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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013 1043

Synthesis of Cross-Coupled Reduced-OrderDispersive Delay Structures (DDSs) With Arbitrary

Group Delay and Controlled MagnitudeQingfeng Zhang, Member, IEEE, Dimitrios L. Sounas, Member, IEEE, and Christophe Caloz, Fellow, IEEE

Abstract—For the first time, a systematic synthesis method forcross-coupled dispersive delay structures (DDSs) with controlledmagnitude for analog signal processing applications is proposed.In this method, the transfer function is synthesized using a poly-nomial expansion approach, which allows to separately controlthe magnitude and group-delay response of the DDS. The synthe-sized transfer function also features a reduced order—namely,half—compared to that of previously reported synthesis tech-niques for linear-phase filters. Once it has been constructed, thetransfer function is transferred into coupling matrices that can beimplemented in arbitrary cross-coupled-resonator technologies.Several design examples are provided for different prescribedgroup-delay responses. An experimental waveguide prototype isdemonstrated. The agreement between the measured and pre-scribed responses illustrates the proposed synthesis method.

Index Terms—Analog signal processing (ASP), cross-coupled,dispersive delay structures (DDSs), group delay, synthesis.

I. INTRODUCTION

A NALOG signal processing (ASP), as opposed to digitalsignal processing (DSP) [1], consists of processing elec-

tronic signals by analog means in real time. It remains indis-pensable at high frequencies even in the current digital age, be-cause of DSP limitations, such as processing speed, high powerconsumption and heat dissipation, low power-handling capa-bility, and limited performance due to A/D converters [2]. Sur-face acoustics wave (SAW) devices [3], which have been ex-tensively used as ASP components, are restricted to frequenciesbelow about 5 GHz, due to resolution limitation in the fabrica-tion of the required interdigital transducers (IDTs). Therefore,new approaches are in high demand at higher microwave fre-quencies to the millimeter-wave range for ASP applications.

Manuscript received October 10, 2012; accepted December 14, 2012. Dateof publication January 28, 2013; date of current version March 07, 2013. Thiswork was supported by the Natural Sciences and Engineering Research Council(NSERC) of Canada under Grant CRDPJ 402801-10 in partnership with Re-search in Motion (RIM).Q. Zhang and C. Caloz are with the Department of Electrical Engineering,

Poly-Grames Research Center, École Polytechnique de Montréal, Montréal,QC, Canada H3T 1J4 (e-mail: [email protected]).D. L. Sounas was with the Department of Electrical Engineering,

Poly-Grames Research Center, École Polytechnique de Montréal, Mon-tréal, QC, Canada H3T 1J4. He is now with the Metamaterials and PlasmonicsResearch Group, The University of Texas at Austin, Austin, TX 78712 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.2013.2241785

The core of a ASP system is a dispersive delay structure(DDS), which is a component exhibiting an arbitrary prescribedgroup-delay response over a given frequency band. Across sucha component, the different frequency components of a broad-band input signal travel at different velocities, and therefore,the frequency-domain information of the input signal is mappedonto time-domain information at the output. The objective whendesigning a DDS is to achieve prescribed group-delay responsesfor specific applications, as for instance, a linear response for areal-time Fourier transformer [4] and a stepped response for aspectrum sniffer [5], as well as an acceptable return loss (usu-ally above 15 dB). DDSs are fundamentally different from fil-ters in that they are designed to follow group-delay specifica-tions, whereas filters follow magnitude specifications. Severalmicrowave applications of various types of DDSs have been re-cently reported in [4] and [6]–[11].DDSs are usually designed as all-pass networks [12], [13].

However, an all-pass network requires that the magnitude beunity at all frequencies, which limits the implementation ofDDSs to coupled transmission-line sections or lattice circuits.Moreover, all-pass networks seem to be implementable only inplanar structures because backward-wave coupling is requiredin the C-sections they use. To the best of our knowledge, nowaveguide all-pass network has ever been reported. This issueessentially constrains the all-pass networks to low frequenciesand low powers. To eliminate this limitation, one may relaxthe all-pass constraint in the frequency band of interest, and asin magnitude filters, design DDSs as non-all-pass networks.This may be achieved in a particularly powerful manner byusing coupling matrix techniques [14], which provide highimplementation flexibility.In non-all-pass DDSs, the challenge is the synthesis of a

transfer function involving both the phase and magnitude. Mostof techniques for simultaneous phase and magnitude designwere reported more than 30 years ago [15]–[18], and a detailedreview of them is available in [19]. These techniques weremainly applied to linear-phase filters, where the main designeffort was set on the magnitude response. Two excellent syn-thesis techniques for simultaneous group-delay and magnitudesynthesis were reported in [20] and [19]. Both techniques arebased on the same principle. First, two kinds of phase polyno-mials of identical group delay are generated using a recurrenceformula for arbitrarily prescribed phase responses. The transferfunction is then expressed as a combination of the two polyno-mials in such a manner that the transfer function has the samegroup delay as the two polynomials, while its magnitude can

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1044 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013

be controlled in the passband. In these techniques, phase isthe major concern, and therefore, is directly controlled by thephase polynomials, whereas the magnitude is of secondary im-portance. Despite their robustness, these techniques suffer fromtwo major drawbacks. First, their phase and magnitude are notcontrolled independently, which may result in poor magnituderesponses. Secondly, the order of their transfer function is twicethat of the phase polynomials, whereas, as will be shown in thispaper, it may be equal to the order of the phase polynomials,which leads to avoidably large DDS size and loss.Here, we propose a novel DDS synthesis technique. In this

technique, the phase and magnitude of the transfer function arecontrolled independently, which provides higher synthesis flex-ibility. Moreover, the order of the synthesized transfer func-tion is the same as that of phase polynomials, which leads tolower loss and higher compactness. Finally, the transfer func-tion is converted into coupling matrices [14], which can be im-plemented with high flexibility following various alternativerouting schematics based on the similarity transform propertyof the coupling matrix.The key contributions of this paper are as follows.1) Cross-coupled DDSs are reported for the time, and it is alsothe first time that the coupling matrix method is appliedto such structures. Compared with conventional all-passDDSs [13], the cross-coupled DDSs have the advantage ofbeing implementable in waveguide technology for high-frequency and/or high-power applications.

2) A novel systematic closed-form synthesis method forcross-coupled DDSs is proposed. Compared to the con-ventional synthesis methods for linear-phase filters [19],[20] (these techniques have never been applied to DDSsto date), the proposed method features a transfer functionwith an order reduced by half, which reduces the size andloss of the components.

3) The transfer function of the proposed method has its re-flection zeros distributed off the imaginary axis. It is thefirst time that such a transfer function is proved compat-ible with the coupling matrix method. Moreover, the re-lation between distribution of the reflection zeros and thesymmetry of the DDS layout is established.

This paper is organized as follows. Section II reviewsexisting techniques for simultaneous magnitude and phasesynthesis. Section III presents the proposed reduced-order andcontrolled magnitude synthesis technique. Section IV providesthe coupling matrix generation method. Section V demonstratesseveral design examples, and Section VI provides an experi-mental illustration. Section VII discusses the root selection andsubsequent design implications. Finally, conclusions are givenin Section VIII.

II. EXISTING PHASE-SYNTHESIS TECHNIQUES

Among the few existing techniques for simultaneous mag-nitude and phase synthesis, only the techniques reported in[19] and [20] can be potentially used for the design of DDSssince they are the only ones accommodating arbitrary phasespecifications.

In these two techniques, the transfer function is obtained interms of two phase polynomials, which can be generated ac-cording to an arbitrary phase specification through simple recur-rence formulas. The two phase polynomials are termed the first-kind phase polynomial, , and second-kind phase polyno-mial, , where is the complex frequency. The ordercorresponds to the number of prescribed phase points over thefrequency band of interest. The phase functions of the two poly-nomials, and , are related by .Once the two polynomials have been constructed, the transferfunction of the DDS is computed by

(1)

This relation shows that is fully determined fromand , which are related only to the specified phase

response. Therefore, there is no extra degree of freedom formagnitude control in this approach. Also note that the orderof is , whereas the order can be reduced to , withmajor benefits in terms of size and loss reduction in practicalimplementations.

III. PROPOSED TRANSFER FUNCTION SYNTHESIS METHOD

A. Reduced-Order Transfer Function

In a passive and reciprocal network, such as a magnitude en-gineered filter, one usually represents and as therational functions

(2a)

(2b)

where is a Hurwitz polynomial with roots in the left half-plane, is conventionally a real polynomial with roots ei-ther on the imaginary axis or symmetrically distributed aboutit, and is conventionally a real polynomial with roots onthe imaginary axis [14]. Note the purely imaginary nature of the

roots ensure the production of reflection zeros at real fre-quencies. Assuming a lossless network, (2) satisfies the energyconservation law , which leads to

(3)

In a DDS, the objective is to generate a transfer function withan arbitrary prescribed response, as illustrated in Fig. 1. In theproposed method, we assume that remains a real polyno-mial, as in conventional magnitude design techniques, so thatthe phase of is entirely determined by . isthen the characteristic phase polynomial of the DDS, compa-rable to or in [19] and [20], to be generated so asto meet the group delay or phase specifications of Fig. 1. Since

must have a lower order than for the system to bephysically realizable, necessarily has the same order as

ZHANG et al.: SYNTHESIS OF CROSS-COUPLED REDUCED-ORDER DDSs WITH ARBITRARY GROUP DELAY AND CONTROLLED MAGNITUDE 1045

Fig. 1. Ideal low-pass response of a DDS.

. Thus, the order of in the proposed approach is halfthe order of in [19] and [20].Another condition to satisfy, in addition to (2) and the phase

specification of Fig. 1, is that the magnitude of shouldbe approximately equal to 1 in the passband, and hence,

(4)

Since is fixed by the prescribed phase response, andmust be adjusted so as to satisfy the relations (3) and (4). If

the order of is , then is the maximal order of and. Since is a real polynomial with roots symmetrically

distributed about the imaginary axis, it can be expressed as

(5)

where denotes the th root of the polynomial, and whereis generally determined by parameters since the ’s

have real and imaginary parts. Moreover, if we follow the con-ventional approaches, can be expressed as

(6)

where denotes the th root of , and is purely imag-inary, so that is also determined by parameters. Thus,there are overall parameters to find for determiningand . Relation (3) provides equations since the poly-nomials involved in the products is of order . Thus, and

would be uniquely determined by (3). However, there isno guarantee that the subsequent and will satisfy (4).Thus, there is a missing degree of freedom to ensure (4) in ad-dition to (3).The missing degree of freedom may be generated by relaxing

the constraints . The polynomial is kept real becauseit would otherwise affect the phase of and complicate thesynthesis procedure. In contrast, the constraint of purely imagi-nary roots for may be relaxed. The number of parametersdetermining subsequently increases to , which leads toa total maximal number of parameters. Thus, in addition to

Fig. 2. Distribution of the roots of in the complex -plane.

the free parameters used to satisfy (3), additional param-eters are now available to satisfy (4). As will be shown later, thesuppression of the imaginary root constraints for may stilllead to a physically realizable DDS, compatible coupling matrixtechniques.The procedure of the proposed transfer function synthesis ap-

proach mainly consists of two steps. The first steps consists inthe generation of following the prescribed phase functionthrough the recurrence formula provided in [20] and [21]. In thesecond step, the transmission polynomial is determined soas to exhibit a magnitude approximating the magnitude offor a flat and approximately unitary response over theentire design frequency band. is then calculated from theenergy conservation equation.

B. Generation of

The prescribed phase function can be computedfrom the prescribed group-delay function by

(7)

where is a constant to be determined later.Since the numerator in (2) is constrained to be a real polyno-

mial, the phase of is simply

(8)

To generate a Hurwitz polynomial with the prescribed phase(8), we use the recurrence algorithm provided in [20] and [21].This algorithm requires that the phase at the origin be zero sincethe phase of a Hurwitz polynomial is always an odd functionof frequency. This condition can be satisfied by settingin (7). The generation procedure starts with discretizing thepassband at the points , where ,

. The corresponding prescribed phases at the discretefrequency points are then , where

. The appropriate th-order Hurwitz polynomialcan then be generated by

(9)


TABLE IPOLYNOMIAL COEFFICIENTS FOR THE POSITIVE-SLOPE DDS

where

. . .

(10)

for , and

(11)

To apply this formula, one first computes the coefficientsfrom to , using (10) and (11), and then obtainsthe polynomials using (9). Note that the order deter-mines the error between the phase response of the synthesizedpolynomial and the specified phase response.

C. Generation of

is found by applying (4), which ensures that the am-plitude of is approximately equal to 1 in the passband.Considering that is a real polynomial with poles on theimaginary axis, (4) may be reformulated as

(12)

One possible approach for finding satisfying (12) is toexpand the right-hand side of (12) in Chebyshev functions sinceChebyshev polynomial approximations provide equal-rippleerrors, which are convenient in filter design. Specifically, wefirst calculate the full Chebyshev expansion in the domain

as [22]

(13)

where

for

for

(14)

Fig. 3. Synthesized coupling matrix and folded-form topology for the cross-coupled DDS with positively linear-sloped group delay.

Fig. 4. Synthesized scattering parameters of the cross-coupled DDSwith linearpositive-slope group delay.

and is the th-order Chebyshev polynomial. isthen constructed by truncating (13) up to th-order term as

(15)


TABLE IIPOLYNOMIAL COEFFICIENTS FOR THE NEGATIVE-SLOPE DDS

where should satisfy so that the transfer functioncan be realized by a passive network. Once has beenformed with (15), it is normalized according to

(16)

so that the magnitude of is less than 1. Since the -do-main coincides with the imaginary axis of the -domain, ,found from by applying the transformation , au-tomatically satisfies the constraint of being a real polynomial.

D. Generation of

Once and have been determined, can be ob-tained from the energy conservation equation (3), alternativelywritten as

(17)

Since is a real polynomial ( andare real polynomials), its roots are symmetrically dis-

tributed about the imaginary axis or lie on the imaginary axisas second-order roots, as shown in Fig. 2. The roots ofcan be selected from the roots of taking one rootfrom each of the symmetrical pairs and the second-order rootsonce. One of the possible choices would be to take all the rootsin the left plane in addition to the roots on the imaginary axis.Other possible choices and subsequent design implications willbe discussed in Section VII. Note that the dominant coefficientof is the square root of that of .

IV. COUPLING MATRIX GENERATION

A. Condition for Applicability of Coupling Matrix Techniques

In Section III, we used an unconventional reflection polyno-mial , whose roots are not constrained to lie on the imagi-nary axis. We have to now examine whether this constraint re-laxation still allows a physical realization and the applicationof the coupling matrix technique [14]. This technique does notimpose any specific condition on . The only condition isthat the system be passive, lossless, and reciprocal. These threerequirements are simultaneously satisfied if the scattering pa-rameters obey the energy conservation relations

(18a)

(18b)

and the orthogonality equation

(19)

Fig. 5. Synthesized coupling matrix and folded-form topology for the cross-coupled DDS with linear negative-slope group delay.

Fig. 6. Synthesized scattering parameters of the cross-coupled DDS with neg-atively linear-sloped group delay.

Expressing as and inserting ittogether with (2) into (19) yields

(20)

Since is a real polynomial, , and then (20)is equivalent to

(21)

Note that this equation only indicates a relation existing betweenand , but does not impose any restriction on the

form of .


TABLE IIIPOLYNOMIAL COEFFICIENTS FOR THE STEP-RESPONSE DDS

Fig. 7. Synthesized coupling matrix and folded-form topology for the cross-coupled DDS with stepped group delay.

It is also interesting to examine the effect of (21) on the re-sulting DDS layout. Conventionally, when the roots of lieon the imaginary axis, (21) dictates that the phase of beequal to . In our case, where the roots of do not lie onthe imaginary axis, the phase of may be totally differentfrom that of , which will lead to an asymmetrical layout inthe physical implementation.

B. Coupling Matrix

The coupling matrix generation technique is given in [14].We will use it here to convert the synthesized transfer functioninto a coupling matrix. Although the general procedure is thesame as that in [14], the generation formula slightly differs.To generate a coupling matrix, the first step is to convert the

scattering matrix into an admittance matrix. By inserting (21)into (2a), one obtains the scattering matrix

(22)

This matrix may be converted into the admittance matrix [23]

(23a)

Fig. 8. Synthesized scattering parameters of the cross-coupled DDS with step-case group delay.

(23b)

(23c)

The numerator and denominator of (23b) are termed the com-plex-even and complex-odd components of , re-spectively [14]. Since the dominant coefficients of and

are usually real, the denominator of (23b) may have alower order than the numerator when the order of is even,which results in unusual terms in the partial fractional expan-sion. To avoid this, both and can be multiplied by ,which is equivalent to exchanging the complex-even and com-plex-odd components of . Note that such an op-eration does not affect the energy conservation law (3).The rest of this procedure is exactly the same as described

in [14]. Once the admittances and have been ob-tained through (23), their partial fraction expansion is formed,corresponding to a transversal array or transversal coupling ma-trix. The transversal array can then be transformed into variouskinds of routing schematics by applying similarity transforma-tions on the coupling matrix.

V. DESIGN EXAMPLES

To illustrate the proposed synthesis method, three designswith different group-delay specifications will be presented inthis section.The first design example is a cross-coupled DDS featuring

a linear positive-slope group-delay response with a swing of


TABLE IVPOLYNOMIAL COEFFICIENTS FOR THE WAVEGUIDE CROSS-COUPLED DDS PROTOTYPE

TABLE VADMITTANCE MATRIX POLYNOMIAL COEFFICIENTS FOR THE WAVEGUIDE CROSS-COUPLED DDS PROTOTYPE

(NOTATION: AND )

Fig. 9. Synthesized couplingmatrix for the waveguide cross-coupled DDS pro-totype over the frequency band of 10–10.05 GHz.

1 s over the band . Table I lists the synthesizedtransfer function polynomials, while Fig. 3 shows the couplingmatrix and corresponding topology. Note that is differentfrom in the coupling matrix, indicating an asymmetricallayout, as discussed in Section IV-A. The calculated responsefor this cross-coupled DDS is plotted in Fig. 4. The synthesizedgroup delay very closely follows the prescribed response overthe entire band of interest with a return loss greaterthan 16 dB. Also note that the response is symmetrical aboutthe origin, as a result of the fact that real-coefficient Hurwitzpolynomials were used in the transfer function.The second design example is a cross-coupled DDS with a

linear negative-slope group-delay response. As in the previousexample, the specified group-delay swing is 1 s over the bandof interest . Table II lists the synthesized transferfunction polynomials, while Fig. 5 shows the coupling matrixand corresponding topology. The calculated response of thiscross-coupled DDS is plotted in Fig. 6. Again, the synthesizedgroup delay very closely follows the prescribed response overthe band with a return loss greater than 19 dB.The third design example is a cross-coupled DDS with a

stepped group-delay response. Specifically, the group delayvaries linearly with a 1-s swing for and remainsconstant for . Table III lists the synthesized transferfunction polynomials and Fig. 7 shows the coupling matrix and

Fig. 10. Configuration of the waveguide cross-coupled DDS prototype. (a) Per-spective view. (b) Half of the fabricated prototype ( , ,

, , , , ,, , , , , ,, , , , ,, , , , , unit:

millimeters).

corresponding topology. The calculated response is plotted inFig. 8. As in the previous two examples, the synthesized groupdelay closely follows the prescribed response very well overthe band with a return loss greater than 17 dB.Note that the step-case group-delay DDS is particularly useful


Fig. 11. Measured and full-wave response of the fabricated DDS. (a) Group delay. (b) Magnitude.

Fig. 12. Different choices for the roots of in the complex -plane (the roots inside the red circle (in online version) are selected).

in the frequency discriminator application, where it maintains aconstant delay within a specific channel while resolving the twochannel in the time domain from the delay difference betweenthem.

VI. EXPERIMENTAL VALIDATION

Consider a cross-coupled waveguide DDS prototype speci-fied to exhibit a linear group-delay response with a 1.8-ns swingover the frequency band of 10–10.05 GHz. The correspondingsynthesized polynomials are given in Table IV. In this example,the roots of are chosen from the left plane among the rootsof for root selection simplicity. The converted ad-mittance matrix and resulting coupling matrix in the bandpassdomain are given in Table V and Fig. 9, respectively.The coupling matrix in Fig. 9 was implemented and fabri-

cated, as shown in Fig. 10. WR-90 (0.9 in 0.4 in) is employedas the host waveguide. The direct couplings are implementedby -plane irises, whereas the cross couplings are realized by-plane square windows on the broadside wall. Due to fabrica-

tion limitations, the cavity corners are not sharp, but rounded,with a minimum radius of 1.6 mm, as shown in the zoomed re-gion of Fig. 10(b).The cross-coupled waveguide DDS is analyzed by full-wave

simulation using the commercial software ANSYS HFSS. The

fabricated prototype is measured via a vector network ana-lyzer with the thru-reflection-line (TRL) calibration method.Fig. 11 compares the synthesized, full-wave, and experimentalresponses. The synthesized response is directly calculated fromthe coupling matrix in Fig. 9. Note that the full-wave andmeasured group delay follows the synthesized response almostperfectly. Also observe that the measured and arein an excellent agreement with the full-wave responses. Themeasured return loss is greater than 20 dB and the maximummeasured insertion loss is around 0.9 dB within the specifiedfrequency band.

VII. ROOT SELECTION

In the design examples of Sections V and VI, we selectedthe roots of lying in the left half-plane as the rootsof . However, this is not the only possible choice. For an-order , different choices exist, and different roots

selection will produce different , resulting in different cou-pling matrices.As mentioned in Section IV, the roots lying off of the imag-

inary axis result into asymmetrical layouts. We define the sym-metry ratio as

(24)


Fig. 13. Coupling matrix corresponding to different choices for the roots of . (a) Roots selection in Fig. 12(b). (b) Roots selection in Fig. 12(c).

Inserting (21) into (24) yields

(25)

where is the phase of . In conventional designs, wherethe roots of are all placed on the imaginary axis, the layoutis symmetrical and is real and unitary. Therefore, the sym-metry ratio of a given implementation is proportional to theproximity of to the real axis. Thus, should be chosen closeto for the imaginary part of to be small. This require-ment can bemet bymaximizing the symmetry in the distributionof the roots of .Fig. 12 shows three different choices for the roots of , re-

sulting in three different coupling matrices, as shown in Figs. 3and 13(a) and (b), respectively. In the case of Fig. 12(a), all theroots of are in the left part of the complex plane. In the caseof Fig. 12(b), some roots are selected in the left plane and othersare in the right plane. In the case of Fig. 12(c), the roots are se-lected according to a zigzag pattern so that they can placed aboutthe imaginary axis as symmetrically as possible. According to(24), the symmetry ratio is minimum in the first case, whereas itis maximum in the third case. This can be verified by checkingthe difference between and in corresponding cou-pling matrices. In the first case, as shown in Fig. 3, the differ-ence is . In the second case, as shownin Fig. 13(a), the difference is , whichis slightly smaller. In the third case, as shown in Fig. 13(a), thedifference is , which is the smallest.Accordingly, different choices of the roots will result in dif-

ferent symmetry ratios in the layout.

VIII. CONCLUSION

A systematic synthesis method has been presented for cross-coupled DDSs with arbitrary prescribed group delay and con-trolled magnitude. Several design examples with different pre-scribed group-delay responses have been presented to illustratethe proposed synthesis method. A waveguide prototype is im-plemented and fabricated. The measured responses are in verygood agreement with the prescribed and full-wave responses.

ACKNOWLEDGMENT

The authors would like to thank all the staff of thePoly-Grames Microwave Research Center, École Poly-

technic de Montréal, Montréal, QC, Canada, for their help andcooperation.

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Qingfeng Zhang (S’07–M’11) was born inChangzhou, China, in December 1984. He receivedthe B.E. degree in electrical engineering from theUniversity of Science and Technology of China(USTC), Hefei, China, in 2007, and is currentlyworking toward the Ph.D. degree in electrical andelectronic engineering at Nanyang TechnologyUniversity, Singapore. His Ph.D. thesis is focusedon dimensional synthesis of wideband waveguidefilters.Since April 2011, he has been a Postdoctoral

Fellow with the Poly-Grames Microwave Research Center, École Polytech-nique de Montréal, Montréal, Canada. His current research interests includefilter synthesis theory, DDSs, ASP systems, leaky-wave antennas, and meta-materials.Mr. Zhang studied under a Nanyang Technology University scholarship from

August 2007 to November 2010.

Dimitrios L. Sounas (M’11) was born in Thes-saloniki, Greece, in September 1981. He receivedthe Diploma and Ph.D. degrees in electrical andcomputer engineering from the Aristotle Universityof Thessaloniki (AUTH), Thessaloniki, Greece, in2004 and 2009, respectively.From August 2010 to October 2012, he was a

Post-Doctoral Fellow with the ElectromagneticTheory and Applications Research Group, ÉcolePolytechnique of Montréal. In November 2012, hejoined the Metamaterials and Plasmonics Research

Group, The University of Texas at Austin, Austin, TX, USA, as a Post-DoctoralFellow. His research interests include analytical and numerical techniques inelectromagnetics, metamaterials, and graphene-based structures.

Christophe Caloz (S’00–A’00–M’03–SM’06–F’10)received the Diplôme d’Ingénieur en Électricité andPh.D. degree from the École Polytechnique Fédéralede Lausanne (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, Los Angeles(UCLA), Los Angeles, CA, USA. In June 2004,he joined the École Polytechnique de Montréal,Montréal, QC, Canada, where he is currently a Full

Professor, a member of the Poly-Grames Microwave Research Center, and theHolder of a Canada Research Chair (CRC). He has authored or coauthoredover 420 technical conference, letter, and journal papers, 12 books, and bookchapters. He holds several patents. His research interests include all fields oftheoretical, computational, and technological electromagnetics engineeringwith a strong emphasis on emergent and multidisciplinary topics, includingparticularly nanoelectromagnetics.Dr. Caloz is a member of the IEEE Microwave Theory and Techniques So-

ciety (IEEEMTT-S) Technical CommitteesMTT-15 (Microwave Field Theory)andMTT-25 (RF Nanotechnology). He is a speaker of the MTT-15 Speaker Bu-reau, the chair of Commission D (Electronics and Photonics), Canadian Unionde Radio Science Internationale (URSI), and an IEEE MTT-S representativeof the IEEE Nanotechnology Council (NTC). He was the recipient of severalawards, including the UCLA Chancellor’s Award for Post-Doctoral Researchin 2004, the IEEE MTT-S Outstanding Young Engineer Award in 2007, andmany Best Paper Awards with his students.

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