New design of recursive Wiener fixed-lag smoother in linear discrete-time stochastic systems
Introduction
The estimation problem given the covariance information has been seen as an important research in the area of the detection and estimation problems for communication systems. In [1], [2], [3], it is assumed that the autocovariance function of the signal is expressed in the semi-degenerate kernel form. The semi-degenerate kernel is the function suitable for expressing a general kind of autocovariance function by a finite sum of nonrandom functions. In the fixed-lag smoother [4], the autocovariance function of the signal is expressed in the degenerate kernel form, not in the semi-degenerate kernel form. The degenerate kernel function cannot express the autocovariance functions of general kinds of stochastic processes. Hence, the fixed-lag smoother using the autocovariance function in the form of the degenerate kernel is not appropriate for estimating the general kinds of stationary or nonstationary signal processes. The expression in the degenerate kernel form of the autocovariance function is obtained through approximating the autocovariance function by the Fourier series transformation. Hence, its approximation error causes the degradation in the estimation accuracy of the fixed-lag smoother. The recursive Wiener fixed-point smoother [5] and filter [6] using the covariance information are designed in linear discrete-time stochastic systems. The estimators require the information of the observation matrix, the system matrix for the state variable, related with the signal, and the cross-variance function of the state variable with the observed value. The information can be obtained from the covariance function of the signal [5]. Also, it is assumed that the variance of the white observation noise is known.
From this respect, this paper newly designs the fixed-lag smoothing and filtering algorithms using the covariance information in linear discrete-time stochastic systems. It is assumed that the signal is observed with additional white observation noise. The estimators require the information of the factorized autocovariance function of the signal. Namely, the estimation algorithms use the observation matrix, the system matrix for the state variable and the cross-variance function between the state variable and the signal. Also, it is assumed that the variance of the white observation noise is known. This factorization of the autocovariance function of the signal leads to the estimation of the stochastic signal generally. The algorithms are derived based on the invariant imbedding method [5]. The fixed-lag smoothing error variance function is also formulated.
A numerical simulation example is shown to investigate the estimation accuracy of the proposed fixed-lag smoother in comparison with the filter.
Section snippets
Fixed-lag smoothing problem
Let an observation equation be given byin discrete-time stochastic systems, where z(k) is an m×1 signal vector, x(k) is an n×1 state variable, H is an m×n observation matrix and v(k) is white observation noise. It is assumed that the signal and the observation noise are mutually independent and that z(k) and v(k) are zero mean. Let the autocovariance function of v(k) be given byHere, δK(·) denotes the Kronecker δ function.
Let a fixed-lag
Algorithm for the fixed-lag smoothing estimate
Let us derive the algorithms for the fixed-lag smoothing and filtering estimates. The algorithms are derived based on the invariant imbedding method [5].
Introducing an auxiliary function, which satisfieswe find from , that
Subtracting the equation obtained by putting k→k−1 in (18) from (18), we haveFrom , with (9), the equation updating J(k,s) is given by
Fixed-lag smoothing error variance function
Let P(k,k+L) represent the fixed-lag smoothing error variance function, which is defined byFrom the orthogonal projection lemma , (52) is written as, in terms of , , ,
A numerical simulation example
Let a scalar observation equation be given byLet the observation noise v(k) be zero-mean white Gaussian noise with the variance R, N(0,R). Let the autocovariance function of the signal z(k) be given byBased on the method in [5], the observation vector H, the cross-variance Kxy(k,k) and the system matrix Φ in the state equation for the state variable x(k) are
Conclusions
In this paper, the Wiener fixed-lag smoother and filter using the covariance information have been designed in linear discrete-time stochastic systems. It is assumed that the covariance information of the signal and the observation noise is known. From the simulation results, for the observation noises N(0,0.72) and N(0,1), the estimation accuracy of the filter is fairly improved by the proposed fixed-lag smoother with the value of L, which minimizes the MSVs of smoothing errors in Fig. 2, L=5
References (7)
Design of predictor using covariance information in continuous-time stochastic systems with nonlinear observation mechanism
Signal Processing
(1998)Design of a fixed-point smoother based on an innovations theory for white Gaussian plus coloured observation noise
International Journal of Systems Science
(1991)Design of estimators using covariance information in discrete-time stochastic systems with nonlinear observation mechanism
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
(1999)
Cited by (3)
Design of RLS fixed-lag smoother using covariance information in linear discrete stochastic systems
2010, Applied Mathematical ModellingRLS fixed-lag smoother using covariance information in linear continuous stochastic systems
2009, Applied Mathematical ModellingDesign of recursive least-squares fixed-lag smoother using covariance information in linear continuous stochastic systems
2007, Applied Mathematical Modelling