Wiener Filter Design Research Articles

Robust Recursive Least-Squares Wiener Filter for Linear Continuous-Time Uncertain Stochastic Systems

For linear continuous-time systems with uncertainties in the system and observation matrices, an original robust RLS Wiener filter is designed in this study. The robust RLS Wiener filter does not assume norm-bounded uncertainty for the system and observation matrices, in contrast to the robust Kalman filter. In the design of the robust RLS Wiener filter, the degraded signal, affected by the uncertainties in the system and observation matrices, is modeled by an autoregressive (AR) model. The system and observation matrices for the degraded signal are formulated from the relationship between the AR model of the degraded signal and the state-space model. Estimation formulas for the system and observation matrices are proposed in Section 2. The robust filtering problem is introduced based on the minimization of the mean-square value of the filtering errors for the system states. The robust filtering estimate is given as an integral transformation of the degraded observations using the impulse response function. The integral equation that an optimal impulse response function satisfies is given in Section 3. Theorem 1 presents a robust RLS Wiener filtering algorithm starting from this integral equation. The proposed robust RLS Wiener filter outperforms the existing robust Kalman filter regarding estimate accuracy, as shown by a numerical simulation example.

High-Efficiency Image Coding via Near-Optimal Filtering

Wiener filtering, which has been widely used in the field of image restoration, is statistically optimal in the sense of mean square error. The adaptive loop filter in video coding inherits the design of Wiener filters, and has been proved to achieve significant improvement on compression performance by reducing coding artifacts and providing high-quality references for subsequent frames. To further improve the compression performance via filtering technique, we explore the factors that may hinder the potential performance of Wiener-based filters, and propose a near-optimal filter learning scheme for high-efficiency image coding. Based on the analyses, we observe that the foremost factor affecting the performance of Wiener-based filters is the divergence of statistical characteristics of training samples, instead of the filter taps or shapes. In view of this, we propose an iterative training method to derive the near-optimal Wiener filter parameters by simultaneously labeling sample categories at the pixel level. These parameters are compressed and transmitted to the decoder side to improve the quality of decoded images by reducing the coding artifacts. Experimental results show that the proposed scheme achieves significant bitrate savings compared with high-efficiency video coding in high-bitrate intra coding scenario.

Causal Restriction and Its Generalization for the Wiener Filter

The classical principles of Wiener filter design consider only the causality constraint. However, additional constraints are often involved in practice. This paper provides a comprehensive theoretical approach for designing Wiener filters with constraint conditions beyond causality. Using integration, we first derive an extension of the Wiener–Hopf equation with an additional term that captures the effect of the constraint conditions on the resulting Wiener filters. Next, we analytically solve the extended Wiener–Hopf equation and develop a new iterative algorithm to obtain its optimum value. We prove the convergence of our iterative algorithm and provide several simulations to illustrate the applicability of our theoretical approach.

Design of RLS Wiener Smoother and Filter from Randomly Delayed Observations in Linear Discrete-Time Stochastic Systems

This paper presents the new algorithm of the recursive least-squares (RLS) Wiener fixed-point smoother and filter based on the randomly delayed observed values by one sampling time in linear discrete- time wide-sense stationary stochastic systems. The observed value

Design of RLS Wiener Smoother and Filter for Colored Observation Noise in Linear Discrete-Time Stochastic Systems

Almost estimators are designed for the white observation noise. In the estimation problems, rather than the white observation noise, there might be actual cases where the observation noise is modeled by the colored noise process. This paper examines to design a new estimation technique of recursive least-squares (RLS) Wiener fixed-point smoother and filter for colored observation noise in linear discrete-time wide-sense stationary stochastic systems. The observation y(k) is given as the sum of the signal z(k)=Hx(k) and the colored observation noise vc(k). The RLS Wiener estimators explicitly require the following information: 1) the system matrix for the state vector x(k); 2) the observation matrix H; 3) the variance of the state vector x(k); 4) the system matrix for the colored observation noise vc(k); 5) the variance of the colored observation noise; 6) the input noise variance in the state equation for the colored observation noise.

Adaptive minimum error-rate filtering design: A review

Adaptive filtering has been an enabling technology and has found ever-increasing applications in various state-of-the-art communication systems. Traditionally, adaptive filtering has been developed based on the Wiener or minimum mean square error (MMSE) approach, and the famous least mean square algorithm with its low computational complexity readily meets the fast real-time computational constraint of modern high-speed communication systems. For a communication system, however, it is the system's bit error rate (BER), not the mean square error (MSE), that really matters. It has been recognised that minimising the MSE criterion does not necessarily produce the minimum BER (MBER) performance. The introduction of the novel MBER design has opened up a whole new chapter in the optimisation of communication systems, and its design trade-offs have to be documented in contrast to those of the classic but actually still unexhausted MMSE and other often-used optimisation criteria. This contribution continues this theme, and we provide a generic framework for adaptive minimum error-probability filter design suitable for the employment in a variety of communication systems. Advantages and disadvantages of the adaptive minimum error-probability filter design are analysed extensively, in comparison with the classic Wiener filter design.

On the behavior of causal projections with applications

This paper investigates the behavior of causal projections. Such projections play an important role in filter design, system approximation and representation, spectral factorization, Wiener filter design, robust estimation, and in many more applications. The paper gives a complete characterization of function spaces for which the causal projections are continuous and bounded. This characterization is done in terms of the modulus of continuity of the functions. It is shown that the Riesz projector has the smallest operator norm among all causal projectors and that the Riesz projector is bounded precisely for those functions for which the modulus of continuity is upper bounded by a regular majorant. Moreover, in the second part of the paper, consequences of these results for the construction of bases, for robust approximations, for the behavior of numerical algorithms, and for filter design applications are discussed in detail.

Behaviour of the spectral factorization for continuous spectral densities

Spectral factorization plays an important role in many applications such as Wiener filter design, prediction, and estimation. It is known that spectral factorization is a non-continuous mapping on the space of all non-negative continuous functions, satisfying the Paley–Wiener condition. Additionally, it will be shown in this paper that every continuous spectrum is a discontinuity point of the spectral factorization. As a consequence, small perturbations in the given data can lead to large errors in the calculated spectral factors. Practical algorithms for the calculation of the spectral factor can only use a finite number of Fourier coefficients of the given spectrum. Consequently, the error in this finite number of coefficients can yield only a finite error in the calculated spectral factor. The paper provides sharp lower and upper bounds for this error. These bounds show that this error grows proportional with the logarithm of the number N of Fourier coefficients which are taken into account.

Wiener design of adaptation algorithms with time-invariant gains

A design method is presented that extends least mean squared (LMS) adaptation of time-varying parameters by including general linear time-invariant filters that operate on the instantaneous gradient vector. The aim is to track time-varying parameters of linear regression models in situations where the regressors are stationary or have slowly time-varying properties. The adaptation law is optimized with respect to the steady-state parameter error covariance matrix for time-variations modeled as vector-ARIMA processes. The design method systematically uses prior information about time-varying parameters to provide filtering, prediction, or fixed lag smoothing estimates for arbitrary lags. The method is based on a transformation of the adaptation problem into a Wiener filter design problem. The filter works in open loop for slow parameter variations, whereas a time-varying closed loop has to be considered for fast variations. In the latter case, the filter design is performed iteratively. The general form of the solution at each iteration is obtained by a bilateral Diophantine polynomial matrix equation and a spectral factorization. For white gradient noise, the Diophantine equation has a closed-form solution. Further structural constraints result in very simple design equations. Under certain model assumptions, the Wiener designed adaptation laws reduce to LMS adaptation. Compared with Kalman estimators, the channel tracking performance becomes nearly the same in mobile radio applications, whereas the complexity is, in general, much lower.

Non-parametric Wiener filter for reducing noise on reproducible pure signals

This paper communicates a novel method of Wiener filtering signals from scientific instruments to reduce their noise content. The main prerequisite for the applicability of this technique is that the pure (i.e. noiseless) signal is reproducible, a condition that is satisfied by a wide range of experimental measurements. The benefits of this Wiener filter design approach are its simplicity, generality and practicality. In particular, signal-dependent or multiplicative noise are accommodated without complication. The non-parametric filter does not require models of pure signal or noise statistics, and is exactly optimal for the observed (i.e. noisy) signal ensemble. Implementation of this Wiener filter only requires measurement of observed signal correlation functions. After deriving the classical stationary signal Wiener filter, the analysis is extended to derive the reproducible stationary pure signal Wiener filter. A distinction is made between non-adaptive Wiener filters derived from correlations computed as ensemble averages, and adaptive Wiener filters derived from correlations computed as time averages. The analysis is extended again to encapsulate non-stationary signal ensembles and further extended to synthesize Wiener filters based on statistically good observed signal correlation estimates for arbitrarily many pure signal reproductions.

Game theory approach to discrete H/sub ∞/ filter design

A finite-horizon discrete H/sub /spl infin// filter design with a linear quadratic (LQ) game approach is presented. The exogenous inputs composed of the hostile noise signals and system initial condition are assumed to be finite energy signals with unknown statistics. The design criterion is to minimize the worst possible amplification of the estimation error signals in terms of the exogenous inputs, which is different from the classical minimum variance estimation error criterion for the modified Wiener or Kalman filter design. The approach can show how far the estimation error can be reduced under an existence condition on the solution to a corresponding Riccati equation. A numerical example is given to compare the performance of the H/sub /spl infin// filter with that of the conventional Kalman filter.

Design of Wiener filters using a cumulant based MSE criterion

This paper proposes a cumulant (higher-order statistics) based mean-square-error (MSE) criterion for the design of Wiener filters when both the given wide-sense stationary random signal x( n) and the desired signal d( n) are non-Gaussian and contaminated by Gaussian noise sources. It is theoretically shown that the designed Wiener filter associated with the proposed criterion is identical to the conventional correlation (second-order statistics) based Wiener filter as if both x( n) and d( n) were noise-free measurements. As the latter, the former can also be obtained by solving a cumulant-based Wiener-Hopf equation associated with a (cumulant-based) orthogonality principle. Then a generalized cumulant projection theorem is proposed which includes the projection of cumulants to correlations associated with the proposed cumulant-based MSE criterion and that associated with Delopoulos and Giannakis' cumulant-based MSE criterion as special cases. Moreover, the proposed cumulant-based MSE criterion and Delopoulos and Giannakis' cumulant-based MSE criterion are equivalent for cumulant order M = 3. Some simulation results for system identification and time delay estimation are then provided to demonstrate the good performance of the proposed cumulant-based Wiener filter. Finally, we draw some conclusions.

Multichannel Wiener filters in color image restoration

Deals with the design of multichannel Wiener filters both in the spatial and frequency domain. FIR and IIR Wiener filters are presented. Color image restoration based on multi-channel autoregressive (AR) image modeling is examined. Detailed discussions on the use of a multichannel Wiener filter in color image restoration incorporating the interchannel correlations and computer simulations are presented also. >

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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Corrections to 'Fast design of direct and ladder Wiener filters with linear phase'

Corrections to 'Fast design of direct and ladder Wiener filters with linear phase'

Fast design of direct and ladder Wiener filters with linear phase

In this paper, the problem of computationally efficient design of finite impulse response Wiener filters with linear phase is considered. For stationary signals these filters are determined by the solution of normal systems of equations whose associated matrix has a Toeplitz-plus (or minus)-Hankel structure. A fast order recursive algorithm for solving these normal equations is developed. The componentparts of this scheme are the Levinson algoritun and an elegant relationship connecting Wiener filters and Wiener filters with linear phase. Finally, it is shown that the well known two multiplier lattice structure introduced by Itakura and Saito can be used in a natural way to realize a ladder Wiener filter with linear phase.

Linear inverse theory and deconvolution

Seismic source wavelet deconvolution can be treated within the framework of the Backus‐Gilbert (BG) inverse theory. A time shift‐invariant version of this theory leads to the Wiener shaping filter, which has enjoyed widespread use for source wavelet deconvolution in exploration seismology. The model of the BG theory is the ground impulse response, the BG mapping kernel is the source wavelet, and the BG resolving kernel is the convolution between the source wavelet and the Wiener shaping filter. BG inversion involves the minimization of an optimality criterion under a set of constraints. The application of the BG “filter energy” or “noise output power” constraint to Wiener filter design leads to the familiar prewhitening parameter that stabilizes the filter on the one hand, but degrades resolution on the other. The BG “unimodular” constraint produces an unbiased estimate of the model, or ground impulse response. These constraints provide novel insights into the performance of deconvolution filters.

A novel approach to seismic signal processing and modeling

This paper demonstrates some results obtained using state‐variable models and techniques on problems for which solutions either cannot be or are not easily obtained via more conventional input‐output techniques. After a brief introduction to state‐variable notions, the following seven problem areas are discussed: modeling seismic source wavelets, simultaneous deconvolution and correction for spherical divergence, simultaneous wavelet estimation and deconvolution, well log processing, design of recursive Wiener filters, Bremmer series decomposition of a seismogram (including suppression of multiples and vertical seismic profiling), and estimating reflection coefficients and traveltimes.

On the general theory of control systems

This paper deals with further advances of the author's recent work on optimal design of control systems and Wiener filters. Specifically, we consider the problem of designing a system to control a plant when (1) not all state variables are measurable, (2) the measured state variables are contaminated with noise, and (3) there are random disturbances. An explicit design procedure (well adapted to digital computation) is presented. In addition, some fundamental new concepts (controllability, observability, etc.) are introduced. A general theory of control systems is outlined which answers many basic questions (what is controllable? why? how?) and gives a highly efficient method of computation. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.

On the general theory of control systems

This paper deals with further advances of the author's recent work on optimal design of control systems and Wiener filters. Specifically, we consider the problem of designing a system to control a plant when (1) not all state variables are measurable, (2) the measured state variables are contaminated with noise, and (3) there are random disturbances. An explicit design procedure (well adapted to digital computation) is presented. In addition, some fundamental new concepts (controllability, observability, etc.) are introduced. A general theory of control systems is outlined which answers many basic questions (what is controllable? why? how?) and gives a highly efficient method of computation. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.