Polynomial Filtering for Linear Discrete Time Non-Gaussian Systems

In this work we propose a new filtering approach for linear discrete time non-Gaussian systems that generalizes a previous result concerning quadratic filtering [A. De Santis, A. Germani, and M. Raimondi, IEEE Trans. Automat. Control, 40 (1995) pp. 1274--1278]. A recursive $\nu$th-order polynomial estimate of finite memory $\Delta$ is achieved by defining a suitable extended state which allows one to solve the filtering problem via the classical Kalman linear scheme. The resulting estimate will be the mean square optimal one among those estimators that take into account $\nu$-polynomials of the last $\Delta$ observations. Numerical simulations show the effectiveness of the proposed method.

[1]  Stuart C. Schwartz,et al.  Robust detection of a known signal in nearly Gaussian noise , 1971, IEEE Trans. Inf. Theory.

[2]  T. Subba Rao,et al.  Linear and non-linear filters for linear, but not gaussian processes† , 1984 .

[3]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[4]  A. Germani,et al.  Optimal quadratic filtering of linear discrete-time non-Gaussian systems , 1995, IEEE Trans. Autom. Control..

[5]  Willem L. De Koning,et al.  Optimal estimation of linear discrete-time systems with stochastic parameters , 1984, at - Automatisierungstechnik.

[6]  Patrick Duvaut,et al.  Optimal linear-quadratic systems for detection and estimation , 1988, IEEE Trans. Inf. Theory.

[7]  A. V. Balakrishnan,et al.  Kalman Filtering Theory , 1984 .

[8]  Patrick Duvaut,et al.  Geometrical properties of optimal Volterra filters for signal detection , 1990, IEEE Trans. Inf. Theory.

[9]  M. Srinath,et al.  Optimum Linear Estimation of Stochastic Signals in the Presence of Multiplicative Noise , 1971, IEEE Transactions on Aerospace and Electronic Systems.

[10]  Rudolf Kulhavý,et al.  Differential Geometry of Recursive Nonlinear Estimation , 1990 .

[11]  B. Picinbono,et al.  Détection d’un signal certain dans un bruit non stationnaire et non gaussien , 1970 .

[12]  Steven I. Marcus,et al.  Optimal nonlinear estimation for a class of discrete-time stochastic systems , 1979 .

[13]  E. Yaz.  Linear state estimators for non-linear stochastic systems with noisy non-linear observations , 1988 .

[14]  Bruce A. Eisenstein,et al.  Detection of weak signals in non-Gaussian noise , 1981, IEEE Trans. Inf. Theory.

[15]  George V. Moustakides,et al.  Robust detection of known signals in asymmetric noise , 1982, IEEE Trans. Inf. Theory.

[16]  L. Kurz,et al.  An Optimal Nonlinear Detector for Digital Data Transmission Through Non-Gaussian Channels , 1966 .

[17]  Benjamin Friedlander,et al.  Asymptotically optimal estimation of MA and ARMA parameters of non-Gaussian processes from high-order moments , 1990 .

[18]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[19]  Engin Yaz,et al.  On the optimal state estimation of a class of discrete-time nonlinear systems , 1987 .

[20]  G. Giannakis.  On the identifiability of non-Gaussian ARMA models using cumulants , 1990 .

[21]  Engin Yaz,et al.  A control scheme for a class of discrete nonlinear stochastic systems , 1987 .

[22]  John B. Thomas,et al.  Detectors for discrete-time signals in non-Gaussian noise , 1972, IEEE Trans. Inf. Theory.