New Trinion and Quaternion Set-Membership Affine Projection Algorithms

This brief introduces new data-selective adaptive filtering algorithms for trinion and quaternion spaces <inline-formula><tex-math notation="LaTeX">$\mathbb{T}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathbb{H}$</tex-math></inline-formula>. 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 (SMTAP and 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.

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