Research Article. Spectral Properties of Chaotic Signals Generated by the Bernoulli Map

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1 Jestr Journal o Engineering Science and Technology Review 8 () (05) -6 Special Issue on Synchronization and Control o Chaos: Theory, Methods and Applications Research Article JOURNAL OF Engineering Science and Technology Review Spectral Properties o Chaotic Signals Generated by the ernoulli Map Raael A. da Costa *,, Murilo. Loiola and Marcio Eisencrat Universidade Federal do AC, Santo André, razil. Polytechnic School o the University o São Paulo, São Paulo, razil. Received September 04; Revised 4 October 04; Accepted 5 October 04 Abstract In the last decades, the use o chaotic signals as broadband carriers has been considered in Telecommunications. Despite the relevance o the requency domain analysis in this ield, there are ew studies that are concerned with spectral properties o chaotic signals. earing this in mind, this paper aims the characterization o the power spectral density (PSD) o chaotic orbits generated by ernoulli maps. We obtain analytic expressions or autocorrelation sequence, PSD and essential bandwidth or chaotic orbits generated by this map as unction o the amily parameter and Lyapunov exponent. Moreover, we veriy that analytical expressions match numerical results. We conclude that the power o the generated orbits is concentrated in low requencies or all parameters values. esides, it is possible to obtain chaotic narrowband signals. Keywords: Chaotic signals, ernoulli map, autocorrelation sequence, spectral analysis, piecewise linear maps.. Introduction A chaotic signal is a deterministic and aperiodic signal that presents sensitivity on initial conditions []. This sensitivity means that the signals obtained with initial conditions near to each other can become very dierent when time passes []. There is a large number o research wors involving applications o chaotic signal in several areas []. In Telecommunications, these researches were intensiied ater the seminal wor by Pecora and Carroll []. Thereater, the ield o communication with chaotic carriers has received a great deal o attention, see e.g. [4,5,6] and the reerences therein. As previously noted by [], chaotic systems can be synchronized, allowing the development o many wors regarding modulation schemes with chaotic signals as Chaos Shit Keying (CSK) and Dierential Chaos Shit Keying (DCSK) [6,7]. They can transmit inormation by the coeicients o combination o sequences generated by chaotic signals. However, ew studies analytically describe the spectral properties o chaotic signals, such as the Power Spectral Density (PSD). As every real world communication channel is bandlimited, to characterize and control the PSD o the generated chaotic signals is o paramount importance [6,8,9]. Thereore, the study o spectral characteristics is a relevant issue when it comes to using chaotic signals in practical communications. The paper is divided into ive sections. Section deines * address: r.costa@uabc.edu.br ISSN: Kavala Institute o Technology. All rights reserved. the ernoulli map and describes its main characteristics. In Section the Autocorrelation Sequence (ACS) and PSD o the chaotic orbits generated by are deduced. In Section 4 the relationship between essential bandwidth and Lyapunov exponents is accessed. Finally, Section 5 presents our conclusion and possible applications o the results.. ernoulli Map In this paper, we consider the map ( )) s( n + ) sn ( () with initial condition s(0) [, ] and [ ] [ ] :,, given by s+, s< + + () s, + s, s which depends on the parameter (, ). Fig. (a) shows () s or 0.8 and in Fig. (b) the orbits sn ( ) or s (0) 0.5 and s (0) are depicted, ()

2 R. A. Costa, et al./journal o Engineering Science and Technology Review 8 () (05) -6 clearly showing the sensitivity on initial conditions characteristic o chaotic signals. stochastic processes. The ollowing development is inspired by [9,]. The ACS R ( ) o a signal sn ( ) or an integer delay is deined by [ ] R () Esnsn ()( + ) (5) where the expected value E[ ] is taen over all initial conditions that generate chaotic signals. To simpliy notation we deine sn ( ) x and sn ( + ) y. (6) The joint density pxy (, ) is then given by * ( ) pxy (, ) p( x) δ y ( x) (7) Fig. (a) ernoulli map () s or 0.8; (b) orbits s(n) or s (0) 0.5 (continuous line) and s (0) (dashed line). As shown in [0,] the invariant density p () * s o orbits o is uniorm. In act, p *() s /, or < s <, or any. This means that the orbit points are uniormly distributed on this interval. Consequently, these orbits are zeromean and their average power is: P E s n [ ( )] () independently o [0,]. The Lyapunov exponent h o almost every orbit generated by is a unction o and can be calculated by []. ( ( 0 )) h ln s n, s p ( s) ds ln ds ln ds + + ( )ln ( )ln (4) where p () * x is the invariant density o and δ is the Dirac unit impulse unction [8]. Substituting (5) and (6) into (4) results in R( ) E[ xy] xyp( x, y) dxdy * ( ) xyp ( x) δ y ( x) dxdy (8) x ( x) dx. To evaluate (7), it is necessary to obtain ( x ) that means -iterations o. The map is composed o two linear segments. The image o each o these segments is U,. Fig. shows ( x ), ( x ), ( x ) the domain [ ] and ( x ), or a generic. We can deduce that when we iterate the number o its linear segments is multiplied by, i.e., ( x ) is ormed by segments. The th m -solution o the equation ( ) a ( m ), with a ( ) m x is represented by. where () is the derivative o. As h > 0 or (, ), the aperiodic signals generated by ( ) are chaotic. The maximum value o h is ln occurring or 0, where the map generates chaotic signals with maximum exponential divergence. In the ollowing section, we derive the ACS and PSD o orbits generated by this map.. Autocorrelation Sequence and Power Spectral Density Chaotic signals generated by a map can be treated as sample unctions o an ergodic stochastic process [0]. For a ixed value o, the signal generated by an initial condition s(0) can be viewed as a sample unction o the stochastic process deined by. We can thus deduce the ACS and the PSD corresponding to as we usually proceed with ordinary Fig. (a) ernoulli map ( x ) ater (b) two iterations, (c) three iterations and (d) -iterations.

3 R. A. Costa, et al./journal o Engineering Science and Technology Review 8 () (05) -6 This way, ( ) given by x in the interval [ a( m ), a( m) ] is x a ( m) a ( m ) ( x), a( m ) x< a( m) (9) a ( m) a ( m ) Substituting (8) into (7), yields a ( m) x a( m) a( m ) R ( ) x dx a ( m ) m a( m) a( m ) (0) y solving the integral in Error! Reerence source not ound., we obtain R ( ) ( a( m) a( m )) to () m The process o iterating the map one time, rom ( x ) + ( x), is illustrated in Fig. where w is the root o ( x ) and is given by ( + ) R ( + ) R ( ) (5) Notice that R(0) E s ( n) is the average power o sn ( ) and is given by (). Solving (4) with the initial condition R (0), we obtain R ( ) +. (6) Figure 4 shows the ACS curves or some values o. The dashed lines indicate the analytical result o (5) and the continuous lines the results o numerical simulation. Clearly, the numerical results agrees with (5). We observe that R ( ) decays monotonically aster to 0 and slower or. These results reveal that chaotic signals do not necessarily have impulsive ACS. w ( a( m) a( m )) + ( a( m) + a( m )) From Fig. the ollowing relations can be inerred: () Fig. 4 Autocorrelation R ( ) or dierent values o. The analytical result o (5) is shown in dashed lines and numerical simulations in continuous lines. Fig. (a) Generic stretch o ( x ) and (b) excerpt o this map ater one iteration + ( x). a+ (m ) a( m ) a + (m ) w () a+ ( m) a( m) The PSD P( ω ) is the discrete-time Fourier transorm (DTFT) o R ( ), considering as the time variable. Calculating the DTFT o (5), we obtain jω P( ω) R( ) e with β ( + β βcos( ω)) (7) Using (0) to evaluate R+ ( ) and substituting () and (), R+ ( ) can be written as: ( + ) R ( + ) a( m) a( m ) ( ) m Comparing () with (0), it ollows that (4) 4 + β (8) Fig. 5 shows the PSD or some values o. The dashed lines indicate the analytical result o (6) and the continuous lines the results o numerical simulation. Clearly, the numerical results agrees with (6). The value o controls the way the power is distributed along the requency axis. The

4 R. A. Costa, et al./journal o Engineering Science and Technology Review 8 () (05) -6 higher the absolute value o, the smaller the requency band where power is concentrated and vice versa.. As h > 0 or all values o, we note that we can generate narrowband chaotic signals. Fig. 5 Power spectral density P( ω ) or dierent values o. The analytical result o (6) is shown in dashed lines and numerical simulations in continuous lines. 4. Essential andwidth Fig. 6 andwidth π as a unction o (a) and (b) Lyapunov exponent h. The essential bandwidth is deined as the length o the requency interval where q 95% o the signal power is concentrated [8]. To calculated or a low-pass signal we must solve P ( ω) d ω q P ( ω) d ω (8) 0 π 0 y using Parseval s Theorem [8] and () π P( ω) d ω (9) 0 Substituting (6) and (8) in (7) and isolating, we obtain qπ β arctan tan β + with β given by (6). Figure 6(a) shows π (0) as a unction o and Fig.6(b) 5. Conclusion In this paper, we have analytically deduced the ACS, PSD and essential bandwidth o the chaotic signals generated by the ernoulli map. Our main conclusion is that by adequately choosing the value o, one can obtain chaotic signals with a desired bandwidth, with its power concentrated in the low requencies range. The analytical expressions presented in this paper conirm the possibility o easy generation o chaotic signals with a speciic essential bandwidth. This means that the usual assumption about chaos implying broadband uncorrelated signals is not always true. We have showed how is related to the essential bandwidth Β or the ernoulli map. Hence, or a required Β, rom (9) the corresponding value o can be determined and, consequently, the piecewise linear map that generates chaotic orbits with this particular Β. As uture wor, we intend to generalize the obtained results to more general piecewise linear maps consisting o an arbitrary number o segments. 6. Acnowledgments shows π as a unction o the Lyapunov exponent h, given The authors would lie to than CNPq-razil (grants 47990/0-9 and 575/0-7), FAPESP (grant by (). We observe an extremely narrowband process or 04/04864-) and UFAC or their support on this research. Reerences. K.T. Alligood, T.D. Sauer, J.A. Yore, Chaos-An ιntroduction to δynamical σystems, Springer, New Yor (996).. S.H. Strogatz, Nonlinear δynamics and ψhaos: With pplications to physics, biology, chemistry and engineering, Perseus oos Group, (00).. L. Pecora and T. Carroll, Synchronization in chaotic systems, Physical Review Letters, vol. 64 (8), pp (990). 4. F.C.M. Lau and C.K. Tse, Chaos-based digital communication systems, Springer, erlin (00) M.P. Kennedy, R. Rovatti, and G. Setti, Chaotic electronics in telecommunications, CRC Press, oca Raton (000). 6. M. Eisencrat, R.R.F. Attux, and R. Suyama, (Eds.) Chaotic signals in digital communications, CRC Press, Inc. (0). 7. G. Kolumbán., M.P. Kennedy, and L.O. Chua, The role o synchronization in digital communications using Chaos - Part II: Chaotic modulation and chaotic synchronization, IEEE Trans. Circuits Syst. I, vol. 45, pp (998). 8..P. Lathi, Modern digital and analog communication systems, Oxord University Press, New Yor (998).

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