Clive Cheong Took

The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes

The quaternion least mean square (QLMS) algorithm is introduced for adaptive filtering of three- and four-dimensional processes, such as those observed in atmospheric modeling (wind, vector fields). These processes exhibit complex nonlinear dynamics and coupling between the dimensions, which make their component-wise processing by multiple univariate LMS, bivariate complex LMS (CLMS), or multichannel LMS (MLMS) algorithms inadequate. The QLMS accounts for these problems naturally, as it is derived directly in the quaternion domain. The analysis shows that QLMS operates inherently based on the so called ldquoaugmentedrdquo statistics, that is, both the covariance E{ xx H} and pseudocovariance E{ xx T} of the tap input vector x are taken into account. In addition, the operation in the quaternion domain facilitates fusion of heterogeneous data sources, for instance, the three vector dimensions of the wind field and air temperature. Simulations on both benchmark and real world data support the approach.

A Quaternion Widely Linear Adaptive Filter

A quaternion widely linear (QWL) model for quaternion valued mean-square-error (MSE) estimation is proposed. The augmented statistics are first introduced into the field of quaternions, and it is demonstrated that this allows for capturing the complete second order statistics available. The QWL model is next incorporated into the quaternion least mean-square (QLMS) algorithm to yield the widely linear QLMS (WL-QLMS). This allows for a unified approach to adaptive filtering of both Q-proper and Q-improper signals, leading to improved accuracies compared to the QLMS class of algorithms. Simulations on both benchmark and real world data support the analysis.

A Quaternion Gradient Operator and Its Applications

Real functions of quaternion variables are typical cost functions in quaternion valued statistical signal processing, however, standard differentiability conditions in the quaternion domain do not permit direct calculation of their gradients. To this end, based on the isomorphism with real vectors and the use of quaternion involutions, we introduce the HR calculus as a convenient way to calculate derivatives of such functions. It is shown that the maximum change of the gradient is in the direction of the conjugate gradient, which conforms with the corresponding solution in the complex domain. Examples in some typical gradient based optimization settings support the result.

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 Unitary Diagonalisation of a Special Class of Quaternion Matrices

Abstract We propose a unitary diagonalisation of a special class of quaternion matrices, the so-called η -Hermitian matrices A = A η H , η ∈ { i , j , κ } arising in widely linear modelling. In 1915, Autonne exploited the symmetric structure of a matrix A = A T to propose its corresponding factorisation (also known as the Takagi factorisation) in the complex domain C . Similarly, we address the factorisation of an ‘augmented’ class of quaternion matrices, by taking advantage of their structures unique to the quaternion domain H . Applications of such unitary diagonalisation include independent component analysis and convergence analysis in statistical signal processing.

An augmented affine projection algorithm for the filtering of noncircular complex signals

An augmented affine projection adaptive filtering algorithm (AAPA), utilising the full second order statistical information in the complex domain is proposed. This is achieved based on the widely linear model and the joint optimisation of the direct and conjugate data channel parameters. The analysis illustrates that the use of augmented complex statistics and widely linear modelling makes the AAPA suitable for the processing of both second order complex circular (proper) and noncircular (improper) signals. The derivation is supported by the analysis of convergence in the energy conservation setting. Simulations on both benchmark and real-world noncircular wind signals support the analysis.

Fast Independent Component Analysis Algorithm for Quaternion Valued Signals

An extension of the fast independent component analysis algorithm is proposed for the blind separation of both \BBQ-proper and \BBQ-improper quaternion-valued signals. This is achieved by maximizing a negentropy-based cost function, and is derived rigorously using the recently developed \mbi\BBH\BBR calculus in order to implement Newton optimization in the augmented quaternion statistics framework. It is shown that the use of augmented statistics and the associated widely linear modeling provides theoretical and practical advantages when dealing with general quaternion signals with noncircular (rotation-dependent) distributions. Simulations using both benchmark and real-world quaternion-valued signals support the approach.

Quaternion-Valued Stochastic Gradient-Based Adaptive IIR Filtering

A learning algorithm for the training of quaternion valued adaptive infinite impulse (IIR) filters is introduced. This is achieved by taking into account specific properties of stochastic gradient approximation in the quaternion domain and the recursive nature of the sensitivities within the IIR filter updates, to give the quaternion-valued stochastic gradient algorithm for adaptive IIR filtering (QSG-IIR). Further, to reduce computational complexity, a variant of the QSG-IIR is introduced, which for small stepsizes makes better use of the available information. Stability analysis and simulations on both synthetic and real world 4D data support the approach.

On Approximate Diagonalization of Correlation Matrices in Widely Linear Signal Processing

The so called “augmented” statistics of complex random variables has established that both the covariance and pseudocovariance are necessary to fully describe second order properties of noncircular complex signals. To jointly decorrelate the covariance and pseudocovariance matrix, the recently proposed strong uncorrelating transform (SUT) requires two singular value decompositions (SVDs). In this correspondence, we further illuminate the structure of these matrices and demonstrate that for univariate noncircular data it is sufficient to diagonalize the pseudocovariance matrix-this ensures that the covariance matrix is also approximately diagonal. The proposed approach is shown to result in lower computational complexity and enhanced numerical stability, and to enable elegant new formulations of performance bounds in widely linear signal processing. The analysis is supported by illustrative case studies and simulation examples.

Augmented Complex Common Spatial Patterns for Classification of Noncircular EEG From Motor Imagery Tasks

A novel augmented complex-valued common spatial pattern (CSP) algorithm is introduced in order to cater for general complex signals with noncircular probability distributions. This is a typical case in multichannel electroencephalogram (EEG), due to the power difference or correlation between the data channels, yet current methods only cater for a very restrictive class of circular data. The proposed complex-valued CSP algorithms account for the generality of complex noncircular data, by virtue of the use of augmented complex statistics and the strong-uncorrelating transform (SUT). Depending on the degree of power difference of complex signals, the analysis and simulations show that the SUT based algorithm maximizes the inter-class difference between two motor imagery tasks. Simulations on both synthetic noncircular sources and motor imagery experiments using real-world EEG support the approach.