LMS Algorithm Based on C-expansion of Quaternion

Nowadays, the quaternion algebra, as an important tool in the multi-dimensional signal processing, has been applied to some fields and the better effects are obtained. In the paper, a least mean square algorithm based on C-expansion of quaternion (LMS-QC) is proposed for adaptive filtering. First, a LMS-QC algorithm for adaptive filtering is derived. The performance of LMS-QC algorithm is analyzed and the select range of stepsize is given. Further, a normalized LMS-QC beamformer is provided. The LMS-QC algorithm overcomes the limit of the least mean square algorithm based on R-expansion of quaternion (QLMS) and is conveniently applied to the multi-dimensional complex-signal processing. Comparing the LMS-QC algorithm with the QLMS algorithm, the computation of weight vector is only half in one iteration. The simulation results show that the LMS-QC algorithm is convergent. In stable state, the mean square error of estimation reaches to the minimum value, and the norm of weight vector and output SINR are close to the optical values. The LMS-QC algorithm performs better than the QLMS algorithm in main measuring range.