一維及二維小波濾波器組之設計

Abstract

傳統設計正交鏡像濾波器組(Quadrature Mirror Filter bank,簡稱QMF)時，不論是一維或二維濾波器組，採用的皆是線性相位有限脈衝響應(linear-phase Finite Impulse Response,簡稱LP-FIR)，這種結構雖然可以完全消除相位失真，但卻有兩大缺點：振幅失真的部分無法完全消除、很難設計陡峭的頻帶邊緣。有鑑於此，文獻上出現了使用IIR全通濾波器(Allpass Filter)作為設計QMF的架構，在論文中我們將介紹這種結構的優點。 在這種結構下，濾波器的設計問題變成了高度非線性最佳化問題，在以往提出的設計方法中，最常見的方式是個別對全通濾波器的相位做最佳化近似，然而這種設計方法卻造成了濾波器組之相位響應在轉態區上有極大的誤差，一般使用相位補償器來解決相位響應之誤差，卻增加了整個系統的延遲。 本篇論文最主要是要探討轉態區上相位響應之誤差的形成原因，提出其解決方法，並在實例設計中，將會指出新的設計方式較以往提出的方式之優點。此外，我們會提出以基於一維全通濾波器建構二維QMF之設計方式，研究成果在於能以更少的濾波器係數設計出較已有之結構更佳的設計結果。

Traditionally, we use linear-phase FIR filter to design the one-dimensional or two-dimensional Quadrature Mirror filter banks(QMF). Although the phase distortion is completely eliminated; this structure has at least two disadvantages: first, it can’t eliminate the amplitude distortion completely, and second, it is hard to obtain a design with sharp band edges. To solve this problem, several papers propose to use the IIR allpass filters to construct an QMF. In this thesis, we will discuss the advantages of QMF constructed by the IIR allpass filters. Filter design problems become a highly non-linear optimization problem in such a structure with IIR filters. Recently, several methods were proposed to design an QMF, and they consider the phase approximation for each allpass filter. However, this design scheme causes larger phase response error in the transition band. As a result, phase compensation is usually used to reduce the phase error with the price of the system delay increased. In this paper, we mainly : I. investigate the cause of phase error in transition band II. solve it by proposing a new design method III. show the advantage of using the method in simulation examples. Besides, we propose a method to design two-dimensional QMF by using one-dimensional allpass filter. This method provides better results with less filter coefficients.