Polynomial Filters Research Articles

Performance Comparison of Several Non-Linear Equalizers in the Context of Mobile Telecommunications

This paper provides a comparative study of several non-linear blind equalizers in terms of computational requirements and of the transient performance, which is the main criterion to be considered in the context of time-varying channels. Computational requirements are estimated as the number of real additions and multiplications associated with the training algorithm, whereas the transient performance is evaluated in terms of convergence time. The impact of local minima on the tracking ability is carefully evaluated, in terms of a suitable criterion proposed in the paper. Simulations involve both single-layer (linear and polynomial filters), as well as multilayer structures (radial basis function, recurrent network, multilayer perceptron). These techniques are applied to the blind equalization of mobile terrestrial and satellite channels. Guidelines are established for the choice of a suitable structure as the major trade-off to be achieved in a mobile context is between computational effort and robustness of tracking capability (with respect to local minima effects).

Effect of nonlinear filters on detrended fluctuation analysis

When investigating the dynamical properties of complex multiple-component physical and physiological systems, it is often the case that the measurable system's output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms. Instead, the output signal is often a linear or nonlinear function of the quantity of interest. Here, we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal, using the detrended fluctuation analysis (DFA) which has been shown to accurately quantify power-law correlations in nonstationary signals. Specifically, we study the effect of three types of transforms: (i) linear ( y(i) =a x(i) +b) , (ii) nonlinear polynomial ( y(i) =a x(k)(i) ) , and (iii) nonlinear logarithmic [ y(i) =log ( x(i) +Delta) ] filters. We compare the correlation and scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power k of the polynomial filter, and (c) the offset Delta in the logarithmic filter. We further apply the DFA method to investigate the "apparent" scaling of three analytic functions: (i) exponential [exp (+/-x+a) ] , (ii) logarithmic [log (x+a) ] , and (iii) power law [ (x+a)(lambda) ] , which are often encountered as trends in physical and biological processes. While these three functions have different characteristics, we find that there is a broad range of values for parameter a common for all three functions, where the slope of the DFA curves is identical. We further note that the DFA results obtained for a class of other analytic functions can be reduced to these three typical cases. We systematically test the performance of the DFA method when estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider.

Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computations

Krylov subspace methods have led to reliable and effective tools for resolving large-scale, non-Hermitian eigenvalue problems. Since practical considerations often limit the dimension of the approximating Krylov subspace, modern algorithms attempt to identify and condense significant components from the current subspace, encode them into a polynomial filter, and then restart the Krylov process with a suitably refined starting vector. In effect, polynomial filters dynamically steer low-dimensional Krylov spaces toward a desired invariant subspace through their action on the starting vector. The spectral complexity of nonnormal matrices makes convergence of these methods difficult to analyze, and these effects are further complicated by the polynomial filter process. The principal object of study in this paper is the angle an approximating Krylov subspace forms with a desired invariant subspace. Convergence analysis is posed in a geometric framework that is robust to eigenvalue ill-conditioning, yet remains relatively uncluttered. The bounds described here suggest that the sensitivity of desired eigenvalues exerts little influence on convergence, provided the associated invariant subspace is well-conditioned; ill-conditioning of unwanted eigenvalues plays an essential role. This framework also gives insight into the design of effective polynomial filters. Numerical examples illustrate the subtleties that arise when restarting non-Hermitian iterations.

Optimal Polynomial Filters

In this paper, we present a family of circular or square optimal polynomial filters for prefiltering two-dimensional polygons and images. The criterion of designing polynomial filters is to maximiz...

A FAMILY OF POLYNOMIAL FILTERS FOR DISCRETE-TIME NONLINEAR STOCHASTIC SYSTEMS

This work presents a family of polynomial filters for discrete-time nonlinear stochastic systems. These filters can be considered the polynomial version of the well known Extended Kalman Filter. The standard EKF consists in the optimal linear filter applied to the linear approximation of systems. The filters presented in this paper are polynomial filters applied to polynomial approximations of nonlinear systems, and therefore each of them is characterized by a pair of integers: the degree of the system approximation and the degree of the filter. The first filter of the family, the one of order (1,1), coincides with the EKF in the standard form. The implementation of the proposed filters does not require the complete knowledge of the noise distributions, but only the moments up to suitable orders. Numerical simulations show the performances of the filters for some values of the degrees.

Problems and Techniques

We are fortunate to have, in this issue's Problems and Techniques section, three papers that should be of interest to all SIAM Review readers. The first paper, "A Fast and Stable Solution Method for the Radiative Transfer Problem," by P. Edstrom, treats the interaction of radiation with turbid media, an important and longstanding scientific problem. (The first citation is a 1905 paper on radiation through a foggy atmosphere.) For readers unfamiliar with the term, in common parlance something "turbid" contains foreign particles that are stirred up or suspended, as in "I leave a white and turbid wake" (Chapter 37 of Moby Dick); in the present context, a turbid medium is highly scattering. Transmission as well as absorption and multiple scattering must be considered in a turbid medium, and several steps are needed to obtain an appropriate mathematical formulation and efficient numerical solution methods. The paper guides the reader through the motivation for and connections among the resulting multiple procedures: application of Fourier analysis to a fundamental integrodifferential equation, discretization using numerical quadrature, transformation to an algebraic eigenvalue problem, and preconditioning strategies for the equations that define boundary and continuity conditions. Special attention is given to techniques for speeding up the computation, and the solution method described has been implemented in a MATLAB code used in the paper and printing industries. This paper features references that are unusually broad in time (starting with an 1871 paper by Lord Rayleigh) and application (ranging over optics, astrophysics, neutron transport, nuclear engineering, and atmospheric science). The second paper, "Reviving the Method of Particular Solutions," by T. Betcke and L. N. Trefethen, presents a readable and engrossing tour of the authors' successful quest to understand and overcome the difficulties experienced by the method of particular solutions (MPS) defined in a classic 1967 paper from the SIAM Journal on Numerical Analysis by L. Fox, P. Henrici, and C. Moler. As carefully explained in the present paper, the problems are caused by the MPS's inability to ensure that an eigenfunction is nonzero in the interior of the domain; ambiguities and ill-conditioning arise from the existence of linear combinations of functions that are close to zero everywhere, even when the associated parameter is not an eigenvalue. A modified method, based on constraining the problem so that admissible functions are bounded away from zero in the interior, is presented that allows reliable detection of spurious approximate eigenfunctions. (The contrast between Figures 4.1 and 5.4 tells the story.) The authors then go on to describe both a geometric interpretation and the connections between the constrained minimization problem and angles between subspaces. Kac's famous 1966 question "Can one hear the shape of a drum?" is used as an example for the modified MPS on several isospectral examples, including an H-shaped domain. The paper also contains a discussion of error bounds, generalizations, open questions, and very recent related work. Krylov subspace methods for linear systems, featured in last issue's Problems and Techniques section, appear again, this time for eigenvalue problems, in our third paper, "Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computations," by C. A. Beattie, M. Embree, and D. C. Sorensen. Like the other two papers, this paper first presents an illuminating summary of both history and state-of-the-art before moving to its heart: analysis of the angles made by an approximating Krylov subspace of a matrix A with a desired invariant subspace, and convergence of polynomial restarting methods that project A onto a Krylov subspace and modify the starting vector using a polynomial in A. Setting an engaging tone by declaring their interest only in the "good" eigenvalues, the paper provides an explanation of how a Krylov space might converge to an associated good invariant subspace, as well as a discussion of how to assess the proximity of subspaces. After defining the "maximal reachable invariant subspace," the authors consider both the gap between this subspace and the $\ell$th Krylov space as $\ell$ increases, and the dependence of the gap on the polynomial that defines a restart filter. Among the results of this analysis are insights into the design of effective polynomial filters and numerical confirmation of the complications that arise when restarting iterations for nonnormal matrices. The reader is encouraged to examine the pseudospectra in Figures 4.8 and 4.10 for visible evidence of these subtleties.

Regional–residual Gravity Field Separation in the Central Andes using Global Geopotential Models

Traditional methods of regional?residual gravity field separation typically involve high-pass filtering, low-order polynomial fitting or computations using isostatic models. All of these methods are susceptible to ambiguity and subjectivity in the selection of filter parameters, polynomial degree and isostatic model. Global Geopotential Models (GGMs), on the other hand, provide long-wavelength gravity data that are based on observations. Therefore, they have the potential to be used to more objectively define the regional gravity field and to give residual anomalies with reduced ambiguity. We have examined residual gravity anomalies in the Central Andes computed using a regional field defined by the recent EIGEN?GRACE01S satellite-only GGM. The resulting residual anomalies show significant correlations with the active volcanic arc and the geometry of the subducting Nazca plate. These results suggest that GGMs have a place in regional gravity interpretation, but the advantages over existing techniques remain to be thoroughly tested.

D02: An algorithm for MSE optimal CSD terms allocation

In the paper easy and smart algorithm for FIR filter coefficients quantization is presented. In the first step digital FIR filter coefficients are computed using any method known from literature. In the next step filter polynomial is factored and the filter is transformed into two cascaded FIR sections. Coefficients of the first section is quantized. In the article simple optimal quantization method is proposed. Coefficients of the next section are computed to minimize MSE. New coefficients of the second section are quantized. The algorithm is illustrated with lowpass, windowed FIR filter.

Polynomial Filtering for Stochastic Non-Gaussian Descriptor Systems

The class of stochastic descriptor systems, also named singular systems, has been widely investigated and many important results in the linear filtering theory have been achieved in the framework of Gaussian processes. Nevertheless, such results could be far from optimal, especially in the case of highly asymmetrical non-Gaussian noises. This paper solves the estimation problem for stochastic singular systems affected by non-Gaussian noises by means of a polynomial filtering algorithm based on the minimum variance criterion. The performance of the polynomial filter can be improved by increasing its degree. The filter structure is such to give back the optimal filter in the case of Gaussian noise, thus yielding a first-order polynomial filter. In the non-Gaussian case, the improvement of the polynomial filter can be highly significative, especially when the noise distribution is strongly asymmetrical. Simulations support theoretical results.

Adaptive-degree polynomial filter for voltammetric signals

The adaptive-degree polynomial filter (ADPF) for data smoothing is further improvement of the least-squares regression formalism introduced to the experimental data processing by Savitzky and Golay. The adaptive approach bases on the statistical testing of the fitting quality of polynomial function to the experimental data in the smoothing window sliding along the curve. It allows the automatic choice of the proper degree of the polynomial function in the various parts of the experimental curve. The sets of the simulated according to the Feldberg model voltammetric curves reflecting calibration process are considered. The normal random noises of different level were added to simulated curves and next processed with ADPF and fixed-degree polynomial filter. The performance of smoothing methods were tested by means of the sum of squared errors, signal to noise ratio and improvement of the analytical parameters such as: detection limit, sensitivity, linearity and other.

Polynomial distance classifier correlation filter for pattern recognition.

We introduce what is to our knowledge a new nonlinear shift-invariant classifier called the polynomial distance classifier correlation filter (PDCCF). The underlying theory extends the original linear distance classifier correlation filter [Appl. Opt. 35, 3127 (1996)] to include nonlinear functions of the input pattern. This new filter provides a framework (for combining different classification filters) that takes advantage of the individual filter strengths. In this new filter design, all filters are optimized jointly. We demonstrate the advantage of the new PDCCF method using simulated and real multi-class synthetic aperture radar images.

Design of linear-phase variable 2-d digital filters using matrix-array decomposition

A novel approach for designing linear-phase variable two-dimensional (2-D) digital filters is described, which is based on the matrix-array decomposition (MAD) of real-valued multidimensional arrays. By using the MAD, the desired variable 2-D magnitude response can be decomposed into the real-valued frequency response specifications of the normal zero-phase constant 2-D filters and the approximation specifications of multidimensional polynomials. Consequently, the problem of approximating the desired variable 2-D magnitude response can be decomposed into sub-problems that involve the design of zero-phase constant 2-D filters and the approximation of multidimensional polynomials. Once the zero-phase constant 2-D filters and the multidimensional polynomials are obtained, interconnecting them yields a zero-phase variable 2-D digital filter. Finally, a linear-phase variable 2-D digital filter can be easily obtained by simply modifying the zero-phase constant 2-D filters (noncausal) to linear-phase ones (causal) through shifting the filter coefficients. Since the sub-problems are much easier to solve than the direct approximation of the given variable 2-D magnitude specification, this MAD-based design approach is extremely efficient and straightforward. In designing zero-phase constant 2-D filters and approximating multidimensional polynomials, we also propose a new objective criterion for selecting appropriate filter orders and polynomial degrees. Furthermore, we also show that the resulting linear-phase variable 2-D filters have highly parallel structures and modularity, which are suitable for high-speed multidimensional signal processing. Two numerical examples are given to demonstrate the effectiveness of the MAD-based design approach.

Polynomial Filtering With Uncertain Observations in Stochastic Linear Systems

In this article we consider the least mean-squared error polynomial estimation problem in systems with uncertain observations. For this purpose we define an augmented system such that the optimal linear estimator of the augmented state based on the augmented observations provides the optimal polynomial estimator for the state of the original system. This augmented system satisfies the necessary conditions to apply the Nahi algorithm, which provides the desired optimal linear filter. Stationary systems are also treated, and we show that the augmented system is asymptotically stationary provided that the original system is asymptotically stable. So, by applying the steady-state form of the Nahi algorithm, we obtain the steady-state polynomial filter of the original state.

A Two-Dimensional Polynomial Filter

The criterion of minimal variance of transformation error is applied to design an nth-order nonlinear polynomial filter for image processing under given moments of order up to 2n of the two-dimensional input signal and additive disturbance. Structurally, the filter is represented by Hammerstein kernels (weight functions) determined from the solution of a system of two-dimensional linear integral equations. For a linear filter, this system is reduced to a two-dimensional Wiener–Hopf equation. The filter accuracy is shown to increase with the filter order n (in any case, it does not decrease). The filter performance is illustrated for practical operation conditions of information transmission channels.

Difference Wavelet - Theory and A Comparison Study

Wavelet methods with polynomial filters are usually favored in applications for their fast wavelet transforms and compact support. However, wavelet methods with rational filters have more freedom to achieve smaller condition numbers, more regularity and better efficiency. Such meth- ods can be attractive if they also possess fast algorithms and have fast decay (as if the corresponding wavelets had compact support). In the first part of this paper, we propose a new wavelet method with rational filters which do have these properties. We call it the difference wavelet method. It is a generalization of Butterworth wavelets. The analysis part is simply averaging and finite differencing. The wavelet coefficients measure the finite differences of the a of an input data sequence. Its synthesis part involves rational filters, which can be performed with linear computational complexity by the cyclic reduction method. Their Riesz basis property, biorthogonality, decay and regularity are investigated. In the second part of this paper, we perform comparison studies of the difference wavelet method (Diff) with three other popular wavelet methods: the Cohen-Daubechies-Feauveau biorthogonal wavelets (CDF), the Daubechies orthogonal wavelets (Daub) and the Chui-Wang semi-orthogonal wavelets (CW). Natural criteria in designing good wavelet methods for representing functions and operators are speed, stability and efficiency. Therefore, the items of our first comparison include (i) operation counts for performing transformations, (ii) condition numbers of the wavelet trans- formations, (iii) compression ratios, by some numerical experiments, for representing (smooth or non-smooth) data sequences and matrices (smooth or non-smooth kernels). The results show that (i) Diff, Daub and CDF have about the same operation counts, and CW has more; (ii) Diff has about the same condition numbers as those of CDF and CW; (iii) Diff has better compression ratio for both (smooth or non-smooth) data sequences and matrices (smooth or non-smooth kernels). The items of our second comparison include regularity, approximation power (the constant in the approximation estimate), approximation errors for non-smooth functions (where Gibbs phenomena appear) and the The results show that Diff has better regularity and better approximation ability with only slightly bigger essential supports. It is evident that the better efficiency of Diff for smooth functions is due to its regularity. It is surprising that, even for non- smooth functions, Diff is comparable to, sometimes even superior to, other methods, despite its infinite-support property. This paper is organized as follows. Sec. 1 is preliminary. Sec. 2 provides the theory of the difference wavelet method. Sec. 3 contains the comparison studies. Experts are suggested to read Sec. 3 directly.

Energy Projection in Polynomial Correlation Filters

In this paper we present a new version of the polynomial correlation filter (PCF) called constrained polynomial correlation filter (CPCF). We investigate the performance of this filter in the presence of clutter noise. The peak-to-sidelobe ratio measure is evaluated for public MSTAR images. The effect of different terms in the polynomial filter is examined by simulation. Then, we introduce a theoretical framework called energy projection to predict the effectiveness of different terms in the CPCF.

Risk evaluation of ventricular tachycardia using wavelet transform irregularity of the high-resolution electrocardiogram.

A new method for analysis of high-resolution ECG signals using a wavelet transform based on a modified Morlet function is presented. A polynomial filter is used to reduce low-frequency, high-amplitude noise components in the analysed signals. The method is tested on test ECG signals with simulated late potentials and finally verified on two post-infarction patient (PP) groups: 62 PPs with ventricular tachycardia and 44 PPs without arrhythmia. A new quantitative parameter, the irregularity factor, is proposed for discrimination between the study groups. The results show a significant difference in the parameter values for tachycardia patients compared with those for patients without arrhythmia. The sensitivity of the proposed method is 85%, and the specificity is 93%.

3D mesh geometry filtering algorithms for progressive transmission schemes

A number of static and multi-resolution methods have been introduced in recent years to compress 3D meshes. In most of these methods, the connectivity information is encoded without loss of information, but user-controllable loss of information is tolerated while compressing the geometry and property data. All these methods are very efficient at compressing the connectivity information, in some cases to a fraction of a bit per vertex, but the geometry and property data typically occupies much more room in the compressed bitstream than the compressed connectivity data. In this paper, we investigate the use of polynomial linear filtering as studied in the Refs. [Taubin G. A signal processing approach to fair surface design. Computer Graphics Proc., Annual Conference Series 1995;351–358; Taubin G, Zhang T, Golub G. Optimal surface smoothing as filter design. IBM Research report RC–20,404, 1996], as a global predictor for the geometry data of a 3D mesh in multi-resolution 3D geometry compression schemes. Rather than introducing a new method to encode the multi-resolution connectivity information, we choose one of the efficient existing schemes depending on the structure of the multi-resolution data. After encoding the geometry of the lowest level of detail with an existing scheme, the geometry of each subsequent level of detail is predicted by applying a polynomial filter to the geometry of its predecessor lifted to the connectivity of the current level. The polynomial filter is designed to minimize the l2-norm of the approximation error but other norms can be used as well. Three properties of the filtered mesh are studied next: accuracy, robustness and compression ratio. The Zeroth Order Filter (unit polynomial) is found to have the best compression ratio. But higher order filters achieve better accuracy and robustness properties at the price of a slight decrease of the compression ratio.

Data smoothing and distortion of X-ray diffraction peaks. I. Theory

A general model of smoothing-induced distortions of an X-ray diffraction (XRD) peak is derived from theoretical analyses by convoluting a Savitzky–Golay (S.–G.) polynomial filter with the Gaussian, Lorentzian and pseudo-Voigt curves. This is based on the assumption that an XRD peak consists of five basic parameters and two additional parameters. Flattening and broadening are the general effects of smoothing. No change in peak area after smoothing is a new feature revealed by this analysis. An unusual change (sharpening with narrowing first, then flattening with broadening as usual) of the peak in a short smoothing range is found for an asymmetric peak. This phenomenon is variable, being governed mainly by the degree of asymmetry of the peak and factors such as peak shifting, curve type, state of the peak top and smoothing range. Formulae describing the smoothing changes of all parameters of a peak are derived. Constants used to link these formulae are set and their properties are analysed with different functions, different states of the peak top, coefficients of shape perfection (`shape-perfect coefficients'), degrees of asymmetry and smoothing ranges.

A Misfire Detection Method Using Nonlinear Moving Polynomial Filtering

The OBD II regulations require for 2002 model year cars a misfire identification rate of 100%. Actual misfire detection methods base on speed measuring but problems remain for high speed and low torque. This paper deals with another approach using knock sensors. Their signal contains strong peaks caused by the pressure distribution.A nonlinear polynomial filter is developed to extract and emphasize these peaks and to design a matched filter for the misfire detection purpose. The efficiency of this method is shown using real knock sensor data of a four-cylinder engine.