Modelling of Random Processes using Orthonormal Bases

In this paper autoregressive (AR) modelling of stationary processes is generalised so that it becomes a special case of modelling using orthonormal bases. Given this interpretation, a general construction of orthonormal basis functions is presented. This construction is such that the AR expansion, and the more recently popular Laguerre expansion emerge as special cases. However, in contrast to these special cases, the bases employed in this paper possess poles that may be arbitrarily set according to prior knowledge of the zeros of the stable spectral factor of the spectral density of interest. By allowing this freedom, the paper shows that it is possible to obtain more accurate approximation of a given spectral density while not increasing the order of the AR-type expansion. The paper also provides a theoretical analysis of the statistical properties of spectral density estimates that employ these generalised bases.

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