Performance of Direct-oversampling Correlator-type Receivers in Chaos-based DS-CDMA Systems Over Frequency Non-selective Fading Channels

Transcription

1 Wireess Persona Communications manuscript No wi be inserted by the editor) Performance of Direct-oversamping Correator-type Receivers in Chaos-based DS-CDMA Systems Over Frequency Non-seective Fading Channes Nguyen Xuan Quyen Pere Baret-Ros Received: 8th Sept 015/ Accepted: 3rd March 017 Abstract In this paper, we present a study on the performance of directoversamping correator-type receivers in chaos-based Direct-sequence code division mutipe access DS-CDMA) systems over frequency non-seective fading channes At the input, the received signa is samped at a samping rate higher than the chip rate This oversamping step is used to precisey determine the deayed-signa components from mutipath fading channes, which can be combined together by a correator for the sake of increasing the SNR at its output The main advantage of using direct-oversamping correator-type receivers is not ony their ow energy consumption due to their simpe structure, but aso their abiity to expoit the non-seective fading characteristic of mutipath channes to improve the overa system performance in scenarios with imited data speeds and ow energy requirements, such as Low-rate wireess persona area networks LR-WPAN) Mathematica modes in discrete-time domain for the conventiona transmitting side with mutipe access operation, the generaized non-seective Rayeigh fading channe, and the proposed receiver are provided and described A rough theoretica bit-error-rate BER) expression is first derived by means of Gaussian approximation We then define the main component in the expression and buid its probabiity mass function PMF) through numerica computation The fina BER estimation is carried out by integrating the rough expression over possibe discrete vaues of the PFM In order to vaidate our findings, PC simuation is performed and simuated performance is compared with the corresponding estimated one Obtained resuts Nguyen Xuan Quyen Schoo of Eectronics and Teecommunications, Hanoi University of Science and Technoogy E-mai: quyennguyenxuan@husteduvn Pere Baret-Ros Department of Computer Architecture, UPC BarceonaTech, Jordi Girona 1-3, Barceona, Spain E-mai: pbaret@acupcedu

2 Nguyen Xuan Quyen, Pere Baret-Ros show that the system performance get better with the increment of the number of paths in the channe Keywords Chaos-based communications Chaos-based DS-CDMA Correator-type receiver Frequency non-seective fading channes BER performance 1 Introduction Pseudo-noise PN) sequences have been widey used in direct-sequence code division mutipe access DS-CDMA) communication systems [1] Despite their good correation properties, the main weakness of these sequences is their imited security For exampe, PN sequences can be easiy reconstructed by inear regression attacks [], due to their sma inear compexity In order to overcome this imitation, severa works [3] have proposed to repace PN sequences by chaotic sequences in DS-CDMA systems Chaotic sequences have an infinite number of states, non-periodicity and wide-band Therefore, they are very difficut to predict, reproduce, and intercept [3]-[5] Interestingy, the generation of an infinitive number of uncorreated chaotic sequences for muti-user operation can be simpy carried out by sighty changing the initia vaues [6]-[8] Instead of using continuous chaotic sequences as in [4,6], the studies in [9,10] proposed the adoption of compex mutieve spreading sequences generated by repeating truncated and quantized chaotic time series for spreading the data symbos Obtained resuts show that DS-CDMA systems using periodic quantized sequences aso have arger capacity and better performance than those using God and m-sequences in a mutipe-access environment Over the ast two decades, mutipe works have been devoted to the design and anaysis of chaos-based DS-CDMA systems These studies can be divided into two main groups: 1) those that addressed the design [11], optimization [1], and synchronization of communication systems for chaotic spreading sequences [13]-[16]; and ) those that provided a theoretica and numerica anaysis of the BER performance for mutipe-access operation under different transmission channes [17]-[7] Most of these previous works used correator-type receivers to study the BER performance over noisy channes [4, 6], [17]-[3], because of their simpe architecture In contrast, the performance of chaos-based DS-CDMA systems over mutipath fading channes has ony been investigated by means of more compicated RAKE receivers, which can combine mutipath components to enhance the signa-to-noise ratio [4]-[7] In this paper, we address the design of a secure physica ayer for wireessbased mutipe-access appications with imited data speed and ow energy consumption in sma transmission areas [31], such as wireess sensor networks WSNs) or ow-rate wireess persona area networks LR-WPANs) [3,33] Due to the short transmission distance and imited data rates in these scenarios, the root-mean-square RMS) deay spread is ess than the symbo duration, hence the wireess transmission environment can be considered as a frequency non-seective fading channe or fat fading channe) [34] We argue that, under

3 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 3 these conditions, direct-oversamping correator-type receivers coud be used as an aternative to the more compex RAKE receivers proposed in previous works [4]-[7] The utiization of direct-oversamping correator-type receivers in these scenarios has two main advantages: 1) they have ow energy consumption due to their simpe architecture, and ) they can better expoit the fat fading characteristic of the channes in order to improve the system performance To the best of our knowedge, this is the first work to study the BER performance of chaos-based DS-CDMA systems over mutipath channes having a frequency non-seective fading characteristic, when direct-oversamping correator-type receivers are used In the receiver, we propose to sampe the incoming signa at a rate higher than the chip rate, before entering the correator This oversamping step aims to specificay determine the deayed-signa components from fading mutipath channes, which can be combined together by the correator in order to increase the signa-to-noise ratio at its output The appication of an oversamping step at the receiver side has been shown to improve the communication features in [8]-[30] The most reevant methods for the theoretica cacuation of the BER performance for chaos-based DS-CDMA systems over noisy and mutipath fading channes are presented in Section II We find that BER expressions of high accuracy methods are very compicated and usuay written in the form of mutipe integra, thus requiring very compex cacuations In order to overcome this drawback, in this paper, we estimate the BER performance in two steps: theoretica derivation and discrete integra Firsty, we derive a rough BER expression in cosed form through Gaussian approximation by assuming that the output signa of the integrator is a random variabe being the sum of statistica independent components Secondy, we take into account the variation of the fading coefficients by computing the probabiity mass function PMF) for the ratio of variabe bit energy to noise power spectra density in the rough expression Finay, we cacuate the estimated BERs by numericay integrating the rough expression over a the discrete vaues of the PMF The main contributions of this paper can be summarized as foows: We propose and investigate the nove idea of appying a direct-samping technique to conventiona correator-type receivers for enhancing the performance of chaos-based DS-CDMA communication systems over frequency non-seective fading channes The operation of the receiver is described by means of buiding and anayzing the mathematica mode of the whoe system in discrete-time domain The performance estimation is carried out by means of both theoretica derivation and numerica computation We verify the proposed mode and estimation through simuation We use a generaized fat Rayeigh fading channe for the first time in the evauation of the receiver performance The discrete-time mode and the typica parameters of the channe are presented The impact of the channe on the system performance is determined in terms of the BER vaue against

4 4 Nguyen Xuan Quyen, Pere Baret-Ros these typica parameters Our resuts show that the more the number of paths in the channe increases, the better the system performs The rest of this paper is organized as foows A review of the reated work is provided in Section II In Section III, we buid and describe the mathematica modeing of the system The methodoogy to estimate the BER performance is presented in Section IV The simuation and estimation resuts are shown and discussed in Section V Finay, the concusions are given in Section VI Reated Work With respect to noisy channes, the performance of a chaos-based DS-CDMA system was investigated for the first time in [4,6] ony by means of numerica simuations Many subsequent studies focused on estimating the BER performance theoreticay for this system The simpest estimation is presented in [17] under the assumption that the bit energy is constant Another approach is proposed in [18,19], where the decision parameter at the output of the correator is considered as a Gaussian variabe These estimation methods are ony vaid when the spreading factors SFs) are very high, whie for ow and moderated SFs they become seriousy imprecise The cause of this inaccuracy is the non-periodic nature of chaotic sequences, as the bit energy of the transmitted signa varies from one bit to another In order to take into account the dynamic behavior of the chaotic sequence, the approach in [0, 1] integrates the BER expression for a given spreading factor over a possibe sequences This method provides an accurate estimation even for sma SFs, but it requires very compex cacuations A method deveoped ater in [, 3] reies on computing the probabiity density function PDF) of the bit energy variabe and then integrating the BER expression over a possibe PDF vaues Athough this method has high accuracy, its BER expression is in the form of a mutipe integra It is important to note that in a estimation methods described above, correator-type receivers are expoited and the reproduced chaotic sequences are assumed to be in a synchronous state Most of the recent studies on chaos-based DS-CDMA systems have focused on investigating the BER performance of the system over fading, mutipath, and mutipath fading channes [4]-[7] In order to overcome the severe consequences of mutipe channes, Rake receivers were used under the assumption that both the sequence synchronization and the channe estimation are perfect A chaos-based DS-CDMA system under a mutipath fading channe was anayzed and measured in [4, 5] The symbos were first separated from inter-symbo and inter-user interference parts The BER expression was then derived in a quite compicated form by using Gaussian approximation Based on the approach of computing the bit energy PDF as in [3], accurate estimations of the BER performance of muti-user systems under mutipath and fading channes were presented in [6] and [7], respectivey However, when the number of users and paths increases, the theoretica determination of the PDF and the cacuation of the BER expression with mutipe integra become

5 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 5 very compicated Severa studies on the design of communication systems at the physica ayer for WSNs and LR-WPANs have been carried out recenty, mainy based on the IEEE Standard [3,33] The studies reated to the design of a moduator/demoduator and a compete transceiver are presented in [35] and [36], respectivey The work in [37] presents an investigation using cooperative orthogona frequency division mutipexing OFDM) for semi distributed detection in WSNs A ow power cooperative mutipe-input mutipe-output MIMO) scheme with ow density parity check channe codes is proposed in [38] The design of the physica ayer for WSNs based on conventiona DS- CDMA systems using the chip intereaving technique is investigated in [31] The atest study on a chaos-based DS-CDMA systems has been pubished in [39] The mutipath fading mode used there is a wide-band channe where both of the fading coefficients and mutipath deays vary according to random distributions The BER expression is given in the cosed form However, simuation resuts are ony shown for one specia case, with significant differences with respect to the theoretica anaysis 3 Mathematica Modeing of The System This section presents the mathematica modeing of the chaos-based DS-CDMA system under study Fig 1 shows the bock diagram of the system The schemes for the transmitting side with K users and the generaized frequency nonseective channe with L deayed paths are dispayed in Fig 1a), whie the scheme for the proposed direct-oversamping correator-type receiver is shown in Fig 1b) 31 Mode of Transmitting Side with Mutipe-access Operation The input bit streams of the K users are spread by K uncorreated chaotic sequences produced by K chaotic generators, which use the same chaotic map with different initia conditions Here, b i) and x i) k, respectivey denote the th bit of the i th user and the k th chip in the th bit of the i th user Both b i) and x i) k, are equa to either +1 or 1 The ratio β = T b/t c is caed the spreading factor, ie, the number of chips per bit, where T b and T c are the bit and chip duration, respectivey The output signa of the i th user in the k th chip duration is given by c i) k, = bi) x i) k, 1) The signa transmitted on the channe is the sum of a output signas from the K transmitters, which can be expressed as foows: K e k, = c i) k, = b i) x i) k, ) i=1 i=1

6 6 Nguyen Xuan Quyen, Pere Baret-Ros 1) b b i) K) b 1) x k, i) x k, i) c k, 1) c k, K) c k, e k, τ 1 τ L α 0 α 1 α L 0) o k, 1) o k, L) o k, η k, rk, r k, K) x k, Samper i) r g,k, a) Correator i) d g,k, λβ Ʃg=1 s i) ) b i) i) x k, b) Fig 1 Bock diagram of the chaos-based DS-CDMA communication system under study: a) The transmitting side with mutipe-access operation and the generaized frequency nonseective fading channe, b) The proposed direct-oversamping correator-type receiver 3 Mode of Generaized Frequency Non-seective Fading Channe The transmitted signa can reach the receiver from various paths with various deays and attenuation The path having the shortest transmission period is considered as the primary path with deay being zero, whie the rest are the secondary paths The channe mode investigated here is assumed to have L secondary paths with impuse response being hn) = L α j δ n τ j ), 3) where α j is the fading coefficient of the j th path, which varies randomy according to a Rayeigh distribution [40] given by fα j ) = α j σj e α j /σ j ), 4) with σ j being the scae parameter of the distribution The mean vaue of each fading coefficient is determined by α j = E[α j ] = σ j π/; τj is the deay of the j th path and δt) is the Dirac impuse Because of the non-seective fading characteristic, the channe deays are assumed to be ess or equa to the chip

7 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 7 k-1) th chip duration k th chip duration k+1) th chip duration 0) o k, T c τ 1 1) o k, L) o k, τ L η k, 0 1 st nd gth λ th Fig Iustration of the signa components at the output of the channe and their samping process in the receiver duration, ie, τ 0 = 0 τ 1 τ j τ L T c The output signa of the channe can be written as a sum of three components as foows: r k, = rk, S + rk, I + η k,, 5) here, rk, S is considered as the beneficia component, which is composed of the signa from the primary path and its deayed versions from the secondary paths; rk, I is the interference component, which is the deayed version from the secondary paths of the previous chip duration; and η k, is additive white Gaussian noise AWGN) The components rk, S and ri k, are respectivey determined by r S k, = α 0 K i=1 b i) x i) k, + α 1 i=1 b i),τ 1 x i) k,,τ α L i=1 b i),τ L x i) k,,τ L, 6) and r I k, = α 1 K i=1 b i),τ 1 x i) k 1,,τ α L i=1 b i),τ L x i) k 1,,τ L, 7) where b i),τ j and x i) respectivey k,,τ j are b i) and x i) k, after being deayed by a period τ j,

8 8 Nguyen Xuan Quyen, Pere Baret-Ros 33 Mode of Proposed Direct-oversamping Correator-type Receiver In the i th receiver, the incoming signa r k, is first oversamped with a rate higher than the chip rate Here, we assume that there are λ sampes per chip The objective of this samping step is to determine more exacty the vaues of the three components above in a bit duration T b The higher the vaue of λ, the higher the achieved accuracy Fig iustrates components of the received signa r k, and the oversamping process within the duration of k th chip in the second interva of th bit The despread-spectrum process is then carried out by mutipying the samping signa r g,k, with the oca chaotic sequence x i) k,, which is reproduced in the same way as in the transmitter and synchronized with the incoming signa by means of the synchronization methods proposed in [13]-[16] Here, the integrate-and-dump bock operates simiary as a sum bock The consecutive sampes at the input are added together in each bit duration to produce, at the output of the correator, the signa s i) as expressed by s i) = λ g=1 where the three components X, Y, Z respectivey are X = L L ) Y = τ j α j and λα j) b i) b i) d i) g,k, = X + Y + Z, 8) x i) k, x i) k, Z = λ + m=1,m i m=1,m i b m) x m) k, xi) k, b m) x m) k, xi) k, + ), 9) m=1,m i 10) η k, x i) k, 11) The detaied deveopment of Eq 8) is presented in Appendix A) In Eqs 9)-11), we can find that Z is the noise component created by the AWGN The sum of X + Y ) can be rewritten in another form as foows: X + Y = S + I, 1) where S is the component of the beneficia signa determined by S = L λ τ j )α j) b i) x i) k,, 13) b m) x m) k 1, xi) k, ),

9 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 9 and I is the interference component between different users determined by L I = τ j )α j) b λ m) x m) k, xi) k, m=1,m i 14) L + τ j α j) b m) x m) k 1, xi) k,, m=1,m i Based on Eqs 11), 13), 14), it is easy to find that the vaues of the components S, I, and Z increase when the number of deay paths L or the number of sampes per chip λ increase However, owing to the ow cross-correation between the different chaotic sequences, and between the chaotic sequences and AWGN, the increasing amount of the components I and Z is much ess than that of the component S Therefore, the SNR of the output signa s i) increases The increment of the SNR eads to the improvement of the system performance The binary vaue of the th bit is finay recovered by the decision circuit using a sign function as foows: { b i) = 1 s i) 0, 1 s i) < 0 15) 4 Estimation of the BER Performance In this section, the two steps proposed to estimate the BER performance of the studied system are presented With the assumptions that the fading coefficients and the bit energy are fixed, a rough BER expression is first derived by means of Gaussian approximation We then define the main component in the expression and buid its PMF by numerica computation The fina BER estimation is carried out by integrating the rough expression over possibe discrete vaues of the PFM 41 Theoretica Derivation Assuming that each bit, either + 1 or 1, appears at the output of the data source with a probabiity of 1/, we first determine the mathematica statistics of the components, ie, X, Y and Z, in the case that a +1 bit is transmitted The mean vaues are determined as foows see Appendix B): E[X b i) = +1, α j ] = E[Y b i) L λα j βe c, 16) L = +1, α j ] = τ j α j βe c, 17)

10 10 Nguyen Xuan Quyen, Pere Baret-Ros and E[Z b i) = +1, α j ] = λe [ η k, x i) k, The mean squared vaues are derived as see Appendix C) E[X b i) = +1, α j ] = E[Y b i) = +1, α j ] = L λα j L τ j α j ] = 0 18) βec4 + ββ + K )E c ), 19) βec4 + ββ + K )E c ), 0) E[Z b i) = +1] = λ N 0 E b 1) [ In the above equations, E c = E x i) ] k,, E c4 = E [x i) 4 ] k,, E b = βe c and N 0 = [ ] E ηk, are the bit energy and noise power spectra density, respectivey Using the mean and mean squared vaues obtained above, the variances of X, Y, Z are cacuated by and V ar[x b i) V ar[y b i) = +1, α j ] = E[X b i) = +1] E [X b i) = +1] L ) ) = λα j βe c4 + βk )Ec, = +1, α j ] = E[Y b i) = +1] E [Y b i) = +1] L ) ) = τ j α j βe c4 + βk )Ec, ) 3) V ar[z b i) = +1] = E[Z b i) = +1] E [Z b i) = +1] = λ E b N 0 4) Due to the physica characteristics of the communication system, in Eqs 9)- 11), the binary data b i) = {±1}, channe deays τ j, number of paths L, number of users K, number of sampes λ, and spreading factor β are constants, whie the parameters, ie, chaotic sequence x i) k,, fading coefficient α j, and Gaussian noise η k,, are assumed to be independent ) random variabes L Under this condition, the components, ie, λα j and m=1,m i b m) x m) k, xi) k, ) of the variabe X, ) L τ j α j and β b i) b i) x i) k, + x i) k,

11 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 11 b m) x m) k, xi) m=1,m i Z = λ β η k, x i) k, k, + β ) m=1,m i b m) x m) k 1, xi) k, ) of the variabe Y, and of the variabe Z, are considered as mutuay independent random variabes Therefore, the variabes, ie, X, Y, and Z are aso statisticay independent The mean vaue and variance of the signa s i) are given as and E[s i) b i) = +1, α j ] = E[X b i) = +1, α j ] + E[Y b i) = +1, α j ] + 0, 5) V ar[s i) b i) = +1, α j ] = V ar[x b i) = +1, α j ] + V ar[y b i) = +1, α j ] + V ar[z b i) = +1] 6) Anaogousy, the case of transmitting a 1 bit is cacuated in the same way: E[s i) b i) V ar[s i) b i) = 1, α j ] = E[s i) b i) = +1, α j ], 7) = 1, α j ] = V ar[s i) b i) = +1, α j ] 8) According to the centra imit theorem CLT) [41], if S is the sum of mutuay independent random variabes, then the distribution function of S is weapproximated by a certain type of continuous function known as a Gaussian or Norma) density function Owing to the output s i) = X + Y + Z with X, Y, Z being mutuay independent random variabes, and according to the CLT, the random variabe s i) approximatey has a Gaussian distribution On the basic of Eqs 15)-7) and the resuts in [18]-[3], [9], the rough BER expression can be derived as foows see Appendix D): BER R = 1 P robsi) = Q 0 b i) V ar[s i) b i) E [s i) b i) = +1, α j ) + 1 P robsi) = +1, α j ] = +1, α j ] > 0 b i) = 1, α j ) ) 1 = Q A + B + 1 ) 1, C where the function Q ) is defined by Qϵ) = 1 π components A, B, C are determined as A = B = Ec4 E c Ec4 E c ) + K 1 β L ϵ / τ j λ α L j 9) expy /)dy and the α j ) + K L / L τ j α j β λ α j 1, 30), 31)

12 1 Nguyen Xuan Quyen, Pere Baret-Ros and L C = α j L τ j λ α j E b N 0 3) This rough expression is vaid ony if the fading coefficients and the bit energy are constant In fact, in our system, the fading coefficients vary randomy with a defined distribution and the bit energy varies according to a given chaotic behavior The previous studies in [4]-[7] show that, in a fading mutipath channe, the variation of bit energy mainy depends upon the fading Moreover, they show that for a arge spreading factor, the effect of chaotic behavior on bit energy is not significant Therefore, in our approach, we assume that the variation of the components within the bracket in Eq 9), ie, A, B, C, is affected mainy by the fading Here, the variabe C is considered as the main component, which is the ratio of variabe bit energy, ie, E vb = ) L α j L τ j λ α j E b, to noise power spectra density N 0 The variabe bit energy E vb fuy depends on the fixed bit energy E b and channe parameters, ie, L, α j, τ j For the sake of simpicity, we consider A, B equa to their average vaues A, B respectivey, which are obtained by repacing α j in A and B by its mean vaue, ie, α j = E[α j ] The rough expression is first approximated by BER R Q A + B + 1 ) 1, 33) C after that, the conditiona BER expression refecting the variation of C is derived as foows: BERC) 0 Q A + B + 1 ) 1 fc)dc, 34) C with fc) being the probabiity density function PDF) of C An appication Exampe: Let s determine the BER vaue of the system in a particuar case, where the studied channe is an one-path Rayeigh fading channe For this case, the number of secondary channes L = 0 and the fading coefficient α 0 varies according to a Rayeigh distribution given by fα 0 ) = α 0 e σ0 α 0 /σ 0 ), with σ 0 being the scae parameter of the distribution [37] Based ) E on Eqs 30)-3), we have A = c4 E + K /β, B = 0 and C = α c 0 E b N 0 The PDF of C is obtained by fc) = 1 C E[C] with E[C] = E[α0] E b σ 0 E b N 0 E[C] e N 0 = [37] The BER vaue is then determined by means of the conditiona

13 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 13 BER expression as foows: BERC) = 0 0 Q Q A + 1 ) 1 1 C E[C] e Ec4 E c + K β C E[C] dc ) α0 E b N 0 e α 0 σ 0 σ0 dα 0 35) In fact, our studied channe is the generaized case of a frequency nonseective fading channe with mutipe primary paths, ie, L 1 Therefore, as the number of secondary paths L increases, the theoretica determination of the fc) becomes more compicated, even impossibe For this reason, another simper approach based on numerica computation to determine the PMF of C, which can be considered as the discrete PDF of C, is presented in the next subsection, where the BER performance is estimated by means of discrete numerica integra 4 Discrete Integra Firsty, the use of numerica computation to buid the PMF of C is described The tota number of sampes M generated for the statistic computation, and the number of points N for potting the PMF, are chosen with proper vaues Since each secondary path has a defined distribution function fα j ) and a given mean vaue α j = E[α j ], we first generate M discrete sampes for each corresponding coefficient, by using a numericay programmabe too eg, Matab), which are expressed by a M-eement vector, #» α j) = and then a vector #» D is created by #» D = L [ α j) 1 α j) α j) M #» α j) L ], 36) τ j α #» j) 37) λ Based on Eq 31), the vector for the M discrete sampes of the variabe C is produced by [ ] #» C = D1 E b D E b D E b M, 38) N 0 N 0 where D p denotes the p th eement of the vector D #» In theory, the variation range of variabe C is 0, ) However, in our numerica approach, the variation is assumed to be in a imited range of 0, C max ], where C max is chosen with a arge enough vaue to guarantee the estimation accuracy The range of 0, C max ] is divided into N equa intervas, ie, ] 0, Cmax N, N 0

14 14 Nguyen Xuan Quyen, Pere Baret-Ros Begin E=0, q=1, p=1 q=q+1 yes q>m no p>n yes End no yes E p =E p +1 p-1)c max <C N q no pc max N p=p+1 Fig 3 Agorithm for the cacuation of N eements in the vector E Cmax N, Cmax N ],, p 1)Cmax N ], pcmax N,, N 1)Cmax N ], NCmax N The number of sampes, denoted by eement E p in vector E, #» from the M sampes in vector ] C #» having a vaue faing into each corresponding interva are cacuated by the agorithm presented in Fig 3 From p 1)Cmax N, pcmax N the vector E #» obtained, the eement P p of vector P #», which denotes the proba- ], is biity to the variabe C having a vaue in the interva computed by p 1)Cmax N, pc max N #» #» E P = M 39) The graph for the PMF of C is created by potting P p versus the centered vaue of each interva, ie, Cp = p 1/)C max N, for p from 1 to N The BER estimation is finay carried out by discretey integrating the approximated BER expression in Eq 33) over N points of the PMF as foows: BER N p=1 Q A + B + 1 ) 1 Pp Cp 40) It is noted that the higher the chosen vaues of M and N, the more accurate the BER estimation, however the arger the computation time is and vice versa Therefore, the choice of the vaues M and N has to guarantee an acceptabe accuracy with an aowabe period of computation time

15 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading L= L= P p 0004 L=3 000 L= Fig 4 Probabiity mass function PMF) of the variabe C in the case of β = 64, λ = 30, E b /N 0 = 10dB with the increment of the number of secondary paths L C * p 5 Simuation Resuts In this section, numerica resuts obtained by computer simuations with different specific parameters are presented to vaidate the estimation resuts derived in Eq 40) The simuation parameters are given as foows: The chaotic map used for generating the chaotic sequences is the Chebyshev poynomia function of order, ie, x k = fx k 1 ) = x k 1 1 with E c = 1/ and E c4 = 3/8 see Appendix E) The number of secondary paths L in the simuated channe changes from 0 to 4 The fading coefficient and deay for the paths are σ 0 = 07, τ 0 = 0, σ 1 = 06, τ 1 = 5, σ = 05, τ = 10, σ 3 = 04, τ 3 = 15, and σ 4 = 03, τ 4 = 0 The spreading factor β varies from 16 to 18, the number of sampes per chip λ changes from 10 to 50, and the number of users K varies from 1 to 10 The parameters used for computing the PMF of C are M = 10 6, N = 10 3 and C max = 180 Fig 4 shows the PMF graph of the variabe D computed according to Subsection 4 in the case of β = 64, λ = 30, E b /N 0 = 10dB aong with a gradua increment of L We can see that when L increases from 1 to 4, the PMF changes More specificay, the vaue region of C p is expanded in the increasing direction, whie the vaue region of P p is reduced This change is refected in Eq 40) where the BER is reduced with the increment of L

16 16 Nguyen Xuan Quyen, Pere Baret-Ros L= BER L= 10 4 L=3 L= L=0 Estimations Simuations E /N db) b 0 Fig 5 BER performance in the case of K = 1, β = 64, λ = 30 with the increment of L 10 1 E b /N 0 = db 10 E b /N 0 =6 db E b /N 0 =10 db 10 3 BER Estimations Simuations K Fig 6 BER vaues versus the number of users K in the case of L = 1, β = 64, λ = 30 with the increment of E b /N 0 The estimated performance of a singe-user system obtained by Eq 40), for the cases of L = 1,, 3, 4, and Eq 35) for L = 0, are presented in Fig 5, together with the corresponding simuated resuts It can be observed that the simuation and the estimation resuts are neary the same, whie the system performance is improved significanty when the number of primary paths in-

17 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading E b /N 0 = db E b /N 0 =4 db BER 10 3 E b /N 0 =6dB E b /N 0 =8 db 10 4 E b /N 0 =10 db 10 5 Estimations Simuations β Fig 7 BER vaues versus the spreading factor β in the case of K = 1, L =, λ = 30 with the increment of E b /N L= L= BER L= L= Estimations Simuations λ Fig 8 BER vaues versus the number of sampes per chip λ in the case of K = 1, β = 64, E b /N 0 = 6dB with the increment of L

18 18 Nguyen Xuan Quyen, Pere Baret-Ros creases For exampe, at the same E b /N 0 = 8dB, the BER vaues for the cases of L = 0, 1,, 3 and 4 are equa to 36 10, , , , and , respectivey The variations of the BER vaues depending on the number of users K and the spreading factor β with different vaues of E b /N 0 are shown in Fig 6 and Fig 7, respectivey The BER performance becomes significanty worse with the increment of K and sighty better with the increment of β Specificay, at the same vaue of E b /N 0 = 10dB, the BERs obtained from simuation increase from to corresponding to K changing from 1 to 10, and decrease from to with β changing from 16 to 18, respectivey In genera, the simuation resuts agree with those obtained by our estimation However, with K = 5 or β = 16, sight differences between them start to be visibe If we continue to increase K 5 or decrease β 16, these differences become more and more cear The reason is that the variabes A, B in Eq 9) are approximated by the fixed vaues A, B in Eq 40) in order to make the BER estimation simper In our chaos-based DS-CDMA system with invariabe deays, the number of sampes per chip λ can be changed by either keeping the chip duration fixed and changing the samping cock in the receiver, or by keeping the samping cock constant and changing the chip duration In the first case, despite the samping cock changing, the ratio, ie, τ j /λ, does not change Note in Eqs 9)-3) that the BER ony depends on the ratio τ j /λ and not on the specific vaues of τ j or λ Therefore, the BER does not change in this case In the second case, the simutaneous change of the chip duration T c in both the transmitter and the receiver eads to a change in the ratio τ j /λ and thus in the BER of the system Fig 8 dispays the BER vaues versus the number of sampes per chip λ for this second case The simuation resuts show that the chaos-based DS-CDMA system performs better when λ is increased For exampe in case of L = 4, the BER vaues are 10, , , and corresponding to λ = 10, 0, 30, 40 and 50 It can be observed that the estimation resuts have a good match with the simuation ones except for the cases of L = 3, 4 with λ = 10 The cause of this mismatch is that the simuation parameters chosen for these cases, ie, τ 3, τ 4 > T c = λ, are not suitabe for the channe condition in theory, ie, a deays T c 6 Concusions In this paper we investigated for the first time the BER performance of directoversamping correator-type receivers in chaos-based DS-CDMA systems over frequency non-seective fading channes The mathematica modes for the transmitting side, the generaized channe, and the proposed receiver are provided and described in detai The system performance is approximatey estimated by using theoretica derivation in combination with discrete numerica integra Our resuts show that the proposed receiver has the foowing noticeabe features: 1) It can perform we with fat fading channes, especiay

19 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 19 the BER performance is significanty enhanced when the number of secondary paths increases; ) Due to its simpe structure, the receiver is a promising aternative to RAKE receivers for the design of a secure physica ayer in WSNs or LR-WPANs; 3) Since the receiver can perform the demoduation process based on discrete-sampe processing, it coud be easiy impemented on programmabe integrated circuits IC) such as FPGA or DSP; 4) The use of an oversamping bock at the input requires that a remaining bocks in the receiver work at higher cock frequencies Athough this increases the power consumption and cost of the soution, with the current deveopment of the IC technoogy, we can easiy impement the receiver with high speed, ow consumption and ow cost ICs The hardware impementation of the proposed receiver is part of our future work Appendix A The expression for the output signa of the correator: s i) = = λ g=1 d i) g,k, = + τ 1 α 1 x i) k, = L λα 0 x i) k, i=1 λ g=1 i=1 d i) g,k, = λ g=1 r g,k, x i) k, b i) x i) k, + λ τ 1)α 1 x i) k, b i) x i) k 1, + + τ Lα L x i) k, b i) x i) k, + λα j) b m) x m) k, xi) k, + m=1,m i i=1 m=1,m i m=1,m i i=1 b i) x i) k, + + λ τ L)α L x i) k, b i) x i) k 1, + λη k,x i) k, ) ) L ) b m) x m) k, xi) k, + τ j α j b m) x m) k 1, xi) k, B The statistica cacuation for the mean vaues of X, Y : E[X b i) = +1, α j ] = = L [ λα j )E L λα j) [ E ] x i) k, x i) k, + E ) + λ m=1,m i ] L = λα j )βe c, η k, x i) k, 41) b m) x m) k, xi) k, 4) i=1 ) b i) x i) k, b i) x i) k,

20 0 Nguyen Xuan Quyen, Pere Baret-Ros E[Y b i) = +1, α j ] = L ) [ τ j α j E + E m=1,m i L ) = τ j α j b m) x m) ] x i) k, k 1, xi) k, [ E x i) k, E ) m=1,m i b m) x m) ] ) L = τ j α j )βe c, C The statistica cacuation for the mean squared vaues of X, Y, Z: k, xi) k, 43) E[X b i) = +1, α j ] = L ) λα j E + E = + L m=1,m i λα j) [ E x i) k, ] K b m) x m) [ E m=1,m i ) x i) k, + E k, xi) k, x i) k, [ E ) 4 ] β 1 + x m) k, ] ) [ E x i) k, x i) β k, ] p=1,p k L ) ) = λα j βe c4 + ββ 1)Ec + βk 1)Ec L ) ) = λα j βe c4 + ββ + K )Ec m=1,m i [ E 44) x i) p, ] b m) x m) k, xi) k,

21 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 1 E[Y b i) = +1, α j ] = L ) τ j α j E + E m=1,m i k,) + E x i) b m) x m) k 1, xi) k, ) m=1,m i b m) x m) L ) ) = τ j α j βe c4 + ββ 1)Ec + βk 1)Ec + βkec L ) ) = τ j α j βe c4 + ββ + K )Ec 45) k, xi) k, E[Z b i) = +1] = λ E η k, x i) k, ) = λ E [ ηk, ] E [x i) ] k, = λ N 0 E b D The rough BER expression obtained by the theoretica derivation: 46) BER R = 1 P robsi) = P robs i) = Q 0 b i) = +1, α j ) + 1 P robsi) > 0 b i) = 1, α j ) ) 1 0 b i) V ar[s i) b i) = +1, α j ] = +1, α j ) = Q E [s i) b i) = +1, α j ] ) ) ) ) L L λα j βe c4 + βk )Ec + τ j α j βe c4 + βk )Ec + λ β N0 E c4 E + K c = Q L τ j ) + β 1 λ α j β L α j = Q A + B + 1 C ) 1 E c4 Ec L ) L λα j L τ j α j β Ec + K ) + α j L τ j λ α j 1 1 L α j L ) τ j λ α Eb j N 0 47) 1 E c 1

22 Nguyen Xuan Quyen, Pere Baret-Ros E The statistica properties of the Chebyshev poynomia function of order : The invariant PDF of x, denoted by ρx) is ρx) = { 1 π 1 x x < 1, 0 otherwise 48) The vaues of E c and E c4 are cacuated as [ E c = E [ E c4 = E x i) k, x i) k, ] = 4 ] = x ρx)dx = x 4 ρx)dx = x 1 π 1 x dx = 1 49) x 4 1 π 1 x dx = ) Acknowedgment This research was carried out in framework of the AREAS+ schoarship program in Erasmus Mundus Action project It was aso partiay supported by the Spanish Ministry of Economy and Competitiveness and EU FEDER under grant TEC C--R SUNSET project) and by the Cataan Government ref 014SGR-147) References 1 Andrew J Viterbi, CDMA: Principes of Spread Spectrum Communication Addison- Wesey, 1 edition, 1995) R L Peterson, R E Zeimer, and D E Borth, Introduction to Spread Spectrum Communications Prentice Ha, New York, NY, USA, 1995) 3 P Stavrouakis, Chaos Appications in Teecommunications CRC Press, 005) 4 G Heidari-Bateni and C D McGiem, Chaotic sequences for spread spectrum: an aternative to PN-sequences, in IEEE Int Conf Se Topics Wireess Communications, Vancouver, Canada, 199, pp J Yu and Y-D Yao, Detection performance of chaotic spreading LPI waveforms IEEE Trans Wireess Commun 4, pp ) 6 G Heidari-Bateni and C D McGiem, A chaotic direct-sequence spread-spectrum communication system IEEE Transactions on Communications 4, pp ) 7 Q Zhang and J Zheng, Choice of chaotic spreading sequences for asynchronous DS- CDMA communication, in Proc IEEE Asia-Pacific Conf CAS, 000, pp L Cong and L Shaoquian, Chaotic spreading sequences with mutipe access performance better than random sequence IEEE Trans Circuits Syst I 47, pp ) 9 G Mazzini, G Setti, and R Rovatti, Chaotic compex spreading sequences for asynchronous DS-CDMA I System modeing and resuts IEEE Tr Circ Syst I 44, pp ) 10 R Rovatti, G Setti, and G Mazzini, Chaotic compex spreading sequences for asynchronous DS-CDMA Part II Some theoretica performance bounds IEEE Trans Circuits Syst I 45, pp ) 11 C Chen, K Yao, K Umeno, E Bigieri, Design of spread-spectrum sequences using chaotic dynamica systems and ergodic theory IEEE Trans Circuits Syst 48, pp )

23 Chaos-based DS-CDMA Performance under Frequency Non-seective Fading 3 1 R Rovatti, G Setti, G Mazzini, Toward sequence optimization for chaos-based asynchronous DS-CDMA systems, in Proc IEEE GLOBECOM, Sydney, Austraia, 1998, pp G Setti, R Rovatti, and G Mazzini, Synchronization Mechanism and Optimization of Spreading Sequences in Chaos-Based DS-CDMA Systems IEICE TRANS on Fundamentas of Eec, Commun and Computer Sciences E8-A, pp ) 14 B Jovic, C Unsworth, G Sandhu, and S Berber, A robust sequence synchronization unit for muti-user DS-CDMA chaos-based communication systems Signa Processing 87, pp ) 15 G Kaddoum, D Roviras, P Charge, and D Fournier-Prunaret, Robust synchronization for asynchronous muti-user chaos-based DS-CDMA Signa Processing 89, pp ) 16 R Vai, S Berber, and S Nguang, Effect of Rayeigh fading on noncoherent sequence synchronization for muti-user chaos based DS-CDMA Signa Processing 90, pp ) 17 S Azou, C Pistre, L L Duff, and G Bure, Sea tria resuts of a chaotic direct sequence spread spectrum underwater communication system in Proc IEEE Oceans, San Diego, Caif, USA, 003, pp W M Tam, F C M Lau, C K Tse, and M M Yip, An approach to cacuating the bit-error rate of a coherent chaos shift-keying digita communication system under a noisy mutiuser environment IEEE Transactions on Circuits and Systems I 49, pp ) 19 S Vitai, R Rovatti, and G Setti, On the performance of chaos-based muticode DS- CDMA systems Circuits, Systems, and Signa Processing 4, pp ) 0 A J Lawrance and G Ohama, Exact cacuation of bit error rates in communication systems with chaotic moduation IEEE Trans on Circuits and Systems I 50, pp ) 1 W Tam, F Lau, C Tse, and A Lawrance, Exact anaytica bit error rates for mutipe access chaos-based communication systems IEEE Trans Circuits Syst II 51, pp ) G Kaddoum, P Charge, D Roviras, D Fournier-Prunaret, A Methodoogy for Bit Error Rate Prediction in Chaos-based Communication Systems Circuits Syst Signa Processing 8, pp ) 3 G Kaddoum, P Charge, and D Roviras, A Generaized Methodoogy for Bit-Error- Rate Prediction in Correation-Based Communication Schemes Using Chaos IEEE Comm Letters 13, pp ) 4 R Rovatti, G Mazzini, and G Setti, Enhanced rake receivers for chaos-based DS- CDMA IEEE Trans Circuits and Sys I 48, pp ) 5 G Mazzini, R Rovatti, and G Setti, Chaos-based asynchronous DS-CDMA systems and enhanced rake receivers: measuring the improvements IEEE Transactions on Circuits and Systems I 48, pp ) 6 G Kaddoum, D Roviras, P Charge, and D Fournier-Prunaret, Accurate bit error rate cacuation for asynchronous chaos-based DS-CDMA over mutipath channe EURASIP Journa on Advances in Signa Processing, vo 009, ID , 1 pages 009) 7 G Kaddoum, M Couon, DRoviras, and P Charge, Theoretica performance for asynchronous muti-user chaos-based communication systems on fading channes Esevier Signa Processing 90, pp ) 8 E N Gibert, Increased Information Rate by Oversamping IEEE Transactions on Information Theory 39, pp ) 9 B E-Khadi, J M Rouvaen, A Menhaj, Y E hiai, Averaging and oversamping correator receiver with input quantization Digita Signa Processing 16, pp ) 30 Y S Lee and B S Seo, OFDM receivers using oversamping with rationa samping ratios IEEE Trans Consumer Eectronics 55, pp ) 31 S Berber and N Chen, Physica Layer Design in Wireess Sensor Networks for Fading Mitigation J Sens Actuator Netw, pp ) 3 IEEE Standard for Loca and Metropoitan Area Networks-Part 154: Low-Rate Wireess Persona Area Networks LR-WPANs) IEEE Standard Revision of IEEE Std ); IEEE: New York, NY, USA, 011; pp 1-94

24 4 Nguyen Xuan Quyen, Pere Baret-Ros 33 A F Moisch, K Baakrishnan, D Cassioi, C-C Chong, S Emami, A Fort, J Kareda, J Kunisch, H Schantz, U Schuster, and K Siwiak, Ieee 80154a channe mode fina report, The Institution of Eectrica Engineers 34 M K Simon and MS Aouini, Fading Channe Characterization and Modeing, in Digita Communication over Fading Channes-A Unified Approach to Performance Anaysis, CRC Press, CC Wang and et a, Zig-Bee 868/915-MHz moduator/demoduator for wireess persona area network IEEE Trans Very-Large Scae IntegrVLSI) Syst 16, pp ) 36 N J Oh, S G Lee, Buiding a 4-GHZ radio transceiver using IEEE IEEE Circuits Devices Mag 1, pp ) 37 M A A-Jarraha, N K A-Ababnehb, M M A-Ibrahimb, R A A-Jarrah, Cooperative OFDM for semi distributed detection in wireess sensor networks AEU Int Jour of Eect and Commu 68, pp ) 38 M R Isam, Y S Han, Cooperative MIMO communication at wireess sensor network: An error correcting code approach J Sens Actuator Netw 11, pp ) 39 S M Berber, Probabiity of error derivatives for binary and chaos-based CDMA systems in wide-band channes IEEE Trans Wireess Comm 13, pp ) 40 M M Siddiqui, Statistica inference for Rayeigh distributions Journa of Research of the Nationa Bureau of Standards Vo 68D, pp ) 41 C M Grinstead and J L Sne, Centra Limit Theorem, Introduction to Probabiity American Mathematica Society, nd Revised edition, Juy 1997), pp