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Chaotic Communication Communication with Chaotic Dynamical Systems Mattan Erez December 2000

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Chaotic Communication. Communication with Chaotic Dynamical Systems Mattan Erez December 2000. 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 - PowerPoint PPT Presentation

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Page 1: Chaotic Communication

Chaotic Communication

Communication with Chaotic Dynamical Systems

Mattan Erez

December 2000

Page 2: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 2

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

Page 3: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 3

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

Page 4: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 4

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

Page 5: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 5

Continuous Time Systems

Described by differential equations dimension 3 for chaotic behavior

Example: Lorenz System

, , and are parameters

zxyz

xzyxy

xyx

)(

Page 6: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 6

Useful Concepts

Attractor: set of orbits to which the system approaches from any initial state (within the attractor basin)

Poincare` Surface of Section

Page 7: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 7

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

Page 8: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 8

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

Page 9: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 9

Example - Lorenz System

XYZ

Xr

Yr

Zr

x(t)

n(t)

s(t) xr(t)

zxyz

xzyxy

xyx

)(

rrr

rrr

rrr

zxyz

xzyxy

xyx

)(

Page 10: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 10

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

Page 11: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 11

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)

Page 12: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 12

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)

Page 13: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 13

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)

Page 14: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 14

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

Page 15: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 15

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

Page 16: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 16

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)

Page 17: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 17

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

Page 18: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 18

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

Page 19: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 19

Example - Double Scroll System

01

01

Page 20: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 20

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

Page 21: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 21

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

Page 22: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 22

Receiver

Threshold to detect 0 and 1

decode the constrained-code

Page 23: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 23

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

Page 24: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 24

Noise Resistance

Force forbidden sequences to form a “noise-gap”

In the example system - translates into stricter RLL constraint

01

Page 25: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 25

Capacity vs. Noise Gap

Devil’s staircase structure

1

.5 1.5+

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December 00 Chaotic Communication – Mattan Erez 26

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

Page 27: Chaotic Communication

December 00 Chaotic Communication – Mattan Erez 27

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.