Inherently lossless structures for eight- and six-channel linear-phase paraunitary filter banks based on quaternion multipliers

Novel factorizations for 8- and 6-channel linear-phase paraunitary filter banks are presented, which are aimed at finite-precision implementation. Using quaternion multipliers as essential building blocks, computational schemes for both critically sampled and oversampled systems, including those with pairwise-mirror-image (PMI) symmetric responses, have been made inherently lossless at the cost of extra operations. Compared to the known dyadic-based solution of this sort, which consists in norm equalization using double-precision scalings, the proposed structures are characterized by similar complexity but are more consistent in terms of wordlength. Additionally, one-regularity (zero DC leakage) constraints can be formulated in terms of hypercomplex coefficients, so that they can be used in the discrete domain, unlike the known method of constraining rotation angles, and an arbitrary stage can be constrained, not only the initial one, as in the known dyadic/lifting-based approach. Even though the quaternion approach is not as general as the mentioned ones, 8-channel systems it applies to are of primary importance in image processing.