Closed-form designs of digital fractional order Butterworth filters using discrete transforms

In this paper, closed-form designs of digital fractional order Butterworth (FOB) filters using discrete transform methods are presented. First, the integer-order Butterworth (IOB) filter is reviewed and the design problem of digital FOB filter is described. Then, the discrete cosine transform (DCT), discrete sine transform (DST) and discrete Fourier transform (DFT) are applied to obtain the transfer functions of FIR fractional order Butterworth filters. Because the closed-form designs are obtained, the filter coefficients are easily computed. Next, the variable FOB filter is designed using the Farrow structure and IIR FOB filter is designed by the Prony method. Finally, application examples in the digital image sharpening are studied to demonstrate the effectiveness of proposed FOB filter design approaches.