Analytical design of maximally flat FIR fractional Hilbert transformers

Abstract

Recently, the classical Hilbert transformer is generalized into the fractional Hilbert transformer which could be implemented optically. This modification 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 filters fractional Hilbert transformer is proposed. The FIR filters are designed in the maximally flat sense. The impulse responses of the filters are uniquely solved and expressed in simple analytic forms. The impulse responses can be exactly expressed as fixed point binary values. The resulting frequency responses approximate the ideal one very well in the middle-frequency band. Efficient hardware realization structures are obtained based on the symmetric properties of the impulse responses. Several design examples with various transform parameters and various filter orders are presented. Some examples of 1-D/2-D edge detection are given. The examples show that the proposed FIR filter can enhance the selected edges very efficiently.