E. Hitzer

**Algebraic foundations of split
hypercomplex nonlinear adaptive filtering**

submitted to
*Math. Methods in the Applied Sciences*,
(2012).

**Abstract:**
A split hypercomplex learning algorithm for the training of nonlinear infinite
impulse response adaptive lters 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.

**AMS Subject Classification:** 60G35, 15A66.

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