Quaternion-valued signal processing has received more and more attentions in the past ten years
due to the increasing need to process three or four-dimensional signals, such as colour images,
vector-sensor arrays, three-phase power systems, dual-polarisation based wireless communica-
tion systems, and wind profile prediction. One key operation involved in the derivation of all
kinds of adaptive signal processing algorithms is the gradient operator. Although there are some
derivations of this operator in literature with different level of details in the quaternion domain,
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 study, we will give a detailed
derivation of the quaternion-valued gradient operator with associated properties and then apply
it to different areas. In particular, it will be employed to derive the quaternion-valued LMS
(QLMS) algorithm and its sparse versions for adaptive beamforming for vector sensor arrays,
and another one is its application to wind profile prediction in combination with the classic
computational fluid dynamics (CFD) approach.
For the adaptive beamforming problem for vector sensor arrays, we consider the crossed-
dipole array and the problem of how to reduce the number of sensors involved in the adap-
tive beamforming process, so that reduced system complexity and energy consumption can be
achieved, whereas an acceptable performance can still be maintained, which is particularly use-
ful for large array systems. The quaternion-valued steering vector model for crossed-dipole
arrays will be employed, and a reweighted zero attracting (RZA) QLMS algorithm is then pro-
posed by introducing a RZA term to the cost function of the original QLMS algorithm. The
RZA term aims to have a closer approximation to the l0 norm so that the number of non-zero
valued coefficients can be reduced more effectively in the adaptive beamforming process.
For wind profile prediction, it can be considered as a signal processing problem and we
can solve it using traditional linear and non-linear prediction techniques, such as the proposed
QLMS algorithm and its enhanced frequency-domain multi-channel version. On the other hand,it using traditional linear and non-linear prediction techniques, such as the proposed
QLMS algorithm and its enhanced frequency-domain multi-channel version. On the other hand,wind flow analysis is also a classical problem in the CFD field, which employs various simulation
methods and models to calculate the speed of wind flow at different time. It is accurate
but time-consuming with high computational cost. To tackle the problem, a combined approach
based on synergies between the statistical signal processing approach and the CFD approach is
proposed. There are different ways of combining the signal processing approach and the CFD
approach to obtain a more effective and efficient method for wind profile prediction. In the
combined method, the signal processing part employs the QLMS algorithm, while for the CFD
part, large eddy simulation (LES) based on the Smagorinsky subgrid-scale (SGS) model will be
employed so that more efficient wind profile prediction can be achieved.