Design of computationally efficient elliptic IIR filters with a reduced number of shift-and-add operations in multipliers

In this paper, a new design method for elliptic IIR filters that provides the implementation of half of the multiplication constants with few shifters and adders is proposed. An IIR filter, when derived by the bilinear transformation from an elliptic minimal Q-factor analog prototype, has its z-plane poles on the circle that is orthogonal to the unit circle and has the center on the real axis of the plane. Due to this property, the center of the circle can be used as a parameter for the representation of a pole, whereas the second parameter is the radius of the pole. It is shown in this paper that the center of the circle is uniquely determined by the frequency for which the filter attenuation is 3 dB. This result is used for the realization based on the parallel connection of two all-pass networks. It is shown that all second-order all-pass sections can be implemented with one common multiplication constant determined by the center of the circle. The design method is presented that, by an appropriate distribution of a margin in the filter performance, predetermines the value of the common constant according to the desired number of shift-and-add operations. This way, half of the multipliers are replaced with a limited number of shifters and adders. Conventional computer programs for IIR elliptic digital filters can be used. The direct approach for the distribution of the z-plane poles among two all-pass functions is developed. The application and efficiency of the proposed design method are demonstrated by examples.