TY - GEN

T1 - The error variance of the optimal linear smoother and maximum-variance fractional pole models

AU - Georgiou, Tryphon T.

PY - 2006

Y1 - 2006

N2 - The variance of the optimal one-step ahead linear prediction error of a discrete-time stationary stochastic process is given by the well-known Szegö-Kolmogorov formula as the geometric mean of the spectral density function. We first derive an analogous expression for the optimal linear smoother which uses the infinite past and the infinite future to determine the present. The least variance turns out to be the harmonic mean of the spectral density function. Building on this, we explore the question of what is the most random power spectrum in the sense of corresponding to the largest variance optimal linear smoother (i.e., least "smoothable"), which is consistent with finitely many covariance moments. It turns out that it can be described by an all-pole model, albeit the poles are fractional.

AB - The variance of the optimal one-step ahead linear prediction error of a discrete-time stationary stochastic process is given by the well-known Szegö-Kolmogorov formula as the geometric mean of the spectral density function. We first derive an analogous expression for the optimal linear smoother which uses the infinite past and the infinite future to determine the present. The least variance turns out to be the harmonic mean of the spectral density function. Building on this, we explore the question of what is the most random power spectrum in the sense of corresponding to the largest variance optimal linear smoother (i.e., least "smoothable"), which is consistent with finitely many covariance moments. It turns out that it can be described by an all-pole model, albeit the poles are fractional.

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U2 - 10.1109/cdc.2006.377324

DO - 10.1109/cdc.2006.377324

M3 - Conference contribution

AN - SCOPUS:39649091153

SN - 1424401712

SN - 9781424401710

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 1685

EP - 1691

BT - Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 45th IEEE Conference on Decision and Control 2006, CDC

Y2 - 13 December 2006 through 15 December 2006

ER -