Nonlinear processing of non-Gaussian stochastic and chaotic deterministic time series

It is often assumed that interference or noise signals are Gaussian stochastic processes. Gaussian noise models are appealing as they usually result in noise suppression algorithms that are simple: i.e. linear and closed form. However, such linear techniques may be sub-optimal when the noise process is either a non-Gaussian stochastic process or a chaotic deterministic process. In the event of encountering such noise processes, improvements in noise suppression, relative to the performance of linear methods, may be achievable using nonlinear signal processing techniques. The application of interest for this thesis is maritime surveillance radar, where the main source of interference, termed sea clutter, is widely accepted to be a non-Gaussian stochastic process at high resolutions and/or at low grazing angles. However, evidence has been presented during the last decade which suggests that sea clutter may be better modelled as a chaotic deterministic process. While the debate over which model is more suitable continues, this thesis investigates whether nonlinear processing techniques can be used to improve the performance of maritime surveillance radar, relative to the performance achievable using linear techniques. Linear and nonlinear prediction of chaotic signals, sea clutter data sets, and stochastic surrogate clutter data sets is carried out. Volterra series filter networks and radial basis function networks are used to implement nonlinear predictors. A novel structure for a forward-backward nonlinear predictor, using a radial basis function network, is presented. Prediction results provide evidence to support the view that sea clutter is better modelled as a stochastic process, rather than as a chaotic process. The clutter data sets are shown to have linear predictor functions. Linear and nonlinear predictors are used as the basis of target detection algorithms. The performance of these predictor-detectors, against backgrounds of sea clutter data and against a background of chaotic noise data is evaluated. The detection results show that linear predictor-detectors perform as well as, or better than, nonlinear predictor-detectors against the non-Gaussian clutter backgrounds considered in this thesis, whilst the reverse is true for a background of chaotic noise. An existing, nonlinear inverse, noise cancellation technique, referred to as Broomhead’s filtering technique in this thesis, is re-investigated using a sine wave corrupted by broadband chaotic noise. It is demonstrated that significant improvements can be obtained using this nonlinear inverse technique, relative to results obtained using linear alternatives, despite recent work which suggested otherwise. A novel bandstop filtering approach is applied to Broomhead’s filtering method, which allows the technique to be applied to the cancellation of signals with a band of interest greater than that of a sine wave. This modified Broomhead filtering technique is shown to cancel broadband chaotic noise from a narrowband Gaussian signal better than alternative linear methods. The modified Broomhead filtering technique is shown to only perform as well as, o

On the estimation of the correlation dimension and its application to radar reflector discrimination

Recently, system theorists have recognized that low order systems of nonlinear differential equations can give rise to solutions which are neither periodic, constant, nor predictable in steady state, but which are nonetheless bounded and deterministic. This behavior, which was first described in the study of weather systems, has been termed 'chaotic.' Much study of chaotic systems has concentrated on analysis of the systems' phase space attractors. It has been recognized that invariant measures of the attractor possess inherent information about the system. One such measure is the dimension of the attractors. The dimension of a chaotic attractor has been shown to be noninteger, leading to the term 'strange attractor;' the attractor is said to have a fractal structure. The correlation dimension has become one of the most popular measures of dimension. However, many problems have been identified in correlation dimension estimation from time sequences. The most common methods for obtaining the correlation dimension have been least squares curves fitting to find the slope of the correlation integral and the Takens Estimator. However, these estimates show unacceptable sensitivity to the upper limit on the distance chosen. Here, a new method is proposed which is shown to be rather insensitive to the upper limit and to perform in a very stable manner, at least in the absence of noise. The correlation dimension is also shown to be an effective discriminant in distinguishing between radar returns resulting from weather and those from the ground. The weather returns are shown to have a correlation dimension generally between 2.0 and 3.0, while ground returns have a correlation dimension exceeding 3.0