以誤差微分之平方作正交鏡像濾波器組之最佳化及使用短模組半頻濾波器設計低複雜性的雙頻道次頻濾波器

Abstract

本論文探討濾波器組設計的三個關鍵性主題，即，次頻濾波器之響應最佳化、低複雜度、以及低誤差實現。 本論文分兩部份，（I）以誤差微分的平方作四分鏡像濾波器組之最佳化；（II）使用短模組半頻/Nyquist(M)濾波器設計低複雜度的一維/二維雙頻道次頻濾波器。 在論文的第一部份中，利用重建誤差的微分，吾人提出新類型的目標函數和四分鏡像濾波器組設計。根據濾波器組之最佳化，吾人開發出效率不錯的低延遲四分鏡像濾波器組演算法及線性相位四分鏡像濾波器組演算法。在和傳統的最小平方誤差設計比較下，實驗結果顯示吾人所提出的新設計有較佳的表現。 在論文的第二部份中，吾人提出新奇且低複雜度的濾波器合成架構方案。根據此架構方案，設計實現出具有狹窄頻帶轉移區和高頻率選擇性的一維半頻濾波器、一維雙頻道濾波器組、二維Nyquist(M)濾波器及二維雙頻道鑽石/象限濾波器組。 現有的次頻濾波器設計，雖或可達到高效益但高複雜度，若求低複雜度則必需犧牲效益，或可達到低複雜度但低效益。吾人所提出的新方案提供簡單有效的方法，在有限精確度實現下，可合成出高效益低複雜度和低誤差實現的次頻濾波器。此種新方法包含頻率響應鮮明化之技巧。 新的低複雜度一維半頻濾波器設計與實現架構是基於一代數遞迴組合的方法，使用可調的短模組半頻濾波器，可合成出高效益低複雜度的半頻濾波器。模組半頻濾波器可由使用者指定。特別地，當模組濾波器架構是無乘法時，整個高階半頻濾波器可無需乘法實現之。 在一維完美重建線性相位濾波器組設計與實現方面，利用半頻濾波器和完美重建之間的關係，吾人推導出一新的低複雜度代數遞迴合成設計。所合成出的次頻濾波器不僅是具有鮮明響應，而且是低複雜度的濾波器。此種次頻濾波器是由幾個短模組半頻濾波器組合成的。 所提出的一維設計，可沿用在具有鮮明響應的二維Nyquist(M)濾波器以及二維雙頻道不可分離鑽石/象限濾波器組之設計上。在使用短模組二維Nyquist(M)濾波器選擇上，設計者可選用無需乘法之濾波器，則可達成無需乘法之濾波運算。 半頻/Nyquist(M)濾波器和一維/二維濾器組均可被合成在一具有相當少算術運算的樹狀多級串接結構。模擬驗證了所提出的設計之有效性。

The dissertation is concerned with three key issues of filter bank design, namely, responses optimization, low computational complexity, and low finite-precision-error realization of subband filters. In particular, this dissertation is divided into two parts: (I) Quadrature-Mirror-Filter (QMF) banks optimization based on derivative information, and (II) low-complexity design and realization of 1-D/2-D two-channel subband filters using short modular half-band/Nyquist(M) filters. The first part focuses on the optimization of QMF banks. New types of objective functions, utilizing derivative information of the reconstruction error in z-domain, are proposed. New designs of QMF banks using the objective functions are studied. Efficient design algorithms for low-delay QMF banks and linear-phase QMF banks are developed. From simulations, the new designs can achieve better results than the conventional design based on the standard least-square-error objective function. The second part focuses on the low-complexity design and realization of subband filters with good numerical properties. We devise novel low-complexity composition schemes for the design and realization of 1-D half-band filters, 1-D two-channel biorthogonal filter banks, 2-D Nyquist(M) filters, and 2-D two-channel diamond/quadrant filter banks, all with narrow transition band and high frequency selectivity. The existing design methods either result in high-performance but high-complexity subband filters or low-complexity but low-performance subband filters. The new schemes provide simple and efficient methods for synthesizing high-performance low-complexity subband filters with good numerical property for finite-precision realization. The synthesis process involves frequency response sharpening. For the low-complexity design and realization of 1-D half-band filters, the proposed scheme is based on an algebraic iterative composition method using adjustable short modular half-band filters. The modular filters can be user selectable as simple ones as desired. Specifically, the designed higher-order half-band filters can be made multiplierless if the modular filters are multiplierless. For the low-complexity design and realization of 1-D biorthogonal linear-phase filter banks, the proposed algebraic iterative composition scheme utilizes the solution of filter bank with two half-band filters. The resulting analysis filters are not only sharp but also low-complexity, which are composed of several short modular half-band filters. The 1-D schemes are extended to the synthesis of 2-D Nyquist(M) filters and two-channel nonseparable diamond/quadrant filter banks with sharp responses. Short modular 2-D Nyquist(M) filters, preferably multiplier-free ones, are used. Based on the proposed schemes, half-band/Nyquist(M) filters and 1-D/2-D filter banks can be synthesized in a tree-like multi-stage cascaded structure with considerably reduced arithmetic operations (that can be made multiplierless). Simulations are shown to validate the effectiveness of the proposed schemes. Chapter 2 Design of QMF Banks Based on Derivative Information Chapter 3 Novel Iterative Synthesis Scheme for Half-Band Filters Chapter 4 Iterative Synthesis Scheme Using Partition of Unity and Binomial Expansion Chapter 5 Design of Low-Complexity Biorthogonal Linear-Phase PR FIR Filter Banks Using Non-PR Strictly Half-Band Complementary Filter Pair Chapter 6 Low-Complexity Design and Realization of 1-D Two-Channel Biorthogonal Filter Banks Using Short Modular Half-Band Filters Chapter 7 Low-Complexity Design and Realization of 2-D Two-Channel Filter Banks Using Short Modular Nyquist(M) Filters Chapter 8 Conclusion and Future Works