Channel Equalization for Chaotic Communications Systems

1
Channel Equalization for Chaotic Communications
Systems

• Mahmut Ciftci
• February, 2nd 2001

2
Outline

• Background
• Chaos in Communications
• Research
• Conclusion and Future Work

3
Introduction

• Edward Lorenz,
• a meterologist in MIT
• trying to predict the weather
• Butterfly Effect
• If a butterfly flaps its wings in China, it could
  change the weather in New York.

4
Dynamical Systems

• Discrete and continuous-time dynamical systems
  are represented as
• and
• A chaotic dynamical system is
• Nonlinear,
• Deterministic, not random
• Irregular
• Never repeats itself

5
Properties of Chaotic Systems

• A continuous/discrete time dynamical system is
  considered chaotic if it has the following
  properties
• Sensitivity to initial conditions
• Small occurences can cause large changes.
• Lyapunov exponents describes the sensitivity
• Dense periodic points
• There exits periodic points in any interval on
  the attractor
• Mixing property
• Starting from
• Strange attractor
• Unstable in a bounded region
• Fractal, i.e. self similar to itself for
  different scales
• Non-integer dimension

6
Sawtooth Map

• Sawtooth Map
• Attractor

7
Logistic Map

• Logistic Map
• Difference between two chaotic sequences with
  initial conditions of x00.2 and x00.2001

8
Lorenz System

• Lorenz System Attractor

9
Symbolic Dynamics

• A means of assigning a finite alphabet of symbols
  to a chaotic signal.
• First, the state space is divided into a finite
  number of partitions and each partition is
  labeled with a symbol. Then instead of
  representing the trajectories by infinite
  sequences of numbers, one watches the alternation
  of symbols.
• The dynamics governing the symbolic sequence is a
  left shift operation.
• Depending on the dynamics, there may be a one-to
  one equivalence between the initial state and the
  infinite sequence of symbols.

10
Symbolic Dynamics (cont.)

• Real dynamics
• Symbolic dynamics
• The relationship between these dynamics
• represents the mapping between the two dynamics

11
Real Life Examples

• Wheather
• Stock Market
• Prices random with a trend
• Trend varies from market-to-market and
  time-to-time
• Irregular Heart Beats
• Controlling heart attacks may mean controlling
  chaotic systems with small perturbations
• Brain Waves
• Dishwashing Machine by Goldstar

12
Properties for Communications

• Why are we interested in chaos?
• Certain properties of chaotic systems are
  appealing for communications such as
• Low power
• Broadband spectra
• Noise-like appearance
• Auto and cross correlation properties
• Self-synchronization property

13
Applications

• Communication systems based on chaos have
  recently been proposed including
• Chaotic modulation and encoding,
• Chaotic masking, and
• Spread spectrum.

14
Block Diagram

• Communication channel distorts the transmitted
  signal by introducing intersymbol interference
  and noise.

15
Motivation

• Most of the proposed systems disregard the
  distortions introduced by typical communication
  channels and fail to work under realistic channel
  conditions.
• Conventional equalization algorithms do not work
  for chaotic communications systems
• Equalization algorithms specifically designed for
  chaotic communications systems are needed

16
Approach

• The goal is to achieve equalization by exploiting
• The knowledge of the system dynamics
• Symbolic-dynamics representation
• Synchronization

17
Proposed Solutions

• Optimal Noise Reduction Algorithm
• Received Signal
• rn xnwn
• The cost function to be minimized
• A trellis diagram based on the system dynamics is
  constructed by exploiting symbolic dynamic
  representation of the chaotic system.
• The Viterbi algorithm is used to estimate the
  chaotic sequence.

18
Proposed Solutions (cont.)

• A dynamics-based blind equalization algorithm
• The knowledge of the system dynamics is exploited
  for equalization
• Sequential channel equalization algorithm
• Dynamics-based trellis diagram is extended to
  accommodate FIR channel model.
• Viterbi algorithm is then used to obtain the
  estimate of the chaotic sequence to update the
  channel coefficients in the NLMS algorithm.

19
Simulation Results

• Estimation

• Equalization

• Multipath fading channel model is used.
• fmT0.005 and SNR15dB

• The proposed Viterbi based algorithm is compared
  with MAP estimator for sawtooth map

20
Conclusion and Future Work

• Optimal estimation and channel equalization
  algorithms have been proposed for chaotic systems
  with symbolic dynamic representation.
• End-to-end chaotic communications systems is to
  be simulated.
• The possibility of extending the results to the
  multi-user communication is to be investigated