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Maximum correntropy criterion based sparse adaptive filtering algorithms for robust channel estimation under non-Gaussian environments
Wentao Ma, Hua Qua, Guan Gui, Li Xu, Jihong Zhaoa, Badong Chen
TL;DR
Sparse adaptive channel estimation methods based on MMSE can lose robustness under non-Gaussian impulsive noise, motivating a method that handles both noise and channel sparsity. The paper combines MCC with a CIM penalty, and simulations confirm desirable performance under impulsive-noise environments alongside convergence analysis and conditions.
Problem
MMSE-based sparse channel estimation methods are robust under Gaussian assumptions but often lose robustness under non-Gaussian impulsive noises.
Method
The paper develops a CIM-penalized MCC sparse adaptive filtering algorithm, using MCC to mitigate impulsive noise and CIM to exploit channel sparsity.
Results
Simulation results confirmed desirable performance of the proposed algorithm under impulsive-noise environments, with derived convergence conditions.
Takeaways & Limitations
The proposed algorithm supports sparse channel estimation under impulsive-noise environments.
Takeaways & Limitations
The convergence condition is an approximation case, although computer simulations are used for evaluation.
Abstract
from arXiv · showhide
Sparse adaptive channel estimation problem is one of the most important topics in broadband wireless communications systems due to its simplicity and robustness. So far many sparsity-aware channel estimation algorithms have been developed based on the well-known minimum mean square error (MMSE) criterion, such as the zero-attracting least mean square (ZALMS), which are robust under Gaussian assumption. In non-Gaussian environments, however, these methods are often no longer robust especially when systems are disturbed by random impulsive noises. To address this problem, we propose in this work a robust sparse adaptive filtering algorithm using correntropy induced metric (CIM) penalized maximum correntropy criterion (MCC) rather than conventional MMSE criterion for robust channel estimation. Specifically, MCC is utilized to mitigate the impulsive noise while CIM is adopted to exploit the channel sparsity efficiently. Both theoretical analysis and computer simulations are provided to corroborate the proposed methods.
1. Introduction
Sparse channel estimation improves exploitation of channel sparsity, but methods developed for Gaussian noise can be unstable at low SNR and sensitive to non-Gaussian impulsive noise. The paper proposes combining MCC and CIM to obtain robust sparse adaptive filtering for impulsive-noise channel estimation.
- Sparse channel estimation methods exploit channel sparsity and can improve estimation performance, particularly in high-SNR environments.
- Existing sparse channel estimation methods may be unstable at low SNR and sensitive to impulsive noise because they were developed under Gaussian noise models.
- The paper addresses the gap by developing a robust sparse adaptive filter for estimating sparse channels in impulsive-noise environments.
- Correntropy replaces traditional MSE to improve robustness against impulsive noise, while a CIM-based penalty exploits channel sparsity.
- The proposed approach is supported by theoretical analysis and numerical simulations, with improved performance when the system is sparse.
2. Correntropy and CIM
Correntropy is a bounded, local similarity measure that is robust to impulsive noise, while CIM provides a nonlinear approximation to the ℓ0-norm for sparsity promotion.
- Correntropy measures nonlinear similarity between two random variables using a kernel-based formulation.
- Correntropy is bounded for any distribution and robust to impulsive noises or outliers.
- The MCC criterion maximizes correntropy between a variable and its estimator for robust adaptive filtering.
- CIM is a nonlinear metric whose suitable kernel-width selection can approximate the ℓ0-norm.
- Because CIM favors sparsity, it can serve as a penalty term in sparse channel estimation.
3 CIMMCC algorithm
CIMMCC combines an MCC estimation criterion with a CIM sparsity penalty to estimate sparse channels under non-Gaussian noise. The resulting algorithm is designed to remain computationally efficient while exploiting sparse channel structure.
- The channel model assumes real-valued channel parameters with sparse structure, where most channel coefficients are zero.
- CIMMCC constructs a cost function that combines MCC for impulsive-noise robustness with CIM for channel-sparsity exploitation.
- The CIM term imposes zero attraction according to the relative magnitudes of the filter coefficients.
- Selecting the CIM kernel width appropriately allows the proposed algorithm to approach an ℓ0-norm penalty.
- CIMMCC requires 3M additions, 2M multiplications, and M+1 exponential calculations per iteration, where M is channel memory size.
4 Performance analysis
The proposed algorithm’s mean and mean-square convergence are analyzed using an approximation approach under stated statistical assumptions. The analysis establishes boundedness, convergence, and a conservative step-size condition for stability.
- Mean convergence: Mean convergence analysis rewrites the proposed algorithm using a generalization approximation and examines the expected weight-vector behavior.The analysis uses the approximated recursion and assumptions concerning input, noise, error nonlinearity, and filter independence.
- Mean-square convergence: Mean-square convergence is derived from the autocorrelation matrix of the filter misalignment vector.The derivation uses the misalignment definition, recursive relations, independence assumptions, and Gaussian fourth-order moment properties.
- Mean convergence: The expected weight vector remains bounded and converges to a limiting vector under the stated assumptions.The boundedness follows from limited expectations of the error nonlinearity and penalty-related terms.
- Mean-square convergence: The filter misalignment autocorrelation matrix converges when the relevant quantities remain bounded, matching conclusions reported for ℓ1-norm and logarithmic penalty terms.The analysis states that the CIM penalty term is bounded because its negative exponential term has a finite maximum.
- Stability condition: The stability condition becomes equivalent to that of standard ZALMS when f(e(n)) equals 1, including the stated limiting case σ→∞.At steady state, the error is approximately equal to the noise, enabling a noise-dependent form of the condition.
- Stability condition: The proposed algorithms converge when the step size satisfies the derived condition, while the corresponding upper bound is conservative because the condition is approximate.Simulations reportedly show convergence for some step sizes slightly larger than the conservative upper bound.
5 Simulation Results
Simulations evaluate the proposed sparse MCC algorithms for time-varying channel estimation under mixed Gaussian and alpha-stable impulsive noise. CIMMCC generally provides fast convergence, low MSD, and robust tracking, although step size and kernel width affect performance.
- Mixed Gaussian noise: Under mixed Gaussian noise, CIMMCC outperforms the other algorithms during the first and second stages and later becomes comparable with RZAMCC.ZALMS and RZALMS have nearly identical performance, as do ZAMCC and RZAMCC.
- Parameter effects: Step size strongly affects convergence: all algorithms perform optimally at 0.005, while CIMMCC has satisfactory convergence below 0.05 and no obvious convergence above 0.1.The conservative theoretical upper bound is 0.085, and MSD becomes very poor when the step size exceeds 0.085.
- Alpha-stable noise: Under alpha-stable noise, sparse MCC algorithms converge faster and achieve better steady-state performance than LMP and MCC, while CIMMCC achieves lower MSD than ZAMCC and RZAMCC.The reported explanation is that CIM better approximates the ℓ0-norm; ZALMS and RZALMS are unstable under impulsive noise.
- Parameter effects: For different alpha values, CIMMCC performs better under less impulsive noise and attains lower MSD when the kernel size is σ_1=1.The simulations also report nonconvergence when the step size exceeds 0.2 in the examined setting.
6 Conclusions
The paper proposes a robust sparse adaptive filtering algorithm that combines MCC with a CIM-based sparsity penalty. Theoretical convergence analysis and simulations support its performance under impulsive noise.
- The proposed algorithm incorporates a CIM-based sparsity penalty into the maximum correntropy criterion (MCC).This combines robust error handling with sparsity exploitation.
- The authors analyze the proposed algorithm’s mean and mean-square convergence.They also derive conditions that guarantee convergence.
- Simulation results confirm the new algorithm’s desirable performance under impulsive noise environments.
1. Sparse MCC with zero-attracting (ℓ1-norm) penalty term(ZAMCC)
This section develops ZAMCC, a sparse MCC algorithm using a zero-attracting ℓ1-norm penalty. The MCC term addresses impulsive noise, while the penalty promotes sparsity through a weighted cost function and update rule.
- ZAMCC is derived by combining MCC with a zero-attracting ℓ1-norm penalty term.The algorithm is explicitly named ZAMCC.
- The MCC term is robust to impulsive noise, whereas the zero-attracting penalty induces sparsity.
- A weight factor λ ≥ 0 balances the MCC estimation error and the sparsity penalty.The weight factor acts as a regularization parameter.
- The resulting ZAMCC update uses the instantaneous error, input vector, and component-wise sign of the estimated parameter vector.The sign function is defined component-wise in the derivation.
2. Sparse MCC with the logarithmic penalty term
This section derives a sparse MCC algorithm with a logarithmic penalty, selected because its behavior is closer to the ℓ0-norm than the ℓ1-norm. A gradient-based adaptive update is then obtained.
- The logarithmic penalty is introduced as an alternative sparsity penalty for the sparse MCC algorithm.
- The log-sum penalty behaves more similarly to the ℓ0-norm than the ℓ1-norm.
- The penalty includes a positive parameter δ, which appears in the logarithmic expression.
- A gradient-based adaptive algorithm is derived from the logarithmic penalty formulation.The derivation is also expressed in vector form.
- The vector-form update uses the error-dependent MCC factor together with a logarithmic-penalty term.The parameters μ and ρ are stated to match those in earlier equations.
Appendix 2：
The appendix analyzes the boundedness of the CIM-related function used in the sparse MCC formulation. It shows that the function remains bounded even when an estimated channel parameter grows without bound.
- The appendix defines the Gaussian-shaped function g(x) = x exp(-x^2/(2σ^2)) for analyzing the CIM term.The function includes the scale parameter σ.
- The CIM-related expression is represented using g(w_i(n)) for an estimated parameter component.
- The analysis evaluates the limiting behavior of g(x) as x approaches infinity using L'Hôpital's rule.
- The limit calculation gives lim g(x) = 0 as x tends to infinity.
- The estimated weight is generally limited, and the CIM vector remains limited even when a channel parameter tends to infinity.