**Likelihood Detector Apparatus and Method**

Sam Gratrix, Robert Jackson and Oleg Zaboronski

*United States Patent Application Publication*,
3 January 2008, US 2008/0002791 A1.

*
A double Viterbi detector that is able to well approximate an optimal detector in terms of bit error
performance, but is readily implementable in hardware with significant silicon area saving and high
throughput.
*

A method and apparatus for receiving a stream of data values from a data medium, wherein the received data values correspond to ideal values but may include added noise that is dependent on previous noise and dependent on data on the data medium, said ideal values being determined by possible values of data on the medium, and for outputting information specifying a sequence of states corresponding to the stream of received data values, said sequence of states corresponding to possible data values on the medium, the apparatus comprising: a first detector, for calculating state sequence likelihood information based on a first noise model and received data values, and for providing said state sequence likelihood information to a second detector; a second detector for calculating weighting values indicating likelihoods that a data value received at a particular time corresponds to a particular state transition, using a second noise model, received data values, and state sequence likelihood information from the first detector; a traceback unit for determining a most likely sequence of data values using said calculated weighting values; and an output for outputting information specifying said determined sequence of states.

**Viterbi Detector for Non-Markov Recording Channels**

Sam Gratrix, Robert Jackson, Tom Parnell and Oleg Zaboronski

*IEEE Transactions on Magnetics*, 1 January 2008, Volume 44, Number 1.

Channel noise in modern perpendicular channels is well approximated by a linear jitter model. In this paper,
we propose a new type of Viterbi detector matched to such a channel model. The differentiating feature of this
detector is recursive evaluation of whitened noise strengths for survivor paths using the on-the-fly Cholesky
factorization of path-dependent correlation matrices. We study the performance of the detector using the
simplest example of low-density channels with intersymbol interference length equal to three. We find that at
bit error rate 10^{-4}, the new detector outperforms the data-dependent autoregressive detector of the
comparable complexity by about 0.9dB.

**Kuramoto-Sivashinsky Equation**

Sam Gratrix

*Encyclopedia of Nonlinear Science*, Routledge, Alwyn Scott Editor, December 2004.

*A short essay contribution on the Kuramoto-Sivashinsky equation. The Encyclopedia of Nonlinear Science
provides complete coverage at the introductory level of the field of nonlinear science, from intuitive
descriptions to mathematical concepts over a wide range of subject matters.*

**Pointwise Dimensions of the Lorenz Attractor**

Sam Gratrix and John N. Elgin

*Physical Review Letters*, 9 January 2004, Volume 92, Number 1, 014101.

We discuss a connection between two complementary views of the Lorenz attractor: the first is the accepted view where the attractor has a smooth measure on a fractal support. This complex system is alternatively manifest as a self-similar curve for the pointwise dimension alpha. We describe why the latter approach is accessible for the analysis of an experimental signal.

*This publication has been reviewed by Thomas C. Halsey and Mogens H. Jensen in the News and Views section
of Nature, titled Hurricanes and Butterflies. Nature, 11 March 2004, Volume 428, Pages 127-128.*

**Spatiotemporal Chaos Analysed Through Unstable Periodic States**

Sam Gratrix

PhD thesis, Imperial College London, 2003.

Multifractal properties of a chaotic attractor are usefully quantified by its spectrum of singularities, f(alpha). Here, alpha is the pointwise dimension of the natural measure at a point on the attractor, and f(alpha) is the Hausdorff dimension of all points with pointwise dimension alpha. Within a more general thermodynamic formalism, the singularity spectrum is one of several ways in which the properties of an attractor can be quantified. The technique used to realize the singularity spectrum is orbit theory. This theory tells one how to take properties of finite time solutions and combine them to approximate the infinite time behaviour, thereby allowing qualitative and quantitative predictions to be made. These techniques are first applied to the Lorenz system, where it is also shown that the variation in the pointwise dimension on a surface of section has self-similar structure.

The general idea of studying the properties of a nonlinear system through the periodic orbits it supports has, to date, been primarily applied to low-dimensional dynamical systems. In the thesis we develop the technique so that it can be applied to the infinite-dimensional Kuramoto-Sivashinsky equation. The continuation and bifurcation package Auto is used to investigate stability and bifurcation properties of different types of special solutions to the Kuramoto-Sivashinsky equation, following an expansion in Fourier modes. One such class of solutions is defined by the Michelson equation, to which a very detailed numerical bifurcation analysis is given.

Orbit theory is applied to regimes of an asymmetric Kuramoto-Sivashinsky equation where complicated behaviour is observed in a manner similar to that used in low-dimensional systems. Each periodic orbit can be considered as a spatiotemporal pattern, in which both qualitative (the structure and bifurcations of) and quantitative (the dimension and spectrum of Lyapunov exponents) aspects are discussed.

**Exact Real Arithmetic in Numerical Computation**

Sam Gratrix

MSc dissertation, Imperial College London, 1999.

Two different systems of real computation, the finite precision arithmetic as implemented by today's computers and the exact real arithmetic system developed at Imperial College, London, are considered. Under certain iterative computations the rounding problems associated with finite precision are shown to be catastrophic, whereas exact real arithmetic gives the correct results. An account of the exact real system is given, accompanied by an original implementation in Mathematica. The implementation clearly displays the mechanisms by which the exact real system works.

The numerical solution of ordinary differential equations by the Runge-Kutta method forms a basis to allow a direct comparison between the systems in an environment where rounding errors arise. The C-LFT Library implementation of exact real arithmetic, developed recently at Imperial College, is compared to both software and hardware finite precision arithmetic. It is shown that finite precision arithmetic provides for a fast, but possibly incorrect solution. Exact real arithmetic removes the rounding error from the solution, although the computer resources required limits its use.