Digital all-pass filter design through discrete Hilbert transform

A simple method for the design of a digital all-pass filter, satisfying the given group delay specification, is presented. The design is based on the discrete Hilbert transform relation, relating the log-magnitude and phase of the Fourier transform of the minimum phase signal. The transfer function of an all-pass filter is completely determined by the coefficients of the denominator polynomial. For the filter to be stable, the denominator polynomial must be minimum phase. From the given group delay specification, the phase corresponding to the pole part of the desired filter is first determined. The magnitude spectrum corresponding to the pole part of the desired filter is obtained from the above phase through the discrete Hilbert transform relation. The method needs just four fast Fourier transform operations. There is no restriction on the order of the filter, and the number of filter coefficients can be selected after the final design, depending on the accuracy desired. The procedure is illustrated through design examples