An adaptive antenna array processor with derivative constraints.
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In antenna array processing it is generally required to enhance the reception or detection of a signal from a particular direction while suppressing noise and interference signals from other directions. An optimisation problem often posed to achieve this result is to minimise the array processor mean output power (or variance) subject to a fixed response in the array look direction. The look direction requirement can be met by imposing a set of linear constraints on the processor weights to yield what is known as the Linearly Constrained Minimum Variance (LCMV) processor. It has been found, however, that LCMV processors are susceptible to errors in the assumed direction of arrival of the desired signal. To achieve robustness against directional mismatch, additional constraints known as derivative constraints can be introduced. These constraints force the first and second order spatial derivatives of the array power response in the look direction to zero. However, constraints corresponding to necessary and sufficient (NS) conditions for these spatial derivatives to be zero are in general quadratic, and the resulting weight vector solution space is non-convex. One approach to this complex problem has been to consider conditions which are only sufficient for the spatial derivatives to be zero. Whilst this results in linear constraints, it exhibits certain anomalous behaviour, for example, dependence on the choice of array phase centre.Recent work in the area of derivative constraints has resulted in a method for efficiently solving the non-convex output power minimisation problem with quadratic derivative constraints. The optimisation problem addressed assumes that the input signal statistics and hence the input signal autocorrelation matrix R are known. In practice, R must be estimated from the receiver data.The main contribution of this thesis is the derivation of a new adaptive algorithm which implements an adaptive array processor with look direction plus 1st and 2nd order NS derivative constraints. The new algorithm is derived from the well-known Recursive Least Squares (RLS) technique but allows linear and quadratic constraints to be incorporated within the recursive framework. The algorithm offers the high performance characteristics associated with RLS methods, namely, fast convergence and high steady-state accuracy. The work encompasses a study of the characteristics of the algorithm in terms of numerical robustness, convergence properties, tracking and computational complexity.The study of the numerical properties of the algorithm has led to the second important contribution of this thesis: the identification of a parameter which is central to the numerical stability of the algorithm in a practical fixed precision environment. We show that this parameter is bounded during stable operation and can therefore be used to detect the onset of numerical instability within the algorithm. In addition, we show how existing techniques can be used to significantly improve the numerical robustness of the algorithm.Another important contribution of the thesis stems from an investigation into the multimodal nature of the quadratic, equality constrained optimisation problem resulting from the use of second order NS derivative constraints. In particular, we show that for a linear antenna array operating under certain conditions, the complex multimodal optimisation problem can be greatly simplified. This has important implications in both optimum and adaptive array signal processing.
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