使用分數雙線性轉換之數位濾波器設計

Abstract

這篇論文我們提出用在類比數位轉換的分數雙線性轉換。此種轉換方式是使用分數性延遲濾波器衍生而來，且包含兩種形式。一種是由拉格蘭治有限脈衝響應分數性延遲濾波器而來；另一種是由席倫無限脈衝響應分數性延遲濾波器而來。它們的低頻振幅響應都呈線性， 但是高頻振幅響應則不相似，為了用在不同的應用上。 根據此種轉換方式，可以容易地設計出數位一階微分器和數位一階積分器。此外，提出一些設計例子來顯示此種類比數位轉換的線性頻率對應性質。所設計的數位微分器和數位積分器用來和現存的微分器和積分器作比較。此種低階數的微分器可適用於使用在即時應用上。 為了改善由雙線性轉換而來，用在動態物體追蹤上的高選擇性三維遞迴濾波器，首先運用二維的實例來討論不同類比數位轉換方式對頻率響應的影響。實驗結果可顯示此種三維遞迴平面濾波器可以擷取出所需要的動態物體。 關鍵詞—雙線性轉換，數位微分器，數位積分器，分數雙線性轉換，分數性延遲濾波器，三維平面共振遞迴濾波器，動態物體追蹤。

The fractional bilinear transform used in analog-to-digital (A/D) conversion is proposed. This transformation is derived by means of fractional delay filter. Two forms are presented. One is approximated by Lagrange FIR fractional delay filter; the other is approximated by Thiran IIR fractional delay filter. Their magnitude responses are both linear in low frequency band, but differ in high frequency band for distinct applications. According to this transformation, first-order digital differentiator and integrator can be easily designed. Additionally, some design examples are illustrated to show the linear frequency mapping property when performing A/D conversion. The designed digital differentiator and integrator are compared with existing differentiators and integrators. The designed low-order differentiator is suitable for real-time applications. To improve highly selective 3-D recursive filters transformed from the bilinear transformation for tracking moving objects, first use 2-D examples to discuss the effects of diverse transformations. The experimental results show that the designed 3-D recursive plane filter can extract the desired moving object. Index terms—Bilinear transformation, digital differentiator, digital integrator, fractional bilinear transformation, fractional delay filter, three-dimensional planar-resonant recursive filter, tracking moving objects.