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Chaotic Communication

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

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Chaotic Communication

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  1. Chaotic Communication Communication with Chaotic Dynamical Systems Mattan ErezDecember 2000

  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 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 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 perfect knowledge of present perfect prediction of future imperfect knowledge of present (practically) no prediction of future Chaotic Communication – Mattan Erez

  5. Continuous Time Systems • Described by differential equations • dimension  3 for chaotic behavior • Example: Lorenz System, r, andb are parameters 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 Chaotic Communication – Mattan Erez

  7. Discrete Time Systems • Described by a mapping function • Can be one-dimensional • Logistic Map • Bernoulli Shift • Tent Map 1 0.5 1 time 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 systemand 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 Chaotic Communication – Mattan Erez

  9. Example - Lorenz System x(t) s(t) xr(t) X Xr Yr Y n(t) Zr Z 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 Chaotic Communication – Mattan Erez

  11. Chaotic Masking • Mask message with noise-like signal • Amplitude of information must be small x(t) s(t) xr(t) + - X Xr Yr Y m(t) n(t) mr(t) Zr Z 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 x(t) s(t) xr(t) + - X Xr Yr Y m(t) n(t) mr(t) Zr Z 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 x(t) s(t) xr(t) + - X Xr Yr Y mr(t) detector n(t) Zr Z m(t) 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 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 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 001101 t0 = 0 t1 = t Dt(0) Dt(1) Dt(2) Dt(3) Dt(4) 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 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 grammarof 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 Chaotic Communication – Mattan Erez

  19. Example - Double Scroll System 0 1 1 0 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)Dx 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 Chaotic Communication – Mattan Erez

  22. Receiver • Threshold to detect 0 and 1 • decode the constrained-code 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 Chaotic Communication – Mattan Erez

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

  25. Capacity vs. Noise Gap • Devil’s staircase structure 1 .5 .5+e 1 Chaotic Communication – Mattan Erez

  26. 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 Direct encoding in chaos neat idea simple circuits? low power? Summary synchronization control (can be done with pseudo-chaos) 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. Chaotic Communication – Mattan Erez

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