REFERENCES 1. V. Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions, J Comput Phys 86 (1990), 414 – 439. 2. R. Coifman, V. Rokhlin, and S. Wandzura, The fast multipole method for the wave equation: A pedestrian prescription, IEEE Antennas Propagat Mag 35 (1993), 7–12. 3. C.C. Lu and W.C Chew, A fast algorithm for solving hybrid integral equation, IEE Proc-H 140 (1993), 455– 460. 4. R.L. Wagner and W.C. Chew, A ray-propagation fast multipole algo- rithm, Microwave Opt Technol Lett 7 (1994), 435– 438. 5. J.M. Song and W.C. Chew, Fast multipole method solution using para- metric geometry, Microwave Opt Technol Lett 7 (1994), 760 –765. 6. J.M. Song and W.C. Chew, Multilevel fastmultipole algorithm for solving combined field integral equations of electromagnetic scatter- ing, Microwave Opt Technol Lett 10 (1995), 14 –19. 7. J.M. Song, C.C. Lu, and W.C. 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Manteuffel, A technique for acceler- ating the convergence of restarted GMRES, SIAM J Matrix Anal Appl, submitted for publication. 13. M. Abromowitz and I.A. Stegun, Handbook of mathematical func- tions, Dover, New York, 1972. 14. H.A. Van Der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Numer Linear Algebra Appl 1 (1994), 369 –386. 15. E. DE Sturler, Nested Krylov methods based on GCR, J Comput Appl Math 67 (1996), 15– 41. 16. S.M. Rao, D.R. Wilton, and A.W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans Antennas Propagat Ap-30 (1982), 409 – 418. 17. H.C. van de Hulst, Light scattering by small particles, Dover Press, New York, 1981. © 2007 Wiley Periodicals, Inc. BANDPASS FILTER OF FORKED STEP IMPEDANCE RESONATOR WITH HARMONIC SUPPRESSION IN DIPLEXER APPLICATIONS Min-Hua Ho, 1 Chung-I G. Hsu, 2 and Chen-Mao Rao 3 1 Department of Electronic Engineering, National Changhua University of Education, 1 Jin-Der Rd., 50007 Changhua City, Taiwan; Corresponding author: ho@cc.ncue.edu.tw 2 Department of Electrical Engineering, Da-Yeh University, 112 Shan- Jiau Rd., Da-Tsuen, 51591 Changhua, Taiwan 3 ZyXEL Corp., 6 Chuang Xin 2nd Rd, Hsinchu County 30076, Taiwan Received 3 April 2007 ABSTRACT: A novel two-stage forked stepped-impedance-resonator (FSIR) bandpass filter design is presented. The FSIR originated from an SIR has its both open ends split as a fork structure. A tunable zero from the coupled lines is created to suppress the second harmonic of the resonator. Two additional zeros obtained from using the asymmetric tapped-line feed are used to improve the filter’s frequency selectivity. A diplexer design based on the FSIR structure is presented. Experiments are conducted to verify the circuit design. Good agreements between the measurements and the simulations are observed. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2665–2668, 2007; Published online in Wiley Inter- Science (www.interscience.wiley.com). DOI 10.1002/mop.22813 Key words: tunable zero; forked SIR; diplexer; harmonic suppression 1. INTRODUCTION The development of modern microwave communication systems, especially in a satellite and mobile communication systems, de- mands channel-select filters having the properties of high selec- tivity, low insertion loss, narrow passband, and possibly a wide stopband bandwidth. In the past, these filters are frequently con- structed by multi-stage coupled resonator structures, which might have considerable losses and circuit size. Recently, a simple two- stage coupled hairpin-resonators together with a novel tap-line asymmetric feed configuration [1– 4] were employed in the filter design to produce two zeros each located by either side of the passband edges. These created zeros might be used to sharpen the passband skirt roll-off. However, the drawback is that they have considerable circuit dimension (corresponding to the half wave- length) and existing spurious harmonic responses, which deterio- rate filter’s stopband performance. In our design, the FSIR is implied in the filter configuration. The FSIR structure can be considered as a deformation of a SIR but still preserves the latter’s characteristics. A proper design of FSIR (or SIR) can reduce the circuit size and control the frequen- cies of the higher order harmonics [5, 6]. Between FSIR and SIR, the former might have a wider range of impedance ratio than that of the latter in practical design. The coupling between two FSIRs might be tighter than that between SIRs for the former having two coupling edges. The coupling scheme between FSIRs in conjunc- tion with the tapped-line asymmetric feed is employed in the filter design to create attenuation poles for eliminating the spurious passband formed by the high order harmonics. The proposed filter exhibits three zeros with one of them being frequency tunable, and the other two zeros, each locating by either side of the passband edges, improving the roll-off at the circuit’s passband skirts. The tunable zero resulted from the coupled-lines effect is analytically analyzed by a capacitively loaded coupled microstrip line model reported by Tsai et al. [7]. The capacitance load is realized by a section of open-end microstrip line. The design curve of the tunable coupling zero’s frequency vs. the length of the load line is Figure 7 Comparison of the residual norm history between LGMRES (42, 8) and GMRES (50) algorithms for scattering from a sphere with radius 3 0 at 600 MHz DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 11, November 2007 2665