Novel 2-D Digital Filter Structures Using Recursive Digital Allpass Filters and Their Applications to Multirate Systems
Date Issued
2007
Date
2007
Author(s)
Yang, Yuan-Hau
DOI
en-US
Abstract
Abstract
The purpose of this dissertation is to devise novel and efficient techniques for optimally designing two-dimensional (2-D) recursive digital filters and 2-D recursive multirate filter banks by employing allpass sections.
First, we review the well-known trust-region method that can efficiently solve the nonlinear optimization problem of designing the proposed 2-D recursive digital filter structure composed of allpass subfilters. Secondly, we develop the efficient optimization algorithms based on the primal affine-scaling variant of Karmarkar's algorithm (PAS algorithm) to iteratively solve the design problems in L1 and L_infinite senses, respectively, when we consider the phase approximation problem. The essences and central ideas of these algorithms are employed throughout.
A novel structure composed of 2-D non-symmetric half-plane (NSHP) digital allpass filters (DAFs) is utilized to design general 2-D recursive digital filters. An appropriate nonlinear objective function is formulated by considering the magnitude, group delay, and stability errors, simultaneously. It is worthy noting that the proposed structure is recursive computable and can be used to design some filters that cannot be accomplished by the existing quarter-plane (QP) allpass-based structures.
According to the results obtained by the novel structure mentioned above, we present the design of 2-D recursive doubly complementary (DC) filters by parallel interconnecting two 2-D allpass sections. The design problem is appropriately formulated to result in a simple linear optimization problem that minimizes the phase error. Thus, the design problem can be efficiently solved by using the PAS algorithm in L1 and L_infinite criteria. It is worthy noting that the 2-D DC filter exhibits very attractive DC symmetric characteristics when the passband and stopband of the 2-D DC filter are symmetric with respect to certain frequency point. Owing to this DC symmetric characteristic, the 2-D DC filter can be designed and implemented very efficiently. Besides, we find that the design of the widely used diamond-shaped filters can be efficiently realized by our proposed DC structure because the diamond-shaped filters possess quadrantal symmetry. This result shows the more general design capability of our design than the design based on 2-D QP allpass filters.
With regard to the 2-D filter bank systems, the application of 2-D DC filter for designing 2-D QMF banks is given. The 2-D recursive DAFs are the fundamental building blocks and we only need to focus on the phase approximation of them. The allpass-based structure will not induce any magnitude distortion. Besides, the phase distortion of the overall QMF system can be compensated by a suitable DAF that plays a role as a phase equalizer. It is shown that the quincunx QMF bank and the parallelogram QMF bank can be easily designed by applying the proposed linear approximation techniques.
Additionally, we deal with the widely considered design example of 2-D recursive circularly symmetric lowpass filter by proposing a novel structure composed of 1-D and 2-D recursive DAFs. The simulation results show very satisfactory performance in comparison with the existing researches.
The minimal realization of digital filters is widely interested because it needs the least hardware requirement and less computational complexity. However, it is not an easy task to develop a minimal realization of a 2-D filter as in the 1-D cases. We consider the realization of a generalized 2-D digital lattice filter by employing the corresponding matrix representation. In addition, the minimal realization of the proposed structure is verified by utilizing the Roesser 2-D state space model.
The corresponding lattice structure of the direct-form 2-D DAF with symmetric-half plane support (SHP) is presented. By solving the backward recursive equations, the reflection coefficient functions of the lattice-form 2-D SHP DAF are obtained. Besides, we present the technique based on the trust-region method to directly calculate the reflection coefficient functions. Thus, the filter structures composed of direct-form 2-D SHP DAFs can be implemented by the this lattice structure. The stability problem of designing 2-D SHP DAF can be easily guaranteed by evaluating the absolute values of the reflection coefficient functions.
Subjects
二維數位全通濾波器
雙重互補
多速率系統
晶格架構
遞迴
穩定性
不對稱半平面
對稱半平面
2-D digital allpass filter
doubly complementary
multirate system
lattice structure
recursive
stability
nonsymmetric half plane
symmetric half plane
Type
thesis
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