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

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

Quaternion valued adaptive signal processing

Recent developments in sensor technology, human centered computing and robotics have brought to light new classes of multidimensional data which are naturally represented as three- or four-dimensional vector-valued processes. Such signals are readily modeled as real vectors in R3 and R4, however, it has become apparent that there are advantages in processing such data in division algebras - the quaternion domain. The progress in the statistics of quaternion variable, particularly augmented statistics and widely linear modeling, has opened up a new front of research in vector sensor modeling, however, there are several key problems that need to be addressed in order to exploit the full power of quaternions in statistical signal processing. The principal problem lies in the lack of a mathematical framework, such as the CR-calculus in the complex domain, for the differentiation of non-holomorphic functions. Since most functions (including typical cost functions) in the quaternion domain are non-holomorphic, as defined by the Cauchy-Riemann-Fueter (CRF) condition, this presents a severe obstacle to solving optimisation problems and developing adaptive filtering algorithms in the quaternion domain. To this end, we develop the HR-calculus, an extension of the CR-calculus, allowing the differentiation of non-holomorphic functions. This is followed by the introduction of the I-gradient, enabling for generic extensions of complex valued algorithms to be derived. Using this unified framework we introduce the quaternion least mean square (QLMS), quaternion recursive least squares (QRLS), quaternion affine projection algorithm (QAPA) and quaternion Kalman filter. These estimators are made optimal for the processing of noncircular data, by proposing widely linear extensions of their standard versions. Convergence and steady state properties of these adaptive estimators are analysed and validated experimentally via simulations on both synthetic and real world signals.Open Acces

Blind extraction of improper quaternion sources

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