Analytical design of maximally flat FIR fractional Hilbert transformers

Recently, the classical Hilbert transformer is generalized into the fractional Hilbert transformer which could be implemented optically. This modi"cation of the Hilbert transform adds an additional degree of freedom on the Hilbert transformer and improves the performance of the transform. In this paper, the design of FIR "lters fractional Hilbert transformer is proposed. The FIR "lters are designed in the maximally #at sense. The impulse responses of the "lters are uniquely solved and expressed in simple analytic forms. The impulse responses can be exactly expressed as "xed point binary values. The resulting frequency responses approximate the ideal one very well in the middle-frequency band. E$cient hardware realization structures are obtained based on the symmetric properties of the impulse responses. Several design examples with various transform parameters and various "lter orders are presented. Some examples of 1-D/2-D edge detection are given. The examples show that the proposed FIR "lter can enhance the selected edges very e$ciently. 2001 Elsevier Science B.V. All rights reserved.