This is my currently most active area of research and deals with communications systems in which messages are transmitted and received using chaotic laser sequences. Such systems are being explored by communication engineering researchers in the USA, Italy, Germany, Switzerland and Japan. In the simplest of these systems binary bits are coded into chaotic sequences by 'plus or minus one multiplication', usually in repeated blocks, and are then transmitted. They are received after distortion by stochastic channel noise and are be decodedby using a laser sequence synchronosed with the transmitter laser. This is compared to the sequence containing the binary message bits in order to determine their values. Statistical research can make significant contributions to the area because of the need for model-based estimation of the bit types and probability calculations relevant to assessing the accuracy of such estimates. The interaction of the engineering and statistical approaches is leading to better understanding and hence to optimization of such systems; earlier difficulties are being resolved, approximations are being replaced by exact results and previously intractable systems are being investigated. The research is bringing new areas into statistics and new statistical aspects of chaos into communications engineering, in mutually beneficial ways.

A key element in the theory is the statistical behaviour of the deterministic chaotic sequences; this has been developed from Kohda's generalizations of Perron-Frobenius theory of invariant distributions and statistical depenedency to bivariate chaos and by the exploration of synchronous chaos in bivariate sequences. The latter is required in communication models in which a copy of the original chaotic sequence has also to be transmitted. Thus statistical behaviour of bivariate chaotic sequences is an important area which has been investigated

Participants in this work include Dr R Hilliam (former EPSRC Research Fellow), Dr G Ohama (former Research Fellow, Shiga University, Japan), Dr N Balakrishna (former Birmingham Commonwealth Visiting Fellow, Cochin University, India) and Dr R Wolff (collaborator, University of Queensland, Australia), Mr Ji Yao (former Research Student and Dr G. Kaddoum (collaborator, University of Quebec). In addition, the programme has benefited from funding by QinetiQ, a Royal Society European Science Exchange Project with the Chaos-Engineering Group at the Universities of Bologna and Ferrara, Italy and connections with researchers in Germany, Japan, Hong Kong and Switzerland.

Selected References

Statistical aspects of chaotic maps with negative dependency in a communications setting, with N. Balakrishna. J. R. Statistic. Soc. B (2001), 62, 843-853.

Binary time series generated by chaotic logistic maps; with R. Wolff, Stochastics and Dynamics, (2003), 3, 529-544.

Exact calculation of bit error rates in communication systems with chaotic modulation, with G Ohama. IEEE Transactions on Circuits and Systems ?I: Fundamental Theory and Applications, (2003), 50, 1391-1400.

The dynamics and statistics of bivariate chaotic maps in communications modelling, with R M Hilliam, Int. J. Bifurcation and Chaos (2004), 14, 4, 1177-1194.

Exact analytical bit-error-rates for multiple access chaos-based communication systems, with W M Tam, F C M Lau, C K Tse, IEEE Transactions on Circuits and Systems II: Fundamental Theory and Applications (2004), 51, 9 473-410,

Bit error rate calculation for multi-user coherent chaos-shift-keying communications systems, with J Yao. IEICE Trans. Fundamentals (2000), E87-A, 9, 2280-2291.

Chaos communication synchronization: Combating noise by distribution transformation, with R Hilliam. Statistics and Computing,( 2005), 15, 43-52.

Bit error probability and bit outage rate in chaos communication, with G.Ohama. Circuits, Systems and Signal Processing, (2005), 24, 5, 519-53

Performance analysis and optimization of multi-user differential chaos-shift-keying communication systems, with J Yao. IEEE Transactions on Circuits and Systems –I: Regular Papers (2006), 53, 9, 2075-2091

Sensitive parameter dependence of autocorrelation functions in piece-wise linear maps; with A L Baranovski, Int. J. Bifurcation and Chaos, (2007) 17, 4, 1185-1197

Statistical dependency in chaos, with N. Balakrishna, Int. J. Bifurcation and Chaos 18, (2008),3207-3219.

Likelihood-based demodulation in multi-user chaos shift keying, with J. Yao. Circuits, Systems and Signal Processing (2008), 27, 847-864.

Chaos communication performance: theory and computation, with G. Kaddoum, P Chargé, and D Roviras, Circuits, Systems and Signal Processing, 30,(2010), 185-208.

Chaos communications: an overview of exact, optimum and approximate results using ststistical theory, in 'Chaotic Systems: Theory and Applkicatins', ed. C H Skiadas & I Dimotikalis, World Scientific, 2010, 155-164

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