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Properties of the Multivariate Cauchy Estimator


In this dissertation, the fundamental structure of a multivariate discrete-time state estimator with Cauchy distributed process noise and measurement noise is discussed in depth. The characteristic function (CF) of the unnormalized conditional probability density function (ucpdf) is found to be a sum of elements that increases at each update of the current measurement. Each term in this sum is composed of a coefficient term which contains the measurement history operating on an exponential term composed of a sum of absolute values whose argument is the inner product of a direction vector with the spectral variable. The objective is to understand the structure of the CF so as to simplify this sum. We shows that directions in the terms of the CF-s are co-aligned only along certain directions which are functions of a unique fundamental basis. Based on the knowledge of combining co-aligned directions, an indexing scheme, called "S" matrix, is developed to indicate which exponential terms can be combined without the necessity of numerical comparison. The S matrix is invariant for systems of the same dimension regardless of the system parameters. The coefficient terms are also restructured and simplified by eliminating all the redundant zero elements. For two-state systems, we show that there are no more than three non-zero elements in each layer of any new coefficient term. Furthermore, with these newly uncovered properties the Cauchy estimator is implemented efficiently using a pre-computational technique. The simulations of three-state and four-state systems illustrate the performance of Cauchy estimator compared with the Kalman Filter.

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