改良式于爾-華克方程式 (Modified Yule-Walker equation) 常被使用去估算自我迴歸程序 (Autoregressive process) 的時域參數。本論文中，我們提出一個設計微波濾波器的新方法。我們可以利用改良式于爾-華克方程式得到具有多頻帶和/或多位階響應的離散時間濾波器。我們利用這個新方法來實現單頻帶、雙頻帶和雙位階的微波濾波器。最後，為了使尺寸更小，我們改變產生高頻衰減極點的兩段並聯開路殘枝的正規化頻率。
這個方法是先提出濾波器在Ｚ域中的系統方程式，再搭配傳輸線鏈散矩陣和改良式于爾-華克方程式，然後以串聯傳輸線、單段式開路殘枝、單段式短路殘枝和兩段式開路殘枝完成。最後提出實驗結果來驗證此方法的可行性。

The modified Yule-Walker equation is commonly used for estimating the time domain parameters of an autoregressive process. In this thesis, we propose a new method to design microwave filters. We employ the modified Yule-Walker equation to obtain discrete-time filters having multi-band and/or multi-level responses. We apply this new method to achieve a single-band bandpass filter, a dual-band bandpass filter, and a two-level bandpass filter. At last, we change the normalized frequency of two-section shunt-open stub, which produce attenuation pole in higher band, to make the size be smaller.
In this method, the system functions of bandpass filters in Z-domain are studied first. The Z-domain chain-scattering matrices of transmission lines and the modified Yule-Walker equation are then derived in the thesis. The filters are implemented with serial lines, open-circuited single-section stubs, short-circuited single-section stubs, and open-circuited two-section stubs. Experimental results are presented to illustrate the validity of this design method.

[1] David M. Pozar, “Microwave Engineering”, 2ndEd
[2] Jia-Shen G. Hong and M.J. Lancaster, “Microstrip Filters for RF/Microwave Applications.”
[3] P. I. Richard, “Resistor-Transmission Line Circuits,” Proc. IRE, vol. 36, pp. 217–220, 1984.
[4] K. Kuroda, “General Properties and Synthesis of Transmission-Line Networks, Microwave Filters and Circuits,” A. Matsumoto, Ed. New York: Academic,
vol. 22, 1970.
[5] Da-Chiang Chang and Ching-Wen Hsue, “Wide-Band Equal-Ripple Filters in Nonuniform Transmission Lines,” IEEE Transactions on Microwave Theory and Techniques, vol. 49, 2001.
[6] Da-Chiang Chang and Ching-Wen Hsue, “Design and implementation of filters using transfer function in the Z domain,” IEEE Trans. Microwave Theory,
vol. 50, pp. 979-985, 2001.
[7] Lin-Chuan Tsai and Ching-Wen Hsue, “Dual-Band Bandpass Filters Using Equal-Length Coupled-Serial-Shunted Lines and Z-Transform Technique,” IEEE Transactions on Microwave Theory and Techniques, vol. 52, 2004.
[8] Friedlander, B., and B. Porat, “The Modified Yule-Walker Method of ARMA Spectral Estimation,” IEEE Transactions on Aerospace Electronic Systems, AES-20, vol. 2, pp. 158-173, 1984.
[9] Michael Jachan, Gerald Matz, and Franz Hlawatsch, “Time-Frequency- Autoregressive Random Process: Modeling and Fast Parameter Estimation” Proc. IEEE ICASSP-03, vol. 6, pp. 125–128, 2003.
[10] Steven M. Crunk, “The Yule-Walker Equations as a Least Squares Problem and the Need for Tapering,” 2005.
[11] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, “Discrete-Time Signal Processing,” 2ndEd.,Prentice Hall, Inc, 1999.
[12] Box, G.E.P. and Jenkins, G.M., “Time Series Analysis, Forecasting and Control,” San Francisco: Holden-Day, 1970.
[13] Dickinson, B.W., Kailath, T., and Morf, M., “Canonical Matrix Fraction and State Space Descriptions for Deterministic and Stochastic Linear Systems,” IEEE Transactions on Automatic Control, AC-19, pp. 656-667, 1974.
[14] Yule, G.U., “On a Method of Investigating Periodicitics in Disturbed Series, with Special Reference to Wolfer's Sunspot Number,” Philosophical Transactions, A 226, pp. 267, 1927.
[15] Walker, G., “On Periodicity in Series of Related Terms,” Proceedings of the Royal Society, A 13, pp. 518, 1931.
[16] de Prony, R. Essai experimentale et analytique. J. Ecole Polvtechnique (Paris), pp. 24-76, 1975.
[17] I. J Bahl and D. k. Trivedi, “A Designer`s Guide to Microstrip Line,” Microwave, 1977.
[18] K. C. Gupta, R. Garg, and I. J. Bahl, “Microstrip Lines and Slotilines,” Artech House, Dedham, Mass.
[19] C.-W. Hsue, C.-W. Ling, and W.-T. Hung, “Discrete-time notch filter and its application to microwave filter,” Microwave and Optical Technology Letters, Vl. 50, vol. 6, pp. 1596-1600, 2008.