近年來為了因應無線通訊演變之需求，雙頻元件研究已經廣受重視。但雙頻帶通濾波器的設計方法分歧，其中利用數學演算法的設計方式與使用雙頻諧振器或頻率轉移函數的方法差異甚鉅，兩者幾乎是各自發展，很少有研究去探討其間的異同。例如利用現有之數學演算法設計雙頻帶通濾波器時，並無法有效地控制兩通帶漣波為等高，這不僅與大部份文獻中的雙頻電路響應不一樣，而且根據我們的研究發現，這還會造成濾波器的截止帶衰減量降低。在本論文中，我們不僅提出修正之演算法來改善上述情形，另一方面也利用此修正方法當作平台，與其他雙頻設計方式搭起橋樑，藉此可以方便地從數學上探討其截止帶響應的差異，並且舉出設計方式來說明如何嚴謹引入額外零點，最後利用實作來證明其可行性。
另一方面，目前文獻中的雙頻濾波器之設計方法，通常無法完整地控制其截止帶響應，僅能在初步設計完成後，透過對整個電路的分析，才能探討其成效，若無法達到規格要求，一切設計都得捨棄重來。為了避免雙頻設計過程的不斷重複，本論文最後提出雙頻濾波器衰減量表，來改善雙頻電路截止帶不能有效設計之問題，就如同使用傳統濾波器衰減量表般，這讓設計者在一開始就能決定截止帶特性，最後論文中並提出一個實作範例來驗證此衰減量表的正確與實用性。

A modified algorithm is proposed in this study to solve the problem of the current syntheses that the ripple levels of both passbands do not match while most realized dual-band filters do have same ripples. The stopband rejection of filters may be compromised because of these uncontrolled ripples. Furthermore, such modification allows stopband responses could be more conveniently compared. Also a structure which could exactly introduce extra transmission zeros is presented. Several rejection diagrams of dual-band filters are generated to help designers to assess the stopband responses at the start of their synthesis procedures.

[1.1] H. Miyake, S. Kitazawa, T. Ishizaki, T. Yamada, and Y. Nagatomi, “A Miniaturized Monolithic Dual Band Filter Using Ceramic Lamination Technique for Dual Mode Portable Telephones,” IEEE MTT-S Int. Microwave Symp. Dig., vol. 2, Jun. 1997, pp. 789-792.
[1.2] S. Sun and L. Zhu, “Coupling Dispersion of Parallel-Coupled Microstrip Lines for Dual- Band Filters with Controllable Fractional Pass bandwidths,” IEEE MTT-S Int. Microwave Symp. Dig., June 2005, pp. 2195-2198.
[1.3] Y. C. Chang, C. H. Kao, M. H. Weng and R. Y. Yang, “Design of the Compact Dual-Band Bandpass Filter With High Isolation for GPS/WLAN Applications,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 12, Dec. 2009, pp. 780-782.
[1.4] H. M. Lee, “Microwave Filter Designs with Different Component Q, Different Mode Velocities, and Dual-Band Characteristics,” Dissertation for Doctor of Philosophy, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., Jan. 2006.
[1.5] Y. L. Chen, “Dual-Band filter for Wireless Communications,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2001.
[1.6] S. Z. Wu, “Dual-Band Filter Design Using Frequency Transformation Methods,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2009.
[1.7] Y. Zhang, K. A. Zaki, J. A. Ruiz-Cruz, and A. E. Atia, “Analytical Synthesis of Generalized Multi-band Microwave Filters,” IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2007, pp. 1273-1276.
[1.8] D. Deslandes and F. Boone, “Iterative Design Techniques for All-Pole Dual-Bandpass Filters,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 11, Nov. 2007.
[1.9] G. Macchiarella, ““Equi-Ripple” Synthesis of Multiband Prototype Filters Using a Remez-Like Algorithm,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 5, pp. 231-233, May 2013.
[2.1] C. M. Tsai, H. M. Lee, and C. C. Tsai, “Planar Filter Design with Fully Controllable Second Passband,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3429-3439, Nov. 2005.
[2.2] H. M. Lee and C. M. Tsai, “Dual-Band Filter Design with Flexible Passband Frequency and Bandwidth Selections,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 5, pp. 1002-1009, May 2007.
[2.3] S. Lee and Y. Lee, “A Uniform Coupled-Line Dual-Band Filter With Different Bandwidths,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 10, pp. 545-547, Oct. 2010.
[2.4] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filter, Impedance-Matching Networks, and Coupling Structures, Artech House, 1980, ch. 8.
[2.5] G. Macchiarella and S. Tamiazzo, “Design Technique for Dual-Passband Filters,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3265-3271, Nov. 2005.
[2.6] R. J. Cameron, M. Yu, and Y. Wang, “Direct-Coupled Microwave Filters With Single and Dual Stopbands,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3288-3297, Nov. 2005.
[2.7] Y. L. Chen, “Dual-Band filter for Wireless Communications,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2001.
[2.8] S. Z. Wu, “Dual-Band Filter Design Using Frequency Transformation Methods,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2009.
[2.9] Y. Zhang, K. A. Zaki, J. A. Ruiz-Cruz, and A. E. Atia, “Analytical Synthesis of Generalized Multi-band Microwave Filters,” IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2007, pp. 1273-1276.
[2.10] D. Deslandes and F. Boone, “Iterative Design Techniques for All-Pole Dual-Bandpass Filters,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 11, Nov. 2007.
[2.11] G. Macchiarella, ““Equi-Ripple” Synthesis of Multiband Prototype Filters Using a Remez-Like Algorithm,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 5, pp. 231-233, May 2013.
[2.12] W. Meng, H. M. Lee, K. A. Zaki, and A. E. Atia, “Synthesis of Multi-Coupled Resonator Filters with Frequency-Dependent Couplings,” IEEE MTT-S Int. Microwave Symp. Dig., May 2010, pp. 1716-1719.
[2.13] Darlington, “Synthesis of Reactance 4-Poles which Produce a Prescribed Insertion Loss Characteristics,” J. Math. Phys., vol. 18, pp. 257-355, Sep. 1939.
[3.1] Y. Zhang, K. A. Zaki, J. A. Ruiz-Cruz, and A. E. Atia, “Analytical Synthesis of Generalized Multi-band Microwave Filters,” IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2007, pp. 1273-1276.
[3.2] D. Deslandes and F. Boone, “Iterative Design Techniques for All-Pole Dual-Bandpass Filters,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 11, Nov. 2007.
[3.3] G. Macchiarella, ““Equi-Ripple” Synthesis of Multiband Prototype Filters Using a Remez-Like Algorithm,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 5, pp. 231-233, May 2013.
[3.4] S. Z. Wu, “Dual-Band Filter Design Using Frequency Transformation Methods,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2009.
[3.5] Y. L. Chen, “Dual-Band filter for Wireless Communications,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2001.
[3.6] B. J. Minnis, Designing Microwave circuits by exact synthesis, Artech House, 1996.
[4.1] G. Macchiarella, ““Equi-Ripple” Synthesis of Multiband Prototype Filters Using a Remez-Like Algorithm,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 5, pp. 231-233, May 2013.
[4.2] Y. Zhang, K. A. Zaki, J. A. Ruiz-Cruz, and A. E. Atia, “Analytical Synthesis of Generalized Multi-band Microwave Filters,” IEEE MTT-S Int. Microwave Symp. Dig., June 2007, pp. 1273-1276.
[4.3] D. Deslandes and F. Boone, “Iterative Design Techniques for All-Pole Dual-Bandpass Filters,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 11, Nov. 2007.
[4.4] R. J. Cameron, M. Yu, and Y. Wang, “Direct-Coupled Microwave Filters With Single and Dual Stopbands,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3288-3297, Nov. 2005.
[4.5] S. Z. Wu, “Dual-Band Filter Design Using Frequency Transformation Methods,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2009.
[4.6] H. M. Lee and C. M. Tsai, “Dual-Band Filter Design with Flexible Passband Frequency and Bandwidth Selections,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 5, pp. 1002-1009, May 2007.
[4.7] Y. L. Chen, “Dual-Band filter for Wireless Communications,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2001.
[4.8] S. Sun and L. Zhu, “Coupling Dispersion of Parallel-Coupled Microstrip Lines for Dual- Band Filters with Controllable Fractional Pass bandwidths,” IEEE MTT-S Int. Microwave Symp. Dig., June 2005, pp. 2195-2198.
[4.9] S. Lee and Y. Lee, “A Uniform Coupled-Line Dual-Band Filter With Different Bandwidths,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 10, pp. 545-547, Oct. 2010.
[5.1] S. Z. Wu, “Dual-Band Filter Design Using Frequency Transformation Methods,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2009.
[5.2] Y. L. Chen, “Dual-Band filter for Wireless Communications,” Thesis for Master of Science, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June 2001.
[5.3] R. J. Cameron, M. Yu, and Y. Wang, “Direct-Coupled Microwave Filters With Single and Dual Stopbands,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3288-3297, Nov. 2005.
[5.4] X. P. Chen, K. Wu, and Z. L. Li, “Dual-Band and Triple-Band Substrate Integrated Waveguide Filters With Chebyshev and Quasi-Elliptic Responses,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 12, pp. 2569-2577, Dec. 2007.
[5.5] H. Di, B. Wu, and C. H. Liang, “Synthesis of Cross-Coupled Tripe-Passband Filters Based on Frequency Transformation,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 8, pp. 432-434, Aug. 2010.
[5.6] A. Papoulis, “Optimum Filters with Monotonic Response,” Proc. IRE, vol. 46, no. 3, pp. 606-609, March 1958.