Performing Modulation Scheme of Chaos Shift Keying with Hyperchaotic Chen System

Transcription

1 6 th Internatonal Advanced echnologes Symposum (IAS 11), May 011, Elazığ, urkey Performng Modulaton Scheme of Chaos Shft Keyng wth Hyperchaotc Chen System H. Oğraş 1, M. ürk 1 Unversty of Batman, Batman/urkey, hdayet.ogras@batman.edu.tr Frat Unversty, Elazg/urkey, mturk@frat.edu.tr Abstract hs paper presents the transmsson of bnary nformaton by usng chaotc swtchng named as Chaos Shft Keyng (CSK) wth a hyperchaotc Chen System. We used hyperchaotc Chen system added a perodc drvng sgnal by Chen. hs new varable (r.sn(cos(φt))) can be thought as a control parameter determnng the chaotc attractor and bfurcatons of the system. he dynamcal behavor of the hyperchaotc system can be perodc, chaotc or hyperchaotc wth the varaton of φ. In the CSK modulaton scheme, chaotc sgnals wth dfferent bt energes are used to transmt the bnary nformaton. In ths paper, the requred chaotc and hyperchaotc sgnals havng dfferent bt energes wll be generated by hyperchaotc Chen system. On the recever sde, the transmtted dgtal nformaton s estmated by usng a non-coherent recever that s not based on synchronzaton but tres to detect the bt energy of the recevng sgnal. Keywords Hyperchaotc system, CSK (Chaos Shft Keyng), Chen system, chaotc communcaton D I. INRODUCION URING the past decade, there has been tremendous nterest worldwde n the possblty of eplotng chaos n the eplotaton of chaos for useful applcatons n engneerng systems. One applcaton area that has attracted a great deal of attenton s communcaton because of the potental benefts that can be ganed from usng chaos for communcatons, ncludng robustness n multpath envronments, ease of spectrum spreadng, added securty etc. In a conventonal communcaton system, the analog sample functon of duraton represents a symbol s a lnear combnaton of snusodal bass functons and the symbol duraton s an nteger multple of the perod of the bass functon. However, n chaos-based communcaton system called chaotc communcaton [1], the analog sample functon of duraton whch represents a symbol s a chaotc bass functons and dgtal bts or symbols are mapped to nonperodc samples of chaotc sgnals derved from chaotc attractors [1, ]. Hyperchaotc system s usually defned as a chaotc system wth more that one postve Lyapunov eponent [3]. Snce hyperchaotc system s more complcated than chaotc systems and has more unstable manfolds, the consttuton of hyperchaotc system has become an etremely sgnfcance and challengng job [4]. Moreover, hyperchaotc system has the characterstcs of hgh capacty, hgh securty and hgh effcency, t has broadly appled potental n nonlnear crcuts, secure communcatons, neural networks, bologcal systems etc [5]. here est some well-known hyperchaotc systems n lterature such as hyperchaotc Rossler systems, hyperchaotc Chua crcut, hyperchaotc Matsumoto-Chua- Kobayash crcut, hyperchaotc amasevcus- Namajunas- Cenys system etc [3]. Recently, Chen et al. [5] presented the hyperchaotc Lü system from the Lü system va state feedback controller. In ths paper, we use hyperchaotc Chen system as a chaotc oscllator to produce chaotc and hyperchaotc sgnals for the transmsson of dgtal nformaton. hs paper s organzed as follows: In secton II, theoretcal nformaton about chaos shft keyng and the constructon of hyperchaotc Chen system wth dynamcal behavors are gven. In secton III, the results of smulatons wll be dscussed and conclusons about transmsson of bnary nformaton are gven wth some graphcal results. A. Chaos Shft Keyng II. HEORY In CSK modulaton scheme, chaotc sgnals wth dfferent bt energes are used to transmt the bnary nformaton. he requred chaotc sgnals havng dfferent bt energes can be generated by dfferent chaotc crcuts as shown n Fgure 1 or they can be produced by the same chaotc crcut and multpled by two dfferent constants. In both cases, the dgtal nformaton to be transmtted s mapped to the bt energes of chaotc sample functons [1, ]. Chaos Generator 1 Chaos Generator g 1(t) g (t) s(t) CSK Modulated Sgnal Fgure 1 : he block dagram of CSK modulator 54

2 z Performng Modulaton Scheme of Chaos Shft Keyng wth Hyperchaotc Chen System Bascally, n CSK, f a +1 s transmtted, the chaotc sgnal g () 1 t, wll be sent. If a -1 s transmtted, dfferent type of chaotc sgnal g () t, s used as the transmtted sgnal [6]. g (t) g (t) when symbol ' 1' s transmtted, 1 s(t) (1) when symbol ' 1' s transmtted, B. Hyperchaotc Chen System In 1999, a novel chaotc dynamcal system, called the Chen system, was created by Chen and Ueta [7]. he system s descrbed by the followng set of ordnary dfferental equatons: a( y ) y ( c a) z cy () z y bz where, y, z are state varables and (a,b,c) are system s parameters. When a=35, b=3, c=8, the system ehbts a chaotc behavor as shown n Fgure Fgure : -z plane of Chen system Assume that the Lyapunov eponents of a system are ( 1,,3,4) and then the dynamcs of system [8] can be characterzed wth. For perodc orbts, 1,,3 0, 4 0 For chaotc attractor, 1 0,,3 0, 4 0 For hyperchaotc attractor, 1, 0, 3 0, 4 0 By addng a perodc drvng sgnal, r.sn(cos(φt)) and couplng t to the second equaton of the Chen system, we get the followng four-dmensonal autonomous system [4]. a( y ) y ( c a) z cy r sn(cos( w)) z y bz w Where wt and φ s a varable parameter, determnng the chaotc attractor, bfurcatons of system. As a result, the new system has four Lyapunov eponents. hs wll lead to a hyperchaotc system. Chaotc attractors can be produced ether by a dynamcal system for dfferent values of a bfurcaton parameter or by a set of completely dfferent dynamcal systems [9]. he dynamcal behavor of the hyperchaotc system can be perodc, chaotc or hyperchaotc wth the varaton of φ. Some dynamcal behavors of hyperchaotc Chen system are nvestgated and demonstrated n Fgure 3 by computer smulatons wth varable φ. A computer software program, MALAB wll be used to calculate Lyapunov eponents of the system wth a=35, b=3, c=8, r=35. Some typcal Lyapunov eponents of the hyperchaotc system are tabulated n able 1 wth respect to φ. For nstance, when φ=17, the system s n state of perodc due to havng three negatve Lyapunov eponents and one zero value. φ Dynamcal Property Chaotc Chaotc Hyperchaotc Perodc Chaotc Chaotc Hyperchaotc able 1 : Dynamcal behavors of the system wth varaton of φ Fg. 3 shows the phase portrats of the hyperchaotc Chen system on the -z plan wth dfferent values of φ. (3) In ths case, the Chen system has one postve Lyapunov eponent , whle the rests are 0 and whch corresponds the chaotc attractor. 55

3 H. Oğraş, M. ürk Fgure 5 : he block dagram of the ransmtter Fg. 5 shows the smple structure of the ransmtter crcut n our model. he changes n frequency of the drvng sgnal depends on the bnary nformaton beng sent and transmtted sgnal correspondng ths bnary nformaton s also shown n Fgure 6. Fgure 3 : Phase portrats of the Hyperchaotc Chen system on the -z plan (a) Perodc attractor (φ=17), (b) Chaotc attractor (φ=19.075), (c) Hyperchaotc attractor (φ=6.) III. SIMULAION RESULS In ths paper, we wll use chaotc and hyperchaotc sgnals as bass functons that generated by usng dfferent values of φ n the same crcut and multplyng them by two dfferent constants (0.04 and 0.6) leads to have completely dfferent bt energy whch can be dstngushed at the recever sde. ypcally, when a bnary nformaton +1 s transmtted, the chaotc sgnal (φ=19.075) wll be sent to the recever and whle -1 s transmtted, a hyperchaotc sgnal (φ=50), s used to transmt the nformaton. hs wll be performed by chaotc swtchng between dfferent values of φ n the same hyperchaotc Chen system. he block scheme of our communcaton model s shown n Fgure 4. Φ= Chaotc Chen System Φ=50 Hyperchaotc 1(t) swtch s(t) (t) ransmtter Channel AWGN Bt duraton Decson Crcut hreshold Recever z Correlator Fgure 4 : Block scheme of the communcaton model Estmated Fgure 6 : (a) Bnary nformaton, (b) r.sn(cos(φt)) correspondng the bnary nformaton, (c) ransmtted sgnal he modulated sgnal conssts of segments of non-perodc chaotc waveforms correspondng to ndvdual bts and the segment of analog waveform correspondng to each bt s unque. herefore, the bt energy of each chaotc waveform s dfferent between each other. he output of the correlator, named observaton sgnal [] z whch s used by the decson crcut s defned by z r ( t) dt s ( t) n( t) dt s ( t) dt s ( t) n( t) dt n ( t) dt (4) Where recevng sgnal s r( t) s( t) n( t) and denotes ntegraton over one sample functon. In nose-free case, the 56

4 Performng Modulaton Scheme of Chaos Shft Keyng wth Hyperchaotc Chen System second and thrd terms are zero. As a result, z assumes the values of s () 1 t and s () t whch represents the bt energes of transmtted sgnals ( Eb( s1), Eb( s ) ) equvalently, bts +1 and -1 respectvely. Due to from the dssmlarty of each chaotc waveforms, the bt energy for each of them s dfferent as shown n Fgure 7 below. Fg. 8 shows the transmsson sgnals between transmtter and recever under AWGN channel. Bt duraton b 0.5 sec, and hreshold level l 0.45 are selected for the smulaton. Fg. 9 shows the BER performance graph of non-coherent CSK modulaton dependng on the dfferent value of r whch means whether the drvng sgnal s used or not. Fgure 7 : (a) ransmtted dgtal sgnal (b) Bt energy of the recevng sgnal In the recever sde, we consder non-coherent detecton type whch synchronzaton s not requred. Correlator estmates the bt energy of the receved nosy sgnal and the decson as to whch symbol was transmtted s made by comparng ths estmate aganst a threshold level. Fgure 8 : (a) ransmtted dgtal nformaton, (b) CSK modulated sgnal, (c) Nosy sgnal n channel (SNR=14 db), (d) Bt energy of the recevng sgnal, (e) Recovered dgtal nformaton Fgure 9 : Performance of non-coherent CSK modulaton wth dfferent values of r IV. CONCLUSION We present a concept of CSK modulator wth hyperchaotc Chen system. Our smulaton results show that the dgtal nformaton can be transmtted over chaotc bass functons generated from hyperchaotc Chen system and can be recovered wth non-coherent detecton by estmatng the bt energy of the recevng sgnal aganst a sutable threshold level. However, ths level requred by the decson crcut depends on the sgnal-to-nose rato (SNR) whch s a sgnfcant drawback of non-coherent CSK. One of the advantages of our communcaton model s that the transmtter s a very smple crcut havng only one chaos generator. hs may causes less power consumpton of the system. Moreover, we also compare the BER performances of CSK dependng on the value of r and observe that when r=35, the performance of the system between db s slghtly better than r=0. However, at 14 db, the system wthout drvng sgnal (r=0) shows better performance than the other. One mportant pont s to decrease the bt duraton n order to ncrease data rate n a per unt tme and the crucal problem that s the bt energy of the chaotc bass functons s not constant whch makes harder to determne the requred threshold level. Our am s to desgn a dynamcal threshold level accordng to the nose n channel for unstable bt energy whch needs to be study further. 57

5 H. Oğraş, M. ürk REFERENCES [1] A. Raz, and M. Al, Chaotc communcatons, ther applcatons and advantages over tradtonal methods of communcaton, IEEE 6 th. Internatonal Symposum Communcaton Systems, pp.1-4, 008. [] M. P. Kennedy, and G. Kolumban, communcatons usng chaos, Sgnal Processng, Vol 80, pp , July [3] Z. Yan, Controllng hyperchaos n the new hyperchaotc Chen system Appled Mathematcs and Computaton, pp , 005. [4] Z. Zh-lang, Z. Q, Y. Ha, G. Jan, A new hyper-chaos generated from Chen s system va an eternal perodc perturbaton, Internatonal workshop on Chaos-Fractals heores and Applcatons, pp , 009. [5] A. Chen, J. Lu, J. Lü, S. Yu, Generatng hyperchaotc Lü attractor va state feedback control, Physca A, pp , 006. [6] M. Hasler, Chaos shft keyng n the presence of nose: A smple dscrete tme eample, In proceedngs of the IEEE Internatonal symposum on Crcuts and Systems, Vol. 3, pp , 1998 [7] N. Smaou, A. Karouma, M. Zrb, Secure Communcatons based on the synchronzaton of the hyperchaotc Chen and the unfed chaotc systems, Communcatons n Nonlnear Scence and Numercal Smulaton, (Artcle n Press), October 010. [8] X. Wu, J. Wang, J. Lu, H. C. Herbert Iu, Hyperchaotc behavor n a non-autonomous unfed chaotc system wth contnuous perodc swtch, Chaos, Soltons & Fractals,Vol. 3, pp , May 1998 [9] M. ürk, and H. Oğraş, Classfcaton of chaos-based dgtal modulaton technques usng wavelet neural networks and performance comparson of wavelet famles, Epert Systems wth Applcatons, Vol. 38, pp ,