Study On Fractional-order Gradient Descent Method And Fractional-order LMS Adaptive Filtering Algorithm

With the development of science and technology,the concept of optimization has become very common.Researchers in related fields have solved many practical problems by establishing optimization problems and searching for the optimal solutions.The gradient descent method is an important algorithm in optimization problems.Its remarkable advantages are simple structure,easy implementation and wide application range.The concept of fractional-order calculus has been developed more than 300 years.By virtue of its characteristics different from integer-order calculus,with the advancement of computer science and technology,the theory of fractional-order calculus has been widely studied by many scholars and applied in different engineering fields.On the other hand,in the field of signal processing,adaptive filtering algorithms have also become the most widely studied filtering algorithms due to their efficient and stable signal processing capabilities.Among them,the Least Mean Square(LMS)algorithm developed based on the gradient descent method has the advantages of a relatively simple structure and a small amount of calculation.Combining fractional calculus with the LMS adaptive algorithm will bring new possibilities for the LMS algorithm to be different from the traditional integer-order LMS algorithm.First of all,this thesis studies the classic gradient descent method,and combines it with fractional-order calculus,to get their extended algorithm,ie.the fractional-order gradient descent method.The reason why the traditional fractional-order gradient descent method is difficult to converge to the true extreme point is analyzed,and the convergence effectiveness of the variable initial value of the fractional-order gradient descent method is tested.In order to simplify the computational complexity of the fractional-order gradient descent method and expand the usage range of the objective function of the fractional-order gradient descent method,a truncated fractional gradient descent method is obtained.Further research and analysis on the convergence performance of different order gradient descent methods are fulfilled.Compared with the conventional gradient descent method,the high-order gradient descent method has faster convergence speed and low convergence accuracy;while the low-order gradient descent method has faster convergence speed and low convergence accuracy,In order to combine the advantages of gradient descent method with different order and solve the contradictory problem of convergence performance requirements in the algorithm,this thesis combines existing research and aims at two situations where the objective function is in univariate form and multivariate form.A variable-order improved optimization algorithm is proposed,and the effectiveness of the improved algorithm proposed in this thesis is verified through a typical example.Then,this thesis introduces the concept of the fractional-order gradient descent method into the LMS adaptive filtering algorithm,and analyzes the filtering performance of the fractional-order LMS adaptive filtering algorithm.In order to improve the filtering effect of the fractional-order LMS adaptive algorithm,the algorithm is improved and named as the variable fractional-order LMS filtering algorithm,which can combine the advantages of high-order and low-order algorithms to achieve both good convergence of the algorithm.The two advantages of speed and convergence accuracy improve the convergence characteristics of the algorithm.By applying it to the noise canceller,it is verified that the fractional-order LMS algorithm has a better filtering effect.Finally,the convergence performance of the fractional-order gradient descent method is summarized,which is different from the characteristics of the integer-order gradient descent method and the filtering characteristics of the fractional LMS adaptive filtering algorithm.Some possible application prospects and scope are pointed out.