臺灣博碩士論文加值系統

本論文將提出一種基於梯度陡降演算法，在極點靈敏度最小化的情形下，再同時將零點靈敏度量度最佳化的無限脈衝數位濾波器結構的設計方法。我們基於稀疏矩陣型式的推導結果，在濾波器已達到極點靈敏度全域最佳化的情形下，再同時考慮濾波器零點靈敏度的最小化問題。藉由矩陣的跡數不等式 (trace inequality) ，推導出一個可以數學處理的成本函數，我們根據此成本函數使用了梯度陡降法調整成本函數中之變數，使最佳化之矩陣參數得以被合成。與過去的研究相比較，本論文除了可以達到目前文獻上相同的濾波性能外，同時可以大幅降低文獻中濾波器實現所需要的計算量，最後我們以數值案例驗證了本論文所提方法的有效性。

This thesis aims to develop an infinite impulse filter synthesis method by using gradient decent algorithm for minimizing a zero sensitivity measure subject to minimal pole sensitivity. Based on the derived result of the sparse normal-form filter realizations, the optimal filter structure can be obtained by considering the zero sensitivity minimization problem subject to the pole-sensitivity function which is summing up all unweighted pole sensitivity measure. By the technique of trace inequalities, a mathematically tractable cost function can be minimized by using gradient decent algorithm to adjust the degree of freedom in the cost function for the minimization problem. Comparing with the existing methods in the literature, our proposed approach may achieve the same performance but less computation consumption under finite precision filter implementations. Finally, numerical examples are performed to illustrate the effectiveness of the proposed approach.

目錄

摘要 iii
ABSTRACT iv
第1章 緒論 1
1.1 研究背景 1
1.2 研究動機 4
1.3 研究目的 7
第2章 問題描述 8
2.1 IIR數位濾波器式數學模型 8
2.2 IIR數位濾波器之極點與零點靈敏度 9
第3章 研究方法 13
2.2 13
3.1 稀疏正規矩陣合成 13
3.2 基於最低極點靈敏度之零點靈敏度最佳化 14
3.3 以矩陣跡數不等式與梯度陡降法進行靈敏度最佳化 16
第4章 數值模擬與分析 18
4.1 可控制典型式濾波器結構 18
4.2 同時考量極零點靈敏度極小化的全參數濾波器 20
4.3 本論文所提出的濾波器結構 22
第5章 結論 24
附錄(本論文程式碼) 25
參考文獻 34

圖目錄

圖 1 1、Direct Form I 數位濾波器架構圖[4] 2
圖 1 2、Direct Form II 數位濾波器架構圖[5] 2
圖 1 3、以串接型式組成之濾波器架構所有可能的組合情形[2] 2
圖 4 1、可控制典型式濾波器在10位元至16位元之頻率響應圖 19
圖 4 2、可控制典型式濾波器在10位元至16位元之頻率響應誤差圖 19
圖 4 3、同時針對及零點極小化之全參數濾波器在10位元至16位元之頻率響應圖 21
圖 4 4、同時針對極零點極小化之全參數濾波器10位元至16位元之頻率響應誤差圖 22
圖 4 5、本論文所提出之濾波器架構在10位元至16位元之頻率響應圖 23
圖 4 6、本論文所提出之濾波器架構在10位元至16位元之頻率響應誤差圖 23

[1]A.V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing, 3rd ed. Prentice-Hall, 2009.
[2]S. K. Mitra, Digital signal processing: a computer-based approach., 2nd ed. McGraw-Hill, 2001.
[3]C. K. Ahn, “Some new results on the stability of direct-form digital filters with finite wordlength nonlinearities,” Signal Processing, vol. 108, pp. 549–557, 2015.
[4]“Digital Filter.” [Online]. Available: https://en.wikipedia.org/wiki/Digital_filter.
[5]“Digital Filter.” .
[6]R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” J. Basic Eng., vol. 82, no. 1, p. 35, 1960.
[7]M. Gevers and G. Li, Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects. Springer-Verlag, 1993.
[8]C. Xiao, “Improved $L_2$-sensitivity for state-space digital system,” IEEE Trans. Signal Process., vol. 45, no. 4, pp. 837–840, Apr.1997.
[9]W.-Y. Yan and J. B. Moore, “On $L^2$-sensitivity minimization of linear state-space systems,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl., vol. 39, no. 8, pp. 641–648, 1992.
[10]G. Li, “On frequency weighted minimal $L_2$ sensitivity of 2-D systems using Fornasini-Marchesini LSS model,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl., vol. 44, no. 7, pp. 642–646, Jul.1997.
[11]G. Li, “Two-dimensional system optimal realizations with $L_2$-sensitivity minimization.,” IEEE Trans. Signal Process., vol. 46, no. 3, pp. 809–813, 1998.
[12]T. Hinamoto, S. Yokoyama, T. Inoue, W. Zeng, and W.-S. Lu, “Analysis and minimization of $L_2$-sensitivity for linear systems and two-dimensional state-space filters using general controllability and observability Gramians,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl., vol. 49, no. 9, pp. 1279–1289, Sep.2002.
[13]T. Hinamoto, H. Ohnishi, and W.-S. Lu, “Minimization of $L_2$-sensitivity for state-space digital filters subject to $L_2$-dynamic-range scaling constraints.,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 52, no. 10, pp. 641–645, Oct.2005.
[14]T. Hinamoto, K. Iwata, and W.-S. Lu, “$L_2$-sensitivity minimization of one- and two-dimensional state-space digital filters subject to $L_2$-scaling constraints,” IEEE Trans. Signal Process., vol. 54, no. 5, pp. 1804–1812, May2006.
[15]T. Hinamoto, T. Oumi, O. I. Omoifo, and W.-S. Lu, “Minimization of Frequency-Weighted $l_2$-Sensitivity Subject to $l_2$-Scaling Constraints for Two-Dimensional State-Space Digital Filters.,” IEEE Trans. Signal Process., vol. 56, no. 10, pp. 5157–5168, Oct.2008.
[16]S. Yamaki, M. Abe, and M. Kawamata, “A closed form solution to $L_2$-sensitivity minimization of second-order state-space digital filters subject to $L_2$-scaling constraints.,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E91–A, no. 7, pp. 1697–1705, 2008.
[17]T. Hilaire, “New $L_2$-dynamic-range-scaling constraints for low parametric sensitivity realizations,” Eur. Signal Process. Conf., no. Eusipco, pp. 988–992, 2009.
[18]T. Hilaire, “Low-Parametric-Sensitivity Realizations With Relaxed $L_{2}$ Dynamic-Range-Scaling Constraints,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 56, no. 7, pp. 590–594, Jul.2009.
[19]S. YAMAKI, M. ABE, and M. KAWAMATA, “Closed Form Solutions to $L_2$-Sensitivity Minimization Subject to $L_2$-Scaling Constraints for Second-Order State-Space Digital Filters with Real Poles,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E93–A, no. 2, pp. 476–487, 2010.
[20]C. K. Ahn, “$l_{2} - l_{\infty}$ Elimination of Overflow Oscillations in 2-D Digital Filters Described by Roesser Model With External Interference,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 60, no. 6, pp. 361–365, Jun.2013.
[21]C. K. Ahn, “$l_{2}-l_{\infty} Suppression of Limit Cycles in Interfered Two-Dimensional Digital Filters: A Fornasini-Marchesini Model Case,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 61, no. 8, pp. 614–618, Aug.2014.
[22]G. Li, “On pole and zero sensitivity of linear systems.,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl., vol. 44, no. 7, pp. 583–590, 1997.
[23]G. Li, “On the structure of digital controllers with finite word length consideration.,” IEEE Trans. Automat. Contr., vol. 43, no. 5, pp. 689–693, May1998.
[24]R. S. H. Istepanian and J. F. Whidborne, Digital Controller implementation and fragility. Springer-Verlag London Ltd., 2001.
[25]J. F. Whidborne, R. S. H. Istepanian, and J. Wu, “Reduction of controller fragility by pole sensitivity minimization.,” IEEE Trans. Autom. Contr., vol. 46, no. 2, pp. 320–325, 2001.
[26]J. Wu, S. Chen, G. Li, R. S. H. Istepanian, and J. Chu, “An improved closed-loop stability related measure for finite-precision digital controller realizations.,” IEEE Trans. Automat. Contr., vol. 46, no. 7, pp. 1162–1166, Jul.2001.
[27]Jun Wu, S. Chen, J. F. Whidborne, and J. Chu, “A unified closed-loop stability measure for finite-precision digital controller realizations implemented in different representation schemes,” IEEE Trans. Automat. Contr., vol. 48, no. 5, pp. 816–822, May2003.
[28]J. Wu, J. Chu, G. Li, and S. Chen, “Constructing sparse realisations of finite-precision digital controllers based on a closed-loop stability related measure.,” IEE Proc. - Control Theory Appl., vol. 150, no. 1, pp. 61–68, Jan.2003.
[29]H.-J. Ko and W.-S. Yu, “Guaranteed Robust Stability of the Closed-Loop Systems for Digital Controller Implementations via Orthogonal Hermitian Transform,” IEEE Trans. Syst. Man Cybern. Part B, vol. 34, no. 4, pp. 1923–1932, Aug.2004.
[30]H.-J. Ko and W.-S. Yu, “Improved eigenvalue sensitivity for finite-precision digital controller realisations via orthogonal Hermitian transform,” IEE Proc. - Control Theory Appl., vol. 150, no. 4, pp. 365–375, Jul.2003.
[31]H.-J. Ko, “Stability analysis of digital filters under finite word length effects via normal-form transformation.,” Asian J. Heal. Inf. Sci., vol. 1, no. 1, pp. 112–121, 2006.
[32]H.-J. Ko and W.-S.Yu, “A novel approach to stability analysis of fixed-point digital filters under finite word length effects,” in 2008 SICE Annual Conference, 2008, pp. 2388–2392.
[33]H.-J. Ko, “The sparse normal-form realization with minimal zero sensitivity measure for finite word-length IIR digital filter implementations,” Submitt. to IEEE Trans. Signal Process., 2015.
[34]T. Hinamoto, A. Doi, and W.-S. Lu, “Minimization of Weighted Pole and Zero Sensitivity for State-Space Digital Filters,” IEEE Trans. Circuits Syst. I Regul. Pap., vol. 63, no. 1, pp. 103–113, Jan.2016.
[35]G. Amit and U. Shaked, “Small roundoff noise realization of fixed-point digital filters and controllers.,” IEEE Trans. Acoust., vol. 36, no. 6, pp. 880–891, Jun.1988.
[36]C. Barnes and A. Fam, “Minimum norm recursive digital filters that are free of overflow limit cycles,” IEEE Trans. Circuits Syst., vol. 24, no. 10, pp. 569–574, Oct.1977.
[37]X. He, G. Li, C. Wan, and T. Wu, “On normal realizations of digital filters with minimum roundoff noise gain,” Signal Processing, vol. 89, no. 2, pp. 226–231, Feb.2009.
[38]G. Li, L. Meng, Z. Xu, and J. Hua, “A novel digital filter structure with minimum roundoff noise.,” Digit. Signal Process., vol. 20, no. 4, pp. 1000–1009, Jul.2010.
[39]R. E. SKELTON and D. A. WAGIE, “Minimal root sensitivity in linear systems,” J. Guid. Control. Dyn., vol. 7, no. 5, pp. 570–574, Sep.1984.
[40]D. Williamson, “Roundoff noise minimization and pole-zero sensitivity in fixed-point digital filters using residue feedback,” IEEE Trans. Acoust., vol. 34, no. 5, pp. 1210–1220, Oct.1986.
[41]B.-S. Chen, H.-C. Lee, and C.-F. Wu, “Pareto Optimal Filter Design for Nonlinear Stochastic Fuzzy Systems via Multiobjective $H_2/H_\infty$ Otimization,” IEEE Trans. Fuzzy Syst., vol. 23, no. 2, pp. 387–399, Apr.2015.