臺灣博碩士論文加值系統

　　以全通濾波器建構的數位濾波器具有通帶振幅低敏感性及雙重互補的優點，因此已被廣泛的應用於數位濾波器的設計中。晶格架構的數位全通濾波器，其頻率響應整體皆對係數的量化誤差具有敏感性低，因此以它建構的數位濾波器除了原有的通帶振幅低敏感性外，止帶振幅以及整體相位敏感性皆降低。晶格架構全通濾波器還有易於檢驗穩定性、結構高重複性等優點，在設計及硬體實現上帶來諸多的方便。
　　目前晶格架構大多應用在一維濾波器中，因為在一維情況下，直接架構與晶格架構存在一對一的轉換，所以兩者理論上可以達到相同的性能，但在二維情況下，由於一般二維多項式無法因式分解，因此一對一轉換的關係不存在。文獻上已提出的二維的晶格架構全通濾波器，只有以FIR多項式作為反射係數的對稱半平面（SHP）晶格架構可達到與直接架構相近的性能，其餘晶格架構，性能皆遠遠不及直接架構。
　　事實上二維晶格架構沒有固定的形式，我們提出了一種廣義的（Generalized）二維晶格架構，將所有可能的變化格式化、參數化，藉由調整結構參數即可做各種的架構變化，針對一個濾波器設計問題，我們嘗試多種不同的結構參數。電腦模擬的結果，在一般濾波器、雙重互補濾波器組、以及正交鏡像濾波器的設計中，均存在能夠達到與直接架構相近性能，甚至超越直接架構性能的晶格架構。
　　此外我們也對實驗時需要用到的最佳化演算法、目標函數進行改良，電腦模擬的結果證實了這些改良確實使設計出來的濾波器性能提升，並降低所需的運算時間。

Using allpass filters as basic building block to construct digital filters has the advantages of passband magnitude low sensitivity and doubly complementary property. Therefore, it has been widely used in digital filter design. Lattice form allpass filter has its entire frequency response less sensitive to round-off error. So, using lattice allpass filter as basic building block can achieve low sensitivity not only in the passband magnitude, but also in the stopband magnitude and the entire phase response. Additionally, the stability of lattice allpass filter can be easily determined by its reflection coefficients, and the structure of lattice allpass filter is highly modular. These advantages bring a lot of convenience to filter design process and hardware implementation.
So far, lattice form is mostly used in one-dimensional filters. Since direct form and lattice form have one-to-one mapping in 1-D case, the two structures can theoretically achieve the same performance. But two-dimensional polynomials are not factorizable in general. Mapping from direct form to lattice form no longer exists in 2-D case. Some 2-D lattice structures have been proposed, but most of them have the performance much worse than direct form. Only the one with symmetric half plane support region using finite impulse response polynomials as its reflection coefficients can achieve the performance close to direct form.
In fact, 2-D lattice form has several different patterns. We propose a generalized 2-D lattice structure, which formats and parameterizes all possible variations of 2-D lattice forms. By adjusting the structural parameters, we can arbitrarily generate new lattice structures. For a specific filter design problem, we try several kinds of lattice structures. Computer simulations show that on the design of general 2-D filters, 2-D doubly complementary filter pairs, and 2-D quadrature mirror filters, there exists some lattice structures which can provide the design performance very close to or even better than direct form.
We also improve the optimization algorithm and objective function used in our experiment. Computer simulations show that the improved algorithm and objective function really enhance the design performance and reduce the required computation time.

口試委員會審定書 i
誌謝 ii
摘要 iii
ABSTRACT iv
目錄 vi
圖目錄 xiii
表目錄 xix
第1章 緒論 1
1.1 研究動機 1
1.2 論文組織架構 2
第2章 最佳化演算法 4
2.1 簡介 4
2.2 線性搜尋法 4
2.3 信賴區間法 7
2.4 原始的WLS演算法 10
2.4.1 基本介紹 10
2.4.2 多誤差向量函數結合之情況 13
2.4.3 多誤差向量結合之問題 15
2.4.4 文獻上的修正方法 20
2.5 改良版WLS演算法 20
2.5.1 多誤差向量結合之正確方式 20
2.5.2 動態改變之權重校正參數 22
2.5.3 二維WLS演算法 25
第3章 以一維晶格架構全通濾波器建構之一維正交鏡像濾波器組的 L∞ 設計 32
3.1 簡介 32
3.2 一維正交鏡像濾波器組之理論分析 33
3.3 一維全通濾波器簡介 35
3.4 以全通濾波器建構之正交鏡像濾波器組 37
3.5 一維晶格架構全通濾波器 39
3.5.1 簡介 39
3.5.2 反射係數與穩定性 41
3.6 目標函數、誤差向量函數及其梯度 42
3.6.1 誤差向量函數 42
3.6.2 目標函數 44
3.6.3 誤差向量函數之梯度 45
3.7 設計流程 46
3.8 實驗列表 47
3.9 實驗結果 51
3.9.1 性能評估參數 51
3.9.2 實驗結果分析 52
3.9.3 濾波器係數列表 56
3.9.4 頻率響應圖 58
3.10 總結 64
第4章 以直接架構NSHP二維全通濾波器建構一般的二維數位濾波器 65
4.1 簡介 65
4.2 NSHP全通濾波器簡介 66
4.2.1 NSHP二維數位全通濾波器基本介紹 66
4.2.2 穩定性與群延遲之關係 68
4.2.3 關鍵頻點 70
4.3 NSHP二維遞迴濾波器穩定條件與穩定性誤差 71
4.3.1 NSHP二維遞迴濾波器穩定條件 71
4.3.2 穩定性誤差 73
4.4 兩個NSHP二維數位全通濾波器建構之濾波器組 76
4.4.1 架構簡介 76
4.4.2 關鍵頻點上的頻率響應 77
4.4.3 振幅反轉次數 78
4.4.4 理想群延遲的選擇 82
4.5 四個NSHP二維數位全通濾波器建構之濾波器組 83
4.5.1 架構簡介 83
4.5.2 有意義的I,M,N設定值 84
4.5.3 理想群延遲的選擇 85
4.6 振幅誤差函數之改良 86
4.6.1 濾波器設計規格 86
4.6.2 原始的誤差函數與潛在問題 86
4.6.3 新的振幅誤差函數 87
4.6.4 雙通濾波器之情況 88
4.7 目標函數、誤差向量函數與其梯度 89
4.7.1 目標函數 89
4.7.2 誤差向量函數 89
4.7.3 梯度矩陣 92
4.8 設計流程 95
4.8.1 實作流程 95
4.8.2 性能評估參數 96
4.9 實驗列表 97
4.9.1 實驗參數設定表 97
4.9.2 頻帶圖 101
4.10 實驗結果 103
4.10.1 權重設置 103
4.10.2 性能評估 104
4.10.3 係數表 106
4.10.4 頻率響應圖 111
4.10.5 極根圖 116
4.11 總結 130
第5章 以晶格架構NSHP二維全通濾波器建構一般的二維數位濾波器 131
5.1 簡介 131
5.2 廣義的二維晶格架構全通濾波器 131
5.3 NSHP限制條件 135
5.3.1 NSHP簡介 135
5.3.2 廣義二維晶格架構之NSHP限制條件 136
5.4 反射係數多項式與穩定性之關係 141
5.4.1 二維晶格架構反射係數與穩定性之關係 141
5.4.2 將任意係數轉為穩定係數 144
5.5 五種可由反射係數判斷穩定性之NSHP具體晶格架構 146
5.6 兩個晶格架構NSHP全通濾波器並聯之結構 150
5.6.1 回顧 150
5.6.2 補償延遲 151
5.6.3 係數支持區間方向 152
5.7 四個晶格架構NSHP全通濾波器組合之結構 154
5.7.1 回顧 154
5.7.2 補償延遲 155
5.7.3 係數支持區間方向 156
5.8 可變參數的選擇 157
5.9 目標函數、誤差向量函數及其梯度 160
5.9.1 目標函數 160
5.9.2 誤差向量函數 161
5.9.3 梯度矩陣 164
5.9.4 不同方向NSHP之誤差及梯度公式 168
5.10 設計流程 169
5.10.1 實作流程 169
5.10.2 性能評估參數 170
5.11 實驗列表 171
5.11.1 實驗參數設置表 171
5.11.2 頻帶圖 176
5.12 實驗結果 178
5.12.1 最佳可變結構參數設定 178
5.12.2 權重參數與映射函數之 δ 參數 180
5.12.3 性能評估 182
5.12.4 量化誤差測試 185
5.12.5 係數表 186
5.12.6 頻率響應圖 192
5.13 總結 197
第6章 以晶格架構NSHP二維全通濾波器建構二維雙重互補濾波器對 198
6.1 簡介 198
6.2 二維雙重互補濾波器對之簡介 198
6.3 DC-HB性質 202
6.3.1 DC-HB性質簡介 202
6.3.2 DC-HB之全通濾波器條件限制 202
6.3.3 廣義晶格架構全通濾波器之半頻對稱限制 204
6.3.4 DC-HB之頻帶限制 206
6.3.5 DC-HB設計問題之簡化 207
6.4 五種晶格架構全通濾波器之半頻對稱版本 207
6.5 可變參數的選擇 211
6.6 將濾波器設計問題轉為個別全通濾波器分母相位近似問題 215
6.7 直接架構下的Least Square Closed Form Solution 216
6.8 穩定性問題 220
6.9 目標函數、誤差向量函數及其梯度 223
6.9.1 目標函數、誤差向量函數 223
6.9.2 梯度矩陣 223
6.9.3 不同方向NSHP之轉換 225
6.10 設計流程 226
6.10.1 實作流程 226
6.10.2 性能評估參數 227
6.11 實驗列表 227
6.11.1 實驗參數設置表 227
6.11.2 頻帶圖 230
6.12 實驗結果 231
6.12.1 可變參數設定值 231
6.12.2 性能評估 233
6.12.3 結果講評 235
6.12.4 係數表 235
6.12.5 頻率響應圖 238
6.13 結論 247
第7章 以晶格架構二維全通濾波器建構二維正交鏡像濾波器組 248
7.1 簡介 248
7.2 二維複速率系統簡介 248
7.3 二維正交鏡像濾波器組之理論分析 249
7.4 平行四邊形正交鏡像濾波器組 250
7.4.1 基本理論分析 250
7.4.2 以全通濾波器建構之PQMF 252
7.4.3 以NSHP晶格架構全通濾波器建構PQMF 254
7.5 鑽石形正交鏡像濾波器組 258
7.5.1 基本理論分析 258
7.5.2 以全通濾波器建構之QQMF 260
7.5.3 以SHP晶格架構全通濾波器建構QQMF 262
7.6 穩定性考量 264
7.7 目標函數、誤差矩陣函數、誤差向量函數及其梯度 265
7.7.1 誤差矩陣函數 265
7.7.2 誤差向量函數 268
7.7.3 目標函數 269
7.7.4 梯度矩陣 269
7.8 設計流程 271
7.8.1 實作流程 271
7.8.2 性能評估參數 272
7.9 實驗列表 273
7.10 實驗結果 275
7.10.1 可變參數選擇 275
7.10.2 性能評估 277
7.10.3 係數表 280
7.10.4 頻率響應圖 281
7.11 總結 295
第8章 Future Work 296
參考資料 311

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