以兩種不同方法進行雙通道正交鏡像濾波器組之設計與最佳化

Abstract

本論文主要討論雙通道正交鏡像濾波器組之設計問題。本論文提出兩套設計方法分別設計兩種雙通道正交鏡像濾波器組，一種係接近完美重建、低階的IIR濾波器組，一種係接近正交、線性相位的FIR 濾波器組。 在接近完美重建、低階的IIR 濾波器組之設計問題中，問題首先被轉化爲等化器設計之最佳化問題。經過多相位分析後，該最佳化問題之變數與係數將被減少一半，有效的減低最佳化演算法之計算量。該雙通道正交鏡像濾波器組之系統延遲值將取決於一個最小平方值之估測。之後，在線性矩陣不等式之架構下形成H2/H∞/混合範數之最小化問題，以求取高階、最佳的FIR 解。最後，低階的IIR 近似解藉著平衡實現與降階技巧來求取。 在接近正交、線性相位的FIR 濾波器組之設計問題中，問題首先被轉化爲低通FIR 濾波器之設計問題。該設計問題將進一步被轉化為具多目標軟性限制之最佳化問題，而各項軟性限制係從各項目標之誤差範數中推導得出。最後，運用現有高效率的計算軟體，該最佳化問題可方便地在線性矩陣不等式之架構中求解。

The two-channel quadrature mirror filter bank design problems are studied in this thesis. Tow different methods for designing a near perfect reconstruction low-order IIR filter bank and a near orthogonal linear phase FIR filter bank are proposed and compared. In the near perfect reconstruction low-order IIR filter bank design, the quadrature mirror filter bank design problem is first converted to an equalizer design optimization problem and described in the polyphase representation, so the optimization variables and coefficients are reduced by half, and significant reduction of computation load is achieved for high-order system. The quadrature mirror filter bank delay in the optimization problem is selected based on the result of a least square estimation problem. High-order optimal FIR solutions are then obtained from the H2/H∞/mixed-norm minimization problems in the linear matrix inequality framework. Finally, approximate low-order IIR solutions are obtained by applying the balanced realization and model order reduction techniques. In the near orthogonal linear phase FIR filter bank design, the filter bank design is first reduced to an FIR lowpass filter design problem. The FIR lowpass filter design problem is then formulated as an optimization problem with soft constraints corresponding to the multiple objectives to fulfill. Constraints for each objective are derived from its error norm. The optimization problem is then formulated in the linear matrix inequality framework so that it can be solved efficiently by the currently available software.