Optimal Linear Filter Research Articles

Optimal linear filter design for systems with correlation in the measurement matrices and noises: recursive algorithm and applications

This paper addresses the optimal least-squares linear estimation problem for a class of discrete-time stochastic systems with random parameter matrices and correlated additive noises. The system presents the following main features: (1) one-step correlated and cross-correlated random parameter matrices in the observation equation are assumed; (2) the process and measurement noises are one-step autocorrelated and two-step cross-correlated. Using an innovation approach and these correlation assumptions, a recursive algorithm with a simple computational procedure is derived for the optimal linear filter. As a significant application of the proposed results, the optimal recursive filtering problem in multi-sensor systems with missing measurements and random delays can be addressed. Numerical simulation examples are used to demonstrate the feasibility of the proposed filtering algorithm, which is also compared with other filters that have been proposed.

Optimal linear phase FIR filter design using particle swarm optimization with constriction factor and inertia weight approach with wavelet mutation

In this paper, Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach is hybridized with Wavelet Mutation (PSOCFIWA-WM) strategy for the optimal design of linear phase FIR filters. Real coded genetic algorithm (RGA), particl

Linear estimation for networked control systems with random transmission delays and packet dropouts

In networked control systems, random delays and packet dropouts are inevitable during data transmissions due to the limited communication capacity. The phenomena of random delays and packet dropouts of both sides from a sensor to an estimator and from a controller to an actuator are modeled via employing two groups of Bernoulli random variables. By defining some new variables, the original system with random delays and packet dropouts is equivalently transformed into a stochastic parameterized system. The optimal linear filter, predictor and smoother in the linear minimum variance sense are presented via the orthogonality projection approach. They depend on the probabilities of the stochastic parameters. The solutions to the optimal linear estimators are given by three equations including a Riccati, a Lyapunov and a simple difference. The stability of the proposed estimators is analyzed. At last, a sufficient condition for the existence of the steady-state estimators is given for time-invariant systems. Two examples are provided to verify the effectiveness of the proposed algorithms.

Optimal Kalman Filtering for a Class of State Delay Systems with Randomly Multiple Sensor Delays

The optimal Kalman filtering problem is investigated for a class of discrete state delay stochastic systems with randomly multiple sensor delays. The phenomenon of measurement delay occurs in a random way and the delay rate for each sensor is described by a Bernoulli distributed random variable with known conditional probability. Based on the innovative analysis approach and recursive projection formula, a new linear optimal filter is designed such that, for the state delay and randomly multiple sensor delays with different delay rates, the filtering error is minimized in the sense of mean square and the filter gain is designed by solving the recursive matrix equation. Finally, a simulation example is given to illustrate the feasibility and effectiveness of the proposed filtering scheme.

Optimal linear estimation for systems with transmission delays and packet dropouts

This study considers a networked system in which the measurement suffers from one-step delay and packet dropouts because of the unreliability of the network. A new model applied to describe the arrival conditions of the measurements is proposed. Based on the new model and using a state augmentation method, optimal linear filter, predictor and smoother are obtained. A sufficient condition for the convergence of the system is given. Finally, the simulation results show the effectiveness of the proposed algorithms.

Linear Unbiased Optimal Filter for Discrete-Time Systems with One-Step Random Delays and Inconsecutive Packet Dropouts

This paper is concerned with the linear unbiased minimum variance estimation problem for discrete-time stochastic linear control systems with one-step random delay and inconsecutive packet dropout. A new model is developed to describe the phenomena of the one-step delay and inconsecutive packet dropout by employing a Bernoulli distributed stochastic variable. Based on the model, a recursive linear unbiased optimal filter in the linear minimum variance sense is designed by the method of completing the square. The solution to the linear filter is given by three equations including a Riccati equation, a Lyapunov equation and a simple difference equation. A sufficient condition for the existence of the steady-state filter is given. A simulation shows the effectiveness of the proposed algorithm. DOI: http://dx.doi.org/10.11591/telkomnika.v11i11.3523

Performance Analysis of the Wiener Optimal Filtering Noise-Reduction Technique

Noise reduction, which aims at extracting the clean speech from noisy observations, has attracted a considerable amount of research attention over the past several decades. Although many methods have been developed to achieve the goal of noise reduction and signal-to-noise ratio (SNR) improvement, there has been remarkably little theoretical justification of their performance due to the difficulty in quantizing the combinatorial effect between noise reduction and speech distortion. This paper attempts to provide a theoretical analysis on the performance of the Wiener optimal filtering noise-reduction technique. We show that the Wiener optimal linear filter can indeed reduce the level of noise. Most importantly, we prove that the output SNR is always greater than, or at least equal to the input SNR, which reveals that the Wiener optimal linear filtering technique is indeed able to make noisy speech signals cleaner.

Transceiver Design for SC-FDE Based MIMO Relay Systems

In this paper, we propose a joint transceiver design for single-carrier frequency-domain equalization (SC-FDE) based multiple-input multiple-output (MIMO) relay systems. To this end, we first derive the optimal minimum mean-squared error linear and decision-feedback frequency-domain equalization filters at the destination along with the corresponding error covariance matrices at the output of the equalizer. Subsequently, we formulate the source and relay precoding matrix design problem as the minimization of a family of Schur-convex and Schur-concave functions of the mean-squared errors at the output of the equalizer under separate power constraints for the source and the relay. By exploiting properties of the error covariance matrix and results from majorization theory, we derive the optimal structures of the source and relay precoding matrices, which allows us to transform the matrix optimization problem into a scalar power optimization problem. Adopting a high signal-to-noise ratio approximation for the objective function, we obtain the global optimal solution for the power allocation variables. We illustrate the excellent performance of the proposed system and compare it to that of conventional orthogonal frequency-division multiplexing MIMO relay systems based on computer simulations.

Cat Swarm Optimization algorithm for optimal linear phase FIR filter design

In this paper a new meta-heuristic search method, called Cat Swarm Optimization (CSO) algorithm is applied to determine the best optimal impulse response coefficients of FIR low pass, high pass, band pass and band stop filters, trying to meet the respective ideal frequency response characteristics. CSO is generated by observing the behaviour of cats and composed of two sub-models. In CSO, one can decide how many cats are used in the iteration. Every cat has its′ own position composed of M dimensions, velocities for each dimension, a fitness value which represents the accommodation of the cat to the fitness function, and a flag to identify whether the cat is in seeking mode or tracing mode. The final solution would be the best position of one of the cats. CSO keeps the best solution until it reaches the end of the iteration. The results of the proposed CSO based approach have been compared to those of other well-known optimization methods such as Real Coded Genetic Algorithm (RGA), standard Particle Swarm Optimization (PSO) and Differential Evolution (DE). The CSO based results confirm the superiority of the proposed CSO for solving FIR filter design problems. The performances of the CSO based designed FIR filters have proven to be superior as compared to those obtained by RGA, conventional PSO and DE. The simulation results also demonstrate that the CSO is the best optimizer among other relevant techniques, not only in the convergence speed but also in the optimal performances of the designed filters.

Optimal and mode-independent filters for generalised Bernoulli jump systems

This paper provides the optimal solution of the filtering design problem for a special class of discrete-time Markov jump linear systems whose transition probability matrix has identical rows. In the two-mode case, this is equivalent to saying that the random variable has a Bernoulli distribution. For that class of dynamic systems we design, with the help of new necessary and sufficient linear matrix inequality conditions, and optimal mode-independent filters with the same order of the plant. As a first proposal available in the literature, for partial information characterised by cluster availability of the mode, we also show it is possible to design optimal full-order linear filters. If some plant matrices do not vary within the same cluster, we show that the optimal filter exhibits the internal model structure. We complete the results with illustrative examples. A realistic practical application considering sensors connected to a network using a communication protocol such as the Token Ring is included in order to put in evidence the usefulness of the theoretical results.

Optimal Linear Filters for Discrete-Time Systems With Randomly Delayed and Lost Measurements With/Without Time Stamps

A novel model is developed to describe possible random delays and losses of measurements transmitted from a sensor to a filter by a group of Bernoulli distributed random variables. Based on the new developed model, an optimal linear filter dependent on the probabilities is presented in the linear minimum variance sense by the innovation analysis approach when packets are not time-stamped. The solution to the optimal linear filter is given in terms of a Riccati difference equation and a Lyapunov difference equation. A sufficient condition for the existence of the steady-state filter is given. At last, the optimal filter is given by Kalman filter when packets are time-stamped.

Filter Design With Secrecy Constraints: The MIMO Gaussian Wiretap Channel

This paper considers the problem of filter design with secrecy constraints, where two legitimate parties (Alice and Bob) communicate in the presence of an eavesdropper (Eve), over a Gaussian multiple-input-multiple-output (MIMO) wiretap channel. This problem involves designing, subject to a power constraint, the transmit and the receive filters which minimize the mean-squared error (MSE) between the legitimate parties whilst assuring that the eavesdropper MSE remains above a certain threshold. We consider a general MIMO Gaussian wiretap scenario, where the legitimate receiver uses a linear Zero-Forcing (ZF) filter and the eavesdropper receiver uses either a ZF or an optimal linear Wiener filter. We provide a characterization of the optimal filter designs by demonstrating the convexity of the optimization problems. We also provide generalizations of the filter designs from the scenario where the channel state is known to all the parties to the scenario where there is uncertainty in the channel state. A set of numerical results illustrates the performance of the novel filter designs, including the robustness to channel modeling errors. In particular, we assess the efficacy of the designs in guaranteeing not only a certain MSE level at the eavesdropper, but also in limiting the error probability at the eavesdropper. We also assess the impact of the filter designs on the achievable secrecy rates. The penalty induced by the fact that the eavesdropper may use the optimal non-linear receive filter rather than the optimal linear one is also explored in the paper.

Optimal Alarm Signal Processing: Filter Design and Performance Analysis

Accuracy and efficiency of alarm systems are of paramount importance in safe operations of industrial processes. Accuracy is measured by false and missed alarm rates (probabilities); while efficiency relates to the detection delay and complexity of the technique used. Moving average filters are often employed in industry for improved alarm accuracy. Can one do better than moving average filters? The following two problems are studied in this paper: First, given both normal and abnormal statistic distributions, how to design an optimal alarm filter (of fixed complexity) for best alarm accuracy, minimizing a weighted sum of false and missed alarm rates? Second, in what cases are moving average filters optimal? For the first problem, design of optimal linear FIR alarm filters is studied, and a numerical optimization based procedure is proposed. For the second problem, a sufficient condition is given under which the moving average filters are optimal.

Solution of inverse variation problem in the theory of optimal linear filtering

It is presented is the possibility to solve the inverse variation problem based on the determination of a primary functional by the results of minimum mean-square error in optimal linear filtering of the signal theory.

A Comparative Study of Amplitude and Timing Estimation in Experimental Particle Physics using Monte Carlo Simulation

Optimal detection of liquid ionization calorimeter signal in experimental particle physics is considered. A few linear and nonlinear approaches for amplitude and arrival time estimation based on the χ2 function are compared in simulation considering the noise sample correlation introduced by the analog pulse shaper. The estimation bias of the first-order approximation, a.k.a linear optimal filtering, is studied and contrasted to those of the second-order as well as the exhaustive search. A gradient-descent technique is presented as an alternative to the exhaustive search with significantly reduced search time and computation complexity. Results from various pulse shapers including the CR-RC2, CR-RC3, and CR2-RC2 are also compared.

Efficient and Accurate Optimal Linear Phase FIR Filter Design Using Opposition-Based Harmony Search Algorithm

In this paper, opposition-based harmony search has been applied for the optimal design of linear phase FIR filters. RGA, PSO, and DE have also been adopted for the sake of comparison. The original harmony search algorithm is chosen as the parent one, and opposition-based approach is applied. During the initialization, randomly generated population of solutions is chosen, opposite solutions are also considered, and the fitter one is selected as a priori guess. In harmony memory, each such solution passes through memory consideration rule, pitch adjustment rule, and then opposition-based reinitialization generation jumping, which gives the optimum result corresponding to the least error fitness in multidimensional search space of FIR filter design. Incorporation of different control parameters in the basic HS algorithm results in the balancing of exploration and exploitation of search space. Low pass, high pass, band pass, and band stop FIR filters are designed with the proposed OHS and other aforementioned algorithms individually for comparative optimization performance. A comparison of simulation results reveals the optimization efficacy of the OHS over the other optimization techniques for the solution of the multimodal, nondifferentiable, nonlinear, and constrained FIR filter design problems.

Optimal Fusion Filtering in Multisensor Stochastic Systems with Missing Measurements and Correlated Noises

The optimal least-squares linear estimation problem is addressed for a class of discrete-time multisensor linear stochastic systems with missing measurements and autocorrelated and cross-correlated noises. The stochastic uncertainties in the measurements coming from each sensor (missing measurements) are described by scalar random variables with arbitrary discrete probability distribution over the interval[0,1]; hence, at each single sensor the information might be partially missed and the different sensors may have different missing probabilities. The noise correlation assumptions considered are (i) the process noise and all the sensor noises are one-step autocorrelated; (ii) different sensor noises are one-step cross-correlated; and (iii) the process noise and each sensor noise are two-step cross-correlated. Under these assumptions and by an innovation approach, recursive algorithms for the optimal linear filter are derived by using the two basic estimation fusion structures; more specifically, both centralized and distributed fusion estimation algorithms are proposed. The accuracy of these estimators is measured by their error covariance matrices, which allow us to compare their performance in a numerical simulation example that illustrates the feasibility of the proposed filtering algorithms and shows a comparison with other existing filters.

Two-Dimensional Tomography from Noisy Projections Taken at Unknown Random Directions.

Computerized tomography is a standard method for obtaining internal structure of objects from their projection images. While CT reconstruction requires the knowledge of the imaging directions, there are some situations in which the imaging directions are unknown, for example, when imaging a moving object. It is therefore desirable to design a reconstruction method from projection images taken at unknown directions. Another difficulty arises from the fact that the projections are often contaminated by noise, practically limiting all current methods, including the recently proposed diffusion map approach. In this paper, we introduce two denoising steps that allow reconstructions at much lower signal-to-noise ratios (SNRs) when combined with the diffusion map framework. In the first denoising step we use principal component analysis (PCA) together with classical Wiener filtering to derive an asymptotically optimal linear filter. In the second step, we denoise the graph of similarities between the filtered projections using a network analysis measure such as the Jaccard index. Using this combination of PCA, Wiener filtering, graph denoising, and diffusion maps, we are able to reconstruct the two-dimensional (2-D) Shepp-Logan phantom from simulative noisy projections at SNRs well below their currently reported threshold values. We also report the results of a numerical experiment corresponding to an abdominal CT. Although the focus of this paper is the 2-D CT reconstruction problem, we believe that the combination of PCA, Wiener filtering, graph denoising, and diffusion maps is potentially useful in other signal processing and image analysis applications.

Optimal Linear Estimation for Networked Systems with One-step Random Delays and Multiple Packet Dropouts

This paper is concerned with optimal linear estimation problem for networked systems with random delays and packet dropouts.Two Bernoulli distributed random variables are employed to describe the phenomena of possible one-step random delay and multiple packet dropouts in data transmission by networks.Based on an innovative analysis approach,the optimal linear filter,predictor and smoother for the state are presented in a linear minimum variance sense.They are solved based on a Riccati equation and a Lyapunov equation.A sufficient condition for the existence of the steady-state estimators is given.A simulation example verifies their effectiveness.

Adaptive Generalized Likelihood Ratio Control Charts for Detecting Unknown Patterned Mean Shifts

Traditional statistical control schemes have been mainly focused on detecting constant mean shifts. In many practical applications, however, the mean of the observed sequence can exhibit a time-varying behavior after the fault occurrence. Shewhart control charts supplemented with sensitizing run rules have been suggested for detecting nonrandom dynamic mean patterns. However, the choice of a particular set of run rules requires some prior knowledge of the possible expected patterns. In addition, the resulting schemes can have poor performance against small shifts. When the mean shift pattern is known in advance, generalized likelihood ratio (GLR), cumulative score (CUSCORE), and optimal general linear filter (OGLF) control charts have also been proposed for change-point detection. Further, some adaptive CUSCORE schemes have been recently developed for detecting an unknown patterned mean shift. However, these adaptive CUSCOREs assume the occurrence of one-sided mean shifts. Hence, their performance may be poor in the case of an oscillatory behavior of the process mean. To overcome these limitations, we propose estimating a possible pattern in the mean using an exponentially weighted moving average (EWMA), which is especially effective with one-sided shifts in the mean, combined with a wavelet smoother, which is effective in estimating oscillatory mean patterns. The two estimates then drive two separate conventional GLR tests for significance. A control chart is proposed for observations from normal distributions and then extended to a completely distribution-free setting. Extensive simulation results demonstrate the efficiency of the proposed charts with a wide variety of patterned mean shifts in both the independent and autocorrelated scenarios and with a wide variety of distributions. In addition, the proposed scheme provides post-signal diagnostic information that estimates the change-point location and the shift pattern of the mean. An R package is available online as supplementary material.