IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007 1163

A Synthesis Method for Dual-PassbandMicrowave Filters

Juseop Lee, Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—This paper describes a synthesis method for sym-metric dual-passband microwave filters. The proposed methodemploys frequency transformation techniques for finding thelocations of poles and zeros of a desired filter. This method canbe used to design dual-passband filters with prescribed passbandsand attenuation at stopbands directly without the need for anyoptimization processes. To validate the procedure a dual-pass-band stripline filter is designed and fabricated. The striplinedual-passband filter is designed with passbands at 3.90–3.95 and4.05–4.10 GHz, and 30-dB attenuation at the stopband. This mea-sured results show a good agreement with the theoretical ones. Thefrequency transformation for symmetric dual-passband filters isalso extended to include asymmetric dual-passband responses.This flexible frequency transformation preserves the attenuationcharacteristics of the low-pass filter prototype. Examples areshown to discuss the flexibility of this transformation.

Index Terms—Circuit synthesis, dual-passband filters, mi-crowave filters.

I. INTRODUCTION

MODERN communication transceivers require high-per-formance microwave filters with low insertion loss, high

frequency selectivity, and small group-delay variation. For highfrequency selectivity, synthesis and design techniques for fil-ters with transmission zeros near passband have been developed[1]–[3]. For those filters, flat group delay in the passband is ac-complished using the external equalizer or the self-equalizationdesign technique [4], [5].

Since modern communications systems, especially satellitecommunications systems, have a complex frequency and spatialcoverage plan, noncontinuous channels might need to be ampli-fied and transmitted through one beam. In this case, comparedwith the power divider/combiner configuration, circulator chainstructures, or manifolds, multiple-passband filter can make thesystem simple. Dual-passband filters of a canonical structurewith a single-mode technique [6], that of an in-line structurewith a dual-mode technique [7], and that of a canonical structurewith a dual-mode technique [8] have been designed and real-ized. A synthesis method of a self-equalized dual-passband filterhas also been presented [9]. These deign methods are all basedon the optimization techniques in the filter synthesis process.

Manuscript received December 14, 2006; revised March 7, 2007.The authors are with the Radiation Laboratory, Department of Electrical En-

gineering and Computer Science, The University of Michigan at Ann Arbor,Ann Arbor, MI 48109-2122 USA (e-mail: ifsnow@ieee.org; saraband@eecs.umich.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2007.897712

To avoid numerical optimization, a method known as thefrequency transformation technique has been introduced in de-signing dual-passband filters [10]. Basically this synthesis fordesigning a dual-passband filter is accomplished by applying afrequency transformation to a low-pass filter prototype. Sincethe stopband response of the dual-passband filter obtainedby using the frequency transformation is not identical to thatof low-pass filter prototype, a few attempts are required tofind a suitable placement of transmission zeros to acquire thedesired attenuation value. One of the frequency transformationin [10] can be adopted for designing an asymmetric dual-pass-band filter. However, this transformation cannot provide theequiripple response in the stopband, which enables high-fre-quency selectivity.

Filters with dual stopbands and the associated frequencytransformation are presented in [11]. Similar to the transforma-tion in [10], this transformation also requires the optimizationto achieve an equiripple response in the stopband and its appli-cations to asymmetric dual-passband or dual-stopband filtersare not discussed.

In this paper, we present a synthesis technique for symmetricdual-passband filters using frequency transformation withoutthe need for optimization. Two frequency transformations aregiven and applied consecutively to the low-pass filter prototypeto obtain the dual-passband filter response. With this method,the dual-passband filters can be designed with prescribed pass-bands and an attenuation value in the stopband since the trans-formation preserves the low-pass filter prototype characteristics.The frequency transformation in general form is also given fordesigning asymmetric dual-passband filters. This transforma-tion is flexible enough to allow for the design of filters with twopassbands of highly different bandwidths. This transformationalso preserves low-pass filter prototype characteristics.

II. DESIGN THEORY

Here, we introduce a procedure for designing dual-passbandmicrowave filters using two frequency transformations. Fig. 1shows the frequency response of the filter in three different fre-quency domains. The domain is the actual frequency domainwhere the filter operates and is a normalized frequency forthe low-pass prototype. Generally, single-passband filters aredesigned in the domain and the frequency transformation isapplied to make the filter operate in the domain. For dual-pass-band filter design, an intermediate normalized frequency isused. The frequency response in the domain can be obtainedby applying two frequency transformations consecutively to thefrequency response in the domain. The coupling matrix of the

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1164 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Fig. 1. Frequency response of the filter in the ; ; and ! domain. The cou-pling matrix of the dual-passband filter is obtained from the transfer function inthe domain.

dual-passband filter is obtained from the transfer function of thefilter in the domain.

A. Low-Pass Filter Prototype

Generally, the transfer function of an th-order low-passfilter prototype can be expressed by

(1)

where with the assumption that the characteristic func-tion only has pure imaginary poles and zeros in thedomain, and is a ripple constant related to the passband returnloss by

(2)

Since here we do not deal with self-equalized dual-passbandfilters, pure imaginary poles and zeros are sufficient for our de-sign. For the case of pure imaginary pole and zeros, the charac-teristic function of the elliptic function filter is given by [12]

(3)

where

(4)

In (4), is the location of the th transmission zero. Note thatthe magnitude of is 1 at for all . Once thetransmission zeros are decided, the poles can be obtained easilyby computing the return loss of the filter. Based on the pole and

zero locations, we can rewrite the characteristic function as arational function

(5)

where

(6)

which makes the magnitude of in (5) 1 at .

B. Frequency Transformation for Dual-Passband Filters

Let us assume that the dual-passband filter has two symmetricpassbands and their passband regions are specified byand (Fig. 1). The coupling matrix is obtained fromthe frequency response in the normalized frequency . Fordual-passband filter synthesis, the previous studies [8], [9] startfrom the domain by finding the locations of poles and zerosof the filters by direct optimization.

In this paper, we obtain the poles and zeros of the dual-pass-band filter by the analytic frequency transformation technique.The frequency transformation from to can be expressed asfollows:

for

for (7)

where and . Since 1 and 1 in the domainare transformed to 1 and in the domain for ,respectively, and 1 and 1 in the domain are transformed to

1 and in for , respectively, we must enforce

(8)

From (8), the constant in (7) can be expressed in terms of theband edge frequency of the dual-passband filter in the domainas follows:

(9)

Similarly, the frequency transformation from to fornarrow bandpass filters is expressed as

(10)

where . Since band edge frequencies in the domainare transformed to those in the domain as

(11)

LEE AND SARABANDI: SYNTHESIS METHOD FOR DUAL-PASSBAND MICROWAVE FILTERS 1165

Fig. 2. Frequency response of a low-pass filter prototype with the transmissionzeros at �j2:0.

therefore, the relationship between the band edge frequencies inthe domain and the coefficients in (10) are as follows:

(12)

From (12), we can express and in terms of band edgefrequencies of the dual-passband filter

(13)

Using (9) and (13), the constant in (7) can also be expressedin terms of band edge frequencies of the dual-passband filter.Therefore, the frequency transformations can be expressed interms of band edge frequencies of the dual-passband filter.

Based on the prescribed narrow passbandsof the dual-passband filter, the frequency transformations

can be defined and, consequently, the transfer functions and cou-pling matrices can be determined. Section III describes the ap-plication of the frequency transformations and then direct appli-cations for designing dual-passband filters.

III. FILTER DESIGN AND MEASUREMENT

Here, the filter with two passbands is designed and realizedto describe the presented filter synthesis theory. The passbandsof the dual-passband filter are chosen to be 3.90–3.95 and4.05–4.10 GHz. Each passband is set to have four poles anda maximum return loss of 20 dB. Minimum attenuation overstopbands is set to be 30 dB.

Since we are seeking a four-pole passband response, we startfrom a four-pole low-pass prototype. Fig. 2 shows the frequencyresponse of the low-pass filter prototype with the transmission

zeros at . This filter prototype has a pair oftransmission zeros at finite frequencies and their locations aredetermined based on the attenuation requirement over the stop-band. Using (1)–(3), we can find the locations of the poles withgiven transmission zeros, return loss, and the number of poles.Poles of the filter are found to be located at

. Since band edge frequencies of the dual-passbandfilter are GHz, GHz, GHz,and GHz, we have from (13). Fromthe frequency transformation in (7), the poles and zeros of thedual-passband filter at finite frequencies in domain can bedetermined as follows:

(14)

Fig. 3 shows the frequency response of the dual-passbandfilter with poles and zeros in (14). With the given poles and zerosof the dual-passband filter in the domain, we can obtain thecharacteristic function in the form given by (5). Expanded formof the transfer function can be expressed as follows [4]:

(15)

where

(16)

1166 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Fig. 3. Frequency response of the dual-passband filter in the domain.

From the transfer function in the domain, the coupling ma-trix can be determined [3]. Since the eight-pole dual-passbandfilter has six zeros in finite frequencies, it can be of a canon-ical structure and its coupling matrix is given by (17), shownat the bottom of this page. The matrix similarity transformationcan be applied to the coupling matrix to obtain different com-binations of positive and negative inter-resonator coupling co-efficients and/or different coupling structures (i.e., asymmetriccanonical structure) [13], [14].

Applying the frequency transformation in (10), we can ob-tain the frequency response of the filter in the domain. Fig. 4shows the frequency response of the filter. Note that the pass-bands of the filter are 3.90–3.95 and 4.05–4.10 GHz and the at-tenuation at the stopbands is 30 dB, which shows the validity ofthis proposed synthesis method for a dual-passband filter usingfrequency transformation with prescribed passbands and atten-uation at stopbands.

The coupling matrix in (17) can be realized for many typesof filter structures. In this paper, we use a stripline structure forthe filter. Fig. 5 shows the conductor layer of the dual-passbandfilter. This conductor layer is positioned in the middle of twometal-backed dielectric layers. The thickness, dielectric con-stant, and loss tangent of the dielectric layers are 1.574 mm,2.2, and 0.0009, respectively. Open-loop resonators can pro-vide both the electric and magnetic couplings between two res-onators and these can be used for realizing both positive andnegative coupling, as required by (17). The tapping position

Fig. 4. Frequency response of the dual-passband filter in the ! domain.

Fig. 5. Conductor layer of stripline structure for the dual-passband filter.

is determined by the external coupling coefficientand the distances between two resonators are determined byinter-resonator coupling coefficients . The design band-width is . The coupling coefficients can be calculatedby the well-known method described in [15] and, hence, is notrepeated in this paper. The full-wave electromagnetic simulatorZeland IE3D is used to calculate the coupling coefficients. Thephysical dimensions of the filter in Fig. 5 are summarized inTable I.

(17)

LEE AND SARABANDI: SYNTHESIS METHOD FOR DUAL-PASSBAND MICROWAVE FILTERS 1167

TABLE IPHYSICAL DIMENSIONS OF THE DUAL-PASSBAND FILTER

Fig. 6. Synthesized and simulated frequency response of the dual-passbandfilter.

The filter has also been simulated using IE3D and Fig. 6compares the simulated and theoretical frequency response ofthe dual-passband filter. Since the filter synthesis does not takeinto account the losses of the filter, the loss factors are not in-cluded in the simulated response in Fig. 6 for clear compar-ison. A good agreement between theoretical and simulated fre-quency responses is shown. The simulated frequency responsehas somewhat lower attenuation at a higher stopband, which isdue to asymmetric locations of transmission zeros. It has beenreported that the asymmetric frequency response is attributedto the frequency-dependent couplings [16]. The measured re-sponse of the fabricated filter (Fig. 7) is compared to the simu-lation one in Fig. 8. The simulated response includes loss factors(conductor loss and dielectric loss) in order to take into accountthe losses of the fabricated filter. Measured response shows areasonably good agreement with the simulated response. Thereis, however, a small frequency shift, which can be attributed tothe fabrication error.

IV. MORE EXAMPLES

In Section III, we dealt with an eight-pole dual-passband filterwith repeated transmission zeros at . Here, a four-poledual-passband filter with repeated transmission zeros at

Fig. 7. Fabricated conductor layer of the stripline structure for the dual-pass-band filter.

Fig. 8. Frequency response of the dual-passband filter. (a) S . (b) S .

and an eight-pole dual-passband filter with no transmissionzeros at are briefly discussed.

Fig. 9 shows a two-pole low-pass prototype filter with notransmission zeros at finite frequencies in the domain. The

1168 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Fig. 9. Frequency response of a low-pass filter prototype with no transmissionzeros at finite frequencies.

return loss and number of the poles of the low-pass prototypefilter determines the location of the poles. Using the frequencytransformation in (7) with , we can obtain the loca-tion of the poles and zeros of the four-pole dual-passband filterin the domain and, therefore, its frequency response can beobtained as shown in Fig. 10. Based on the locations of the polesand zeros in the domain, the transfer function and couplingmatrix can be obtained as described in Section III. The transferfunction is

(18)

where

(19)

and the coupling matrix is

(20)

The filter with no transmission zeros at can alsobe synthesized. This kind of filter can be designed with thelow-pass prototype filter having no transmission zeros at infi-nite frequencies in the domain. Fig. 11 shows a four-polelow-pass prototype filter with the transmission zeros atand . Using the frequency transformation in (7), we canobtain the locations of the poles and zeros of the dual-passbandfilter in the domain and, therefore, its frequency response canbe obtained as shown in Fig. 12. Based on the locations of thepoles and zeros in the domain, the transfer function and cou-pling matrix can be obtained as described in Section III. Since

Fig. 10. Frequency response of a dual-passband filter whose low-pass proto-type is shown in Fig. 9.

Fig. 11. Frequency response of a low-pass filter prototype with no transmissionzeros at infinite frequencies.

Fig. 12. Frequency response of a dual-passband filter whose low-pass proto-type is shown in Fig. 11.

the eight-pole filter has eight zeros in finite frequencies, it mightemploy the coupling between source and load.

V. ASYMMETRIC DUAL-PASSBAND FILTER DESIGN

We have dealt with a synthesis method for designing sym-metric dual-passband filters. The frequency transformation for

LEE AND SARABANDI: SYNTHESIS METHOD FOR DUAL-PASSBAND MICROWAVE FILTERS 1169

Fig. 13. Frequency transformation from the domain to the domain using(21).

Fig. 14. Frequency responses of asymmetric dual-passband filters in the domain. (a) = 0:1; = 0:7; = 0:5. (b) = �0:8; =

0:2; = �0:3.

asymmetric dual-passband filters is given in [10], and it is pos-sible to obtain two desired passband characteristics. However, itis not clear whether equiripple responses both in the passbandand stopband, which enables high-frequency selectivity, can orcannot be achieved.

Here, we briefly explain the frequency transformation forasymmetric dual-passband filters. The advantage of this fre-quency transformation is that the attenuation characteristics ofthe low-pass prototype is preserved. Therefore, it is possible to

obtain equiripple responses both in the passband and stopband.Fig. 13 shows the frequency transformation from the domainto the domain. In the domain, we have asymmetricpassbands. The frequency transformation shown in Fig. 13 canbe obtained by rewriting (7) in general form as follows:

for

for (21)

where and are determined by the band edge fre-quencies in the domain, which are also determined by ar-bitrarily prescribed passbands of the filter. The upper frequencyregion and the lower one is bisected by , which can be chosenarbitrarily between and .

Fig. 14 shows the frequency response in the domain usingthe frequency transformation in (21) and a low-pass filter proto-type in Fig. 2. It should be noted that the frequency transforma-tion in (21) is very flexible in designing asymmetric dual-pass-band filters while preserving the attenuation characteristics ofthe low-pass prototype.

Based on the frequency transformation given in (21), the lo-cations of the poles and zeros can be found in the domain,which makes it possible to obtain the transfer function and cou-pling matrix.

VI. CONCLUSION

This paper has described a synthesis method for a symmetricdual-passband filter. Frequency transformations have been es-tablished and applied to the low-pass filter prototype in orderto obtain the frequency response of the dual-passband filter.For analytic filter synthesis, the frequency transformations havebeen given in terms of the prescribed passbands of the dual-pass-band filter.

To validate the presented synthesis method, the eight-poledual-passband filter with passbands of 3.90–3.95 and4.05–4.10 GHz has been designed and measured. The fre-quency response of the designed filter has shown a goodagreement with the synthesized frequency response.

The frequency transformation for symmetric dual-passbandfilters has been generalized for asymmetric dual-passband fil-ters. This transformation is found to be flexible enough to allowfor designing bandpass filters with two passbands of signifi-cantly different bandwidths.

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Juseop Lee (A’02–M’03) received the B.E. and M.E.degrees in radio science and engineering from KoreaUniversity, Seoul, Korea, in 1997 and 1999, respec-tively, and is currently working toward the Ph.D. de-gree at The University of Michigan at Ann Arbor.

In 1999, he joined LG Electronics (formerlyLG information and Communications), where hisresearch activities included reliability analysis ofRF components for code-division multiple-access(CDMA) cellular systems. In 2001, he joined Elec-tronics and Telecommunications Research Institute

(ETRI), where he was involved in designing passive microwave equipment forKu- and Ka-band communications satellites. In 2005, he joined The Univer-sity of Michigan at Ann Arbor, where he is currently a Research Assistant withthe Radiation Laboratory. His research interests include RF and microwavecomponents, satellite transponders, and electromagnetic theories.

Kamal Saranbandi (S’87–M’90–SM’92–F’00) re-ceived the B.S. degree in electrical engineering fromSharif University of Technology, Tehran, Iran, in1980, and the M.S. degree in electrical engineeringand the M.S. degree in mathematics and Ph.D.degree in electrical engineering from The Universityof Michigan at Ann Arbor, in 1986, 1989, and 1989,respectively.

He is currently Director of the Radiation Lab-oratory and a Professor with the Department ofElectrical Engineering and Computer Science, The

University of Michigan at Ann Arbor. He possesses 22 years of experiencewith wave propagation in random media, communication channel modeling,microwave sensors, and radar systems, and is leading a large research groupincluding two research scientists, 12 doctoral students, and two mastersstudents. He has graduated 24 doctoral students and has supervised numerouspostdoctoral students. He has served as the Principal Investigator on numerousprojects sponsored by the National Aeronautics and Space Administration(NASA), Jet Propulsion Laboratory (JPL), Army Research Office (ARO),Office of Naval Research (ONR), Army Research Laboratory (ARL), NationalScience Foundation (NSF), Defence Advanced Research Projects Agency(DARPA), and numerous industries. He has authored or coauthored numerousbook chapters and over 145 papers in refereed journals on miniaturized andon-chip antennas, metamaterials, electromagnetic scattering, wireless channelmodeling, random media modeling, microwave measurement techniques,radar calibration, inverse scattering problems, and microwave sensors. He alsohas had over 340 papers and invited presentations in numerous national andinternational conferences and symposia on similar subjects. He is listed inAmerican Men and Women of Science, Who’s Who in America, and Who’sWho in Science and Engineering. His research areas include microwave andmillimeter-wave radar remote sensing, metamaterials, electromagnetic wavepropagation, and antenna miniaturization.

Dr. Sarabandi is a member of the NASA Advisory Council appointed bythe NASA Administrator. He has also served as a vice president of the IEEEGeoscience and Remote Sensing Society (GRSS) and as a member of the IEEETechnical Activities Board Awards Committee. He serves as an associate editorfor PROCEEDINGS OF THE IEEE and has served as an associate editor for theIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE SENSORS

JOURNAL. He is also a member of Commissions F and D of URSI. He was therecipient of the Henry Russel Award presented by the Regent of The Universityof Michigan at Ann Arbor. He was the recipient of the 1999 German AmericanAcademic Council (GAAC) Distinguished Lecturer Award presented by theGerman Federal Ministry for Education, Science, and Technology, which isgiven to approximately ten individuals worldwide in all areas of engineering,science, medicine, and law. He was a recipient of a 1996 Electrical Engineeringand Computer Science (EECS) Department Teaching Excellence Award and a2004 College of Engineering Research Excellence Award. He was a recipientof the IEEE Geoscience and Remote Sensing Distinguished AchievementAward and The University of Michigan at Ann Arbor Faculty RecognitionAward, both in 2005. He was also a recipient of the Best Paper Award presentedat the 2006 Army Science Conference. Over the past several years, jointpapers presented by his students at numerous international symposia (IEEEAPS’95,’97,’00,’01,’03,’05,’06; IEEE IGARSS’99,’02, IEEE MTT-S IMS’01,USNC URSI’04,’05,’06, AMTA’06) have received Student Paper Awards.