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

• This is a research branch of fractional calculus and FIR filter design.
• The design accuracy of the FIR filter is improved by using fractional derivative constraint.
• The designs of band-pass filter, Hilbert transformer and low-pass filter have been studied in this paper.
• The performance comparison with conventional integer derivative constrained method has been demonstrated.

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.