Elsevier

Signal Processing

Volume 203, February 2023, 108792
Signal Processing

Performance analysis of the augmented complex-valued least mean kurtosis algorithm

https://doi.org/10.1016/j.sigpro.2022.108792Get rights and content

Abstract

The augmented complex-valued least mean kurtosis (ACLMK) algorithm was proposed to enhance the performance of adaptive filters by utilizing the negated kurtosis of the error signal. This paper provides a comprehensive physical insight into its stochastic behaviors and a more thorough performance analysis. By using Isserlis’ theorem, the evolutions of both the covariance and complementary covariance matrices of the weight error vector are established, based on which the expressions for transient mean-square deviation (MSD) and excess mean-square error (EMSE) are derived. Moreover, the higher-order moments of the measurement noise are also taken into account. Different from the existing framework which treats the ACLMK as a variable step-size augmented complex-valued least mean square (ACLMS) algorithm, the steady-state mean-square performance of the ACLMK is analyzed through the approximate uncorrelating transform (AUT) to yield the theoretical steady-state MSD and EMSE. The proposed framework of performance analysis is shown to provide more accurate models for the theoretical MSD and MSE. Simulation results are provided to validate the theoretical findings for second-order noncircular input signals in second-order circular and noncircular noisy environments.

Introduction

The well-known least mean-square (LMS) algorithm may be the most widely used adaptive algorithm owing to its simplicity and robustness [1]. However, stemming from the feature of the minimum mean-square error (MSE) criterion, the LMS is optimal only in Gaussian noise environments, and it may perform poorly in sub-Gaussian noise environments, e.g., in uniform or binary noise environments. Various efforts have been devoted to achieve a better performance in such cases, and the least mean fourth (LMF) algorithm [2], which was developed by minimizing the fourth-order moment of the estimation error, has attracted much attention due to its effectiveness and ease of implementation. Unfortunately, the LMF outperforms the LMS only in some strong sub-Gaussian noise environments, and it will lose its advantage in Gaussian or non-strong sub-Gaussian noise environments [3].

To achieve good robustness for a wider range of types of noise, based on the minimization of the negated kurtosis of the estimation error, the least mean kurtosis (LMK) algorithm was proposed by Tanrikulu and Constantinides [4]. By appropriately adjusting the design parameters, the LMK can outperform the LMS even for Gaussian noise, which cannot be achieved by the LMF. Thus, although suffering from a slightly higher computational complexity, the LMK has been widely studied, applied, and extended in recent years [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].

Over last two decades complex adaptive algorithms have received much attention, for instance, the complex-valued LMS (CLMS) algorithm [16], because a complex number can carry more information than a real number. Besides, signals in some applications are more easily handled in their complex-valued form. For example, the input signals are complex arising when modelling narrow-band signals modulating a high frequency carrier. After the CLMS, complex adaptive algorithms have been extensively studied, such as the complex-valued normalized LMS (CNLMS) algorithm, complex-valued affine projection (CAP) algorithm, and complex-valued recursive least-squares (CRLS) algorithm [1]. Because of the superior performance of the LMK, it was also extended to complex domain, leading to the complex-valued LMK (CLMK) algorithm [17]. However, like many other complex-valued adaptive algorithms which are straightforward extensions of their real-valued counterparts, the CLMK is only optimal for second-order circular (proper) signals which exhibit rotation-invariant probability distributions [18].

In some practical applications such as adaptive beamforming [19], [20], stereophonic acoustic echo cancellation [21], self-interference cancellation [22], and frequency estimation [23], [24], where noncircularity appears due to the underlying generating physics, an additional performance gain can be achieved when both the complex-valued signals and their conjugates are jointly processed [25]. This model, called the widely linear (WL) model, has inspired various adaptive algorithms [26], [27], [28], [29], [30], [31]. Particularly, the augmented CLMK (ACLMK) algorithm was proposed recently [32], which outperforms other MSE-based methods because its cost function that builds on the negated kurtosis of the complex-valued error signal is nearly free from the effect of noise statistics. However, the performance analysis provided in [32] was carried out by treating the ACLMK as a variable step-size augmented CLMS (ACLMS). It assumes that the variable step-size is independent of other parts of the update equation. This prevents the derived theoretical steady-state MSE from obtaining highly accurate results.

To address the above issue, this work presents a comprehensive performance analysis of the ACLMK, which covers the theoretical characterization of the transient mean and mean-square performance as well as a more accurate steady-state MSE. The contributions of this work are summarized as follows. Firstly, using the Isserlis’ theorem [33], we develop the evolution formulas of both the covariance and complementary covariance matrices of the weight error vector. This serves to provide theoretical results of the transient mean-square deviation (MSD) and excess MSE (EMSE), which is absent in [32]. Secondly, instead of viewing the ACLMK as a variable step-size ACLMS by neglecting the dependence of the step-size on the output error, we exploit the approximate uncorrelating transform (AUT) [34] and facilitate an alternative steady-state performance analysis, throughout which the order of moments is consistent with that of the kurtosis. Furthermore, the impact of the noncircularity rate of the noise is taken into account in both the transient and steady-state performance analysis. These contributions yield a more accurate prediction of the steady-state MSD and EMSE than that in [32], as demonstrated by Monte Carlo simulations.

Organization: In Section 2 the ACLMK is briefly reviewed. The performance of this algorithm is analyzed in Section 3, covering the transient mean weight and mean-square analysis as well as a more accurate steady-state analysis. In Section 4 simulation results are given to demonstrate the consistency between the simulated results and the theoretical findings. The paper is concluded in Section 5.

Notation: Lowercase letters, boldface letters, and boldface uppercase letters denote respectively scalars, column vectors, and matrices. (·)T, (·)*, (·)H, and (·)1 denote transpose, complex conjugate, Hermitian transpose, and matrix inversion, respectively. E[·] is the expectation operator. tr{·} takes the trace of a matrix. IM and 0M represent the M×M identity and zero matrices, respectively. 1M is the M×1 vector with unit entries; [·] and [·] extract the real and imaginary parts of a complex number, respectively. j=1 denotes the imaginary unit.

Section snippets

Review of the ACLMK algorithm

Consider the problem of system identification. The desired response dn of the unknown system is generated by the widely linear model given by [25]dn=hoHxn+goHxn*+vnwhere ho=[h1o,h2o,,hNo]T and go=[g1o,g2o,,gNo]T are the standard and conjugate weight vectors of the unknown system, respectively, xn=[xn,xn1,,xnN+1]T is the input vector which consists of the most recent N samples of the input signal xn, and vn is the measurement noise of variance σv2=E[|vn|2] and complementary variance σ˜v2=E[v

Performance analysis

Define the weight error vector asw˜n=wown.Then, the estimation error in (7) can be rewritten asen=zn+vnwherezn=w˜nHunrepresents the noise-free estimation error. Following the procedure in [35], we defineθ^n=σ^e,n2|en|2=λσ^e,n12=λθ^n1+λ|en1|2.Then, (6) changes intown+1=wn+μ(3θ^n+2|en|2)en*un.Subtracting both sides of (12) from wo and using (9), we havew˜n+1=w˜nμ(3θ^n+2|zn+vn|2)(zn*+vn*)un=w˜nμ(3θ^nzn*+2znzn*2+4vn*znzn*+2vnzn*2+2vn*2zn+4vnvn*zn*+3θ^nvn*+2vnvn*2)un.

To proceed, we define the

Simulation results

The unknown weight vectors ho and go of length N=5 were randomly selected and normalized to unit energy. The adaptive filter weights were initialised with zeros. All simulation results were obtained by averaging over 100 independent trials. To show the effectiveness of the presented methods, the measurement noise vn was assumed to be second-order circular and noncircular, respectively.

Conclusion

A comprehensive analysis of the transient and steady-state performance of the ACLMK has been provided. The transient analytical model has been first established to predict the transient mean and mean-square behavior of the ACLMK. Then, based on this model we have derived the theoretical expressions for steady-state MSD and EMSE, which have been shown to more accurately predict the steady-state behavior of the algorithm than current analysis in the literature. Through the analysis, we have found

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the Natural Science Foundation of Jiangsu Province under grant BK20191419 and in part by the National Natural Science Foundation of China under Grant 61901291.

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