Designs of matrix fractional order differentiators

• This is a research branch of fractional calculus and matrix filter design.
• Closed-form designs are obtained by DST and DCT methods.
• Optimal designs are obtained by convex optimization method.
• Image sharpening and signal de-noising applications are shown.

In this paper, the designs of matrix fractional order differentiator (MFOD) for differentiating digital signals are presented. First, the definitions of fractional derivatives are reviewed briefly and design problem of MFOD is stated. Then, three kinds of methods for designing MFOD are described including the conventional FIR and IIR filter methods, the discrete sine transform (DST) and discrete cosine transform (DCT) methods, and optimization methods. Next, numerical examples are demonstrated to compare the performances of these three design methods and the variable MFOD design is also studied. Finally, the image sharpening application and signal de-nosing application are used to show the effectiveness of the proposed matrix fractional order differentiators.