Normalized Subband Adaptive Filter Research Articles

Constrained Normalized Subband Adaptive Filter Algorithm and Its Performance Analysis

AbstractConstrained least mean square (CLMS) algorithm is the most popular constrained adaptive filtering algorithm due to its simple structure and easy implementation. However, its convergence slows down when the input signal is colored. To address this issue, this paper firstly introduces the normalized subband adaptive filter (NSAF) into the constrained filtering problem and derives a constrained NSAF (CNSAF) algorithm using the Lagrange multiplier method. Benefiting from the good decorrelation capability of the NSAF, the proposed CNSAF algorithm significantly improves the convergence performance of the CLMS algorithm under colored inputs. Then, the mean and mean-square stability of the CNSAF algorithm is analyzed, and the theoretical models to characterize the transient and steady-state mean square deviation (MSD) behaviors of the CNSAF algorithm are derived utilizing the Kronecker product property and vectorization method. Further to extend the CNSAF algorithm to the problem of sparse system identification, a sparse version of the CNSAF algorithm (S-CNSAF) is derived. Finally, the validity of the derived theoretical MSD prediction models and the superiority of the proposed algorithms are confirmed by extensive computer simulations on system identification with colored inputs.

Robust Normalized Subband Adaptive Filter Algorithm Based on Logarithmic and Total Least Squares for Correlated Input Signals and Impulsive Noise

The total least squares normalized subband adaptive filter (TLS-NSAF) algorithm proposed in recent years has shown excellent performance in processing the errors-in-variables (EIV) model of correlated input signals. However, when the system is Interferenced by the impulsive noise, the convergence of TLS-NSAF algorithm will be seriously deteriorated. To address this problem, this paper improves the TLS-NSAF algorithm by using the logarithmic function, and proposes a robust NSAF algorithm based on logarithmic and total least squares method (RNSAF-LTLS). The algorithm performs well in an environment where input signal is a correlated input signal and output signal contains impulsive noise. In addition, there are few studies in the existing literature that apply the TLS method to subbands, and there are few performance analyses associated with it. To address this problem, this paper analyzes the local stability, derives the step size that guarantees the stability of the RNSAF-LTLS algorithm, and calculates the steady-state mean squared deviation (S-MSD). Finally, the RNSAF-LTLS algorithm is verified in system identification (SI) and acoustic echo cancellation (AEC) applications. The simulation results prove the superiority of the proposed algorithm and the correctness of the theoretical analysis.

Alternate Implementation of NSAF and NLMS Learning Rules for Adaptive Filters

From the perspective of the original time-domain, the normalized subband adaptive filtering (NSAF) algorithm leads to the freeze of the filter update in N-1 out of every N iterations so that slowing convergence of the filter. To fill this flaw, this paper proposes to perform the normalized least mean square (NLMS) algorithm in this frozen interval and then obtain the NSAF-NLMS algorithm with faster convergence. Moreover, we analyze the mean-square behavior of the NSAF-NLMS algorithm and give its stability condition, transient model, and steady-state model. Then, to handle sparse systems, the proportionate NSAF-NLMS algorithm is developed further to speed up the filter’s convergence. Simulation results have demonstrated the performance analysis and the superiority of the proposed algorithms over the existing counterparts.

Robust Subband Adaptive Filter Algorithms-Based Mixture Correntropy and Application to Acoustic Echo Cancellation

To acquire an improvement of the performance of the subband adaptive filter with impulsive interference, the normalized subband adaptive filter (NSAF) algorithm-based maximum correntropy criterion (MCC), called MCC-NSAF, has been developed. However, it is hard for the MCC criterion to take into account both the fast convergence rate and low steady-state error with a fixed kernel bandwidth. To handle this dilemma, in this paper, the normalized subband adaptive filter algorithm-based maximum mixture correntropy criterion (MMC) algorithm, called MMC-NSAF, is proposed. The MMC-NSAF algorithm integrates two correntropies with complementary control sizes of Gaussian kernel and can not only converge quickly but also has a small misalignment. Yet the MMC-NSAF still has some limitations because of its fixed step size. Therefore, a convexly combined MMC-NSAF algorithm, namely MMC-CNSAF, is proposed. The MMC-CNSAF algorithm utilizes two Gaussian kernels of different size and can combine overall improvement between convergence rate and steady-state error simultaneously. Eventually, the convergence and the computational complexity are analyzed. To achieve the test and verification of the robustness of proposed algorithms, simulations are carried out to discuss the effect of the parameters algorithm under colored input signals in impulsive interference environments for system identification. Meanwhile, the performances of the proposed algorithms under the influence of the acoustic echo cancellation in practical application are studied.

Sparsity-Aware Logarithmic Hyperbolic Cosine Normalized Subband Adaptive Filter Algorithm With Step-Size Optimization

To enhance the filtering accuracy of the traditional sign subband adaptive filter (SSAF) algorithm and its individual-weighting-factors variant, this brief proposes the logarithmic hyperbolic cosine normalized subband adaptive filter algorithm (LHCNSAF) and its sparsity-aware version through minimizing the LHC cost function. As the combination of mean-square-error and mean-absolute-error criterion, the presented LHCNSAF demonstrates higher filtering accuracy under white Gaussian background noise and impulse noise environments than SSAF-type algorithms, with almost no increase in computational complexity. To tackle the trade-off between filtering accuracy and convergence behavior caused by the fixed step-size, an effective variable step-size scheme is also devised based on the transient model of proposed algorithms through minimizing the mean-square deviation (MSD) at each iteration with some reasonable assumptions. Additionally, the stability and complexity analysis are also provided. Finally, the computer simulations certify the proposed algorithms possess good performance in terms of convergence speed, steady-state error, and tracking capability via contrasting with the considering algorithms under system identification application.

A Novel Normalized Subband Adaptive Filter Algorithm Based on the Joint-Optimization Scheme

Herein, we propose a normalized subband adaptive filter (NSAF) algorithm that adjusts both the step size and regularization parameter. Based on the random-walk model, the proposed algorithm is derived by minimizing the mean-square deviation of the NSAF at each iteration to calculate the optimal parameters. We also propose a method for estimating the uncertainty in an unknown system. Consequently, the proposed algorithm improves performance in terms of tracking speed and misalignment. Simulation results show that the proposed NSAF outperforms existing algorithms in system identification scenarios.

Improved multiband structured subband adaptive filter algorithm with L0-norm regularization for sparse system identification

The improved multiband structured subband adaptive filter (IMSAF) algorithm improves the performance of normalized subband adaptive filter (NSAF) algorithm by employing the recent regressors at each subband. The present study introduces the IMSAF algorithm for sparse system identification. The L0-norm regularization term is applied to the proposed cost function of IMSAF and the L0-IMSAF is established. The L0-IMSAF has significantly better convergence speed than conventional IMSAF. In the following, the theoretical steady-state performance analysis of the L0-IMSAF is presented. To reduce the computational complexity of the L0-IMSAF, the selective regressor (SR) and the dynamic selective regressor (DSR) strategies are utilized and L0-SR-IMSAF and L0-DSR-IMSAF are proposed. The approaches in L0-SR-IMSAF and L0-DSR-IMSAF algorithms are based on the selection of the regressors at each subband. The L0-SR-IMSAF and L0-DSR-IMSAF have good convergence speed, low steady-state error, and low computational complexity features. The good performances of the proposed algorithms are demonstrated through several simulation results in sparse system identification.

A full mean-square analysis of CNSAF algorithm for noncircular inputs

A full performance analysis of complex normalized subband adaptive filter (CNSAF) algorithm will provide guidelines for designing the adaptive filter. However, because of the noncircular characteristic of complex-value signal, the complementary mean-square performance analysis of the CNSAF algorithm has not been presented in the literature. In order to give the detailed theoretical expressions of the CNSAF algorithm, the present study first analyzes the mean-square deviation (MSD) with the energy-conservation method, and then the complementary mean-square derivation (CMSD) behavior is given using pseudo-energy-conservation method. Analytical expressions are obtained for the transient MSD and CMSD of the CNSAF algorithm. Also, the steady-state MSD and CMSD are predicted based on the closed-form expressions. Besides, the analysis results are not constrained by the distribution of input signals. Finally, simulation results obtained for diffferent inputs present a good consistence with the analytical results.

Robust Normalized Subband Adaptive Filter Algorithm with a Sigmoid-Function-Based Step-Size Scaler and Its Convex Combination Version

In this paper, by inserting the logarithm cost function of the normalized subband adaptive filter algorithm with the step-size scaler (SSS-NSAF) into the sigmoid function structure, the proposed sigmoid-function-based SSS-NSAF algorithm yields improved robustness against impulsive interferences and lower steady-state error. In order to identify sparse impulse response further, a series of sparsity-aware algorithms, including the sigmoid L0 norm constraint SSS-NSAF (SL0-SSS-NSAF), sigmoid step-size scaler improved proportionate NSAF (S-SSS-IPNSAF), and sigmoid L0 norm constraint step-size scaler improved proportionate NSAF (SL0-SSS-IPNSAF), is derived by inserting the logarithm cost function into the sigmoid function structure as well as the L0 norm of the weight coefficient vector to act as a new cost function. Since the use of the fix step size in the proposed SL0-SSS-IPNSAF algorithm, it needs to make a trade-off between fast convergence rate and low steady-state error. Thus, the convex combination version of the SL0-SSS-IPNSAF (CSL0-SSS-IPNSAF) algorithm is proposed. Simulations in acoustic echo cancellation (AEC) scenario have justified the improved performance of these proposed algorithms in impulsive interference environments and even in the impulsive interference-free condition.

A Class of Diffusion Proportionate Subband Adaptive Filters for Sparse System Identification over Distributed Networks

This paper aims to extend the proportionate adaptation concept to the design of a class of diffusion normalized subband adaptive filter (DNSAF) algorithms. This leads to four extensions of the algorithm associated with different step-size variations, namely diffusion proportionate normalized subband adaptive filter (DPNSAF), diffusion \(\mu \)-law PNSAF (DMPNSAF), diffusion improved PNSAF (DIPNSAF) and diffusion improved IPNSAF (DIIPNSAF). Subsequently, steady-state performance, stability conditions and computational complexity of the proposed algorithms are investigated. For each extension the performance has been evaluated using both real and simulated data, where the outcomes demonstrate the accuracy of the theoretical expressions and effectiveness of the proposed algorithms.

Variable step-size normalized subband adaptive filtering algorithm for self-interference cancellation

Self-interference (SI) is usually generated by the simultaneous transmission and reception of the same system, which has a negative effect on the subsequent processing of the received signal. To solve this problem, a variable step-size normalized subband adaptive filtering (NSAF) algorithm is proposed to eliminate the SI. First, the optimal weight vector of each subband is obtained based on minimizing the mean-square deviation between the optimal weight vector and its estimation, and then the power of the residual SI is estimated by the shrinkage method. To eliminate the influence of input noise estimation bias on the algorithm, we estimate the input noise power of each subband and replace the fixed input noise power at each iteration. Then, the convergence, steady-state behavior and computational complexity are analyzed to prove the feasibility of the proposed algorithm. Simulation results demonstrate that the proposed algorithm is superior to cited algorithms for SI cancellation. Moreover, for the direction of arrival estimation, the received signal processed by the proposed algorithm is more accurate than those cited algorithms.

Robust Adaptive Filtering Algorithm for Self-Interference Cancellation with Impulsive Noise

Self-interference (SI) is usually generated by the simultaneous transmission and reception in the same system, and the variable SI channel and impulsive noise make it difficult to eliminate. Therefore, this paper proposes an adaptive digital SI cancellation algorithm, which is an improved normalized sub-band adaptive filtering (NSAF) algorithm based on the sparsity of the SI channel and the arctangent cost function. The weight vector is hardly updated when the impulsive noise occurs, and the iteration error resulting from impulsive noise is significantly reduced. Another major factor affecting the performance of SI cancellation is the variable SI channel. To solve this problem, the sparsity of the SI channel is estimated with the estimation of the weight vector at each iteration, and it is used to adjust the weight vector. Then, the convergence performance and calculation complexity are analyzed theoretically. Simulation results indicate that the proposed algorithm has better performance than the referenced algorithms.

Proximal Normalized Subband Adaptive Filtering for Acoustic Echo Cancellation

In this paper, we propose a novel normalized subband adaptive filter algorithm suited for sparse scenarios, which combines the proportionate and sparsity-aware mechanisms. The proposed algorithm is derived based on the proximal forward-backward splitting and the soft-thresholding methods. We analyze the mean and mean square behaviors of the algorithm, which is supported by simulations. In addition, an adaptive approach for the choice of the thresholding parameter in the proximal step is also proposed based on the minimization of the mean square deviation. Simulations in the contexts of system identification and acoustic echo cancellation verify the superiority of the proposed algorithm over its counterparts.

Sparsity-aware normalized subband adaptive filters with jointly optimized parameters

The normalized subband adaptive filter (NSAF) has a faster convergence rate than the NLMS adaptive filter when the input signal is correlated. Recently some sparsity-aware NSAFs (SA-NSAFs) were presented, which make use of the sparsity of the unknown system to accelerate convergence rate or reduce the steady-state misalignment. However, like the NSAF they also need to take a tradeoff between fast convergence rate and small steady-state misalignment. To address this problem, this paper proposes to jointly optimize the step-size and intensity factor of the SA-NSAFs. Another advantage of the presented method is that it can solve the problem of selecting the optimal intensity factor for the SA-NSAFs with repeated manual attempts. The parameter optimization is achieved by minimizing the mean-square deviation (MSD). Simulation results are provided to show the superiority of the proposed algorithms.

Robust normalized subband adaptive filter algorithm against impulsive noises and noisy inputs

This paper proposes a robust normalized subband adaptive filter (RNSAF) algorithm, which has robust performance for impulsive noise environments and noisy inputs. Although the M-estimate normalized subband adaptive filter (MNSAF) algorithm achieves robustness against impulsive noises, it generates biased estimates when the input is noisy. Based on the unbiasedness criterion, we propose a bias-compensation vector added in the RNSAF algorithm to compensate for the bias resulting from input noises. The statistical analysis reveals that the RNSAF algorithm can provide unbiased estimates. The stability analysis is also performed and the stability conditions are obtained. Moreover, by minimization of the mean-square deviation, a variable step size scheme is derived to achieve better performance. Simulation results in the context of system identification demonstrate that the proposed algorithm not only obtains robust performance in the impulsive noise environment but also achieves improved performance under noisy inputs.

Scheduled Step-Size Subband Adaptive Filter Algorithm With Implemental Consideration

This article proposes a scheduled step-size normalized subband adaptive filter (NSAF) algorithm in stationary system environment. The mean-square deviation of the NSAF according to the step size is analyzed geometrically to construct a pre-designed trajectory. The mean-square deviation learning curve of the NSAF algorithm is forced to follow the pre-designed trajectory. This method removes the need for the NSAF algorithm to introduce tuning parameters and does not add any additional online computation. The table of the scheduled step sizes can be reconstructed online in proportion to not only the number of taps but also the number of subbands once they are scheduled offline. The novel memory-efficient scheduling scheme minimizes the memory space required and simplifies operation without performance degradation. Because of these features, the proposed algorithm performs as well as the variable-step-size NSAFs studied previously, and is very suitable for chip level implementation in terms of computational complexity and memory space. Simulation results show that the proposed algorithm is robust against external environment change and has good performance compared to the existing variable step-size algorithms without any additional online computation and tuning parameter.

Diffusion ACNSAF Algorithm for Secure Distributed Estimation Under Complex-Valued Attacks

In order to protect the distributed network from complex-valued attack, a secure diffusion augmented complex-valued normalized subband adaptive filter algorithm is proposed in this article, which is derived from a novel complex-valued detection method. This complex-value detection method can be considered to consist of two parts. First, a non-cooperative augmented complex-valued normalized subband adaptive filter algorithm is used to provide reliable reference estimations under complex-valued attacks, and then the threshold test is constructed by using the reliable reference estimations to detect the trustworthy neighbors of each node. The second part is to reassemble the information from the trustworthy neighbors by using the diffusion augmented complex-valued normalized subband adaptive filter algorithm. Then, the theoretical analyses of the mean and mean-square performance of the proposed algorithm are derived based on the energy conservation framework. Finally, some experiments are performed to show the effectiveness of the proposed algorithm under non-time-varying and time-varying complex-valued attacks, respectively.

Influence of input noises on the mean-square performance of the normalized subband adaptive filter algorithm

The normalized subband adaptive filter (NSAF) algorithm has gained a large amount of attention owning to its fast convergence rate for correlated inputs. However, the convergence analysis of the NSAF algorithm has not been researched extensively, especially in the presence of noisy inputs which is frequently encountered in the applications of system identification and channel estimation. In this paper, we perform the performance analysis of the NSAF algorithm under noisy inputs. According to several reasonable assumptions and approximations, the expressions of the transient-state and steady-state mean-square deviation (MSD) are derived. Simulations under different kinds of environments confirm the accuracy of our theoretical expressions.

A Joint Optimized Robust Acoustic Echo Cancellation Algorithm

When the input signals for acoustic echo cancellation (AEC) are related signals, the convergence speed of the traditional normalized least mean square (NLMS) algorithms is significantly reduced. In this paper, a joint optimization robust AEC algorithm is proposed to solve this problem. Based on the analysis of the convergence of the normalized subband adaptive filtering (NSAF) algorithm, the algorithm is optimized by minimizing the mean square error (MSE) of the NSAF algorithm, combining sub-band time-varying step factor and time-varying regularization parameter to update the filter weight vectors. And when the impulse noise occurs, the sub-band cut-off parameter is updated in a recursive manner, which makes the algorithm achieve fast convergence speed and low steady-state error, and has strong robustness to impulse noise. In a series of experiments on AEC, simulation results show that the performance of the algorithm is better than the existing algorithms.

Widely Linear Complex-Valued Diffusion Subband Adaptive Filter Algorithm

The adaptive algorithms applied to distributed networks are usually real-valued diffusion subband adaptive filter algorithms. However, it cannot be used for processing the complex-valued signals. In this paper, a novel augmented complex-valued diffusion normalized subband adaptive filter (D-ACNSAF) algorithm is proposed for distributed estimation over networks. In order to deal with the noncircular complex-valued signals, the D-ACNSAF algorithm uses the widely linear model for a diffusion network. Due to the second-order statistical properties of signal, the D-ACNSAF algorithm can process the circular and non-circular complex-valued signals simultaneously. Moreover, the stability and mean-square steady-state analysis of the proposed algorithm are derived based on the spatial–temporal energy conservation principle. Computer simulation experiments on complex-valued system identification and prediction show that the proposed algorithm has better performance (lower mean-square deviation and faster convergence rate) than diffusion complex least-mean-square and diffusion augmented complex least-mean-square algorithms. And the simulation results are consistent with the analysis results.