New design of recursive Wiener fixed-lag smoother in linear discrete-time stochastic systems

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Abstract

This paper newly designs the Wiener fixed-lag smoother and filter using covariance information 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, the cross-variance function of the state variable with the observed value and the variance of the white observation noise. It is assumed that the signal is observed with additive white noise.

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 byy(k)=z(k)+v(k),z(k)=Hx(k),in 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 byE[v(k)vT(s)]=R(k)δK(k−s),R(k)>0.Here, 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 satisfiesJ(k,s)R(s)=Φ−sKxy(s,s)−∑i=1kJ(k,i)HKxy(i,s),we find from , thath(k,s)=ΦkJ(k,s),0⩽s⩽k.

Subtracting the equation obtained by putting kk−1 in (18) from (18), we have(J(k,s)−J(k−1,s))R(s)=−J(k,k)HKxy(k,s)−∑i=1k−1(J(k,i)−J(k−1,i))HKxy(i,s).From , 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 byP(k,k+L)=E[(z(k)−ẑ(k,k+L))(z(k)−ẑ(k,k+L))T].From the orthogonal projection lemma z(k)−ẑ(k,k+L)⊥ẑ(k,k+L), (52) is written as, in terms of , , ,P(k,k+L)=HKxy(k,k)−Hi=1kg(k,i)E[υ(i)ẑT(k,k+L)]−H∑i=k+1k+Lg(k,i)E[υ(i)ẑT(k,k+L)]=HKxy(k,k)−HE[x̂(k,k)ẑT(k,k+L)]−∑i=k+1k+L(HKxy(k,i)−HS(k)(ΦT)i−kHT)R−1(i)E[υ(i)ẑT(k,k+L)]=HKxy(k,k)−HE[x̂(k,k)(ẑ(k,k)+f(k+1,k+L))T]−∑i=k+1k+L(HKxy(k,i)−HS(k)(ΦT)i−kHT)R−1

A numerical simulation example

Let a scalar observation equation be given byy(k)=z(k)+v(k).Let 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 byK(0)=σ2,K(m)=σ2122−1)α1m/[(α2−α1)(α2α1+1)]−α212−1)α2m/[(α2−α1)(α1α2+1)]},0<m,α12=−a1±a12−4a22,a1=−0.1,a2=−0.8,σ=0.5.Based 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

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