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Quaternion-valued signal processing has received increasing attentionrecently. One key operation involved in derivation of all kinds of adaptivealgorithms is the gradient operator. Although there have been some derivationsof this operator in literature with different level of details, it is still notfully clear how this operator can be derived in the most general case and howit can be applied to various signal processing problems. In this work, we willgive a general derivation of the quaternion-valued gradient operator and thenapply it to two different areas. One is to combine with the classiccomputational fluid dynamics (CFD) approach in wind profile prediction and theother one is to apply the result to the adaptive beamforming problem for vectorsensor arrays.

A General Quaternion-valued Gradient Operator and Its Applications to Computational Fluid Dynamics and Adaptive Beamforming