Maximum Complex Correntropy Criterion Research Articles

Simplified Set-Membership Augmented Affine Projection Generalized Maximum Complex Correntropy Criterion Algorithm

Simplified Set-Membership Augmented Affine Projection Generalized Maximum Complex Correntropy Criterion Algorithm

Maximum Complex Correntropy Criterion Adaptive IIR Filtering Based on Gauss-Newton Approach

Adaptive infinite impulse response (IIR) filters have received noticeable consideration for its lower computational complexity compared with adaptive finite impulse response (FIR) filters recently. At present, the adaptive IIR algorithms are almost using the well-established stochastic gradient descent method, but the convergence rate is less than satisfactory. In this paper, to pursue the fast convergence speed of Gauss-Newton (GN) method, we present a GN algorithm for adaptive IIR filters with complex coefficients based on the complex correntropy measure. Besides, the steady-state excess mean square error is provided. Simulations are performed to show the eximious performance of this algorithm.

A robust iterative receiver for single carrier underwater acoustic communications under impulsive noise

Achieving reliable underwater acoustic (UWA) communications is challenging due to the rapid changes of UWA channels and presence of impulsive noise. The soft decision feedback equalization (SDFE) has attracted much attention due to its capability of tackling the rapid changes of UWA channels. However, the existing SDFE exhibits error floor and even divergence due to the presence of severe impulsive noise. To overcome this problem, an improved SDFE (ISDFE) is proposed, which incorporates impulsive noise detection, clipping level estimation and damping to handle the impulsive noise. As the performance of the ISDFE also depends on the accuracy of channel estimation (CE), we propose a variable kernel width maximum complex correntropy criterion Kalman filter (VKW-MCCC-KF) algorithm to achieve robust and accurate CE under impulsive noise. With the proposed ISDFE and VKW-MCCC-KF, an iterative receiver for single carrier (SC) UWA mobile communication under impulsive noise is proposed, which incorporates the adaptive broadband Doppler compensation, VKW-MCCC-KF based CE and ISDFE, working in an iterative manner. Experimental results show that, compared with existing receivers, the proposed receiver achieves better performance and robustness to impulsive noise under fast changing UWA channels.

Generalized Maximum Complex Correntropy Augmented Adaptive IIR Filtering.

Augmented IIR filter adaptive algorithms have been considered in many studies, which are suitable for proper and improper complex-valued signals. However, lots of augmented IIR filter adaptive algorithms are developed under the mean square error (MSE) criterion. It is an ideal optimality criterion under Gaussian noises but fails to model the behavior of non-Gaussian noise found in practice. Complex correntropy has shown robustness under non-Gaussian noises in the design of adaptive filters as a similarity measure for the complex random variables. In this paper, we propose a new augmented IIR filter adaptive algorithm based on the generalized maximum complex correntropy criterion (GMCCC-AIIR), which employs the complex generalized Gaussian density function as the kernel function. Stability analysis provides the bound of learning rate. Simulation results verify its superiority.

Widely Linear Maximum Complex Correntropy Criterion Affine Projection Algorithm and Its Performance Analysis

Recently, affine projection algorithm has been extensively studied in the Gaussian noise environment. However, the performance of affine projection algorithm will deteriorate rapidly in the presence of impulsive noise and other non-Gaussian noise. To address this issue, this paper proposes a novel affine projection algorithm based on the complex Gaussian kernel function, called widely linear maximum complex correntropy criterion affine projection algorithm (WL-MCCC-APA). With the maximum complex correntropy criterion (MCCC), the robustness of affine projection algorithm is improved significantly in the non-Gaussian noise environment, especially in the presence of impulsive noise. Meanwhile, a variable step size WL-MCCC-APA (VSS-WL-MCCC-APA) method based on fixed-point theory is proposed to further balance the convergence rate and steady-state misalignment of the algorithm. Finally, the step size boundary analysis and steady-state analysis are presented for theoretical analysis. Simulation results verify the correctness of obtained theoretical results and the superiority of the proposed WL-MCCC-APA and VSS-WL -MCCC-APA.

Complex Shannon Entropy Based Learning Algorithm and Its Applications

Nowadays, adaptive filters in complex domain have received increasing attention since they can be applied to a wider scenario than real valued cases. As two typical information theoretic criteria, the minimum complex kernel risk-sensitive loss (MCKRSL) and maximum complex correntropy criterion (MCCC) have shown robustness to non-Gaussian noises in complex domain adaptive filtering. However, since both criteria cannot consider the probability distribution of error, they are not optimal in the presence of some non-Gaussian noises. This paper firstly defines a complex Shannon entropy in terms of the probability distribution of error. Then, a novel adaptive filter is proposed using the minimum complex Shannon entropy (MCSE) criterion. More significantly, the convergence behavior of the MCSE algorithm is discussed, and thus the steady-state excess mean square error (EMSE) is calculated for theoretical analysis. Finally, simulated results on system identification and channel estimation validate the obtained theoretical results and the merits of the proposed MCSE algorithm.

Generalised maximum complex correntropy‐based DOA estimation in presence of impulsive noise

Most existing methods for direction-of-arrival (DOA) estimation are severely influenced by impulsive noise due to their Gaussian noise assumption. As a typical non-linear similarity measure, the maximum correntropy criterion (MCC) has been considered to restrain impulsive noise, because of its ability to exploit the high-order statistics of signal. However, Gaussian kernel-based MCC method is not always the optimal choice and is only suitable for the real-valued signal, which certainly limits its applications. To solve the aforementioned problems, in this study, the authors proposed a novel generalised maximum complex correntropy criterion (GMCCC)-based complex-valued quasi-Newton method to restrain impulsive noise. GMCCC adopts the generalised complex Gaussian density function as the kernel function with more flexible parameters. Besides, it can extend the benefit to the complex-valued signal. Furthermore, its properties are formalised. The complex-valued quasi-Newton method guarantees the positive definite Hessian matrix to achieve the alternate minimisation of signal subspace and signal matrix. GMCCC achieves the accurate DOA estimation from the received data which does not require the covariance matrix. Stability performance and convergence are analysed. Experiment results show that the proposed GMCCC algorithm possesses the robustness and outperforms the state-of-the-art algorithms.

Bias-Compensated MCCC Algorithm for Widely Linear Adaptive Filtering With Noisy Data

Recently, widely linear adaptive filtering with noisy input and output has attracted much attention. As an unbiased estimation method, the widely linear complex-valued estimated input least mean square (WLC-EILMS) algorithm has shown its superiority in the noisy inputs and outputs. However, WLC-EILMS is not robust to impulsive noises. To overcome this defect, this brief uses the maximum complex correntropy criterion (MCCC) to generate a novel widely linear complex-valued estimated-input MCCC (WLC-EIMCCC) algorithm. The proposed WLC-EIMCCC algorithm exhibits superior performance especially in dealing with impulsive noises, and its performance analysis is also given for theoretical analysis. Simulations on system identification and channel equalization validate the correctness of the theoretical steady-state excess mean square error (EMSE) and the superior performance of WLC- EIMCCC.

Complex Correntropy with Variable Center: Definition, Properties, and Application to Adaptive Filtering.

The complex correntropy has been successfully applied to complex domain adaptive filtering, and the corresponding maximum complex correntropy criterion (MCCC) algorithm has been proved to be robust to non-Gaussian noises. However, the kernel function of the complex correntropy is usually limited to a Gaussian function whose center is zero. In order to improve the performance of MCCC in a non-zero mean noise environment, we firstly define a complex correntropy with variable center and provide its probability explanation. Then, we propose a maximum complex correntropy criterion with variable center (MCCC-VC), and apply it to the complex domain adaptive filtering. Next, we use the gradient descent approach to search the minimum of the cost function. We also propose a feasible method to optimize the center and the kernel width of MCCC-VC. It is very important that we further provide the bound for the learning rate and derive the theoretical value of the steady-state excess mean square error (EMSE). Finally, we perform some simulations to show the validity of the theoretical steady-state EMSE and the better performance of MCCC-VC.

Performance evaluation of the maximum complex correntropy criterion with adaptive kernel width update

The complex correntropy is a recently defined similarity measure that extends the advantages of conventional correntropy to complex-valued data. As in the real-valued case, the maximum complex correntropy criterion (MCCC) employs a free parameter called kernel width, which affects the convergence rate, robustness, and steady-state performance of the method. However, determining the optimal value for such parameter is not always a trivial task. Within this context, several works have introduced adaptive kernel width algorithms to deal with this free parameter, but such solutions must be updated to manipulate complex-valued data. This work reviews and updates the most recent adaptive kernel width algorithms so that they become capable of dealing with complex-valued data using the complex correntropy. Besides that, a novel gradient-based solution is introduced to the Gaussian kernel and its respective convergence analysis. Simulations compare the performance of adaptive kernel width algorithms with different fixed kernel sizes in an impulsive noise environment. The results show that the iterative kernel adjustment improves the performance of the gradient solution for complex-valued data.

A Robust Adaptive Filter for a Complex Hammerstein System.

The Hammerstein adaptive filter using maximum correntropy criterion (MCC) has been shown to be more robust to outliers than the ones using the traditional mean square error (MSE) criterion. As there is no report on the robust Hammerstein adaptive filters in the complex domain, in this paper, we develop the robust Hammerstein adaptive filter under MCC to the complex domain, and propose the Hammerstein maximum complex correntropy criterion (HMCCC) algorithm. Thus, the new Hammerstein adaptive filter can be used to directly handle the complex-valued data. Additionally, we analyze the stability and steady-state mean square performance of HMCCC. Simulations illustrate that the proposed HMCCC algorithm is convergent in the impulsive noise environment, and achieves a higher accuracy and faster convergence speed than the Hammerstein complex least mean square (HCLMS) algorithm.

Convergence Analysis of a Fixed Point Algorithm Under Maximum Complex Correntropy Criterion

With the emergence of complex correntropy, the maximum complex correntropy criterion (MCCC) has been applied to the complex-domain adaptive filtering. The MCCC uses the fixed point method to find the optimal solution, which provides good robustness in the non-Gaussian noise environment, especially for the impulse noise. However, the convergence analysis for the fixed point method is limited to the real-domain filtering. In this letter, we provide the convergence analysis of fixed point based MCCC algorithm in complex-domain filtering. First, by using the matrix inversion lemma, we rewrite the MCCC algorithm to a gradient-like version. In addition, we provide two computationally efficient versions of MCCC. Then, we provide the stability analysis and obtain the excess mean square error for MCCC. Finally, simulation results confirm the correctness of the convergence analysis in this letter.

Recursive Minimum Complex Kernel Risk-Sensitive Loss Algorithm.

The maximum complex correntropy criterion (MCCC) has been extended to complex domain for dealing with complex-valued data in the presence of impulsive noise. Compared with the correntropy based loss, a kernel risk-sensitive loss (KRSL) defined in kernel space has demonstrated a superior performance surface in the complex domain. However, there is no report regarding the recursive KRSL algorithm in the complex domain. Therefore, in this paper we propose a recursive complex KRSL algorithm called the recursive minimum complex kernel risk-sensitive loss (RMCKRSL). In addition, we analyze its stability and obtain the theoretical value of the excess mean square error (EMSE), which are both supported by simulations. Simulation results verify that the proposed RMCKRSL out-performs the MCCC, generalized MCCC (GMCCC), and traditional recursive least squares (RLS).

Complex Kernel Risk-Sensitive Loss: Application to Robust Adaptive Filtering in Complex Domain

Recently, the maximum complex correntropy criterion (MCCC) algorithm has shown its superiority for adaptive filter in complex domain. Compared with the traditional mean square error (MSE) criterion-based algorithms, MCCC uses complex correntropy as similarity measure, which is robust even in the presence of impulse noise. However, the error performance surface of complex correntropic loss (CC-loss) is not optimal, which is steep nearby the optimal solution and flat far from the optimal solution. In this paper, we propose a new similarity measure in complex domain, namely, complex kernel risk-sensitive loss (CKRSL). Based on CKRSL, we derive a new adaptive filter algorithm in the complex domain by using the gradient-based method, i.e., the CKRSL algorithm. Compared with the MCCC algorithm, the CKRSL algorithm has faster convergence rate and higher filtering accuracy. Meanwhile, it is robust against outlier. In addition, we provide the stability analysis and the steady-state excess MSE for the CKRSL algorithm. Simulation results confirm the correctness of the theoretical results and the superiority of the CKRSL algorithm.

Complex Correntropy: Probabilistic Interpretation and Application to Complex-Valued Data

Recent studies have demonstrated that correntropy is an efficient tool for analyzing higher order statistical moments in non-Gaussian noise environments. Although correntropy has been used with complex data, no theoretical study was pursued to elucidate its properties, nor how to best use it for optimization. By using a probabilistic interpretation, this work presents a novel similarity measure between two complex random variables, which is defined as complex correntropy. A new recursive solution for the maximum complex correntropy criterion is introduced based on a fixed-point solution. This technique is applied to a system identification, and the results demonstrate prominent advantages when compared against three other algorithms: the complex least mean square, complex recursive least squares, and least absolute deviation. By the aforementioned probabilistic interpretation, correntropy can now be applied to solve several problems involving complex data in a more straightforward way.