基於創新的晶格架構全通濾波器之二維遞迴式數位濾波器與濾波器組的最佳化設計

Abstract

以全通濾波器建構的數位濾波器具有通帶振幅低敏感性及雙重互補的優點，因此已被廣泛的應用於數位濾波器的設計中。晶格架構的數位全通濾波器，其頻率響應整體皆對係數的量化誤差具有敏感性低，因此以它建構的數位濾波器除了原有的通帶振幅低敏感性外，止帶振幅以及整體相位敏感性皆降低。晶格架構全通濾波器還有易於檢驗穩定性、結構高重複性等優點，在設計及硬體實現上帶來諸多的方便。 目前晶格架構大多應用在一維濾波器中，因為在一維情況下，直接架構與晶格架構存在一對一的轉換，所以兩者理論上可以達到相同的性能，但在二維情況下，由於一般二維多項式無法因式分解，因此一對一轉換的關係不存在。文獻上已提出的二維的晶格架構全通濾波器，只有以FIR多項式作為反射係數的對稱半平面（SHP）晶格架構可達到與直接架構相近的性能，其餘晶格架構，性能皆遠遠不及直接架構。 事實上二維晶格架構沒有固定的形式，我們提出了一種廣義的（Generalized）二維晶格架構，將所有可能的變化格式化、參數化，藉由調整結構參數即可做各種的架構變化，針對一個濾波器設計問題，我們嘗試多種不同的結構參數。電腦模擬的結果，在一般濾波器、雙重互補濾波器組、以及正交鏡像濾波器的設計中，均存在能夠達到與直接架構相近性能，甚至超越直接架構性能的晶格架構。 此外我們也對實驗時需要用到的最佳化演算法、目標函數進行改良，電腦模擬的結果證實了這些改良確實使設計出來的濾波器性能提升，並降低所需的運算時間。

Using allpass filters as basic building block to construct digital filters has the advantages of passband magnitude low sensitivity and doubly complementary property. Therefore, it has been widely used in digital filter design. Lattice form allpass filter has its entire frequency response less sensitive to round-off error. So, using lattice allpass filter as basic building block can achieve low sensitivity not only in the passband magnitude, but also in the stopband magnitude and the entire phase response. Additionally, the stability of lattice allpass filter can be easily determined by its reflection coefficients, and the structure of lattice allpass filter is highly modular. These advantages bring a lot of convenience to filter design process and hardware implementation. So far, lattice form is mostly used in one-dimensional filters. Since direct form and lattice form have one-to-one mapping in 1-D case, the two structures can theoretically achieve the same performance. But two-dimensional polynomials are not factorizable in general. Mapping from direct form to lattice form no longer exists in 2-D case. Some 2-D lattice structures have been proposed, but most of them have the performance much worse than direct form. Only the one with symmetric half plane support region using finite impulse response polynomials as its reflection coefficients can achieve the performance close to direct form. In fact, 2-D lattice form has several different patterns. We propose a generalized 2-D lattice structure, which formats and parameterizes all possible variations of 2-D lattice forms. By adjusting the structural parameters, we can arbitrarily generate new lattice structures. For a specific filter design problem, we try several kinds of lattice structures. Computer simulations show that on the design of general 2-D filters, 2-D doubly complementary filter pairs, and 2-D quadrature mirror filters, there exists some lattice structures which can provide the design performance very close to or even better than direct form. We also improve the optimization algorithm and objective function used in our experiment. Computer simulations show that the improved algorithm and objective function really enhance the design performance and reduce the required computation time.