Algebraic foundations of split hypercomplex nonlinear adaptive filtering
This work addresses a domain-specific problem in signal processing for hypercomplex data, offering incremental improvements by correcting algebraic oversights in existing methods.
The authors tackled the problem of training nonlinear adaptive filters for hypercomplex signals by proposing a split hypercomplex learning algorithm that strictly adheres to hypercomplex algebra and calculus laws, which were previously neglected, and they predict performance improvements, such as for quaternions, with rigorous convergence analysis.
A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Keywords: Quaternionic adaptive filtering, Hypercomplex adaptive filtering, Nonlinear adaptive filtering, Hypercomplex Multilayer Perceptron, Clifford geometric algebra