2394 ieee transactions on microwave theory and … · and transmission-line all-pass networks [12],...

2394 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 8, AUGUST 2012

Synthesis of Narrowband Reflection-Type PhasersWith Arbitrary Prescribed Group Delay

Qingfeng Zhang, Member, IEEE, Shulabh Gupta, Student Member, IEEE, and Christophe Caloz, Fellow, IEEE

Abstract—An exact closed-form synthesis method is proposedfor the design of narrowband reflection-type (mono-port) phaserswith arbitrary prescribed group-delay responses. The proposedsynthesis technique consists in three steps. First, it transforms thephase problem from the bandpass domain to the low-pass domainusing a one-port ladder network, where a mathematical synthesisis performed via a Hurwitz polynomial. Second, it transforms thesynthesized low-pass network back to the bandpass domain for im-plementation in a specific technology. Third, it uses an iterativepost-distortion correction technique to compensate for distributedeffects over the broader bandwidth required. The proposed syn-thesis method is verified by both full-wave analysis and experimentwhere the synthesized bandpass network is realized in an iris-cou-pled waveguide configuration.

Index Terms—Analog signal processing, dispersive delay struc-ture (DDS), group-delay engineering, Hurwitz polynomial, phaser.

I. INTRODUCTION

T HE demand for communication and sensor systemswith ever-increasing data throughput and reliability

has spurred considerable interest for new ultra-widebandtechnologies. In particular, analog signal-processing tech-niques [1], little exploited thus far at microwaves, have beenidentified as attractive real-time alternatives to purely digitalsignal-processing techniques for detecting and monitoringultra-wideband microwave signals. Digital approaches aremost appropriate at low frequencies, where their great flex-ibility, compact size, low cost, and high reliability provideclear benefits. However, at higher frequencies, they sufferfrom reduced performance, excessive power consumption, andhigh cost due to analog-to-digital (A/D) and digital-to-analog(D/A) converters. Analog devices and systems are thereforeincreasingly attractive at these frequencies. Recently, reportedapplications in this area include analog real-time spectrumanalyzers for the characterization and monitoring of complexnonstationary signals [2], [3], tunable impulse delay lines [4],

Manuscript received February 09, 2012; accepted April 21, 2012. Date ofpublication June 11, 2012; date of current version July 30, 2012. This workwas supported by the Natural Sciences and Engineering Research Council ofCanada (NSERC) under Grant CRDPJ 402801-10 in partnership with Researchin Motion (RIM).Q. Zhang and C. Caloz are with the Department of Electrical Engineering,

PolyGrames Research Center, École Polytechnique de Montréal, Montréal, QC,Canada H3T 1J4 (e-mail: [email protected]).S. Gupta was with the Department of Electrical Engineering, PolyGrames

Research Center, École Polytechnique deMontréal, Montréal, QC, Canada H3T1J4 . He is now with the Department of Electrical Computer and Energy Engi-neering, University of Colorado at Boulder, Boulder, CO 80309–0425 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMTT.2012.2198486

Fig. 1. Reflection-type DDS system using a one-port to two-port conversionmechanism appended to the phaser. (a) Using a circulator. (b) Using a 3-dBhybrid coupler.

compressive receivers [5], real-time Fourier transformers [6],and inverse Fourier transformers [7].The core of an analog real-time signal processor is a disper-

sive delay structure (DDS), which is a component providing adesired group-delay response versus frequency. A DDS may betransmission-type or reflection-type.Transmission-type DDSs include surface-acoustic-wave

(SAW) devices [8], magnetostatic-wave devices [9], multi-section coupler-based super-conductive delay lines [10], [11],and transmission-line all-pass networks [12], [13]. Thus far, toour knowledge, no systematic synthesis method is available todesign arbitrarily prescribed group-delay responses in trans-mission-type DDSs. Reflection-type DDSs may be realized bycombining a one-port phaser with a circulator or two phaserswith a 3-dB hybrid coupler, as illustrated in Fig. 1. The responseof the resulting two-port DDS system is thus composed of thesum of the group-delay response of the circulator or 3-dBhybrid and the phaser. Since the dispersion of the componentsexternal to the phaser, the circulator, or the hybrid coupler, canbe accounted for in the group-delay response of the phaser, thegroup-delay synthesis can be restricted to the phaser part ofthe DDS system. Compared with transmission-type DDSs, thereflection-type DDS, with a one-port phaser, is easier to syn-thesize because the reflection coefficient of a one-port phaser,neglecting losses in the first approximation, is unity. One-portphasers typically include Bragg gratings at optical frequencies[14] and chirped dispersive delay lines at microwave frequen-cies [15]–[17]. In particular, microwave chirped delay-linestructures are realized using spatial impedance profiles, but thisapproach is restricted to specific kinds of implementation [16].A coupled-resonators network approach is more general forthe design of phasers. Such an approach was first proposed in[18] and later improved in [19]. However, the correspondingsynthesis techniques reported to date have been based onoptimization procedures.This paper presents an exact synthesis technique for realizing

one-port phasers with an arbitrary prescribed group-delay re-sponse versus frequency over a specified bandwidth. The pro-posed technique is based on synthesizing a generalized one-

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ZHANG et al.: SYNTHESIS OF NARROWBAND REFLECTION-TYPE PHASERS WITH ARBITRARY PRESCRIBED GROUP DELAY 2395

Fig. 2. Equivalent circuit representation for a one-port phaser. (a) Distributed-element bandpass network. (b) Lumped-element low-pass network with K-in-verters. (c) Lumped-element LC ladder network.

port low-pass ladder network obtained by transforming the pre-scribed group delay response to the low-pass domain. The pa-rameters of the elements of the resulting lowpass ladder networkare next computed using an efficient iterative polynomial gener-ation procedure, and the synthesis results are then transformedback to the bandpass domain to achieve the desired response. Incontrast to conventional optimization techniques, the proposedsynthesis is exact and closed-form for narrowband phaser de-signs. An iterative post-distortion correction technique is furtherproposed for broader bandwidths.This paper is organized as follows. Section II presents the

synthesis theory for realizing phasers with arbitrary prescribedgroup-delay response in a narrowband frequency range andprovides a narrowband design example. Section III extends thesynthesis using the proposed post-distortion correction tech-nique for broader bandwidths. Several illustrative exampleswith broader bandwidths using both ideal circuit and wave-guide implementations are provided. Section IV provides anexperimental validation. Finally the conclusions are providedin Section VI.

II. SYNTHESIS THEORY

A one-port phaser can be modeled by a short-circuiteddistributed-element bandpass network [20], as shown inFig. 2(a). The objective is to synthesize an arbitrary pre-scribed phase versus frequency response in the frequencyrange extending from to , as shown in Fig. 3(a), usingthe distributed-element bandpass network of Fig. 2(a). Theprinciple of the proposed synthesis technique is as follows.First, the original bandpass phase function of Fig. 3(a)is transformed into the low-pass phase function ofFig. 3(b) through a mapping function , illustrated inFig. 3(c), which is derived from the transformation betweenthe distributed-element bandpass network of Fig. 2(a) and thelumped-element low-pass network of Fig. 2(b). This network isnext transformed into the low-pass LC ladder network shownin Fig. 2(c) so that the phase problem of the low-pass networkof Fig. 2(b) simplifies to a problem expressed in terms of a Hur-

Fig. 3. Transformation of the bandpass phase function into the low-passphase function through a mapping function . (a) Phaseversus bandpass frequency function to synthesize. (b) Phase versus

low-pass frequency function used as an auxiliary step for the synthesis of(a). (c) Bandpass frequency to low-pass frequency mapping function

. (d) Low-pass frequency identity function.

witz polynomial.1 In essence, the proposed synthesis techniqueconsists in transforming the phase problem of the bandpassdomain into the low-pass domainproblem, performing mathematical synthesis via a Hurwitzpolynomial in the low-pass domain, and then transforming theresult back to the bandpass domain for implementation. Thevarious steps of the synthesis procedure are summarized inFig. 4 and will be established in the forthcoming sections.

A. Bandpass to Low-Pass Transformation

The one-port phaser is represented by the equivalent circuit inFig. 2(a), where the distributed reactances are usually realizedby half-wavelength transmission lines. It is generally the groupdelay that is specified in analog signal-processing applica-tions, and the corresponding phase is then obtained from

as

(1)

where is an arbitrary phase constant. As shown in Fig. 3, thelow-pass phase function is obtained from the bandpassphase function via a mapping function

(2)

and it may be expressed as

(3)

1A Hurwitz polynomial is a polynomial whose coefficients are positive realnumbers or, equivalently, whose zeros are located in the left half-plane of thecomplex Laplace plane [21], [22]. The ratio of the even part to the odd part of aHurwitz polynomial can be expanded as a continued fraction whose coefficientsare all positive.


Fig. 4. Flowchart for the proposed phaser synthesis procedure.

The networks of Fig. 2(a) and (b) differ only in terms of theelements interconnecting the -inverters. Their bandpass-to-low-pass transformation reads

(4)

and is subject to the conditions

(5a)

(5b)

where represents the reactances in Fig. 2(a) and repre-sents the inductances in Fig. 2(b). The ratio of (4) to (5b) yieldsthen the required mapping function

(6)

where it is noted that the mapping function depends on thespecific implementation of the reactance functions.The distributed resonators in Fig. 2(a) are typically realized

by transmission lines, which can be approximated by series re-actances [23] when the length of the transmission lines is closeto half the guided wavelength. The corresponding series reac-tance is

(7)

where , , and are the characteristics impedance, length,and guided wavelength of the th transmission lines, respec-tively. Inserting (7) into (5a), we obtain

(8)

which, upon substitution into (7), yields

(9)

It should be noted from (8) that the length of the transmissionline is half the guided wavelength at the lowest frequency of the

specified bandwidth, . Inserting (9) into (5b), we obtain theequivalent inductance

(10)

and inserting (9) into (6) yields the mapping function

(11)

which is to be used in Fig. 3(c) to provide the transformed low-pass phase function of Fig. 3(b).The low-pass network of Fig. 2(b) can be further transformed

into the LC ladder network of Fig. 2(c), which is a simplifiedlow-pass network without K-inverters [20], [24]. The LC laddernetwork of Fig. 2(c) is employed because it provides a directconnection to a Hurwitz polynomial. The corresponding nor-malized input impedance may be written

. . .

(12)

where is the complex frequency, and and arethe even and odd parts of a Hurwitz polynomial [21], [22],respectively. The corresponding reflection coefficient expressedin terms of this impedance reads

(13)

Since the magnitude of the Hurwitz polynomial is aneven function and its phase is an odd function (becausethe roots of a Hurwitz polynomial are distributed as conjugatepairs in the complex Laplace plane), we may write in the low-pass domain

(14)


Assuming the ladder network is lossless, as conventionally donein the magnitude synthesis of filters, we have at the same time

(15)

Comparing (15) and (14) indicates that the phase of the Hurwitzpolynomial is related to the phase of the ladder network by thesimple relation

(16)

At this point, we need to generate the Hurwitz polynomial withthe calculated phase .

B. Generation of Arbitrary-Phase Hurwitz Polynomialin the Low-Pass Domain

An arbitrary-phase polynomial, whose phase is zero at theorigin (as required from the fact that the phase of a Hurwitzpolynomial is an odd function) and exhibits specified valuesat a given set of frequencies, can be generated using the recur-rence procedure presented in [25]. First, one specifies a set offrequency points and calculates the corresponding prescribedphase values for the Hurwitz polynomial using (16). Let

and , where, , and are the frequency and phase

sets. The condition can be satisfied by settingin (1). Then, we can generate the th-order polynomialcorresponding to this phase using the recurrence formula [25]

(17)

where

. . .

(18)

for and

(19)

To apply this formula, one first computes the coefficients ,from to , using (18) and (19), and next obtainsthe polynomials in (17). These polynomials correspondto the function in (14) with the order . We can next buildthe input impedance of the ladder network by taking theratio of the odd part over the even part of ,according to (12). The lumped-element values in the laddernetwork of Fig. 2(c) are then straightforwardly identified afterwriting the result in the form of a continued fraction expansionas in (12).

C. Low-Pass-to-Bandpass Transformation

Once the lumped-element parameters in Fig. 2(c) havebeen determined, we can transform the LC ladder network backto the distributed-element bandpass network of Fig. 2(a) via the

Fig. 5. H-plane-iris-coupled waveguide reflection-type phaser: (a) structureand (b) full-wave results.

lumped-element low-pass network of Fig. 2(b). Using the equiv-alence condition between the -inverters of Fig. 2(b) and theparameters of Fig. 2(c), as well as the equivalent inductancesin (10), the -inverters in Fig. 2(a) are found as

(20a)

(20b)

(20c)

where . These formulas are the same as the well-known formulas for two-port networks [20], [24], except for thelast inverter , since the one-port network in Fig. 2(b) isterminated by an inductance instead of a resistor as intwo-port networks. The whole synthesis procedure is summa-rized in Fig. 4.

D. Design Example

To verify the proposed synthesis technique, an H-plane-iris-coupled waveguide phaser, shown in Fig. 5(a), is designed forlinear group delay with a swing of 10 ns. A WR90 (22.86 mm10.16 mm) waveguide is chosen as the housing waveguide,


Fig. 6. Illustration of the procedure providing the new mapping functionin the proposed correction technique using (a) the bandpass phase curve

and (b) the corresponding low-pass phase curve .

and the thickness of all of the irises is set to 1.5 mm. The di-mensions of the waveguide phaser are listed in Table II. Thefull-wave response of the resulting phaser (computed using themode-matching commercial software Mician ) is shownin Fig. 5(b) within the frequency range from 10 to 10.05 GHz(0.5% fractional bandwidth). The response of the synthesizedstructure closely follows the prescribed group-delay response.The maximum error between the realized and prescribed re-sponses is approximately 2%,which is acceptable for most prac-tical applications.

III. POSTDISTORTION TECHNIQUE FOR BROADER BANDWIDTHS

A. Theory

The synthesis method presented in the previous section workswell for bandwidths below 1%. In the case of broader band-widths, however, the realized group-delay response is degradeddue to the imperfect modeling of the ideal circuit network ofFig. 2(a) by practical microwave structures. Specifically, thedegradation is due to the two following factors. Firstly, a half-wavelength transmission line, which is really a -network [23],cannot be modeled by a series reactance in a wide frequencyband. Secondly, the ideal -inverters in Fig. 2(a) cannot be real-ized by practical microwave structures, which unavoidably ex-hibit dispersion over a wide frequency range. While these twoaforementioned factors generally restrict the fractional band-width to less than 10% for conventional magnitude filter designs[20], [24], they restrict the bandwidth to much smaller ranges,typically of less than 1%, on phasers, due to the high sensi-tivity of the group delay to the approximations involved. For thisreason, this section provides an iterative correction procedure.Let us reconsider a prescribed group-delay response

over the bandwidth . The corresponding phase, obtained by (1), is first transformed to the low-pass do-

main using the initial mapping function (11). This phasefunction is next used to generate the Hurwitz polynomial forconstructing the low-pass ladder network of Fig. 2(c) with acorresponding phase that we shall note . The low-passladder network is then transformed back into the bandpass net-work of Fig. 2(a) using (20). The corresponding reactance func-tions and -inverters are finally implemented using practicalmicrowave structures. Due to distributed effects, the synthe-sized phase includes a phase error

Fig. 7. Flowchart of the correction procedure.

. In the narrowband designs , this error is gener-ally negligible, and the synthesis thus ends here. In the designswith wider bandwidths, however, the error between the realizedphase and the prescribed phase is large and possiblyunacceptable.Due to the aforementioned degraded factors, the mapping

function (11) derived based on the bandpass network of Fig. 2(a)cannot be used. Therefore, we develop here a new mappingfunction, which takes into account the nonidealities of the band-pass network of Fig. 2(a), to correct the degraded bandpassphase. It is derived based on the degraded bandpass phase func-tion and low-pass phase function , which can bemathematically represented by the auxiliary functions and

, respectively, depicted in Fig. 6 and generated as

(21)

(22)

In practice, these auxiliary functions can be obtained nu-merically using interpolation or curve-fitting methods [26].The bandpass phase corresponding to each value is

, as shown in Fig. 6(a). This phase value inthe low-pass domain is related to the low-pass frequencythrough , i.e., . Therefore, using the auxiliaryfunctions and , the value of corresponding toeach can be calculated, and the desired mapping function

can be built sequentially. This can be mathemat-ically expressed as , which may bewritten as

(23)

The procedure of Fig. 2 is repeated using the newmapping func-tion to construct a new bandpass network, which resultsin a refined phase response , with the corresponding phaseerror . If is less than the acceptableerror , the synthesis is complete, otherwise a second iteration ismade, and so on. At each th iteration, a new mapping function


Fig. 8. Group-delay responses of different phasers based on ideal circuits. (a) Sixth-order, 1–1.05 GHz. (b) Tenth-order, 1–1.05 GHz. (c) 10-degree, 20–20.1 GHz.

TABLE ICALCULATED -PARAMETER FOR THE RESPONSE IN FIG. 8

is computed based on the low-pass phaseand the bandpass phase , using (21)–(23), until the de-sired accuracy is achieved, i.e., . The overall iterativeprocess is summarized in Fig. 7.

B. Design Examples

Fig. 8 shows three examples of linear-slope group-delay re-sponses based on the circuit model of Fig. 2(a) with ideal -in-verters. The first two examples are for a swing of 10ns within the frequency band from 1 to 1.05 GHz (5% frac-tional bandwidth). The third example is high-frequency designexhibiting a swing of 1.2 ns from 20 to 21 GHz (5% frac-tional bandwidth). The corresponding calculated parameters ofthe -inverter are listed in Table I.The figures compare the synthesized responses before and

after the group-delay correction. A large distortion is seen in the

group-delay response before correction, which indicates that thegroup delay is very sensitive to the narrowband approximationof half-wavelength transmission lines. Furthermore, this distor-tion increases with increasing order of the phaser, as seen bycomparing Fig. 8(a) and (b). Using the postdistortion techniqueof Fig. 7, the distortions in the realized group-delay responsesare dramatically reduced, leading to an enhanced-bandwidth de-sign reaching here 5% (about more than for the purelyclosed-form synthesis of Section II).A waveguide implementation with the same configuration as

in Fig. 5 is also provided to verify the proposed postdistortiontechnique. The dimensions of the waveguide phaser are listed inTable II. The full-wave response of the resulting phaser (com-puted using the mode-matching commercial software Mician

) is shown in Fig. 9. It can be seen that the realizedgroup-delay curve after the post-distortion technique has been


TABLE IIDIMENSION OF THE WAVEGUIDE PHASER IN FIG. 10 (UNIT: MM)

Fig. 9. Full-wave comparison of the group delay before and after correction.

Fig. 10. Photograph of the fabricated prototype with round corners.

significantly improvedwith amaximum error about 2%, therebyillustrating the proposed postdistortion technique.

IV. EXPERIMENTAL RESULTS

Awaveguide prototype, as shown in Fig. 10, is fabricated cor-responding to the response in Fig. 9. Whereas in the ideal designthe irises are sharply configured with 90 corners, in the fabri-cated prototype they are connected to the sides of the waveguidewith rounded corners (with radius of 3 mm), due to fabricationconstraints. The modified dimensions, taking this effect into ac-count, are listed in Table II. The fabricated prototype is mea-sured with a vector network analyzer using offset short-short-load (SSL) calibration. The measured -parameter response is

Fig. 11. Simulated and measured response for the waveguide phaser of Fig. 10.

shown in Fig. 11 and is compared with the response of theideal bandpass circuit of Fig. 2(a) and the full-wave responseobtained using the commercial software FEM-HFSS. Althoughthe agreement of the measured response with the prescribed re-sponse is acceptable, discrepancies are observed. Consideringthe good agreement in Fig. 9 between the synthesized and pre-scribed responses for the case of irises without corners, the dis-crepancy in Fig. 11 between the full-wave analysis and the idealcircuit is attributed to the imperfect modeling of the roundedcorners in the waveguide. Moreover, a discrepancy is observedbetween experimental and full-wave results. In order to find itscause, a comparison over a larger frequency range is shown inFig. 13. It is noted that the peak of the experimental responseis slightly shifted, by 0.025 GHz (about 0.25%), below the full-wave curve. This may be due to over-drilling of rounded cornersin the fabrication process, as suggested by the correspondinggreen curve of full-wave results in Fig. 13, with 2.7-mm-radiusinstead of 3.0-mm-radius rounded corners (within the precisionof the drilling machine).

V. DISCUSSION

The proposed postdistortion technique can be used to im-prove the response obtained by synthesis (Section II) forbandwidths extended beyond the synthesis bandwidth. How-ever, it cannot be applied to bandwidths larger than 5%–7%,according to empirical observations. The postdistortion tech-nique (Section III) consists in introducing new mappingfunction to take into account the degrading factorsover a bandwidth greater than the initial one (Section II).This function is initially obtained from the low-pass phase


Fig. 12. Comparison of experimental and full-wave results for the waveguidephaser of Fig. 10 over an extended frequency range.

Fig. 13. Convergence (error ) of the postdistortion technique for phasers withthe different bandwidths indicated in the legend.

and degraded bandpass phase responses and substituted to theoriginal mapping function in the synthesis (Section II), yieldingnew -parameters for the low-pass and bandpass networks.This substitution procedure is repeated iteratively with a newfunction at each iteration step until convergence hasbeen reached. However, this iterative procedure might fail toconverge in very broadband designs. To illustrate this, Fig. 13shows the convergence of three designs with different band-widths. This convergence is expressed in terms of the error

(24)

where and indicate the obtained and prescribedgroup-delay responses, respectively. It is noted that the design

procedure converges quickly for the designs with 5.0% and6.0% bandwidths, while it fails to converge for the design with7.3% bandwidth.

VI. CONCLUSION

An exact closed-form synthesis method has been proposedfor the design of reflection-type phasers with an arbitraryprescribed group-delay response. The proposed synthesistechnique consists of transforming the phase problem in thebandpass domain into the low-pass domain, performing math-ematical synthesis via a Hurwitz polynomial in the low-passdomain, and then transforming the result back to the bandpassdomain for implementation. In addition, an iterative correctiontechnique has also been proposed to extend the proposed syn-thesis method to broader bandwidths. Several examples basedon both the ideal circuit and the waveguide implementationshas been presented to verify the proposed synthesis method,where the realized group-delay responses show a good agree-ment with the prescribed group delays.

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Qingfeng Zhang (S’07–M’11) was born in De-cember 1984, in Changzhou, China. He receivedthe B.E. degree in electrical engineering from theUniversity of Science and Technology of China(USTC), Hefei, China, in 2007, and the Ph.D.degree in electrical and electronic engineeringfrom Nanyang Technology University, Singapore,in 2010. His dissertation focused on dimensionalsynthesis of wide-band waveguide filters withoutglobal optimization.Since April 2011, he has been a Postdoctoral

Fellow with Poly-Grames Microwave Research Center, École Polytechniquede Montréal, Montréal, QC, Canada. His current research interests includefilter synthesis, dispersive delay structures, analog signal-processing systems,and leaky-wave antennas.

Shulabh Gupta (S’09) was born on December14, 1982, in Etah, India. He received the B.Tech.degree in electronics from the Indian School ofMines, Dhanbad, India, in 2004, the M.S. degree intelecommunications from INRS-EMT, Universitédu Québec, Montréal, QC, Canada, in 2006, and thePh.D. degree in electrical engineering from the ÉcolePolytechnique de Montréal, Montréal, Montréal,QC, Canada. His M.S. thesis research concernedoptical signal processing related to the propagationof light in linear and nonlinear optical fibers and

fiber Bragg gratings. His Ph.D. research was about the analog signal processingtechniques using dispersion engineered structures.From December 2009 to May 2010, he was a Visiting Research Fellow with

the Tokyo Institute of Technology, Tokyo, Japan, where he was involved withthe application of artificial magnetic surfaces for oversized slotted waveguideantennas. He is currently a Postdoctoral Fellow with the University of Col-orado at Boulder. His current research interests are high-power ultrawidebandantennas, traveling-wave antennas, dispersion engineered structures for UWBsystems and devices, nonlinear effects, and Fourier optics inspired leaky-wavestructures and systems.Mr. Gupta was a recipient of the Young Scientist Award of EMTS Ottawa’07,

URSI-GA, Chicago’08 and ISAP Jeju’11. He was the finalist in the Most Cre-ative and Original Measurements Setup or Procedure Contest of the 2008 IEEEMicrowave Theory and Techniques Society (IEEE MTT-S) International Mi-crowave Symposium, Atlanta, GA, in 2008.

Christophe Caloz (S’00–A’00–M’03–SM’06–F’10)received the Diplôme d’Ingénieur en Électricité andPh.D. degree from École Polytechnique Fédérale deLausanne (EPFL), Lausanne, Switzerland, in 1995and 2000, respectively.From 2001 to 2004, he was a Postdoctoral Re-

search Engineer with the Microwave ElectronicsLaboratory, University of California at Los Angeles(UCLA). In June 2004, he joined École Poly-technique de Montréal, Montréal, QC, Canada,where he is now a Full Professor, a member of

the Poly-Grames Microwave Research Center, and the holder of a CanadaResearch Chair (CRC). He has authored and coauthored over 420 technicalconference, letter and journal papers, 12 books and book chapters, and heholds several patents. His research interests include all fields of theoretical,computational and technological electromagnetics engineering, with strongemphasis on emergent and multidisciplinary topics, including particularlynanoelectromagnetics.Dr. Caloz is a member of the IEEE Microwave Theory and Techniques

Society (MTT-S) Technical Committees MTT-15 (Microwave Field Theory)and MTT-25 (RF Nanotechnology), a Speaker of the MTT-15 Speaker Bureau,the Chair of the Commission D (Electronics and Photonics) of the CanadianUnion de Radio Science Internationale (URSI) and an MTT-S representativeat the IEEE Nanotechnology Council (NTC). He received several awards,including the UCLA Chancellor’s Award for Post-doctoral Research in 2004,the IEEE MTT-S Outstanding Young Engineer Award in 2007, and many bestpaper awards with his students.

2394 ieee transactions on microwave theory and … · and transmission-line all-pass networks [12],...

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