Discrete and Continuous Estimation in Correlated Noise with Finite Observation Time

Citation

Skidmore, Lionel Joseph (1964) Discrete and Continuous Estimation in Correlated Noise with Finite Observation Time. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1V5T-2J86. https://resolver.caltech.edu/CaltechETD:etd-10102002-090435

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis analytic formulas are derived for the elements of the inverse covariance matrix of sampled rational noise. It is shown that the number of terms composing these formulas is dependent only on the order of the noise and not on the dimension of the covariance matrix. Some special cases are worked out in detail. The estimation of the parameter [theta] in the process y(t) = [theta]S(t) + n(t), where t is in the interval [0,L], n(t) is rational noise, and S(t) is deterministic, is considered in detail for first and second order noise. A minimum variance continuous filter, f(t), which gives an estimate of [....] through [....] and its associated variance are computed. Also computed is a discrete minimum variance estimate of the form, [....] where the [....] are the "weights" for the sampled data and T is the sampling period. It is shown that the discrete weighting function and its variance approaches the continuous weighting function and its variance when the density of observations approaches infinity. It is seen that in general the discrete weighting function does not create the equivalent of a delta function and its derivatives by a simple differencing operation through the use of Kronecker deltas. Asymptotic properties of the variance of the discrete estimate are considered. The asymptotic term is defined as the first order term in the power series expansion of the variance. It is seen that for a smooth S(t) and first order noise, the asymptotic term is zero. In the special case of S(t) equal to a constant and second order noise it is shown that the asymptotic term is zero if the noise has zeros in its spectral density and nonzero if the noise is all pole. The connection between autoregressive noise and rational noise is considered in detail for second order noise. It is seen that rational noise will have autoregressive properties only for a special pole-zero configuration and a particular sampling rate. The advantages of sampling at this rate are discussed and a special case is considered. It is shown that the results obtained for one parameter, one signal, and one noise can be easily extended to a vector of parameters, a matrix of signals, and a vector of noises. The only restriction is that the components of the noise vector be uncorrelated.

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