Design of Gaussian Approximate Filter and Smoother for Nonlinear Systems with Correlated Noises at One Epoch Apart

In this study, the authors investigate the filtering and smoothing problems of nonlinear systems with correlated noises at one epoch apart. A pseudomeasurement equation is firstly reconstructed with a corresponding pseudomeasurement noise, which is no longer correlated with the process noise. Based on the reconstructed measurement model, new Gaussian approximate (GA) filter and smoother are derived, from which Kalman filter and smoother can be obtained for linear systems. For nonlinear systems, different GA filters and smoothers can be developed through utilizing different numerical methods for computing Gaussian-weighted integrals involved in the proposed solution. Numerical examples concerning univariate nonstationary growth model, passive ranging problem, and target tracking show the efficiency of the proposed filtering and smoothing methods for nonlinear systems with correlated noises at one epoch apart.     

Stack filters, stack smoothers, and mirrored thresholddecomposition

Stack smoothers have received considerable attention in signal processing in the past decade. Stack smoothers define a large class of nonlinear smoothers based on positive Boolean functions (PBF) applied in the binary domain of threshold decomposition. Although stack smoothers can offer some advantages over traditional linear FIR filters, they are in essence smoothers lacking the flexibility to adequately address a number of signal processing problems that require bandpass or highpass filtering characteristics. In this paper, mirrored threshold decomposition is introduced, which, together with the associated binary PBF, define the significantly richer class of stack filters. Using threshold logic representation, a number of properties of stack filters are derived. Notably, stack filters defined in the binary domain of mirrored threshold decomposition require the use of double weighting of each sample in the integer domain. The class of recursive stack filters and the corresponding recursive weighted median (RWM) filters in the integer domain admitting negative weights are introduced. The new stack filter formulation leads to a more powerful class of estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior stack smoother structures     

Exponential stability of filters and smoothers for hidden Markovmodels

We address the problem of filtering and fixed-lag smoothing for discrete-time and discrete-state hidden Markov models (HMMs), with the intention of extending some important results in Kalman filtering, notably the property of exponential stability. By appealing to a generalized Perron-Frobenius result for non-negative matrices, we are able to demonstrate exponential forgetting for both the recursive filters and smoothers; furthermore, methods for deriving overbounds on the convergence rate are indicated. Simulation studies for a two-state and two-output HMM verify qualitatively some of the theoretical predictions, and the observed convergence rate is shown to be bounded in accordance with the theoretical predictions     

Wiener filter design using polynomial equations

A simplified way of deriving realizable and explicit Wiener filters is presented. Discrete-time problems are discussed in a polynomial equation framework. Optimal filters, predictors, and smoothers are calculated by means of spectral factorizations and linear polynomial equations. A tool for obtaining these equations, for a given problem structure, is described. It is based on the evaluation of orthogonality in the frequency domain, by means of canceling stable poles with zeros. Comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical Wiener solution. The simplicity of the proposed derivation method is particularly evident in multistage filtering problems. To illustrate, two examples are discussed: a filtering and a generalized deconvolution problem. A new solvability condition for linear polynomial equation appearing in scalar problems is also presented     

Estimation of nonstationary EEG with Kalman smoother approach: an application to event-related synchronization (ERS)

An adaptive spectrum estimation method for nonstationary electroencephalogram by means of time-varying autoregressive moving average modeling is presented. The time-varying parameter estimation problem is solved by Kalman filtering along with a fixed-interval smoothing procedure. Kalman filter is an optimal filter in the mean square sense and it is a generalization of other adaptive filters such as recursive least squares or least mean square. Furthermore, by using the smoother the unavoidable tracking lag of adaptive filters can be avoided. Due to the properties of Kalman filter and benefits of the smoothing the time-frequency resolution of the presented Kalman smoother spectra is extremely high. The presented approach is applied to estimation of event-related synchronization/desynchronization (ERS/ERD) dynamics of occipital alpha rhythm measured from three healthy subjects. With the Kalman smoother approach detailed spectral information can be extracted from single ERS/ERD samples.     

New nonlinear algorithms for estimating and suppressing narrowbandinterference in DS spread spectrum systems

It has been shown that the narrowband (NB) interference suppression capability of a direct-sequence (DS) spread spectrum system can be enhanced considerably by processing the received signal via a prediction error filter. The conventional approach to this problem makes use of a linear filter. However, the binary DS signal, that acts as noise in the prediction process, is highly non-Gaussian. Thus, linear filtering is not optimal. Vijayan and Poor (1990) first proposed using a nonlinear approximate conditional mean (ACM) filter of the Masreliez (1975) type and obtained significant results. This paper proposes a number of new nonlinear algorithms. Our work consists of three parts. (1) We develop a decision-directed Kalman (DDK) filter, that has the same performance as the ACM filter but a simpler structure. (2) Using the nonlinear function in the ACM and the DDK filters, we develop other nonlinear least mean square (LMS) filters with improved performance. (3) We further use the nonlinear functions to develop nonlinear recursive least squares (RLS) filters that can be used independently as predictors or as interference identifiers so that the ACM or the DDK filter can be applied. Simulations show that our nonlinear algorithms outperform conventional ones     

Robust frequency-selective filtering using weighted myriad filtersadmitting real-valued weights

Weighted myriad smoothers have been proposed as a class of nonlinear filters for robust non-Gaussian signal processing in impulsive noise environments. However, weighted myriad smoothers are severely limited since their weights are restricted to be non-negative. This constraint makes them unusable in bandpass or highpass filtering applications that require negative filter weights. Further, they are incapable of amplifying selected frequency components of an input signal. In this paper, we generalize the weighted myriad smoother to a richer structure: a weighted myriad filter admitting real-valued weights. This involves assigning a pair of filter weights (one positive and the other negative) to each of the input samples. Equivalently, the filter can be described as a weighted myriad smoother applied to a transformed set of samples that includes the original input samples as well as their negatives. The weighted myriad filter is analogous to a normalized linear FIR filter with real-valued weights whose absolute values sum to unity. By suitably scaling the output of the weighted myriad filter, we extend it to yield the so-called scaled weighted myriad filter, which includes (but is more powerful than) the traditional unconstrained linear FIR filter. Finally we derive stochastic gradient-based nonlinear adaptive algorithms for the optimization of these novel myriad filters under the mean square error criterion     

Information fusion Wiener filter for the multisensor multichannel ARMA signals with time-delayed measurements

For the multisensor multichannel autoregressive moving average (ARMA) signals with time-delayed measurements, a measurement transformation approach is presented, which transforms the equivalent state space model with measurement delays into the state space model without measurement delays, and then using the Kalman filtering method, under the linear minimum variance optimal weighted fusion rules, three distributed optimal fusion Wiener filters weighted by matrices, diagonal matrices and scalars are presented, respectively, which can handle the fused filtering, prediction and smoothing problems. They are locally optimal and globally suboptimal. The accuracy of the fuser is higher than that of each local signal estimator. In order to compute the optimal weights, the formulae of computing the cross-covariances among local signal estimation errors are given. A Monte Carlo simulation example for the three-sensor target tracking system with time-delayed measurements shows their effectiveness.     

A new image smoothing method based on a simple model of spatialprocessing in the early stages of human vision

The difficulty of preserving edges is central to the problem of smoothing images. The main problem is that of distinguishing between meaningful contours and noise, so that the image can be smoothed without loss of details. Substantial efforts have been devoted to solving this difficult problem, and a plethora of filtering methods have been proposed in the literature. Non-linear filters have proved to be more efficient than their linear counterparts. Here, a new nonlinear filter for noise smoothing is introduced. This filter is based on the psychophysical phenomenon of human visual contrast sensitivity. Results on real images are presented to demonstrate the validity of our approach compared to other known filtering methods.     

Optimal demodulation of PAM signals

Kalman filtering theory is applied to yield an optimal causal demodulator for pulse-amplitude-modulated (PAM) signals in the presence of white Gaussian noise. The discrete-time data (or sampled continuous-time data) are assumed to be either a stationary or non-stationary Gaussian stochastic process, in general nonwhite. Optimal demodulation with delay is also achieved by application of Kalman filtering theory. The resulting demodulators (fixed-lag smoothers) are readily constructed and their performance represents in many cases a significant improvement over that for the optimal demodulator without delay. The fixed-lag smoothing results are in contrast to those for amplitude-modulated signals (AM) where only approximate fixed-lag smoothing is possible, and this with considerable design effort. The performance of the optimal PAM demodulator is shown to be equivalent to that of an optimal discrete filter for the discrete data.     

Fuzzy rank LUM filters.

The rank information of samples is widely utilized in nonlinear signal processing algorithms. Recently developed fuzzy transformation theory introduces the concept of fuzzy ranks, which incorporates sample spread (or sample diversity) information into the sample ranking framework. Thus, the fuzzy rank reflects a sample's rank, as well as its similarity to the other sample (namely, joint rank order and spread), and can be utilized to improve the performance of the conventional rank-order-based filters. In this paper, the well-known lower-upper-middle (LUM) filters are generalized utilizing the fuzzy ranks, yielding the class of fuzzy rank LUM (F-LUM) filters. Statistical and deterministic properties of the F-LUM filters are derived, showing that the F-LUM smoothers have similar impulsive noise removal capability to the LUM smoothers, while preserving the image details better. The F-LUM sharpeners are capable of enhancing strong edges while simultaneously preserving small variations. The performance of the F-LUM filters are evaluated for the problems of image impulsive noise removal, sharpening and edge-detection preprocessing. The experimental results show that the F-LUM smoothers can achieve a better tradeoff between noise removal and detail preservation than the LUM smoothers. The F-LUM sharpeners are capable of sharpening the image edges without amplifying the noise or distorting the fine details. The joint smoothing and sharpening operation of the general F-LUM filters also showed superiority in edge detection preprocessing application. In conclusion, the simplicity and versatility of the F-LUM filters and their advantages over the conventional LUM filters are desirable in many practical applications. This also shows that utilizing fuzzy ranks in filter generalization is a promising methodology.     

An efficient numerical technique for the solution of the monodomain and bidomain equations

Most numerical schemes for solving the monodomain or bidomain equations use a forward approximation to some or all of the time derivatives. This approach, however, constrains the maximum timestep that may be used by stability considerations as well as accuracy considerations. Stability may be ensured by using a backward approximation to all time derivatives, although this approach requires the solution of a very large system of nonlinear equations at each timestep which is computationally prohibitive. In this paper we propose a semi-implicit algorithm that ensures stability. A linear system is solved on each timestep to update the transmembrane potential and, if the bidomain equations are being used, the extracellular potential. The remainder of the equations to be solved uncouple into small systems of ordinary differential equations. The backward Euler method may be used to solve these systems and guarantee numerical stability: as these systems are small, only the solution of small nonlinear systems are required. Simulations are carried out to show that the use of this algorithm allows much larger timesteps to be used with only a minimal loss of accuracy. As a result of using these longer timesteps the computation time may be reduced substantially.     

Elimination of interference terms of the discrete Wignerdistribution using nonlinear filtering

Methods for interference reduction in the Wigner distribution (WD) have traditionally relied on linear filtering. This paper introduces a new nonlinear filtering approach for the removal of cross terms in the discrete WD. Realizing that linear smoothing kernels are unable to completely cancel the cross-terms without compromising time-frequency concentration and resolution of the auto-terms, a nonlinear filtering algorithm is devised where the filter automatically adapts to the rapidly changing nature of the WD plane. Varying the filter behavior from an identity operation at one extreme to a lowpass linear filter at the other, a near-optimal removal of cross terms is achieved. Unlike traditional smoothing and optimal kernel design techniques, this algorithm does not reduce the time-frequency resolution and concentration of the auto-terms and performs equally well for a very large variety of signals     

Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature

In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.     

B-spline signal processing. I. Theory

The use of continuous B-spline representations for signal processing applications such as interpolation, differentiation, filtering, noise reduction, and data compressions is considered. The B-spline coefficients are obtained through a linear transformation, which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering. The same property also applies for the indirect B-spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines. The filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for B-splines of any order. Applications to differentiation, filtering, smoothing, and least-squares approximation are examined. The extension of such operators for higher-dimensional signals such as digital images is considered     

Robust Wiener- Kolmogorov theory

A minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete-time wide-sense stationary signals when knowledge of the signal and/or noise spectra is inexact. The solution is given (under mild regularity conditions) in terms of a least favorable pair of spectra, thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily. It is noted that this design method leads, in particular, to robustn-step predictors, robust causal filters, and robustn-lag smoothers. The method of design is illustrated by a thorough development of the special case of one-step noiseless prediction. Further, solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise, and numerical examples are given for this case which illustrate the effectiveness of this design.     

Remarks on linear and nonlinear filtering

This communication tries to give some insight into relationships existing between Viterbi and the forward-backward algorithm (used in the context of hidden Markov models) on the one hand and Kalman filtering and Rauch-Tung Striebel smoothing on the other. We give a unifying view which shows how those algorithms are related and give an example of a nonlinear hybrid system that can be filtered through a mixed algorithm     

Filtering of colored noise for speech enhancement and coding

Scalar and vector Kalman filters are implemented for filtering speech contaminated by additive white noise or colored noise, and an iterative signal and parameter estimator which can be used for both noise types is presented. Particular emphasis is placed on the removal of colored noise, such as helicopter noise, by using state-of-the-art colored-noise-assumption Kalman filters. The results indicate that the colored noise Kalman filters provide a significant gain in signal-to-noise ratio (SNR), a visible improvement in the sound spectrogram, and an audible improvement in output speech quality, none of which are available with white-noise-assumption Kalman and Wiener filters. When the filter is used as a prefilter for linear predictive coding, the coded output speech quality and intelligibility are enhanced in comparison to direct coding of the noisy speech     

Edge-aware smoothing through adaptive interpolation

The goal of edge-aware filtering is to smooth out small-scale structures while preserving large object boundaries. A fundamental idea to design such filters is to avoid smoothing across strong edges. In this paper, we explore a new approach which iteratively adds the edge information back to a smoothed image. We study the smoothed image as the starting point of the iteration and the optimal stopping criterion. We demonstrate that in a wide range of applications the proposed technique can produce competitive results as those of state-of-the-art edge-aware filters. In particular, the proposed algorithm has the best performance in texture smoothing.     

大失准角下SINS的KF/EKF2混合滤波对准

针对捷联惯导系统(SINS)大失准角下滤波对准过程中非线性滤波器状态维数过大的问题,提出了一种基于模型分解的卡尔曼滤波/二阶扩展卡尔曼滤波(KF/EKF2)混合滤波方法,将基于欧拉平台误差角的非线性滤波模型分解为线性部分和非线性部分,分别采用线性KF滤波和非线性EKF2滤波处理,并且设计了混合滤波的滤波步骤。实验结果表明,KF/EKF2混合滤波算法在计算量、实时性及精度等方面优于最常用的无迹卡尔曼滤波(UKF)和EKF2滤波。