Performance analysis of the augmented complex-valued least mean kurtosis algorithm
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. , , , and denote transpose, complex conjugate, Hermitian transpose, and matrix inversion, respectively. is the expectation operator. takes the trace of a matrix. and represent the identity and zero matrices, respectively. is the vector with unit entries; and extract the real and imaginary parts of a complex number, respectively. denotes the imaginary unit.
Section snippets
Review of the ACLMK algorithm
Consider the problem of system identification. The desired response of the unknown system is generated by the widely linear model given by [25]where and are the standard and conjugate weight vectors of the unknown system, respectively, is the input vector which consists of the most recent samples of the input signal , and is the measurement noise of variance and complementary variance
Performance analysis
Define the weight error vector asThen, the estimation error in (7) can be rewritten aswhererepresents the noise-free estimation error. Following the procedure in [35], we defineThen, (6) changes intoSubtracting both sides of (12) from and using (9), we have
To proceed, we define the
Simulation results
The unknown weight vectors and of length 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 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.
References (47)
- et al.
Robust frequency estimation of unbalanced power system using a phase angle error based least mean kurtosis algorithm
Int. J. Elect. Power Energy Syst.
(2019) - et al.
A widely linear model for stereophonic acoustic echo cancellation
Signal Process.
(2013) - et al.
An augmented affine projection algorithm for the filtering of noncircular complex signals
Signal Process.
(2010) - et al.
Selective partial-update augmented complex-valued LMS algorithm and its performance analysis
Signal Process.
(2021) - et al.
Stochastic analysis of the least mean kurtosis algorithm for Gaussian inputs
Digit. Signal Process.
(2016) - et al.
Variable step-size widely linear complex-valued NLMS algorithm and its performance analysis
Signal Process.
(2019) - et al.
Stochastic behavior of the nonnegative least mean fourth algorithm for stationary Gaussian inputs and slow learning
Signal Process.
(2016) - et al.
Performance analysis of diffusion least mean fourth algorithm over network
Signal Process.
(2017) Adaptive Filters
(2008)- et al.
The least mean fourth (LMF) adaptive algorithm and its family
IEEE Trans. Inf. Theory
(1984)
Dependence of the stability of the least mean fourth algorithm on target weights non-stationarity
IEEE Trans. Signal Process.
Least-mean kurtosis: a novel higher-order statistics based adaptive filtering algorithm
Electron. Lett.
A novel kurtosis driven variable step-size adaptive algorithm
IEEE Trans. Signal Process.
Blind multiuser detector based on LMK criterion
Electron. Lett.
A gradient-based variable step-size scheme for kurtosis of estimated error
IEEE Signal Process. Lett.
An improved least mean kurtosis (LMK) algorithm for sparse system identification
Int. J. Inform. Electron. Eng.
Kernel least mean kurtosis based online chaotic time series prediction
Chin. phys. lett.
Improved filtered-x least mean kurtosis algorithm for active noise control
Circuit Syst. Signal Process.
Novel quaternion-valued least-mean kurtosis adaptive filtering algorithm based on the GHR calculus
IET Signal Process.
Kurtosis-based CRTRL algorithms for fully connected recurrent neural networks
IEEE Trans. Neural Netw. Learn. Syst.
Widely linear least mean kurtosis-based frequency estimation of three-phase power system
IET Gener. Transmiss. Distrib.
Widely linear quaternion-valued least-mean kurtosis algorithm
IEEE Trans. Signal Process.
The complex LMS algorithm
Proc. IEEE
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