ON THE INFLUENCE

OF TRANSMISSION
LINE ON COMMUNICATIONUSING CHAOS SYNCHRONIZATION.Junja KA WATA 1
1Dept.

Akio USHIDA3

Yoshifwni NISH102of Information

Science and Systems Eng., Tokushima
Bunri University
kawata@me.bunri-u.
ac.jp
2’3Dept. of Electrical and ElectronicEng., Tokushima
University
nishio@ee. tokushima-u.ac.jp,
ushida@ee. tokushima-u.ac.jp

frequently

ABSTRACTIn this study, a transmitter-receiversystem utilizing chaos
synchrontizatimr
is investigated,
where transmitterand receiver subsystems
are coupled by a transmissionline, assuming practicalimplementation.
The influence of bothloss of transmission
line and mismatchingon synchronization is examined by computer simulation using the methodof characteristics.
1.

SYSTEMINTRODUCTION

Recently
chaos synchronization
has received a great dealof attention
since its discovery by Prxmra and Carroll [1].
A number of recent papers have been published in the areaand many secure communication
techniques [2]- [4] also havebeen developed,
which may be useful for practical application. However the transmitterand receiver are only directly
coupled through voltage buffer in [2]- [3]. For practical implementationof these secure communication
techniques, thesubsystems
need to be intercmmccted
by a transmission
line. On the other hand, a digital secure communication
system via DSP irnplemcntat ion [4] has been developed,
where the transmitterand receiver are interconnectedbya ciigital transmission
line. The authors [4] have also discussed the disadvantage
with the usc of analog transmission
line: the influence of noise and gain distortion.
It seems
very importantthat the effect of transmission
line has oncommunicant ion system is investigated,for practical implementation.
Accordingly,
in this article, we investigate the influenceof the loss of an analog transmission
line and mismatchingon synchronization
in a transmitter-receiversystem based
cm the concept of Pecora and Carroll. The system is shownin Fig.
1. .4s widely used in [2]- [3], we also use Chua’s
circuit as the transmitterand receiver, which are coupled
by an analog transmission
line with matching impedance.
We analyze phenomena observed in the system numerically,
because it includes a transmission
line. For the purpose, we
usc the method of characteristics[5] which has been pr~
posed for the transient simulationof Iossless transrni.ssicmlines. In the method of characteristics
a lossless transmission line is modeled as the characteristic
model. For lossy
transrnission lines, a brute-force approach [6] is used, where
the 10SSY transmission
line is modeled by several cawaded
cells. F,acb cell consists of lumped resistors and a losslessline which is replaced by its characteristic
model.
Thus
the transient response can be calculated easily by computer
simulation
with the numerical integration
method such as
Runge-Kutta
rnethod.
We first show by computer
calculationthat the subsystems synchronize
when the transmission
line is lossless.
Then the influence of loss on chaos synchronization
is investigated for lossy transmission
line. Further the influcnrx
of matchingerror on that is alsa examined.
It is shownby numerical experirnents that synchronization
breaks morethan we had expected,

as loss increases.

CIRCUITMODELSYNCHRONIZATION

2.

ANDFig.
1 shows a transmitter-receiversystem bassed on thePecora-Carroll
concept.
Each subsystem is based on Chua’s
circuit whose chaotic behavior haa been widely studied (seereferences in [8]). The transmitterand receiver are interconnected by a transmissionline, where a matching impedanceis connected
at the receiving side to avoid signal distortiondue to reflection.
to the reccivcr through
Chaotic signal VCI is transmitted
the trarrsrnission lirm. After some time delay ~, the transmitted signal reaches at the receivingend.
The voltage
Ocl (t) at the end will be equal to Vcl (t – ~) if the transmission line is lossless and its characteristic
impedance
iscompletely
matched to the matching impedance.
The first(second) receiver subsystem then reproduce&2(t) (&l (t))
which will be a copy of vc2(t – T) (WCI (t — T)), only when
componentsof the subsystcm
are exactly identical to thetransmitter(first receiver) subsystem.
As this system includes the transmissionline, we can notinvestigate
the system anal ytically.
Therefore
the system
will be just analyze numerically using the method of characteristics.
.4 krssless transmission
line can be modeled using
the characteristicmode] [.5]. Replacing
the IOSSIMS transmission
line by the characteristic
model, the equivalent circuit as shown in Fig. 2 can be obtained,
which yields thefollowing system equations:dvcl=G(vc2dvcz=

G(VCI – VC2) + iL,__vc2

c, ~C2~@(1)– VC1) – g(VCl),(2)(3)

dt–Ocl=q(t – T) . zm/(zo+ Zm),

cl(t)

=

2ticl

e2(t)

=

2VC1 –el(t–dn>2

C:2
*

‘
– ez(t – r),(4)
(5)
(6)

T),

= G(ficl– v&2) + ii,(7)
(8)

dv~ ,
c1 ~

=

where g(. ) is a piecewiseg(v)= 7r~1v+ ~(moG(&2(9)– V&l) – g(v&l),

linear function

definedby:– 7rLl)[lv+BPl– Iv – DPI].

(lo)

Eqs. (1)-(3),(7)-(8) and (9) describe the dynamics of therransrnittcr, the first receiver, and the seconci receiver subsystem, respectively,
while Eqs.
(4)-(6)are rlerivcd fromthe characteristic
model [5].

Copyright 1997 IEEE897

For the case of lossy transmissionline, Eqs. (4)-(6) must
be replaced by the corresponding
state equations,becausethe lossy transmission line is approximately
modeled by several cascaded cells. Each cell consists of lumped resistors
and a lossless line which is further replaced with its characteristic model (see [6] for detail).
In [7] a theoretic just ification for this approach as well as an error analysis in thefrequency domain have been reported.
Bssed on the error
analysis we can decide the number of cells we need for aprescribed modeling error.
To investigate
the synchronizationin the transmitterreceiver system, let us fix the parameters of the transmitterso that it exhibits a chaotic attractor.
From Chua’s circuit,the double scroll attractor can be observed for the following
parameters:

R = l/G=1700[Q],
Cl = 10[nF],Cz = 100[nF],
L.= 18[mH],rno = –0.75[mS],
ml = –0.41[mS],l?p = 1 [IT].
These values are frxed throughout
the following discussion.
We suppose here that the parameters of both transmitter and receiver subsystems
are completelyidentical. First,
let’s examine the system for the case that the transmission
line is lossless and has the following parameters.
ZO = 50[$)],

7 = 0.5[msec]

Using the method of characteristics[5], the computer simulation is carried out, where we use 4th order Runge-K ut ta
method (stcpsim
is 0.00001 [msec]) throughout
the folkrwing simulations.
Fig. 3 shows the phase portrait.
In thefigure, it. should be noted that the horizontal
axis in (b)and the vertical axis in (c) are the time-delayed
voltage
I&l. and the vertical axis in (b) is the tinlc-dela~ed
voltageI&.
As we had expected,
the transmitted
signal VC;I ancl
the reproduced signal I,& synchronizedcompletely.
For arbitrary values of ZO, ~ different from above example, theyalso synchronizedcompletely.
3.

INFLUENCE

OFLOSS

Since a transmission
line such as a coaxial cable has loss
normally, we need to investigate the system for the case oflossy transmission
line. In this case the time-delayed
signalOcl at the receiving end is different from the transmitted
signal VC1 because of attrmuation.
In the simulation
weused the following set of parameters.

Rd = 20[Q/km],
Gd = OIS/km]

Ld = 0.25[rnH/km],

cd = 100[nF/krn],

not found.
In order to investigate the effect of increasing
loss fully, thus, we examined the time variation of difference between the voltages VC1 and I&l for various length.
Some results are shown in Fig. 5. As we had expected,
itis clear that synchronization
breaks more frequently as loss
increases.
4.

INFLUENCEOFMISMATCHING

Here we will investigate
the effect of rnisrnatchinghas cmsynchronization,
because it may be very difficult that Z“, iscompletely
matched to ZKKin real circuit.
Adding an error
factor to the matching impedance,
the inflrrencc of nlatching error is examined.
In this case the reflection duc tomismatchingtakes place at the receiving end and cmwequently the transmitted
signal is affected by the distortion.
The difference of voltages UC] and V&l is plotted in Fig. 6for 5% error.
For lossless transmissionline, synchronization largely breaks at times, while remarkable difference bymatching error is not observed for 10SSY transmissionline(CompareFig. 5(a) ,(b) with Fig. 6(b),(c)).
This reason
will be that the reflection waveforms induced by mismatching is also attenuatedbecause of loss.
5.

CONCLUSION

In this article we have investigated
the influence of transmission line on a transmitter-receiversystem using chaos
synchronization.
It is found by numerical experiments
thatthe transmitterand receiver synchronize
roughly for relatively smaller loss and that clesyllcllroIliziitiorl
occurs more
frequently
as 10ss increases.
It also secrns that for Iossy
transmission
line synchronizatiorl
suffers from loss than mismatching due to attenuation.
We will investigate
whether the subsystems
synchronize
independent ly of 10SS, when t,he attenuatedsignal Oc;I isproperly amplified
using an amplifier at the rcccivcr subsystem.
We also intend to study how dcrnodulatecl
signalis affected by both loss and mismatching,when a information signal is added to our system and then modulated
bychaotic signal.
REFERENCES[1] L. M. Pecora and T. L. Carroll, “Synchronization in chaoticsystems,” J%ys. Rev. Lett., vol. 64,no. 4,pp. 821–824, 1990.

[2]L. Kocarev

et al., “Experimental demonstration of secure
Int.
.J. Bi@rcommunication via chaotic synchronization,”
cution Chaos, vol. 2, pp. 709713, 1992.

[3] H. Dedieu, M. P. Kennedy and M. Hssler, “Chaos shift key-

The per-unit-lengthpararnet.ers of a real coaxial cable isprobably the almost same as above one. Based on the error
analysis [7] we evaluated the modeling error and then decided N so that it is within 0.1!70 in a prescribedfrequency
range. We calculated
the value of matching impedance
bythe following equation:

ing: Modulation and demodulation of a chaotic character
using self-synchronizing Chua’s circuits,” IEEE ‘7km.9. Carcuits Syst. II, vol. 40, no. 10, pp. 634--642, Oct. 1993.

[4]H. Kamata, T. Endo and Y. Ishida, “Secure communication
system using chaos via DSP implementation,”IS(2’AS ’96, Vol. 3, pp. 112-115, 1996.

Proc. IEEE

[5]F. H. Branin, Jr., “Tkansient analysis of lossless trmx+nlission line,” Proc. IEEE, vol. 55, pp. 2012-2013,Nov. 1967.

[6] H. W. Donunel,
where the value of w is supposed to be the natural angularfrequency of Chua’s circuit decided by L and Cz. See Fig. 4.
Roughly speaking, the subsystems synchronize yet. Further
we exarninc the system for larger loss. To invest igatc the effect of loss, we fix the per-unit-lengthparameters anti thensimulate the system by varying the line length. In this casethe total loss of transmission
line increases with the length.
For larger loss, the subsystems
did not synchronize
unfortunately, and obvious differences in the phase plain were‘;Digital computer solutiurr of electromagnetic transients in a single or nndtiphasc networks,” IEEE
‘Trans. Pourer App. Syst., vol. PAS-88, pp. 388-399, Apr.
1969.

[7] R. Gupta and L. T. Pileggi,‘iModeling Iossy transmission
lines using the method of characteristics,”
IEEE Ikans. Lb=cuits Syst., vol. 43, no. 7, pp. 580-582, July 1996.

p] L. O. Chua, “Chua’s circuit: Ten years later,” IEICE Tr-arrs.
Fundamentals, vol. E77-A,

no. 11, pp.

1811 18’21, Nov.

1994.

Copyright 1997 IEEE

898

iL

R

L
—
Transmitter

Figure‘iL

Second receiver subsystem
First receiver subsystem

subsystem
1. Transmitter-receiversystemusing

Chua’scircuit.

RL—
Figure2. Equivalentcircuit

for the system

shownin Fig.

1.
—

4.0.$.:

0.0.2,” ~-4.00.0(a)Fieure 3. Synchronizationin-(a) vcl-&zplane, (b)4, -2.o~-4.0

Vc 10.0

40.0(b)of the svstem
for the case thatV&l-v&z pjane, (c) VCI-V&l plane,-4.04,00.0&l(c)the transmission
respectively.

line is lossless:

Vcl4()

Phase

portrait

2.0.$

c
30.0

0.0-2.0 -.—
-, 0I

0.0
(a)

Figureplane,

4. Synchronization(b) V&l-v&z plane,

L‘WI, -2.0 ~-4.0-4.f4~?&,4o4.

(b)of the system for 0.01 [km] 10SSY transmission(c) VCI-V&l plane, respectively.

Copyright 1997 IEEE

(c)

line:

Phaseportrait

Vclin (a) VC1-7JCZ

-8.0/,-“-()

Figure5. Timeevolutiont – T[msec]
500/.

t – r[msec](h)of VCI – V&l withthe variationof length(loss).

‘(JiJ(a) 0.1 [km],.:(b) 0.5[km],(c) 1 [km].

II:0.0-&o~ 8.0—--....—

-~..

0(a)

t – 7[msec]

5M

t – r[msec]

500”

t – T[nlsec]~oo~-.2UIu~ 0.0-8.00.(b). .

>8.0
—+.?
I.

$ ().0.--8.00.,.

(c)
Figure 6. The
0.5[km].

influenceof matchingerror

5~0 for (a) Iossless line, and lossy line with length

Copyright 1997 IEEE(b) O.l[km],(c)