Robust least mean logarithmic square adaptive filtering algorithms

A robust diffusion algorithm using logarithmic hyperbolic cosine cost function for channel estimation in wireless sensor network under impulsive noise environment

In this paper, we investigate the convergence performance of a sparsified kernel least mean square (KLMS) algorithm in which the input is added into the dictionary only when the prediction error in amplitude is larger than a preset threshold. Under certain conditions, we derive an approximate value of the steady-state excess mean square error (EMSE). Simulation results confirm the theoretical predictions and provide some interesting findings, showing that the sparsification can not only be used to constrain the network size (hence reduce the computational burden) but also be used to improve the steady-state performance in some cases.

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.

Robust kernel adaptive filtering for nonlinear time series prediction

The adaptive filtering algorithms based on the arctangent cost function framework have shown robustness against impulsive noise. In this brief, the standard least mean square (LMS) algorithm under this framework is concentrated on, which is called the arctangent LMS (ATLMS) algorithm. The steady-state excess mean square error (EMSE) and mean square deviation (MSD) of the ATLMS algorithm are analyzed using the energy conservation relation. In stationary environment, both Gaussian and non-Gaussian situations are discussed. The closed-form expressions of the steady-state EMSE and MSD are obtained using Taylor’s expansion. In non-stationary environment, a first-order random-walk model is used for modeling the time-varying optimal weight. Theoretical steady-state performance and the optimal step size are derived. Simulation results under different noise environments verify the validity of our theoretical findings.

Merging the adaptive networks with the free space optical (FSO) communication technology is a very interesting field of research because by adding the benefits of this technology, the adaptive networks become more efficient, cheap and secure. This is due to the fact that FSO communication uses unregistered visible light bandwidth instead of the overused radio spectrum. However, in spite of all the benefits of FSO communication, this technology suffers from optical noise and turbulence. In this paper, we investigate the exact effect of the negative exponential and k-distribution modeled very strong turbulence conditions on the performance of diffusion adaptive networks. The simulation and theoretical results based on the steady state Mean square deviation (MSD) and Excess mean square Error (EMSE) vlaues show the deteriorating effects of these link models on diffusion networks. The FSO communication technology, while very profitable and applicable, is not always a suitable means of implementing wireless networks. For this reason, we suggested the channel estimation for these conditions.

Smoothed least mean p-power error criterion for adaptive filtering

Based on the theory of simple preconditioned iterative linear equation solvers, we recently introduced a framework within which all major adaptive filter algorithms can be viewed as special cases through the specification of a few parameters. This resulted in two versions of a generic adaptive filter update equation which we in this paper use in deriving general explicit expressions for the learning curve, the steady state excess mean square error (EMSE) and the steady state mean square coefficient deviation (MSD) that are applicable to many families of adaptive filter algorithms. We present experimental results supporting the usefulness and validity of our approach.

Adaptive filters play an important role in the fields related to digital signal processing and communication, such as system identification, noise cancellation, channel equalization, and beamforming. In practical applications, the computational complexity of an adaptive filter is an important consideration. The Least Mean Square (LMS) algorithm is widely used because of its low computational complexity ($O(N)$) and simplicity in implementation. The least squares algorithms, such as Recursive Least Squares (RLS), Conjugate Gradient (CG), and Euclidean Direction Search (EDS), can converge faster and have lower steady-state mean square error (MSE) than LMS. However, their high computational complexity ($O(N^2)$) makes them unsuitable for many real-time applications. A well-known approach to controlling computational complexity is applying partial update (PU) method to adaptive filters. A partial update method can reduce the adaptive algorithm complexity by updating part of the weight vector instead of the entire vector or by updating part of the time. In the literature, there are only a few analyses of these partial update adaptive filter algorithms. Most analyses are based on partial update LMS and its variants. Only a few papers have addressed partial update RLS and Affine Projection (AP). Therefore, analyses for PU least-squares adaptive filter algorithms are necessary and meaningful. This monograph mostly focuses on the analyses of the partial update least-squares adaptive filter algorithms. Basic partial update methods are applied to adaptive filter algorithms including Least Squares CMA (LSCMA), EDS, and CG. The PU methods are also applied to CMA1-2 and NCMA to compare with the performance of the LSCMA. Mathematical derivation and performance analysis are provided including convergence condition, steady-state mean and mean-square performance for a time-invariant system. The steady-state mean and mean-square performance are also presented for a time-varying system. Computational complexity is calculated for each adaptive filter algorithm. Numerical examples are shown to compare the computational complexity of the PU adaptive filters with the full-update filters. Computer simulation examples, including system identification and channel equalization, are used to demonstrate the mathematical analysis and show the performance of PU adaptive filter algorithms. They also show the convergence performance of PU adaptive filters. The performance is compared between the original adaptive filter algorithms and different partial-update methods. The performance is also compared among similar PU least-squares adaptive filter algorithms, such as PU RLS, PU CG, and PU EDS. In addition to the generic applications of system identification and channel equalization, two special applications of using partial update adaptive filters are also presented. One application uses PU adaptive filters to detect Global System for Mobile Communication (GSM) signals in a local GSM system using the Open Base Transceiver Station (OpenBTS) and Asterisk Private Branch Exchange (PBX). The other application uses PU adaptive filters to do image compression in a system combining hyperspectral image compression and classification.

Maximum Correntropy Criterion (MCC) based adaptive filters had received much attention due to its robustness against impulsive noise. In this paper the convergence analysis of MCC adaptive filter based on white input is performed. The condition for stability is analyzed and the steady state mean square error (MSE) and mean square deviation (MSD) error is derived in terms of step size, variance of noise source, length of the system and kernel width. Moreover a criteria for parameter selection to obtain improved performance of MCC adaptive filter is also proposed. Simulations in the context of unknown system identification scenario were performed to prove the validity of the theoretical analysis made.

Performance analysis of the deficient length augmented CLMS algorithm for second order noncircular complex signals

A novel methodology to bound the steady-state mean square performance of the diffusion complex least mean square (D-CLMS) and the diffusion widely linear (augmented) CLMS (D-ACLMS) algorithm is proposed. This is achieved by exploiting the almost identical nature of the steady-state filter weights at all nodes. The proposed approach allows for the consideration of the second-order terms in the recursion for the weight error covariance matrix, without compromising the mathematical tractability of the problem. The closed form expressions for the mean square deviation (MSD) and excess mean square error (EMSE) for both the D-CLMS and D-ACLMS allow for the performance of the algorithms to be quantified as a function of the noncircularity of the input data.

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

Constrained adaptive filtering algorithms have been extensively studied in many applications. Most existing constrained adaptive filtering algorithms are developed under the mean square error (MSE) criterion, which is an ideal optimality criterion under Gaussian noises. This assumption however fails to model the behavior of non-Gaussian noises found in practice. Motivated by the robustness and simplicity of maximum correntropy criterion (MCC) for non-Gaussian impulsive noises, this paper proposes a new adaptive filtering algorithm called constrained maximum correntropy criterion (CMCC). Specifically, CMCC incorporates a linear constraint into a MCC filter to solve a constrained optimization problem explicitly. The proposed adaptive filtering algorithm is easy to implement, has low computational complexity, and can significantly outperform those MSE based constrained adaptive algorithms in heavy-tailed impulsive noises. Additionally, the mean square convergence behaviors are studied under energy conservation relation, and a sufficient condition to ensure the mean square convergence and the steady-state mean square deviation (MSD) of the CMCC algorithm are obtained. Simulation results confirm the theoretical predictions under both Gaussian and non-Gaussian noises, and demonstrate the excellent performance of the novel algorithm by comparing it with other conventional methods.

This paper proposes a new regularized transform domain normalized LMS (R-TDNLMS) algorithm and studies its mean and mean square convergence performances. The proposed algorithm extends the conventional TDNLMS algorithm by imposing a regularization term on the filter coefficients to reduce the variance of estimators due to the lacking of excitation in a certain frequency band or in the presence of modeling errors. Difference equations describing the mean and mean square convergence behaviors of this algorithm are derived so as to characterize its convergence condition and steady-state excess mean square error (MSE). It shows that regularization can help to reduce the MSE by trading slight bias for variance. Based on this analysis, a new formula to select the regularization parameter for white Gaussian inputs is proposed, which leads to a new variable regularized TDNLMS (VR-TDNLMS) algorithm. Computer simulations are conducted to examine the improved convergence performance, steady-state MSE and robustness to power-varying inputs of the proposed algorithm and verify the effectiveness of the theoretical analysis. Furthermore, the application of the proposed VR-TDNLMS algorithm to the design and implementation of acoustic system identification and active noise control (ANC) systems show that they considerably outperforms traditional TDNLMS algorithms at low excitation or in the presence of modeling errors. Moreover, the theoretical analysis provides simple design formulas for achieving a given excess MSE (EMSE) and step-size bound for stable operation.