Research On Adaptive Filtering Algorithms For Quaternion

As the extension of real and complex numbers,quaternion expands the two-dimensional space into the four-dimensional space.Its own orthogonal characteristics can describe the relationship between the signal components well,which are loaded into each part of quaternion and can been kept the internal structure.Therefore,quaternion adaptive filtering theory is widely used in the processing of multi-dimensional signal such as wind field signal and vector array signal.As an important index of measurement,steady-state performance determines the performance of the filter for all kinds of adaptive filtering algorithms.For this reason,it is of theoretical and practical significance to analyze the steady state performance of quaternion adaptive filtering algorithm for multidimensional signals processing.In this thesis,the energy conservation relation of quaternion adaptive processing during successive iteration update is deduced.Then based on separation principle,some analytical expressions of the steady state performance of quaternion least mean square(QLMS)algorithm are deduced in stationary and non-stationary environment,including steady state mean square error(MSE),the tracking MSE,the optimal step-size and the minimum MSE.Respectively,the restrictive conditions for these expressions are also given.Simulation experiments verify the theoretical results.Secondly,in order to deal with the deteriorating performance of QLMS algorithm for non-Gaussian signal,quaternion maximum correlation entropy criterion(QMCC)adaptive algorithm is proposed.Compared with QLMS,simulation results demonstrate the higher stability of QMCC algorithm in non-Gaussian environment,especially at low signal to noise ratio.At the same time,based on the energy conservation relation and complex form Taylor series expansions,the expressions of the steady-state MSE of QMCC algorithm are further derived in steady environment.The simulation results show that the theoretical values fit the simulation values well.Finally,the quaternion wireless communication system model is constructed.The quaternion adaptive filtering algorithms is successfully applied to the channel equalization system,including QLMS equalization algorithm and the quaternion constant modulus blind equalization algorithm(QCMA).Aiming at the phase ambiguity problem of QCMA algorithm,a concurrent quaternion equalizer is proposed,in which a decision-directed QLMS(DD_QLMS)equalizer operates cooperatively with QCMA equalizer.The new algorithm not only can compensate phase offset,but also has lower steady-state MSE.