Second-order Noncircularity Research Articles

Full Mean Square Performance Bounds on Quaternion Estimators for Improper Data

Novel physical insights are provided into the mean square performance bounds of quaternion-valued widely linear (WL), semi-widely linear (SWL), and strictly linear estimators for the generality of quaternion-valued Gaussian data. This is achieved by first defining three kinds of complementary mean square errors (CMSEs) of these estimators and by further exploiting the corresponding degrees of $\mathbb {H}$ -improperness (second-order noncircularity). Next, the investigation of the bounds of the attainable CMSEs by these classes of estimators shows that only a joint consideration of the proposed CMSE analysis and the standard MSE analysis provides enough degrees of freedom for a detailed account of the MSE performance. The so-established framework for the analysis of estimators for $\mathbb {H}$ -improper data is shown to be capable of measuring error power distribution for each data channel, an important finding, which is not possible to obtain through the standard MSE analysis only. Simulations in the system identification setting support the analysis.

Sparse Bayesian learning for off-grid DOA estimation with Gaussian mixture priors when both circular and non-circular sources coexist

In this paper, the problem of off-grid direction of arrival (DOA) estimation for the more general case of coexisting circular and non-circular signals is investigated from the perspective of sparse Bayesian learning (SBL). To utilize the second-order non-circularity of received signals, we carry out the DOA estimation by jointly representing the covariance and pseudo-covariance vectors. Although the sparse coefficient vectors of the covariance and pseudo-covariance vectors share common joint sparsity in the angular domain of non-circular sources, they have additional individual sparsity accounts for circular sources. Thus, the existing SBL methods based on joint sparsity will inevitably induce undesirable biases. To deal with this problem, a novel SBL method with the Gaussian mixture priors is developed. The proposed method can automatically identify the non-circular sources from the candidate angle grid and align the directional information of the non-circular sources in both the covariance and pseudo-covariance vectors. Moreover, the closed-form expressions for the perturbations of the covariance and pseudo-covariance vectors are also re-derived. Simulation results demonstrate that the proposed method achieves a significant performance improvement over existing methods.

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.

ESPRIT-like two-dimensional direction finding for mixed circular and strictly noncircular sources based on joint diagonalization

In this paper, a two-dimensional (2-D) direction-of-arrival (DOA) estimation method for a mixture of circular and strictly noncircular signals is presented based on a uniform rectangular array (URA). We first formulate a new 2-D array model for such a mixture of signals, and then utilize the observed data coupled with its conjugate counterparts to construct a new data vector and its associated covariance matrix for DOA estimation. By exploiting the second-order non-circularity of incoming signals, a computationally effective ESPRIT-like method is adopted to estimate the 2-D DOAs of mixed sources which are automatically paired by joint diagonalization of two direction matrices. One particular advantage of the proposed method is that it can solve the angle ambiguity problem when multiple incoming signals have the same angle θ or β. Furthermore, the theoretical error performance of the proposed method is analyzed and a closed-form expression for the deterministic Cramer-Rao bound (CRB) for the considered signal scenario is derived. Simulation results are provided to verify the effectiveness of the proposed method.

Quaternion-valued echo state networks.

Quaternion-valued echo state networks (QESNs) are introduced to cater for 3-D and 4-D processes, such as those observed in the context of renewable energy (3-D wind modeling) and human centered computing (3-D inertial body sensors). The introduction of QESNs is made possible by the recent emergence of quaternion nonlinear activation functions with local analytic properties, required by nonlinear gradient descent training algorithms. To make QENSs second-order optimal for the generality of quaternion signals (both circular and noncircular), we employ augmented quaternion statistics to introduce widely linear QESNs. To that end, the standard widely linear model is modified so as to suit the properties of dynamical reservoir, typically realized by recurrent neural networks. This allows for a full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances, and a rigorous account of second-order noncircularity (improperness), and the corresponding power mismatch and coupling between the data components. Simulations in the prediction setting on both benchmark circular and noncircular signals and on noncircular real-world 3-D body motion data support the analysis.

Robust Capon beamforming exploiting the second-order noncircularity of signals

This paper extends the worst-case performance optimization based robust Capon beamformer (RCB) to the generalized case with possible noncircular signal-of-interest (SOI) and/or noncircular interferences. The proposed widely linear (WL) beamformer exploits the second-order noncircularity of the SOI and interferences simultaneously, and it is robust against the errors in the steering vector, sample covariance matrix and estimated noncircularity coefficient of the SOI. Further, a theoretical performance analysis for the class of WL beamformers using generalized loading in the presence of steering vector errors is presented. The superior performance of the proposed beamformer over the RCB is demonstrated by the theoretical results and various numerical experiments.

A class of quaternion Kalman filters.

The existing Kalman filters for quaternion-valued signals do not operate fully in the quaternion domain, and are combined with the real Kalman filter to enable the tracking in 3-D spaces. Using the recently introduced HR-calculus, we develop the fully quaternion-valued Kalman filter (QKF) and quaternion-extended Kalman filter (QEKF), allowing for the tracking of 3-D and 4-D signals directly in the quaternion domain. To consider the second-order noncircularity of signals, we employ the recently developed augmented quaternion statistics to derive the widely linear QKF (WL-QKF) and widely linear QEKF (WL-QEKF). To reduce computational requirements of the widely linear algorithms, their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3-D data, further reducing the computational requirements. Simulations using both synthetic and real-world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion Kalman filters.

Widely Linear MVDR Beamformers for Noncircular Signals Based on Time-Averaged Second-Order Noncircularity Coefficient Estimation

The optimal widely linear (WL) minimum variance distortionless response (MVDR) beamformer, which has a powerful performance for the reception of a noncircular signal, was proposed by Chevalier in 2009. Nevertheless, in spectrum monitoring or passive listening, the optimal WL MVDR beamformer is hard to implement due to an unknown second-order (SO) noncircularity coefficient. This paper aims at estimating the time-averaged SO noncircularity coefficient of a desired noncircular signal, whose waveform is unknown but whose steering vector is known, in the context of the optimal WL MVDR beamformer. The proposed noncircularity coefficient estimator can process 2N - 1 rectilinear signals at most using an array of N sensors. Moreover, a frequency-shift WL MVDR beamforming algorithm is proposed for a noncircular signal having a nonnull frequency offset or carrier residue, jointly with the estimation of the frequency offset of the rectilinear signal. Due to the inevitable estimation error of the time-averaged SO noncircularity coefficient, a diagonal loading technique is used to enhance the robustness of the optimal WL beamformers. Simulations are shown to verify the effectiveness of the proposed algorithms.

GLRT-based array receivers for the detection of a known signal with unknown parameters corrupted by noncircular interferences

The detection of a known signal with unknown parameters in the presence of noise plus interferences (called total noise) whose covariance matrix is unknown is an important problem which has received much attention these last decades for applications such as radar, satellite localization or time acquisition in radio communications. However, most of the available receivers assume a second order (SO) circular (or proper) total noise and become suboptimal in the presence of SO noncircular (or improper) interferences, potentially present in the previous applications. The scarce available receivers which take the potential SO noncircularity of the total noise into account have been developed under the restrictive condition of a known signal with known parameters or under the assumption of a random signal. For this reason, following a generalized likelihood ratio test (GLRT) approach, the purpose of this paper is to introduce and to analyze the performance of different array receivers for the detection of a known signal, with different sets of unknown parameters, corrupted by an unknown noncircular total noise. To simplify the study, we limit the analysis to rectilinear known useful signals for which the baseband signal is real, which concerns many applications.

An adaptive approach for the identification of improper complex signals

A real-time approach for the identification of second-order noncircularity (improperness) of complex valued signals is introduced. This is achieved based on a convex combination of a standard and widely linear complex adaptive filter, trained by the corresponding complex least mean square (CLMS) and augmented CLMS (ACLMS) algorithms. By providing a rigorous account of widely linear autoregressive modelling the analysis shows that the monitoring of the evolution of the adaptive convex mixing parameter within this structure makes it possible to both detect and track the complex improperness in real time, unlike current methods which are block based and static. The existence and uniqueness of the solution are illustrated through the analysis of the convergence of the convex mixing parameter. The analysis is supported by simulations on representative datasets, for a range of both proper and improper inputs.

Quaternion-Valued Nonlinear Adaptive Filtering

A class of nonlinear quaternion-valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions. To circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion-valued estimation models, we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space. It is shown that the quaternion-valued exponential function is locally analytic, and, since local analyticity extends to polynomials, products, and ratios, we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models. This provides a unifying framework for the derivation of gradient-based learning algorithms in the quaternion domain, and the derived algorithms are shown to have the same generic form as their real- and complex-valued counterparts. To make such models second-order optimal for the generality of quaternion signals (both circular and noncircular), we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters. This allows full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances to cater rigorously for second-order noncircularity (improperness), and the corresponding power mismatch in the signal components. Simulations over a range of circular and noncircular synthetic processes and a real world 3-D noncircular wind signal support the approach.

On the asymptotic distribution of GLR for impropriety of complex signals

In this paper, the problem of testing impropriety (i.e., second-order noncircularity) of a sequence of complex-valued random variables (RVs) based on the generalized likelihood ratio test (GLRT) for Gaussian distributions is considered. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that RVs are proper or improper, and under the true, not necessarily Gaussian distribution of the RVs. The considered RVs are independent but not necessarily identically distributed: assumption which has never been considered until now. This enables us to deal with the practical important situations of noncircular RVs disturbed by residual frequency offsets and additive circular noise. The receiver operating characteristic (ROC) of this test is derived as byproduct, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.