Chaotic Communication
Communication with Chaotic Dynamical Systems
Mattan Erez
December 2000
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Chaotic Communication
Not an oxymoron Chaos is deterministic
Two chaotic systems can be synchronized
Chaos can be controlled
Communicating with chaos Use chaotic instead of periodic waveforms
Control chaotic behavior to encode information
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Outline
What is chaos
Synchronizing chaos
Using chaotic waveforms
Controlling chaos
Information encoding within chaos
Capacity
Summary: Why (or why not) use chaos?
References and links
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What is Chaos?
Non-linear dynamical system Deterministic Sensitive to initial conditions
( - Lyapunov exponent)
Dense Infinite number of trajectories in finite region of phase space
)0()( xetx t
perfect knowledge of present
perfect prediction of future
imperfect knowledge of present
(practically) no prediction of future
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Continuous Time Systems
Described by differential equations dimension 3 for chaotic behavior
Example: Lorenz System
, , and are parameters
zxyz
xzyxy
xyx
)(
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Useful Concepts
Attractor: set of orbits to which the system approaches from any initial state (within the attractor basin)
Poincare` Surface of Section
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Discrete Time Systems
Described by a mapping function Can be one-dimensional
Logistic Map
Bernoulli Shift
Tent Map
time
0.5 1
1))(1)(()( nxnxnx
101mod21 xxx nn
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Chaos Synchronization
Non-trivial problem sensitivity to initial conditions + density initial state never accurate in a real system trivial if dealing with finite precision simulations
Chaotic Synchronization (Pecora and Carrol Feb. 1990) Couple transmitter and receiver by a drive signal Build receiver system with two parts
response system and regenerated signal Response system is stable (negative Lyapunov exp.) Converges towards variables of the drive system Can synchronize in presence of noise and parameter
differences
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Example - Lorenz System
XYZ
Xr
Yr
Zr
x(t)
n(t)
s(t) xr(t)
zxyz
xzyxy
xyx
)(
rrr
rrr
rrr
zxyz
xzyxy
xyx
)(
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Chaotic Waveforms in Comm.
Chaotic signals are a-periodic
Spread spectrum communication Instead of binary spreading sequences
Directly as a wideband waveform
Code-division techniques Replaces binary codes
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Chaotic Masking
Mask message with noise-like signal Amplitude of information must be small
XYZ
Xr
Yr
Zr
x(t)
n(t)
s(t) xr(t)
m(t)
+-
mr(t)
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Dynamic Feedback Modulation
Mask message with chaotic signal Removes restriction on small message amp. Care must be taken to preserve chaos
XYZ
Xr
Yr
Zr
x(t)
n(t)
s(t) xr(t)
m(t)
+-
mr(t)
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Chaos Shift Keying
Modulate the system parameters with the message Similar concept to FSK but for a different parameter
Suitable mostly for digital communication Shift to a different attractor based on information symbol
Also DCSK to simplify detection
XYZ
Xr
Yr
Zr
x(t)
n(t)
s(t)
m(t)
xr(t) +-
detectormr(t)
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Problems in Conventional CDMA Binary m-sequences
good auto-correlation bad cross-correlation few codes
Binary gold sequences good cross-correlation acceptable auto-correlation few codes
Binary random maps good auto-correlation good cross-correlation many codes very large maps (storage)
Very long and complex (re)synchronization
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Chaotic Sequences for CDMA
Simple description of chaotic systems one dimensional maps
Very large number of codes many useful maps many initial states (sensitivity to initial conditions)
Good spectral properties a-periodic with a flat (or tailored) spectrum
Good auto/cross correlation mostly based on numerical results “Checbyshev sequences” yield 15% more users
Fast synchronization If based on self-sync chaotic systems
Low probability of intercept chaotic sequence are real-valued and not binary
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Chaos in Ultra WB - CPPM Impulse communication
uses PN sequences and PPM PN spectrum has spectral peaks
Chaotic Pulse Position Modulation
Circuit implementation simple tent map and time-voltage-time converters extremely fast synchronization (4 bits) Low power
))1(()( oninforamatitntFnt 001101t0 = 0t1 = t
t(0) t(1) t(2) t(3) t(4)
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Controlling Chaos
Chaotic attractor (usually) consists of infinite number of unstable periodic orbits
Small perturbation of accessible system param forces the system from one orbit to a more desirable one (Ott, Grebogi, and Yorke - Mar. 1990)
the effect of the control is not immediate each intersection of the phase-space coordinate eith
the surface of section a control signal is given the exact control is pre-determined to shift the orbit to
the desired one, such that a future intersection will occur at the desired point
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Encoding in Chaos
Use symbolic dynamics to associate information with the chaotic phase-space phase space is partitioned into r regions each region is assigned a unique symbol the symbol sequences formed by the trajectories of the
system are its symbolic dynamics Identify the grammar of the chaotic system
the set of possible symbol sequences (constraint) depends on the system and symbol partition
Exercise chaos control to encode the information within the allowed grammar
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Example - Double Scroll System
01
01
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Symbolic Dynamics Transmission
Use previous regions for two symbols Build coding function - r(x)
for each intersection point (region) - record the following n-bit sequence
Build an inverse coding function s(r) define a region as the mean state-space point
corresponding to the n-bit sequence r.
Build a control function d(r) small perturbations: p = d(r)x
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Transmission (2)
Encode user information to fit the grammar use a constrained-code based on the grammar for the experimental setup demonstrated, the constraint
is a RLL constraint Transmit the message
load the n-bit sequence of r(x0) into a shift register shift out the MSB and shift in the first message bit (LSB) the SR now holds the word r1’ with the desired information bit
the next intersection occurs at x1=s(r1) of the original system at that point we apply the control pulse to correct the trajectory:
p=d(r1)(x1-s(r1’))
repeat
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Receiver
Threshold to detect 0 and 1
decode the constrained-code
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Capacity of Chaotic Transmission
The capacity of the system is its topological capacity define a partition and assign symbols w count the number of n-symbol sequences the system
can then produce N(w,n)
Additional restrictions on the code (for noise resistance) decrease capacity
nnwN
nwtopH ),(limsup
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Noise Resistance
Force forbidden sequences to form a “noise-gap”
In the example system - translates into stricter RLL constraint
01
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Capacity vs. Noise Gap
Devil’s staircase structure
1
.5 1.5+
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Summary
Chaos in spread-spectrum (and CDMA)
spectral properties synchronization can be
fast and simple compact and efficient
representation good multi-user
performance worse single-user
performance loss of synchronization mismatched parameters
low power circuits enhanced security
LPI + numerous codes(can be done with pseudo-chaos)
Direct encoding in chaos neat idea
simple circuits?
low power?
synchronization control
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References and Links
http://rfic.ucsd.edu/chaos Communication based on synchronizing chaos
L. Pecora and T. Carroll, “Synchronization in Chaotic Systems,” Physical Review Letters,Vol. 64, No. 8, Feb. 19th, 1990
L. Pecora and T. Carroll, “Driving Systems with Chaotic Signals,” Physical Review A, Vol. 44, No. 4, Aug. 15th, 1991
K. Cuomo and A. Oppenheim, “Circuit Implementation of Synchronized Chaos with Application to Communication,” Physical Review Letters, Vol. 71, No. 1, July 5th, 1993
G. Heidari-Bateni and C. McGillem, “A Chaotic Direct-Sequence Spread-Spectrum Communication System,” IEEE Transactions on Communications, Vol. 42, No. 2/3/4, Feb./Mar./Apr. 1994
G. Mazzini, G. Setti, and R. Rovatti, “Chaotic Complex Spreading Sequences for Asynchronous DS-CDMA-Part I: System Modeling and Results,” IEEE Transactions on Circuits and Systems-I, Vol. 44, No. 10, Oct. 1997
Communication based on controlling chaos E. Ott, C. Grebogi, and J. Yorke, “Controlling Chaos,” Physical Review Letters, Vol. 64, No. 11, Mar. 12th, 1990 S. Hayes, C. Grebogi, and E. Ott, “Communicating with Chaos,” Physical Review Letters, Vol. 70, No. 20, May
17th, 1993 S. Hayes, C. Grebogi, E. Ott, and A. Mark, “Experimental Control of Chaos for Communication,” Physical Review
Letters, Vol. 73, No. 13, Sep. 26th, 1994 E. Bollt, Y-C Lai, and C. Grebogi, “Coding, Channel Capacity, and Noise Resistance in Communicating with
Chaos,” Physical Review Letters, Vol. 79, No. 19, Nov. 10th, 1997 J. Jacobs, E. Ott, and B. Hunt, “Calculating Topological Entropy for Transient Chaos with an Application to
Communicating with Chaos,” Physical Review E, Vol. 57, No. 6, June 1998. I. Marino, E. Rosa, and C. Grebogi, “Exploiting the Natural Redundancy of Chaotic Signals in Communication
Systems,” Physical Review Letters, Vol 85, No. 12, Sep. 18th, 2000.