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Channel Equalization for Chaotic Communications Systems

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**1. **Channel Equalization for Chaotic Communications Systems Mahmut Ciftci
February, 2nd 2001

**2. **Georgia Institute of Technology Center for Signal and Image Processing Outline
Background
Chaos in Communications
Research
Conclusion and Future Work

**3. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing Discrete and continuous-time dynamical systems are represented as
and
A chaotic dynamical system is
Nonlinear,
Deterministic, not random
Irregular
Never repeats itself Dynamical Systems

**5. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing Sawtooth Map

**7. **Georgia Institute of Technology Center for Signal and Image Processing Logistic Map Logistic Map
Difference between two chaotic sequences with initial conditions of x[0]=0.2 and x[0]=0.2001

**8. **Georgia Institute of Technology Center for Signal and Image Processing Lorenz System Lorenz System Attractor

**9. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing Symbolic Dynamics (cont.) Real dynamics
Symbolic dynamics
The relationship between these dynamics
represents the mapping between the two dynamics

**11. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing Applications Communication systems based on chaos have recently been proposed including
Chaotic modulation and encoding,
Chaotic masking, and
Spread spectrum.

**14. **Georgia Institute of Technology Center for Signal and Image Processing Block Diagram

**15. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing Approach The goal is to achieve equalization by exploiting
The knowledge of the system dynamics
Symbolic-dynamics representation
Synchronization

**17. **Georgia Institute of Technology Center for Signal and Image Processing Proposed Solutions Optimal Noise Reduction Algorithm
Received Signal
r[n]= x[n]+w[n]
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. **Georgia Institute of Technology Center for Signal and Image Processing 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. **Georgia Institute of Technology Center for Signal and Image Processing Simulation Results

**20. **Georgia Institute of Technology Center for Signal and Image Processing 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