Design of optimum recursive digital filters with double zeros on the unit circle leading to symmetric ladder wave digital filter structures

A class of odd-order low-pass recursive digital filters leading to symmetric ladder wave digital (LWD) filter implementations is introduced. In the pass-band, these filters have the same equiripple performance as elliptic filters of the same order, whereas in the stop-band there are the maximum numbers of double zeros. This implies that if the odd filter order is 2M+1, then, in addition to a single zero at z = 1, the proposed filter has M/2 complex-conjugate double zero pairs [(M-1)/2 double and one single complex-conjugate zero pair] on the unit circle for M even [odd]. By slightly modifying an algorithm proposed earlier by the second author, a very fast iterative technique is developed for designing these filters to exhibit also an equiripple stop-band performance. When implemented using LWD filter structures, the proposed filters have the following two benefits compared to elliptic filters. First, the resulting structure becomes symmetric, which reduces the number of distinct multipliers by a factor of two. Second, the maximum amplitude value remains to be unity as well as the two outputs of the LWD structure remain power complementary also after quantizing the coefficient values (without additional scaling operations being needed when implementing elliptic filters.) Examples are included illustrating that the price to be paid for these attractive properties is only a 3- to 5-dB reduction in the stop-band attenuation.