Stephen McLaughlin, Department of Electrical Engineering, University of Edinburgh.
In December Professional Group E5 of the IEE organised a colloquium examining the impact of nonlinear dynamics upon a variety of applications. This meeting brought together people from the mathematics, signal processing and circuit theory communities.
The first talk, on Fractal Channel Equalisation, was presented by Professor David Broomhead of the Applied Mathematics Department of UMIST. In his all to brief talk Professor Broomhead presented results, developed by himself with Jerry Huke and Mark Muldoon. In the talk he showed how the response of a linear channel to a random digital input resulted in a stochastic system of the sort described by Barnsley as an Iterated Function System. It was then shown that modern geometric methods developed for deterministic time series analysis could be used in this genuinely stochastic setting. The approach was shown to lead to a more general proof of the convergence of fractionally spaced equalisers.
The next talk was presented by Dr. Bernie Mulgrew of the Department of Electrical Engineering at the University of Edinburgh. Dr. Mulgrew discussed the the implications of nonlinear dynamics for noise cancellation. The work reported in this talk had originally been suggested by work by Broomhead and Huke at a previous meeting. In his talk Dr. Mulgrew delineated the roles of linear and nonlinear prediction in forming the inverse of a linear system. In a variety of illustrative applications it was shown how a nonlinear inverse of a linear system can produce better results than a linear inverse. The key point is that constructing a causal inverse to a linear system leads to an estimation or approximation problem. If the problem is restricted to an MMSE estimate then the expected value of the signal of interest conditioned upon the observed signal provides the optimal or Bayesian estimate. If the signal of interest or the noise is non-Gaussian then the optimal estimate is in general a nonlinear function of the observed signal.
The third talk by James Heald of University College London discussed a methodology for reducing the impact of observation noise when observing nonlinear dynamical systems. He also illustrated the difficulty with such algorithms. The key point made was that multiple solutions can be present in any nonlinear inference problem. The wider point is that it may not be possible to uncover them simply by re-optimising from new sets of starting conditions. More systematic strategies are clearly needed to give confidence that solutions are unique.
Ronan Farrell then presented a talk on joint work with Orla Feely of University College Dublin. In his talk he discussed the use of techniques from nonlinear dynamics in understanding and designing sigma-delta modulators. They demonstrated how a range of nonlinear techniques provide more accurate descriptions of the behaviour of a variety of sigma-delta analogue to digital converters.
The first talk after lunch reported work at the Department of Electrical Engineering at the University of Edinburgh and was presented by Iain Mann, discussing the use of Poincare maps for detection of pitch in voiced speech. The talk demonstrated how Poincare maps could be used as a basis for pitch contour detection. However, it also illustrated the limitations of the technique, in particular its lack of robustness in a variety of applications.
The next paper was presented by Dr Jerry Huke of UMIST. He discussed state space reconstruction using inter spike intervals. This represents the beginning of a more detailed study of state dependent sampling and posed a number of open questions. In particular how these techniques may be applied to experimental signals was seen as key to any future work.
The final paper by J.Y. Lee and Asoke Nandi of the University of Strathclyde applied a variety of techniques from nonlinear dynamics to time series from mechanical oscillating systems where impacting is likely to occur. They showed how a blind deconvolution technique was used to clean up the signal prior to analysis for evidence of chaotic behaviour.
The papers presented at this workshop are available in a short digest which can be obtained from the IEE. Contact firstname.lastname@example.org.