Sliding Window Evaluation of the Wiener-Hopf Equation

2006

In this paper a nonlinear extension to the Wiener filter is presented. A direct approach has been devised of replacing the autocorrelation function with a novel function called correntropy, derived from ideas on kernel-based learning theory and information theoretic learning. The linear Wiener filter, widely used because of its simplicity and optimality for linear systems and Gaussian distribution, is no longer effective when dealing with nonlinear time series data. The proposed method incorporates higher order moments in the general form of autocorrelation and improves upon the linear filter. Moreover, the computation cost is still lower than some kernel based methods and has a closed form solution to the problem unlike neural network based methods.

2007

Power Spectral Density (PSD) computed by taking the Fourier transform of auto-correlation functions (Wiener-Khintchine Theorem) gives better result, in case of noisy data, as compared to the Periodogram approach. However, the computational complexity of Wiener-Khintchine approach is more than that of the Periodogram approach. For the computation of short time Fourier transform (STFT), this problem becomes even more prominent where computation of PSD is required after every shift in the window under analysis. In this paper, recursive version of the Wiener-Khintchine theorem has been derived by using the sliding DFT approach meant for computation of STFT. The computational complexity of the proposed recursive Wiener-Khintchine algorithm, for a window size of N, is O(N).

2003

Signal Processing Group, Universitat de Vic, Sagrada Fam´ilia 7, 08500, Vic, [email protected], [email protected], [email protected] Wiener system is a linear time-invariant filter,followed by an invertible nonlinear distortion. As-suming that the input signal is an independent andidentically distributed (iid) sequence, we proposean algorithm for estimating the input signal onlyby observing the output of the Wiener system. Thealgorithm is based on minimizing the mutual infor-mation of the output samples, by means of a steep-est descent gradient approach.1. INTRODUCTIONWhen linear models fail, nonlinear models are power-ful tools for modeling practical situations. Many re-searches have been done in the identification and/orthe inversion of nonlinear systems. These works usu-ally assume that both the input and the output of thedistortion are available, and are based on higher-orderinput/outputcross-correlation [1] or on the applicationof the Bussgang and Prices the...

Latin American applied research Pesquisa aplicada latino americana = Investigación aplicada latinoamericana

In this paper we propose a Wiener-like approximation scheme that uses Rational Wavelets for the linear dynamical structure and Orthonormal High Level Canon-ical Piecewise Linear functions for approxi-mating the nonlinear static part. This struc-ture allows to approximate any nonlinear, time-invariant, causal dynamic systems with fading memory and has the following advantages: ca-pability of time-frequency location, design of the linear dynamic part taking into account the a priori knowledge of the system, and min-imum number of parameters of Orthonormal High Level Canonical Piecewise Linear func-tions determined straightforwardly.

2016

This paper proposed a reproducing kernel method for solving Wiener-Hopf equations of the second kind. In order to eliminate the singularity of the equation, a transform is used. The advantage of this numerical method is the representation of exact solution in reproducing kernel Hilbert space and accuracy in numerical computation is higher. On the other hand, by improving the traditional reproducing kernel method and the definition of the operator of W Hilbert space, the solutions of Wiener-Hopf equation of the second kind are obtained. The approximate solution converges uniformly and rapidly to the exact solution. Numerical examples indicate that this method is efficient for solving these equations. The validity of the method is illustrated with two examples.

2017

Abstract—The Cholesky decomposition plays an important role in finding the inverse of the correlation matrices. As it is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition. As different methods exist to find the Cholesky decomposition of a given matrix, this paper presents the comparative study of a proposed RChol algorithm with the conventional methods. The RChol algorithm is an explicit way to estimate the modified Cholesky factors of a dynamic correlation matrix.

ISA Transactions, 2017

In this paper, an online identification algorithm is presented for nonlinear systems in the presence of output colored noise. The proposed method is based on extended recursive least squares (ERLS) algorithm, where the identified system is in polynomial Wiener form. To this end, an unknown intermediate signal is estimated by using an inner iterative algorithm. The iterative recursive algorithm adaptively modifies the vector of parameters of the presented Wiener model when the system parameters vary. In addition, to increase the robustness of the proposed method against variations, a robust RLS algorithm is applied to the model. Simulation results are provided to show the effectiveness of the proposed approach. Results confirm that the proposed method has fast convergence rate with robust characteristics, which increases the efficiency of the proposed model and identification approach. For instance, the FIT criterion will be achieved 92% in CSTR process where about 400 data is used.

Sensors, 2019

The concept presented in this paper is based on previous dynamical methods to realize a time-varying matrix inversion. It is essentially a set of coupled ordinary differential equations (ODEs) which does indeed constitute a recurrent neural network (RNN) model. The coupled ODEs constitute a universal modeling framework for realizing a matrix inversion provided the matrix is invertible. The proposed model does converge to the inverted matrix if the matrix is invertible, otherwise it converges to an approximated inverse. Although various methods exist to solve a matrix inversion in various areas of science and engineering, most of them do assume that either the time-varying matrix inversion is free of noise or they involve a denoising module before starting the matrix inversion computation. However, in the practice, the noise presence issue is a very serious problem. Also, the denoising process is computationally expensive and can lead to a violation of the real-time property of the s...

Geophysical Prospecting, 1986

For a new approach to designing the time-varying Wiener filter, the input is first divided into sections and then the time-varying filter is determined from the entire input and the desired output. The technique differs from the existing one in which the time-invariant filter is determined from each section. Hence, the main difference, between the proposed and the existing technique lies in the arrangement of input data. The proposed technique requires fewer computational operations and performs better than the time-invariant Wiener filter, as illustrated by numerical examples.

Complexity, 2020

Wiener, Hammerstein, and Wiener–Hammerstein structures are useful for modelling dynamic systems that exhibit a static type nonlinearity. Many methods to identify these systems can be found in the literature; however, choosing a method requires prior knowledge about the location of the static nonlinearity. In addition, existing methods are rigid and exclusive for a single structure. This paper presents a unified approach for the identification of Wiener, Hammerstein, and Wiener–Hammerstein models. This approach is based on the use of multistep excitation signals and WH-EA (an evolutionary algorithm for Wiener–Hammerstein system identification). The use of multistep signals will take advantage of certain properties of the algorithm, allowing it to be used as it is to identify the three types of structures without the need for the user to know a priori the process structure. In addition, since not all processes can be excited with Gaussian signals, the best linear approximation (BLA) w...

IEEE Transactions on Signal Processing, 1992

Various signal processing applications require the computation of an estimate of the autocorrelation sequence from available data. The evaluation of the autocorrelation sequence typically represents the bulk of the computation time required in each application. This work investigates frequency domain techniques for the evaluation of the autocorrelation in the case of multidimensional sequences. The bidimensional case is treated in detail not only because it is of special interest for the applications, but also because the reasoning used for this case can be applied to higher dimension signals. The analysis confirms that the use of the proposed frequency domain techniques leads to significant computation time savings.

IEEE Signal Processing Letters, 2008

In this work, a non-iterative identification approach is presented for estimating a Single Input Single Output (SISO) Wiener model, comprising an Infinite Impulse Response (IIR) discrete transfer function followed by static non-linearity. Global Orthogonal Basis Functions and orthogonal Hermite polynomials are used as expansion bases for the linear subsystem and the non-linearity, respectively. A multi-index based method is used to transform the non-convex optimization over the parameter values into an over-parametrized linear regression. A Singular Value Decomposition based method is then used to project the result of the over-parametrized linear regression onto the class of Linear Non-linear (LN) models. The advantages obtained by using orthogonal polynomials are illustrated using a series of simulation examples.

ACM Transactions on Mathematical Software, 1998

We discuss the computational study of curves of Hopf and double-Hopf points in the software package CONTENT developed at CWI, Amsterdam. These are important points in the numerical study of dynamical systems characterized by the occurrence of one or two conjugate pairs of pure imaginary eigenvalues in the spectrum of the Jacobian matrix. The bialternate product of matrices is extensively used in three codes for the numerical continuation of curves of Hopf points and in one for the continuation of curves of double-Hopf points. In the double-Hopf and two of the single-Hopf cases this is combined with a bordered matrix method. We use this software to find special points on a Hopf curve in a model of chemical oscillations and by computing a Hopf and a double-Hopf curve in a realistic model of a neuron.

IEEE Access, 2019

This paper presents a forward-path, novel, two-dimensional (2-D) sliding discrete Fourier transform (SDFT) algorithm based on the column-row 2-D DFT concept and the shifted window property. After applying a descending dimension method (DDM), a mixed-radix and butterfly-based structure can be further employed to effectively implement the proposed algorithm. Conceptually, it has many advantages, including greater stability, more accuracy, and less computational complexity because there are no extra feedback loops in the calculation. The evaluation results are based on the following conditions: (1) the window size (N) to 16 × 16; (2) the test pattern is an SVC grayscale video, and the formats are CIF and 4CIF with 30fps; (3) one multiplication involves four real multiplications and two real additions. The proposed 2-D SDFT method clearly reduced the number of multiplications by 43.8% and only increased the number of additions by 33%, compared with the state-of-the-art Park's method. Additionally, for the first 100 frames of the CIF and 4CIF sequences, the proposed method saves 10.8% and 10.9% of the processing time, respectively, on average. Overall, the proposed DDM-based 2-D SDFT algorithm can be applied to calculate not only 1-D but also 2-D SDFT spectrum, and are especially appropriate for hybrid applications. INDEX TERMS Sliding discrete Fourier transform (SDFT), two-dimension SDFT (2-D SDFT), column-row 2-D DFT, descending dimension method (DDM), DDM-based 2-D SDFT.