA systems only code acquisition is analyzed[12–16,18–22].
Jovic et al.presented in[23]a new method for achieving and maintaining synchronization for synchro-nous multi-user chaos-based DS-CDMA.This method called code aided synchronization(CAS)is proposed and evaluated in the presence of additive white Gaussian noise (AWGN)and multi-user interferences.Code acquisition and tracking phases are studied and analyzed in[23].The synchronization system proposed in[23]uses a single pseudo-random binary sequence as pilot signal to achieve and maintain the synchronization.The use of a binary periodic pilot signal(PPS)for chaos-based DS-CDMA synchronization system in[23]is to show that robust synchronization of a chaos-based DS-CDMA system is possible.They also show in[23]that in terms of code acquisition,the binary pilot signal outperforms the logistic and Bernoulli chaotic maps.The authors have proposed in their paper a multi-user chaos based DS-CDMA system with a synchronization unit to achieve and maintain?ne synchronization of chaotic sequences.
In many papers,chaos-based communication systems are mainly studied for showing the attractive properties of chaotic sequences in the spreading spectrum framework [24–27].A lower attention is put on the implementation techniques.In order to implement practical chaos-based DS-CDMA systems,it is necessary to develop robust synchronization techniques which are able to work in low SNR environment.In our paper we have focused our attention on such robust synchronization.
In our paper we are interested in the synchronization system proposed in[23].We proposed here two synchro-nization systems.The?rst system is the extension of the synchronous CAS method of[23]to an asynchronous multi-user case.In this system,a PN signal will be used for synchronization purpose,like in[23],as an additive periodic pilot sequence.This synchronization procedure is called asynchronous CAS with additive pilot sequence (ACAS-A).
In the second system,the PN code is used also for the synchronization purpose but instead of being an additive sequence as in ACAS-A and in[23],we have used it
as a multiplicative one.This synchronization procedure is called asynchronous CAS with multiplicative pilot sequence(ACAS-M).This second approach outper-forms the ACAS-A in terms of synchronization and BER performances.
In our paper we have focused on the?rst synchroni-zation phase(acquisition)of the chaotic sequence. The mathematical model of the code tracking loop is presented for the chaos-based DS-CDMA system in[23].
The paper is organized as follows.In Section2we have ?rst presented chaos-based DS-CDMA system with the synchronization unit.First of all,the initial synchroniza-tion is presented and analyzed in terms of probability of detection(P d)and probability of false alarm(P fa).Simula-tion results together with some conclusive remarks are then given.In Section3the chaotic communication system with the ACAS-M unit is presented.Then,the synchronization performance is shown in Section 3.3, simulation results and comparisons with the ACAS-A system are provided.The?nal section reports some conclusive remarks.
2.Chaos-based DS-CDMA system with additive pilot signals(ACAS-A)
2.1.Chaotic generator
Throughout the paper,a Chebyshev polynomial func-tion of order2is chosen as chaotic generator:
x kt1?1à2x2
k
(1) The choice of this map is related to its simplicity for generating chaotic sequences.Moreover,it is shown in [23,28,29]that it allows better performances than many other maps for chaos-based DS-CDMA systems.Chaotic sequences are normalized such that their mean values are all zero and their mean squared values are unity,i.e.,
E?x k ?0and E?x2
k
?1.
2.2.Transmitter structure
The studied system is a DS-CDMA communication system with M asynchronous users.This system is the extension of the system studied in[23].The system of[23] is used only for the chaotic sequence synchronization in synchronous mode,but our system can be used for asynchronous mode.As shown in Fig.1(a),a stream
of data symbols from user m(semT
i
)with period T s are spreaded by a chaotic signal xemTetTgenerated from Eq.(1) at the emitter side.Symbols of different users are independent of one another.Chaotic sequences of all users are generated using the same chaotic generator with different initial conditions.A new chaotic sample (or chip)is generated every time interval equal to T c
(xemT
k
?xemTekT cT).
As shown in Fig.1(a),system a Gold code pemTetTis added to each user,and is used as the PPS with period equal to T.Gold codes have been chosen for PPS because of their good properties concerning cross-correlation in
G.Kaddoum et al./Signal Processing89(2009)807–818 808
asynchronous systems.They also have a single auto-correlation peak at zero,just like ordinary PN sequences. Gold sequences(codes)are constructed from the modulo-2addition of two maximum length preferred pair PN sequences.By shifting one of the two PN sequence,we get a different Gold sequence.This property can be use to generate codes which will permit multiple access on the channel.The use of Gold sequences permits the transmis-sion to be asynchronous.The receiver can synchronize using the auto-correlation property of the Gold sequence [30].At each emitter,the PPS and the chaotic sequence are generated with synchronous generators every time inter-val T c.These two generators are using the same master clock.The period of the PPS is equal to T?NT c.For asynchronous systems in Fig.1(b)a pilot signal is added to each user before all user are summed up for transmission, which then warrants the use of a pilot signal for each user. The PPS and the spreaded data of each user are summed and transmitted through an AWGN channel.As shown in Fig.1(b),chaotic generators are initialized every LT (L integer)time interval in order to let the receiver know the starting and the ending samples of every spreading chaotic data frame.Furthermore,periodic initialization of the chaotic sequence can prevent a real system from divergences due to computation precision.If a different precision is used at the emitter and receiver side,it is possible to observe a divergence of the chaotic sequences generated at the two sides after a certain number of computed samples.The divergence will be emphasized if the Lyapunov exponents of the chaotic map are very high.Periodic initialization of chaotic sequence will solve the above problem.In addition,the power of the pilot signal is lower than the power of the spreading data signal to avoid degradations in term of BER coming from the PPS.In fact,the PPS acts as a noise for transmitted informa-tions.The PPS power level must be suf?ciently high for detection and synchronization purposes but suf?ciently low for the BER performance of the chaotic transmission. The emitted signal by user m,uemTetTis given by
uemTetT?demTetTtpemTetT
demTetT?
???
P
p X
i
X bà1
k?0
semT
i
xemT
i btk
getàei btkTT cT
pemTetT?
?????
P p
q X
l
X
Nà1
l?0
pemT
l
getàel NtlTT cT(2)
where xemT
i btk
are the chaotic samples corresponding to
data symbol semT
i
;pemT
i
are binary??1 elements of the PPS sequence,getTis the pulse shaping?lter commonly found in communication systems.This pulse shaping?lter can take different forms like raised-cosine?lter,Gaussian ?lter,etc.In this paper we have chosen a rectangular pulse of unit amplitude on?0;T c .P(respectively,P p)are the power of the spreaded data signal(respectively,the PPS). The parameter b is equal to the number of chaotic samples in a symbol duration and,by analogy with DS-CDMA,we have called this parameter the spreading factor b?T s=T c.
2.3.Receiver structure
The additive noise has a power spectral density equal to N0=2.In order to recover transmitted symbols,an exact replica of the chaotic sequence of interest must be generated at the receiver.We assume that the initial conditions of the M chaotic generators used by the M users at the receiver side are known and equal to the ones used at the transmitter side.In this paper,we are mainly
Fig.1.(a)Chaotic communication system with the synchronization unit,(b)structure of the transmitted signal.
809
interested in the code acquisition process,thus,despread-ing and demodulation process will not be developed.Spreaded data signals can be seen as interferences during the synchronization process,since only PPS are of interest here.
The received signal can be written as r et T?
X M n ?1
u en Tet àt en TTtn et T
(3)
where t en Tis the asynchronous delay associated to user n ,
and n et Tis the AWGN.
Acquisition of the PPS signal within the chaotic communication system is possible due to the fact that the various PPS and the chaotic signal are non-correlated with low magnitude peaks of their cross-correlation functions [23].
As shown in Fig.1(a),the PPS and the chaotic sequence are generated synchronously every time interval T c using the same master clock.To establish and maintain
chronization for user m at the receiver side,the local must be synchronized.As shown in Fig.2,when sequence time offset (or delay)of user m is known receiver (end of acquisition phase)the chaos time is also de facto found and the synchronous detection of the chaos-based DS spreading symbol can be performed.This is the ACAS-A method.
The classical CDMA acquisition method is presented in [11].In our paper we have applied classical serial search mode.In this method,the acquisition circuit attempts to cycle through and test all possible phases one by one (serially).Our main objective in this paper is to test the synchronization method,the choice of this acquisition strategy returns to a low circuit complexity.To improve the synchronization performance of the system,we can apply other synchronization strategies (multidwell detec-tion)[11].For the serial search strategy,the received signal of a given user m is multiplied by a locally generated pilot signal (p em Tet àd T),where d is an arbitrary delay.We integrate the product of r et Tby p em Tet àd Tover a period named acquisition time integration as shown in Fig.2.Without loss of generality,we have taken the acquisition time integration equal to (T ?NT c ).Then,we compare the output decision variable to a predetermined
threshold y in order to know if the acquisition is accomplished or not.
Each receiving user require its own PPS tracking system.The PPS tracking unit of user m (Fig.3)starts its operation only after initial acquisition has been achieved.The goal of code tracking is to ?ne align the approximate offset acquired between the received and the local despreading sequences and to maintain the synchroniza-tion [8,11].The numeric voltage controlled oscillator (NCO)of Fig.3commands together the clocks of the PPS generator and the chaotic generator.Finally when the tracking phase is achieved thanks to the PPS signal,the chaotic sequence generated by the generator of the user m will be perfectly synchronized with the received chaotic sequence.
2.4.Theoretical expression of P d and P fa
In this section,theoretical expressions of the prob-ability of detection (P d )and the probability of false alarm (P fa )are evaluated.The received signal multiplied by the local PPS of user m is expressed,after integration,by D em T?z em Tta em Ttg em Ttc
em T
(4)
The term of interest in the decision variable is z
em T?Z
NT c
p em Tet àt em TTp em Tet àd Td t
z
em T
?
R em Tp e
t em Tàd T
where R em Tp et em T
àd Tis the auto-correlation function of the PPS code.a em T
is a zero mean Gaussian noise due to the additive noise n et T:
a
em T
?
Z
NT c
n et Tp em Tet àd Td t
g em Tis the multi-user interference due to the chaotic
spreaded signals:
g
em T
?
Z
NT c X M n ?1
d en T
et àt en TTp em Tet àd Td t
Fig.2.Synchronization unit,acquisition procedure of user m .
G.Kaddoum et al./Signal Processing 89(2009)807–818
810
c em Tis the interference due to other PPS sequences:c
em T
?
Z
NT c X
M n ?1
n a m
p en Tet àt en TTp em Tet àd Td t
The noise n et Tand PPS are independent.The variance of a em Tis given by
s 2a em T?
P p
2
N 0NT c with s 2p em T?P p
(5)
Thanks to the central limit theorem,the multi-user
interference g em Tis zero mean and Gaussian with variance:
s 2g em T?E
Z NTc X M
n ?1
d em Tet àt em TTp em Tet àd Td t !22435s 2
g em T?
PP p MNT 2c
(6)
Expression (6)relies on the fact that chaotic sequences
and the PPS of user m are uncorrelated with zero mean values.Furthermore,for a given chaotic sequence,all samples have a low correlation value [31].
Referring to Gold codes properties [30],the integral Z ?R
NT c p en Tet àt en TTp em Tet àd Td t can only take three va-lues m 1,m 2,m 3.A probability of appearance x is associated to each of these values x 2f x 1;x 2;x 3g ,for a given set of delays.The mean and variance of Z are given by E ?Z ?
X 3i ?1
m i x i
(7)
Expression (7)is obtained by averaging the values taken
by Z for all possible delays t en Tthat are supposed to be uniformly distributed on ?0;T c
s 2Z ?E ?Z 2 àE ?Z 2
s 2Z ?P 2p NT 2c àE ?Z
2(8)
For M à1interfering and independent users,the mean
and the variance of c em T
are given by E ?c
em T
?eM à1TE ?Z
(9)
and
s 2c em T
?eM à1Ts 2Z (10)Thanks to the central limit theorem,c
em T
follows a
Gaussian distribution.Finally we have
E ?D em T ?R em Tp et
em T
àd TtE ?c em T
(11)
and
s 2D em T
?s 2a em Tts 2g em Tts 2
c em T(12)
2.4.1.Probability of detection and probability of false alarm
The main motivation of this paper is to present a new method for robust synchronization for asynchronous chaos-based DS-CDMA.For this reason we are interested in computing the probability of detection P d and the
Fig.3.PPS tracking system of user m .
G.Kaddoum et al./Signal Processing 89(2009)807–818
811
probability of false alarm P fa to demonstrate the feasibility of this method.
Probability of detection for user m is given by Pr eD em TX y Twhen d ?t em T(y is the threshold):
P d ?Q
y àR em T
p e0TàE ?c
em T s D em T
!
(13)
where Q ex T?R t1x e1=??????2p p Te eàu 2
=2Td u .
The probability of false alarm of user m and for an offset d 0is given by Pr eD em TX y Twhen T c o j t em Tàd 0j o eN à1TT c .
The variable of interest is z em T?R em Tp et
em T
àd 0T?m j .In fact,for Gold codes the auto-correlation function can take four values:one for a zero lag and m 1;m 2;m 3for other lags [30].
The P fa ,with respect to all possible values of d 0is given by
P fa ?X 3j ?1
x j Q y àE ?c em T àm j
s D em T
!(14)
2.5.Simulation results
For our simulations,the spreading factor is taken equal to b ?16,the number of users is M ?3and each transmitted signal contains L ?256sequences of PPS,the time integration is T ?127T c .Emitted powers of the PPS sequence and the chaotic sequence for each user are P p ?0:2and P ?1,respectively.The performance of this system is examined for different ratio of the PPS chip energy to noise power spectral density eE c =N 0T;where E c =N 0is equal to
E c =N 0?10log 10
P p T c
0 (15)First of all,we have compared in Fig.4(a)theoretical expressions of the P d given by (13)with simulation results for different thresholds (y ?30;20;10),looking at Fig.4(a)that we have a perfect ?t between the theoretical probability of detection and simulation results.
In Fig.4(b)we have plotted the probability of false
alarm for ?xed probabilities of detection.The threshold is computed from expression (13)for a given P d .If we need to increase the synchronization performance for ?xed values of E c =N 0,T and P d ,we must increase the power of the PPS.On the other hand,increasing the power of the PPS will degrade the BER performance.After simulation of various PPS powers,we have seen that P p equal to 20%of P can give a good trade off between BER and synchroni-zation performances.
https://www.doczj.com/doc/ab7382508.html,parison and discussion
In order to evaluate and compare the performance of the system of the proposed the ACAS-A system,we have plotted the BER curves for the system with and without pilots signals.The channel used for transmission is an AWGN channel,the power of the pilot signal is 20%of the power of the spreaded signal,and we assume the perfect synchronization of chaotic sequence.The BER curves are plotted for a spreading factor b of 64,and for different number of users (M ?2;4;8;16).From Fig.5,we can observe that the PPS signal degrades the performance of the system in terms of BER.From Fig.5it can be observed that the degradation between the two system is approxi-mately 0.5–1dB for M ?2and 1–8dB for M ?16.This degradation relies on the fact that the PPS signal can be seen as an additive noise for the spreaded data signals.To overcome this problem and to improve the perfor-mance of the system both for synchronization and BER,a new system will be introduced in the next section.3.Chaos-based DS-CDMA system with a multiplicative pilots signal (ACAS-M)
For this system we have focused also our study in the ?rst synchronization phase (acquisition)of the chaotic sequence.A PPS sequence will be also used for synchronization purposes but instead of being an additive sequence as in [23]or in our ?rst system,we have used it as a multiplicative sequence [32].This new
?20
?18?16?14?12?10?8
0.75
0.80.850.90.951E c /N 0 [dB]
P d
10?3
10?2
10?1100
?20
?18?16?14?12?10?8
P fa
E c /N 0
Fig.4.(a)Comparison between theoretical expression of P d and simulations;(b)false alarm probabilities for a ?xed probability of detection.
G.Kaddoum et al./Signal Processing 89(2009)807–818
812
acquisition procedure has several advantages.First of all,in this procedure the PPS sequence acts no more as a noise for the chaotic spreaded signal like our ?rst synchroniza-tion system.The second advantage is that the chaotic sequence power can be exploited in the synchronization process in order to increase the decision variable in acquisition.
3.1.Transmitter structure
The M asynchronous users are spreaded by the same type of chaotic sequences described as in Section 2.1.The frame structure of each user is shown in Fig.6(b).A PPS code p em Tet Tis associated to each user,with period T s where T s ?NT c .The PPS and the chaotic sequence are
0246
8101214161820
10?8
10?710?610?510?410?310?210?1100Theoretical and simulation performances of different digital chaotic sequences
E b /N 0 [dB]
B E R
Fig.5.Performance of the chaos-based DS-CDMA with and without PPS.
Fig.6.(a)Chaotic communication system with the synchronization unit;(b)structure of the transmitted signal.
G.Kaddoum et al./Signal Processing 89(2009)807–818
813
synchronously generated with the same rate T c .The PPS signal and the chaotic sequence of each user multiply the binary data stream of user.The chaotic generator is initialized every LT s time interval to let the receiver know the starting and the ending samples of every spreading data frame.Without loss of generality we have taken the spreading factor b ?T s =T c ?N .
In Fig.6(b),p em Tk
are the chips of the PPS signal,x em T
k are the chaotic samples of the user m ,L is an integer and u em Tis the output frame.This frame is composed from the multiplication of the PPS signal with the chaotic sequence.The frames of all asynchronous users are transmitted over an AWGN channel.
This second system ACAS-M is presented in Fig.6(a).After spreading the data symbols by the chaotic sequence x em Tet T,the spreaded signal is multiplied by the PPS signal.The main motivation to use the PPS signals in our system is to give reliable information both for transmitter and receiver to initialize the chaotic generator every LT s time interval.
3.2.Receiver structure
The received signal can be written as r et T?
X M n ?1
u en Tet àt en T
Ttn et T
(16)
where t en Tis the delay associated to user n ,and n et Tis the AWGN with power spectral density equal to N 0=2.u en Tet Tis the emitted signal by user n .In the second system,all emitted signals have the same power P .
Since the PPS and the chaotic sequence are synchro-nously generated with the same rate T c ,the drift between the two sequences at the transmitter and receiver side for user m is corrected by the acquisition process.The correction of the drift allows the chaotic generator at the receiver to have the time offset.
For the system of Fig.6(a)we have applied the classical serial search mode as shown in Fig.7.The received signal of a given user m is multiplied by a locally generated pilot signal (p em Tet àd T)and by the corresponding chaotic sequence (x em Tet àd T),where d is an arbitrary delay.Then,we integrate the correlation product of r et Tby p em Tet àd Tand x em Tet àd Tover the period denoted time integration.Without loss of generality,we take the time integration equal to T ?NT c .Then we compare this decision variable
to a predetermined threshold y to know if the acquisition is accomplished or not.
3.3.Theoretical expression of P d and P fa
This section is devoted to the computation of the theoretical expressions of probability of detection and probability of false alarm.The received signal multiplied by the local PPS and chaotic sequence of user m is expressed by
D em T?z em Tta em Ttg em T
z em T?Z NT c
u em Tet àt em TTp em Tet àd Tx em Tet àd Td t
(17)
We are interested here in the acquisition performance,
without loss of generality we take s em T
i
?1z em T?Z NT c
v em Tet àt em TTv em Tet àd Td t
where v em Tet T?p m et Tx em Tet T
z em T?R em Tv
et em T
àd Ta em T
?
Z
NT c
n et Tp em Tet àd Tx em Tet àd Td t
g
em T
?
Z
NT c X
M n ?1n a m
u en Tet àt en TTp em Tet àd Tx em Tet àd Td t
where p em Tet àd Tis the PPS generated locally x em Tet àd Tis the chaotic sequence of user m ,d is an arbitrary delay,and NT c is the time integration.z em Tis the term of interest of the decision variable D em T.
a em Tis a Gaussian noise with zero mean because noise n et Tand PPS are independent and zero mean.The variance of a em Tis given by
s 2a em T?P
2
N 0NT c
(18)
g em Tis the multi-user noise.Thanks to the central limit
theorem,this noise is zero mean and Gaussian with variance:
s 2g em T?P 2eM à1TNT 2
c
(19)
Expression (19)is derived by referring to the properties of chaotic sequences:Chaos of different users are uncorre-lated and,for a given chaotic sequence,all samples have a low correlation value [31].
Fig.7.Chaotic synchronization system.
G.Kaddoum et al./Signal Processing 89(2009)807–818
814
Finally we have
E?DemT ?RemT
v
etemTàdT(20)
s2 DemT?
P
2
N0NT ctP2eMà1TNT2
c
(21)
3.3.1.P d and P fa expressions
The P d for user m is given by PreDemTX yTfor an offset value of d?temT,where zemT?RemTve0T.
The auto-correlation function is:RemT
v e0T?T c P
P Nà1
k?0
?pemT
k 2?xemT
k
2?T c P
P Nà1
k?0
?xemT
k
2.
Since chaotic signals are given by deterministic equations and for a low time integration,the term
P Nà1
k?0?xemT
k
2is not constant[29].Moreover,the auto-
correlation function RemT
v
e0Tis not constant and the probability of detection is
P d?
Zt1
0Q
yàu
s
DemT
peuTd u(22)
where peuTis the probability density function of auto-correlation values corresponding to the offset errors equal to zero.
For a high integration time,the auto-correlation function RemT
v
e0Tcan be seen as a constant[29]and the probability of detection will be
P d%Q yàRemT
v
e0T
s
DemT
!
(23)
The probability of false alarm of user m and for an offset d0 is given by PreDemTX yTwhen T c o j temTàd0j oeNà1TT c:
PemTfa ?Q
yàRemT
v
etemTàd0T
s
DemT
!
(24)
The whole P fa is given by
PemTfa ?
Zt1
à1
Q
yàm
s
DemT
pemTd m(25)
where pemTis the probability density function of auto-
correlation values corresponding to the offset errors
greater than T c.
3.3.2.Simulation results
For our simulations,the length of each transmitted
frame is:L?6000and the performance of this system is
examined for an emitted power P?1and for different
ratio of the chip energy to noise power spectral density
eE c=N0T;where E c=N0is equal to
E c=N0?10log10
PT c
(26)
First of all we will compare the performance in terms of P d
the system between the?rst system ACAS-A and the
second system ACAS-M in the mono user case when
the threshold is?xed to y?21and for an integration time
equal to63chips.In the?rst system the PPS power is20%
of the spreaded signal with.It is clear,looking at Fig.8,
that the second system outperforms the?rst system.
In Fig.9we have plotted the performance in term of P d
in the multi-user case with M?5.Simulation results are
compared to Eq.(23)where RemT
v
e0Tis assumed to be
constant.For a suf?cient long integration time,RemT
v
e0Tis
very close to a constant equal to NT c[29].The threshold y
is taken equal to8in Fig.9.It is clear,looking at Fig.9,that
we have a perfect?t between theoretical expression and
simulations when the integration time is greater than
N?15.For N equal to15the difference between
simulations and theoretical expression is related to the
non-constant value of RemT
v
e0T.For an integration time less
than15chips the energy of the chaotic sequence can be no
more considered as a constant equal to NT c[29].
We are now interested in the computation of the P fa.
Eq.(25)has been computed as follows(27):
PemT
fa
?
X c
i?1
Pem iTQ
yàm i
s
DemT
(27)
where c is the number of histogram classes of m and Pem iT
is the probability of being in the i th class.Pem iTis obtained
?22?20?18?16?14?12?10?8
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
E c/N0
P
d
https://www.doczj.com/doc/ab7382508.html,parison between the P d of the two systems for a?xed threshold y?10and N?31.
G.Kaddoum et al./Signal Processing89(2009)807–818815
by plotting the histogram of the energy distribution of the chaotic sequence for a given time integration.
In Fig.10the computed expression of the P fa given by (27)and the simulation for a set of acquisition times and thresholds are in good agreement for any integration time NT c .
Finally in Fig.11we have plotted the probability of false alarm for ?xed probabilities of detection.The threshold is obtained from expression (23)for a given P d .
4.Conclusion
In this paper we have presented two systems in order to achieve synchronization for chaos-based DS-CDMA system.In order to compare and to evaluate the additive PPS degradation for synchronization process in asynchronous multi-user.We extend synchronization method of [23]
applied for the synchronous multi-user case to asynchronous multi-user case (ACAS-A)in the ?rst chaos-based-DS-CDMA system.The synchronization performance of the chaotic communication system is evaluated in presence of noise and multi-user interferences.Conventional ideas of CDMA syn-chronization process have been applied to the chaos based DS-CDMA using a PPS https://www.doczj.com/doc/ab7382508.html,ing Gold sequences,the code acquisition phase has been evaluated in terms of probability of detection and probability of false alarm.Theoretical expressions of the probability of detection and the probability of false alarm have been proposed with very good agreement with simulation results.In the second system (ACAS-M),the conventional ideas of DS-CDMA synchronization systems have also been applied to the chaos based DS-CDMA using a PPS signal.The main goal of the PPS is to give the time reference for chaos generator initialization.Theoretical expressions of the probability of detection and the probability of false alarm are determined.Simulation results con?rm the
?20
?15?10
?50
0.70.75
0.80.850.90.951
E c /N 0
P d
https://www.doczj.com/doc/ab7382508.html,parison between theoretical expression of P d and simulations for N ?15;31;63,threshold y ?8and M ?5.
?15
?10
?50
0.1
0.150.2
0.250.30.350.40.45E c /N 0
P f a
https://www.doczj.com/doc/ab7382508.html,parison between the computed and simulated expression of P fa for N ?15;31;63,y ?5;10;21and M ?5.
G.Kaddoum et al./Signal Processing 89(2009)807–818
816
exactitude of our theoretical expressions.The proposed method using a multiplicative PPS sequence as a timing reference for each user brings major advantages.First of all,this method can be applied in asynchronous transmissions.Second advantage is that the PPS sequence is no more noise for the chaotic spreaded signal.Extensions of this method in jamming environments together with multipath channels are currently under study.References
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