二維遞迴式數位濾波器之穩定性與晶格系統架構之理論及設計

Abstract

二維(2-D)數位全通濾波器(DAF)擁有只在相位上變化的性質，並且已用於補償相位失真的訊號。它的結構有許多令人期望的優點，例如:低硬體複雜度與低的係數量化誤差。它也可以被用於設計多範圍的濾波功能。在本論文中，我們提出二維因果且實係數或負數係數皆被限制於第一象限(QP)之數位全通濾波器的單調遞減相位響應性質。在實係數的情形下，我們也證明出之前文獻所提出的有限輸入有限輸出(BIBO)之相位穩定性條件只是對於QP數位全通濾波器的充分條件，而此穩定性條件對於二維可分離(separable)數位全通濾波器是充分且必要條件。相對於之前文獻所提出的相位穩定性條件，我們提出的相位響應穩定性性質擁有增加相位設計自由度的優點。我們提出的相位響應穩定性性質有一個重要的應用，即是可用來選擇適當且期望的相位規格來設計出穩定的QP數位全通濾波器。 二維非對稱半平面(NSHP)數位全通濾波器擁有較一般化的因果特性，且效能比二維QP數位全通濾波器來的要好。因此，我們也提出二維因果且實係數或負數係數皆被限制於NSHP支撐區域之數位全通濾波器的相位響應性質來決定BIBO穩定性。再者，我們也考慮二維NSHP數位全通濾波器之分子多項式對於穩定性的影響。我們提出的相位響應穩定性性質有許多應用。其中一個重要的應用，即是可用來選擇適當且期望的相位規格來設計出穩定的NSHP數位全通濾波器。我們也提出一個特徵濾波器(eignfilter)設計方法來設計二維NSHP數位全通濾波器來套用上述的應用。 一維(1-D)晶格(lattice)結構數位濾波器擁有低通帶敏感度與對量化誤差的抗性。它的模組化性質更造就了工業上的應用。另外，一維晶格結構數位濾波器對於一維直接形式(direct form)數位濾波器來說有較低的執行複雜度。一維直接形式數位全通濾波器的係數與一維晶格結構數位全通濾波器的反射係數有著一對一映射的性質，然而二維晶格結構數位全通濾波器並無法擁有這個特性。因此，我們提出一個二維晶格架構來實現有一般化因果特性的二維數位全通濾波器。我們採用四種基本晶格區塊來實現擁有類似NSHP係數支撐區域的楔形係數支撐區域之二維數位全通濾波器。我們也提出此二維晶格數位全通濾波器的兩種變異形式之二維晶格結構。我們利用Rosser狀態空間模型來驗證我們所提出的二維晶格數位全通濾波器之最小實現性質。我們提出一個最小平方設計方法(least-squares design)與一個最小最大誤差設計方法(minimax design)來求解所提出二維晶格數位全通濾波器架構產生的非線性最佳化問題。我們所提出二維晶格數位全通濾波器架構的新穎性在於不但繼承了一維Gray-Markel晶格數位全通濾波器架構的優點並且擁有比現存所有二維晶格數位全通濾波器架構的效能要好的優點。接著，我們提出一個並排連結(parallel-combination)結構並使用我們所提出的二維晶格數位全通濾波器架構來設計一般的二維遞迴式濾波器。此提出的二維遞迴式濾波器的新穎性在於不但繼承了所提出二維晶格數位全通濾波器的優秀特性並且擁有比現存使用直接形式NSHP數位全通濾波器來建構的二維遞迴式濾波器的效能要好的優點。

A two-dimensional (2-D) digital allpass filter (DAF) has a property of varying only phase with constant magnitude and it has mainly been used as a phase compensator for distorted signals. It is a structure that has some desirable attributes such as low complexity and low coefficient quantization error. It also can be used to design a wide range of filtering functions. In this doctoral dissertation, we present the monotone phase-response property of a two-dimensional (2-D) causal digital allpass filter (DAF) with real coefficients or complex coefficients in the quarter-plane (QP) support region. Regarding the circumstance of real coefficients, we also prove that the previously proposed bounded-input bounded-output (BIBO) stability criterion on the viewpoint of unwrapped phase is necessary and sufficient for 2-D separable DAFs, but is only sufficient for QP DAFs. The resultant property possesses the advantage of increasing the freedom of phase design over the previously proposed one. A remarkable application of the presented property is choosing an appropriate specification for the desired phase response of a 2-D QP DAF design. A 2-D nonsymmetric half-plane (NSHP) recursive DAF possesses more general causality and performs better than a 2-D quarter-plane (QP) recursive DAF. Hence, we also present the phase-response property for the BIBO stability of a 2-D causal recursive DAF with NSHP support region. Both cases of filters with real coefficients and complex coefficients are explored. Moreover, the effect of the numerator polynomial of a 2-D NSHP DAF on stability is also considered. The presented phase-response property has several applications. A remarkable application is that it can be utilized to enforce stability for a 2-D NSHP DAF design by choosing an appropriate phase specification. The eigenfilter design of 2-D NSHP DAFs for this application is also presented. The 1-D lattice filter structure exhibits the attractive advantages of low passband sensitivity and robustness to quantization error. The modularity of this structure makes industrial application. Additionally, 1-D digital lattice filter structure requires lower computational cost than 1-D direct form digital filter. The filter coefficients of 1-D direct-form allpass filter and the reflection coefficients of 1-D lattice allpass filter have a one-to-one mapping relationship. However, 2-D lattice allpass structures always do not have this relationship. Hence, we present a lattice structure for the realization of 2-D recursive DAFs with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a NSHP support region. Two variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Then, we present a parallel-combination structure composed of the 2-D lattice DAFs for the design of 2-D recursive filters. The novelty of the 2-D recursive filter is that it not only inherits the desirable attributes of lattice filters but also possesses the advantage of better performance over the 2-D recursive NSHP filters.