Filter-bank transforms with exact inverses

• Main Author: Parra, Paulo Mario
• Other Authors: Marcellin, Michael
• Language: en_US
• Published: The University of Arizona. 2003
• Subjects: Applied Mechanics.
• Online Access: http://hdl.handle.net/10150/289984

Uniformly sampled filter-bank transforms and their inverses are introduced and the conditions to obtain perfect reconstruction upon inversion are explored. It is shown that perfect reconstruction requires both filter addition and multiplication and the necessary and sufficient conditions for these operations are given. Examples indicate how to use the conditions to construct perfect-reconstruction synthesis filters from a given set of analysis filters. Additionally, an iterative scheme is presented that achieves exact inversion to an arbitrary accuracy. The methods to obtain synthesis filters are applied to discretizations of the continuous wavelet transform using both finite and infinite impulse response filters. If exact reconstruction is not a requisite, it is possible to improve imperfect-reconstruction filter banks so that their inverse is closer to the input signal. Two methods to achieve such improvement are described. To better understand the discretizations, one has to look at the continuous case. Therefore the discrete-time filter-bank transforms definitions are extended to continuous-time signal processing. It is shown that the Gabor and continuous wavelet transforms are special cases of the continuous-time extension. The methods introduced in the discrete-time case are used to derive all the linear time-invariant synthesis functions of these two transforms. A straightforward generalization of the Gabor and wavelet transforms generates filter banks whose bandwidths can vary arbitrarily with center frequency. These filters are used to create a cochlear transform, i.e., a "mixed" transform that behaves like a Gabor transform at low center frequencies and like a continuous wavelet transform at high center frequencies. The methodology described in this thesis is implemented in a set of algorithms whose complete documentation are given in chapter 4.