  Quaternion-valued signal processing has received increasing attention
recently. One key operation involved in derivation of all kinds of adaptive
algorithms is the gradient operator. Although there have been some derivations
of this operator in literature with different level of details, it is still not
fully clear how this operator can be derived in the most general case and how
it can be applied to various signal processing problems. In this work, we will
give a general derivation of the quaternion-valued gradient operator and then
apply it to two different areas. One is to combine with the classic
computational fluid dynamics (CFD) approach in wind profile prediction and the
other one is to apply the result to the adaptive beamforming problem for vector
sensor arrays.
