Design of Allpass Fractional Delay Filter and Fractional Hilbert Transformer Using Closed-Form of Cepstral Coefficients

Abstract

The paper describes a cepstrum-based approach to design the maximally flat allpass fractional delay filter and the fractional Hilbert transformers. The maximal flatness criteria on the phase/group delay responses are formulated as a system of linear equations in terms of cepstral coefficients. The solution to the cepstral coefficients has closed-form expression. Moreover, it's very attractive that the resultant cepstral coefficients are directly proportional to the design parameters: the fractional delay and phase shift in the delay filters and Hilbert transformers, respectively. This virtue implies a simple scheme for updating new filter coefficients corresponding to various delay value or phase shift amount. Only one set of cepstral coefficients needs to be designed and stored and the new set is obtained simply by multiplying the old set with a ratio. A fixed module is used to update the filter coefficients from the cepstral coefficients.