Unbiased Finite Impulse Response Research Articles

Review of Unbiased FIR Filters, Smoothers, and Predictors for Polynomial Signals

Extracting an estimate of a slowly varying signal corrupted by noise is a common task. Examples can be found in industrial, scientific and biomedical instrumentation. Depending on the nature of the application the signal estimate is allowed to be a delayed estimate of the original signal or, in the other extreme, no delay is tolerated. These cases are commonly referred to as filtering, prediction, and smoothing depending on the amount of advance or lag between the input data set and the output data set. In this review paper we provide a comprehensive set of design and analysis tools for designing unbiased FIR filters, predictors, and smoothers for slowly varying signals, i.e. signals that can be modeled by low order polynomials. Explicit expressions of parameters needed in practical implementations are given. Real life examples are provided including cases where the method is extended to signals that are piecewise slowly varying. A critical view on recursive implementations of the algorithms is provided.

Linear Optimal Fusion of Local Unbiased FIR Filters

This paper presents a multi-sensor decentralized fusion unbiased finite impulse response (UFIR) filter for discrete time-invariant state-space models. Fusion is provided in the minimum variance sense. By calculating the cross covariance between any of two local filters for the extended state-space model, linear optimal weights are derived to fuse local UFIR estimates. Simulation conduced for a two-state polynomial model shows that the proposed fusion UFIR filter has higher robustness than the fusion Kalman filter against errors in the noise statistics and temporary model uncertainties.

GPS-based Tracking over WSNs with Delayed and Missing Data using UFIR Filtering

Vehicles tracking is organized to increase safety in smart cities by localizing cars using the Global Positioning System (GPS). The GPS-based system provides accurate tracking, but is also required to be reliable and robust. As a main estimator, we propose using the unbiased finite impulse response (UFIR) filter, which meets these needs as a more robust alternative to the Kalman filter (KF). The UFIR filter is developed for vehicle tracking in discrete-time state-space over wireless sensor networks (WSNs) with time-stamped discretely delayed on k-step-lags and missing data. The state-space model is represented in a way such that the UFIR filter, KF, and H∞ filter can be used universally. Applications are given for measurement data, which are cooperatively transferred from a vehicle to a central station through several nodes with k-step-lags. Better tracking performance of the UFIR filter is shown experimentally.

Single-Agent Finite Impulse Response Optimizer for Numerical Optimization Problems

This paper introduces a new single-agent metaheuristic optimization algorithm, named single-agent finite impulse response optimizer (SAFIRO). This proposed algorithm is inspired by the estimation ability of the ultimate iterative unbiased finite impulse response (UFIR) filter. The UFIR filter is one of the variants of the finite impulse response (FIR) filter, whereby in state space models, the FIR filter can be used as an option other than the Kalman filter (KF) for state estimation. Unlike the KF, the UFIR filter does not require any noise covariance, error covariance, and initial condition to calculate the state estimate. The UFIR filter also provides an iterative Kalman-like form to improve the estimation process. In the SAFIRO algorithm, the agent works as an individual UFIR to find an optimal or a near-optimal solution, where the agent needs to perform two main tasks; measurement and estimation. The performance of the SAFIRO algorithm is evaluated using the CEC 2014 Benchmark Test Suite for single-objective optimization and statistically compared with the several well-known metaheuristic optimization algorithms, such as Particle Swarm Optimization algorithm, Genetic Algorithm, and Grey Wolf Optimization algorithm. The experimental results show that the proposed SAFIRO algorithm is able to converge to the optimal and the near-optimal solutions, and significantly outperform all the aforementioned state-of-the-art metaheuristic algorithms.

Inlet NOx and NH3 Concentration Estimation for Diesel-engine SCR Systems by Combining Data-Driven Model and Unbiased FIR Filter

Selective catalytic reduction (SCR) systems have been widely used in diesel engine applications. In an SCR system, input NOx and NH3 concentration information are of critical importance for the urea dosage controller design and system fault detection. Generally, the NOx and NH3 concentration are obtained by physical sensors. However, the physical sensors do not only increase the cost of overall system, but also induce measurement delays. To deal with this issue, an input observer combining a data-driven model and an unbiased finite impulse response (FIR) filter is proposed. The structure of data-driven model is auto-regressive exogenous (ARX) model and partial least square (PLS) is utilized to identify the parameters in the ARX model. Nevertheless, fuzzy c-means (FCM) is also employed to partition the data and obtain multiple local linear models for describing the nonlinearities of the system. At last, an unbiased FIR filter is adopted to estimate the input NOx and NH3 concentration simultaneously due to its strong robustness against the noise. The comparisons between the unbiased FIR filter algorithm and Kalman filter algorithm are carried out in MATLAB/SIMULINK. The simulation results demonstrate that the performance of proposed estimator is outstanding.

Bayesian state estimation on finite horizons: The case of linear state–space model

The finite impulse response (FIR) filter and infinite impulse response filter including the Kalman filter (KF) are generally considered as two different types of state estimation methods. In this paper, the sequential Bayesian philosophy is extended to a filter design using a fixed amount of most recent measurements, with the aim of exploiting the FIR structure and unifying some basic FIR filters with the KF. Specifically, the conditional mean and covariance of the posterior probability density functions are first derived to show the FIR counterpart of the KF. To remove the dependence on initial states, the corresponding likelihood is further maximized and realized iteratively. It shows that the maximum likelihood modification unifies the existing unbiased FIR filters by tuning a weighting matrix. Moreover, it converges to the Kalman estimate with the increase of horizon length, and can thus be considered as a link between the FIR filtering and the KF. Several important properties including stability and robustness against errors of noise statistics are illustrated. Finally, a moving target tracking example and an experiment with a three degrees-of-freedom helicopter system are introduced to demonstrate effectiveness.

Distributed Unbiased FIR Filtering With Average Consensus on Measurements for WSNs

Industrial wireless sensor networks (WSNs) often operate under harsh conditions that require robustness from an estimator of a measured quantity. We propose a novel distributed unbiased finite-impulse response (UFIR) filter called micro-UFIR filter that, unlike the micro-Kalman filter (micro-KF), is robust against modeling errors in uncertain noise environments. The micro-UFIR filter is derived based on average consensus on measurements and, unlike the micro-KF, requires only one consensus filter. Better robustness of the micro-UFIR filter is shown analytically and confirmed by simulations of a WSN and a vehicle travelling along a circular trajectory under unpredictable impacts, impulsive noise, and errors in the noise statistics.

New discrete orthogonal moments for signal analysis

The paper addresses some computational aspects and applications of the polynomial unbiased finite impulse response functions originally derived by Shmaliy as a new class of a one-parameter family of discrete orthogonal polynomials. We present two new explicit formulas to compute these polynomials directly. They are based on the discrete Rodrigues’ representation and hypergeometric form, respectively. These straightforward calculations lead us to propose another discrete signal representation in moment domain. Experimental results show a comparison between the proposed moments and discrete Chebyshev moments in terms of noise-free/noisy signal feature extraction and reconstruction error.

General Unbiased FIR Filter With Applications to GPS-Based Steering of Oscillator Frequency

The general unbiased finite-impulse response (UFIR) filter proposed in this brief has important structural advantages against its basic predecessor. It can be applied to systems with or without the control input. We derive this filter in a batch form and then design its fast iterative Kalman-like algorithm using recursions. The iterative UFIR algorithm proposed is applied to the three-state polynomial model which is basic in clock synchronization. We test it by the global positioning system-based frequency steering of an oven-controlled crystal oscillator. Better robustness and higher accuracy of the UFIR filter against the Kalman filter are shown experimentally.

On the Iterative Computation of Error Matrix in Unbiased FIR Filtering

It is proved that the iterative computation form for the mean square error (MSE) matrix of the batch unbiased finite impulse response (UFIR) filter exactly equals that of the iterative UFIR filter form, unlike what was previously thought. Based on the iterative MSE matrix form, we suggest two strategies for defining the optimal horizon length for the UFIR filter. The results are verified using the two-state polynomial and harmonic models.

Fusion Kalman/UFIR Filter for State Estimation With Uncertain Parameters and Noise Statistics

In this paper, we fuse the Kalman filter (KF) that is optimal but not robust with the unbiased finite-impulse response (UFIR) filter which is more robust than KF but not optimal. The fusion filter employs the KF and UFIR filter as subfilters and produces smaller errors under the industrial conditions. In order to provide the best fusion effect, the operation point where UFIR meets Kalman is determined by applying probabilistic weights to each subfilter. Extensive simulations of the three degree of freedom (3-DOF) hover system have shown that the fusion filter output tends to range close to that by the best subfilter. Experimental verification provided for a 1-DOF torsion system has confirmed validity of simulation.

Blind Robust Estimation With Missing Data for Smart Sensors Using UFIR Filtering

Smart sensors are often designed to operate under harsh industrial conditions with incomplete information about noise and missing data. Therefore, signal processing algorithms are required to be unbiased, robust, predictive, and desirably blind. In this paper, we propose a novel blind iterative unbiased finite impulse response (UFIR) filtering algorithm, which fits these requirements as a more robust alternative to the Kalman filter (KF). The tradeoff in robustness between the UFIR filter and KF is learned analytically. The predictive UFIR algorithm is developed to operate in control loops under temporary missing data. Experimental verification is given for carbon monoxide concentration and temperature measurements required to monitor urban and industrial environments. High accuracy and precision of the predictive UFIR estimator are demonstrated in a short time and on a long baseline.

Extended least square unbiased FIR filter for target tracking using the constant velocity motion model

This paper proposes a new nonlinear state estimator that has a finite impulse response (FIR) structure. The proposed state estimator is called the extended least square unbiased FIR filter (ELSUFF) because it is derived using a least square criterion and has an unbiasedness property. The ELSUFF is a special FIR filter designed for the constant velocity motion model and does not require noise information, such as covariance of Gaussian noise. In situations where noise information is highly uncertain, the ELSUFF can provide consistent performance, while existing nonlinear state estimators, such as the extended Kalman filter (EKF) and the particle filter (PF), often exhibit degraded performance under the same condition. Through simulations, we demonstrate the robustness of the ELSUFF against noise model uncertainty.

Unified Maximum Likelihood Form for Bias Constrained FIR Filters

In this letter, the maximum likelihood (ML) finite-impulse response (FIR) filter is proposed for discrete time-variant state-space models with nonsingular system matrix. The ML FIR filter has the deadbeat property and its form is universal for all known bias constrained FIR filters. By the identity weighting matrix, the ML FIR filter becomes the unbiased FIR filter, which ignores the noise statistics and the initial error statistics. Otherwise, the ML FIR filter is equivalent to the optimal FIR filter with embedded unbiasedness and to the minimum variance unbiased FIR filter. An example of a stochastic resonator demonstrates higher immunity of the ML FIR filter against errors in the imprecisely defined noise statistics than in the kalman filter.

Ultimate iterative UFIR filtering algorithm

Measurements are often provided in the presence of noise and uncertainties that require optimal filters to estimate processes with highest accuracy. The ultimate iterative unbiased finite impulse response (UFIR) filtering algorithm presented in this paper is more robust in real world than the Kalman filter. It completely ignores the noise statistics and initial values while demonstrating better accuracy under the mismodeling and temporary uncertainties and lower sensitivity to errors in the noise statistics.

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.

Improved UFIR Tracking Algorithm for Maneuvering Target

Maneuvering target tracking is a target motion estimation problem, which can describe the irregular target maneuvering motion. It has been widely used in the field of military and civilian applications. In the maneuvering target tracking, the performance of Kalman filter(KF) and its improved algorithms depend on the accuracy of process noise statistical properties. If there exists deviation between process noise model and the actual process, it will generate the phenomenon of estimation error increasing. Unbiased finite impulse response(UFIR) filter does not need priori knowledge of noise statistical properties in the filtering process. The existing UFIR filters have the problem that generalized noise power gain(GNPG) does not change with measurement of innovation. We propose an improved UFIR filter based on measurement of innovation with ratio dynamic adaptive adjustment at adjacent time. It perfects the maneuvering detect-ability. The simulation results show that the improved UFIR filter has the best filtering effect than KF when process noise is not accurate.

Fast Kalman-Like Optimal Unbiased FIR Filtering With Applications

In this paper, an optimal unbiased finite impulse response (OUFIR) filter is proposed as a linking solution between the unbiased FIR (UFIR) filter and the Kalman filter (KF). We first derive the batch OUFIR estimator to minimize the mean square error (MSE) subject to the unbiasedness constraint and then find its fast iterative form. It is shown that the OUFIR filter is full horizon (FH) and that its estimate converges to the KF estimate by increasing the horizon length $N$ . As a special feature, we note that the FH OUFIR filter operates almost as fast as the KF. Several other critical properties of the OUFIR filter are illustrated based on simulations and practical applications. Similar to the UFIR filter, and contrary to the KF, the OUFIR filter is highly insensitive to the initial conditions. It has much better robustness than KF against temporary model uncertainties. Finally, the OUFIR filter allows for ignoring system noise, which is typically not well known to the engineer.

Improving Reliability of Particle Filter-Based Localization in Wireless Sensor Networks via Hybrid Particle/FIR Filtering

The need for accurate, fast, and reliable indoor localization using wireless sensor networks (WSNs) has recently grown in diverse areas of industry. Accurate localization in cluttered and noisy environments is commonly provided by means of a mathematical algorithm referred to as a state estimator or filter. The particle filter (PF), which is the most commonly used filter in localization, suffers from the sample impoverishment problem under typical conditions of real-time localization based on WSNs. This paper proposes a novel hybrid particle/finite impulse response (FIR) filtering algorithm for improving reliability of PF-based localization schemes under harsh conditions causing sample impoverishment. The hybrid particle/FIR filter detects the PF failures and recovers the failed PF by resetting the PF using the output of an auxiliary FIR filter. Combining the regularized particle filter (RPF) and the extended unbiased FIR (EFIR) filter, the hybrid RP/EFIR filter is constructed in this paper. Through simulations, the hybrid RP/EFIR filter demonstrates its improved reliability and ability to recover the RPF from failures.

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.