Designs of Variable Fractional Delay Digital Filter and Fractional Order Differintegrator

[[abstract]]近年來，由於可調式數位濾波器在系統上自我調整的能力，可調式數位濾波器的設計變成在數位訊號處理裡最重要的分支之一。可調式數位濾波器通常可分成二類。一類是具有可調整振幅響應的濾波器。例如，可調變截止頻率的濾波器以及分數階可調式微分器或積分器。另一類是具有分數延遲之可調式濾波器。在本論文中，我們提出加權最小平方差逼近法來設計可調式數位濾波器。通常，目標誤差能以線性函數來表示時，那麼，我們會直接以加權最小平方差逼近法來求最佳解。相反地，當它是非線性最佳化問題時，我們則使用遞迴二次式的方法。此外，若想將最大目標誤差最小化，則可使用遞迴式的加權最小平方差逼近法的技術。 在本論文中，上述的方法將用在下列之研究： 用遞迴加權式最小平方差逼近法設計分數延遲可調式FIR數位濾波器。（第二章） 一種設計分數延遲可調式FIR數位濾波器之新準則。（第三章） 一種分數延遲可調式FIR數位濾波器之新架構及其設計。（第四章） 用遞迴加權式最小平方差逼近法設計分數延遲可調式Allpass數位濾波器。（第五章） 一種設計分數延遲可調式Allpass數位濾波器的新準則。（第六章） 一種設計分數延遲可調式IIR數位濾波器的新方法。（第七章） 以遞迴式之方法設計分數階可調式FIR數位微積分器。（第八章） 一種分數階可調式寛頻帶FIR數位微分器之新架構及其設計。（第九章） 以遞迴式之方法設計分數階可調式IIR數位微積分器。（第十章）[[abstract]]For the past decade, the design of variable digital filters became one of the most important branches in digital signal processing because of the self-adjustable ability of a variable digital filter online. The variable digital filters are generally classified into two categories. One is the filters with adjustable magnitude response such as the filters with variable cut-off frequencies and the variable fractional-order differentiators/integrators. The other is the filters with variable fractional-delay response. In this dissertation, the weighted least-squares method will be proposed to design variable digital filters. Generally, a general weighted least-squares method can be applied directly to find the optimal solution when the objective error can be formulated in a linear function. On the contrary, when the problem concerns a nonlinear optimization, an iterative quadratic method is applied. Furthermore, if it is desirable to minimize a specified maximum error, the technique of iterative weighted least-squares method will be used which constitutes the inner loop of the overall procedures while the iterative method stated above makes up the outer loop. In this dissertation, the stated method will be applied to the following topics: Minimax design of variable fractional-delay FIR digital filters by iterative weighted least- squares approach (Chapter 2). A new criterion for the design of variable fractional-delay FIR digital filters (Chapter 3). A new structure for the design of variable fractional-delay FIR digital filters (Chapter 4). Minimax phase error design of allpass variable fractional-delay digital filters by iterative weighted least-squares method (Chapter 5). A new method for least-squares and minimax group-delay error design of allpass variable fractional-delay digital filters (Chapter 6). A new method for the design of variable fractional-delay IIR digital filters (Chapter 7). An iterative method for the design of variable fractional-order FIR differintegrators (Chapter 8). A new structure for the design of wideband variable fractional-order FIR differentiators (Chapter 9). Iterative design of variable fractional-order IIR differintegrators (Chapter 10)