ANALOG signal processing (ASP), as opposed to digital

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1 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH Synthesis of Cross-Coupled Reduced-Order Dispersive Delay Structures (DDSs) With Arbitrary Group Delay and Controlled Magnitude Qingfeng Zhang, Member, IEEE, Dimitrios L. Sounas, Member, IEEE, and Christophe Caloz, Fellow, IEEE Abstract For the first time, a systematic synthesis method for cross-coupled dispersive delay structures (DDSs) with controlled magnitude for analog signal processing applications is proposed. In this method, the transfer function is synthesized using a polynomial expansion approach, which allows to separately control the magnitude and group-delay response of the DDS. The synthesized transfer function also features a reduced order namely, half compared to that of previously reported synthesis techniques for linear-phase filters. Once it has been constructed, the transfer function is transferred into coupling matrices that can be implemented in arbitrary cross-coupled-resonator technologies. Several design examples are provided for different prescribed group-delay responses. An experimental waveguide prototype is demonstrated. The agreement between the measured and prescribed responses illustrates the proposed synthesis method. Index Terms Analog signal processing (ASP), cross-coupled, dispersive delay structures (DDSs), group delay, synthesis. I. INTRODUCTION ANALOG signal processing (ASP), as opposed to digital signal processing (DSP) [1], consists of processing electronic signals by analog means in real time. It remains indispensable at high frequencies even in the current digital age, because of DSP limitations, such as processing speed, high power consumption and heat dissipation, low power-handling capability, and limited performance due to A/D converters [2]. Surface acoustics wave (SAW) devices [3], which have been extensively used as ASP components, are restricted to frequencies below about 5 GHz, due to resolution limitation in the fabrication of the required interdigital transducers (IDTs). Therefore, new approaches are in high demand at higher microwave frequencies to the millimeter-wave range for ASP applications. Manuscript received October 10, 2012; accepted December 14, Date of publication January 28, 2013; date of current version March 07, This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada under Grant CRDPJ in partnership with Research 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 ( qfzhang@ieee.org). D. L. Sounas was with the Department of Electrical Engineering, Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC, Canada H3T 1J4. He is now with the Metamaterials and Plasmonics Research Group, The University of Texas at Austin, Austin, TX USA. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMTT The core of a ASP system is a dispersive delay structure (DDS), which is a component exhibiting an arbitrary prescribed group-delay response over a given frequency band. Across such a component, the different frequency components of a broadband input signal travel at different velocities, and therefore, the frequency-domain information of the input signal is mapped onto time-domain information at the output. The objective when designing a DDS is to achieve prescribed group-delay responses for specific applications, as for instance, a linear response for a real-time Fourier transformer [4] and a stepped response for a spectrum sniffer [5], as well as an acceptable return loss (usually above 15 db). DDSs are fundamentally different from filters in that they are designed to follow group-delay specifications, whereas filters follow magnitude specifications. Several microwave applications of various types of DDSs have been recently 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 be unity at all frequencies, which limits the implementation of DDSs to coupled transmission-line sections or lattice circuits. Moreover, all-pass networks seem to be implementable only in planar structures because backward-wave coupling is required in the C-sections they use. To the best of our knowledge, no waveguide all-pass network has ever been reported. This issue essentially constrains the all-pass networks to low frequencies and low powers. To eliminate this limitation, one may relax the all-pass constraint in the frequency band of interest, and as in magnitude filters, design DDSs as non-all-pass networks. This may be achieved in a particularly powerful manner by using coupling matrix techniques [14], which provide high implementation flexibility. In non-all-pass DDSs, the challenge is the synthesis of a transfer function involving both the phase and magnitude. Most of techniques for simultaneous phase and magnitude design were reported more than 30 years ago [15] [18], and a detailed review of them is available in [19]. These techniques were mainly applied to linear-phase filters, where the main design effort was set on the magnitude response. Two excellent synthesis techniques for simultaneous group-delay and magnitude synthesis were reported in [20] and [19]. Both techniques are based on the same principle. First, two kinds of phase polynomials of identical group delay are generated using a recurrence formula for arbitrarily prescribed phase responses. The transfer function is then expressed as a combination of the two polynomials in such a manner that the transfer function has the same group delay as the two polynomials, while its magnitude can /$ IEEE

2 1044 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013 be controlled in the passband. In these techniques, phase is the major concern, and therefore, is directly controlled by the phase polynomials, whereas the magnitude is of secondary importance. Despite their robustness, these techniques suffer from two major drawbacks. First, their phase and magnitude are not controlled independently, which may result in poor magnitude responses. Secondly, the order of their transfer function is twice that of the phase polynomials, whereas, as will be shown in this paper, 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 are controlled independently, which provides higher synthesis flexibility. Moreover, the order of the synthesized transfer function is the same as that of phase polynomials, which leads to lower loss and higher compactness. Finally, the transfer function is converted into coupling matrices [14], which can be implemented with high flexibility following various alternative routing schematics based on the similarity transform property of the coupling matrix. The key contributions of this paper are as follows. 1) Cross-coupled DDSs are reported for the time, and it is also the first time that the coupling matrix method is applied to such structures. Compared with conventional all-pass DDSs [13], the cross-coupled DDSs have the advantage of being implementable in waveguide technology for highfrequency and/or high-power applications. 2) A novel systematic closed-form synthesis method for cross-coupled DDSs is proposed. Compared to the conventional synthesis methods for linear-phase filters [19], [20] (these techniques have never been applied to DDSs to date), the proposed method features a transfer function with an order reduced by half, which reduces the size and loss of the components. 3) The transfer function of the proposed method has its reflection zeros distributed off the imaginary axis. It is the first time that such a transfer function is proved compatible with the coupling matrix method. Moreover, the relation between distribution of the reflection zeros and the symmetry of the DDS layout is established. This paper is organized as follows. Section II reviews existing techniques for simultaneous magnitude and phase synthesis. Section III presents the proposed reduced-order and controlled magnitude synthesis technique. Section IV provides the coupling matrix generation method. Section V demonstrates severaldesignexamples,andsectionviprovidesanexperimental illustration. Section VII discusses the root selection and subsequent design implications. Finally, conclusions are given in Section VIII. II. EXISTING PHASE-SYNTHESIS TECHNIQUES Among the few existing techniques for simultaneous magnitude and phase synthesis, only the techniques reported in [19] and [20] can be potentially used for the design of DDSs since they are the only ones accommodating arbitrary phase specifications. In these two techniques, the transfer function is obtained in terms of two phase polynomials, which can be generated according to an arbitrary phase specification through simple recurrence formulas. The two phase polynomials are termed the firstkind phase polynomial,, and second-kind phase polynomial,,where is the complex frequency. The order corresponds to the number of prescribed phase points over the frequency band of interest. The phase functions of the two polynomials, and, are related by. Once the two polynomials have been constructed, the transfer function of the DDS is computed by This relation shows that is fully determined from and, which are related only to the specified phase response. Therefore, there is no extra degree of freedom for magnitude control in this approach. Also note that the order of is, whereas the order can be reduced to,with major benefits in terms of size and loss reduction in practical implementations. III. PROPOSED TRANSFER FUNCTION SYNTHESIS METHOD A. Reduced-Order Transfer Function In a passive and reciprocal network, such as a magnitude engineered filter, one usually represents and as the rational functions (1) (2a) (2b) where is a Hurwitz polynomial with roots in the left half -plane, is conventionally a real polynomial with roots either on the imaginary axis or symmetrically distributed about it, and is conventionally a real polynomial with roots on the imaginary axis [14]. Note the purely imaginary nature of the roots ensure the production of reflection zeros at real frequencies. Assuming a lossless network, (2) satisfies the energy conservation law, which leads to In a DDS, the objective is to generate a transfer function with an arbitrary prescribed response, as illustrated in Fig. 1. In the proposed method, we assume that remains a real polynomial, as in conventional magnitude design techniques, so that the phase of is entirely determined by. is then the characteristic phase polynomial of the DDS, comparable to or in [19] and [20], to be generated so as to meet the group delay or phase specifications of Fig. 1. Since must have a lower order than for the system to be physically realizable, necessarily has the same order as (3)

3 ZHANG et al.: SYNTHESIS OF CROSS-COUPLED REDUCED-ORDER DDSs WITH ARBITRARY GROUP DELAY AND CONTROLLED MAGNITUDE 1045 Fig. 2. Distribution of the roots of in the complex -plane. Fig. 1. Ideal low-pass response of a DDS.. Thus, the order of in the proposed approach is half the 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 should be approximately equal to 1 in the passband, and hence, Since is fixed by the prescribed phase response, and must 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 (4) the free parameters used to satisfy (3), additional parameters are now available to satisfy (4). As will be shown later, the suppression of the imaginary root constraints for may still lead to a physically realizable DDS, compatible coupling matrix techniques. The procedure of the proposed transfer function synthesis approach mainly consists of two steps. The first steps consists in the generation of following the prescribed phase function through the recurrence formula provided in [20] and [21]. In the second step, the transmission polynomial is determined so as to exhibit a magnitude approximating the magnitude of for a flat and approximately unitary response over the entire design frequency band. is then calculated from the energy conservation equation. where denotes the th root of the polynomial, and where is generally determined by parameters since the s have real and imaginary parts. Moreover, if we follow the conventional approaches, can be expressed as (5) B. Generation of The prescribed phase function from the prescribed group-delay function can be computed by (7) where denotes the th root of,and is purely imaginary, so that is also determined by parameters. Thus, there are overall parameters to find for determining and. Relation (3) provides equations since the polynomials involved in the products is of order.thus, and would be uniquely determined by (3). However, there is no guarantee that the subsequent and will satisfy (4). Thus, there is a missing degree of freedom to ensure (4) in addition to (3). The missing degree of freedom may be generated by relaxing the constraints. The polynomial is kept real because it would otherwise affect the phase of and complicate the synthesis procedure. In contrast, the constraint of purely imaginary roots for may be relaxed. The number of parameters determining subsequently increases to, which leads to a total maximal number of parameters. Thus, in addition to (6) where is a constant to be determined later. Since the numerator in (2) is constrained to be a real polynomial, the phase of is simply 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 since the phase of a Hurwitz polynomial is always an odd function of frequency. This condition can be satisfied by setting in (7). The generation procedure starts with discretizing the passband at the points,where,. The corresponding prescribed phases at the discrete frequency points are then,where. The appropriate th-order Hurwitz polynomial can then be generated by (8) (9)

4 1046 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013 TABLE I POLYNOMIAL COEFFICIENTS FOR THE POSITIVE-SLOPE DDS where... (10) for,and (11) To apply this formula, one first computes the coefficients from to, using (10) and (11), and then obtains the polynomials using (9). Note that the order determines the error between the phase response of the synthesized polynomial and the specified phase response. Fig. 3. Synthesized coupling matrix and folded-form topology for the crosscoupled DDS with positively linear-sloped group delay. C. Generation of is found by applying (4), which ensures that the amplitude of is approximately equal to 1 in the passband. Considering that is a real polynomial with poles on the imaginary axis, (4) may be reformulated as (12) One possible approach for finding satisfying (12) is to expand the right-hand side of (12) in Chebyshev functions since Chebyshev polynomial approximations provide equal-ripple errors, which are convenient in filter design. Specifically, we first calculate the full Chebyshev expansion in the domain as [22] (13) where Fig. 4. Synthesized scattering parameters of the cross-coupled DDS with linear positive-slope group delay. for and is the th-order Chebyshev polynomial. is then constructed by truncating (13) up to th-order term as for (14) (15)

5 ZHANG et al.: SYNTHESIS OF CROSS-COUPLED REDUCED-ORDER DDSs WITH ARBITRARY GROUP DELAY AND CONTROLLED MAGNITUDE 1047 TABLE II POLYNOMIAL COEFFICIENTS FOR THE NEGATIVE-SLOPE DDS where should satisfy so that the transfer function can be realized by a passive network. Once has been formed with (15), it is normalized according to (16) so that the magnitude of is less than 1. Since the -domain coincides with the imaginary axis of the -domain,, found from by applying the transformation,automatically satisfies the constraint of being a real polynomial. D. Generation of Once and have been determined, can be obtained from the energy conservation equation (3), alternatively written as (17) Since is a real polynomial ( and are real polynomials), its roots are symmetrically distributed about the imaginary axis or lie on the imaginary axis as second-order roots, as shown in Fig. 2. The roots of can be selected from the roots of taking one root from each of the symmetrical pairs and the second-order roots once.oneofthepossiblechoiceswouldbetotakealltheroots in the left plane in addition to the roots on the imaginary axis. Other possible choices and subsequent design implications will be discussed in Section VII. Note that the dominant coefficient of is the square root of that of. Fig. 5. Synthesized coupling matrix and folded-form topology for the crosscoupled DDS with linear negative-slope group delay. IV. COUPLING MATRIX GENERATION A. Condition for Applicability of Coupling Matrix Techniques In Section III, we used an unconventional reflection polynomial, whose roots are not constrained to lie on the imaginary axis. We have to now examine whether this constraint relaxation still allows a physical realization and the application of the coupling matrix technique [14]. This technique does not impose any specific condition on. The only condition is that the system be passive, lossless, and reciprocal. These three requirements are simultaneously satisfied if the scattering parameters obey the energy conservation relations and the orthogonality equation (18a) (18b) (19) Fig. 6. Synthesized scattering parameters of the cross-coupled DDS with negatively linear-sloped group delay. Expressing as and inserting it together with (2) into (19) yields (20) Since is a real polynomial,,andthen(20) is equivalent to (21) Note that this equation only indicates a relation existing between and, but does not impose any restriction on the form of.

6 1048 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013 TABLE III POLYNOMIAL COEFFICIENTS FOR THE STEP-RESPONSE DDS Fig. 8. Synthesized scattering parameters of the cross-coupled DDS with stepcase group delay. Fig. 7. Synthesized coupling matrix and folded-form topology for the crosscoupled DDS with stepped group delay. (23b) (23c) It is also interesting to examine the effect of (21) on the resulting DDS layout. Conventionally, when the roots of lie on the imaginary axis, (21) dictates that the phase of be equal to. In our case, where the roots of do not lie on the imaginary axis, the phase of may be totally different from that of, which will lead to an asymmetrical layout in the physical implementation. B. Coupling Matrix The coupling matrix generation technique is given in [14]. We will use it here to convert the synthesized transfer function into a coupling matrix. Although the general procedure is the same 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 The numerator and denominator of (23b) are termed the complex-even and complex-odd components of,respectively [14]. Since the dominant coefficients of and are usually real, the denominator of (23b) may have a lower order than the numerator when the order of is even, which results in unusual terms in the partial fractional expansion. To avoid this, both and can be multiplied by, which is equivalent to exchanging the complex-even and complex-odd components of. Note that such an operation 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 obtained through (23), their partial fraction expansion is formed, corresponding to a transversal array or transversal coupling matrix. The transversal array can then be transformed into various kinds of routing schematics by applying similarity transformations on the coupling matrix. This matrix may be converted into the admittance matrix [23] (22) (23a) V. DESIGN EXAMPLES To illustrate the proposed synthesis method, three designs with different group-delay specifications will be presented in this section. The first design example is a cross-coupled DDS featuring a linear positive-slope group-delay response with a swing of

7 ZHANG et al.: SYNTHESIS OF CROSS-COUPLED REDUCED-ORDER DDSs WITH ARBITRARY GROUP DELAY AND CONTROLLED MAGNITUDE 1049 TABLE IV POLYNOMIAL COEFFICIENTS FOR THE WAVEGUIDE CROSS-COUPLED DDS PROTOTYPE TABLE V ADMITTANCE MATRIX POLYNOMIAL COEFFICIENTS FOR THE WAVEGUIDE CROSS-COUPLED DDS PROTOTYPE (NOTATION: AND ) Fig. 9. Synthesized coupling matrix for the waveguide cross-coupled DDS prototype over the frequency band of GHz. 1 s over the band. Table I lists the synthesized transfer function polynomials, while Fig. 3 shows the coupling matrix and corresponding topology. Note that is different from in the coupling matrix, indicating an asymmetrical layout, as discussed in Section IV-A. The calculated response for this cross-coupled DDS is plotted in Fig. 4. The synthesized group delay very closely follows the prescribed response over the entire band of interest with a return loss greater than 16 db. Also note that the response is symmetrical about the origin, as a result of the fact that real-coefficient Hurwitz polynomials 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 previous example, the specified group-delay swing is 1 s over the band of interest. Table II lists the synthesized transfer function polynomials, while Fig. 5 shows the coupling matrix and corresponding topology. The calculated response of this cross-coupled DDS is plotted in Fig. 6. Again, the synthesized group delay very closely follows the prescribed response over the 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 delay varies linearly with a 1-s swing for and remains constant for. Table III lists the synthesized transfer function polynomials and Fig. 7 shows the coupling matrix and Fig. 10. Configuration of the waveguide cross-coupled DDS prototype. (a) Perspective view. (b) Half of the fabricated prototype (,,,,,,,,,,,,,,,,,,,,,,, unit: millimeters). corresponding topology. The calculated response is plotted in Fig. 8. As in the previous two examples, the synthesized group delay closely follows the prescribed response very well over the band with a return loss greater than 17 db. Note that the step-case group-delay DDS is particularly useful

8 1050 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013 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 a constant delay within a specific channel while resolving the two channel in the time domain from the delay difference between them. VI. EXPERIMENTAL VALIDATION Consider a cross-coupled waveguide DDS prototype specified to exhibit a linear group-delay response with a 1.8-ns swing over the frequency band of GHz. The corresponding synthesized polynomials are given in Table IV. In this example, the roots of are chosen from the left plane among the roots of for root selection simplicity. The converted admittance matrix and resulting coupling matrix in the bandpass domain are given in Table V and Fig. 9, respectively. ThecouplingmatrixinFig.9wasimplementedandfabricated, as shown in Fig. 10. WR-90 (0.9 in 0.4 in) is employed as the host waveguide. The direct couplings are implemented by -plane irises, whereas the cross couplings are realized by -plane square windows on the broadside wall. Due to fabrication limitations, the cavity corners are not sharp, but rounded, with a minimum radius of 1.6 mm, as shown in the zoomed region 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 analyzer with the thru-reflection-line (TRL) calibration method. Fig. 11 compares the synthesized, full-wave, and experimental responses. The synthesized response is directly calculated from the coupling matrix in Fig. 9. Note that the full-wave and measured group delay follows the synthesized response almost perfectly. Also observe that the measured and are in an excellent agreement with the full-wave responses. The measured return loss is greater than 20 db and the maximum measured insertion loss is around 0.9 db within the specified frequency band. VII. ROOT SELECTION In the design examples of Sections V and VI, we selected the roots of lying in the left half-plane as the roots of. However, this is not the only possible choice. For an -order, different choices exist, and different roots selection will produce different, resulting in different coupling matrices. As mentioned in Section IV, the roots lying off of the imaginary axis result into asymmetrical layouts. We define the symmetry ratio as (24)

9 ZHANG et al.: SYNTHESIS OF CROSS-COUPLED REDUCED-ORDER DDSs WITH ARBITRARY GROUP DELAY AND CONTROLLED MAGNITUDE 1051 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) ACKNOWLEDGMENT The authors would like to thank all the staff of the Poly-Grames Microwave Research Center, École Polytechnic de Montréal, Montréal, QC, Canada, for their help and cooperation. where is the phase of. In conventional designs, where the roots of are all placed on the imaginary axis, the layout is symmetrical and is real and unitary. Therefore, the symmetry ratio of a given implementation is proportional to the proximity of to the real axis. Thus, should be chosen close to for the imaginary part of to be small. This requirement can be met by maximizing the symmetry in the distribution of the roots of. Fig. 12 shows three different choices for the roots of,resulting in three different coupling matrices, as shown in Figs. 3 and 13(a) and (b), respectively. In the case of Fig. 12(a), all the roots of are in the left part of the complex plane. In the case of Fig. 12(b), some roots are selected in the left plane and others are in the right plane. In the case of Fig. 12(c), the roots are selected according to a zigzag pattern so that they can placed about the imaginary axis as symmetrically as possible. According to (24), the symmetry ratio is minimum in the first case, whereas it is maximum in the third case. This can be verified by checking the difference between and in corresponding coupling matrices. In the first case, as shown in Fig. 3, the difference is. In the second case, as shown in Fig. 13(a), the difference is,which is slightly smaller. In the third case, as shown in Fig. 13(a), the difference is, which is the smallest. Accordingly, different choices of the roots will result in different symmetry ratios in the layout. VIII. CONCLUSION A systematic synthesis method has been presented for crosscoupled DDSs with arbitrary prescribed group delay and controlled magnitude. Several design examples with different prescribed group-delay responses have been presented to illustrate the proposed synthesis method. A waveguide prototype is implemented and fabricated. The measured responses are in very good agreement with the prescribed and full-wave responses. REFERENCES [1] A. Oppenheim, R. Schafer, and J. Buck et al., Discrete-Time Signal Processing. New York, NY, USA: Prentice-Hall, [2] M. Lewis, SAW and optical signal processing, in IEEE Proc. Ultrason. Symp., Sep. 2005, vol. 24, pp [3] C. Campbell, Surface Acoustic Wave Devices and Their Signal Processing Applications. New York, NY, USA: Academic Press, [4] S. Gupta and C. 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10 1052 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 3, MARCH 2013 [17] M. Fahmy, The use of padé approximants in the derivation of distributed low-pass filters with simultaneous flat amplitude and delay characteristics, Int. J. Circuit Theory Appl., vol. 8, no. 3, pp , [18] M. Fahmy, On the problem of filters with arbitrary amplitude and phase constraints, Int. J. Circuit Theory Appl., vol. 5, no. 1, pp , [19] M. Abo-zahhad and T. Henk, Design of selective lowpass sampleddata and digital filters exhibiting equiripple amplitude and phase error characteristics, Int. J. Circuit Theory Appl., vol. 23, no. 1, pp , Jan [20] J. Rhodes, Filters approximating ideal amplitude and arbitrary phase characteristics, IEEE Trans. Circuit Theory, vol. CT-20, no. 2, pp , Mar [21] T. Henk, The generation of arbitrary-phase polynomials by recurrence formulae, Int. J. Circuit Theory Appl., vol. 9, no. 4, pp , Oct [22] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. New York, NY, USA: Dover, 1964, vol. 55, no [23] D. Pozar, Microwave Engineering. New York, NY, USA: Wiley, Qingfeng Zhang (S 07 M 11) was born in Changzhou, China, in December He received the B.E. degree in electrical engineering from the University of Science and Technology of China (USTC), Hefei, China, in 2007, and is currently workingtowardtheph.d.degreeinelectricaland electronic engineering at Nanyang Technology University, Singapore. His Ph.D. thesis is focused on dimensional synthesis of wideband waveguide filters. Since April 2011, he has been a Postdoctoral Fellow with the Poly-Grames Microwave Research Center, École Polytechnique de Montréal, Montréal, Canada. His current research interests include filter synthesis theory, DDSs, ASP systems, leaky-wave antennas, and metamaterials. Mr. Zhang studied under a Nanyang Technology University scholarship from August 2007 to November Dimitrios L. Sounas (M 11) was born in Thessaloniki, Greece, in September He received the Diploma and Ph.D. degrees in electrical and computer engineering from the Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in 2004 and 2009, respectively. From August 2010 to October 2012, he was a Post-Doctoral Fellow with the Electromagnetic Theory and Applications Research Group, École Polytechnique of Montréal. In November 2012, he joined the Metamaterials and Plasmonics Research Group, The University of Texas at Austin, Austin, TX, USA, as a Post-Doctoral Fellow. His research interests include analytical and numerical techniques in electromagnetics, metamaterials, and graphene-based structures. Christophe Caloz (S 00 A 00 M 03 SM 06 F 10) received the Diplôme d Ingénieur en Électricité and Ph.D. degree from the École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer with the Microwave Electronics Laboratory, 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 the Holder of a Canada Research Chair (CRC). He has authored or coauthored over 420 technical conference, letter, and journal papers, 12 books, and book chapters. He holds several patents. His research interests include all fields of theoretical, computational, and technological electromagnetics engineering with a strong emphasis on emergent and multidisciplinary topics, including particularly nanoelectromagnetics. Dr. Caloz is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Technical Committees MTT-15 (Microwave Field Theory) and MTT-25 (RF Nanotechnology). He is a speaker of the MTT-15 Speaker Bureau, the chair of Commission D (Electronics and Photonics), Canadian Union de Radio Science Internationale (URSI), and an IEEE MTT-S representative of the IEEE Nanotechnology Council (NTC). He was the recipient of 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 Best Paper Awards with his students.