Widely linear dynamic quaternion valued least mean square algorithm for linear filtering

The Recent developments in sensor technology; human centered computing and robotics have brought to light new classes of multidimensional data which are naturally represented as quaternion three and four-dimensional vector-valued processes. Such signals are readily modeled as real normal vectors in R3 and R4; however; it has become obvious that there are advantages in processing multidimensional data in division algebras (quaternion domain). The progress in the statistics of quaternion variable, particularly augmented statistics and widely linear modeling; has opened up a new front of research in channel equalization, vector sensor modeling and system identification. However, prediction gain, tracking ability and convergence speed of quaternion adaptive filters still need to be improved due to the fixed step size of those types of algorithms. Choosing the right value of step size is very important for the adaptation process of the algorithm. There is a tradeoff between the convergence speed and the missadjustment of the system. Using large step size value will produce high convergence speed and high missadjustment while using small step size value will produce slow convergence speed and low missadjustment. since in real scenario the input signal power does not remain constant, that will change the step-size according the changes of the input signal of the algorithm which increase the tradeoff between the convergence speed and the missadjustment. This changing will cause noise amplification and affects the convergence speed. In this thesis, a new quaternion gradient based adaptive algorithm for FIR adaptive filter is developed. The proposed algorithm is capable of processing the generality of quaternion and complex data signals in both noisy and noise-free environments. The new adaptive algorithm is called dynamic quaternion least mean square algorithm (DQLMS) because of the normalization process of the filter input and the variable step-size. Those techniques proved to be very useful to enhance the trade-off between the convergence speed and the steady-state MSE and achieve small misadjustment and fast convergence speed. The sign function has been implemented in the process of filter coefficients adjustments in order to get faster adaptation processes, for high speed communication. The DQLMS algorithm is extended to the widely linear model forming the WL-DQLMS algorithm in order for the algorithm to be able to capture the full second order statistics. Prediction gain, tracking ability and convergence speed of the proposed algorithms are analyzed and validated experimentally by various simulations on both synthetic and real world multidimensional data. The performance of the proposed algorithms are compared with quaternion least mean square QLMS, zero-attract quaternion least mean square ZA-QLMS, and widely linear quaternion least mean square WL-QLMS algorithms. In noise cancellation, the DQLMS and WL-DQLMS algorithms were able to recover the input signal in 30 and 50 samples respectively while the QLMS and ZA-QLMS needed 250 and 200 samples respectively in order to recover the same data. A superior performance is achieved by the proposed algorithms in system modeling where the DQLMS was able to track the correct weights values of the different modeled systems 430 sample faster than the QLMS and ZA-QLMS algorithms while the WL-DQLMS was faster than the WLQLMS algorithm by 950 samples. In prediction setting the proposed algorithms showed 4dp to 8dp higher prediction gain than other algorithms. Thus, the proposed algorithms proved to be superior over the other algorithms in all aspects.

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