數位整數微分器，分數希爾伯特轉換器，及分數微分器的時域設計及皮亞諾被積函數

Design of Digital Integer order Differintegrator, Fractional Hilbert Transformer and Differentiator in Time Domain with Peano Kernel

林家卉(Chia-Huei Lin)

指導教授 ： 貝蘇章

國立臺灣大學/電機資訊學院/電信工程學研究所/碩士(2008年)

https://doi.org/10.6342/NTU.2008.02036

摘要

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。

關鍵字

整數微分/積分器 ； 希爾伯特轉換器 ； 分數積分器 ； 泰勒級數 ； 皮亞諾被積函數

並列摘要

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.

並列關鍵字

IIR filter ； fractional differentiator ； differentiator ； integrator ； Hilbert transformer ； Taylor series ； Peano kernel

參考文獻

[3] T.B. Deng and Y. Lian, “Weighted-least-squares design of variable fractional-delay FIR filters using coefficient symmetry,” IEEE Trans. Signal Processing, vol. 54, no.8, pp.3023-3038, Aug.2006.

[5] P.H.Wang, “Peano kernels of fractional delay systems,” IEEE International Conference on Acoustics, Speech and Signal Processing, 2007. ICASSP 2007. Volume: 3, page(s): III-1477-III- 1480, April 2007.

[6] B. Carlsson, ”Maximum flat digital differentiator,” Electron. Lett., vol. 27, pp.675-677, Apr.1997.

[7] S. Sunder and V. Ramachandran, “Design of equiripple nonrecursive digital differentiator and Hilbert transformers using a weighted least-squares technique,” IEEE Trans. Signal Process., vol. 42, no. 9, pp.2504-2509, Sep. 1994.

[8] M.A.Al-Alaoui, “Novel digital integrator and differentiator,” Electron. Lett., vol. 29, no. 4, pp. 376-378, Feb. 1993.