Study of the quaternion LMS and four-channel LMS algorithms

The recently proposed quaternion least-mean-square (QLMS) algorithm for adaptive filtering of three- and four-dimensional signals has been analysed in the context of multi-step ahead prediction. For rigour, the relationship between multichannel LMS (MLMS) and QLMS is examined, and their differences are highlighted. This is achieved both in terms of the input-output relationship and in terms of the dynamics of weight updates. The convergence of QLMS is investigated and stability bounds confirm that QLMS and MLMS are fundamentally different. Simulations on both synthetic and real world multidimensional signals support the analysis.