Quaternionic Lattice Structures for Four-Channel Paraunitary Filter Banks
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Research Article Quaternionic Lattice Structures for Four-Channel Paraunitary Filter Banks Marek Parfieniuk and Alexander Petrovsky Department of Real-Time Systems, Faculty of Computer Science, Bialystok Technical University, Wiejska 45A street, 15-351 Bialystok, Poland Received 31 December 2005; Revised 1 October 2006; Accepted 9 October 2006 Recommended by Gerald Schuller A novel approach to the design and implementation of four-channel paraunitary filter banks is presented. It utilizes hypercomplex number theory, which has not yet been employed in these areas. Namely, quaternion multipliers are presented as alternative paraunitary building blocks, which can be regarded as generalizations of Givens (planar) rotations. The corresponding quaternionic lattice structures maintain losslessness regardless of coefficient quantization and can be viewed as extensions of the classic twoband lattice developed by Vaidyanathan and Hoang. Moreover, the proposed approach enables a straightforward expression of the one-regularity conditions. They are stated in terms of the lattice coefficients, and thus can be easily satisfied even in finite-precision arithmetic. Copyright © 2007 M. Parfieniuk and A. Petrovsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
Paraunitary filter banks (PUFBs) can be considered the most important among multirate systems [1]. This results from the fact that such filter banks are lossless in addition to guaranteeing perfect reconstruction. A clear relation between the fullband and subband signal energies greatly simplifies theoretical considerations, and hence makes PUFBs useful for applications such as image coding. The paraunitary property means that the basis functions related to the subbands of a filter bank are orthogonal. However, it is more convenient to work with the analysis polyphase transfer matrix E(z), which is paraunitary if EH (z−1 )E(z) = cIM , where c is a nonzero constant and M denotes the number of channels [2]. Thus, instead of constraining the impulse response coefficients, the usual way to obtain a PUFB is to compose its polyphase matrix from suitable building blocks. From a different point of view, the matrix is appropriately factorized. In this way, other properties of the filter frequency responses can be simultaneously imposed, such as linear phase (LP), pairwisemirror-image (PMI) symmetry, and regularity. The selection and arrangement of factorization components are decisive.
Lattice and dyadic-based factorizations of paraunitary polyphase matrices can be distinguished. The first approach utilizes Givens (planar) rotations [2]. They are implemented with the help of a specific structure, whose shape is the reason for using the name “lattice.” The second technique is based on Householder reflections and degree-one building blocks, which are of a different nature [3]. The lattice structures are more f
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