# 臺灣博碩士論文加值系統

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 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 iiABSTRACT iiiDesign of Digital Integer order Differintegrator, Fractional Hilbert Transformer and Differentiator in Time Domain with Peano Kernel 1Chapter 1 Introduction 1Chapter 2 Fractional Delay Filter and Its Peano Kernel 42.1 Introduction 42.2 Derivation of Fractional Delay filter 52.2.1 Preliminaries 52.2.2 Derivation of Fractional delay filter 52.3 Peano Kernel of the FD system 72.4 Design Examples 82.5 Conclusion 11Chapter 3 IIR Integer Order Differintegrator and Its Peano Kernel 123.1 Introduction 123.2 Derivation of Discrete Integer order differentiator and Integrator 123.2.1 Integer order Differentiator 123.2.2 Integer order Integrator 153.3 Derivation of the Peano Kernel 153.4 Design examples 173.5 Stability 193.6 Conclusion 20Chapter 4 Fractional Hilbert Transformer and Its Peano Kernel 224.1 Introduction 224.2 Derivation of fractional Hilbert Transformer 234.3 Peano Kernel of the Fractional Hilbert transformer 264.4 Design examples 284.5 Conclusion 30Chapter 5 Design of Discrete Fractional Differentiator with Peano Kernel 325.1 Introduction 325.2 Derivation of the Discrete Fractional Differentiator 335.2.1 Discretization of the fracional differentiation 335.2.2 System approximation with Taylor series 345.3 Peano Kernel of the fractional differentiator 365.4 Design examples 375.5 Conclusion 40Chapter 6 Conclusion 41REFERENCE 45
 [1]T. I. Laakso, V. Valimali, M. Karjalainen, and U.K. Laine, “splitting the unit delay,” IEEE Signal Processing Mag., pp. 30-60, Jan. 1996.[2]A. V. Oppenhein, R.W. Schafer, and J.R. Buck, Discrete-Time Signal Processing, 2nd. Ed., Prentice-Hall,Inc., 1999.[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.[4]S. Samadi, M. O. Ahmad, and M. N. S. Swamy, “Results on maximally flat fractional delay systems,” IEEE trans. Circuits syst. I, vol. 51, no.11, pp. 2271-2286, Nov. 2004.[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.[9]N. Q. Ngo, “A new approach for the design of wideband digital integrator and differentiator,” IEEE Trans. Circuits Syst. II, Express Briefs, vol. 53, no. 9, pp. 936-940, Sep. 2006.[10]M.A.Al-Alaoui, ”Linear phase low-pass IIR digital differentiators,” IEEE Trans Signal Process., vol. 55, no.2, pp. 691-706, Feb. 2007.[11]C.C.Tseng, “Digital differentiator design using fractional delay filter and limit computation,” IEEE trans. Circuits Syst. I, Regular Papers, Vol. 52, no. 10, pp. 2248-2259, Oct. 2005.[12]Soo-Chang Pei and Peng-Hua Wang,” Closed-form design of maximally flat FIR Hilbert transformers, differentiators, and fractional delayers by power series expansion,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume: 48, Issue: 4page(s): 389-398, Apr 2001.[13]Le Bihan, J. , “Coefficients of FIR digital differentiators and Hilberttransformers for midband frequencies,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Volume: 43, Issue: 3, page(s): 272-274, Mar 1996.[14]A.W.Lohmann, D. Mendlovic and Zalevsky, “Fractional Hilbert transform,” Optic letters, vol.21, no.4, pp. 281-283, Feb 1996.[15]S.C. Pei and M.H.Yeh, “Discrete fractional Hilbert transform,” Proc. IEEE Int. Symp. on Circuits an Systems, Montery, California, May 1998.[16]Soo-Chang Pei; Huei-Shan Lin; Peng-Hua Wang, "Design of Allpass Fractional Delay Filter and Fractional Hilbert Transformer Using Closed-Form of Cepstral Coefficients," Circuits and Systems, 2007. ISCAS 2007. IEEE International Symposium on , vol., no., pp.3443-3446, 27-30 May 2007[17]Chien-Chen Tseng,” Design of FIR and IIR fractional order simpson digital integrators,” Signal Processing, Volume 87, Issue 5, May 2007, Pages 1045-1057[18]C.C. Tseng, S.C. Pei and S.C. Hsia, Computation of fractional derivatives using Fourier transform and digital FIR differentiator, Signal Process. 80 (2000), pp. 151–159.[19]R.S. Barbosa, J.A.T. Machado and I.M. Ferreira, Least-squares design of digital fractional-order operators, 1st IFAC Workshop on Fractional Differentiation and its Applications (July 2004), pp. 434–439.[20]Keith B. Oldham, Jerome Spanier, “The fractional calculus”,Academic Press, New York and London, 1974.[21]D.Kincaid and w.Cheney, Numerical Analysis, 2nd. Ed. Brooks/Cole Publishing Co., 1996
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