Variable Step-size NLMS Algorithm Research Articles

New variable step-size fast NLMS algorithm for non-stationary systems

New variable step-size fast NLMS algorithm for non-stationary systems

A new and effective nonparametric variable step-size normalized least-mean-square algorithm and its performance analysis

This paper presents a new and effective nonparametric variable step-size normalized least-mean-square (VSS-NLMS) algorithm which provides excellent performance in various signal-to-noise ratio (SNR) scenarios. The step-size update expressions are based on steady-state step-size which derives from the filter weight update equations. Specifically, the step-size adjustment rules employ the square root of the estimated error power and the system noise power to enhance the effectiveness of the proposed algorithm. Detailed transient-state and steady-state analyses are presented for the Gaussian white input signal and suggest that the steady-state performance of the proposed algorithm is independent of the smoothing parameter (used in the step-size update expressions). In addition, performance comparisons between the proposed algorithm and other well-known VSS-NLMS algorithms show the improvements of the proposed algorithm. Furthermore, theoretical analyses and simulated behaviors of the proposed algorithm are in good agreement for both transient-state and steady-state. Through simulation results, the irrelevance between the steady-state performance and the smoothing parameter is verified.

A Denoising Method of Micro-Turbine Acoustic Pressure Signal Based on CEEMDAN and Improved Variable Step-Size NLMS Algorithm

The acoustic pressure signal generated by blades is one of the key indicators for condition monitoring and fault diagnosis in the field of turbines. Generally, the working conditions of the turbine are harsh, resulting in a large amount of interference and noise in the measured acoustic pressure signal. Therefore, denoising the acoustic pressure signal is the basis of the subsequent research. In this paper, a denoising method of micro-turbine acoustic pressure signal based on the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and Variable step-size Normalized Least Mean Square (VSS-NLMS) algorithms is proposed. Firstly, the CEEMDAN algorithm is used to decompose the original signal into multiple intrinsic mode functions (IMFs), based on the cross-correlation coefficient and continuous mean square error (CMSE) criterion; the obtained IMFs are divided into clear IMFs, noise-dominated IMFs, and noise IMFs. Finally, the improved VSS-NLMS algorithm is adopted to denoise the noise-dominated IMFs and combined with the clear IMF for reconstruction to obtain the final denoised signal. Adopting the above principles, the acoustic pressure signals generated by a micro-turbine with different rotation speeds and different states (normal turbine and fractured turbine) are denoised, respectively, and the results are compared with the axial flow fan test (ideal interference-free signal). The results show that the denoising method proposed in this paper has a good denoising effect, and the denoised signal is smooth and the important features are well preserved, which is conducive to the extraction of acoustic pressure signal characteristics.

A Variable Step Size Normalized Least-Mean-Square Algorithm Based on Data Reuse

The principal issue in acoustic echo cancellation (AEC) is to estimate the impulse response between the loudspeaker and microphone of a hands-free communication device. This application can be addressed as a system identification problem, which can be solved by using an adaptive filter. The most common one for AEC is the normalized least-mean-square (NLMS) algorithm. It is known that the overall performance of this algorithm is controlled by the value of its normalized step size parameter. In order to obtain a proper compromise between the main performance criteria (e.g., convergence rate/tracking versus accuracy/robustness), this specific term of the NLMS algorithm can be further controlled and designed as a variable parameter. This represents the main motivation behind the development of variable step size algorithms. In this paper, we propose a variable step size NLMS (VSS-NLMS) algorithm that exploits the data reuse mechanism, which aims to improve the convergence rate/tracking of the algorithm by reusing the same set of data (i.e., the input and reference signals) several times. Nevertheless, we involved an equivalent version of the data reuse NLMS, which provides the convergence modes of the algorithm. Based on this approach, a sequence of normalized step sizes can be a priori scheduled, which is advantageous in terms of the computational complexity. The simulation results in the context of AEC supported the good performance features of the proposed VSS-NLMS algorithm.

Research on digital predistortion technology based on improved NLMS algorithm

In modern communication systems, digital predistortion (DPD) improve the linearity of PA and reduce the influence of nonlinear behavior on signal transmission. When identifying the inverse model of PA with traditional variable step size algorithm, the rate of convergence is slower, easy to be effected by noise and convergence instability exists at the optimal solution, which affects the optimization degree of the whole predistortion system. In this article, a digital predistortion technique based on improved NLMS algorithm is proposed to improve the accuracy of inverse model identification, convergence rate and anti-noise performance of the predistorter by changing the iteration function and introducing the autocorrelation matrix of the error of adjacent moments. The instability of the system at the optimal solution is reduced by introducing a new weight coefficient to update the correlation term. The simulation results show that the anti-noise performance, convergence rate and stability of the predistortion system based on New variable step size NLMS algorithm(NVNLMS) are obviously better than NLMS, and the out-of-band suppression of predistortion is optimized by 10 dB compared with the original system EVM is improved by 0.0031, ACPR is improved 9.1 dB.

Variable Step-Size Sparsity-Induced Augmented Complex-Valued NLMS Algorithm

The widely linear model has attracted much attention due to its good features for non-circular adaptive signal processing in recent years. In this paper, a sparsity-induced augmented complex-valued NLMS algorithm is proposed to promote the performance of the adaptive filter for estimating sparse systems, which is established by incorporating the $$l_0$$ -norm regularization into the squared error normalized by the input vector. To address the problem of trade-off between fast convergence rate and low steady-state misalignment, we minimize the variance of the a posteriori error to derive an optimal step-size and then some practical problems are considered. Simulation results are provided to verify the superior performance of the proposed algorithm.

Hybrid PSO-NLMS (HPSO-NLMS) algorithm for blind speech quality enhancement in time domain

Speech intelligibility enhancement by adaptive two-channel filtering algorithms in the time domain is a fundamental problem in the areas of signal processing application. In this paper, we address this problem and suggest a new adaptive hybrid algorithm combined with Forward blind source separation (FBSS) structure to retrieve a speech signal from noisy observations. Usually, the Normalized Least Mean Square algorithm (NLMS) is widely used in adaptive filtering application, however conventional NLMS presents difficulties to solve the trade-off between the speed of convergence and the low steady state error. Recently, numerous variable step-size NLMS (VNLMS) algorithms were proposed to solve this trade-off, our proposed algorithm falls within the same type of algorithm. The proposed Hybrid PSO-NLMS (HPSO-NLMS) algorithm is a combination of the NLMS and the Particle Swarm Optimization algorithms (PSO), where we use PSO to obtain the appropriate NMLS step size at each iteration. We integrate the new HPSO-NLMS in the FBSS structure, and compare their performances to conventional NLMS and PSO algorithms. We show that the proposed algorithm can produce better noise reduction performance, in terms of several objective metric such as the system misalignment (SM), the Segmental signal to noise Ratio (SegSNR) and the segmented mean square error (SegMSE).

A Study on Noise Reduction of Φ-OTDR System Based on VSS-NLMS Algorithm

To improve the signal-to-noise ratio (SNR) of vibration location information in $\Phi $ -OTDR system, a variable step size normalized least mean square (VSS-NLMS) denoising method is proposed in the article. First, the mathematical principle of VSS-NLMS method is established. The method can adaptively filter the input signal and adjust the parameters. It can also update the step factor adaptively. The simulation results show that the SNR of this method is improved from −1.16 dB to 12.01 dB. Second, a heterodyne coherent optical system is built. After the detected beat signal is demodulated by $\textit {I/Q}$ quadrature demodulation, the amplitude signal and phase signal of the Rayleigh backscattered light are acquired. The amplitude signal determines the location information and the phase signal determines the vibration signal. Finally, the wavelet denoising method, the empirical mode decomposition soft threshold (EMD-soft) denoising method and the variational mode decomposition nonlinear wavelet threshold (VMD-NWT) denoising method are compared with the VSS-NLMS denoising method in the experiment. The experimental results show that the location information SNR of the VSS-NLMS denoising method is improved to 59.31 dB, 46.81 dB, 50.14 dB, 34.00 dB and 67.09 dB for 5 groups of 100 Hz - 500 Hz vibration events at the location of the 10.14 km sensing fiber. At the same time, the vibration signals of 5 groups are demodulated. This method has laid a foundation for the application of oil and gas pipelines.

Variable tap-length non-parametric variable step-size NLMS adaptive filtering algorithm for acoustic echo cancellation

In teleconferencing and communication systems, the use of Loudspeaker Enclosure Microphone device leads to undesired echoes. To reduce these echoes, an acoustic echo canceller is used. In this paper, we propose a Variable Tap-length Non-Parametric Variable Step-Size NLMS (VT-NPVSS-NLMS) algorithm based on adaptive filtering for acoustic echo cancellation. The step-size is adjusted without the need for tuning too many parameters, using only the square of the average autocorrelation of a priori and a posteriori estimates of error. Moreover, the tap-length is varied to facilitate a high convergence speed and a small bias in tap-length from the optimum length. Hence, such a combination of the variable step-size algorithm with a variable tap-length provides faster convergence and reduced steady-state mean square error. The performance of the proposed algorithm is evaluated in terms of steady-state and transient mean square error. The advantages of the proposed algorithm, as compared to other adaptive algorithms, are presented using simulation results. From the results, we infer that for the VT-NPVSS-NLMS, the convergence speed is increased and the steady-state mean square error is reduced when compared with the conventional NLMS and the variable-step-size NLMS algorithm with a fixed tap-length.

Adaptive beamformers using 2D-novel ULA for cellular communication

This paper presents the use of a novel two dimensional (2D) uniform linear array for adaptive beamforming algorithms. We used this novel antenna configuration to study the popular adaptive beamforming algorithms, namely, least mean square algorithm (LMS) and least normalized mean square algorithm (NLMS) and also proposed a variable step-size NLMS (VSSNLMS) algorithm. These beamformers are named as 2D-LMS, 2D-NLMS and 2D-VSSNLMS algorithms respectively. These algorithms not only improve the convergence rate but also mitigates the effect of interference by producing deep nulls in the interference directions. These algorithms are suitable for low-power, low-cost, anti-interference MIMO-WLAN, WiMAX, 4G LTE and other advanced communication systems.

On the stochastic modeling of a VSS-NLMS algorithm with high immunity against measurement noise

This paper presents a comprehensive study of the variable step-size normalized least-mean-square (VSS-NLMS) algorithm introduced by Zipf, Tobias, and Seara [IEEE International Telecommunications Symposium (September 2010)]. Specifically, taking into account white and correlated Gaussian input data, a stochastic model is developed to predict the algorithm behavior for both transient and steady-state phases. Based on the proposed model, some interesting characteristics of the considered algorithm are verified, ratifying the robustness of the step-size adjustment rule against uncorrelated measurement noise. In addition, the impact of the smoothing parameter (used in the step-size adjustment rule) on the algorithm performance is discussed, aiming to provide some useful design guidelines. Through simulation results, the accuracy of the proposed model and some features of the algorithm are verified for different operating scenarios. Moreover, performance comparisons between the considered VSS-NLMS algorithm and other important and recent algorithms from the literature are presented.

A new combined-step-size normalized least mean square algorithm for cyclostationary inputs

For cyclostationary input signals, the normalized least mean square (NLMS) algorithm suffers from large steady-state errors. The average mean square deviation (MSD) analysis of the NLMS and least mean square (LMS) algorithms shows that NLMS has good transient response which is independent of the reference inputs, whereas the steady-state MSD of LMS does not depend on the periodic input power. In this paper, therefore, the combined-step-size NLMS (CSSNLMS) algorithm is proposed to reduce steady-state misalignment as compared to those in the conventional variable step-size (VSS) NLMS algorithms while achieving similar convergence rate for cyclostationary input signals. The proposed CSSNLMS algorithm combines and utilizes the different performances of the NLMS and LMS algorithms, in which NLMS is with large step-size and LMS uses small step-size. The mixing parameter is indirectly adjusted by use of the shrinkage denoising method in accordance with the estimated noise-free a priori error. The mean square performance analysis indicates that the proposed CSSNLMS algorithm can achieve the merits of NLMS and LMS under cyclostationary inputs. Finally, simulation results on system identification and echo cancellation verify the theoretical analysis and the efficiency of the proposed algorithm.

Excision of Ocular Artifacts from EEG Using NVFF-RLS Adaptive Algorithm

This paper presents a technique for the removal of ocular artifacts from electro-encephalogram (EEG) by using adaptive filtering. The major concern is electro-oculogram (EOG) signal present in a recorded EEG signal, which appears due to abrupt eye movements. In the presented method, we use separately recorded horizontal EOG (HEOG) and vertical EOG (VEOG) signals as two reference inputs, which are processed using finite impulse response (FIR) filters. The linear filter coefficients are adaptively updated using a numerical variable forgetting factor (NVFF) recursive least squares (RLS) algorithm, which tracks nonstationary EOG signals. Subsequently, the processed HEOG and VEOG signals are subtracted from recorded EEG signal to obtain an artifact-free EEG signal. Simulation is conducted using synthetic EEG signal corrupted by noise, synthetic HEOG and VEOG signals. The real-time recorded EEG signal (corrupted by EOG and noise) is also refined using the separately recorded reference EOG signals and FIR filtering technique. For synthetic and real-time signals, the simulation results are presented to demonstrate that linear NVFF-RLS algorithm-based artifact and noise excision technique outperforms conventional fixed forgetting factor RLS, fixed step-size NLMS and generalized variable step-size NLMS algorithms, in terms of the reduction in mean-squared error, under low as well as high signal-to-noise ratio conditions.

Improved variable step-size NLMS adaptive filtering algorithm for acoustic echo cancellation

Acoustic echo canceller (AEC) is used in communication and teleconferencing systems to reduce undesirable echoes resulting from the coupling between the loudspeaker and the microphone. In this paper, we propose an improved variable step-size normalized least mean square (VSS-NLMS) algorithm for acoustic echo cancellation applications based on adaptive filtering. The steady-state error of the NLMS algorithm with a fixed step-size (FSS-NLMS) is very large for a non-stationary input. Variable step-size (VSS) algorithms can be used to decrease this error. The proposed algorithm, named MESVSS-NLMS (mean error sigmoid VSS-NLMS), combines the generalized sigmoid variable step-size NLMS (GSVSS-NLMS) with the ratio of the estimation error to the mean history of the estimation error values. It is shown from single-talk and double-talk scenarios using speech signals from TIMIT database that the proposed algorithm achieves a better performance, more than 3 dB of attenuation in the misalignment evaluation compared to GSVSS-NLMS, non-parametric VSS-NLMS (NPVSS-NLMS) and standard NLMS algorithms for a non-stationary input in noisy environments.

Variable is better than invariable: sparse VSS-NLMS algorithms with application to adaptive MIMO channel estimation.

Channel estimation problem is one of the key technical issues in sparse frequency-selective fading multiple-input multiple-output (MIMO) communication systems using orthogonal frequency division multiplexing (OFDM) scheme. To estimate sparse MIMO channels, sparse invariable step-size normalized least mean square (ISS-NLMS) algorithms were applied to adaptive sparse channel estimation (ACSE). It is well known that step-size is a critical parameter which controls three aspects: algorithm stability, estimation performance, and computational cost. However, traditional methods are vulnerable to cause estimation performance loss because ISS cannot balance the three aspects simultaneously. In this paper, we propose two stable sparse variable step-size NLMS (VSS-NLMS) algorithms to improve the accuracy of MIMO channel estimators. First, ASCE is formulated in MIMO-OFDM systems. Second, different sparse penalties are introduced to VSS-NLMS algorithm for ASCE. In addition, difference between sparse ISS-NLMS algorithms and sparse VSS-NLMS ones is explained and their lower bounds are also derived. At last, to verify the effectiveness of the proposed algorithms for ASCE, several selected simulation results are shown to prove that the proposed sparse VSS-NLMS algorithms can achieve better estimation performance than the conventional methods via mean square error (MSE) and bit error rate (BER) metrics.

Robust Auxiliary-Noise-Power Scheduling in Active Noise Control Systems With Online Secondary Path Modeling

This paper deals with the auxiliary noise-based methods for active noise control (ANC) systems with online secondary path modeling (SPM). The proposed method comprises two adaptive filters: the modified Filtered-X normalized least-mean-square algorithm-based ANC filter, and the normalized least-mean-square algorithm-based SPM filter excited by auxiliary noise. The auxiliary noise injected for online SPM, degrades the noise-reduction performance of the ANC system. A two-stage gain scheduling strategy is proposed to vary power of the auxiliary noise. In the first stage the gain is varied on the basis of power of the error signal of SPM filter, and in the second stage the gain is varied on the basis of the correlation estimate of the two adjacent samples of the error signal of SPM filter. The main idea is to inject large-power auxiliary noise at the start up or when a change in the acoustic paths is detected, and to reduce the power as the system converges. The proposed method achieves a fast convergence of the SPM filter and gives a robust performance in the presence of strong perturbation in acoustic paths. Furthermore, the proposed method improves the noise-reduction performance at steady-state even in the presence of an uncorrelated disturbance at the error microphone. Moreover, the improved performance is achieved at a lower computational cost as compared with a recent method proposed in [A. Carini, and S. Malatini, “Optimal variable step-size NLMS algorithms with auxiliary noise power scheduling for feedforward active noise control,”IEEE Trans. Audio, Speech Lang. Process., vol. 16, no. 8, pp. 1383-1395, Nov. 2008]. Extensive simulations are carried out to verify the effectiveness of the proposed method.

A New Variable Step-Size NLMS Algorithm and Its Performance Analysis

Numerous variable step-size normalized least mean-square (VSS-NLMS) algorithms have been derived to solve the dilemma of fast convergence rate or low excess mean-square error in the past two decades. This paper proposes a new, easy to implement, nonparametric VSS-NLMS algorithm that employs the mean-square error and the estimated system noise power to control the step-size update. Theoretical analysis of its steady-state behavior shows that, when the input is zero-mean Gaussian distributed, the misadjustment depends only on a parameter β controlling the update of step size. Simulation experiments show that the proposed algorithm performs very well. Furthermore, the theoretical steady-state behavior is in very good agreement with the experimental results.

A Novel Variable Step-Size NLMS Algorithm and Its Analysis

By building the generalized Sigmoid function relationship between normalized step-size and the power of error signal, a novel variable step-size NLMS algorithm is proposed. It is proved that the step-size of NPVSS-NLMS changes as the new algorithm does when A=σv -m and B=2. The physical meanings of the parameters in this algorithm are explored. The theoretical analysis illustrate that this algorithm combine the virtues of NPVSS-NLMS and Sigmoid function, and it leads to faster convergence rate and lower final misalignment. The computer simulation results support the theoretical analysis.

A New Variable Step-Size NLMS Algorithm and its Performance Evaluation in Echo Cancelling Applications

In this work we introduce a variable step-size normalized LMS algorithm for adaptive echo cancellation in a FIRstructure. In the proposed scheme, the step-size adjustment is controlled by using the square of the cross-correlationbetween the squared output error and the adaptive filter output. The proposed algorithm (that we call VSSSC aftervariable step size based on the squared cross-correlation) was evaluated using white noise and speech signals.Simulation results show that our proposal achieves better performance than similar algorithms in single and doubletalk. The proposed algorithm can be used in a number of applications such as in echo reduction for long-haul voicecommunications.

A Variable Step-Size Proportionate NLMS Algorithm for Identification of Sparse Impulse Response

Recently, proportionate adaptive algorithms have been proposed to speed up convergence in the identification of sparse impulse response. Although they can improve convergence for sparse impulse responses, the steady-state misalignment is limited by the constant step-size parameter. In this article, based on the principle of least perturbation, we first present a derivation of normalized version of proportionate algorithms. Then by taking the disturbance signal into account, we propose a variable step-size proportionate NLMS algorithm to combine the benefits of both variable step-size algorithms and proportionate algorithms. The proposed approach can achieve fast convergence with a large step size when the identification error is large, and then considerably decrease the steady-state misalignment with a small step size after the adaptive filter reaches a certain degree of convergence. Simulation results verify the effectiveness of the proposed approach.