臺灣博碩士論文加值系統

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。

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.

List of Figures ………………………………………………………………………... ix

List of Tables ………………………………………………………………………… xv

Chapter 1. Introduction and Motivations …………………………………………... 1
1.1 Motivations ………………………………………………………………….. 1
1.2 Overview and Contribution of the Dissertation ……………………………... 2

Chapter 2. Optimization Methods for Nonlinear Minimization Problems ……….. 5
2.1 Introduction …………………………………………………………………. 5
2.2 Unconstrained Nonlinear Optimization Problems …………………………... 6
2.3 Line-Search Methods ………………………………………………………... 7
2.4 Trust-Region Methods ………………………………………………………. 8
2.5 Conclusion …………………………………………………………………. 11

Chapter 3. Optimization Methods for Linear Minimization Problems ………….. 13
3.1 Introduction ………………………………………………………………... 13
3.2 Linear Minimization Problems …………………………………………….. 14
3.2.1 Formulation of Linear Minimization Problems …………..………….. 14
3.2.2 Solutions …...…..……………………………………………….. 15
3.2.3 Linear Programming Formulations of the and Problems …. 16
3.3 Primal Affine-Scaling Variant of Karmarkar’s Algorithm (PAS Algorithm)..18
3.4 Iterative Procedures ………………………………………………………... 22
3.4.1 Minimization Algorithm ……..…………………………………. 22
3.4.2 Minimization Algorithm ……..………………………………… 24
3.5 Conclusions ………………………………………………………………... 25
3.6 Appendix ………………………………………………………………...… 26

Chapter 4. Design of Two-Dimensional Recursive Digital Filters Using
Nonsymmetric Half-Plane Allpass Filters …………………………. 31
4.1 Introduction ………………………………………………………………... 31
4.2 2-D Recursive NSHP Digital Allpass Filter ……………………………….. 34
4.2.1 Conventional 2-D Recursive Digital Allpass Filter ……………….…. 34
4.2.2 2-D Recursive NSHP Digital Allpass Filter ……………………….… 34
4.2.3 Stability of 2-D Recursive NSHP Allpass Filter …………………….. 38
4.3 2-D Recursive Filter Structure Based on 2-D NSHP DAFs ……………….. 41
4.3.1 Proposed Filter Structure .……………………………………………. 42
4.3.2 Examples of Applications ……………………………………………. 48
4.3.2.1 Fan Filters …………………………………………………….. 48
4.3.2.2 Circularly Symmetric Lowpass Filters ……………………….. 52
4.3.2.3 Diamond-Shaped Lowpass Filters ……………………………. 52
4.4 Design Technique ………………………………………………………….. 52
4.4.1 Formulation of the Design Problem …………………………………. 52
4.4.2 Iterative Design Algorithm …………………………………………... 54
4.5 Computer Simulation Examples …………………………………………… 55
4.6 Conclusion …………………………………………………………………. 60

Chapter 5. Remarks on the Design of Two-Dimensional Recursive Digital Filters
Using Nonsymmetric Half-Plane Allpass Filters …………………... 71
5.1 Preface ……………………………………………………………………... 71
5.2 Support Region and Recursive Computability …………………………….. 72
5.3 Choice of Initial Guess …………………………………………………….. 76
5.4 Strategy for Choosing Relative Weights …………………………………… 79
5.5 Phase Linearity of the Designed Filters ……………………………………. 80
5.6 Comparisons with Levenberg-Marquardt Algorithms ……………………... 82
5.7 Analytic Derivatives of the Objective Function …………………………… 83
5.7.1 The Proposed Design Procedure ……………………………………... 83
5.7.2 Computer Simulation Examples ……………………………………... 85
5.8 Conclusion …………………………………………………………………. 88
5.9 Appendix …………………………………………………………………... 89

Chapter 6. Design of Two-Dimensional Doubly Complementary Filters
Composed of Two Nonsymmetric Half-Plane Digital Allpass Filters
……………………………………………………………………….. 105
6.1 Introduction ………………………………………………………………. 105
6.2 A Brief Description of 2-D NSHP Allpass Filters ………………………... 108
6.2.1 Frequency Characteristics …………………………………………... 108
6.2.2 Stability Constraints on Phase Response ............................................ 109
6.3 Doubly Complementary Filter Pair ………………………………………. 110
6.4 Least-Squares Design Technique …………………………………………. 114
6.5 Application in Sampling Structure Conversion …………………………... 116
6.5.1 Sampling Structure Conversion …………………………………….. 116
6.5.2 2-D Doubly Complementary Half-Band Property ………………….. 119
6.6 Design Example …………………………………………………………... 120
6.7 Conclusion ………………………………………………………………... 125
6.8 Appendix …………………………………………………………………. 125

Chapter 7. Design of Two-Dimensional Doubly Complementary Filters Using a
Pure Delay Section and a Symmetric Half-Plane Digital Allpass
Filters ……………………………………………………………….. 131
7.1 Introduction ………………………………………………………………. 131
7.2 2-D Doubly Complementary Filter Pair ...................................................... 133
7.2.1 Modified 2-D DC Structure ……………………...…………………. 133
7.2.2 Relationships between Magnitude Deviations and Phase Deviations. 136
7.2.3 Characteristics of Symmetric Half-Plane Digital Allpass Filters …... 137
7.2.4 2-D Doubly Complementary Half-Band Recursive Filters ………… 139
7.3 Norm Design Techniques ……………………………………………. 140
7.4 Computer Simulation Results …………………………………………….. 144
7.5 Conclusion ………………………………………………………………... 148

Chapter 8. Design of Two-Channel Quadrature Mirror Filter Banks Using
Two-Dimensional Digital Allpass Filters .………………………… 167
8.1 Introduction ………………………….…………………………………… 167
8.2 A Brief Review of Conventional 2-D Quadrature Mirror Filter Banks …... 169
8.2.1 Input/Output Relationship for Quincunx QMF Banks ……………... 171
8.2.2 Input/Output Relationship for Parallelogram QMF Banks …….…… 174
8.3 Proposed QMF Structure …………………………………………….…… 176
8.3.1 A Brief Description of the DCF-Based Structure ……………….….. 176
8.3.2 2-D Doubly Complementary Symmetric Properties …………….…. 180
8.3.2.1 QQMF Bank ………………………………………………… 180
8.3.2.2 PQMF Bank …………………………………………………. 183
8.3.3 Equivalent Analysis/Synthesis Systems ……………………………. 184
8.3.3.1 QQMF Bank ………………………………………………… 184
8.3.3.2 PQMF Bank …………………………………………………. 186
8.3.4 Compensation for Phase Distortion ………………………………… 187
8.4 Formulation of the Design Problems ……………………………………... 188
8.4.1 Problem Formulation for Designing QQMF Bank …………………. 188
8.4.2 Problem Formulation for Designing PQMF Bank …………………. 190
8.4.3 Proposed Design Techniques ……………………………………….. 192
8.5 Design Examples …………………………………………………………. 192
8.6 Conclusion ………………………………………………………………... 197

Chapter 9. Design of 2-D Circularly Symmetric Lowpass Filter Using 1-D and
2-D Digital Allpass Subfilters ……………………………………… 225
9.1 Introduction ………………………………………………………………. 225
9.2 Proposed Structure with 1-D and 2-D Recursive Digital Allpass Filters … 227
9.3 Formulation of the Design Problem …………………………………….... 231
9.4 Design Examples ………………………………………………………..... 234
9.5 Conclusion ………………………………………………………………... 238

Chapter 10. Generalized 2-D Digital Lattice Filters with Minimal Delay and State
Space Realization ……………………………………….………….. 243
10.1 Introduction ………………………………………………………………. 243
10.2 Conventional 2-D Lattice Filters …………………………………………. 244
10.2.1 FIR QP Lattice Structure ………………………………………….. 245
10.2.2 IIR QP Lattice Structure …………………………………………... 248
10.3 Proposed 2-D Lattice Structure …………………………………………... 250
10.3.1 Proposed FIR Lattice Structure …………………………………… 250
10.3.2 Proposed IIR Lattice Structure ……………………………………. 254
10.4 Conclusion ………………………………………………………………... 258

Chapter 11. Lattice-Form 2-D Recursive Digital Allpass Filter with Symmetric
Half-Plane Support Region …………….………………………… 259
11.1 Introduction ………………………………………………………………. 259
11.2 Characteristics of Direct-Form Symmetric Half-Plane Digital Allpass Filter …………………………………………………………………………….. 261
11.3 Lattice-Form 2-D Symmetric Half-Plane Digital Filters ..………………... 263
11.3.1 Lattice-From 2-D FIR Symmetric Half-Plane Digital Filter ……… 263
11.3.2 Lattice-From 2-D Recursive Symmetric Half-Plane Digital Allpass
Filter ................................................................................................. 264
11.3.3 Transformation from Direct Form to Lattice Form ……….………. 266
11.3.4 State Space Realization ……………………………………………. 267
11.4 Proposed Technique for Designing the Lattice-From 2-D SHP Digital Allpass
Filter ………………………………………………………………………. 269
11.4.1 Formulation of the Design Problem ………………………………. 269
11.4.2 Iterative Design Algorithm ………………………………………... 270
11.5 Computer Simulation Results …………………………………………….. 271
11.6 Conclusion ………………………………………………………………... 275

Chapter 12. Conclusions and Future Research Works ………………………... 287

Bibliography ……………………………………………………………………….. 289

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