16 research outputs found

    A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence

    Full text link
    A novel framework for a unifying treatment of quaternion valued adaptive filtering algorithms is introduced. This is achieved based on a rigorous account of quaternion differentiability, the proposed I-gradient, and the use of augmented quaternion statistics to account for real world data with noncircular probability distributions. We first provide an elegant solution for the calculation of the gradient of real functions of quaternion variables (typical cost function), an issue that has so far prevented systematic development of quaternion adaptive filters. This makes it possible to unify the class of existing and proposed quaternion least mean square (QLMS) algorithms, and to illuminate their structural similarity. Next, in order to cater for both circular and noncircular data, the class of widely linear QLMS (WL-QLMS) algorithms is introduced and the subsequent convergence analysis unifies the treatment of strictly linear and widely linear filters, for both proper and improper sources. It is also shown that the proposed class of HR gradients allows us to resolve the uncertainty owing to the noncommutativity of quaternion products, while the involution gradient (I-gradient) provides generic extensions of the corresponding real- and complex-valued adaptive algorithms, at a reduced computational cost. Simulations in both the strictly linear and widely linear setting support the approach

    Widely Linear State Space Filtering of Improper Complex Signals

    Get PDF
    Complex signals are the backbone of many modern applications, such as power systems, communication systems, biomedical sciences and military technologies. However, standard complex valued signal processing approaches are suited to only a subset of complex signals known as proper, and are inadequate of the generality of complex signals, as they do not fully exploit the available information. This is mainly due to the inherent blindness of the algorithms to the complete second order statistics of the signals, or due to under-modelling of the underlying system. The aim of this thesis is to provide enhanced complex valued, state space based, signal processing solutions for the generality of complex signals and systems. This is achieved based on the recent advances in the so called augmented complex statistics and widely linear modelling, which have brought to light the limitations of conventional statistical complex signal processing approaches. Exploiting these developments, we propose a class of widely linear adaptive state space estimation techniques, which provide a unified framework and enhanced performance for the generality of complex signals, compared with conventional approaches. These include the linear and nonlinear Kalman and particle filters, whereby it is shown that catering for the complete second order information and system models leads to significant performance gains. The proposed techniques are also extended to the case of cooperative distributed estimation, where nodes in a network collaborate locally to estimate signals, under a framework that caters for general complex signals, as well as the cross-correlations between observation noises, unlike earlier solutions. The analysis of the algorithms are supported by numerous case studies, including frequency estimation in three phase power systems, DIFAR sonobuoy underwater target tracking, and real-world wind modeling and prediction.Open Acces

    Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing

    Full text link
    Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.Comment: 41 pages, single column, 10 figure

    Adaptive signal processing algorithms for noncircular complex data

    No full text
    The complex domain provides a natural processing framework for a large class of signals encountered in communications, radar, biomedical engineering and renewable energy. Statistical signal processing in C has traditionally been viewed as a straightforward extension of the corresponding algorithms in the real domain R, however, recent developments in augmented complex statistics show that, in general, this leads to under-modelling. This direct treatment of complex-valued signals has led to advances in so called widely linear modelling and the introduction of a generalised framework for the differentiability of both analytic and non-analytic complex and quaternion functions. In this thesis, supervised and blind complex adaptive algorithms capable of processing the generality of complex and quaternion signals (both circular and noncircular) in both noise-free and noisy environments are developed; their usefulness in real-world applications is demonstrated through case studies. The focus of this thesis is on the use of augmented statistics and widely linear modelling. The standard complex least mean square (CLMS) algorithm is extended to perform optimally for the generality of complex-valued signals, and is shown to outperform the CLMS algorithm. Next, extraction of latent complex-valued signals from large mixtures is addressed. This is achieved by developing several classes of complex blind source extraction algorithms based on fundamental signal properties such as smoothness, predictability and degree of Gaussianity, with the analysis of the existence and uniqueness of the solutions also provided. These algorithms are shown to facilitate real-time applications, such as those in brain computer interfacing (BCI). Due to their modified cost functions and the widely linear mixing model, this class of algorithms perform well in both noise-free and noisy environments. Next, based on a widely linear quaternion model, the FastICA algorithm is extended to the quaternion domain to provide separation of the generality of quaternion signals. The enhanced performances of the widely linear algorithms are illustrated in renewable energy and biomedical applications, in particular, for the prediction of wind profiles and extraction of artifacts from EEG recordings

    Performance Analysis of Shrinkage Linear Complex-Valued LMS Algorithm

    Get PDF
    The shrinkage linear complex-valued least mean squares (SL-CLMS) algorithm with a variable step size overcomes the conflicting issue between fast convergence and low steady-state misalignment. To the best of our knowledge, the theoretical performance analysis of the SL-CLMS algorithm has not been presented yet. This letter focuses on the theoretical analysis of the excess mean square error transient and steady-state performance of the SL-CLMS algorithm. Simulation results obtained for identification scenarios show a good match with the analytical results
    corecore