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

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.

A smart antenna system combines multiple antenna elements with a signal processing capability to optimize its radiation and/or reception pattern automatically in response to the signal environment through complex weight selection. The weight selection process to get suitable Array factor with low Half Power Beam Width (HPBW) and Side Lobe Level (SLL) is a complex method. The aim of this task is to design a new approach for smart antennas to minimize the noise and interference effects from external sources with least number of iterations. This paper presents Heuristics based adaptive step size Complex Least Mean Square (CLMS) model for Smart Antennas to speedup convergence. In this process Benveniste and Mathews algorithms are used as heuristics with CLMS and the improvement of performance of Smart Antenna System in terms of convergence rate and array factor are discussed and compared with the performance of CLMS and Augmented CLMS (ACLMS) algorithms.

A novel hybrid filter combining the complex least mean square (CLMS) and augmented CLMS (ACLMS) algorithms for complex domain adaptive filtering is introduced. The ACLMS has been shown to have improved performance in terms of prediction of non-circular complex data compared to that of the CLMS. By taking advantage of this along with the faster convergence of the CLMS, the hybrid filter is shown to give improved performance over both algorithms for both circular and non-circular data. Simulations on complex-valued synthetic and real world data support the effectiveness of this approach.

Current approaches to the mean-square analyses of the complex-least-mean-square (CLMS) and augmented CLMS (ACLMS) algorithms can be challenging due to the difficulty in diagonalising the augmented covariance matrix. By employing the recently introduced approximate uncorrelating transform (AUT), which diagonalizes the covariance and pseudocovariance matrices with a single singular value decomposition (SVD), we derive closed form expressions for both transient and steady-state mean square stability for the CLMS and ACLMS. Relationships between the degree of circularity of the input signal and the bound on the step-sizes of the CLMS and ACLMS are also established. We also show that for both CLMS and ACLMS, the steady-state misadjustment increases with the degree of non-circularity of the input signal. Simulations in the context of frequency estimation in power grid support the analyses.

An adaptive beam former is a device, which is able to steer and modify 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.It needs to have a low SLL and low beam width to reduce the antenna's radiation/reception ability in unintended directions.The weights can be chosen to minimize the SLL and to place nulls at certain angles.A vast number of possible window functions that are available to provide the weights to be used in Smart Antennas.This paper presents various traditional windowing techniques such as Binomial, Kaiser-Bessel, Blackman, Gaussian, and so on for computing weights for adaptive beam forming and also neural based methods like, Least Mean Square (LMS), Complex LMS (CLMS) [5], and Augmented CLMS (ACLMS) [1] algorithms.This paper discusses about various observations on signal processing techniques of Smart Antennas, that compromise between SLL and beam width (Directivity), to improve the base station capacity in Cellular and Mobile Communications and also the performance analysis of CLMS and ACLMS in terms of SLL and beam width, error convergence rate.

We examine the problem of estimating the frequency of a three-phase power system in an adaptive and low-cost manner when the voltage readings are contaminated with observational error and noise. We assume a widely-linear predictive model for the αβ complex signal of the system that is given by Clarke's transform. The system frequency is estimated using the parameters of this model. In order to estimate the model parameters while compensating for noise in both input and output of the model, we utilize the notions of total least-squares fitting and gradient-descent optimization. The outcome is an augmented gradient-descent total least-squares (AGDTLS) algorithm that has a computational complexity comparable to that of the complex least mean square (CLMS) and the augmented CLMS (ACLMS) algorithms. Simulation results demonstrate that the proposed algorithm provides significantly improved frequency estimation performance compared with CLMS and ACLMS when the measured voltages are noisy and especially in unbalanced systems.

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.

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

Operating frequency is one of the most important parameters in power system. This paper proposes a new algorithm for power system frequency estimation. The algorithm is based on the Hilbert-transform phase-locked loops (HTPLL) and the complex least mean square (CLMS) method. The three-phase voltage data is input to the HTPLLs to obtain an approximate frequency estimation, and a complex voltage vector is formed using an a s -transform. By then, combining the complex voltage vector and the approximate estimate, the CLMS algorithm provides a refined frequency estimation. Simulation results of the proposed algorithm are presented and the performance of the algorithm is compared to other frequency tracking techniques.

Real world complex-valued signals typically exhibit rotation-dependent distributions (noncircularity), and significant performance gains in learning algorithms can be obtained by accounting for information beyond the standard second-order noncircularity (impropriety). To this end, we introduce a new closed form definition of complex correntropy which is general enough to cater for both circular and noncircular distributions in complex data, and serves as a basis for a novel cost function for widely linear adaptive filtering, termed the maximum improper complex corren-tropy criterion (MICCC). A stochastic gradient adaptive filtering algorithm is developed based on the MICCC, and its standard and complementary convergence and stability analyses are conducted with respect to both the circularity of the estimation error and the kernel size in the underlying Parzen estimator. Performance advantages over the strictly linear correntropy algorithm (MCCC) and the mean square error based complex least mean square (CLMS) and augmented CLMS (ACLMS) are demonstrated through analysis and simulations.

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.

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.

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.

In this paper, a fractional order calculus based least mean square algorithm is proposed for complex system identification. The proposed algorithm, named as, fractional complex least mean square (FCLMS), successfully deals with the problem of complex error due to negative weights or complex input/output in the FLMS. For the evaluation purpose a complex linear system is considered. The FCLMS algorithm successfully identifies the complex system and achieve high convergence rate without compromising the steady state error. The convergence rate of the proposed FCLMS is two times better than that of the complex least mean square (CLMS).

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