Augmented Complex Least Mean Square Research Articles

Augmented Complex Least Lncosh Algorithm for Adaptive Frequency Estimation

In this paper, an augmented complex least mean lncosh (ACLlncosh) algorithm is proposed to estimate the frequency of the three-phase unbalanced power system in impulsive noise environment. By introducing the Clarke’s αβ transformation and widely linear (WL) model into the Llncosh cost function, and using CR Calculus, the ACLlncosh algorithm is derived. Moreover, a variable parameter method is proposed to improve the performance of ACLlncosh. The mean and mean-square stability of the proposed ACLlncosh algorithm are analyzed, and the corresponding stable step size ranges are also given. Computer simulations under some typical power system operating situations verify the superiority of the proposed ACLlncosh algorithm than the complex least mean lncosh (CLlncosh) and augmented complex least mean square (ACLMS) algorithms in power system frequency estimation.

An affine combination of two augmented CLMS adaptive filters for processing noncircular Gaussian signals

This paper investigates the statistical behavior of an affine combination of two augmented complex least mean-square (ACLMS) adaptive filters for processing noncircular Gaussian signals. First, the optimal affine combiner is studied and its steady state mean behavior is explicitly quantified. This makes it possible to reveal the impact of the input second-order noncircularity coefficient on the combiner. Next, the mean square error analysis of the proposed affine-combined ACLMS filters is provided, proving that the combined scheme performs no less than the best of the individual ACLMS subfilters. Besides, to facilitate physically realization of the combination strategy, an error power based algorithm that can iterate the affine combiner to the optimal one is also proposed. Finally, illustrative numerical simulations are provided to support the theoretical results.

Online censoring based complex-valued adaptive filters

A class of complex-valued adaptive filtering algorithms is proposed, with the aim to reduce the cost of data processing in the complex domain. This is achieved by leveraging the advantages of the online censoring (OC) strategy and complex-valued adaptive signal processing (ASP). The proposed algorithms, namely the OC based complex-valued least mean square (OC-CLMS), OC based augmented CLMS (OC-ACLMS), OC based hybrid (OC-Hybrid), OC based complex-valued recursive least square (OC-CRLS), and OC based augmented CRLS (OC-ACRLS) algorithms, censor the less informative complex-valued data under certain rules, that is, they use only the most informative data for updating weight vectors. This is shown to considerably reduce the cost of data processing for processing both circular and noncircular complex-valued signals. Moreover, to censor possible outliers in the complex domain, we also develop the robust versions of the proposed algorithms, called the ROC-CLMS, ROC-ACLMS, ROC-Hybrid, ROC-CRLS, and ROC-ACRLS algorithms. Simulation results over both system identification and real-world prediction scenarios verify the attractive properties of the proposed OC strategy based complex-valued algorithms. Moreover, this study paves the way for using the proposed algorithms to process streaming big data in the complex domain.

Variants of Partial Update Augmented CLMS Algorithm and Their Performance Analysis

Naturally complex-valued information or those presented in complex domain are effectively processed by an augmented complex least-mean-square (ACLMS) algorithm. In some applications, the ACLMS algorithm may be too computationally- and memory-intensive to implement. In this paper, a new algorithm, termed partial-update ACLMS (PU-ACLMS) algorithm is proposed, where only a fraction of the coefficient set is selected to update at each iteration. Doing so, two types of partial-update schemes are presented referred to as the sequential and stochastic partial-updates, to reduce computational load and power consumption in the corresponding adaptive filter. The computational cost for full-update PU-ACLMS and its partial-update implementations are discussed. Next, the steady-state mean and mean-square performance of PU-ACLMS for non-circular complex signals are analyzed and closed-form expressions of the steady-state excess mean-square error (EMSE) and mean-square deviation (MSD) are given. Then, employing the weighted energy-conservation relation, the EMSE and MSD learning curves are derived. The simulation results are verified and compared with those of theoretical predictions through numerical examples.

Widely Linear Adaptive Instantaneous Frequency Estimation in Vector Hydrophones

To solve the problem that the time-frequency resolution of Short-Time Fourier Transform (STFT) is constrained by the window length and the moving step of the short time window, and to utilize the merits of a widely linear method, a novel instantaneous frequency estimation method in vector hydrophone was proposed. In this paper, a complex variable was constructed. It is composed of sound pressure and particle velocity as its real part and imaginary part, respectively. The constructed variable was approved to be second order noncircular (improper). For the modelling of noncircular signals, the standard linear estimation is not adequate and the pseudo-covariance matrix should also be taken into consideration. As a result, a widely linear adaptive instantaneous frequency estimation algorithm and its three solutions based on the augmented complex least mean square (ACLMS) method are presented to estimate the instantaneous frequency in vector hydrophones. The results of simulations and laboratory experiments prove that this approach based on a widely linear model performs better compared to STFT and strict linear filter methods.

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.

Complementary Mean Square Analysis of Augmented CLMS for Second Order Noncircular Gaussian Signals

A novel physical insight is provided into the behavior and performance of the augmented complex least mean square (ACLMS) algorithm for widely linear adaptive estimation of general second-order noncircular (improper) Gaussian signals, whereby the off-diagonal elements of the covariance and complementary covariance matrices are nonzero. This is achieved through a novel complementary mean square analysis, a counterpart to the standard mean square analysis, and which focuses on the behavior of the complementary second-order statistics of the output error and the augmented weight error vector. We next establish the effect of the degree of input noncircularity on the evolution for these two key parameters that govern the ACLMS. Both transient and steady-state performances are addressed and a stability bound on the step-size for their convergence is established. Simulations in the system identification setting support the analysis.

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.

Tracking analysis of augmented complex least mean square algorithm

SummaryThe augmented complex least mean‐square (ACLMS) algorithm is a suitable algorithm for the processing of both second‐order circular (proper) and noncircular (improper) signals. In this paper, we provide tracking analysis of the ACLMS algorithm in the non‐stationary environments. Using the established energy conservation argument, we derive a variance relation that contains moments that represent the effects of non‐stationary environment. We evaluate these moments and derive closed‐form expressions for the excess mean‐square error (EMSE) and mean‐square error (MSE). The derived expressions, supported by simulations, reveal that unlike the stationary case, the steady‐state EMSE, and MSE curves are not monotonically increasing functions of the step‐size parameter. We also use this observation to optimize the step‐size learning parameter. Simulation results illustrate the theoretical findings and match well with theory. Copyright © 2015 John Wiley & Sons, Ltd.

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.

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.

An Adaptive Diffusion Augmented CLMS Algorithm for Distributed Filtering of Noncircular Complex Signals

An adaptive diffusion augmented complex least mean square (D-ACLMS) algorithm for collaborative processing of the generality of complex signals over distributed networks is proposed. The algorithm enables the estimation of both second order circular (proper) and noncircular (improper) signals within a unified framework of augmented complex statistics. The analysis shows that the performance advantage of the widely linear D-ACLMS over the strictly linear D-CLMS increases with the degree of noncircularity while maintaining similar performance for proper data. Simulations on both synthetic benchmark and real world noncircular data support the approach.

Complex-valued prediction of wind profile using augmented complex statistics

This paper presents a novel approach for the simultaneous modelling and forecasting of wind whereby the wind field is considered as a vector of its speed and direction components in the field of complex numbers C . To account for the intermittency and coupling of wind speed and direction, we propose to use the recently introduced framework of augmented complex statistics. The augmented complex least mean square (ACLMS) algorithm is introduced and its usefulness in wind forecasting is analysed. Simulations over different wind regimes support the approach.