TECHNICAL PHYSICS LETTERS VOLUME 25, NUMBER 8 AUGUST 1999

Information transmission using a synchronous chaotic response with filteringin the communication channel

A. S. Dmitriev and L. V. Kuz’min

Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow~Submitted January 28, 1999!Pis’ma Zh. Tekh. Fiz.25, 71–77~August 26, 1999!

A principle is proposed for obtaining a chaotic synchronous response in a driven nonlinearoscillatory system under conditions where the signal of the driving chaotic system is filtered by thecoupling channel. Music and speech signals are used to demonstrate the efficiency ofapplying this principle to transmit information through a channel having a limited frequencyband. © 1999 American Institute of Physics.@S1063-7850~99!02708-1#

arn

s-e-.in

lifio

eithcarethriefexafi-iv

yn

tuea

roenne

b-t

nr-elin

er

verd byr

be

the

or-

er,

The use of dynamic chaos to transmit information, pticularly confidential information, has attracted close attetion among researchers.1–7 Among the various proposed sytems, success was achieved with a system for nonlinmixing of an information signal with a chaotic one, introduced in Ref. 3 using a ring oscillator8,9 as the chaos sourceThis system proved effective for transmitting informationthe low-frequency10 and radio-frequency ranges,11,12and wasalso used in experiments in the optical range,13 where thechaos source was an oscillator consisting of a laser ampand a fiber-optic cable as a delayed-feedback element. Hever, the system3 also exhibits fairly high sensitivity toamplitude-frequency distortions of the signal in the chann

In order to eliminate perturbations associated wamplitude-frequency distortions in the channel, it was logito use correcting elements in the form of filters with asponse the reciprocal of the effective channel filter, atentrance to the receiving devices. However, studies carout in Refs. 14 and 15 showed that this approach is ineftive since, for many of the most interesting cases, an esolution of the correction problem is either lacking or difcult to achieve, and an approximate solution does not gthe desired improvement in the quality of the chaotic schronous response.

In the present paper we propose a new approach toextraction of information by means of a chaotic synchronoresponse in a nonlinear oscillatory system in the presencthe filtering properties of the channel, and we report the mresults of an investigation of the information extraction pcess. The idea of the approach is to incorporate elemhaving filtering properties equivalent to those of the chanin the transmitter~driving chaotic system! and receiver~driven nonlinear oscillatory system!.

A functional diagram demonstrating the principle of otaining a chaotic synchronous response and extractinginformation component from the chaotic signal~Fig. 1! con-sists of a transmitter I, a detector II, and a coupling chanIII which filters the incoming chaotic signal from the infomation component. The transmitter incorporates a nonlinsubsystem1 consisting of an instantaneous-response nonear converter and a first-order low-pass filter~Fig. 1b! and a

6651063-7850/99/25(8)/3/$15.00

--

ar

erw-

l.

l-ed

c-ct

e-

hesofin-tsl

he

el

ar-

linear subsystem2 comprising a second-order low-pass filt~Fig. 1c!, a summator and filter3 ~Fig. 1d! whose propertiesare equivalent to those of the channel filter. The receiconsists of the same elements with the summator replacea subtractor. If element3 is eliminated from the transmitteand receiver, and the coupling channel is considered toideal, the system is reduced to that introduced in Ref. 3.

In the presence of a chaotic synchronous signal andabsence of an input information signal, i.e.,S50, the signalleaving the subtractor in the detector isSf5U2Z250. If weconsider the propagation of an information signalSÞ0through the circuit elements, we can confirm that the infmation signal at the detector exitSf is a copy of the signalS

FIG. 1. Schematic of transmission system: I — transmitter, II — receivand III — coupling channel.

© 1999 American Institute of Physics

nal

r

666 Tech. Phys. Lett. 25 (8), August 1999 A. S. Dmitriev and L. V. Kuz’min

FIG. 2. Power spectra and signal fragments for a sigpower/chaos power ratio of26 dB: a — information signalS, b — sum U of chaotic and information signals aftechannel filter, and c — information signalSf .

-o

nse-

c

qu

ethe

ub-us-

heg atherthn of

at

orss

d a

-thens:

passed through the filter3. It thus remains for us to determine: ~a! whether the ring oscillator remains a sourcechaos when the additional filter3 is introduced;~b! whetherit will be possible to obtain a chaotic synchronous respoin the detector; and~c! whether the chaotic synchronous rsponse is stable in the strong or weak sense~this determinesthe presence or absence of on–off intermittence16–18and alsothe level of distortions of the information signal in the detetor!.

Let us assume, to be specific, that III and therefore3 aresecond-order bandpass filters. Then, the normalized etions for the information transmission system are:

TX11X15F~Z1!,

Y11a1Y11Y15X1 ,

Z11a2Z11v2Z15v2~Y11S!,

U11a2U11v2U15v2~Y11S!, ~1!

TX21X25F~U !,

Y21a1Y21Y25X2 ,

Z21a2Z21v2Z25v2Y2 .

f

e

-

a-

The first three equations in~1! describe the transmitter, thfourth describes the channel, and the last three describedetector. The first equation describes the nonlinear ssystem 1 consisting of a series-connected instantaneoresponse nonlinear converter having the characteristic

F~U,m!5m@ uU1Su2uU2Su1~ uU21u2uU11u!/2#, ~2!

wherem is a parameter, and a first-order low-pass filter; tnext equation describes the linear subsystem comprisinsecond-order low-pass filter; the third equation describessecond-order bandpass filter of the oscillator; the fouequation describes the same filter but for the propagatiothe signal through the coupling channel,S is the informationsignal, andY11S is the signal entering the channel andthe entrance of the transmitter bandpass filter.

Thus, the driving system in Fig. 1 is an the oscillatwith a first-order low-pass filter, a second-order low-pafilter, and a second-order bandpass filter. We know19 that inthese oscillators there is a region of chaotic oscillations anregion of hyperchaos, so that condition~a! is satisfied.

The system of equations~1! has a solution whose trajectories in twelve-dimensional phase space lie entirely inseven-dimensional hyperplane determined by the conditio

asyeot

s

re

cere

tothtoth

ofhe

1iono

ereioin

arineem

beaos,en

n-

-

os

r.

os

667Tech. Phys. Lett. 25 (8), August 1999 A. S. Dmitriev and L. V. Kuz’min

X15X2 , Y15Y2 , Y15Y2 , Z15U, Z15U. ~3!

If these solutions are stable under the action of smperturbations, the steady-state oscillations of the driventem induced by the chaotic action of the driving system archaotic synchronous response. In order to obtain a chasynchronous response and extract information, we usedhyperchaos regime of the transmitter with the parameterm535, T50.2, a150.3, a250.2, andv52.5. A necessarycondition for the existence of a chaotic synchronoussponse is that the first Lyapunov exponentl1

2 of the detectorshould be negative. Direct calculations show that the nesary condition for locking stability is satisfied, and therefocondition ~b! is satisfied. However, this is not sufficientensure stability of the chaotic synchronous response instrong sense: the attractor of the system may have trajecsections which are unstable in the transversal direction tohyperplane of the locking attractor, which leads to on–intermittence, i.e., randomly aperiodic disruptions of tlocking regime.

Calculations made using a technique used in Ref.showed that in this case the locking attractor has no sectwith local instability transversal to the hyperplane, the sychronous response is stable in the strong sense, and nooff intermittence occurs.

Computer experiments on information transmission wcarried out using music and speech audio signals, and donstrated the high precision of reconstructing the informatcomponent of the signal. Figure 2 shows correspondpower spectra and signal fragments.

To conclude, the proposed principle for obtaining a chotic synchronous response in the presence of the filteproperties of the channel is effective. The principle has bdemonstrated for a transmission system where the infor

lls-atiche

-

s-

erye

f

7ns-n–

em-ng

-gna-

tion signal is mixed with the chaotic one but it cannotused in other information transmission systems, using chfor example in transmission systems with switching betwechaotic regimes.

This work was supported by the Russian Fund for Fudamental Research~Grant No. 97-01-00800!.

1L. Kocarev, K. S. Halle, K. Eckertet al., Int. J. Bifurcation Chaos Appl.Sci. Eng.2, 709 ~1992!.

2K. Cuomo and A. Oppenheim, Phys. Rev. Lett.71~1!, 65 ~1993!.3A. R. Volkovski� and N. F. Rul’kov, Pis’ma Zh. Tekh. Fiz.19~2!, 71~1993! @Tech. Phys. Lett19, 97 ~1993!#.

4Yu. L. Bel’ski� and A. S. Dmitriev, Radiotekh. Elektron.38, 1310~1993!.5A. K. Kozlov and V. D. Shalfeev, Pis’ma Zh. Tekh. Fiz.19~23!, 83 ~1993!@Tech. Phys. Lett.19, 769 ~1993!#.

6M. Hasler, H. Dedieu, M. Kennedyet al., in Proceedings of InternationalSymposium on Nonlinear Theory and Applications, Hawaii, USA, 1993,pp. 87–92.

7S. Heyes, C. Grebogi, and E. Ott, Phys. Rev. Lett.70~20!, 87 ~1993!.8A. S. Dmitriev and V. Ya. Kislov, Radiotekh. Elektron.29, 2389~1984!.9A. S. Dmitriev and V. Ya. Kislov,Stochastic Oscillations in Radio Physics and Electronics@in Russian#, Nauka, Moscow~1989!.

10A. S. Dmitriev, A. I. Panas, and S. O. Starkov, Int. J. Bifurcation ChaAppl. Sci. Eng.5, 371 ~1995!.

11A. S. Dmitriev, A. I. Panas, S. O. Starkovet al., Int. J. Bifurcation ChaosAppl. Sci. Eng.7, 32511~1997!.

12A. S. Dmitriev, L. V. Kuz’min, A. I. Panaset al., Radiotekh. E´ lektron.43,1115 ~1998!.

13G. D. VanWiggeren and R. Roy, Phys. Rev. Lett.81, 3547~1998!.14T. L. Carroll, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.42~3!,

105 ~1995!.15T. L. Carroll and G. A. Johnson, Phys. Rev. E57, 1555~1998!.16L. V. Kuzmin and A. I. Panas, inProceedings of Workshop Nonlinea

Dynamics of Electronic Systems, NDES’97, Moscow, Russia, 1997, pp485–490.

17E. Ott et al., Phys. Rev. Lett.71, 4134~1993!.18A. Chenys and H. Lustfeld, J. Phys. A29, 11 ~1996!.19A. S. Dmitriev, A. I. Panas, and S. O. Starkov, Int. J. Bifurcation Cha

Appl. Sci. Eng.6, 851 ~1996!.

Translated by R. M. Durham