Research on Variable Step Size Technique of SPE for Complex Environmental Conditions

To address the challenge of local optimization and step size factor parameter setting in the full attraction model of the firefly algorithm (FA), this paper introduces level-based attraction and variable step size to FA. In the level-based attraction model, fireflies are firstly grouped in different levels according to the brightness, and each firefly randomly selects two fireflies from a higher level to learn from. By using the variable step size strategy, the searching step size decreases with the number of iterations. Herein, the level-based attracting model can increase the diversity of individual learning and improve the ability to jump out of local optimization. Meanwhile, dynamic adjustment of step size can balance the detection and development ability of the algorithm and improve the optimization accuracy of the algorithm. To evaluate the effectiveness of the proposed algorithm, comparisons are drawn against optimized FA, particle swarm optimization algorithm, artificial bee colony algorithm and differential evolution algorithm on a variety of test function sets.

A variable step size total least squares affine-projection-like algorithm: Formula derivation and performance analysis

Summary form only given. This paper presents a Variable Step Size (VSS) technique that is integrated with a Stop-And-Go (SAG) approach for blind adaptive equalization of Quadrature Amplitude Modulation (QAM) signals. The proposed technique is applied to the Constant-Modulus Algorithm (CMA) and the Multi-Modulus Algorithm (MMA). The SAG approach allows the VSS technique to either vary the step size or to hold the previous one depending on a specific error criterion. This leads to the reduction of the amount of computations involved Simulation results show that algorithms with the VSS technique result in less steady-state Mean-Squared Errors (MSEs) than those obtained for algorithms without the VSS technique for the same speed of convergence.

An adaptive system is a well-known method for removing noise from noise-corrupted speech. The problem arises in many situations such as airplane cockpits and automobiles. The variable step size normalized least mean squares (VSS-NLMS) algorithm incorporating a variable step size parameter whose values are based on the ratio of signal-to-noise power has very good convergence speed and low steady-state misadjustment. This paper investigates the rate of convergence by the wavelet based LMS (WLMS) algorithm to the noise speech problem. Simulation results are presented to compare the performance of the variable step size NLMS algorithm, the wavelet based LMS Algorithm, and other commonly used fixed step size LMS algorithms.

A modified wide-angle parabolic equation (MWAPE) [St. Mary et al., J. Comput. Phys. 71, 304–315 (1987)] is studied for a uniform plane-parallel waveguide with pressure-release surface and rigid bottom. In this situation, the MWAPE is separable, and the resulting fourth-order boundary value problem is obtained and solved, which leads to an exact solution of the MWAPE. It is shown that the depth-dependent eigenfunctions in this instance are identical to those resulting from both the standard parabolic equation and Helmholtz equation in the same channel. Modal wavenumbers for the MWAPE are shown to be equivalent to the first three terms of the Taylor expansion for the Helmholtz wavenumbers, which is one term better than the standard PE wavenumbers. This exact solution can serve as an accuracy benchmark for numerical implementations of the MWAPE.

In Chapter 2, we have seen how the parabolic equation could be interpreted in terms of Fourier transforms for the case of propagation in vacuum. The situation gets more complex when refractive index variations are present, since they cannot be dealt with directly with Fourier transform techniques. Section 3.2 introduces the powerful split-step/Fourier technique pioneered by Hardin and Tappert, which replaces the original problem with propagation through a sequence of phase screens. At this stage the discussion is limited to the narrow-angle case, starting from the standard parabolic equation.

The parabolic approximation to the wave equation is examined within the context of normal mode theory. In a layered waveguide, the horizontal propagation constants and modal amplitudes of the field Ψ satisfying the standard parabolic equation (SPE) approximation can be mapped exactly into the amplitudes and wave numbers of the normal modes for the field p satisfying the Helmholtz wave equation. However, this is not the case for certain other parabolic approximations, such as the wide-angle parabolic equation (WAPE) approximation. Approximate mode functions for the WAPE approximation are developed. These mode functions are then used to decompose range-independent sound-pressure fields computed using the WAPE approximation. The resulting modal coefficients and eigenfunctions obtained using the WAPE mode functions are compared with those obtained using standard normal mode theory.

This paper proposes a new application of fuzzy logic to variable step size least mean square (VS-LMS) adaptive beamforming algorithm in CDMA systems. The proposed algorithm adjusts the step size of the least mean square (LMS) by using the application of fuzzy logic in which the increase or decrease of step size depends on the fuzzy inference results of the mean square error (MSE). Computer simulation results show that the proposed algorithm has a better capacity of tracking compared with the conventional LMS algorithms and other variable step size LMS algorithms.

This paper incorporates a correntropy-induced metric (CIM)-based sparsity constraint into variable step size normalized least mean square (VSSNLMS) algorithm for identification of sparse system corrupted by additive noise. The proposed CIM-VSSNLMS algorithm makes a good trade-off between convergence characteristics, filter stability, and steady state error. The proposed (CIM-VSSNLMS) algorithm incorporates a variable step size that accelerates the convergence characteristics and lowers the normalized misalignment (NMSA) error with respect to CIM-NLMS with constant value of step size. An expression for variant step size is derived under the stability condition of proposed algorithm. Finally, the implementation of the proposed algorithm is carried out in MATLAB software to manifest the improved estimated behavior in the identification of sparse system.

A wide‐angle parabolic wave equation solution, using shift‐map and finite‐difference techniques, is presented. The corresponding split‐step Fourier solution is well known. The solution using finite‐difference technique, where the standard parabolic wave equation is modified into the so‐called Claerbout equation allowing propagation angles up to 45° from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift‐map technique is incorporated into the finite‐difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split‐step solution, which is believed to perform well for terrain slopes up to 10°–15°, and discontinuous slope changes on the order of 15°–20°. This solution is of first order with respect to the terrain slope. The resulting difference scheme is for solving a tridiagonal system. When using the finite‐difference technique, it is also possible to find a second order solution with respect to the terrain slope. The resulting difference scheme in this case is for solving a pentadiagonal system. This new solution performs well for slopes up to about 15°–20° and discontinuous slope changes up to about 30°–40°, which is an improvement.

Delta Sigma Analog to Digital Converters (ADCs) are superior to their counterparts in achieving high resolution for low to medium bandwidth applications. This paper discusses an improved digital calibration technique for enhancing Signal to Noise ratio (SNR) of Delta Sigma ADC which is directly related to Effective Number of Bits (ENOB). The proposed method uses the Adaptive Line Enhancer (ALE) for improving the SNR of the modulator. Adaptive Line Enhancer filters out the estimate of signal of interest from noisy input signal by updating filter coefficients using Least Mean Square (LMS) Algorithm. Variable step size (VSS) LMS is used for updating filter coefficients in Adaptive Line Enhancer instead of Fixed step size (FSS) LMS to reduce the required number of iterations. The simulation results for second order single loop Delta Sigma modulator show approximately 20dB improvement in SNR after calibration and substitution of VSS LMS with FSS LMS has reduced approximately 4000 iterations.

The step size of least mean square (LMS) algorithm is significant for its performance. To be specific, small step size can get small excess mean square error but results in slow convergence. However, large step size may cause instability. Many variable step size least mean square (VSSLMS) algorithms have been developed to enhance the control performance. In this paper, a new VSSLMS was proposed based on Kwong’s algorithm to evaluate the robustness. The approximate analysis of dynamic and steady-state performance of this developed VSSLMS algorithm was given. An active vibration control system of piezoelectric cantilever beam was established to verify the performance of the VSSLMS algorithms. By comparing with the current VSSLMS algorithms, the proposed method has better performance in active vibration control applications.

Myosin VI dimer walks toward the minus end of the actinfilamentwith a large and variable step size of 25–36 nm. Two competingmodels have been put forward to explain this large step size. TheSpudich model assumes that the myosin VI dimer associates at a distaltail near the cargo-binding domain, which makes two full-length singleα-helix (SAH) domains serve as long legs. In contrast, the Houdusse–Sweeneymodel assumes that the association occurs in the middle (between residues913 and 940) of the SAH domain and that the three-helix bundles unfoldto ensure the large step size. Their consistency with the observationof stepping motion with a large and variable step size has not beenexamined in detail. To compare the two proposed models of myosin VI,we computationally characterized the free energy landscape experiencedby the leading head during the stepping movement along the actin filamentusing the elastic network model of two heads and an implicit modelof the SAH domains. Our results showed that the Spudich model is moreconsistent with the 25–36 nm step size than the Houdusse–Sweeneymodel. The unfolding of the three-helix bundles gives rise to thefree energy bias toward a shorter distance between two heads. Besides,the stiffness of the SAH domain is a key factor for giving strongenergetic bias toward the longer distance of stepping. Free energyanalysis of the stepping motion complements the visual inspectionof static structures and enables a deeper understanding of underlyingmechanisms of molecular motors.

A variable step size Least Mean Square (LMS) algorithm based on the difference of the error signal (ESD-VSS) is proposed.The convergence and steady state behavior of the ESD-VSS is analyzed and the key parameters are given.Simulations are conducted and the results show that, compared with the Modified Variable Step Size (MVSS) algorithm, modified LMS algorithm and the standard LMS algorithm, the ESD-VSS provides faster convergence speed and smaller steady state misadjustment.Index Terms -LMS algorithm, variable step size, adaptive filter, MSE I . IntroductionThe LMS algorithm proposed by Widrow and Hoff has been widely used in channel equalization [1] and system identification [2], due to its robustness and simplicity.However the tradeoff between the convergence speed and the steady state misadjustment is a challenging problem.To achieve faster convergence speed and small misadjustment, a number of modified algorithms have been proposed.Ref. [3] utilized the self-correlation to control the step size, and suppress the interference of irrelevant noise effectively, but the convergence speed and the tracking capability are still expected to be improved.Ref. [4] gave a variable step size LMS algorithm using estimate signal error and thus it is of relatively faster tracking speed and smaller steady state error.However, the computational complexity of the algorithm is larger due to the tangent and exponent arithmetic, which makes it difficult to implement.

This paper presents a novel approach to Lyapunov stability theory-based adaptive filter (LAF) design. The proposed design is based on the minimization of the Euclidean norm of the difference weight vector under negative definiteness constraint defined over a novel linear Lyapunov function. The proposed fixed step size LAF (FSS-LAF) algorithm is first obtained by using the method of Lagrangian multipliers. The FSS-LAF satisfying asymptotic stability in the sense of Lyapunov provides a significant performance gain in the presence of a measurement noise. The stability of the FSS-LAF algorithm is also statistically analyzed in this study. Moreover, gradient variable step size (VSS) algorithms are adapted to the FSS-LAF algorithm to further enhance the performance for the first time in this paper. These VSS algorithms are Benveniste (BVSS), Mathews and Farhang–Ang (FVSS) algorithms. Simulation results on system identification problems show that the bounds of step size for the FSS-LAF algorithm are verified, and especially, the BVSS-LAF and FVSS-LAF algorithms provide a better trade-off between steady-state mean square deviation error and convergence rate than other proposed algorithms.