New Trinion and Quaternion Set-Membership Affine Projection Algorithms

This brief introduces new data-selective adaptive filtering algorithms for trinion and quaternion spaces T and H. The work advances the set-membership trinion-and quaternion-valued normalized least mean square (SMTNLMS and SMQNLMS) and the set-membership trinion-and quaternion-valued affine projection (SMTAPand SMQAP) algorithms. We derive set-membership trinion algorithms and then, as special cases, obtain trinion algorithms not employing the set-membership strategy. Prediction simulations based on recorded wind data are provided, showing the improved performance of the proposed algorithms in terms of reduced computational complexity. Then, the quaternion-based SMQAP and SMQNLMS algorithms are derived, and their improved performances are verified in an adaptive beamforming problem.