Chaotic Communication Communication with Chaotic Dynamical Systems Mattan Erez December 2000

December 00Chaotic Communication – Mattan Erez2 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

December 00Chaotic Communication – Mattan Erez3 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

December 00Chaotic Communication – Mattan Erez4 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 perfect knowledge of present perfect prediction of future imperfect knowledge of present (practically) no prediction of future

December 00Chaotic Communication – Mattan Erez5 Continuous Time Systems Described by differential equations dimension  3 for chaotic behavior Example: Lorenz System , , and  are parameters

December 00Chaotic Communication – Mattan Erez6 Useful Concepts Attractor: set of orbits to which the system approaches from any initial state (within the attractor basin) Poincare` Surface of Section

December 00Chaotic Communication – Mattan Erez7 Discrete Time Systems Described by a mapping function Can be one-dimensional Logistic Map Bernoulli Shift Tent Map time

December 00Chaotic Communication – Mattan Erez8 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

December 00Chaotic Communication – Mattan Erez9 Example - Lorenz System X Y Z XrXr YrYr ZrZr x(t)x(t) n(t)n(t) s(t)s(t)xr(t)xr(t)

December 00Chaotic Communication – Mattan Erez10 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

December 00Chaotic Communication – Mattan Erez11 Chaotic Masking Mask message with noise-like signal Amplitude of information must be small X Y Z XrXr YrYr ZrZr x(t)x(t) n(t)n(t) s(t)s(t)xr(t)xr(t) m(t)m(t) + - mr(t)mr(t)

December 00Chaotic Communication – Mattan Erez12 Dynamic Feedback Modulation Mask message with chaotic signal Removes restriction on small message amp. Care must be taken to preserve chaos X Y Z XrXr YrYr ZrZr x(t)x(t) n(t)n(t) s(t)s(t)xr(t)xr(t) m(t)m(t) + - mr(t)mr(t)

December 00Chaotic Communication – Mattan Erez13 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 X Y Z XrXr YrYr ZrZr x(t)x(t) n(t)n(t) s(t)s(t) m(t)m(t) xr(t)xr(t) + - detector mr(t)mr(t)

December 00Chaotic Communication – Mattan Erez14 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

December 00Chaotic Communication – Mattan Erez15 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

December 00Chaotic Communication – Mattan Erez16 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 t 0 = 0 t 1 = t  t(0)  t(1)  t(2)  t(3)  t(4)

December 00Chaotic Communication – Mattan Erez17 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

December 00Chaotic Communication – Mattan Erez18 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

December 00Chaotic Communication – Mattan Erez19 Example - Double Scroll System

December 00Chaotic Communication – Mattan Erez20 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

December 00Chaotic Communication – Mattan Erez21 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(x 0 ) into a shift register shift out the MSB and shift in the first message bit (LSB) the SR now holds the word r 1 ’ with the desired information bit the next intersection occurs at x 1 =s(r 1 ) of the original system at that point we apply the control pulse to correct the trajectory: p=d(r 1 )(x 1 -s(r 1 ’ )) repeat

December 00Chaotic Communication – Mattan Erez22 Receiver Threshold to detect 0 and 1 decode the constrained-code

December 00Chaotic Communication – Mattan Erez23 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

December 00Chaotic Communication – Mattan Erez24 Noise Resistance Force forbidden sequences to form a “noise-gap” In the example system - translates into stricter RLL constraint 0 1

December 00Chaotic Communication – Mattan Erez25 Capacity vs. Noise Gap Devil’s staircase structure 

December 00Chaotic Communication – Mattan Erez26 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

December 00Chaotic Communication – Mattan Erez27 References and Links Communication based on synchronizing chaos L. Pecora and T. Carroll, “Synchronization in Chaotic Systems,” Physical Review Letters,Vol. 64, No. 8, Feb. 19 th, 1990 L. Pecora and T. Carroll, “Driving Systems with Chaotic Signals,” Physical Review A, Vol. 44, No. 4, Aug. 15 th, 1991 K. Cuomo and A. Oppenheim, “Circuit Implementation of Synchronized Chaos with Application to Communication,” Physical Review Letters, Vol. 71, No. 1, July 5 th, 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 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 Communication based on controlling chaos E. Ott, C. Grebogi, and J. Yorke, “Controlling Chaos,” Physical Review Letters, Vol. 64, No. 11, Mar. 12 th, 1990 S. Hayes, C. Grebogi, and E. Ott, “Communicating with Chaos,” Physical Review Letters, Vol. 70, No. 20, May 17 th, 1993 S. Hayes, C. Grebogi, E. Ott, and A. Mark, “Experimental Control of Chaos for Communication,” Physical Review Letters, Vol. 73, No. 13, Sep. 26 th, 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. 10 th, 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 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. 18 th, 2000.