Granular limit-cycles of L∞-scaled second-order all-pass sections

The granular limit cycles of three second-order all-pass digital filters are investigated. Two of them are well-known two-port wave digital all-pass filters, and the third is a combination of a two-port adaptor and a direct-form structure. When implementing a complex conjugate pole pair, this third structure has the nice property of being optimally scaled in the L_∞ sense with no regard to the coefficient values. It is proved that the structures considered are free of granular limit cycles for coefficients with absolute value lower than one half and a two´s-complement rounding. For coefficient values out of this range, the limit-cycle amplitudes are compared by means of measurements. The third structure is briefly compared to the other two with respect to round-off noise and forced-response stability