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This paper proposes a new adaptive algorithm for the second-order blind signal separation (BSS) problem with convolutive mixtures by utilising a combination of an accelerated gradient and a conjugate gradient method. For each iteration of the adaptive algorithm, the search point and the search direction are obtained based on the current and the previous iterations. The algorithm efficiently calculates the step size for the accelerated conjugate gradient algorithm in each iteration. Simulation results show that the proposed accelerated conjugate gradient algorithm with optimal step size converges faster than the accelerated descent algorithm and the steepest descent algorithm with optimal step size while having lower computational complexity. In particular, the number of iterations required for convergence of the accelerated conjugate gradient algorithm is significantly lower than the accelerated descent algorithm and the steepest descent algorithm. In addition, the proposed system achieves improvement in terms of the signal to interference ratio and signal to noise ratio for the dominant speech outputs.

The location of the grounding grid conductors is critical for performing corrosion diagnosis and maintenance work. An improved magnetic field differential method to locate the unknown grounding grid based on truncation errors and the round-off errors analysis is presented in this paper. It was proven that a different order of the magnetic field derivative can be used to determine the position of the grounding conductor according to the peak value of the derivative. Due to the accumulative error of higher differentiation, the truncation error and rounding error were used to analyze to accumulative error and to determine the optimal step size to measure and calculate the higher differentiation. The possible range and probability distribution of the two kinds of errors at each order are described, and the index of peak position error was derived, which can be used to locate the grounding conductor in the power substation. [ABSTRACT FROM AUTHOR]

This paper proposes a new adaptive algorithm for the second-order blind signal separation (BSS) problem with convolutive mixtures by utilising a combination of an accelerated gradient and a conjugate gradient method. For each iteration of the adaptive algorithm, the search point and the search direction are obtained based on the current and the previous iterations. The algorithm efficiently calculates the step size for the accelerated conjugate gradient algorithm in each iteration. Simulation results show that the proposed accelerated conjugate gradient algorithm with optimal step size converges faster than the accelerated descent algorithm and the steepest descent algorithm with optimal step size while having lower computational complexity. In particular, the number of iterations required for convergence of the accelerated conjugate gradient algorithm is significantly lower than the accelerated descent algorithm and the steepest descent algorithm. In addition, the proposed system achieves improvement in terms of the signal to interference ratio and signal to noise ratio for the dominant speech outputs. [ABSTRACT FROM AUTHOR]

The location of the grounding grid conductors is critical for performing corrosion diagnosis and maintenance work. An improved magnetic field differential method to locate the unknown grounding grid based on truncation errors and the round-off errors analysis is presented in this paper. It was proven that a different order of the magnetic field derivative can be used to determine the position of the grounding conductor according to the peak value of the derivative. Due to the accumulative error of higher differentiation, the truncation error and rounding error were used to analyze to accumulative error and to determine the optimal step size to measure and calculate the higher differentiation. The possible range and probability distribution of the two kinds of errors at each order are described, and the index of peak position error was derived, which can be used to locate the grounding conductor in the power substation.

The inverse problems as a classical mathematical model arise frequently in practical applications. While the split equality problem is an inverse problem arising frequently in signal processing, phase retrieval and medical image reconstruction etc. which is also an important extension of the split feasibility problem. In this paper, motivated and inspired by an idea for solving a variational inequality, we proved that several two-step methods and relaxed two-step methods can be applied to solve the split equality problem. What's more, we proved that the technique of optimal step size for solving the variational inequality is still efficient for the split equality problem. Meanwhile, we also confirmed that these provided methods also apply to solving the split feasibility problem. [ABSTRACT FROM AUTHOR]

The algebraic reconstruction technique (ART) and its generalization the block-iterative ART are often used to solve large scale linear systems Ax=b which arise in many applications, such as image processing. Two ingredients, i.e. stepsizes and blocks, play a key role in block-iterative ART. In this paper we address the problem of how to choose stepsizes and blocks of A so as to achieve optimal convergence rate of the block-iterative ART. We introduce a random cyclic selection method for selecting blocks for the block-iterative ART in order to reduce computational complexity. Our numerical experiments demonstrate the efficiency of our selections of stepsizes and blocks in the block-iterative ART. [ABSTRACT FROM AUTHOR]

The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention in many areas because of its efficiency and easy implementation. To theoretically guarantee convergence of the linearized ADMM, the step size for the linearized subproblems, or the reciprocal of the linearization parameter, should be sufficiently small. On the other hand, small step sizes decelerate the convergence numerically. Hence, it is interesting to probe the optimal (largest) value of the step size that guarantees convergence of the linearized ADMM. This analysis is lacked in the literature. In this paper, we provide a rigorous mathematical analysis for finding this optimal step size of the linearized ADMM and accordingly set up the optimal version of the linearized ADMM in the convex programming context. The global convergence and worst-case convergence rate measured by the iteration complexity of the optimal version of linearized ADMM are proved as well. [ABSTRACT FROM AUTHOR]

For generalized gamma probability densities, this paper studies the monotonicity of step sizes of optimal symmetric uniform scalar quantizers with respect to mean squared-error distortion. The principal results are that for the special cases of Gaussian, Laplacian, two-sided Rayleigh, and gamma densities, optimal step size monotonically decreases when the number of levels $N$ increases by two, and that for any generalized gamma density and all sufficiently large $N$ , optimal step size again decreases when $N$ increases by two. Also, it is shown that for a Laplacian density and sufficiently large $N$ , optimal step size decreases when $N$ increases by just one. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm for the second-order blind signal separation problem with convolutive mixtures. An iterative first order gradient method based on the accelerated gradient is developed for solving the optimization problem. For each search direction, the question becomes how to effectively calculate the optimal step size in each iteration. Here, we propose an efficient algorithm for obtaining the step size by first reformulating the objective function as a fourth order polynomial in terms of the step size, where the polynomial coefficients are required to be calculated only once per iteration. An optimal step size search procedure using the Newton’s method is developed with the step size is efficiently obtained for each iteration. Simulation results in a simulated room environment and a real environment show that the proposed algorithm converges faster than the existing methods with a lower number of iterations and a lower computational complexity. In addition, the proposed algorithm can separate the speech signals and reduce the background noise simultaneously. [ABSTRACT FROM AUTHOR]

This paper investigates the convexity of the mean squared-error distortion of symmetric uniform scalar quantization with respect to step size. The principal results include proofs for odd numbers of levels that distortion is not convex for any symmetric density and that it is convex for even numbers of levels for densities, such as Gaussian, Laplacian, and gamma, but is not, in general for two-sided Rayleigh. For the latter case, an interval is derived that includes the optimal step size and over which the distortion is convex. The proofs of convexity use the Euler–Maclaurin formula applied to the second derivative of distortion, with upper bounds on the remainder term. These results imply that a zero of the derivative of the distortion for these densities, which has been previously conjectured optimal, is indeed the optimal step size, because the distortion is convex either globally or locally over a sufficiently wide interval to ensure a global minimizer. [ABSTRACT FROM AUTHOR]

This paper proposes an optimized least mean absolute third (OPLMAT) algorithm to improve the capability of the adaptive filtering algorithm against Gaussian and non-Gaussian noises when the unknown system is a time-varying parameter system under low signal-noise rate. The optimal step size of the OPLMAT is obtained based on minimizing the mean-square deviation at the current time. In addition, the mean convergence and steady-state error of the OPLMAT algorithm are derived theoretically, and the computational complexity of OPLMAT is analyzed. Furthermore, the simulation experimental results of system identification presented illustrate the principle and efficiency of the OPLMAT algorithm. Simulation results demonstrate that the proposed algorithm performs much better than the LMAT and NLMAT algorithms. [ABSTRACT FROM AUTHOR]

Although the linear minimum mean square error (LMMSE) filter removes speckle in polarimetric synthetic aperture radar (PolSAR) images, it has the disadvantage of losing edge detail. In this paper, we propose a new filter based on robust independent component analysis (RICA) and LMMSE. This approach describes edge features in a span image by selecting the adaptive direction window and calculating the edge weight value of the spatial domain, and improves the objective function by using a step polynomial to extract the estimate of the source image with minimum noise. This technique preserves not only the edge information in the images, but also the polarimetric information. Experiments were conducted on the NASA/JPL AIRSAR L-band of the San Francisco area, and evaluated by means of the speckle reduction index and the edge preservation index. The experimental results show that the proposed method effectively reduces speckle, retains edges, and preserves the polarimetric scattering mechanisms. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method to improve the efficiency of the perturbation and observation (P&O) maximum power point tracking (MPPT) algorithm. The proposed method can improve the dynamic tracking response of P&O, reduce the steady state oscillation of P&O and eliminate the possibility of the algorithm to lose its tracking direction. To estimate the optimal duty ratio of the boost converter at the maximum power point, an optimal duty ratio estimation algorithm based on a four-parameter model of a photovoltaic (PV) module is proposed. Furthermore, the ratio of the power-voltage curve slope to the current in the PV module is defined as K pvi . Because K pvi has a small variation range at different irradiations, a piecewise linear function dependent on K pvi is developed to generate the optimal step size of P&O. Finally, K pvi is utilized to avoid drift phenomenon in case of fast change in irradiation. To prove its effectiveness, the proposed P&O algorithm is compared with conventional P&O and adaptive P&O using the sudden step change test, Ropp test, and four one-day irradiation and temperature profile tests in Simulink. The proposed algorithm is also benchmarked by MPPT efficiency ( η MPPT ) calculation. The results show that the η MPPT of the proposed P&O is increased by 0.65% in one-day irradiation and temperature profile with light cloud test. In addition, the η MPPT of the proposed P&O can be further improved up to 1.27% and 1.24% in one-day profile with heavy cloud test and one-day profile with dark cloud test, respectively. Moreover, it needs neither additional sensors nor expensive microcontrollers. [ABSTRACT FROM AUTHOR]

On the basis of RobustICA, this paper proposes a speckle reduction of PolSAR images by deploying step size polynomials to improve the objective function, obtaining the global optimal step size, and finally exacting the source image estimations with less noise. These experiments were conducted using ESAR data, whose effects were evaluated using the new methods from the coherent speckles index, mean square error, flatness index, the equivalent number of looks and the edge keeping index. The results show that new filter has better denoising effects and edge keeping ability, as well as better time efficiency. [ABSTRACT FROM AUTHOR]

This paper introduces an optimal variable step-size affine projection algorithm for the modified filtered-x active noise control systems. First, the recursion form of the error covariance from the tap weight update equation is constructed, not ignoring the dependency between the estimation error and the secondary noise signal. Such consideration has not been concerned previously for the analysis of the modified filtered-x affine projection algorithm. Second, a recursion form of the mean square deviation is derived from that of the error covariance. From the recursion form, an optimal step size is decided to get the fastest convergence rate. Both the recursion forms of the mean square deviation and the optimal step size require scalar additions and multiplications that do not contribute to the overall complexity seriously. The simulation results on the active noise control environments show both fast convergence rate and low steady-state error. [ABSTRACT FROM AUTHOR]

This letter proposes a new algorithm for solving the optimal zero-forcing beamforming problem that maximizes the user achievable rate with restriction on the per-antenna element power constraints. An accelerated gradient method with step size search is proposed for solving the problem. For each iteration of the gradient approach, a quick one dimensional search is employed to obtain the step size. The advantage of the step-size search is that it is relatively fast and requires only a few calculations of the objective function. Design examples show that the proposed algorithm converges faster than the gradient approach and the accelerated gradient approach with a constant step size while achieving a low computational complexity. [ABSTRACT FROM AUTHOR]

This paper proposes a new computational procedure for solving the optimal zero-forcing beamforming problem in multiple antenna channels that maximizes user achievable rate with restriction on the per-antenna element power constraints. An interior point method with optimal step size procedure is developed in which the step size for the line search in the Newton search direction is calculated exactly for each iteration. This significantly enhances the efficiency associated with the line search. Design examples show that the proposed algorithm converges rapidly to the optimal solution with low computational complexity. [ABSTRACT FROM AUTHOR]

Abstract: The explicit Euler's method is known to be very easy and effective in implementation for many applications. This article extends results previously obtained for the systems of linear differential equations with constant coefficients to arbitrary systems of ordinary differential equations. Optimal (providing minimum total error) step size is calculated at each step of Euler's method. Several examples of solving stiff systems are included. [Copyright &y& Elsevier]

Abstract: This paper proposes a new computational procedure for solving the second-order gradient-based blind signal separation (BSS) problem with convolutive mixtures. The problem is formulated as a constrained optimization problem where the time domain constraints on the unmixing matrices are added to ease the permutation effects associated with convolutive mixtures. A linear transformation using QR factorization is developed to transform the constrained optimization problem into an unconstrained problem. A conjugate gradient procedure with the step size derived optimally at each iteration is then proposed to solve the optimization problem. The advantage of the procedure is that it has low computational complexity, as it does not require multiple evaluations of the objective function. In addition, fast convergence of the conjugate gradient algorithm makes it suitable for online implementation. The convergence of the conjugate gradient algorithm with optimal step size is compared to the fixed step size case and the optimal step size steepest descent algorithm. Evaluations are performed in real and simulated environments. [Copyright &y& Elsevier]

Independent component analysis (ICA) aims at decomposing an observed random vector into statistically independent variables. Deflation-based implementations, such as the popular one-unit FastICA algorithm and its variants, extract the independent components one after another. A novel method deflationary ICA, referred to as RobustICA, is put forward in this paper. This simple technique consists of performing exact line search optimization of the kurtosis contrast function. The step size leading to the global maximum of the contrast along the search direction is found among the roots of a fourth-degree polynomial. This polynomial rooting can be performed algebraically, and thus at low cost, at each iteration. Among other practical benefits, RobustICA can avoid prewhitening and deals with real- and complex-valued mixtures of possibly noncircular sources alike. The absence of prewhitening improves asymptotic performance. The algorithm is robust to local extrema and shows a very high convergence speed in terms of the computational cost required to reach a given source extraction quality, particularly for short data records. These features are demonstrated by a comparative numerical analysis on synthetic data. RobustICA's capabilities in processing real-world data involving noncircular complex strongly super-Gaussian sources are illustrated by the biomedical problem of atrial activity (AA) extraction in atrial fibrillation (AF) electrocardiograms (ECGs), where it outperforms an alternative ICA-based technique. [ABSTRACT FROM AUTHOR]

Abstract: Two iterative algorithms are presented in this paper to solve the minimal norm least squares solution to a general linear matrix equations including the well-known Sylvester matrix equation and Lyapunov matrix equation as special cases. The first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual form. Necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions. Sufficient condition that is easy to compute is also given. Moreover, two methods are proposed to choose the optimal step sizes such that the convergence speeds of the algorithms are maximized. Between these two methods, the first one is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained. The second method is to minimize the square sum of the F-norm of the error matrices produced by the algorithm and it is shown that the optimal step size exits uniquely and lies in an interval. Several numerical examples are given to illustrate the efficiency of the proposed approach. [Copyright &y& Elsevier]

Abstract: In this study, commutation error (CE) is defined in adaptive infinite impulse response (IIR) filter-based ANC systems. CE is subsequently introduced into a new residual error to develop a new LMS-based ANC algorithm in an aim to liberate the restriction of slow adaptation posed on traditional ANC algorithms. A new deterministic analysis based on a linear time-varying system is performed to investigate convergence properties of the developed algorithm: (1) An optimal step size for the fastest convergence rate can be derived. (2) Given a persistent excitation condition and a step-size constraint, we find that the algorithm is uniformly asymptotically stable. Computer simulations indeed demonstrate a greatly improved convergence rate and efficient ANC performance for the developed algorithm as compared with that using the conventional algorithms. Experimental results verify the enhanced ANC performance in real applications. These together support the new IIR filter-based adaptive algorithm that includes CE for superior ANC performance with respect to the convergence rate and noise reduction level. [Copyright &y& Elsevier]

This paper focuses on the constant power (CP) criterion for blind linear equalization of digital communication channels. This recently proposed criterion is specially designed for the extraction of q-ary phase shift keying (q-PSK) signals using finite impulse response equalizers. When zero-forcing equalizers exist, the CP cost function accepts exact analytic solutions that are unaffected by undesired local extrema and spare costly iterative optimization. A subspace-based method exploiting the Toeplitz-like structure of the solution space is put forward to recover the minimum-length equalizer impulse response from the overestimated-length solutions. The proposed method is more robust to the relative weights of the minimum-length equalizer taps than existing techniques. In less ideal scenarios where the analytic solutions are only approximate minimizers of the criterion, a gradient-descent algorithm is proposed to minimize the cost function. To reduce the detrimental effects of suboptimal equilibria and accelerate convergence, the iterative algorithm is initialized with the approximate closed-form solution, and an optimal step size is incorporated into its updating rule. This optimal step size, which globally minimizes the cost function along the search direction, can be computed algebraically. A semi-blind implementation, which is useful when training data are available, further reduces the impact of undesired local extrema and enhances the convergence characteristics (particularly the robustness to the equalizer initialization) of the iterative algorithm from just a few pilot symbols. All these beneficial features are demonstrated with an experimental study of the proposed CP-based methods in a variety of channels and simulation conditions. [ABSTRACT FROM AUTHOR]

In this paper, a new frequency-domain active noise control (ANC) system is proposed in which an optimal step size is used for updating at each frequency. The proposed system achieves convergence faster than in the case of the conventional ANC system with the Filtered-x LMS method updated with the step size converging at the fastest speed. This is because the step size with the fastest convergence can be adapted at each frequency in the proposed system. In the conventional system, convergence is not possible unless the phase error of the secondary path model satisfies the stability condition. In the proposed system, on the other hand, excellent silencing characteristics without divergence can be obtained by letting the step size become 0 at frequencies not satisfying the stability condition, because a different step size at each frequency can be used. In this paper, the theoretical equation for the optimal step size is derived by using only the information obtainable during system operation. Next, a configuration of the ANC system that can make use of the optimal step size derived from this theoretical equation is explained. Further, the method for identifying the frequency causing instability and its control method are explained. Finally, by computer simulation, the effectiveness of the proposed system is verified. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(7): 51–61, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10000 [ABSTRACT FROM AUTHOR]

More and more new types of observational data provide many new opportunities for improving numerical weather forecasts. Among these, the GPS (Global Positioning System) bending angle is undoubtedly very important. There are many advantages of the GPS bending angle, such as high resolution, availability in all weather conditions, and global data coverage. Thus it is very valuable to assimilate GPS bending angle data into numerical weather models. This paper introduces how to obtain and assimilate the GPS bending angle. There are two methods of assimilation: the indirect method and direct method, and they are both introduced in this paper. During the minimizing process of variational assimilation, calculation efficiency is very important and the optimal step size greatly influences the algorithm efficiency. Based on the characteristics of the minimizing algorithm, we obtain an adaptive method for calculating the optimizing step suitable for all kinds of minimization algorithms through mathematical deduction. Finally, a numerical variational assimilation experiment is performed using the GPS bending angle data of 11 October 1995. The numerical results indicate the validity of the variational assimilation method and the adaptive method introduced here. [ABSTRACT FROM AUTHOR]

In four—dimensional variational data assimilation (4DVAR) technology, how to calculate the optimal step size is always a very important and indeed difficult task. It is directly related to the computational efficiency. In this research, a new method is proposed to calculate the optimal step size more effectively. Both nonlinear one—dimensional advection equation and two—dimensional inertial wave equation are used to test and compare the influence of different methods of the optimal step size calculations on the iteration steps, as well as the simulation results of 4DVAR processes. It is in evidence that the different methods have different influences. The calculating method is very important to determining whether the iteration is convergent or not and whether the convergence rate is large or small. If the calculating method of optimal step size is properly determined as demonstrated in this paper, then it can greatly enlarge the convergence rate and further greatly decrease the iteration steps. This research is meaningful since it not only makes an important improvement on 4DVAR theory, but also has useful practical application in improving the computational efficiency and saving the computational time. [ABSTRACT FROM AUTHOR]

• A neural-network-based Lagrange multiplier selection model is formulated. • A two-stage algorithm is designed to solve the nonlinear and nonsmooth programming. • Some special designs of the neural network are proposed. • Price response feature is found to be nonlinear and temporally coupled. As a general difficult problem, the slow convergence of existing distributed demand response methods greatly hinders the reliable applications in smart grid. To overcome this problem, this paper proposes a new distributed method, namely the neural-network-based Lagrange multiplier selection (NN-LMS), to prominently reduce the iterations and avoid an oscillation. The key improvement lies in the forecast strategy of a load serving entity (LSE), who applies a specially designed neural network to capture the users' price response features. A novel Lagrange multiplier selection model, the NN-LMS model, is formulated to optimize the iterative step sizes. This complex model is then solved by a two-stage NN-LMS algorithm, containing the interval bisection method and improved sensitivity method in the first and second stages, respectively. In addition, the data selection, batch normalization and additional network are applied to boost the performance of neural networks. Case studies validate the optimality, improved convergence and numerical stability of the proposed method, and demonstrate its great potential in distributed demand response and other smart grid applications. [ABSTRACT FROM AUTHOR]