導數限制下線性相位有限脈衝響應濾波器的一般化結構

在這篇博士論文中，我們提出可滿足任意頻率下的導數限制之第一型至第四型線性相位有限脈衝濾波器的新設計結構。此結構是由所謂的基本濾波器(cardinal filters) 的線性組合構成，而目標頻率響應在目標頻率下的導數值即為線性組合基本濾波器的加權係數。由於基本濾波器的組成係數可由遞迴公式獲得且與目標頻率響應無關，我們所提出的結構可泛用於任意導數限制下的線性相位有限脈衝濾波器設計。推導基本濾波器的組成係數之主要關鍵，在於選定能夠滿足特定微分性質之特定三角函數並運用其泰勒展開式。經由理論證明我們所選定的特定三角函數滿足特定的微分方程性質，我們亦推導出基本濾波器的組成係數之遞迴公式，大幅提升基本濾波器組成係數的數值計算穩定度。最後，我們以範例實證我們所提出的新結構，亦可結合最小平均平方誤差法則，進一步改善目標頻率響應的設計準確度。

關鍵字

有限脈衝響應濾波器；線性相位濾波器；導數限制

並列摘要

In this dissertation, novel structures of types I, II, III, and IV linear-phase FIR filters, whose frequency responses satisfy given derivative constraints imposed upon an arbitrary frequency, are proposed. It is comprised of a linear combination of parallelly connected sub-filters, called the cardinal filters, with weighted coefficients being the successive derivatives of the desired frequency response at the constrained frequency. Since the cardinal filters can be synthesized via recursive closed-form expressions, regardless of the desired system amplitude response, the proposed structure provides a universal design for arbitrary derivative-constrained linear-phase FIR filters. The key to derive the coefficients of cardinal filters is the determination of the power series expansion of certain trigonometric-related functions. By showing the elaborately chosen trigonometric-related functions satisfy specific differential equations, recursive formulas for the coefficients of cardinal filters are subsequently established, which make stable their computations. At last, a simple enhancement of the cardinal filters design by incorporating the mean square error (MSE) minimization is presented through examples.