  Quaternion-valued wireless communication systems have been studied in the
past. Although progress has been made in this promising area, a crucial missing
link is lack of effective and efficient quaternion-valued signal processing
algorithms for channel equalization and beamforming. With most recent
developments in quaternion-valued signal processing, in this work, we fill the
gap to solve the problem by studying two quaternion-valued adaptive algorithms:
one is the reference signal based quaternion-valued least mean square (QLMS)
algorithm and the other one is the quaternion-valued constant modulus algorithm
(QCMA). The quaternion-valued Wiener solution for possible block-based
calculation is also derived. Simulation results are provided to show the
working of the system.
