Design of generalized fractional order gradient descent method

This paper focuses on the convergence problem of the emerging fractional order gradient descent method, and proposes three solutions to overcome the problem. In fact, the general fractional gradient method cannot converge to the real extreme point of the target function, which critically hampers the application of this method. Because of the long memory characteristics of fractional derivative, fixed memory principle is a prior choice. Apart from the truncation of memory length, two new methods are developed to reach the convergence. The one is the truncation of the infinite series, and the other is the modification of the constant fractional order. Finally, six illustrative examples are performed to illustrate the effectiveness and practicability of proposed methods.Comment: 8 pages, 16 figure

Designs of fractional derivative constrained 1-D and 2-D FIR filters in the complex domain

[[abstract]]In this paper, the designs of fractional derivative constrained one-dimensional (1-D) and two-dimensional (2-D) FIR filters in the complex domain are investigated. First, the definition of fractional derivative is reviewed briefly. Then, the 1-D FIR filters with complex-valued frequency responses are designed by minimizing the integral squares error or maximum absolute error under the constraint that the actual response and ideal response have several same fractional derivatives at the prescribed frequency point. Next, the proposed method is extended to design fractional derivative constrained 2-D FIR filters with complex-valued frequency responses. Finally, design and application examples are demonstrated to show that the proposed method has larger design flexibility than the conventional integer derivative constrained methods