Channel Equalization for Chaotic Communications Systems

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