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Gaussian noise: prediction based on its value and N derivatives

Gaussian noise: prediction based on its value and N derivatives

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An explicit form for the Slepian model of Gaussian noise X(t + τ) conditioned on the value X(t) and any desired number N of its derivatives X'(t), X"(t), …, X(N)(t) at a given time t is obtained by using determinants for the Gram- Schmidt orthogonalisation of linear combinations of random variables, and then applying them to the least-mean-squared-error estimation of any zero-mean stationary random process. In this way a number of widely useful results are unified, clarified, simplified and extended. Finally, the application to a random process with a spectral density of Gaussian shape is studied.

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