@article{Bris:13b:Group-theoretic-structure,
title = {Group-theoretic structure of linear phase multirate filter banks},
author = {Christopher M. Brislawn},
url = {https://datascience.dsscale.org/wp-content/uploads/sites/3/2016/06/Group-TheoreticStructureOfLinearPhaseMultirateFilterBanks.pdf},
doi = {http://dx.doi.org/10.1109/TIT.2013.2259292},
year = {2013},
date = {2013-08-06},
journal = {IEEE Transactions on Information Theory},
volume = {59},
number = {9},
abstract = {Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For Dinvariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group C is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group S is the semidirect product of C by the diagonal gain scaling matrix group D. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group, W, that is in fact equal to its own scaled lifting group, W = SW, the results of this paper characterize the group-theoretic structure of W up to isomorphism. Although the half-sample symmetric class H does not form a group, it can be partitioned into cosets of its lifting cascade group, CH, or, alternatively, into cosets of its scaled lifting group, SH. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a 'noncommutative vector space.'},
note = {LA-UR-12-20858},
keywords = {Filter bank, free product, group, group lifting structure, JPEG 2000., Lifting, Linear phase filter, Polyphase matrix, semidirect product, Unique factorization, Wavelet},
pubstate = {published},
tppubtype = {article}
}