Geometric algebra based least-mean absolute third and least-mean mixed third-fourth adaptive filtering algorithms

With regards to the problem of multidimensional signal processing in the field of adaptive filtering, geometric algebra based higher-order statistics algorithms were proposed. For instance, to express a multidimensional signal as a multi-vector, the adaptive filtering algorithms described in this work leverage all of the benefits of GA theory in multidimensional signal processing. GA space is employed to extend the traditional least-mean absolute third (LMAT) and newly deduced least-mean mixed third-fourth (LMMTF) adaptive filtering methods for multi-dimensional signal processing. The objective of the presented GA-based least-mean absolute third (GA-LMAT) and GA-based least-mean mixed third-fourth (GA-LMMTF) algorithms is to minimize the cost functions by using higher-order statistics of the error signal e(n) in GA space. The simulation's results revealed that at significantly smaller step size, the given GA-LMAT algorithm is better than the others in terms of steady-state error and convergence rate. Besides, the defined GA-LMMTF algorithm mitigates for the instability of GA-LMAT as the step size increases and illustrates an improved performance relative to mean absolute error and convergence rate.