Maximally flat allpass fractional Hilbert transformers

Recently, a generalization of the Hilbert transformer, the fractional Hilbert transformer, was defined and developed. In this paper, we propose a design of the allpass filter to realize the fractional Hilbert transformer based on the maximally flat approximation to the desired phase response. The coefficients are solved analytically for the traditional Hilbert transformer which is a special case of the fractional Hilbert transformer. Based on the closed-form coefficients, we show that the maximally flat allpass Hilbert transformers are stable. Design examples indicate that the proposed filters exhibit good approximation to the desired frequency response.