Unconstrained Filter Research Articles

Distributed state estimation with state equality constraints in the presence of packet dropping

In this paper, we investigate distributed Kalman consensus filter with state equality constraints in the presence of packet dropping. Firstly, the unconstrained distributed Kalman filter is designed by utilizing the local measurement information of each sensor node, and then the consensus term is added to it to derive the unconstrained distributed Kalman consensus filter. Secondly, we research the equality constrained distributed Kalman consensus filter based on the projection operator to achieve better filter performance, where the unconstrained estimation gain is calculated by solving a modified algebraic Riccati equation and a Lyapunov equation. Additionally, we design the second type of equality constrained distributed Kalman consensus filter by using time-stamping technology and projection operator to eliminate the effect of the Lyapunov equation in the infinite time domain on the convergence analysis, and its unconstrained estimation gain only requires solving a modified algebraic Riccati equation. Finally, the simulation experiments demonstrate that the equality constrained distributed Kalman consensus filter outperforms the unconstrained filter and that the second type of estimator exhibits superior performance compared to the first.

On the asymptotic behavior of linearly constrained filters for robust multi-channel signal processing

The Kalman filter (KF) is known to loose its optimality properties when the model used does not perfectly match the true system. Extending the use of linear constraints to this filter recently proved to be efficient to mitigate a large class of parametric model mismatch, through the linearly constrained Kalman filter and minimum variance filter (LCKF and LCMVF). However, the asymptotic performances of these new filters are still an open question. In this work, we bring a first answer to the latter problem in the case of measurement model mismatch. We show that both LCKF and LCMVF are equivalent to unconstrained filters in which the directions of the constraints are cancelled by projection, allowing a better understanding of their asymptotic properties. In particular, the steady-state mean square error, when it exists, is derived. The consistency of the filters with respect to nonlinear mismatches is also improved via new constraints. An array processing example is provided to assess the derived formulas, the consistency and performance of the filters.

Color Sparse Representations for Image Processing: Review, Models, and Prospects

Sparse representations have been extended to deal with color images composed of three channels. A review of dictionary-learning-based sparse representations for color images is made here, detailing the differences between the models, and comparing their results on the real and simulated data. These models are considered in a unifying framework that is based on the degrees of freedom of the linear filtering/transformation of the color channels. Moreover, this allows it to be shown that the scalar quaternionic linear model is equivalent to constrained matrix-based color filtering, which highlights the filtering implicitly applied through this model. Based on this reformulation, the new color filtering model is introduced, using unconstrained filters. In this model, spatial morphologies of color images are encoded by atoms, and colors are encoded by color filters. Color variability is no longer captured in increasing the dictionary size, but with color filters, this gives an efficient color representation.

State estimation and fault detection and identification for constrained stochastic linear hybrid systems

In this study, a fault detection and identification (FDI) scheme are proposed for constrained stochastic linear hybrid systems (CSLHSs) based on our proposed constrained innovation hybrid estimator (CIHE). CSLHSs are stochastic linear hybrid systems that have state equality constraints on the dynamics of each mode. The CIHE is derived to estimate the CSLHS in such a way that the expectation of the innovations of the mode-matched filters satisfy the constraints. Therefore the constraint information is available in the residual of the CIHE for FDI of not only system dynamic faults but also constraint violations. Each constrained, mode-matched filter is proved to be unbiased, satisfy the constraint, and have smaller estimation error than corresponding unconstrained filters. It is also proved that the CIHE is unbiased and that three residuals from the CIHE have zero mean with a given covariance for different cases of system faults. By monitoring the statistical properties of these residuals, both constraint and discrete faults are detected and identified. The estimation and FDI characteristics of the CIHE algorithm are illustrated using a water tank system as an example.

Quaternion Estimation from Vector Observations using a Matrix Kalman Filter

A novel two-stage quaternion estimator from vector observations that is a synthesis between Wahba's approach and the Kalman filtering approach is presented. The first stage features an optimal denoising procedure of the elements of a time-varying noisy K-matrix. The second stage produces a quaternion estimate from the filtered K-matrix via any eigenvalue-eigenvector solver. This work's contribution consists in performing the denoising via Kalman filtering. For that purpose, a matrix Kalman filter (MKF) is developed, which has the advantage of preserving the natural formulation of the matrix plant equations. As a result, two aspects of a previous algorithm, called Optimal-REQUEST (OPREQ), are improved: the K-matrix update estimation stage uses a matrix gain rather than a scalar gain, and that gain is optimized with respect to the classical minimum-variance cost. This work assumes that the sensed lines of sight (LOS) are time invariant as seen in the chosen reference frame. This assumption fits in various operational mission architectures. An exact Kalman filter is developed that accounts for the state-multiplicative noise in the process equation. A reduced estimator is also developed assuming simple expressions for the filter covariance matrices. A constrained estimator, which enforces the symmetry and null-trace of the estimated matrix, is designed using the pseudomeasurement (PM) technique. Extensive Monte-Carlo simulations illustrate the performance of the novel filters with a spinning and nutating spacecraft (SC) as a case study. Extensive Monte-Carlo simulations show that the proposed estimator outperforms OPREQ. As illustrated by additional Monte-Carlo simulations, the constrained MKF exhibits a better transient and a better steady-state accuracy than the unconstrained filter for large initial disturbances in the symmetry and null-trace properties.

Improving Land Data Assimilation Performance with a Water Budget Constraint

AbstractA weak constraint is introduced in ensemble Kalman filters to reduce the water budget imbalance that occurs in land data assimilation. Two versions of the weakly constrained filter, called the weakly constrained ensemble Kalman filter (WCEnKF) and the weakly constrained ensemble transform Kalman filter (WCETKF), are proposed. The strength of the weak constraint is adaptive in the sense that it depends on the statistical characteristics of the forecast ensemble. The resulting filters are applied to assimilate synthetic observations generated by the Noah land surface model over the Red Arkansas River basin. The data assimilation experiments demonstrate that, for all tested scenarios, the constrained filters produce analyses with nearly the same accuracy as unconstrained filters, but with much smaller water balance residuals than unconstrained filters.

Constrained particle filter approach to approximate the arrival cost in Moving Horizon Estimation

Moving Horizon Estimation (MHE) is an efficient state estimation method used for nonlinear systems. Since MHE is optimization-based it provides a good framework to handle bounds and constraints when they are required to obtain good state and parameter estimates. Recent research in this area has been directed to develop computationally efficient algorithms for on-line application. However, an open issue in MHE is related to the approximation of the so-called arrival cost and of the parameters associated with it. The arrival cost is very important since it provides a means to incorporate information from the previous measurements to the current state estimate. It is difficult to calculate the true value of the arrival cost; therefore approximation techniques are commonly applied. The conventional method is to use the Extended Kalman Filter (EKF) to approximate the covariance matrix at the beginning of the prediction horizon. This approximation method assumes that the state estimation error is Gaussian. However, when state estimates are bounded or the system is nonlinear, the distribution of the estimation error becomes non-Gaussian. This introduces errors in the arrival cost term which can be mitigated by using longer horizon lengths. This measure, however, significantly increases the size of the nonlinear optimization problem that needs to be solved on-line at each sampling time. Recently, particle filters and related methods have become popular filtering methods that are based on Monte-Carlo simulations. In this way they implement an optimal recursive Bayesian Filter that takes advantage of particle statistics to determine the probability density properties of the states. In the present work, we exploit the features of these sampling-based methods to approximate the arrival cost parameters in the MHE formulation. Also, we show a way to construct an estimate of the log-likelihood of the conditional density of the states using a Particle Filter (PF), which can be used as an approximation of the arrival cost. In both cases, because particles are being propagated through the nonlinear system, the assumption of Gaussianity of the state estimation error can be dropped. Here we developed and tested EKF and eight different types of sample based filters for updating the arrival cost parameters in the weighted 2-norm approach (see Table 1 for the full list). We compare the use of constrained and unconstrained filters, and note that when bounds are required the constrained particle filters give a better approximation of the arrival cost parameters that improve the performance of MHE. Moreover, we also used PF concepts to directly approximate the negative of the log-likelihood of the conditional density using unconstrained and constrained particle filters to update the importance distribution. Also, we show that a benefit of having a better approximation of the arrival cost is that the horizon length required for the MHE can be significantly smaller than when using the conventional MHE approach. This is illustrated by simulation studies done on benchmark problems proposed in the state estimation literature.

Optimal Filtering in Pilot‐Aided Carrier Recovery

The paper deals with carrier recovery based on pilot symbols in single‐carrier systems. Wiener′s method is used to determine the optimal unconstrained filter in estimation of phase noise assuming that a sequence of equally spaced pilot symbols is available. Our analysis allows to capture two effects that are not considered in the existing literature: the impact of aliasing due to sampling of the phase noise sequence at the pilot rate and the cyclostationary nature of the estimate hence of its performance. Experimental results are derived also for the case, where the filter is constrained to the cascade of two moving averages. These results show that, in the considered example, the mean‐square phase error of the constrained filter is within 0.35 dB from the MSE of the optimal filter.

Design of smallest size two-dimensional linear-phase FIR filters with magnitude error constraint

This paper presents an online procedure that produces the smallest feasible size of two-dimensional FIR filters with prescribed magnitude error constraint. The procedure uses the mean square normalized error of constrained and unconstrained least-square filters to produce the initial and the subsequent sizes that converge to the smallest feasible one in a few iterations, where the constrained least-square filters are defined as the least-square filters satisfying the magnitude error constraint. The procedure finally returns a smallest size filter that satisfies the magnitude error constraint and has least total squared magnitude error. Design examples of diamond-shaped, rectangular, and elliptic filters are provided, and comparisons with an exhaustive search are given.

Kalman Filter Constraint Switching for Turbofan Engine Health Estimation

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints are often neglected because they do not fit easily into the structure of the Kalman filter. Recently published work has shown a new method for incorporating state variable inequality constraints in the Kalman filter, which has been shown to generally improve the filter s estimation accuracy. However, the incorporation of inequality constraints poses some risk to the estimation accuracy as the Kalman filter is theoretically optimal. This paper proposes a way to tune the filter constraints so that the state estimates follow the unconstrained (theoretically optimal) filter when the confidence in the unconstrained filter is high. When confidence in the unconstrained filter is not so high, then we use our heuristic knowledge to constrain the state estimates. The confidence measure is based on the agreement of measurement residuals with their theoretical values. The algorithm is demonstrated on a linearized simulation of a turbofan engine to estimate engine health.

An adaptive coherence filter

This paper derives an adaptive coherence filter for canceling interference from a signal of interest whose power spectral density is symmetric. A basic property of the Fourier transform of real signals is that their spectra are Hermitian symmetric. This property is exploited to determine which part of a spectrum is interference and which part is the signal of interest. An unconstrained Wiener filter is derived that exploits the frequency domain symmetry of the signal of interest. While the adaptive coherence filter is based on the Fourier transform property of real signals, an extension of the algorithm is provided so the filter can be used on any signal that displays spectral symmetry. A practical method for implementing the filter is provided. The filter has application in the area of telecommunications, but is applicable in wireless communication applications where a signal, that displays spectral symmetry, is corrupted by interfering signals within the signal of interest's bandwidth.

A Lattice-Based Minimal Gray-Scale Switching Algorithm for Obtaining the Optimal Increasing Filter from the Optimal Filter

For binary window-based filters, the optimal increasing filter is often derived from the optimal unconstrained (nonincreasing) filter by iteratively switching the filter values at pixels from 0 to 1 or from 1 to 0 so as to make the resulting filter be the optimal increasing filter. This paper gives a corresponding switching algorithm for gray-scale nonlinear filters, and it does so in the context of finite lattices, which makes the algorithm applicable to computational morphology on lattices. The algorithm is minimal in the sense that it involves a minimal search if one wishes to be certain to obtain the optimal increasing filter when beginning with the optimal unconstrained filter.

Realizable Linear and Decision Feedback Equalizers: Properties and Connections

Recently, there has been renewed interest in the use of infinite impulse response (IIR) linear equalizers (LEs) for digital communication channels as a means for both improving performance and blindly initializing decision feedback structures (DFEs). Theoretical justification for such an approach is usually given assuming unconstrained filters, which are not causal and therefore not implementable in practice. We present an analysis of realizable (i.e., causal, stable, and of finite degree) minimum mean square error (MMSE) equalizers for single-input multiple-output channels, both in the LE and DFE cases, focusing on their structures and filter orders, as well as the connections between them. The DFE resulting from rearranging the MMSE LE within a decision feedback loop is given special attention. It is shown that although this DFE does not in general coincide with the MMSE DFE, it still enjoys certain optimality conditions. The main tools employed are the Wiener theory of minimum variance estimation and Kalman filtering theory, which show interesting properties of the MMSE equalizers not revealed by previous polynomial approaches.

Design of IIR digital all-pass filters using least pth phase error criterion

In this brief, the least pth-phase error criterion is adopted to design digital infinite-impulse-response all-pass filters to have an arbitrarily prescribed phase response. First, an iterative reweighted least-squares method is used to design an unconstrained all-pass filter whose optimal filter coefficients are obtained by solving a set of linear simultaneous equations in each iteration. Then, the proposed method is extended to design a constrained all-pass filter. Finally, a design example of the Hilbert transformer is demonstrated to illustrate the effectiveness of the new design method.

Pattern Recognition Theory in Nonlinear Signal Processing

A body of research has developed within the context of nonlinear signal and image processing that deals with the automatic, statistical design of digital window-based filters. Based on pairs of ideal and observed signals, a filter is designed in an effort to minimize the error between the ideal and filtered signals. The goodness of an optimal filter depends on the relation between the ideal and observed signals, but the goodness of a designed filter also depends on the amount of sample data from which it is designed. In order to lessen the design cost, a filter is often chosen from a given class of filters, thereby constraining the optimization and increasing the error of the optimal filter. To a great extent, the problem of filter design concerns striking the correct balance between the degree of constraint and the design cost. From a different perspective and in a different context, the problem of constraint versus sample size has been a major focus of study within the theory of pattern recognition. This paper discusses the design problem for nonlinear signal processing, shows how the issue naturally transitions into pattern recognition, and then provides a review of salient related pattern-recognition theory. In particular, it discusses classification rules, constrained classification, the Vapnik-Chervonenkis theory, and implications of that theory for morphological classifiers and neural networks. The paper closes by discussing some design approaches developed for nonlinear signal processing, and how the nature of these naturally lead to a decomposition of the error of a designed filter into a sum of the following components: the Bayes error of the unconstrained optimal filter, the cost of constraint, the cost of reducing complexity by compressing the original signal distribution, the design cost, and the contribution of prior knowledge to a decrease in the error. The main purpose of the paper is to present fundamental principles of pattern recognition theory within the framework of active research in nonlinear signal processing.

Relations between thek-dimensional unconstrained and linear phase FIR Wiener filters

In this note, we show that thek-dimensional linear phase FIR Wiener filter can be obtained from the unconstrained filter with a simple “reverse and add” operation. The relation between the linear phase and unconstrained phase filters is obtained also in the case of a multichannelk-dimensional system. In this case, the properties of the autocorrelation matrix do not allow the simplifications noted for the single channel case.

Restoration and reconstruction of AVHRR images

Describes the design of small convolution kernels for the restoration and reconstruction of Advanced Very High Resolution Radiometer (AVHRR) images. The kernels are small enough to be implemented efficiently by convolution, yet effectively correct degradations and increase apparent resolution. The kernel derivation is based on a comprehensive, end-to-end system model that accounts for scene statistics, image acquisition blur, sampling effects, sensor noise, and postfilter reconstruction. The design maximizes image fidelity subject to explicit constraints on the spatial support and resolution of the kernel. The kernels can be designed with finer resolution than the image to perform partial reconstruction for geometric correction and other remapping operations. Experiments demonstrate that small kernels yield fidelity comparable to optimal unconstrained filters with less computation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Small convolution kernels for high-fidelity image restoration

An algorithm is developed for computing the mean-square-optimal values for small, image-restoration kernels. The algorithm is based on a comprehensive, end-to-end imaging system model that accounts for the important components of the imaging process: the statistics of the scene, the point-spread function of the image-gathering device, sampling effects, noise, and display reconstruction. Subject to constraints on the spatial support of the kernel, the algorithm generates the kernel values that restore the image with maximum fidelity, that is, the kernel minimizes the expected mean-square restoration error. The algorithm is consistent with the derivation of the spatially unconstrained Wiener filter, but leads to a small, spatially constrained kernel that, unlike the unconstrained filter, can be efficiently implemented by convolution. Simulation experiments demonstrate that for a wide range of imaging systems these small kernels can restore images with fidelity comparable to images restored with the unconstrained Wiener filter. >

Optimal Filters for Binary Detectors with Comparator Hysteresis

Methods that increase the speed of digital detection comparators can also produce hysteresis, wherein the effective threshold in a given signaling interval is influenced by the previous output. If the comparator is fed by a peak-limited IF detector output followed by a detection filter with unity dc gain, the hysteresis can be a significant fraction of the peak comparator input. This concise paper derives the detection filter impulse response for which the resulting eye-opening-to-rms-noise ratio is maximized. The impulse response is taken to be either unconstrained in time or confined to a signaling interval (as in filter-and-dump detection). The results for each case are applied to 2- and 4-level coherent phase shift keying (CPSK). Expressions are obtained for the maximized eye-opening-to-rms-noise ratio as a function of comparator hysteresis and phase modulation pulse shape. These expressions are normalized to represent the excess in required power above the theoretical minimum, and graphical results are given. They show the suboptimality of filter-and-dump detection when hysteresis is high, the insensitivity of performance to hysteresis when unconstrained optimal filters are used, and the relative effects of phase modulation pulse shape on required power.