正交鏡像濾波器組(Quadrature Mirror Filter banks, QMF)為小波理論中的重要課題，也是影像處理及資料壓縮上的應用工具之一；傳統上使用具有線性相位有限脈衝(Linear Phase Finite Impulse Response)的高低通濾波器來建構QMF；近年來許多文獻以數位全通濾波器(Digital Allpass Filter, DAP)來實現QMF，這種設計架構除了可以消除濾波器組振幅失真的問題外，亦可使用較少的階數達成所需的規格要求。

在此架構下，我們面臨的最佳化問題為一個高度非線性之環境；解決此問題的演算法一般而言有兩種；第一是列舉法(Enumeration)，也就是在所有可行解空間中，將所有可行解列舉出並逐一檢驗，再從中挑選出最佳解。此解法的優點在於保證可以找到全域最佳解(Global Optimum)，然而所需的運算時間與運算量極大，並不符合實際需求；第二種是線性近似化(linearized algorithm)演算法，將原先所面臨的高度非線性化的問題做線性近似，以期望能在較短的搜索時間內找到最佳解；此解法的優點在於運算時間短且運算量較小，卻容易陷於局部最佳解(Local Optimum)。因此，本論文中提出使用粒子群演算法作為最佳化法則，期望能在列舉法與線性近似法中取得平衡，並讓搜尋過程不容易受限於局部最佳解，並且能找出優於線性近似演算法的解。

Quadrature Mirror Filter banks is an important topic of wavelet theory, image processes and data compression; Traditionally a linear-phase finite impulse response(LP-FIR) low-pass filter and high-pass filter are used to construct QMF, recently several reports suggest that implement QMF by using IIR digital allpass filter, this structure could solve some problem such as amplitude distortion, also, be able to achieve the same specifications as using FIR filter with less filter order.

Under this structure, the design problem we face is a highly nonlinear optimization problem. In general, there are two traditional algorithms to solve this problem. The first one is called enumeration, it lists all the feasible solutions in the search space and test them, select the best solution from all the candidates. The advantage of enumeration is that it is guaranteed to find the global optimum; however, it costs too much search time and computation loading. The second is linearized algorithm, which linearized the nonlinear problem so that less computation time and loading are needed, at the same time, it is easier trapped in local optimum. Based on the above concept, in this paper we propose a type of evolution algorithm—Particles Swarm Optimization (PSO) algorithm to be optimizer, which is expected be balanced between enumeration and linearized algorithm, not to trapped in local optimum and find a better solution than the solution solved by linearized algorithm.

口試委員會審定書 #
中文摘要 i
Abstract ii
論文目錄 iii
圖目錄 vi
表目錄 ix
第一章 序論 1
1.1 研究動機 1
1.2 論文架構 2
第二章 最佳化問題與PSO演算法 3
2.0 簡介 3
2.1 粒子群最佳化演算法(Particles Swarm Optimization ,PSO) 4
2.2 混亂型PSO (Chaos Particles Swarm Optimization ,CPSO) 8
2.3 混和型PSO (Proposed Chaotic PSO, CHOPSO) 10
第三章 實係數IIR全通濾波器與雙通道雙重互補濾波器組設計 12
3.0 簡介 12
　 3.1一維數位全通濾波器 12
　 3.2基於數位全通濾波器之雙重互補濾波器組理論 13
　 3.3 穩定性與理想相位之選擇 15
　　 3.3.1 完美重建條件 15
　　 3.3.2 理想相位設定條件 16
　3.4 基於 準則之PSO演算法設計 18
　　 3.4.1 基於Minimax準則之設計 18
　　 3.4.2 基於MMSE+WLS準則之設計 20
　3.5 設計實例與討論 23
　　 3.5.1 設計範例一 24
　　 3.5.2 結果討論 40
第四章 一維雙通道正交鏡像濾波器組架構與設計問題 41
　 4.0 簡介 41
　4.1一維多速率系統 42
　4.2一維雙通道正交鏡像濾波器組架構 43
　4.3以全通濾波器為基礎之一維雙通道正交鏡像濾波器組架構 45
　4.4理想相位之選定 47
　4.5 最佳化目標之設定 51
　　 4.5.1 以群延遲為主的最佳化目標 51
　　　　4.5.2 以相位誤差為主的最佳化目標 53
　　4.6基於 的準則之最佳化設計 53
　　　　4.6.1以Minimax準則為基礎之PSO設計 53
　　　　4.6.2 以MMSE+WLS準則為基礎之PSO設計 54
　　4.7 設計實例與討論 56
　　　　4.7.1 設計範例一 57
　　　　4.7.2 設計範例二 75
　　　　4.7.3 設計範例三 93
　　　　4.7.4 結果討論 111
第五章　一維非均勻濾波器組之設計 112
　　5.0 簡介 112
　　5.1 雙通道線性相位非均勻濾波器組之架構 113
　　5.2 NDF架構及其完美重建條件 114
　　5.3 基於IIR濾波器之NDF設計及最佳化目標函數 120
　　　　5.3.1 基於一維全通濾波器之NDF設計 120
　　　　5.3.2 滿足PR條件之以相位響應為基礎的目標函數 125
　　　　5.3.3 滿足PR條件之以群延遲響應為基礎的目標函數 126
　　5.4 基於L∞之最佳化設計 127
　　 5.4.1 以fitphase(w)為目標函數之設計 127
　　 5.4.2 以fitGD(w)為目標函數之設計 128
　　5.5 設計實例與討論 129
　　　　5.5.1 設計範例一 130
　　　　5.5.2 設計範例二 141
　　　　5.5.3 結果討論 153
第六章 二維Quincunx QMF及Parallelogram QMF之設計 154
　　6.0 簡介 154
　　6.1 使用頻譜遮蓋建構Quincunx QMF 155
　　6.2 二維Quincunx QMF濾波器組架構 160
　　　　6.2.1 傳統二維正交鏡像濾波器組架構 160
　　　　6.2.2 以全通濾波器建構二維QQMF 161
　　　　6.2.3 以一維全通濾波器建構二維QQMF 164
　　6.3 以一維QMF建構二維Parallelogram QMF 166
　　　　6.3.1 使用頻譜遮蓋建構PQMF 166
　　6.4 二維Parallelogram QMF濾波器組架構 169
　　　　6.4.1 PQMF之完美重建條件與其理想相位 169
　　　　6.4.2以一維全通濾波器建構二維PQMF 171
　　6.5 設計實例與討論 173
　　　　6.5.1 設計範例一 174
　　　　6.5.2 設計範例二 199
　　　　6.5.3 設計範例三 223
　　　　6.5.4 結果討論 248
第七章　結論 249

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