Research in electronic communications has developed chaos-based modelling to enable messages to be carried by chaotic broad-band spreading sequences. When such systems are used it is necessary to simultaneously know the spreading sequence at both the transmitting and receiving stations. This is possible using the idea of synchronization with bivariate maps, providing there is no noise present in the system. When noise is present in the transmission channel, recovery of the spreading sequence may be degraded or impossible. Once noise is added to the spreading sequence, the result may no longer lie within the boundary of the chaotic map. A usual and obvious method of dealing with this problem is to cap iterations lying outside the bounds at their extremes, but the procedure amplifies loss of synchronization. With a minimum of technical details and a computational focus, this paper first develops relevant dynamical and communication theory in the bivariate map context, and then presents a better way of improving synchronization by distribution transformation. The transmission sequence is transformed, using knowledge of the invariant distribution of the spreading sequence, and before noise corrupts the signal in the transmission channel. An ‘inverse’ transformation can then be applied at the receiver station so that the noise has a reduced impact on the recovery of the spreading sequence and hence its synchronization. Statistical simulations illustrating the effectiveness of the approach are presented.
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Hilliam, R.M., Lawrance, A.J. Chaos communication synchronization: Combatting noise by distribution transformation. Stat Comput 15, 43–52 (2005). https://doi.org/10.1007/s11222-005-4788-6
- bivariate maps
- invariant distribution
- Perron-Frobenius theory