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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
SYNCHRONIZATION AND COMMUNICATION USING CHAOTIC
FREQUENCY MODULATION
A. R. VOLKOVSKII*s AND L. S. TSIMRINGt
Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402, U.S.A.
SUMMARY
In this paper we propose a chaotic frequency modulation (CFM) technique for spread spectrum communications.
Existence and stability of the synchronous mode is demonstrated analytically and numerically. Unlike synchronization of
two conventional chaotic oscillators via an FM channel, the proposed method provides selectivity with respect to
interference within the frequency range of chaotic frequency oscillations. This method provides a viable alternative to
a recently proposed &interpolated' frequency hopping for spread-spectrum communication systems. Copyright ( 1999
John Wiley & Sons, Ltd.
KEY WORDS: chaos; synchronization; communication
1. INTRODUCTION
In the last decades spread spectrum communication systems have become of interest for use in many military
and commercial applications. In these systems the bandwidth of the transmitted signal is much wider than
the bandwidth of the information signal, which leads to the improved performance in tough multi-user
multi-path environments and reduces jamming, interference and probability of intercept.1,2
One of the most popular methods of spectrum spreading is frequency hopping. In a typical scheme of
non-coherent frequency hopping, the transmitter carrier signal frequency is modulated by a pseudo-random
hopping code. It is assumed that both transmitter and receiver share the code and are synchronized (usually
by exchanging special synchronizing codes). Then the frequency of the transmitter is shifted by an information signal and by detecting this relative frequency shift the data are extracted at the receiver. In coherent FH
systems, the receiver and transmitter maintain exact phase synchronization of the carrier, and the information is encoded in the phase variations of the transmitted signal at a current carrier frequency.
If the receiver is unable to maintain exact carrier phase for demodulation (which is di$cult because of
discreet frequency hops), a di!erential phase encoding is used (see Reference 3). In this scheme, the "rst part of
the frequency hop serves as a phase reference, and the subsequent parts within the same hop are phase shifted
with respect to it according to the information sequence. In Reference 4 it has been suggested to use chaotic
signals for frequency modulation in the di!erential phase shift keying scheme. In that method, the carrier
frequency is modulated according to a chosen chaotic signal. It should be noted that this method does not
exploit speci"c deterministic features of the chaotic signals. Indeed, frequency-modulating signal does not
have to be chaotic*any random or pseudo-random signal (as in conventional di!erentially coherent
modulation techniques) would do as well.
* Correspondence to: A. R. Volkovskii, Institute for Nonlinear Science, University of California, San Diego, La Jolla, California
92093-0402, U.S.A.
s e-mail: volkovsk@routh.ucsd.edu
t e-mail: lev@io!e.ucsd.edu
Contract/grant sponsor: US Army Research O$ce; contract grant/number: MURI grant DAAG55-98-1-0269
Contract/grant sponsor: US Department of Energy; contract grant/number: DOE/DE-FG03-95-R14516
CCC 0098}9886/99/060569}08$17.50
Copyright ( 1999 John Wiley & Sons, Ltd.
Received September 1998
Revised May 1999
570
A. R. VOLKOVSKII AND L. S. TSIMRING
Figure 1. Standard FM synchronization scheme
Fast discrete frequency hopping, with at least several hops per information bit, provides a high level of
redundancy and therefore high resistance against interference and fading. It is known that the phase lock
loop (PLL)-based frequency hopping has two main problems: spectral splatter and transient mismatch
between the transmit and receive synthesizers. Besides the common solutions used to reduce these e!ects,
such as voltage controlled oscillator (VCO) pretuning, swapping (&ping-ponging') multiple synthesizers, and
transient hop interval dwell and guard times (see Reference 5 and references therein), the alternative
&interpolated frequency hopping' (IFH) technique was recently proposed in Reference 5. The hopping code in
IFH-transceiver is interpolated by the digital "lter, so instead of abrupt hops, frequency varies smoothly in
time, which results in better synchronization and therefore lower BER.
Analog chaotic oscillators provide a natural way of generating smoothly varying frequency modulation.
An additional bene"t of employing analog chaotic oscillators is that they may exhibit self-synchronization,
thus eliminating the need for special synchronizing sequences. A traditional way of using frequency
modulation consists of modulating the transmitter VCO by the signal from the chaos generator, and
demodulating this signal at the receiver before feeding it into the receiver's chaos generator (see Figure 1).
Demodulation (frequency detection) is most e!ectively performed using a phase lock loop.6 The gain of the
PLL must be chosen large enough to provide frequency variation of VCO within the frequency range of the
3
transmitted signal. The problem with this approach is that any interfering signal with frequency close enough
to the current carrier frequency of the transmitted signal, will cause a large perturbation of the frequency of
the receiver VCO. This may cause an instability and de-synchronization of the PLL.
In this paper we propose a di!erent chaotic FM communication scheme based on including the phase lock
loop and the chaotic generator in the feedback loop of the receiver VCO. We show that stable synchronization can be achieved for a certain class of chaotic FM systems. In computer simulations we demonstrate that
the synchronous regime is much more robust against additive disturbances in a channel than a simple FM
scheme (Figure 1). We also demonstrate that within this scheme, a binary information signal can be
transmitted via phase modulation and coherent detection at the receiver.
2. SYNCHRONIZATION SCHEME
The block diagram of the proposed synchronization scheme is shown in Figure 2. In the transmitter, one of
the state variables x (t) of the chaotic oscillator (CH ) is used for the frequency modulation of the voltage
5
5
controlled oscillator (VCO ) to get a spread spectrum signal. The receiver consists of chaotic oscillator CH ,
5
3
voltage-controlled oscillator VCO , phase discriminator (?), low-pass "lter (LPF) and adder (AD). The
3
chaotic oscillators CH and voltage-controlled oscillators VCO are assumed identical. The sum of the
5,3
5,3
state variable x (t) and the output signal from the low-pass "lter of the PLL u(t) modulates the frequency of
3
VCO . When the state variables x (t) and x (t) are close to each other, the phase lock loop
3
5
3
(?PLPFPADPVCO ) locks on the frequency of the transmitted signal. In this regime the output
3
voltage u of LPF is proportional to the open-loop frequency o!set and therefore to the di!erence x (t)!x (t).
5
3
The LPF output u feeds the CH , providing uni-directional dissipative coupling between the chaotic
3
oscillators in the transmitter and receiver. This coupling leads to chaotic synchronization. When the chaotic
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
CHAOTIC FREQUENCY MODULATION
571
Figure 2. CFM synchronization scheme
oscillators are synchronized and the voltage controlled oscillators are phase-locked, switching the phase of
the FM signal can be used for binary information transmission (see Section 4 below).
For the analysis of the proposed scheme suppose that x (t) and x (t) are normalized state variables:
5
3
maxDx (t) D"maxDx (t) D"1. The frequencies of VCOs are modulated by the chaotic signals:
5
3
uR , u "u (1!m x )
5
5
0
1 5
uR , u "u (1!m x !m u(t))
3
3
0
1 3
2
(1)
where u is the &natural' frequency of VCO, m and m are the modulation gain coe$cients, u(t) is the LPF
0
1
2
output signal normalized by the maximum output voltage of phase discriminator.
Denote u"u !u and consider the "rst-order low pass "lter with the transfer function
3
5
K(s)"1/(1#¹s). Then the equations for PLL can be written in the form:
uR " u m (x (t)!x (t))!u m u
0 1 5
3
0 2
¹uR " /(u)!u
(2)
where the function /(u) is the normalized characteristic of phase discriminator (max(/(u)"1)). If a multiplier is used as a phase discriminator, /(u),sinu.
The chaotic systems in transmitter and receiver can be described by the following equations:
q xR "f (x , X ) , q xR "f (x , X )#eu
#) 5
5 5
#) 3
3 3
q XQ "F(x , X ), q XQ "F(x , X )
#) 5
5 5
#) 3
3 3
(3)
where: x , x are state variables used for frequency modulation; X , X are vectors of other variables;
5 3
5
3
f and F are non-linear functions; q is a characteristic time of chaotic oscillations; e is a coupling
#)
parameter.
Combining (2) and (3) and changing time to a dimensionless form: q"tJu m /¹, one can derive the
0 2
following set of equations for both transmitter and receiver:
xR "a f (x , X )
5
5 5
XQ "aF(x , X )
5
5 5
xR "a( f (x , X )#eu)
3
3 3
(4)
XQ "aF(x , X )
3
3 3
juQ "b(x (t)!x (t))!u
5
3
1
uR "/(u)!u
j
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
572
A. R. VOLKOVSKII AND L. S. TSIMRING
Figure 3. Phase portrait of the system (4) for a;1. & indicates planes of fast locking of the phase lock loop, and line m shows a slow
synchronization of the chaotic generators
where: a"(J¹/u m )/q "q /q is the relative speed of chaotic frequency modulation; b"m /m is the
0 2 #)
1-- #)
1 2
relative depth of frequency modulation; j"1/Ju m ¹ is the PLL damping parameter.
0 2
For small a;1, the dynamics of system (4) can be separated into fast and slow motions. Phase lock
loop dynamics (last two equations of (4)) is fast, and the dynamics of chaotic generators is slow. So,
started from arbitrary initial condition, the system "rst approaches the manifold of slow motions in which
PLL dynamics is slaved to the dynamics of the chaotic oscillators (see Figure 3). On the slow motion
manifold, one can neglect time derivatives uQ and uR , so u"b(x (t))!x (t), u"/~1(u). Then the remaining
5
3
equations read
xR "a f (x , X )
5
5 5
XQ "aF(x , X )
5
5 5
(5)
xR "a ( f (x , X )#eb(x (t)!x (t)))
3
3 r
5
3
XQ "aF(x , X )
3
3 3
This system describes the dynamics of two dissipatively coupled chaotic oscillators. If the subsystem
XQ "F(x, X) is stable (as determined by the conditional Lyapunov exponents7), then for large enough
coupling strength e, system (5) exhibits stable synchronization (x (t)!x (t)P0, X (t)!X (t)P0) as tPR
5
3
5
3
for (almost) any initial conditions (see, for example, 8}10). Thus in the limit a;1 the problem of synchronization of the CFM system is reduced to the synchronization of the low-frequency chaotic oscillators CH .
5,3
However, if parameter a is of the order of 1, the dynamics of the full system (4) is more complicated and
synchronization may not occur.
Frequencies of the VCO change within the range *)"m u around the central frequency u . However,
5,3
1 0
0
in the neighbourhood of the synchronized state, the frequency of the VCO is close to that of VCO ,
3
5
and therefore the bandwidth of LPF can be made small as compared to the bandwidth of the chaotically
modulated carrier *) (or, equivalently, ¹<(m u )~1). In fact, the signal which should pass through
1 0
the "lter, u, has the bandwidth of the low-frequency chaotic oscillators CH , and so the bandwidth of
5,3
LPF should be determined by the bandwidth of the chaotic oscillator itself (¹(q ). This provides selectivity
#)
of the proposed scheme with respect to in-band interference, since signals with frequencies not close to
the current transmitter frequency, are e!ectively "ltered out by the low-pass "lter in PLL. Only during
short intervals when frequencies of the carrier and the interference are su$ciently close, is the PLL
disturbed by the latter, and some deviation from the synchronized state occurs. Unlike the simple
FM modulation scheme (Figure 1), the gain of the phase lock loop m can be made much smaller
2
than m , and therefore even large in-band interference is signi"cantly attenuated in the PLL and does not
1
destroy synchronization.
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
573
CHAOTIC FREQUENCY MODULATION
Figure 4. Region of stable CFM synchronization in the parameter plane (j, e) for system (4), (6), (7) for b"5
3. NUMERICAL SIMULATIONS
We performed numerical simulations of the synchronization scheme shown in Figure 2 for two third-order
chaotic systems described earlier,10 with X"y, z, and
f (x, y, z)"y,
G
F(x, y, z)"
!x!dy#z,
c(kG(x)!z)!py
The parameters of the systems are d"0)43, k"24)7, p"0)72, c"0)1, and
G
G(x)"
H
(6)
x(1!x2), at!1)2(x(1)2,
0)528,
at x(!1)2,
!0)528,
at x'1)2
(7)
The phase discriminator function was chosen /(u),sin(u). We indeed found a large parameter region for
a stable synchronized operation of the CFM system. Figure 4 shows the region of lock-in (convergence from
any arbitrary initial conditions to the synchronous state) in the parameter plane (j, e) for b"5 and two
di!erent values of a. For small a"0)01, the lock-in occurs for e'0)15 except for very small j. For larger
a"0)1, the lock-in region shrinks.
We investigated the selectivity of the proposed method by adding a sinusoidal component at the frequency
u within the CFM range to the transmitted signal. The amplitude of the interference component was chosen
1
to be 20% of the amplitude of the transmitted signal. The results of the simulations are shown in Figure 5.
Small splashes of the PLL output signal u are produced when the frequency of the transmitted signal is close
to u . Nevertheless, since m (m , frequency of VCO remains close to VCO , and the system does not lose
1
2
1
3
5
synchronization. In contrast, similar interference in the conventional scheme (Figure 1) produced large
perturbations of the frequency of VCO , and loss of synchronization. In this scheme, the receiver PLL must
3
have the same large gain m as the transmitter in order to change the frequency of VCO within the range of
1
3
operation. Therefore, all in-band interference signals directly a!ect the chaotic oscillator at the receiver
(Figure 6).
4. COMMUNICATION USING CHAOTIC FM SYSTEM
The proposed method of chaotic frequency modulation can be readily utilized for information transmission.
Indeed, since the PLL provides phase synchronization between transmitter and receiver, binary-phase switch
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
574
A. R. VOLKOVSKII AND L. S. TSIMRING
Figure 5. CFM synchronization in the presence of in-band interference. Dashed line in the top panel indicates the frequency of the
interference signal u "23
1
Figure 6. Synchronization of chaotic oscillators via standard FM link in the presence of in-band interference at the frequency u "23
1
keying (BPSK) at the transmitter can be easily detected at the receiver. In order to maintain synchronization
irrespective to phase switching, the phase lock loop must operate at the second harmonic of the carrier
frequency, so a multiplier and a high-pass "lter should be added to the scheme (see Figure 7). An example of
the binary information transmission using this system, is shown in Figure 8. After initial transient, phase
switching is readily detected at the receiver without any loss of synchronization.
It is important to emphasize that this synchronization scheme is robust against small detunings
of parameters. We investigated the in#uence of the parameter mismatch between transmitter and receiver
chaos generators on the quality of synchronization. We varied parameter c in (6) of the receiver while
3
keeping c "0)1 at the transmitter. Figure 9 shows RMS values of the di!erence x (t)!x (t) and frequency
5
5
3
di!erence u (t)!u (t) as functions of the parameter mismatch *"(c !c )/c . For small values of the
3
5
3
5 5
mismatch, the di!erence x (t)!x (t) is small, and PLL is able to adjust the frequency of VCO to keep exact
5
3
3
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
575
CHAOTIC FREQUENCY MODULATION
Figure 7. BPSK communication scheme using CFM synchronization
Figure 8. Digital communication using BPSK in conjunction with CFM: upper panel*frequencies of transmitter and receiver VCOs,
lower panel*phase variation at the transmitter (dashed line) and at the receiver (solid line)
Figure 9. RMS values of x (t)!x (t) (squares) and u (t)!u (t) (circles) as functions of the parameter mismatch *"(c !c )/c
5
3
5
3
3
5 5
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
576
A. R. VOLKOVSKII AND L. S. TSIMRING
synchronization (u "u ). At larger values of *'6%, PLL is unable to maintain synchronization, since the
3
5
open-loop PLL frequency detuning m u Dx (t)!x (t)D occasionally exceeds the PLL hold range m u .
1 0 5
3
2 0
5. CONCLUSIONS
An this paper we presented a method for communication using chaotic frequency modulation. This method
di!ers from a simple modulation/demodulation technique, in which a phase lock loop directly reconstructs
the low-frequency chaotic signal in the receiver. In the proposed method frequencies of voltage-controlled
oscillators in both transmitter and receiver are modulated by the chaotic generators. The phase lock loop
detects the error (di!erence) between the chaotic signals, which is used for synchronization of the chaotic
oscillators. This allows a reduced PLL gain and provides good selectivity of this scheme with respect to the
in-band interference. We demonstrated that this method combined with phase shift keying can be used for
binary spread spectrum communication. Quantitative analysis of the channel capacity for the proposed
scheme and extending it for multi-user communication remains an important topic for future research.
ACKNOWLEDGEMENTS
The authors acknowledge numerous discussions on the subject with H. Abarbanel, L. Larson, N. Rulkov,
and M. Sushchik. This work was supported by the U.S. Army Research O$ce under MURI grant
DAAG55-98-1-0269 and by the U.S. Department of Energy under grant DOE/DE-FG03-95R14516.
REFERENCES
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5. N. M. Filiol, C. Plett, T. Riley and M. A. Copeland, &An Interpolated Frequency-Hopping Spread-Spectrum Transceiver', IEEE
¹rans. Circuits System, 45, 3}12 (1998).
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9. V. S. Afraimovich, N. N. Verichev and M. I. Rabinovich, &Stochastic synchronization of oscillation in dissipative systems',
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Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 27: 569}576 (1999)
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