臺灣博碩士論文加值系統

在這篇論文中，我們提出了一個數位整數微分/積分器、ㄧ個希爾伯特轉換器及一個分數積分器。利用一個本來用於設計分數延遲濾波器的時域分析方法，我們可以相當直接地推導出以上所述的濾波器。這個時域分析方法是對系統所需的理想輸入及輸出訊號作泰勒級數的展開，接著利用這些泰勒級數作為一個線性非時變系統的輸入及輸出訊號，並以此系統來模擬所需的濾波器。藉由一些簡單的簡化步驟，我們可以由該線性非時變系統得到ㄧ組聯立方程式，並由此解出濾波器的係數。這個時域分析方式除了可以得到我們所提出的濾波器外，我們還可以得到每個線性非時變系統的皮亞諾被積函數。皮亞諾被積函數可以代表系統在設計時與理想值的誤差。我們在這篇論文中也呈現了各種濾波器的設計實例。

In this dissertation, we propose an integer order differentiator, an integer order integrator, a fractional order differentiator and a Hilbert transformer. Using a time domain analysis method which is originally used in the derivation of a fractional delay filter, we find the derivations of these proposed filters are quite straightforward. By expanding the predefined ideal input and output signals of the filter into their Taylor series, we can approximate the desired filter with these Taylor series as input and output signals of a LTI system. After some simplification, we derive a set of linear equations which can be solved numerically for the filter’s coefficients. Aside form the derivations of these filters, this time domain analysis allows us to derive a Peano kernel for each proposed filter. A Peano kernel can represents the approximation error of the filter. Some designing examples are also demonstrated in this dissertation.

口試委員會審定書
誌謝 i
中文摘要 ii
ABSTRACT iii
Design of Digital Integer order Differintegrator, Fractional Hilbert Transformer and Differentiator in Time Domain with Peano Kernel 1
Chapter 1 Introduction 1
Chapter 2 Fractional Delay Filter and Its Peano Kernel 4
2.1 Introduction 4
2.2 Derivation of Fractional Delay filter 5
2.2.1 Preliminaries 5
2.2.2 Derivation of Fractional delay filter 5
2.3 Peano Kernel of the FD system 7
2.4 Design Examples 8
2.5 Conclusion 11
Chapter 3 IIR Integer Order Differintegrator and Its Peano Kernel 12
3.1 Introduction 12
3.2 Derivation of Discrete Integer order differentiator and Integrator 12
3.2.1 Integer order Differentiator 12
3.2.2 Integer order Integrator 15
3.3 Derivation of the Peano Kernel 15
3.4 Design examples 17
3.5 Stability 19
3.6 Conclusion 20
Chapter 4 Fractional Hilbert Transformer and Its Peano Kernel 22
4.1 Introduction 22
4.2 Derivation of fractional Hilbert Transformer 23
4.3 Peano Kernel of the Fractional Hilbert transformer 26
4.4 Design examples 28
4.5 Conclusion 30
Chapter 5 Design of Discrete Fractional Differentiator with Peano Kernel 32
5.1 Introduction 32
5.2 Derivation of the Discrete Fractional Differentiator 33
5.2.1 Discretization of the fracional differentiation 33
5.2.2 System approximation with Taylor series 34
5.3 Peano Kernel of the fractional differentiator 36
5.4 Design examples 37
5.5 Conclusion 40
Chapter 6 Conclusion 41
REFERENCE 45

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