Circuits Syst Signal Process (2009) 28: 925–944 DOI 10.1007/s00034-009-9124-5
A Methodology for Bit Error Rate Prediction in Chaos-based Communication Systems G. Kaddoum · P. Chargé · D. Roviras · D. Fournier-Prunaret
Received: 2 April 2008 / Revised: 22 October 2008 / Published online: 20 August 2009 © Birkhäuser Boston 2009
Abstract This paper is devoted to the derivation of an exact analytical expression of the bit error rate for chaos-based DS-CDMA systems. For the studied transmission system, we suppose that synchronization is achieved perfectly, coherent reception is considered, and an Additive White Gaussian Noise channel (AWGN) is assumed. In the first part of the paper, performance of a mono-user system with different chaotic sequences is evaluated and compared in terms of the error probability. This comparison is realized thanks to the probability density function of the bit energy of a chaotic sequence. The bit error rate can be easily derived by numerical integration. In some particular cases, for certain chaotic sequences with known probability density function of bit energy, we propose an analytical expression of the bit error. In the second part of the paper, the performance of a chaos-based DS-CDMA system is evaluated in the multi-user case. A general conclusion is that probability density function of chaos bit energy, for a given spreading factor, can give a clear idea about how to choose a “good” chaotic sequence for improving the performance of the chaos-based CDMA system. Keywords Chaos-based DS-CDMA · Energy distribution · Bit error rate · Multi-user interference G. Kaddoum () · D. Roviras Signal & Communication, IRIT/ENSEEIHT, University of Toulouse, Toulouse, France e-mail:
[email protected] D. Roviras e-mail:
[email protected] G. Kaddoum · P. Chargé · D. Fournier-Prunaret Nonlinear Dynamical Systems, LATTIS/INSA, University of Toulouse, Toulouse, France P. Chargé e-mail:
[email protected] D. Fournier-Prunaret e-mail:
[email protected]
926
Circuits Syst Signal Process (2009) 28: 925–944
1 Introduction In the last decade, many researches showed high interest in the application of chaotic signals to communication systems [23]. This interest is related to the advantages that chaotic sequences can offer, such as robustness in multipath environments, resistance to jamming, low probability of interception, and security of transmission [15]. One of these applications is the spread spectrum technique using chaotic sequences instead of classical antipodal pseudo-noise (PN) sequences. Nowadays, PN sequences are the most popular sequences for direct sequence spread spectrum (DS/SS). These sequences have good correlation properties but they have limited security; they can be reconstructed by a linear regression attack because of their short linear complexity [20]. On the other hand, using chaotic sequences increases transmission security because chaotic signals can be seen as non-periodic signals with an infinite number of states [10, 23]. In addition, stringent statistical properties concerning the magnitude of the auto- and cross-correlation functions of chaotic sequence [11] have motivated researchers to study performances of chaos-based communication systems [23]. Many schemes have been proposed and studied like Chaos Shift Keying (CSK) [6], Differential Chaos Shift Keying (DCSK) [12, 13] and chaos-based Direct Sequence Code Division Multiple Access (DS-CDMA) systems [14, 23]. Coherent systems like CSK and chaos-based DS-CDMA require coherent correlators with the assumption that the receiver is able to generate a locally synchronous chaotic signal. In the following, we will focus our study of Bit Error Rate (BER) performance on coherent chaotic systems like chaos-based DS-CDMA. In BER performance computation, many results have been presented using various assumptions. Because of these assumptions, computed BER are generally different from their true value. The simplest approximation used in [2], for example, is to consider the transmitted chaotic bit energy being constant. This approximation can be reasonable when the considered spreading factors are very large (symbol duration much greater than chaotic chip duration). Nevertheless, for small or moderated spreading factors, these assumption yields to very imprecise BER performance. In fact, because of the non-periodic nature of chaotic signals, the transmitted bit energy of chaos-based DS-CDMA systems varies from one bit to another. Another classical assumption is to use the “Gaussian approximation” for the decision parameter at the correlator output [15, 24, 25], based upon the Central Limit Theorem. In that case, the correlator output is supposed to follow a Gaussian law. Tam et al. [25] have proposed a simple way of deriving the BER of the CSK system by computing numerically the first two moments of chaotic signal correlation functions. This approximation is also valid for very high spreading factors but suffers from precision for small ones [26]. In [16] and [27], a third method for computing BER performance has been developed by Lawrance and Ohama. They did not use the constant bit energy approximation neither the Gaussian assumption. Only additive channel noise and multiple access interference noise follow, in their study, a Gaussian distribution. Their approach enables full dynamics of the chaotic sequence by integrating the BER expression for a given chaotic map over all possible chaotic sequences for a given spreading factor. This latter method is compared to the BER computation under Gaussian assumption
Circuits Syst Signal Process (2009) 28: 925–944
927
in [26] and seems more realistic to match the exact BER. But, as it is said in [16], the aim of the method was not to give implementable procedures for realistic sized systems. The main goal of our paper is to study the BER performance of coherent chaosbased DS-CDMA systems when using short spreading factors. Because previous presented approximations are not valid for such small spreading factors, we have developed a new method for computing the BER performance. This new method gives very accurate results even for very small spreading factors together with a very low computation charge. The idea is to compute the Probability Density Function (PDF) of the chaotic bit energy and to integrate BER over all possible values of the PDF. For certain chaotic maps it is possible to find an analytical expression of the PDF bit energy, and therefore, after integration, an analytical expression of the BER. The shape of the PDF bit energy is a qualitative indication concerning BER performances. As we will see, the sharper the PDF is, the better will the BER performance be. Even though a large literature exists on chaotic spreading sequences design [4], optimization [22], and modeling [18, 19], this new approach can give more insight in the choice of a chaotic map. The paper is organized as follows. In Sect. 2, we first present the different chaotic maps that will be used throughout the paper together with the transmission system (receiver and transceiver architecture). Section 3 is devoted to the BER performance analysis in mono-user case of chaotic sequences by using the PDF of the bit energy. The influence of the shape of the PDF bit energy is presented. In Sect. 4, BER derivation methodologies for BER computation are presented. In the first part, numerical BER computation is proposed by numerical integration of the BER over all possible values of the bit energy. This method is always possible and precision and computation charge is independent form the chaotic map neither the spreading factor. In the second part, an analytical method for BER computation is derived using the PDF of the bit square energy. This last analytical method is applied in Sect. 5 for the exact analytical expression of the BER for a class of chaotic sequences generated by a piecewise-linear map. Finally, in Sect. 6, BER performance is evaluated in the multi-user case. The last section reports some conclusive remarks.
2 System Description 2.1 Chaotic Generators In order to better describe the influence of the shape of a chaotic map, in this study, two kinds of one-dimensional maps have been chosen, namely a smooth function and a piecewise function. The first one is the Chebyshev polynomial function of order 2 (CPF), and the second one is a piecewise linear map (PWL). 2.1.1 CPF Map The CPF map is given by 2 − 1. xk = 2xk−1
(1)
928
Circuits Syst Signal Process (2009) 28: 925–944
The choice of this map is related to its simplicity for generating a chaotic sequence. Moreover, it is shown in [9], that it allows better performances than many other maps for chaos-based DS-CDMA systems. 2.1.2 Piecewise Linear Map The PWL map is given in [3] by zk = K|xk | + φ [mod 1], xk+1 = sign(xk )(2zk − 1).
(2)
This map depends on K and φ parameters. K is a positive integer and φ (0 < φ < 1) is a real number. Both can be changed to produce different sequences, and the initial condition x0 will be chosen to satisfy the condition 0 < x0 < 1/K. In addition, this map has a known PDF for the energy distribution. This PDF will be used later in this paper in order to derive the exact analytical BER expression. Throughout the paper, the PWL parameters are fixed as follows: K = 3, φ = 0.1. 2.2 Transmitted Signal The studied system is a DS-CDMA communication system with M users. As shown in Fig. 1, a stream of data symbols from user m(Si(m) = ±1) with period TS are spread by a chaotic signal x (m) (t) generated from (1) or (2) at the emitter side. Symbols of different users are independent of one another. The chaotic sequences of all users are generated using the same chaotic generator with different initial conditions. A new chaotic sample (or chip) is generated for every time interval equal to (m) TC (xk = x (m) (kTC )). The mean value of each chaotic sequence is equal to zero. The spreading factor β is equal to the number of chaotic samples in a symbol duration (β = TS /TC ). g(t) is the pulse shaping filter commonly found in communication systems. This pulse shaping filter can take different forms like Raised-cosine filter, Gaussian filter, etc. In this paper, we have chosen a rectangular pulse of unit amplitude on [0, TC ]. 2.3 Received Signal As shown in Fig. 2(a), channels of all users are flat fading; therefore, all frequency components of the signal will experience the same magnitude of fading. We assume also that we are in a slow fading transmission which means that amplitude and phase Fig. 1 Emitter number m
Circuits Syst Signal Process (2009) 28: 925–944
929
Fig. 2 (a) Multi-user received signal; (b) Receiver for user n
changes imposed by the channel can be considered roughly constant over a long sequence of transmitted symbols. Furthermore, all users are asynchronous. The multi-user received signal is given by
v(t) =
M ∞ β−1
(n) Cn Si(n) xiβ+k g t − (iβ + k)TC − τ (n) + ϑ(t),
(3)
n=1 i=0 k=0 (n)
where xiβ+k are the chaotic samples corresponding to the ith data symbol of user (n)
number n (Si ), τ (n) is the propagation delay and ϑ(t) is an Additive White Gaussian Noise with power spectral density equal to N0 /2. We assume that we have a perfect power control which means that the channel gains Cn are taken equal to 1 for the rest of the paper. The perfect chaos synchronization is assumed at the receiver side. This means that the sampling at the output of the matched filter takes in consideration the propagation delay τ (n) . For the demodulation of user number n, an exact replica of the (n) emitted chaotic sequence xiβ+k is generated locally at the receiver side. The offset k is computed by taking into account the propagation delay τ (n) . Without loss of generality, we will take τ (n) = 0 in the rest of the paper. Chaotic sequences being low cross-correlated for every lag, this hypothesis is not restrictive. For the demodulation process shown in Fig. 2(b), the received signal is first despread by the local chaotic sequence and then integrated on a symbol duration TS .
930
Circuits Syst Signal Process (2009) 28: 925–944
3 Performance Analysis in Mono-user Case In this section, we present a new approach for computing the BER for the chaosbased DS-CDMA system. This approach is based on the bit energy distribution of the spreading chaotic sequence. This methodology leads to an accurate performance prediction with low computing charge. In the mono-user case, we take for simplification (n) = xiβ+k ). (xiβ+k At the correlator output, the decision variable associated to the ith symbol can be written as D i = S i TC
β−1
(xiβ+k )2 + TC
k=0
β−1
(i)
ϑk xiβ+k = Si Ebc + ni ,
(4)
k=0
(i) where Ebc is the chaotic energy corresponding to the ith symbol interval (i)
Ebc = TC
β−1
(xiβ+k )2
(5)
k=0
and ni is the additive Gaussian noise after de-spreading and correlation with the chaotic sequence. In the constant energy case (classical antipodal DS-CDMA or BPSK cases), the BER is given by the following expression [21]: 2Eb BERBPSK = Q , (6) N0 +∞ −u2 where Eb is the constant bit energy and Q(x) = x √1 e( 2 ) du. In the chaos2π based DS-CDMA system the energy of (5) does not have a zero variance for all transmitted bits. For the ith symbol, the mean and variance of the decision variable are derived as follows:
(i) (i) E[Di ] = E Si Ebc + ni = Ebc Si . (7) For the symbol duration i, the variance is given by Var[Di ] = E
TC
β−1 k=0
2 ϑk xiβ+k
=
N0 (i) E . 2 bc
(8)
According to the fact that data symbols are equally distributed on the set {+1, −1}, and using (7) and (8), it comes that the detection probability of the symbol i (with (i) energyEbc ) is given by (i) 2Ebc Per(i) = Q . (9) N0
Circuits Syst Signal Process (2009) 28: 925–944
931
In this case, the BER of the system is given by the integral of (9) over all possible values of the bit energy +∞ 2Ebc Q (10) p(Ebc ) dEbc , BERchaos = N0 0 where Ebc is the chaos energy computed on a symbol duration TS , and p(Ebc ) is the corresponding probably density function (PDF) . 3.1 Energy Distribution of Chaotic Sequences In order to compute (10), it is necessary to get the bit energy distribution. First of all, we have fitted the histogram of the energy distribution for the chaotic sequences under study. Figure 3 shows the histogram of the bit energy for three different spreading sequences and for a spreading factor equal to 10. Two of them are described in Sects. 2.1.1 and 2.1.2, while the third one is a Gaussian distributed random sequence with uncorrelated samples. The histogram is obtained here by using one million chaotic samples. From these samples, energies of successive bits are calculated for a given spreading factor. The bit energy is assumed to be the output of a stationary random process [8]; hence, the histogram obtained in Fig. 3 can be considered as a good estimation of the PDF. Looking carefully to the PDF distribution in Fig. 3, we can find that the CPF sequence will give better results in terms of BER than the two other sequences because the CPF sequence has lower distribution values for low energies. In contrast, the PDF energy of the Gaussian distributed sequence has higher distribution values for low energy. This will give a worse result in terms of BER for this
Fig. 3 Histogram of bit energy for various spreading sequences
932
Circuits Syst Signal Process (2009) 28: 925–944
sequence. In other words, it appears that performance provided by chaotic sequences will lay between two bounds: a lower bound given by antipodal PN sequences (BPSK case) and a higher bound given by the Gaussian distributed sequence. 3.2 Lower and Upper Bound of the BER Before starting to study performances of chaotic sequences, let us see the lower and upper bound performances of a chaos-based DS-CDMA system. The lower bound of the BER for chaos-based DS-CDMA system is given by (6) when the bit energy is constant. As shown in (10), the BER is a function of the distribution of the bit energy. The degradation in performance is more important when the distribution of the bit energy is high for low values of the energy. In [16], the Gaussian sequence appears to be a very bad sequence for spreading spectrum and the BER is close to be an upper bound of the BER for all types of chaotic maps (nevertheless, there is no demonstration of this behavior). This sequence is made of independent samples with zero mean and variance equal to σX2 . The BER for Gaussian sequence is computed in [16] as σX BERGauss = E Q − , (11) χβ2 σ where χβ2 is a chi-square random variable with β degrees of freedom, and σ 2 the variance of the Gaussian additive noise.
Fig. 4 BER lower and upper bound simulation
Circuits Syst Signal Process (2009) 28: 925–944
933
Fig. 5 Histogram of two successive square samples of Chebyshev chaotic map
Figure 4 shows BER lower and upper bounds of the system obtained in (6) and (11), respectively, with the simulation performance of CPF, PWL and the Gaussian sequence for β = 1, 2, 5, 10, and a number of bits equal to 104 . As we can see from Fig. 4, the performance obtained by using the piecewise linear map is increasing regularly when the spreading factor is increasing, too. On the other hand, for the Chebyshev map, the improvement is very important between a spreading factor equal to 1 and a spreading factor equal to 2. This improvement in the BER performance can be explained by the correlation between successive squared samples in the Chebyshev sequence. 2 ] In Fig. 5, the histogram of the couple of two consecutive square samples [xk2 , xk+1 is plotted for the Chebyshev case for one million chaotic samples. As we can see from Fig. 5, two consecutive samples are highly correlated. The probability of having two samples very close to zero is very low. This means that having bit energy very low, for β = 2, has a very low probability. On the contrary, the probability of having two samples of the form (0, +1) or (+1, +1) is high, yielding a high bit energy (almost 1 of the two values is high). This is the reason for the great improvement of BER from a spreading factor equal to 1 to a spreading factor equal to 2 for the Chebyshev map. For the piecewise linear map, we have less dependency between consecutive samples and, in addition, the probability to find two successive samples close to zero is high, as shown in Fig. 6. This behavior explains why the improvement for PWL is lower than the CPF one when the spreading factor goes from 1 to 2.
4 BER Derivation In order to compute (10), we can adopt two approaches. The first one is always applicable and relies on a numerical integration of (10). The second approach is based on an analytical resolution of (10) in order to get an analytical expression of the BER.
934
Circuits Syst Signal Process (2009) 28: 925–944
Fig. 6 Histogram of two successive square samples of piecewise linear map for K = 3, φ = 0.1
4.1 Numerical BER Integration When the PDF of the bit energy is not close to a classical distribution (example of CPF PDF in Fig. 3), the analytical integration of (10) is impossible and the only way is to make a numerical integration. The numerical integration expression is given by
BERchaos ≈
m i=1
Q
(i)
2Ebc N0
(i) P Ebc ,
(i)
(12)
where m is the number of histogram classes and P (Ebc ) is the probability of having (i) the energy in intervals centered on Ebc . This approach can be applied for any type of chaotic sequence with quite simple operations: histogram of the bit energy followed by a numerical integration. In addition, this approach explores the dynamics properties of chaotic sequence and gives results with very high accuracy. Moreover, due to its low computation change, this method can be implemented for realistically-sized systems. Figure 7 gives simulation results together with the numerical integration method of the BER. BER of two sequences (CPL, PWL (K = 3)) for β = 10 plus the lower bound corresponding to the BPSK case are given in Fig. 7. Computed BER in Fig. 7 are obtained by using the histograms of Fig. 3 together with (12). The perfect match between simulation results and our numerical method confirms the accuracy of this approach. For a lower spreading factor (β = 5 in Fig. 8) performance of different sequences tends to worsen compared to the BPSK lower bound because when the spreading factor is lower the energy distribution tends to be broader. For a DS-CDMA system, a low spreading factor has a limited benefit. To improve the BER in a chaos based-DS-CDMA, we can increase the spreading factor. When the spreading factor is very high (β = 50 in Fig. 9) all BER tend to be equal to the lower BER bound (BPSK case). This means that the energy distribution tends to be very
Circuits Syst Signal Process (2009) 28: 925–944
Fig. 7 Computed and simulated BER performance for different chaotic sequences (β = 10)
Fig. 8 Computed and simulated BER performance for different chaotic sequences (β = 5)
935
936
Circuits Syst Signal Process (2009) 28: 925–944
Fig. 9 Computed BER performance for different chaotic sequences (β = 50)
sharp around the mean energy when the spreading factor is high. This is illustrated in Fig. 9. 4.2 Analytical Approach In order to develop analytically (10), two solutions can be proposed: +∞ 2Y1 Q 1-BERchaos = p(Y1 ) dY1 ; Y1 = Ebc , N0 0 +∞ 2Y22 Q 2-BERchaos = p(Y2 ) dY2 ; Y2 = Ebc . N0 0
(13)
(14)
To compute (13) (respectively, (14)), it is fundamental to have an expression of the PDF of the bit energy (respectively, of the root square of the bit energy (RSE)). The analytical expression from (13) seems more difficult to derive. In the meantime, referring to the intensive work on the analytical expression of (14), the second solution seems to be more tractable. In the framework of mobile radio channels, the BER expression is given by +∞ 2λ2 Eb BER Radio = Q p(λ) dλ, (15) N0 channel 0 where λ is the channel attenuation. Analytical expressions of (15) are derived in the cases of channels following Rayleigh [5], Nakagami [7] or Rice distribution [17].
Circuits Syst Signal Process (2009) 28: 925–944
937
Fig. 10 Estimation of the root square energy distribution for PWL map (K = 3, φ = 0.1, β = 10)
Because (14) and (15) have the same form, we have mainly focused our interest on (14) in order to get an analytical expression of the BER. In order to derive (14), we need the PDF of RSE. Deriving analytically the PDF seems intractable because chaotic samples are not totally independent. A second solution is to approximate the histogram of the RSE with a known PDF (Rayleih, Nakagami or Rice). This will allow us to determine an analytical expression of (14). Looking at Fig. 3, the CPF sequence has an irregular shape of its PDF, and computations to get this analytical expression have appeared intractable. On the other hand, the PWL sequence seems to be an interesting candidate to compute the analytical BER expression. Fig. 10 shows the RSE distribution of PWL sequence obtained for 106 samples and β = 10. Looking at the shape of the PDF of RSE in Fig. 10, two possible candidate laws have been tested: Rice and Nakagami PDF. Moreover, (15) has an analytical expression for these two laws. Rice and Nakagami estimated PDF are plotted and compared to the PDF of RSE. The Chi-Square Goodness-of-Fit test confirms an advantage for the Rice distribution.
5 Analytical BER Derivation in the PWL Case The RSE approximation by Rice distribution leads to an analytical BER expression for PWL chaotic map. The general Rice distribution function is given in Appendix A.
938
Circuits Syst Signal Process (2009) 28: 925–944
5.1 Parameters Estimation of Rice Distribution In order to obtain an analytical BER expression, the parameters of the Rice distribution (30) must be derived from the parameters of the PWL sequence. Since the bit energy is connected to the Rice variable via (i)
R 2 = Ebc = TC
β−1
(xiβ+k )2 ,
(16)
k=0
parameters Ω and γ of (30) can be identified (see Appendix A). 2 ) for one chaotic sample is equal to 1/3. The derived E(xiβ+k Then, the scale parameter Ω is given by TC Ω = E R 2 = TC E (xiβ+k )2 = β . 3 β−1
(17)
k=0
The variance of R 2 is then
2 4 TC 2 Var R = E R − β , 3
(18)
βTc2 2 2 + 2Tc2 E R4 = , (β − n)E xiβ+k xiβ+k+n 5
(19)
with β−1 n=1
2 2 xiβ+k+n ) is estimated using the PWL chaotic sequence. Then, γ can where E(xiβ+k be obtained using (30) as
γ=
β−1 2 β2 9 2 +2 − 1. (β − n)E xiβ+k xiβ+k+n 5β 9
(20)
n=1
5.2 Analytical BER Expression Analytical BER expression derived from (15) when the PDF of the root square energy is a Rice distribution is [17] 2 u + v2 d 1 exp − I0 (uv), (21) BERPWL = Q(u, v) − 1 + 2 1+d 2 where Q(u, v), u, v, d are given in Appendix B. Figure 11 compares the BER obtained using the analytical expression of (21) and the ones given by Monte Carlo simulations. The lower bound BER of (6) is also plotted for reference. It clearly appears in Fig. 11 that we have a perfect match between simulations and the analytical results. Expression (21) can thus be used for all types of spreading factors even for very small ones.
Circuits Syst Signal Process (2009) 28: 925–944
939
Fig. 11 Analytical and simulated BER for β = 5, 10, 30 of PWL map with K = 3, φ = 0.1
6 BER Expression in the Multi-user Case In the previous sections, we have detailed our approach to derive the BER numerically or analytically. In this section, we generalize to a multiple access chaos-based communication system, we study the effect of the multi-user interference noise, and we show that our approach can be applied easily in the multi-user case. The channel is an AWGN channel, and we assume that we use a coherent chaotic demodulator. Perfect synchronization of the chaos is assumed for the user of interest. The received signal is expressed in (3). 6.1 Multi-user Interference for Multiple Access Chaos-based DS-CDMA The decision variable at the output of the correlator, associated to the ith symbol of user m is (m)
Di
(m)
= Si
TC
β−1 k=0
where yiβ+k =
M n=1 n=m
(m) (m) 2 xiβ+k + TC xiβ+k (yiβ+k + ϑk ), β−1
(22)
k=0
(n) (n)
Si xiβ+k , and ϑk is the additive Gaussian noise component
during the kth chip of the ith data symbol. According to the fact that yiβ+k is the sum of M − 1 samples of M − 1 independent zero mean chaotic sequences (from different independent users), we will consider that
940
Circuits Syst Signal Process (2009) 28: 925–944
yiβ+k is a normal random sequence with the following statistics: E[yiβ+k ] = 0,
(23)
Var[yiβ+k ] = (M − 1)σx2 ,
(24) (n)
where σx2 is the mean power of a spreading chaotic sequence xiβ+k . It can be also demonstrated that the mean energy of a transmitted chaotic chip of the sequence (n) xiβ+k is EC = TC σx2 . ϑk and yiβ+k are then two independent zero mean Gaussian random sequences. Di can then be considered as a normal random variable with the following moments:
(m) (m) (i,m) = Si Ebc , E Di
(m) (i,m) Var Di = Ebc (M − 1)EC + N0 /2 . The general expression of the BER in multi-user case is given by (i,m) Ebc BERi,m = Q . (M − 1)EC + N0 /2
(25) (26)
(27)
The BER is given by the integration of (27) over all possible values of the bit energy +∞ 2Ebc BERchaos = Q (28) p(Ebc ) dEbc , N0 + 2(M − 1)EC 0 (m)
where p(Ebc ) is the corresponding probability density function (PDF). As shown before, (28) can be derived numerically or analytically. In the multi-user case, we also have a perfect match between the three BER as shown in Fig. 12. These simulation results validate the exactitude of our assumption by considering the multi-user interference as a Gaussian noise in (26). In the multi-user case, a higher spreading factor will reduce the multi-user interference noise by spreading these interference powers over a larger band (see Fig. 13).
7 Conclusion Since chaotic sequences are deterministic, neither the Gaussian approximation for the decision variable nor the constant bit energy assumptions are very accurate. In this paper, we have presented a new simple methodology for computing the BER expression. In our study, an AWGN channel is used and a perfect synchronization of the chaos is assumed for mono- and multi-user cases at the receiver side. The new approach to derive the BER is based on the bit energy distribution and gives accurate results in perfect match with simulations. Furthermore, computation charge is very small and independent of the chaotic map and spreading factor. When different
Circuits Syst Signal Process (2009) 28: 925–944
941
Fig. 12 Analytical expression, numerical computation, and simulated BER for β = 5 and M = 1, 2, 3 of PWL map with K = 3, φ = 0.1
Fig. 13 Analytical, computed, and simulated BER for β = 64 and M = 1, 4, 8 of PWL map with K = 3, φ = 0.1
942
Circuits Syst Signal Process (2009) 28: 925–944
chaotic maps are compared, the best performance (lower bound) in terms of BER corresponds to constant bit energy (classical antipodal PN sequence, BPSK case). An upper bound for chaos-based DS-CDMA systems corresponds to a Gaussian sequence that gives the higher distribution of the bit energy for low energy values. In addition, the histogram of bit energy gives qualitative results concerning the BER performance of chaotic sequences used for spreading spectrum. The best chaotic sequence for a chaos-based DS-CDMA system is the sequence that has a small variance of energy around its mean value. When the PDF of the bit energy has an irregular shape as it is the case with the CPF sequence, only numerical integration methods are possible in order to compute the BER. In special cases, when the distribution of the root square bit energy has a known distribution (Rice, Nakagami or Rayleigh), we can derive a full exact analytical BER expression. For the PWL chaotic map, such an analytical expression has been obtained which perfectly matched with simulation results, even for small spreading factors. In the multi-user case, the multi-user interference was computed and approximated by a Gaussian distribution. Simulation results confirmed the validity of our Gaussian multi-user interference assumption. Appendix A General Rice Distribution Function (i) General Rice distribution function is given by (29) [1], where R = Ebc is a random variable following the Rice distribution function 2(Kr +1)r (Kr +1)r 2 Kr (Kr +1) exp(−K − )I (2 r) if r ≥ 0, r 0 Ω Ω Ω pR (r) = 0 if r < 0, where r ≥ 0, Kr ≥ 0, Ω ≥ 0.
(29)
I0 (x) is the zero order modified Bessel function of the first kind, and Ω is the scale parameter. Also
2 γ = Var R 2 / E R 2 . (30) Ω = E R2 ; The shape parameter K can be expressed in terms of γ explicitly as [1] √ 1−γ Kr = . √ 1− 1−γ
(31)
Let us define the following variables: σ2 =
Ω 2(Kr + 1)
and α 2 =
Kr Ω . (Kr + 1)
The PDF is then obtained by a simple change of variables in (29) 2 +α 2 r αr exp(− r 2σ if r ≥ 0, 2 )I0 ( σ 2 ) pR (r) = σ 2 0 elsewhere.
(32)
Circuits Syst Signal Process (2009) 28: 925–944
943
Appendix B The parameters of BER over Rician channel √ γr2 [1 + 2d − 2 d(d + 1)] , u= 2(1 + d) √ γr2 [1 + 2d + 2 d(d + 1)] v= , 2(1 + d) γr2 =
α2 , 2σ 2
d = σ2
Eb , N0
(33)
(34)
where Eb is the constant bit energy before spreading, and Q(α, β) is the Marcum Q-function [17] given by ∞ 1 2 2 x M e−(x +α )/2 IM−1 (αx) dx, (35) Q(α, β) = M−1 α β and In (x) is a modified Bessel function of the first kind.
References 1. A. Abdi, C. Tepedelenlioglu, M. Kaveh, G. Giannakis, On estimation of K parameter for Rice fading distribution. IEEE Commun. Lett. 5, 92–94 (2001) 2. S. Azou, C. Pistre, G. Burel, A chaotic direct sequence spread-spectrum system for underwater communication, in Proc. IEEE-Oceans, Biloxi, Mississippi, 29–31 October 2002, pp. 2409–2415 3. P. Chargé, D. Fournier-Prunaret, V. Guglielmi, Features analysis of a parametric PWL chaotic map and its utilization for secure transmissions. Chaos Solitons Fractals 38, 1411–1422 (2008) 4. C.C. Chen, K. Yao, K. Umeno, E. Biglieri, Design of spread-spectrum sequences using chaotic dynamical systems and ergodic theory. Proc. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 1110–1114 (2001) 5. J. Cheng, N.C. Beaulieu, Accurate DS-CDMA bit-error probability calculation in rayleigh fading. IEEE Trans. Wirel. Commun. 1, 3–15 (2002) 6. H. Dedieu, M.P. Kennedy, M. Hasler, Chaos shift keying: Modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits. IEEE Trans. Circuits Syst. II 40, 634–642 (1993) 7. R. Esposito, Error probabilities for the Nakagami channel. IEEE Trans. Inf. Theory 13, 145–148 (1967) 8. S.H. Isabelle, G.W. Wornell, Statistical analysis and spectral estimation techniques for onedimensional chaotic signals. IEEE Trans. Signal Process. 45, 1495–1997 (1997) 9. G. Kaddoum, P. Chargé, D. Roviras, D. Fournier-Prunaret, Comparison of chaotic sequences in a chaos based DS-CDMA system, in Proc. International Symposium on Nonlinear Theory and Its Applications (NOLTA), Vancouver, Canada, 16–19 September 2007 10. M.P. Kennedy, R. Rovatti, G. Setti, Chaotic Electronics in Telecommunications (CRC Press, Boca Raton, 2000), pp. 151–180, Chap. 6 11. T. Khoda, A. Tsuneda, Pseudo noise sequences by chaotic nonlinear maps and their correlations properties. Fundam. Electron. Commun. Comput. Sci. 76, 855–862 (1993) 12. G. Kolumbán, G. Kis, Z. Jákó, M.P. Kennedy, FM-DCSK: A robust modulation scheme for chaotic communications. Trans. Fundam. Electron. Commun. Comput. Sci. 81, 1798–1802 (1998) 13. G. Kolumbán, B. Vizvári, W. Schwarz, A. Abel, Differential chaos shift keying: A robust coding for chaotic communication, in Proc. Nonlinear Dynamics of Electronic Systems, Seville, Spain, June 1996, pp. 87–92
944
Circuits Syst Signal Process (2009) 28: 925–944
14. A.P. Kurian, S. Puthusserypady, S.M. Htut, Performance enhacment of DS/CDMA system using chaotic complex spreading sequence. IEEE Trans. Wirel. Commun. 4, 984–989 (2005) 15. F.C.M. Lau, C.K. Tse, Chaos-based Digital Communication Systems (Springer, Berlin, 2003), pp. 17– 145, Chaps. 2, 3, 4, 5, 6 16. A.J. Lawrance, G. Ohama, Exact calculation of bit error rates in communication systems with chaotic modulation. IEEE Trans. Circuits Syst I, Fundam. Theory Appl. 50, 1391–1400 (2003) 17. W.C. Lindsey, Error probabilities for Rician fading multichannel reception of binary and N -ary signals. IEEE Trans. Inf. Theory 10, 339–350 (1964) 18. G. Mazzini, G. Setti, R. Rovatti, Chaotic complex spreading sequences for asynchronous DSCDMA—I: System modeling and results. IEEE Trans. Circuits Syst. I 44, 937–947 (1997) 19. G. Mazzini, G. Setti, R. Rovatti, Chaotic complex spreading sequences for asynchronous DSCDMA—II: Some theoretical performance bounds. IEEE Trans. Circuits Syst. I 45, 496–506 (1998) 20. R.L. Peterson, R.E. Zeimer, D.E. Borth, Introduction to Spread Spectrum Communications (PrenticeHall, New York, 1995) 21. J.G. Proakis, Digital Communications (McGraw-Hill, New York, 2001), pp. 233–254, Chap. 5 22. R. Rovatti, G. Setti, G. Mazzini, Toward sequence optimization for chaos-based asynchronous DSCDMA systems, in Proc. IEEE GLOBECOM, Sydney, Australia, 8–12 November 1998, pp. 2174– 2179 23. P. Stavroulakis, Chaos Applications in Telecommunications (CRC Press, New York, 2006), pp. 14– 123, Chaps. 2, 3 24. M. Sushchik, L.S. Tsimring, A.R. Volkovskii, Performance analysis of correlation-based communication schemes utilizing chaos. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 1684–1691 (2000) 25. W.M. Tam, F.C.M. Lau, C.K. Tse, An approach to calculating the bit error rate of a coherent chaosshift-keying digital communication system under a noisy multiuser environment. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49, 210–223 (2002) 26. W.M. Tam, F.C.M. Lau, C.K. Tse, A.J. Lawrance, Exacte analytical bit error rate for multiple access chaos-based communication systems. IEEE Trans. Circuits Syst. II, Express Briefs 51, 473–481 (2004) 27. J. Yao, A.J. Lawrance, Bit error rate calculation for multi-user chaos-shift keying communication system. IEICE Trans. Fond. E 87, 2280–2291 (2004)
