Steady-state Mean Square Error Research Articles

Theoretical convergence analysis of the FXLMS-based feedforward hybrid active noise control system

Real noise typically includes both broadband and narrowband components, such as noise generated by rotating machines. The feedforward hybrid active noise control (FFHANC) system reduces both components separately by arranging broadband and narrowband controllers in parallel. Although considerable efforts have been made to improve the FFHANC system, a theoretical analysis of its performance is still lacking. Consequently, the application of the FFHANC system lacks a solid foundation. Furthermore, the stability bounds of step sizes that guarantee convergence are not available in the literature. To address these problems, a convergence analysis of the FFHANC system based on the filtered-x least-mean-square algorithm is proposed. The analysis of the mean and mean-square convergence behaviors of the weight errors theoretically illustrates the advantages of the FFHANC system in terms of fast convergence and high reduction. The mutual coupling of parallel controllers in the mean-square sense is derived and discussed. By predicting the steady-state mean-square error and stability bounds, the proposed model can also guide the choice of step sizes. Our analysis lays the theoretical foundation for the practical application of the FFHANC system. Extensive simulations are performed to demonstrate the validity of the proposed theoretical theory.

An Improved Recursive Least Square Algorithm For Adapting Fuzzy Channel Equalizer

Adaptive filters have been thoroughly investigated in digital communication. They are especially exploited as equalizers, to compensate for channel distortions, although equalizers based on linear filters perform poorly in nonlinear distortion. In this paper, a nonlinear equalizer based on a fuzzy filter is proposed and a new algorithm for the adaptation parameters is presented. The followed approach is based on a regularization of the Recursive Least Square (RLS) algorithm and an incorporation of fuzzy rules in the adaptation process. The proposed approach, named Improved Fuzzy Recursive Least Square (IFRLS), enhances significantly the fuzzy equalizer performance through the acquisition of more convergence properties and lower steady-state Mean Square Error (MSE). The efficiency of the IFRLS algorithm is confirmed through extensive simulations in a nonlinear environment, besides the conventional RLS, in terms of convergence abilities, through MSE, and the equalized signal behavior. The IFRLS algorithm recovers the transmitted signal efficiently and leads to lower steady-state MSE. An improvement in convergence abilities is noticed, besides the RLS.

Analysis of the LMS and NLMS algorithms using the misalignment norm

This work describes the convergence of the misalignment square norm (MSN) of the NLMS and LMS algorithms. It is shown that the MSN decrease is almost proportional to the mean square error (MSE). This allows obtaining simple expressions for the steady-state MSE. Also, it allows limiting the amount of time that the MSE takes large values and a curve that limits the MSE of LMS at any given time, independent of the input and background noise signals’ properties. Finally, it is also shown that many complications in the analysis of the LMS and NLMS algorithms can come from variations in the input vector square norm. The proposed analysis becomes very simple for long filters or constant power signals.

Estimating and Tracking Wireless Channels Under Carrier and Sampling Frequency Offsets

This article addresses the challenge of estimating and tracking wireless channels under carrier and sampling frequency offsets, which also incorporate phase noise and sampling time jitter. We propose a novel adaptive filter that explicitly estimates the channel impulse response, carrier frequency offset, and sampling frequency offset by minimizing the mean-square error (MSE) and, when the estimated parameters are time-varying, inherently performs tracking. The proposed filter does not have any requirements for the structure of the waveform, but the digital transmitted waveform must be known to the receiver in advance. To aid practical implementation, we derive upper bounds for the filter's step sizes. We also derive expressions for the filter's steady-state MSE performance, by extending the well-known energy conservation relation method to account for the self-induced nonstationarity and coupling of update equations that are inherent in the proposed filter. Theoretical findings are verified by comparison to simulated results. Proof-of-concept measurement results are also provided, which demonstrate that the proposed filter is able to estimate and track a practical wireless channel under carrier and sampling frequency offsets.

The $\mathcal{H}_2$-optimal Control Problem of CSVIU Systems: Discounted, Counterdiscounted, and Long-Run Solutions

The paper deals with stochastic control problems associated with $H_2$ performance indices such as energy or power norms or energy measurements. The control applies to a class of systems for which a stochastic process conveys the underlying uncertainties, known as control and state variation increase uncertainty (CSVIU). These indices allow various emphases, focusing on the transient behavior with the discounted norm to stricter conditions on stability and steady-state mean-square error and convergence rate, using the optimal overtaking criterion. The long-run average power control stands as a midpoint in this respect. A critical advance regards the explicit form of the optimal control law, expressed in two forms. One takes a perturbed affine Riccati-like form of feedback solution; the other comes from a generalized normal equation that arises from the nondifferentiable local optimal problem. They are equivalent, but the former allows optimality and stability, and the latter develops into a search method to attain the optimal law. A detectability notion and a Riccati solution grant stochastic stability from the behavior of the norms. The energy overtaking criterion requires a further constraint on a matrix spectral radius. With these findings, the paper revisits the rising of the inaction solution, a prominent feature of CSVIU models to deal with the uncertainty inherent to poorly known models. Besides the optimal solution, it provides the tools to pursue it.

Diffusion augmented complex-valued LMS algorithm with shared measurements and its performance analysis

The diffusion augmented complex-valued LMS (DACLMS) algorithm was proposed for distributed estimation over networks to deal with noncircular signals. This paper develops an improved DACLMS by sharing measurements between nodes within the same neighborhood. To predict its stochastic behavior, including the transient mean weight error, standard mean-square error (MSE), and complementary MSE (CMSE), is analyzed under some frequently-used statistical assumptions. A joint consideration of the MSE and CMSE performance is also analyzed to provide extra insight to the stochastic behavior of the proposed algorithm. Furthermore, the closed-form expressions for the steady-state MSE and CMSE are subsequently derived based on the transient analysis results. Since the proposed steady-state performance analysis does not assume that the network has the same data statistics at all nodes, it is applicable to the DACLMS without shared measurements and more general than the existing steady-state analysis. Simulation results are finally provided to verify the accuracy of the theoretical expressions and the effectiveness of the proposed algorithm.

Steady-State Mean-Square Error Analysis for Adaptive Joint Process Filter Using Pipelined Feedforward Second-Order Volterra Architecture

Steady-State Mean-Square Error Analysis for Adaptive Joint Process Filter Using Pipelined Feedforward Second-Order Volterra Architecture

Steady-State Mean-Square Error Analysis for Non-Negative Least Lncosh Algorithm

Lncosh cost function is regarded as a natural logarithm obtaining from a hyperbolic cosine function, which has drawn growing attention due to its robust to impulsive noise. In this brief, a nonnegative adaptive algorithm is proposed based on lncosh function, named as NNLlncosh, which is derived by incorporating the nonnegativity constraint into lncosh cost function to deal with a nonnegativity constraint optimization problem under impulsive system noises. The steady-state excess mean square error (EMSE) for the newly constructed NNLlncosh algorithm is presented. Obtained results from the computer simulation validate the theoretical analysis and verify the excellent characteristics of the NNLlncosh over various non-Gaussian system noises.

Adaptive modulation based on steady-state mean square error for underwater acoustic communication

To improve the transmission efficiency and facilitate the realization of the scheme, an adaptive modulation (AM) scheme based on the steady-state mean square error (SMSE) of blind equalization is proposed. In this scheme, the blind equalization is adopted and no training sequence is required. The adaptive modulation is implemented based on the SMSE of blind equalization. The channel state information doesn’t need to be assumed to know. To better realize the adjustment of modulation mode, the polynomial fitting is used to revise the estimated SNR based on the SMSE. In addition, we also adopted the adjustable tap-length blind equalization detector to obtain the SMSE, which can adaptively adjust the tap-length according to the specific underwater channel profile, and thus achieve better SMSE performance. Simulation results validate the feasibility of the proposed approaches. Simulation results also show the advantages of the proposed scheme against existing counterparts.

A variable-scale S-type kernel fractional low-power adaptive filtering algorithm

The adaptive kernel algorithms usually achieve a good convergence performance and a tracking performance due to the universal approximator, offering an excellent solution to many problems with nonlinearities. However, as is well known, the convergence rate and steady-state error of adaptive filtering algorithm are a pair of inherent contradictions, and the kernel method is not exceptional. For this problem, a robust kernel adaptive filtering algorithm, called the variable-scaling factor kernel fractional lower power adaptive filtering algorithm based on the Sigmoid function, is developed by creating a new framework of cost function which combines the kernel fractional low power error criterion with the Sigmoid function for system identification of different noise environments. This new cost framework incorporates a scaling factor into the cost function of the Sigmoid kernel fractional lower power adaptive filtering algorithm (VS-SKFLP) in this paper. One of the main features in the new framework is its scaling factor. This scaling factor is used to control the steepness of the Sigmoid function, and the steepness can affect the convergence speed of filtering algorithm. The scaling factor provides a tradeoff between the convergence rate and the steady-state mean square error (MSE), which improves the convergence rate under the same steady-state mean square error. However, it is also an important problem to choose an appropriate scale factor. Therefore, a variable-scale factor SKFLP algorithm is also proposed to improve the convergence rate and steady-state MSE, simultaneously. The proposed variable-scale factor structure consists of a function of error, featuring the adaptive updates of their parameter estimated by making discerning use of the error. In this paper, the nonlinear saturation characteristic of the Sigmoid function and low order norm criterion are used to overcome the performance degradation of training data destroyed by non-Gaussian impulse noise and colored noise. Through the convergence analysis, the parameter estimation sequence of our proposed algorithm proves convergent. Simulation results show that the proposed algorithm (VS-SKFLP) outperforms other kernel adaptive filtering algorithms in system recognition with different noise environments.

Projected Kernel Least Mean $p$ -Power Algorithm: Convergence Analyses and Modifications

Sparsified kernel adaptive filters (SKAFs) is an attractive filtering solution with low memory and computational complexity. Most of existing SKAFs are based on the mean square error (MSE) criterion under Gaussian noise assumption for its simplicity and convenience. When the assumption deviates largely from the underlying truth, the performance of these methods could degrade significantly. In this paper, we propose a novel SKAF, named as projected kernel least mean $p$ -power algorithm (PKLMP), based on the mean $p$ -power error (MPE) criterion and vector projection (VP) method. We provide convergence analyses in terms of the stead-state MSE, based on a Taylor expansion method, and derive the lower and upper bounds for the steady-state excess MSE. We also conduct mean convergence analysis for PKLMP, and derive convergence conditions. To exploit the information in the desired outputs, we further derive a modified PKLMP by smoothing the desired signal. Finally, a simple and effective online variable kernel centers strategy is proposed to improve the filtering performance of the proposed KAFs. Simulation results under a static function estimation, a chaotic time-series prediction, and two real-world time-series predictions are conducted and validate the effectiveness of the proposed PKLMP algorithms.

Steady-State Mean-Square Error Performance Analysis of the Tensor LMS Algorithm

In some system identification problems, the weight vector to be estimated can be represented by a tensor product of two low dimensional component vectors. This feature is useful for the design of adaptive filters with fast convergence rate and/or low computational complexity. The tensor LMS algorithm has been proposed under this scenario and its recent extensions find applications in several contexts. This brief analyzes the steady-state mean-square error (MSE) performance of the tensor LMS algorithm, which can provide performance prediction and design guideline for engineers. Since the updates of the weight-vectors of the component adaptive filters (CAFs) are coupled, it is challenging to perform steady-state MSE analysis. To address this problem, this brief first establishes the variance relation in steady state for the CAFs and then decouples their excess MSEs by solving a system of linear equations. Finally, the steady-state MSE of the reproduced adaptive filter is composed of the excess MSEs of the CAFs and the variance of the system noise. Simulations are performed to verify the accuracy of the theoretical findings.

Feature Adaptive Filtering: Exploiting Hidden Sparsity

We have been witnessed a growing research activity to advance new strategies to detect and exploit underlying sparsity in the parameters of physical models. In many cases, the sparsity is not explicit in the relations among the parameter coefficients requiring some suitable tools to reveal the potential sparsity. This work proposes a family of adaptive filtering algorithms, aimed at exposing some hidden features of the unknown parameters. Although the basic idea applies to any algorithm, we will concentrate the work in the LMS-type algorithms, giving rise to a family collectively named as Feature LMS (F-LMS) algorithms. These algorithms increase the convergence speed and reduce the steady-state mean-squared error, in comparison with the classical LMS solution. The main idea is to apply linear transformations, through the so-called feature matrices, to reveal the sparsity hidden in the coefficient vector, followed by a sparsity-promoting penalty function to exploit the exposed sparsity. For illustration, a few F-LMS algorithms for lowpass, bandpass, and highpass systems are introduced by using simple feature matrices that require either only simple operations or can learn the features. Simulations and real-life experiments demonstrate that the F-LMS algorithms bring about several performance improvements whenever the unknown sparsity of parameters is exposed.

The Extended Feature LMS Algorithm: Exploiting Hidden Sparsity for Systems with Unknown Spectrum

The feature least-mean-square (F-LMS) algorithm has already been introduced to exploit hidden sparsity in lowpass and highpass systems. In this paper, by proposing the extended F-LMS (EF-LMS) algorithm, we boosted the F-LMS algorithm to exploit hidden sparsity in more general systems, those which are neither lowpass nor highpass. To this end, by means of the so-called feature matrix, we reveal the hidden sparsity in coefficients and utilize the $$l_1$$-norm to exploit the exposed sparsity. As a result, the EF-LMS algorithm will improve the convergence rate and the steady-state mean-squared error (MSE) as compared to the traditional least-mean-square algorithm. Moreover, in this work, we analyze the convergence behavior of the coefficient vector and the steady-state MSE performance of the EF-LMS algorithm. Through synthetic and real-world experiments, it has been seen that the EF-LMS algorithm can improve the convergence rate and the steady-state MSE whenever the hidden sparsity is revealed.

On the Stochastic Modeling of the NSVR–IAF–PNLMS Algorithm for Correlated Gaussian Input Data

This paper presents a stochastic model for the normalized smoothed variation rate individual-activation-factor proportionate normalized least-mean-square (NSVR–IAF–PNLMS) algorithm. Specifically, taking into account correlated Gaussian input data, model expressions are derived for predicting the mean weight vector, gain distribution matrix, NSVR metric, learning curve, weight-error correlation matrix, and steady-state excess mean-square error. Such expressions are obtained by considering the time-varying characteristics of the gain distribution matrix. Simulation results are shown confirming the accuracy of the proposed model for different operating conditions.

Distributed Separated-Decorrelation LMS Algorithms Over Sensor Networks With Noisy Inputs

When the input is a highly correlated and noisy signal over sensor networks, it will lead to the severe performance degeneration of traditional distributed algorithms in terms of convergence rate and steady-state error. To tackle such an issue, this paper proposes distributed bias-compensated separated-decorrelation least mean-square (BC-SDLMS) algorithms. Due to the adoption of new separated-decorrelation structure and bias-compensated term, the steady-state mean-square error with the proposed algorithms can be reduced in comparison with the previous decorrelation schemes. The mean-square analysis is also carried out, which indicates that the proposed algorithms can converge to an unbiased solution. Moreover, an effective estimate for the noise variance at the input terminal of every node is designed. Finally, simulation comparisons are made between the proposed BC-SDLMS algorithms and the competing methods in terms of convergence rate and steady-state error for different colored and noisy inputs.

Tracking Analysis of Gaussian Kernel Signed Error Algorithm for Time-Variant Nonlinear Systems

This brief establishes a novel kernel-based model with a random walk variation of the optimum weight coefficients to characterize the time-variant nonlinear system. Then, the steady-state tracking performance of the kernel signed error algorithm (KSEA) with Gaussian kernel is analyzed for the proposed time-variant nonlinear system in the presence of non-Gaussian impulsive noise. The theoretical findings enable us to determine the optimal step-size that minimizes the steady-state excess mean-square error under this non-stationary environment. Simulation results illustrate the usefulness and accuracy of the derived analytical models for characterizing the steady-state tracking behavior of Gaussian KSEA.

An Adaptive Upper Threshold Based Gain Function for the ZA-PNLMS Algorithm

The recently proposed Zero-Attracting Proportionate Normalized Least Mean Square (ZA-PNLMS) algorithm improves the performance of the PNLMS algorithm for identifying sparse systems. In particular, it keeps the fast initial convergence rate of the PNLMS algorithm, and improves its transient performance by arresting the fall in the convergence rate at the later stage of the adaptation process, and it also improves the steady-state mean square error (MSE). However, the improvement in the steady-state performance is marginal. In this brief, we propose a novel gain function for the ZA-PNLMS algorithm by introducing an adaptive upper threshold parameter. It has two fold implications. First, the proposed threshold parameter truncates the proportional gains of highly active taps when they approach to their steady-states and helps the lesser active taps to converge faster by providing them more gains, and thereby the overall transient performance is further improved. Secondly, it also improves steady-state MSE significantly by reducing the fluctuation of the active taps at the steady-state. Extensive simulation studies have been carried out to justify the improvement obtained by the proposed algorithm.

Multichannel Active Noise Control Based on Filtered-x Affine Projection-Like and LMS Algorithms with Switching Filter Selection

Affine projection (AP) algorithms have been demonstrated to have faster convergence speeds than the conventional least mean square (LMS) algorithms. However, LMS algorithms exhibit smaller steady-state mean square errors (MSEs) when compared with affine projection (AP) algorithms. Recently, several authors have proposed alternative methods based on convex combinations to improve the steady-state MSE of AP algorithms, even with the increased computational cost from the simultaneous use of two filters. In this paper, we present an alternative method based on an affine projection-like (APL-I) algorithm and least mean square (LMS) algorithm to solve the ANC under stationary Gaussian noise environments. In particular, we propose a switching filter selection criteria to improve the steady-state MSE without increasing the computational cost when compared with existing models. Here, we validate the proposed strategy in a single and a multichannel system, with and without automatically adjusting the scaling factor of the APL-I algorithm. The results demonstrate that the proposed scheme exploits the best features of each filter (APL-I and LMS) to guarantee rapid convergence with a low steady-state MSE. Additionally, the proposed approach demands a low computational burden compared with existing convex combination approaches, which will potentially lead to the development of real-time ANC applications.

Normalized LMS algorithm and data-selective strategies for adaptive graph signal estimation

This work proposes a normalized least-mean-squares (NLMS) algorithm for online estimation of bandlimited graph signals (GS) using a reduced number of noisy measurements. As in the classical adaptive filtering framework, the resulting GS estimation technique converges faster than the least-mean-squares (LMS) algorithm while being less complex than the recursive least-squares (RLS) algorithm, both recently recast as adaptive estimation strategies for the GS framework. Detailed steady-state mean-squared error and deviation analyses are provided for the proposed NLMS algorithm, and are also employed to complement previous analyses on the LMS and RLS algorithms. Additionally, two different time-domain data-selective (DS) strategies are proposed to reduce the overall computational complexity by only performing updates when the input signal brings enough innovation. The parameter setting of the algorithms is performed based on the analysis of these DS strategies, and closed formulas are derived for an accurate evaluation of the update probability when using different adaptive algorithms. The theoretical results predicted in this work are corroborated with high accuracy by numerical simulations.