Effect of embedded unbiasedness on discrete-time optimal FIR filtering estimates

Unbiased estimation is an efficient alternative to optimal estimation when the noise statistics are not fully known and/or the model undergoes temporary uncertainties. In this paper, we investigate the effect of embedded unbiasedness (EU) on optimal finite impulse response (OFIR) filtering estimates of linear discrete time-invariant state-space models. A new OFIR-EU filter is derived by minimizing the mean square error (MSE) subject to the unbiasedness constraint. We show that the OFIR-UE filter is equivalent to the minimum variance unbiased FIR (UFIR) filter. Unlike the OFIR filter, the OFIR-EU filter does not require the initial conditions. In terms of accuracy, the OFIR-EU filter occupies an intermediate place between the UFIR and OFIR filters. Contrary to the UFIR filter which MSE is minimized by the optimal horizon of N opt points, the MSEs in the OFIR-EU and OFIR filters diminish with N and these filters are thus full-horizon. Based upon several examples, we show that the OFIR-UE filter has higher immunity against errors in the noise statistics and better robustness against temporary model uncertainties than the OFIR and Kalman filters

Fast bias-constrained optimal FIR filtering for time-invariant state space models

This paper combines the finite impulse response filtering with the Kalman structure (predictor/corrector) and proposes a fast iterative bias-constrained optimal finite impulse response filtering algorithm for linear discrete time-invariant models. In order to provide filtering without any requirement of the initial state, the property of unbiasedness is employed. We first derive the optimal finite impulse response filter constrained by unbiasedness in the batch form and then find its fast iterative form for finite-horizon and full-horizon computations. The corresponding mean square error is also given in the batch and iterative forms. Extensive simulations are provided to investigate the trade-off with the Kalman filter. We show that the proposed algorithm has much higher immunity against errors in the noise covariances and better robustness against temporary model uncertainties. The full-horizon filter operates almost as fast as the Kalman filter, and its estimate converges with time to the Kalman estimate

Improving state estimates over finite data using optimal FIR filtering with embedded unbiasedness

In this paper, the optimal finite impulse response (OFIR) with embedded unbiasedness (EU) filter is derived by minimizing the mean square error (MSE) subject to the unbiasedness constraint for discrete time-invariant state-space models. Un like the OFIR filter, the OFIR-EU filter does not require the initial conditions. In terms of accuracy, the OFIR-EU filter occupies an intermediate place between the UFIR and OFIR filters. With a two-state harmonic model, we show that the OFIR-UE filter has higher immunity against errors in the noise statistics and better robustness against temporary model uncertainties than the OFIR and Kalman filters

Improved Distributed Estimation Method for Environmental\ud time-variant Physical variables in Static Sensor Networks

In this paper, an improved distributed estimation scheme for static sensor networks is developed. The scheme is developed for environmental time-variant physical variables. The main contribution of this work is that the algorithm in [1]-[3] has been extended, and a filter has been designed with weights, such that the variance of the estimation errors is minimized, thereby improving the filter design considerably\ud and characterizing the performance limit of the filter, and thereby tracking a time-varying signal. Moreover, certain parameter optimization is alleviated with the application of a particular finite impulse response (FIR) filter. Simulation results are showing the effectiveness of the developed estimation algorithm

Iterative Unbiased FIR State Estimation: A Review of Algorithms

In this paper, we develop in part and review various iterative unbiased finite impulse response (UFIR) algorithms (both direct and two-stage) for the filtering, smoothing, and prediction of time-varying and time-invariant discrete state-space models in white Gaussian noise environments. The distinctive property of UFIR algorithms is that noise statistics are completely ignored. Instead, an optimal window size is required for optimal performance. We show that the optimal window size can be determined via measurements with no reference. UFIR algorithms are computationally more demanding than Kalman filters, but this extra computational effort can be alleviated with parallel computing, and the extra memory that is required is not a problem for modern computers. Under real-world operating conditions with uncertainties, non-Gaussian noise, and unknown noise statistics, the UFIR estimator generally demonstrates better robustness than the Kalman filter, even with suboptimal window size. In applications requiring large window size, the UFIR estimator is also superior to the best previously known optimal FIR estimators

Unbiased, optimal, and in-betweens: the trade-off in discrete finite impulse response filtering

In this survey, the authors examine the trade-off between the unbiased, optimal, and in-between solutions in finite impulse response (FIR) filtering. Specifically, they refer to linear discrete real-time invariant state-space models with zero mean noise sources having arbitrary covariances (not obligatorily delta shaped) and distributions (not obligatorily Gaussian). They systematically analyse the following batch filtering algorithms: unbiased FIR (UFIR) subject to the unbiasedness condition, optimal FIR (OFIR) which minimises the mean square error (MSE), OFIR with embedded unbiasedness (EU) which minimises the MSE subject to the unbiasedness constraint, and optimal UFIR (OUFIR) which minimises the MSE in the UFIR estimate. Based on extensive investigations of the polynomial and harmonic models, the authors show that the OFIR-EU and OUFIR filters have higher immunity against errors in the noise statistics and better robustness against temporary model uncertainties than the OFIR and Kalman filters