Research Of Quaternion Adaptive Filtering Algorithms Based On HR Differential Operator

In order to process the hypercomplex signal,a quaternion adaptive filtering algorithm is proposed.In the Quaternion Least Mean Square(QLMS)adaptive filtering algorithm,there are many different versions of the QLMS algorithm due to the difference in calculation of the quaternion gradient.However,all the versions have their insufficient.Some versions borrow the derivative rule from complex and real domain directly.And although some versions have introduced the quaternion differential operator,they have ignored the non-commutativity of quaternion.It is worth noting that the rigorous processing of gradients is a prerequisite for the derivation of the stochastic gradient optimization algorithm.However,due to the lag of traditional quaternion differential operator theory,the development of quaternion adaptive filtering algorithm has been constrained.This problem has been improved by the introduction of the HR differential operator,with which the quaternion adaptive filtering algorithm can be derived and proved strictly.The paper mainly studied the quaternion differential operator theory and quaternion adaptive filtering algorithm.Because of the non-commutativity of quaternion multiplication,the product rule and chain rule of quadratic functions have essential differences from the traditional rules in the real and the complex domain.Although previous scholars have given the conclusions,the derivation is too complex and cumbersome,and the relationship with real domain differential is not clear enough.Therefore,by the equivalence relation between HR calculus and multivariate real calculus,we provide a new provement for HR calculus,and we have introduced the new prove of product rule,chain rule and the first order Taylor expansion.This method not only can simplify the process of proof,but also be wellunderstand.Then,three QLMS adaptive filtering algorithms are derived based on HR calculus operator: strict linear,semi-wide-linear and wide-linear QLMS algorithm.And,compared with the previous results,it illustrated the validity of the quaternion adaptive filtering algorithm based on HR calculus operator.Then,we analyzed the minimum error of these QLMS algorithms in steady state and compared their advantages and disadvantages.The conclusion is that the widely linear one is better than the semi-widely linear one and the semi-widely linear one is better than the strict linear one.At the same time,we also give thedirect minimum errors comparison between the widely linear and the strict linear QLMS in steady state,and the superiority of the wide linear algorithm is further proven.However,the computational complexity of the widely linear QLMS algorithm is very large,so in order to reduce the computational complexity,a new low complexity linear algorithm is proposed in this paper.