基於遞迴數位全通濾波器之新穎的二維數位濾波器及其多速率系統應用

Novel 2-D Digital Filter Structures Using Recursive Digital Allpass Filters and Their Applications to Multirate Systems

楊元豪(Yuan-Hau Yang)

指導教授 ： 李枝宏

國立臺灣大學/電機資訊學院/電信工程學研究所/博士(2007年)

https://doi.org/10.6342/NTU.2007.01159

摘要

本論文的目的在於發展新穎且有效率的技術來最佳化設計基於全通架構的二維遞迴數位濾波器與二維遞迴多速率濾波器組。
首先，我們回顧將用以解決非線性最佳化問題之廣為人知的信賴區間方法。接著，為了迭代地解決L_1與L_infinite準則的線性最小化問題，我們發展了基於PAS演算法之有效率的最佳化方法。這些演算法將成為後續進行濾波器設計時主要的演算法。
由二維不對稱半平面數位全通濾波器所構成的新穎架構被用來設計一般的二維遞迴數位濾波器。藉由考慮振幅、群組延遲與穩定性誤差建構出合適的非線性目標函數，並使用信賴區間法來求得其最佳解。
藉由並接兩個數位全通結構，我們探討了二維遞迴雙重互補濾波器的設計問題。由於該二維遞迴雙重互補濾波器由數位全通濾波器所組成，所以以相位近似問題為出發點可以推導出合適的線性最佳化問題。於是該設計問題可以透過符合L_1與L_infinite準則的PAS演算法來有效率地解決。值得注意的是二維遞迴雙重互補濾波器當其通帶與止帶相互對稱於某個頻率點時，會具有相當吸引人的雙重互補對稱性質;而此性質將可使二維遞迴雙重互補濾波器的設計與實現變得相當容易且有效率。
接著我們考慮二維遞迴雙重互補濾波器於二維多速率濾波器系統的相關應用。這種基於全通濾波器的濾波器組可以完全避免一般濾波器組會遭受到的振幅失真的問題。此外，相位失真的問題則可以藉由額外附加的全通濾波器來予以補償。
二維圓形對稱低通濾波器的設計問題是個廣為討論的課題。基於前幾個章節所建立的基礎知識，我們提出了以一維與二維數位全通濾波器所構成的新架構來設計二維圓形對稱低通濾波器。與現有的設計相較之下，我們所提出的架構與設計方法具有更佳的表現。
因為數位濾波器的最小化實現可以降低硬體需求與運算複雜度，所以是個廣為討論的研究課題。然而，二維數位濾波器的最小化實現，並不像一維數位濾波器的最小化實現那樣的容易。藉由矩陣表示法，我們考慮了一般化的二維數位濾波器之最小化實現。此外，我們利用Rosser二維穩態空間模型來驗證此架構確實可達成最小化實現。最後我們展示了直接型式二維對稱半平面數位全通濾波器所對應的晶格架構。藉由解反向遞迴式，二維對稱半平面晶格數位全通濾波器的反射係數函數可以由直接型式二維對稱半平面數位全通濾波器轉換而來。此外，我們亦可藉由基於信賴區間法的設計方法，來直接求得反射係數函數。於是，前面所提及的基於二維對稱半平面數位全通濾波器的架構均可以其對應的晶格架構來實做。而且我們可以直接由反射係數函數的絕對值，來確保二維對稱半平面數位全通濾波器的穩定性。

關鍵字

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

並列摘要

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.

並列關鍵字

2-D digital allpass filter ； doubly complementary ； multirate system ； lattice structure ； recursive ； stability ； nonsymmetric half plane ； symmetric half plane

參考文獻

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