Complex Least Mean Square Research Articles

Suppression of synchronous current of active magnetic bearings considering radial displacement inconsistency

Suppressing rotor synchronous current is vital for achieving precise control in active magnetic bearing (AMB) rotor systems. The displacement amplitude inconsistency in orthogonal channels presents new challenges for synchronous current suppression. This paper proposes an improved complex least mean square (CLMS) algorithm with error compensation to effectively mitigate rotor unbalanced vibration. Firstly, a dynamic model of the AMB-rotor system with an unbalanced mass is established, and the displacement amplitude inconsistency within the orthogonal channels of the radial magnetic bearings is analyzed. Subsequently, the rotor displacement signal from the orthogonal channels is converted into a complex plane signal. By calculating the negative sequence weight component, the displacement signal is reconstructed, and an error compensation channel based on the CLMS is introduced to enable real-time tracking and compensation of displacement amplitude inconsistencies in the orthogonal channels. Moreover, system stability across the full speed range is ensured through the application of the phase compensation angle method. Finally, experimental validation on a magnetic levitation molecular pump prototype confirms the effectiveness of the proposed algorithm.

A novel quantum calculus-based complex least mean square algorithm (q-CLMS)

The Least Mean Square (LMS) algorithm has a slow convergence rate as it is dependent on the eigenvalue spread of the input correlation matrix. In this research, we solved this problem by introducing a novel adaptive filtering algorithm for complex domain signal processing based on q-derivative. The proposed algorithm is based on Wirtinger calculus and is called as q- Complex Least Mean Square (q-CLMS) algorithm. The proposed algorithm could be considered as an extension of the q-LMS algorithm for the complex domain. Transient and steady-state analyses of the proposed q-CLMS algorithm are performed and exact analytical expressions for mean analysis, mean square error (MSE), excess mean square error (EMSE), mean square deviation (MSD) and misadjustment are presented. Extensive experiments have been conducted and a good match between the simulation results and theoretical findings is reported. The proposed q-CLMS algorithm is also explored for whitening applications with satisfactory performance. A modification of the proposed q-CLMS algorithm called Enhanced q-CLMS (Eq-CLMS) is also proposed. The Eq-CLMS algorithm eliminates the need for a pre-coded value of the q-parameter thereby automatically adapting to the best value. Extensive experiments are performed on system identification and channel equalization tasks and the proposed algorithm is shown to outperform several benchmark and state-of-the-art approaches namely Complex Least Mean Square (CLMS), Normalized Complex Least Mean Square (NCLMS), Variable Step Size Complex Least Mean Square (VSS-CLMS), Complex FLMS (CFLMS) and Fractional-ordered-CLMS (FoCLMS) algorithms.

Complex multi-kernel random Fourier adaptive algorithms under the complex kernel risk-sensitive p-power loss

The complex least mean square (CLMS) adaptive algorithm based on the minimum mean square error (MSE) criterion has been widely used for linear applications with Gaussian noises in complex domain. However, the MSE criterion suffers from performance degeneration in the presence of non-Gaussian noise. To address this issue, a novel complex kernel risk-sensitive p-power loss (CKRSP-L) criterion is first constructed to combat non-Gaussian noises. Then, based on the constructed CKRSP-L criterion, a novel complex kernel risk-sensitive p-power (CKRSP) algorithm is proposed to provide robustness to non-Gaussian noises and performance improvement for linear systems, simultaneously. Further, to extend the CKRSP algorithm into nonlinear systems, a novel complex multi-kernel random Fourier mapping (CMRFM) is proposed to transform the original input data into a complex multi-kernel random Fourier features space (CMRFFS), and thus another novel complex multi-kernel random Fourier kernel risk-sensitive p-power (CMRFKRSP) algorithm is presented for nonlinear applications in complex domain. Finally, the steady-state excess mean square errors (SEMSEs) of CKRSP and CMRFKRSP are also calculated for theoretical analysis of performance. Monte Carlo simulations conducted in different noise environments validate the correctness of the obtained SEMSEs and performance advantages of CKRSP and CMRFKRSP.

Comments on “Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization”

The purpose of this note is to discuss some aspects of recently proposed fractional-order variants of complex least mean square (CLMS) and normalized least mean square (NLMS) algorithms in Shah et al. (Nonlinear Dyn. 88(2):839–858, 2017). It is observed that these algorithms do not always converge, whereas they have apparently no advantage over the CLMS and NLMS algorithms whenever they converge. Our claims are based on analytical reasoning and are supported by numerical simulations.

Steady-State Mean-Square Deviation Analysis of the Zero-Attracting CLMS Algorithm With Circular Gaussian Input

When the complex least mean-square (CLMS) algorithm is used to estimate a complex sparse system, its convergence rate is relatively slow. To make full use of the sparsity of the unknown system to improve convergence rate, a zero-attracting CLMS (ZA-CLMS) algorithm has recently been proposed. In order to predict the statistical behavior of the complex adaptive filter after converging, this paper analyzes the steady-state mean-square deviation (MSD) performance of ZA-CLMS with circular Gaussian input. For mathematical tractability, in the analysis, the unknown system weights are divided into two sets according to their values and some statistical assumptions are used. The simulation results are provided to show the superior performance of ZA-CLMS for complex sparse system identification and to verify the validity of the theoretical expressions for predicting the steady-state MSD performance.

Complex correntropy function: Properties, and application to a channel equalization problem

The use of correntropy as a similarity measure has been increasing in different scenarios due to the well-known ability to extract high-order statistic information from data. Recently, a new similarity measure between complex random variables was defined and called complex correntropy. Based on a Gaussian kernel, it extends the benefits of correntropy to complex-valued data. However, its properties have not yet been formalized. This paper studies the properties of this new similarity measure and extends this definition to positive-definite kernels. Complex correntropy is applied to a channel equalization problem as good results are achieved when compared with other algorithms such as the complex least mean square (CLMS), complex recursive least squares (CRLS), and least absolute deviation (LAD).

A perspective on CLMS as a deficient length augmented CLMS: Dealing with second order noncircularity

Mean and mean square performance analyses of the strictly linear complex least mean square (CLMS) algorithm are addressed for widely linear estimation (WLE) of second order noncircular (improper) Gaussian inputs, for which both the covariance and pseudo-covariance matrices contain nonzero off-diagonal elements. A detailed performance analysis of standard CLMS in this ‘suboptimal’ context is not trivial but is important for practical applications. To this end, we here consider the strictly linear CLMS as a ‘deficient length’ version of the widely linear augmented CLMS (ACLMS) algorithm, which is second order optimal for improper inputs. Rigorous performance analysis is provided, which also statistically quantifies the suboptimality of CLMS in both the transient and steady state stages. The recently introduced approximate uncorrelating transform (AUT) is employed to derive closed-form expressions for the mean square stability and the steady-state performance of CLMS. In addition, since WLE with second order noncircular inputs is a general linear estimation problem in the complex domain C, these results provide a generalised framework from which current statistical descriptions of CLMS for strictly linear estimation (SLE) can be deduced as special cases. Simulations in system identification settings validate the findings.

Adaption Penalized Complex LMS for Sparse Under-Ice Acoustic Channel Estimations

Accurate channel information is usually required in under-ice acoustic (UIA) communication, which exhibits a sparse characteristic. A type of norm-constrained least mean square (LMS) algorithm performs well in estimating the real-valued (passband) channels but cannot be directly applied to complex-valued (baseband) channels. This paper generalizes norm-constrained complex LMS. Complex-valued zero-attracting LMS and $l_{0}$ -norm LMS (C $l_{0} $ -LMS) are mentioned first. These two algorithms render a fixed penalty to each coefficient, which may deteriorate the performance. Then, we propose a complex-valued adaption penalized LMS (CAP-LMS) to further utilize the sparsity of the UIA channels. The adaption penalty is achieved by dividing $p$ -norm-like constraints into two separate groups according to the $l_{1} $ -norm of each coefficient. For the dominant coefficients in the large group, the norm constraint disappears to reduce the estimation bias. For the small coefficients, the adaption penalty aims to accelerate the convergence speed. Simulation results are presented to demonstrate the superior performance of the CAP-LMS algorithm. Data processing results from two under-ice experiments show the feasibility and validity of the proposed algorithms in practical UIA applications.

A Full Mean Square Analysis of CLMS for Second-Order Noncircular Inputs

A full mean square transient and steady-state analysis of the complex least mean square (CLMS) algorithm is provided for strictly linear estimation of general second-order noncircular (improper) Gaussian inputs. To this end, we also consider the performance assessment in terms of the evolution of the complementary mean square error (CMSE) and the complementary covariance (pseudocovariance) matrix of the weight error vector of CLMS. This makes it possible to measure the degrees of noncircularity of the output error and the weight error vector, which arise due to second-order noncircularity (improperness) of the system input and system noise. The recently introduced approximate uncorrelating transform, which allows for joint direct diagonalization of both the input covariance and complementary covariance matrices with a single singular value decomposition, is then employed in order to derive a unified bound on the step-size, which guarantees the convergence of both the standard MSE and the proposed CMSE. A joint consideration of the standard mean square performance analysis and the proposed complementary performance analysis is shown to provide full second order, closed form, statistical descriptions of both the transient and steady state performances of CLMS for second-order noncircular (improper) Gaussian input data. Simulations in the system identification setting support the analysis.

Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization

Equalization filtering is an effective technique applied to minimize the inter-symbol interference (ISI) in multipath fading channels; the problem gets worse for higher-order constellations which are required for high data rates in today’s communication systems. The least mean square (LMS) filter is a computationally efficient and easily implementable algorithm but suffers from slow convergence; highly complex filters are required to nullify the effects of ISI. In this paper, we develop complex modified fractional-order (FO) nonlinear variants of the LMS and the NLMS algorithms and apply in adaptive channel equalization, in both feed-forward and decision feedback configurations. In addition to the standard first-order derivative, the update in the modified LMS also depends on the FO derivative of the mean square error, the final update is formed using a combination of conventional update term and a nonlinear term obtained through Riemann–Liouville fractional derivative. The step size of the FNLMS scheme in fractional part is not only a function of the input energy but also the FO. The differintegral operator working as differentiator helps improve the convergence rate because the algorithm becomes nonlinear; the fractional algorithms provide more parameters to control the rate of convergence and have simple implementation with almost similar complexity. The performances of the schemes are validated through extensive simulation results for block fading channels (frequency flat and selective) to evaluate the symbol error rate for higher-order quadrature amplitude modulation schemes, mean square error and combined channel and equalizer responses to show the improved inverse modeling of the channel. Simulation experiments confirm the superiority of the proposed algorithm over the traditional counterparts.

Quantized augmented complex least-mean square algorithm: Derivation and performance analysis

Augmented adaptive filters provide superior performance over their conventional counterparts when dealing with noncircular complex signals. However, the performance of such filters may change considerably when they are implemented in finite-precision arithmetic due to round-off errors. In this paper, we study the performance of recently introduced augmented complex least mean-square (ACLMS) algorithm when it is implemented in finite-precision arithmetic. To this aim, we first derive a model for the finite-precision ACLMS updating equations. Then, using the established energy conservation argument, we derive a closed-form expression, in terms of the excess mean-square error (EMSE) metric which explains how the quantized ACLMS (QACLMS) algorithm performs in the steady-state. We further derive the required conditions for mean stability of the QACLMS algorithm. The derived expression, supported by simulations, reveals that unlike the infinite-precision case, the EMSE curve for QACLMS is not monotonically increasing function of the step-size parameter. We use this observation to optimize the step-size learning parameter. Simulation results illustrate the theoretical findings.

Frequency Estimation of Unbalanced Three-Phase Power System using a New LMS Algorithm

This paper presents a simple and easy implementable Least Mean Square (LMS) type approach for frequency estimation of three phase power system in an unbalanced condition. The proposed LMS type algorithm is based on a second order recursion for the complex voltage derived from Clarke’s transformation which is proved in the paper. The proposed algorithm is real adaptive filter with real parameter (not complex) which can be efficiently implemented by DSP. In unbalanced situations, simulation experiments show the advantages and drawbacks of the proposed algorithm in comparison to Complex LMS (CLMS) and Augmented Complex LMS (ACLMS) methods. Index Terms-Frequency estimation, LMS algorithm, adaptive filter, power systems.

Accurate parameter estimation for unbalanced three-phase system.

Smart grid is an intelligent power generation and control console in modern electricity networks, where the unbalanced three-phase power system is the commonly used model. Here, parameter estimation for this system is addressed. After converting the three-phase waveforms into a pair of orthogonal signals via the α β-transformation, the nonlinear least squares (NLS) estimator is developed for accurately finding the frequency, phase, and voltage parameters. The estimator is realized by the Newton-Raphson scheme, whose global convergence is studied in this paper. Computer simulations show that the mean square error performance of NLS method can attain the Cramér-Rao lower bound. Moreover, our proposal provides more accurate frequency estimation when compared with the complex least mean square (CLMS) and augmented CLMS.

A Novel Approach for Hybrid of Adaptive Amplitude Non-Linear Gradient Decent (AANGD) and Complex Least Mean Square (CLMS) Algorithms for Smart Antennas

An adaptive beam former is a device, which is able to steer and modifies an array's beam pattern in order to enhance the reception of a desired signal, while simultaneously suppressing interfering signals through complex weight selection. However, the weight selection is a critical task to get the low Side Lobe Level (SLL) and Low Beam Width. One needs to have a low SLL and low beam width to reduce the antenna's energy radiation/reception ability in unintended directions. The weights can be chosen to minimize the SLL and to place nulls at certain angles. The convergence of the array output towards desired signal is also very important for a good signal processing tool of an adaptive beam former. A vast number of possible window functions are available to calculate the weights for Smart Antennas. From the analysis of many of these algorithms, it is observed that there is a compromise between HPBW and SLL. But in case of smart antennas, both of these parameters must have low values to get good performance. In our earlier work it is proposed that Complex Least Mean Square (CLMS) and Augmented Complex Least Mean Square ( ACLMS) algorithms gives low beam width and side lobe level in noisy environment. Another neural algorithm Adaptive Amplitude Non Linear Gradient Decent algorithm (AANGD) has the advantage of more number of control parameters over CLMS and ACLMS algorithms. In this paper the hybrid of CLMS and AANGD is presented and this novel hybrid algorithm has outperformed the hybrid algorithm of CLMS and ACLMS in the aspect of convergence towards the desired signal.

Complex Adaptive LMS Algorithm Employing the Conjugate Gradient Principle for Channel Estimation and Equalization

The Complex Block Least Mean Square (LMS) technique is widely used in adaptive filtering applications because of its simplicity and efficiency from a theoretical and implementation standpoint. However, the limitations of the Complex Block LMS technique are slow convergence and dependence on the proper choice of the stepsize or convergence factor. Moreover, its performance degrades significantly in time-varying environments. In this paper, a novel adaptive LMS technique named the Complex Block Conjugate LMS algorithm, CBC-LMS, is presented. Based on the Conjugate Gradient Principle, the proposed technique searches orthogonal directions to update the filter coefficients instead of the negative gradient directions used in the Complex Block LMS algorithm. In addition, the CBC-LMS algorithm derives optimal stepsizes to adjust the adaptive system coefficients at each iteration. As a result, the developed method overcomes the inherent limitations of the existing Complex Block LMS algorithm. The performance of the CBC-LMS technique is tested in wireless channel estimation and equalization applications, using both computer simulations and laboratory experiments. Furthermore, the developed technique is compared to the Complex Block LMS method and a recently proposed method, which is called Complex Optimal Block Adaptive LMS (OBA-LMS). The experimental and simulation results confirm that the proposed CBC-LMS technique achieves faster convergence with comparable accuracy and reduced computational complexity, relative to the existing techniques.

Nonlinear system modeling and identification using Volterra‐PARAFAC models

SUMMARYDiscrete‐time Volterra models are widely used in various application areas. Their usefulness is mainly because of their ability to approximate to an arbitrary precision any fading memory nonlinear system and to their property of linearity with respect to parameters, the kernels coefficients. The main drawback of these models is their parametric complexity implying the need to estimate a huge number of parameters. Considering Volterra kernels of order higher than two as symmetric tensors, we use a parallel factor (PARAFAC) decomposition of the kernels to derive Volterra‐PARAFAC models that induce a substantial parametric complexity reduction. We show that these models are equivalent to a set of Wiener models in parallel. We also show that Volterra kernel expansions onto orthonormal basis functions (OBF) can be viewed as Tucker models that we shall call Volterra‐OBF‐Tucker models. Finally, we propose three adaptive algorithms for identifying Volterra‐PARAFAC models when input–output signals are complex‐valued: the extended complex Kalman filter, the complex least mean square (CLMS) algorithm and the normalized CLMS algorithm. Some simulation results illustrate the effectiveness of the proposed identification methods. Copyright © 2011 John Wiley & Sons, Ltd.

Convergence analysis of a complex LMS algorithm with tonal reference signals

Often one encounters the presence of tonal noise in many active noise control applications. Such noise, usually generated by periodic noise sources like rotating machines, is cancelled by synthesizing the so-called antinoise by a set of adaptive filters which are trained to model the noise generation mechanism. Performance of such noise cancellation schemes depends on, among other things, the convergence characteristics of the adaptive algorithm deployed. In this paper, we consider a multireference complex least mean square (LMS) algorithm that can be used to train a set of adaptive filters to counter an arbitrary number of periodic noise sources. A deterministic convergence analysis of the multireference algorithm is carried out and necessary as well as sufficient conditions for convergence are derived by exploiting the properties of the input correlation matrix and a related product matrix. It is also shown that under convergence condition, the energy of each error sequence is independent of the tonal frequencies. An optimal step size for fastest convergence is then worked out by minimizing the error energy.

A global gradient-noise covariance expression for stationary real Gaussian inputs

Supervised parameter adaptation in many artificial neural networks is largely based on an instantaneous version of gradient descent called the least-mean-square (LMS) algorithm. This paper considers only neural models which are linear with respect to their adaptable parameters and has two major contributions. First, it derives an expression for the gradient-noise covariance under the assumption that the input samples are real, stationary, Gaussian distributed but can be partially correlated. This expression relates the gradient correlation and input correlation matrices to the gradient-noise covariance and explains why the gradient noise generally correlates maximally with the steepest principal axis and minimally with the one of the smallest curvature, regardless of the magnitude of the weight error. Second, a recursive expression for the weight-error correlation matrix is derived in a straightforward manner using the gradient-noise covariance, and comparisons are drawn with the complex LMS algorithm.

Performance advantage of complex LMS for controlling narrow-band adaptive arrays

In narrow-band adaptive-array applications, the mean-square convergence of the discrete-time real least mean-square (LMS) algorithm is slowed by image-frequency noises generated in the LMS loops. The complex LMS algorithm proposed by Widrow et al. is shown to eliminate these noises, yielding convergence of the mean-squared error (MSE) at slightly over twice the rate. This paper includes a comprehensive analysis of the MSE of adaptation for LMS. The analysis is based upon the method developed in the 1968 dissertation by K. D. Senne, and it represents the most complete treatment of the subject published to date.

Performance advantage of complex LMS for controlling narrow-band adaptive arrays

In narrow-band adaptive-array applications, the mean-square convergence of the discrete-time real least mean-square (LMS) algorithm is slowed by image-frequency noises generated in the LMS loops. The complex LMS algorithm proposed by Widrow et aL is shown to eliminate these noises, yielding convergence of the mean-squared error (MSE) at slightly over twice the rate. This paper includes a comprehensive analysis of the MSE of adaptation for LMS. The analysis is based upon the method developed in the 1968 dissertation by K. D. Senne, and it represents the most complete treatment of the subject published to date.