Chaotic Communication

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

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

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
5
Continuous Time Systems

• Described by differential equations
• dimension ? 3 for chaotic behavior
• Example Lorenz System?, r, and b are
  parameters

6
Useful Concepts

• Attractor set of orbits to which the system
  approaches from any initial state (within the
  attractor basin)
• Poincare Surface of Section

7
Discrete Time Systems

• Described by a mapping function
• Can be one-dimensional
• Logistic Map
• Bernoulli Shift
• Tent Map

1
0.5
1
time
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

9
Example - Lorenz System
x(t)
s(t)
xr(t)
X
Xr
Y
Yr
n(t)
Zr
Z
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

11
Chaotic Masking

• Mask message with noise-like signal
• Amplitude of information must be small

x(t)
s(t)
xr(t)

-
X
Xr
Y
Yr
n(t)
m(t)
mr(t)
Z
Zr
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
Y
Yr
n(t)
m(t)
mr(t)
Z
Zr
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
Y
Yr
mr(t)
detector
n(t)
Z
Zr
m(t)
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

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

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)
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

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

19
Example - Double Scroll System
0
1
1
0
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

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 x1s(r1) of the
  original system
• at that point we apply the control pulse to
  correct the trajectory pd(r1)(x1-s(r1))
• repeat

22
Receiver

• Threshold to detect 0 and 1
• decode the constrained-code

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

24
Noise Resistance

• Force forbidden sequences to form a
  noise-gap
• In the example system - translates into stricter
  RLL constraint

0
1
25
Capacity vs. Noise Gap

• Devils staircase structure

1
.5
1
.5e
26
Summary
synchronization
control

• Direct encoding in chaos
• neat idea
• simple circuits?
• low power?

• 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)
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.