Removing Cyclostationary Properties in a Chaos ... - Springer Link
Circuits Syst Signal Process (2011) 30:1391–1400 DOI 10.1007/s00034-010-9232-2 B R I E F C O M M U N I C AT I O N
Removing Cyclostationary Properties in a Chaos-Based Communication System Georges Kaddoum · Samuel Gagné · François Gagnon
Received: 10 July 2010 / Revised: 5 November 2010 / Published online: 30 November 2010 © Springer Science+Business Media, LLC 2010
Abstract Chaotic signals are used in digital communications primarily in a bid to increase the security of transmissions. Moreover, second-order cyclostationary characteristics can easily be identified in chaotic signals used in communication systems. The detection of the cyclostationary properties in the transmitted signal decreases the security level for such systems. In this paper, we focus our attention on the eradication of cyclostationary properties present in chaotic signals, and to that end, we introduce a new method based on symbol period randomization to eliminate the spectral lines corresponding to the multiples of the baud rates. Finally, we compare our proposed method with another existing method in order to show the efficiency of ours in eliminating the cyclostationary properties. Keywords Chaos-based communication system · Secure communications · Cyclostationary signature suppression
1 Introduction The use of chaotic signals in digital communications was first proposed in [9], and since then, various modulation and demodulation schemes have been suggested and
This work has been supported in part by Ultra Electronics TCS and the Natural Science and Engineering Council of Canada as part of the “High Performance Emergency and Tactical Wireless Communication Chair” at École de technologie supérieure. G. Kaddoum () · S. Gagné · F. Gagnon LACIME Laboratory, University du Québec, Montréal, Canada e-mail: [email protected] S. Gagné e-mail: [email protected] F. Gagnon e-mail: [email protected]
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analyzed [6]. When used in digital communications or as encryption keys, chaotic codes can easily ensure a low probability of detection of transmitted symbols. The reason we use the real value of the chaotic signal in our paper is because it provides a number of advantages, which may compensate for some performance degradation of chaos-based communication systems. These advantages include the extreme simplicity of chaotic signals generation and other important aspects involving security. The received signal remains demodulated mainly due to the chaotic synchronization of the generator on the receiver side. Pecora and Carroll proposed a chaotic synchronization method in [10]. In real cases, the chaotic synchronization method is still difficult to achieve in noisy environments, and demodulating the received signal in coherent receivers with an acceptable error rate is a real challenge. To overcome this problem, a promising system is proposed in [3, 11], and [7] is based on the symbolic sequence associated with the chaotic map. On the receiver side, a maximum likelihood estimator based on a Viterbi algorithm with soft decision is used to estimate the trajectory of the transmitted sequence. This proposed communication system is interesting because the chaotic synchronization is not needed in order to decode the transmitted message. In this paper, we analyze the cyclostationary properties of the transmitted signal of the system in [7], but without OFDM modulation. This paper concentrates solely on the second-order (wide-sense) cyclostationary properties introduced by the transmitted chaotic signal. Therefore, when discussing cyclostationary properties, we do so in the wider sense of the theory (“wide-sense theory deals with moments, whereas strict-sense theory deals with probability distributions” [5]). A well-known method is used to detect the presence of cyclostationarity in a signal by computing the spectral coherence, which is essentially the same as calculating the autocorrelation function of the Fourier transform of the signal, but substituting frequency shifts for time lags. When the spectral coherence shows a peak, it means two frequencies are coherent. In other words, this means that the first term in the spectral coherence calculation corresponds to the first frequency, and the second term corresponds to the second frequency. According to the Wiener–Khintchine theorem, the autocorrelation of the Fourier transform for zero frequency shifts is just the Fourier transform of the signal squared, so simply squaring the chaotic signal may be enough to observe cyclostationarity. It has been shown in [8] that the cyclostationary properties of some chaotic signals used in digital communication systems can be detected. The use of cyclostationary properties of chaotic signals for communication is observed in [2]. With this type of signal used in secure transmissions, the cyclostationary properties must be removed. A study aimed at removing the cyclostationary properties of the chaotic signals used in [1]. The authors use the phase randomization technique to reduce the detectability of cyclostationary features introduced by the constant symbol rate of the chaotically modulated signal. In this paper, we will first show that the cyclostationary process cannot be eliminated in [1], but rather, attenuated, and then we present another technique based on the randomization of the symbol period, which can eliminate the cyclostationary process. The channel used for the transmission is an additive white Gaussian channel (AWGN). The paper is organized as follows. Section 2 presents a brief explanation of cyclostationarity, and is followed by an overview of the chaos-based communication system in Sect. 3. The proposed technique for cyclostationary suppression is also given in Sect. 3, while some numerical results are presented in Sect. 4, and the paper is concluded in Sect. 5.
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2 Introduction to Cyclostationarity A signal x(t) is cyclostationary if the autocorrelation function is periodic in time with period T : Rx (t, τ ) = R x (t + T , τ ), Rx (t, τ ) = x(t)x ∗ (t + τ ) ,
(1)
where ∗ denotes the complex conjugate and · is the time averaging operation. Thanks to the periodicity of the autocorrelation function, (1) can be decomposed into Fourier coefficients 1 T /2 α Rx (τ ) = lim Rx (t, τ )e−j 2παt dt, (2) T →∞ T −T /2 α is called the cyclic frequency. The function Rxα (τ ) is called the cyclic autocorrelation function (CAF). By using the Winner relation states, the spectral correlation function (SCF) can be obtained from the Fourier transform of the cyclic autocorrelation of (2). ∞ Sxα (f ) = Rxα (τ )e−j 2πf τ dτ. (3) −∞
For a fixed number of N samples, the estimate of Sxα (f ) becomes 1 1 S˜xα (f ) = NT
N/2
XT
n=−N/2
where
XT (n, f ) =
α α ∗ n, f + XT n, f − , 2 2
n+T /2
(4)
x(u)e−j 2πf u du.
n−T /2
Sxα (f )
denotes the spectral correlation function, which represents the correlation of two spectral components at frequencies f + α/2 and f − α/2. The SCF is considered as an efficient tool for analyzing cyclostationary signals. Physically, a signal is considered having no cyclostationary properties if the CAF or the SCF of the signal are under a defined threshold for all α = 0. Furthermore, if α = 0, CAF and SCF are reduced, respectively, to the conventional autocorrelation function and the power spectral density function. The cyclic frequencies α are typically related to the signal symbol rate and carrier frequency [4]. 3 Chaos-Based Communication System 3.1 Chaotic Modulator Based on Symbolic Dynamics In this paper, we use the symbolic dynamics method to obtain an equivalent chaotic sequence rather than generating the chaotic signal by directly iterating (5). x[n] = f x[n − 1] , x[n] ∈ I, (5)
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where f (·) is a nonlinear function defined into the interval I , and x[n] represents the state of the system. By using symbolic dynamics modulation, the binary symbols are modulated into the chaotic map. By partitioning a chaotic phase space into a finite number of partitions, I = I0 , I1 , . . . , IK−1 , and labeling each region with a specific symbol si , (si ∈ Ii ), the trajectories can be converted to a symbolic sequence. The idea is to divide the phase state into a finite number of partitions, I = I0 , I1 , . . . , IK−1 and assign each partition a symbol si , (si ∈ Ii ), where each symbol si is assigned to an interval Ii and x[i] visits during its time evolution. In this paper, the backward iteration of the chaotic map is used. This process introduced in [7] consists in iterating from the final condition x[n] onto the inverse of (5) −1 −n (Ii ) (f −1 ), following which the initial condition is contained in the set N n=0 f [7]. When N is very large, the set contains only the initial condition. By using the backward iteration guided by the sequence of symbols, the chaotic map converges toward the initial condition x[0] independently of x[N ]. With this method applied, the sensitivity of the chaotic map to the initial conditions is eliminated, and the chaotic demodulation is simpler to achieve without the need for a complex synchronization mechanism. The inverse chaotic map used in this paper to modulate the transmitted symbols (si ) is ⎧ (1−p)x−(1+p) ⎪ , if s = 1; ⎪ 2 ⎨ fs−1 (x) = px, ⎪ ⎪ ⎩ (1−p)x+(1+p) 2
if s = 2;
(6)
, if s = 3;
here p is a control parameter of the map, 0 < p < 1. The parameter p controls the width of the middle region. We use only the first and the third regions to transmit the symbols, and use of the second region is prohibited in order to avoid some problems associated with the numerical instability of the chaotic sequence (simply s[n] = 2 is not used). The motivation for using this chaotic map lies in the adjustable guard region, which allows us to control the security of transmissions. Note that the lower the control parameter, the higher the security of the transmission system will be. For the input bits (b ∈ 0, 1) b = [b[1], b[2], . . . , b[N ]]T , the symbolic sequence is s[n] = 1 + 2b[n],
1 ≤ n ≤ N.
(7)
The output chaotic signal generated from (6) and guided by the input signal of (7) can only belong to the first and the third regions, and the middle region is used as a guard interval to ensure a minimum distance between the two different waveforms associated with the bits 0 and 1. At the output of the AWGN channel, the received signal is x[n]h(t − nT ) + w(t), (8) r(t) = n
where T is the transmission symbol period, x(t) is the chaotic sequence, h(t) is the impulse response of the transmission filter, and w(t) is a white Gaussian noise with two side power spectral densities equal to N0 /2.
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3.2 Cyclic Frequencies Analysis Before presenting our approach to remove the cyclostationary properties, it is important to show the cyclic frequencies characteristic of this signal. As defined before, the transmitted baseband signal is given as follows: e(t) = x[n]h(t − nT ), (9) n
where e(t) is the signal at the output of the modulator. Assuming that the chaotic symbolic symbols are centered and i.i.d. with the variance σx2 = x[n]x ∗ [n]. Therefore, by using (1) and (9), the time varying autocorrelation of this signal can be simplified to 2 Rx (t, τ ) = σx h(t − nT )h(t − nT + τ ) . (10) n
It is seen that Rx (t, τ ) is periodic in time t with the period equal to T , thus symbolic dynamics modulation exhibits second-order cyclostationarity with cyclic frequencies α = Tms , where m is an integer. Next section shows a new approach to remove the second-order cyclostationarity of the transmitted signal. 3.3 Cyclostationary Signature Suppression This part is dedicated to explaining our technique for eliminating the cyclostationary properties from the transmitted signal. Once it is explained, we then compare our method to the phase randomization technique used in [1]. We show that the phase randomization technique of [1] reduces, but does not eliminate, the cyclostationary. Moreover, our proposed method can eliminate the cyclostationary peaks from the corresponding spectra equal to multiples of the baud rate for any symbol period. Our method aims to break the periodicity related to the symbol period T , and to that end, we introduce a variable θ , which allows us to have a new symbol period for each transmitted symbol s. As shown in Fig. 1, the resultant signal has a non-constant amplitude as a result of the non-periodic nature of chaotic signal and the non-periodic symbol period due to time randomization. This waveform shape leads to high security level required for secure transmissions (e.g. military transmission). After passing the symbol randomization block, the received signal is r(t) = x[n]h t − n T + θ [n] + w(t), (11) n
where θ is an integer pulse, (−A < θ < A; A = {a1 , a2 , . . . , ak−1 , ak }) and 0 ≤ A < T . By randomizing the symbol period, the cyclostationarity features introduced by the constant symbol rate are eliminated. This can be a simple and efficient way to ensure a very low probability of interception. The variable A is perfectly known on the receiver side, giving the receiver basic information about the period length of each symbol period, and without this information, the receiver cannot decode the received signal.
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Fig. 1 Chaotic waveform before and after the time randomization
3.4 Receiver Structure It is shown in [7] that the transmitted message modulated according to the chaotic symbolic dynamics modulation in (8) can be recovered using the Viterbi algorithm with a trellis of only two states. In order to demodulate the received signal r(t) of the chaotic communication system given in (11), the trellis diagram can be modified to accommodate the impulse response h(t) of the transmission filter and the deterministic random sequence θ [n], which is well known on the receiver side. The number of states of the resulting trellis diagram is directly related to the signal memory induced by the impulse response h(t) and θ [n]. For a filter of length L symbols, the total number of states increases to 2L , and each symbol is affected by the interference of L/2 past and future symbols. Each state represents a possible symbolic sequence S[m] of length L, and the branch metrics are cij [n] = r[n] − yij [n], where cij [n] is the cost of taking the j th branch starting from the ith node (1 ≤ i, j ≤ 2) at the nth time instant and yij [n] is the sample resulting from the interference induced by h(t), θ [n] as yij [n] =
L+1
zij m T + θ [n] h t − m T + θ [n] ,
(12)
m=1 (−1)
where zij [m] = f(v[m]) (xi [m]) and v[m] = [S[m], j ]. The Viterbi algorithm is then used to search through the 2L states of the trellis for the most likely transmitted chaotic sequence.
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4 Simulations and Discussions In this section, simulation results are discussed in order to prove our methodology for eliminating cyclostationary peaks. The power spectral density function of the square samples of the chaotic signal at the output of the modulator is shown in Fig. 2. To that end, the squared samples of the chaotic signal are computed with the fast Fourier algorithm for 4096 points of resolution and for a symbol T = 8Tc (Tc is time chip). With the phase randomization technique, the phase varies within the fixed symbol period, and for the symbol period randomization, the symbol period varies randomly, depending on the value of the parameter A. In Fig. 2(a), the first spectral lines represent the cyclic frequency α = 2/T . We can clearly see that the spectral line has been attenuated by the phase randomization method in Fig. 2(b), and has totally disappeared with the symbol period randomization technique in Fig. 2(c), where A = {2Tc , 4Tc }. Figures 3 and 4 show the relation between the number of different symbol period values and the stationary properties of the chaotic signal. In Fig. 3(c), we show that for three different values of the symbol period (T = {4Tc , , 8Tc , 12Tc }), the spectral line at the first harmonic remains there; however, by a small increasing of the value of T (five different values), the spectral lines are removed. In Fig. 4, we set the parameter A, (A = {Tc , 2Tc }), and plot the power spectral density for three different initial values of the symbol period (T = 4Tc , T = 8Tc , T = 16Tc ). We can see that after time randomization, the spectral line appears when the deviation compared to the value of T is small (i.e., after randomization (T = {15Tc , 16Tc , 17Tc }).
Fig. 2 (a) PSD of the squared chaotic samples without phase randomization, (b) PSD with phase randomization, (c) PSD with symbol period randomization
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Fig. 3 PSD with T = 8Tc and parameter: (a) A = {Tc , 2Tc , 3Tc , 4Tc }; (b) A = {2Tc , 4Tc }; (c) A = {4Tc }
Fig. 4 PSD with A = {Tc , 2Tc } and parameter: (a) T = 4Tc ; (b) T = 8Tc ; (c) T = 16Tc
Figure 5 shows the performance of the communication system when the maximum likelihood estimator based on the Viterbi algorithm is used on the receiver side. The bit error rate curves aim to show that the receiver can demodulate correctly—with an acceptable level of performance—when the time randomization is used. In this
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Fig. 5 BER performances of BPSK with the chaotic communication system for different values of the parameter p
simulation, the symbol periods are (T = {6Tc , , 8Tc , 12Tc }), and for different values of the parameter p. In fact, when the value of p increases, the distance between the two different waveforms grows, and it thus becomes easier to distinguish between samples belonging to the first region from those belonging to the third region of the chaotic map. Overall, this leads to a better performance of the system, but also at lower level of security.
5 Conclusion In this paper, a new technique is presented for eliminating the cyclostationary properties in a chaos-based communication system. A modified Viterbi algorithm is presented in order to demodulate the received signal. Once the cyclostationary features are eliminated, the security of the transmission is increased because the probability of interception becomes low. Simulation results have been used to illustrate and to compare the results to the phase randomization technique, and they show that our proposed method eliminates the spectral line corresponding to a multiple of symbol rate. Future works will look at an analytical study of the relation between the parameter of randomization θ .
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References 1. S. Atwal, F. Gagnon, C. Thibeault, An LPI wireless communication system based on chaotic modulation, in Proc. Milcom 09, Boston, USA, 2009 2. T.L. Carroll, Using the cyclostationary properties of chaotic signals for communications. IEEE Trans. Circuits Syst. 49, 357–362 (2002) 3. M. Cifti, D.B. Williams, Optimal estimation and sequential channel equalization algorithms for chaotic communications systems. EURASIP J. Appl. Signal Process. 4, 249–256 (2001) 4. W. Gardner, Exploitation of spectral redundancy in cyclostationary signals. IEEE Trans. Signal Process. 8, 14–36 (1991) 5. W. Gardner, D. Cochran, Cyclostationarity in Communications and Signal Processing (IEEE Press, New York, 1943) 6. G. Kolumbán, M.P. Kenedy, L.O. Chua, The role of synchronization in digital communications using chaos-part II: Chaotic modulation and chaotic synchronization. IEEE Trans. Circuits Syst. 45, 1129– 1140 (1998) 7. D. Luengo, I. Santamaria, Secure communications using OFDM with chaotic modulation in the subcarriers, in Proc. VTC 2005-Spring, Stocholm, Sweden, 2005 8. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd edn. (McGraw, New York, 1984) 9. U. Parlitz, L.O. Chua, L. Kocarev, K.S. Halle, A. Shang, Transmission of digital signals by chaotic synchronization. Int. J. Bifrurc. Chaos 2, 973–947 (1992) 10. L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems. Phys. Rev. A 64, 821–823 (1990) 11. J. Schweizer, T. Schimming, Symbolic dynamics for processing chaotic signals—II: communication and coding. IEEE Trans. Circuits Syst. 48, 1283–1295 (2001)
Removing Cyclostationary Properties in a Chaos-Based Communication System Georges Kaddoum · Samuel Gagné · François Gagnon
Received: 10 July 2010 / Revised: 5 November 2010 / Published online: 30 November 2010 © Springer Science+Business Media, LLC 2010
Abstract Chaotic signals are used in digital communications primarily in a bid to increase the security of transmissions. Moreover, second-order cyclostationary characteristics can easily be identified in chaotic signals used in communication systems. The detection of the cyclostationary properties in the transmitted signal decreases the security level for such systems. In this paper, we focus our attention on the eradication of cyclostationary properties present in chaotic signals, and to that end, we introduce a new method based on symbol period randomization to eliminate the spectral lines corresponding to the multiples of the baud rates. Finally, we compare our proposed method with another existing method in order to show the efficiency of ours in eliminating the cyclostationary properties. Keywords Chaos-based communication system · Secure communications · Cyclostationary signature suppression
1 Introduction The use of chaotic signals in digital communications was first proposed in [9], and since then, various modulation and demodulation schemes have been suggested and
This work has been supported in part by Ultra Electronics TCS and the Natural Science and Engineering Council of Canada as part of the “High Performance Emergency and Tactical Wireless Communication Chair” at École de technologie supérieure. G. Kaddoum () · S. Gagné · F. Gagnon LACIME Laboratory, University du Québec, Montréal, Canada e-mail: [email protected] S. Gagné e-mail: [email protected] F. Gagnon e-mail: [email protected]
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analyzed [6]. When used in digital communications or as encryption keys, chaotic codes can easily ensure a low probability of detection of transmitted symbols. The reason we use the real value of the chaotic signal in our paper is because it provides a number of advantages, which may compensate for some performance degradation of chaos-based communication systems. These advantages include the extreme simplicity of chaotic signals generation and other important aspects involving security. The received signal remains demodulated mainly due to the chaotic synchronization of the generator on the receiver side. Pecora and Carroll proposed a chaotic synchronization method in [10]. In real cases, the chaotic synchronization method is still difficult to achieve in noisy environments, and demodulating the received signal in coherent receivers with an acceptable error rate is a real challenge. To overcome this problem, a promising system is proposed in [3, 11], and [7] is based on the symbolic sequence associated with the chaotic map. On the receiver side, a maximum likelihood estimator based on a Viterbi algorithm with soft decision is used to estimate the trajectory of the transmitted sequence. This proposed communication system is interesting because the chaotic synchronization is not needed in order to decode the transmitted message. In this paper, we analyze the cyclostationary properties of the transmitted signal of the system in [7], but without OFDM modulation. This paper concentrates solely on the second-order (wide-sense) cyclostationary properties introduced by the transmitted chaotic signal. Therefore, when discussing cyclostationary properties, we do so in the wider sense of the theory (“wide-sense theory deals with moments, whereas strict-sense theory deals with probability distributions” [5]). A well-known method is used to detect the presence of cyclostationarity in a signal by computing the spectral coherence, which is essentially the same as calculating the autocorrelation function of the Fourier transform of the signal, but substituting frequency shifts for time lags. When the spectral coherence shows a peak, it means two frequencies are coherent. In other words, this means that the first term in the spectral coherence calculation corresponds to the first frequency, and the second term corresponds to the second frequency. According to the Wiener–Khintchine theorem, the autocorrelation of the Fourier transform for zero frequency shifts is just the Fourier transform of the signal squared, so simply squaring the chaotic signal may be enough to observe cyclostationarity. It has been shown in [8] that the cyclostationary properties of some chaotic signals used in digital communication systems can be detected. The use of cyclostationary properties of chaotic signals for communication is observed in [2]. With this type of signal used in secure transmissions, the cyclostationary properties must be removed. A study aimed at removing the cyclostationary properties of the chaotic signals used in [1]. The authors use the phase randomization technique to reduce the detectability of cyclostationary features introduced by the constant symbol rate of the chaotically modulated signal. In this paper, we will first show that the cyclostationary process cannot be eliminated in [1], but rather, attenuated, and then we present another technique based on the randomization of the symbol period, which can eliminate the cyclostationary process. The channel used for the transmission is an additive white Gaussian channel (AWGN). The paper is organized as follows. Section 2 presents a brief explanation of cyclostationarity, and is followed by an overview of the chaos-based communication system in Sect. 3. The proposed technique for cyclostationary suppression is also given in Sect. 3, while some numerical results are presented in Sect. 4, and the paper is concluded in Sect. 5.
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2 Introduction to Cyclostationarity A signal x(t) is cyclostationary if the autocorrelation function is periodic in time with period T : Rx (t, τ ) = R x (t + T , τ ), Rx (t, τ ) = x(t)x ∗ (t + τ ) ,
(1)
where ∗ denotes the complex conjugate and · is the time averaging operation. Thanks to the periodicity of the autocorrelation function, (1) can be decomposed into Fourier coefficients 1 T /2 α Rx (τ ) = lim Rx (t, τ )e−j 2παt dt, (2) T →∞ T −T /2 α is called the cyclic frequency. The function Rxα (τ ) is called the cyclic autocorrelation function (CAF). By using the Winner relation states, the spectral correlation function (SCF) can be obtained from the Fourier transform of the cyclic autocorrelation of (2). ∞ Sxα (f ) = Rxα (τ )e−j 2πf τ dτ. (3) −∞
For a fixed number of N samples, the estimate of Sxα (f ) becomes 1 1 S˜xα (f ) = NT
N/2
XT
n=−N/2
where
XT (n, f ) =
α α ∗ n, f + XT n, f − , 2 2
n+T /2
(4)
x(u)e−j 2πf u du.
n−T /2
Sxα (f )
denotes the spectral correlation function, which represents the correlation of two spectral components at frequencies f + α/2 and f − α/2. The SCF is considered as an efficient tool for analyzing cyclostationary signals. Physically, a signal is considered having no cyclostationary properties if the CAF or the SCF of the signal are under a defined threshold for all α = 0. Furthermore, if α = 0, CAF and SCF are reduced, respectively, to the conventional autocorrelation function and the power spectral density function. The cyclic frequencies α are typically related to the signal symbol rate and carrier frequency [4]. 3 Chaos-Based Communication System 3.1 Chaotic Modulator Based on Symbolic Dynamics In this paper, we use the symbolic dynamics method to obtain an equivalent chaotic sequence rather than generating the chaotic signal by directly iterating (5). x[n] = f x[n − 1] , x[n] ∈ I, (5)
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where f (·) is a nonlinear function defined into the interval I , and x[n] represents the state of the system. By using symbolic dynamics modulation, the binary symbols are modulated into the chaotic map. By partitioning a chaotic phase space into a finite number of partitions, I = I0 , I1 , . . . , IK−1 , and labeling each region with a specific symbol si , (si ∈ Ii ), the trajectories can be converted to a symbolic sequence. The idea is to divide the phase state into a finite number of partitions, I = I0 , I1 , . . . , IK−1 and assign each partition a symbol si , (si ∈ Ii ), where each symbol si is assigned to an interval Ii and x[i] visits during its time evolution. In this paper, the backward iteration of the chaotic map is used. This process introduced in [7] consists in iterating from the final condition x[n] onto the inverse of (5) −1 −n (Ii ) (f −1 ), following which the initial condition is contained in the set N n=0 f [7]. When N is very large, the set contains only the initial condition. By using the backward iteration guided by the sequence of symbols, the chaotic map converges toward the initial condition x[0] independently of x[N ]. With this method applied, the sensitivity of the chaotic map to the initial conditions is eliminated, and the chaotic demodulation is simpler to achieve without the need for a complex synchronization mechanism. The inverse chaotic map used in this paper to modulate the transmitted symbols (si ) is ⎧ (1−p)x−(1+p) ⎪ , if s = 1; ⎪ 2 ⎨ fs−1 (x) = px, ⎪ ⎪ ⎩ (1−p)x+(1+p) 2
if s = 2;
(6)
, if s = 3;
here p is a control parameter of the map, 0 < p < 1. The parameter p controls the width of the middle region. We use only the first and the third regions to transmit the symbols, and use of the second region is prohibited in order to avoid some problems associated with the numerical instability of the chaotic sequence (simply s[n] = 2 is not used). The motivation for using this chaotic map lies in the adjustable guard region, which allows us to control the security of transmissions. Note that the lower the control parameter, the higher the security of the transmission system will be. For the input bits (b ∈ 0, 1) b = [b[1], b[2], . . . , b[N ]]T , the symbolic sequence is s[n] = 1 + 2b[n],
1 ≤ n ≤ N.
(7)
The output chaotic signal generated from (6) and guided by the input signal of (7) can only belong to the first and the third regions, and the middle region is used as a guard interval to ensure a minimum distance between the two different waveforms associated with the bits 0 and 1. At the output of the AWGN channel, the received signal is x[n]h(t − nT ) + w(t), (8) r(t) = n
where T is the transmission symbol period, x(t) is the chaotic sequence, h(t) is the impulse response of the transmission filter, and w(t) is a white Gaussian noise with two side power spectral densities equal to N0 /2.
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3.2 Cyclic Frequencies Analysis Before presenting our approach to remove the cyclostationary properties, it is important to show the cyclic frequencies characteristic of this signal. As defined before, the transmitted baseband signal is given as follows: e(t) = x[n]h(t − nT ), (9) n
where e(t) is the signal at the output of the modulator. Assuming that the chaotic symbolic symbols are centered and i.i.d. with the variance σx2 = x[n]x ∗ [n]. Therefore, by using (1) and (9), the time varying autocorrelation of this signal can be simplified to 2 Rx (t, τ ) = σx h(t − nT )h(t − nT + τ ) . (10) n
It is seen that Rx (t, τ ) is periodic in time t with the period equal to T , thus symbolic dynamics modulation exhibits second-order cyclostationarity with cyclic frequencies α = Tms , where m is an integer. Next section shows a new approach to remove the second-order cyclostationarity of the transmitted signal. 3.3 Cyclostationary Signature Suppression This part is dedicated to explaining our technique for eliminating the cyclostationary properties from the transmitted signal. Once it is explained, we then compare our method to the phase randomization technique used in [1]. We show that the phase randomization technique of [1] reduces, but does not eliminate, the cyclostationary. Moreover, our proposed method can eliminate the cyclostationary peaks from the corresponding spectra equal to multiples of the baud rate for any symbol period. Our method aims to break the periodicity related to the symbol period T , and to that end, we introduce a variable θ , which allows us to have a new symbol period for each transmitted symbol s. As shown in Fig. 1, the resultant signal has a non-constant amplitude as a result of the non-periodic nature of chaotic signal and the non-periodic symbol period due to time randomization. This waveform shape leads to high security level required for secure transmissions (e.g. military transmission). After passing the symbol randomization block, the received signal is r(t) = x[n]h t − n T + θ [n] + w(t), (11) n
where θ is an integer pulse, (−A < θ < A; A = {a1 , a2 , . . . , ak−1 , ak }) and 0 ≤ A < T . By randomizing the symbol period, the cyclostationarity features introduced by the constant symbol rate are eliminated. This can be a simple and efficient way to ensure a very low probability of interception. The variable A is perfectly known on the receiver side, giving the receiver basic information about the period length of each symbol period, and without this information, the receiver cannot decode the received signal.
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Fig. 1 Chaotic waveform before and after the time randomization
3.4 Receiver Structure It is shown in [7] that the transmitted message modulated according to the chaotic symbolic dynamics modulation in (8) can be recovered using the Viterbi algorithm with a trellis of only two states. In order to demodulate the received signal r(t) of the chaotic communication system given in (11), the trellis diagram can be modified to accommodate the impulse response h(t) of the transmission filter and the deterministic random sequence θ [n], which is well known on the receiver side. The number of states of the resulting trellis diagram is directly related to the signal memory induced by the impulse response h(t) and θ [n]. For a filter of length L symbols, the total number of states increases to 2L , and each symbol is affected by the interference of L/2 past and future symbols. Each state represents a possible symbolic sequence S[m] of length L, and the branch metrics are cij [n] = r[n] − yij [n], where cij [n] is the cost of taking the j th branch starting from the ith node (1 ≤ i, j ≤ 2) at the nth time instant and yij [n] is the sample resulting from the interference induced by h(t), θ [n] as yij [n] =
L+1
zij m T + θ [n] h t − m T + θ [n] ,
(12)
m=1 (−1)
where zij [m] = f(v[m]) (xi [m]) and v[m] = [S[m], j ]. The Viterbi algorithm is then used to search through the 2L states of the trellis for the most likely transmitted chaotic sequence.
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4 Simulations and Discussions In this section, simulation results are discussed in order to prove our methodology for eliminating cyclostationary peaks. The power spectral density function of the square samples of the chaotic signal at the output of the modulator is shown in Fig. 2. To that end, the squared samples of the chaotic signal are computed with the fast Fourier algorithm for 4096 points of resolution and for a symbol T = 8Tc (Tc is time chip). With the phase randomization technique, the phase varies within the fixed symbol period, and for the symbol period randomization, the symbol period varies randomly, depending on the value of the parameter A. In Fig. 2(a), the first spectral lines represent the cyclic frequency α = 2/T . We can clearly see that the spectral line has been attenuated by the phase randomization method in Fig. 2(b), and has totally disappeared with the symbol period randomization technique in Fig. 2(c), where A = {2Tc , 4Tc }. Figures 3 and 4 show the relation between the number of different symbol period values and the stationary properties of the chaotic signal. In Fig. 3(c), we show that for three different values of the symbol period (T = {4Tc , , 8Tc , 12Tc }), the spectral line at the first harmonic remains there; however, by a small increasing of the value of T (five different values), the spectral lines are removed. In Fig. 4, we set the parameter A, (A = {Tc , 2Tc }), and plot the power spectral density for three different initial values of the symbol period (T = 4Tc , T = 8Tc , T = 16Tc ). We can see that after time randomization, the spectral line appears when the deviation compared to the value of T is small (i.e., after randomization (T = {15Tc , 16Tc , 17Tc }).
Fig. 2 (a) PSD of the squared chaotic samples without phase randomization, (b) PSD with phase randomization, (c) PSD with symbol period randomization
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Fig. 3 PSD with T = 8Tc and parameter: (a) A = {Tc , 2Tc , 3Tc , 4Tc }; (b) A = {2Tc , 4Tc }; (c) A = {4Tc }
Fig. 4 PSD with A = {Tc , 2Tc } and parameter: (a) T = 4Tc ; (b) T = 8Tc ; (c) T = 16Tc
Figure 5 shows the performance of the communication system when the maximum likelihood estimator based on the Viterbi algorithm is used on the receiver side. The bit error rate curves aim to show that the receiver can demodulate correctly—with an acceptable level of performance—when the time randomization is used. In this
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Fig. 5 BER performances of BPSK with the chaotic communication system for different values of the parameter p
simulation, the symbol periods are (T = {6Tc , , 8Tc , 12Tc }), and for different values of the parameter p. In fact, when the value of p increases, the distance between the two different waveforms grows, and it thus becomes easier to distinguish between samples belonging to the first region from those belonging to the third region of the chaotic map. Overall, this leads to a better performance of the system, but also at lower level of security.
5 Conclusion In this paper, a new technique is presented for eliminating the cyclostationary properties in a chaos-based communication system. A modified Viterbi algorithm is presented in order to demodulate the received signal. Once the cyclostationary features are eliminated, the security of the transmission is increased because the probability of interception becomes low. Simulation results have been used to illustrate and to compare the results to the phase randomization technique, and they show that our proposed method eliminates the spectral line corresponding to a multiple of symbol rate. Future works will look at an analytical study of the relation between the parameter of randomization θ .
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