Chaotic dynamics of oscillators based on circuits with ... - IEEE Xplore

place nulls at 0, = M",O2 = (17". ;inti O3 = 68" by using the abnvc d u ] projectors ~'~ technique with thc iiorin IIP(i)ll = [j.: I P ( i ) ( ~ ] ~ ~ (tlic Ui- and U*, were implcmcntcd by using the mntheinatical resdts described in [ 5 ] ) . The synthcsiscd pallern is shown in 1;ig. I h . A sccoiirl cxaiiiple refers to a circular ring array or radius X = 1.8X, consisting or A' = 30 equally spaccrl isolropic elements. The array lies in the plane of n Cnrtcsian system O(x,y,z) and its cciitrc coinciidcs wilh 1he origin 0.The gcncric far-freltl paltern in tlie plane or the army is indicaicd by P(i)(+),where (I is the azitiiulli angle. Fig. 2a shows llic pallern P(io)($) (thick linc) obtaincd by approximating tlic pattern FO($)(Ihiii line) in the so ns to placc nulls at mean-square sensc [7]. We modified /'(io)($) $1 = 49", $2 = 4 8 " , $3 = - 47", $4 = 47", $5 = 48", aiId $6 = 49", by tising tlic above technique with the norm 111'(i)ll = 1j-i lP(i)($)12d$]i'2 (the projectors U, and U, were implemented hy tlie techniques descrikd in [3] and [7], rcspectively). The sytithesised pattern is shown in Fig. 2A. Notc that the palleriis in Fig. Ih and Fig. 26 have deep nulls in the required directions and sutisfrictorily approximate the patterns in Fig. la arid Fig. 2cr, rccspcctivcly. In monclusion, we have described an itcrahe leclmiqiie that allows nulls to be placed in thc radialioii pattern of an antenna army by modifying only thc excitation phases. The technicpc is based 011 the mcthod of projcclions, is easy to iinplcnicnt and gives reasonably satisfactory resulls in nll of the c a m cxamincd, including thosa illuslraled here. S-JJ

as possible; (ii) the bwndwidth of the chaotic signml must bc suitable Tor limited-baiidwidtli communication chaniiels; (iii) the chaos gcncration must bc of such 21 nature as to allow chnotic synchronisation at the receiver and extraction of the mcssiigc itironnation [41. Most clcctronic chaos generators developed thus f i r have used nonlincar circiiils wilh Cliua diodes or hy.qteresis-type elements (as an example, scc IS, 61). Thcy generate rather low-dimetision chaotic signals: tl~ereitre usually less than three non-negirtivc Lyapunov cxponenls, corresponding to n dimension of < 5 . The low complcxily of the chaos yields a low degree o f confideddily, and an eavesdropper can defeat the chaotic key relativcly easily. In contrast, oscillators einploying dclaycd nonliiicar feedback caii generate high-dimcnsional clinos ( h c nuiiibcr of positive Lyapunov cxponciits can be > 10). Optical systems with nonlincar fedhick, oiie employing R ring fibre laser [2] and thc other a wavelength tunable laser diode 171, have recently bccn demonslrnled. The dimension d of the chaos thus produccd is given by rl = 0.4&71~,where Tis die time deltry introduccd by thc feedback loop, T is its response timc, and p is thc bifurcation parameter. In the experiment dcscribcd in [7] thc maximum dimension was reported to bc 250, i.a. the numkr of posilive Lyapitiiov cxpuiicnts WAY -125. In this Lctter wc report B nonlinear-feedback chaotic electronic oscillator that has potcnlial application Lo secure radiocomiminicalions. nonllnearily -1lowpass

Q

tilter]

A Y

detector

I tlmedelay

Rcfcrcnces 1

2 3 4 5

6

PRRSAD, s ~ ,and CCIARAN,IC.: 'On the constrained synthesis oC array pitterns with applic.arioiisto circular and IIC lirrays', LWfl Trm?,~., 1984, AP-32, (7). pp. 725-730 NG, u.P., ER, M.II., and KOT. c.: 'Lincar w i l y geonielry synthcsis with minimum sidc Iobc lcvel and null control', Pvnc,, M/c~(JIv. Anrennm Propug., 1994, 141, (3, pp. 162-166 wscovo. R.: 'I'atfcrn synthcsis with null constraitits for circular arrays of cqually ~ p l i c e disotropic elenicntp', ILL PI.oE.,Yicroiv. Anfewwl.x Proprig., 1996, 143, (Z), pp, 103 -106 IIAUIT. K.I..: 'Phasc-only adnptive nulling with a gcnctic algorithm', imui "VIS., 1997, AP-45, (GI,pp. 1009-ioi~ VFSCOVD. R.: 'Null control io linear array i n the presence of 1111 uppcr hound for the cxcittltion dynamics'. Fimc. Ini. Conf. E1cctroin:ig. in Advaucetl Applications, Torino, Italy, September 1999, pp. 433-435 LEVI. A., ~ i i dSI',\lw, H.: 'Iinagc rcvloriition by the mcthod of

generalized projections with npplicatioti lo rcstorution l'rom mngnitudc', J. Opt. SUC.A t w r : , 1984, I, (9), pp. 932-943 (Pnrl A) 7 v~scovo,R . : 'Constrained nnd unconstraiiicd synthesis or m i r y Factor Tor circular arrays', IEE'nB Trotis., 1995, AP-43, (]I),pi). 1405- I 4 10

Chaotic dynamics of oscillators based on circuits with VCO and nonlinear delayed feedback

An cleclronic oscillator that produccs clws with n diniensioiiality st least 20 r i m s Ihighcr tliaii the dimensionality obtaincd with prcviously publishcd elcctroiit cliilos gceiicratnrs is rcportcd

hlro&c/ion: ?'he idea of hiding a messagc sigiial in chaos was first proposcd carly in the 1990s [l]. Sincc thcn, chaos-based coding ant1 decoding have been dc~nonstraltdsuccessfiilly [2, 31. ldcally, ihe chaos generators uscd in thcse systems should havc thc following ctiaractcristics: (i) llie chaotic signal should be of tlie highest possiblc dimcnsionality (hypercliaosj, in ordcr lo ensure the gentcst complexity possible and, tlicrcforc, as high a degree of sccurity

3rd February 2000

otid

schcrruiir diagrams oJ clrriotic osciliaror

n Block diagram h Sclicinatic diagr:iin

Pritic/ji/c of operation: A funclional block diagram for tlic systcm and il sclicinatic diagram Tor the circuit are shown in Fig. 1. The saurcc is R voltage-controlled oscillator (VCO), which produces a frcqii~ii~lq-motlulate~ (FM) sigiial with arnplitude A. nild fieqiiency/(t). This FM signal is sent into ii filtcr N L (nonlinear in frequency) and an envelope detector, which detccts the envelope of the output signal from the fillcr. This cnvclopc signal is delnycd by T and wves as tlic driving signal V(r) of the VCO. The responsc time of thc lowpnss filler corresponds to the feedback loop rime conshiill t.

L. Largcr, V.S. Udaltsov, 9.P. Goedgcbucr and W.T. Rhodes

ELECTRONICS LETTERS

b

Big. 1 8 t w k

Vo/.36

Driven ns shown through tlie feedback circuit, thc VCO generates a chaotic FM signal. The frequency fof the signal is iclatcd to thc dtiviug volkgc V(t)of the VCO by&) =so + S. Ut), where S = r#MV is the tuning ratc of thc VCO, and is the frequency wheti V = 0 can be adjustcrl via an additional DC voltagc). Thc N L device is formed by n set of m parallel RLC filteis with rcsonaiiw frcqucncics,i;, cqually spaced by AJ aut1 with quality hclocs Ql, where d is the index number of thc tiltcr. For filrers with gain G! (assuming for sitnplicity that Q,G, I) thc iionlinearity is

vu

described by

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199

The envelope detector consists of a full-wavc rcctificr circuit (operational amplifiers AD844) and au integrator (OP262, cutoff frequency 100kHz). The gain K of the detector cm be adjusted electrically by use of il multiplier (AD633). The dynamics of the circuit in Fig. la can bc dcscribed by the so-called Ikeda equstion [SI: z(t)

+ T-Wt) =p dt

’

in communication systems. The implcmcntation disciisscd above is suitable for analoguc voicc coding and decoding by use OF n s p cific synchronislitionmcthod that will be rcported in a loiiger ai-tide. The chaotic signal obtaiiied is a baseband signal that cnn modultite a carricr al a Dqucncy wiitable for conventional communication systems. 2

F[:c(t- T)]

where ,U(() is related to the input voltagc V(t) of thc VCO by x(t) = n,SV(t)/Afand where /3 = x.K-A,yS/4“ The experimental implemenPdtioi1IS rcp~-csc~~lOd in Fig. lb. The VCO is a model MAX038 froin Maxim. The ceiilral frequency .h can be adjusted from IOOkfIz to IMHz. The tuning rate of the VCO is S = 919kHzi\r, and the amplitude of its output signal is A. = IV. A sct of threc parallel RLC filters with operational amplificrs is realised with 95pH inductors, capacitors C, = 2.17nP,C, = 1.27iiF, C, = 89SpF, and resistors RI = 6.7!&, X2= 6 . 8 k Q R, = 8.9kR, yielding rewniince frcqucncics: f i = 35Oki-Iz, h = 458kHz,and& = 545kHz spa& by A$= 100kEIz. The measured values of the qudity factors are: Ql= 15.0, Q2 = 15.2, and Q3= 15.6. The gain of each filter is adjusted such that QiCi = I. The delay linc is an intcgratcd circuit inodel RD5106 from EGG Reticon. It is driven by an extemal clcck of frequency f,! = MIkHz, yielding a delay T = 544~. The cutoff frcqucncy of thc first-order lowpass filter isf, = 8.16kHz, corresponding to R tiinc response T = L9.5ps, and ratio T/T = 28. Tliosc pardmeters were also used in the numerical simulatioiisfor solving eqn. 2 by applying the RungsKutta praccdurc.

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bifurcation paramster p b

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0

Expximctit

b Simulation

2% Ilme, ms

= 362kI-Iz)

a

Note also that the complexity of chaos can be increased casily by illcreasing the ratio UTand the number of resonmce filters. Work is in progress to develop a circuit matched to the digital encddingldecodingof a voicc. 48’

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98

frequency, kHz b

D

-0.04

0.04

time, ms c

Fig. 2 Cl~ooticfluGtiru/ionv of V(t), aswciiaied spectrtrrn and auiocorrclur ion fit~criori

fo = 362kHz, p = l l , d =

120

Chaotic fluctuations of V(t) b Associated spectrum

2 December I993 0 1nu 2000 E i e c ~ o n i aLeiters Onlha No: 20000203 DOL 10.104We1:20000203 L. Larger, V.S. Udnltsov, J.P. Gaedgcbucr and W.T. Rhodcs (Lahorrrloire d’Opliqw PM Dufferrx, U M R CNRS 6603, Uviveruitt de bi-anche-Comtk, I6 route de Grrry, 25030 Bwanpn Cedw, Fiance) L. Larger, V.S. Udaltsov, I.P. Goedgcbuer and W.T. Rhodes: also with GTL-CNRS Telecom, UMR CNRS 6603, Georgia Tech Lorraine, 2 rue Marconi, 57070 Meiz, Franco

Referenas

c Autocorrelation function

R e w h Results obtained froin numerical simulations and experirnentr, are shown in Figs. 2 and 3. The chaotic signal V‘l) gcncrated by the oscillator (Fig. 2 4 , its expcriiiicntal FET spectrum (Fig. 2b), and the mlculated autmrrchtion function (ACF) (Fig. 2c), wcrc obtained withh = 362kHu, p = 11. The spectrum of the signal is that of white noise. The ACF fcdturcs a sin& pcak typical of high-dimensional chaotic signals. The dimensionality of the chaos is estimatcd to bz d = 0.4p.T/z 120, which IS more than 20 times highcr than diincnsionaliticspreviously reported. The bifurcation diagrtrms (Fig. 3n and b) show excellent a m r d belween the numerical simulations and experiments, illustrating Ihat the numerical simulations we used can be applied to circuits with different parameters and different t y p of nonlincaritics.

Conclu,~io~ We have described a nonlinear time-delay4 fccdback circuit, mnstructed of standard clcctronic components, for the generatioil of high-dimcnsionnl chaos. Thc bandwidth of the chaotic signal generated can be easily matched to the standards used

200

AchoivledXmew The participation of V.S. Udaltsov was supported by the Centre Natiatial dc la Rcclicrche Scientifique (CNRS), Prance.

I

OTT, E., GREBOOI.c., and YORKE, LA.: ‘Controlling cliaos’, Phyx Rev. Left., 1940, 64, (ll), pp. 119&1199

2

VANWIGCIIKEN, GI>., and ROY. 11.: ‘Communication with chaotic lasers’, Science, 1998, 279, pp. 1198-1200 PAKLITL, U,, CIIUA. L O , KOCAREV. I.., HAI.15, K.S., and SHANG. A.: ‘Transmission of digital signals by chaotic synclironizalian’, IN. J Bijiircutiun Clluos, 1992, 2, pp. 973-977 PECORA. L.M., and CARIIOLL.TL.: ‘Synclironization in chaotic systems’, Phyx Rev. Lerr., 1990, 64,(8), pp. 821-824 SL‘ORACH,M.: ’Sccurc communication l>y hystcrccxis-based chaotic circuit’, Electron. Lerr., 1998, 34, (ll), pp. 1077-.1076 TAMASWICIIIS, A., CENYS. A,, MYKOLAITIS, G., NAMAJUNAS, A., rind LINDDERG, E.: ‘Hyperchaotic oscillnlor with gyrators’, Eieciron. LCK, 1997, 33, (7), pp. 542-544 LARGER, I,,, COFIUGI:IIUI~.R, ],I>,, and MAROLLA,J.M.: ‘Chaotic oscillator in wavelength: n IICW nctup for invcstigating differential diffcrciw equations describiug nonlinenr dynemics’, IEEE J. Q i m r u m E/ec/ruiz., 1998, QE-.M, (4),pp. 59460 1 IKEDA, K,, and MATSUMOM, K.: ‘High-dimensional chaos behavior in systems with he-delayed fcedhnck’, Physica D.. 1987, 29, pp.

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