Transition between flat magnitude and flat group delay low pass recursive digital filters

Many a paper has been written on the characteristics of the continuous-time and discrete-time low-pass filters which are designed to have either maximally flat magnitude characteristics or maximally flat group delay responses. The polynomial (all pole) low pass recursive digital filters which are characterised by characterise the transition between a flat magnitude and flat group delay characteristics, named Transitional Butterworth Thiran (TBTh) recursive digital filters, is considered. The characteristics of the resulting filter change gradually from the characteristics of the well known Butterwort’s filter to the characteristics of the Thiran’s filter with the maximally flat group delay characteristic. Poles of the TBTh filter are obtained by interpolating between poles Butterworth and Thiran’s filters by varying a parametar which controls the pole positions that enables a tradeoff between the steepness magnitude characteristic at passband edge and group delay deviation.