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352 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 2, FEBRUARY 2009

Highly Efficient Grouping Strategy for theAnalysis of Two-Port Arbitrarily Shaped

H -Plane Waveguide DevicesÁngel Belenguer Martínez, Héctor Esteban González , Member, IEEE ,

Vicente E. Boria Esbert , Senior Member, IEEE , Carmen Bachiller, and José Vicente Morro

Abstract—A new grouping strategy for the analysis of two-portarbitrarily shaped -plane devices in rectangular waveguidesis presented. The new grouping strategy accelerates a previousmethod based on the method of moments and the Green’s functionof an infinite line source placed inside two parallel plates. Thecomputational cost of analyzing several -plane filters of differentgeometries is reduced by about 50%, while maximum accuracy is

maintained, as a result of using the new grouping strategy. IndexTerms—Cavity resonator filters, Green’s function, method

of moments (MoM), rectangular waveguides, scattering.

I. INTRODUCTION

T HE WELL-KNOWN method of moments (MoM) [1] istypically employed for the analysis of antennas and scat-

tering problems in free space [2], [3] where no boundary condi-tions are imposed on the corresponding Green’s function. How-ever, the MoM can also be applied to the analysis of electromag-

netic problems with boundary conditions, such as -plane de-vices in waveguide technology [4]–[6], which are widely used intelecommunication applications, as can be inferred from severalrecent publications [7]–[11]. To apply the MoM to the analysisof rectangular waveguide -plane devices, the Green’s func-tion of an infinite line source between two parallel plates mustbe used [12]. This Green’s function, defined as the summationof a series of guided modes [13], has been previously used tosolve the scattering of single [14], [15] or multiple [16], [17]

-plane posts in a rectangular waveguide.The convergencerateof the summation of this Green’s function has been accelerated,either using the Kummer’s transformation [14], [17], or by em-

ploying particular basis functions in the application of MoM[16]. Recently [18], the convergence rate of the MoM with theGreen’s function of a line source between parallel plates has

Manuscript received July 28, 2008; revised October 26, 2008. First publishedJanuary 19, 2009; current version published February 06, 2009. This work wassupported by the Universidad Politécnica de Valencia.

Á. Belenguer Martínez is with the Departamento de Ingeniería eléctrica,electrónica, automática y comunicaciones, Universidad de Castilla-La Mancha,Castilla-La Mancha, Spain (e-mail: [email protected]).

H. Esteban González, V. E. Boria Esbert, C. Bachiller, and J. V. Morroare with the Departamento de Comunicaciones, Universidad Politécnicade Valencia, 46022 Valencia, Spain (e-mail: [email protected];[email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TMTT.2008.2011199

been improved by first evaluating the integrals related to the ap-plication of MoM, and then summing the resulting terms, in-stead of first summing the Green’s function and then solvingthe integral equations of MoM, as this was traditionally han-dled. This new approach simultaneously reduces the computa-tional cost and increases accuracy, as the distributed sources can

now be placed along the surface of the scatterer, and no singu-larity arises because it disappears through integration before theGreen’s function is summed. In this paper, we significantly im-prove the numerical efficiency of the technique presented in [18]by grouping the radiation of neighboring MoM contour cells.

The idea of grouping the electromagnetic fields produced bycertain groups of cells is not new. This technique has been typ-ically employed in the fast multipole method (FMM) [19]–[22]and in the multilevel fast multipole algorithm [23]–[25], inboth cases providing important efficiency improvements. Thesemethods accelerate the solution of the matrix system providedby the MoM, by using an iterative technique such as conjugate

gradients, biconjugate gradients, biconjugate gradients stabi-lized, and others [22], [26]–[29]. These iterative techniquesdo not directly solve the matrix system of the MoM,but instead they approximate the solution iteratively throughmultiple matrix-vector products of the form . Again theseproducts are not computed directly, but they are evaluated moreefficiently thanks to a strategy based on three steps, whichare: 1) aggregation of the radiation of neighboring cells into asingle radiation pattern; 2) translation; and 3) disaggregationof the incoming field into each MoM cell. Those groupingstrategies imply a cost reduction of the matrix-vector productfrom to for the FMM, and forthe multilevel fast multipole algorithm, where represents the

number of discretization cells considered along the scatterercontours.

Due to this important cost reduction, the FMM and the mul-tilevel fast multipole algorithm are widely used in the analysisof complex and large scattering structures. Recent studies arefocused on improving the accuracy by refining several subpro-cesses of the grouping strategy, e.g., by the anterpolation/inter-polation of radiation patterns of groups [30], [31], or by usingbetter sets of basis functions in the MoM current expansion [32],[33]. Other studies try to improve the efficiency by acceleratingthe convergence of the iterative algorithm used in the matrixsystem solution [34]–[36] or trying to parallelize the computa-

tions [37]–[41]. Finally, there are other papers that are focused0018-9480/$25.00 © 2009 IEEE

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BELENGUER MARTÍNEZet al.: GROUPING STRATEGY FOR ANALYSIS OF TWO-PORT ARBITRARILY SHAPED -PLANE WAVEGUIDE DEVICES 353

on using these algorithms for analyzing new 2-D or 3-D struc-tures in free space or in free semispaces, as in [42]–[46].

In this paper, we propose a method for analyzing two-portarbitrarily shaped -plane devices in rectangular waveguides.This new method uses the technique proposed in [18] for accel-erating theconvergence andimproving theaccuracy of the MoM

using the Green’s function of a line source among parallel-platewaveguide. The numerical efficiency of this technique is im-proved here with the use of a new grouping strategy inspired bythe FMM.

II. ACCELERATION PROCEDURE

The arbitrarily shaped -plane waveguide devices are ana-lyzed by applying MoM (pulses as basis functions and deltas astest functions, i.e., point matching) [1], which leads to the fol-lowing matrix system:

(1)

Once this matrix system is solved, we obtain a set of coeffi-cients , which can be used to reconstruct the current inducedover the scatterers contours due to the excitation . Using thecoefficients , it is possible to compute the generalized scat-tering matrix of the global structure according to the algorithmrecently presented in [18].

The most time-consuming process of the MoM is solvingthe matrix system. An alternative to directly solving the ma-trix system, especially with large matrix systems, is the useof an iterative algorithm such as conjugate gradients, biconju-gate gradients, biconjugate gradients stabilized, and others [22],

[26]–[29]. These algorithms approximate the solution itera-tively through multiple matrix-vector products of the form .These matrix products are also time consuming, but we canobtain an accurate approximation for them if we take into ac-count that they represent the field scattered by the whole struc-ture with current distribution at the center of each MoM cell.Using this physical interpretation for the product , we pro-pose a grouping strategy similar to that used in the FMM. Themain difference between those techniques andour method is thatthey are typically applied to open space problems (i.e.,antennas)while our method will be applied to -plane devices in a rectan-gular waveguide (see Fig. 1). As in the FMM, we approximate

the matrix product with a grouping strategy in the followingthree independent steps [20], [21].Step 1) First, grouping of neighbor cells with the proper

combination of their scattered fields into a single“modal radiation pattern.”

Step 2) Next, the “modal radiation pattern” is translated tothe reference plane of each target group.

Step 3) Finally, the summation of all the “modal radiationpatterns” incident over each target group is evalu-ated at each one of the cells belonging to that group,and this completes a process which is equivalent tothe matrix-vector computation.

In the FMM terminology, the first process is named “ag-gregation,” the second one “translation,” and the third and last

Fig. 1. Rectangular waveguide with line source. (a) 3-D view. (b) Top view.

one “disaggregation.” We must adapt this grouping strategy toclosed problems in order to achieve an important improvementin the global numerical efficiency.

A. Aggregation

In open space, where the field is expanded into open-spacemodes that propagate radially, the grouping of the radiation of neighbor cells is based on a 2-D or 3-D proximity criterion.However, inside an -plane device in rectangular waveguide,the geometry is invariant in height ( dimension in Fig. 1). Con-sidering that the exciting field is the fundamental mode,which is also invariant along , then all the scattered fields bythe -plane structure are also invariant inheight ( modes).Therefore, the 3-D structure of Fig. 1(a) can be reduced to theequivalent 2-D problem of Fig. 1(b).

In this case, we have to choose another criterion to makegroups. Since our field spectrum is a set of guided modespropagating either to the right or to the left , we haveto make groups according to their separation along the propa-gation axis so we apply a 1-D proximity criterion, and groupthe MoM cells (line sources) into groups, as shown in theexample of Fig. 2. The groups are always ordered from left toright.

As in the FMM, we should now find a way to combine thefield scattered by all the cells of the same group. Since the fieldscattered by a line source inside the parallel plates is expandedas a summation of modes propagating either to the right or left,we must aggregate the total field scattered by all the cells of a

group to the right (propagating toward ), and to the left (prop-agating toward ). We must specify two reference planes, one


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Fig. 2. Example of the 1-D grouping strategy for a four-pole coupled cavitiesH -plane filter with rounded corners in the coupling windows. The MoM cellsare grouped into G = 5 groups.

Fig. 3. Position of the reference planes for right and left propagating modes forthe g th group in an H -plane coupled cavities filter. The coordinates of the n thMoM cell are also shown.

for the set of modes propagating to the right, and another for themodes propagating to the left (see Fig. 3).

Let be the group that is being processed, be the po-

sition in the axis of the reference plane for left propagatingmodes, and be for the right propagating modes. Let bethe number of cells in the group , and be the weight of the MoM basis function of the th cell belonging to that group.Taking everything into consideration and applying the resultsof [18], we can compute the left and right propagatingspectra as

(2)

(3)

where identifies the MoM cell and identifies theguided mode, being the higher mode considered

for the analysis. The values of and can be found in[18].

To simplify the expressions, we can put them in matrix form.For that purpose, we define two new matrices of ele-ments

(4)(5)

Now the left and right propagating spectra of group can becomputed in matrix form as

(6)

(7)

where is a vector of size and stores thecurrent weights of the MoM cells of the th group

.

B. Translation

In the aggregation step, we have combined the field scatteredby all the cells of a group into two spectra, one propagating tothe left and another one to the right. The spectra of each groupare related to their own reference planes, so to compute the totalfield incident to a particular group , we must translate all the in-coming spectra from other groups to the same reference planes;one for the spectra that come from the left and another forthe spectra that come from the right .

The translation of a guided modes spectrum from one to an-other reference plane only requires a phase shift for each mode,and that depends on the distance from one to another refer-ence plane. To translate the right propagating spectrum of group

, whose reference plane is , to the left referenceplane of group ( must be to the right of ), i.e., to ,we just multiply by a diagonal matrix whose ele-ments are

(8)

Similarly, if we want to translate the left propagating spec-trum of group to the right reference plane of group , wemust use the diagonal matrix , whose elements are

(9)

Equations (8) and (9) are identical. Therefore,

(10)

and, therefore, we only need to compute left or right translationmatrices.

Once we have computed the translation matrices, we canwrite an expression for the total field incident to a particulargroup coming from the MoM cells of the rest of the groups

(11)

(12)

where and are, respectively, the incoming field spectrafrom the right and left. The groups are ordered from left to right,

so for a particular group , the groups are atthe left side of group and the groups are


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placed at the right side of group , and being the total numberof groups.

C. Disaggregation

Once we have computed for each group the incomingspectra from the right and from the left , we must

translate each spectrum and evaluate the field at the centerof each MoM cell of group . This step is called

disaggregation.The right and left incoming spectra that propagate, respec-

tively, to the left and to the right can be translatedand evaluated at the center of the th cell of group

in matrix form using

(13)

(14)

where and are m atrices o f elements of the f orm

(15)

(16)

The total field over each cell in group due to thescattering of the rest of the groups is

(17)

where is a vector of elements. Each element repre-sents the field scattered over the th cell in group by all the

cells, except those in group .Substituting (13) and (14) in (17),

(18)

Substituting (11) and (12) in the former equation

(19)

and finally substituting (6) and (7), we obtain

(20)

D. Intra-Group Scattering

Equation (20) provides the field scattered over the cells of group by all the cells of the other groups. To compute the totalfield incident for these cells, we must still consider the field thatcomes from the other cells of the same group

(21)

where is the submatrix of the entire MoM coefficients ma-

trix of (1) that contains the interactions among cells of group .In [18], there is a complete study of the value of the different pa-

rameters involved that have to be used to obtain a good tradeoff between accuracy and efficiency.

Finally, the total field incident over the cells of group is

(22)

where the entire vector with the field incident to all the cellsof the MoM problem is

...(23)

With the new grouping strategy, we avoid the computation of the whole MoM coefficients matrix since we only computethe interactions among cells of the same group. Besides, we donot compute the matrix product . The computational cost isreduced and the only loss of accuracy comes from the truncation

of the Green’s function, as in [18]. The new grouping strategydoes not add any additional loss of accuracy since we obtainexactly the same vector , but using (22), instead of computing

and multiplying by .

E. Number of Modes

The number of guided modes considered for theaggregation, translation, and disaggregation stages cannotbe infinite. The highest order used in all calculations is

. The optimum value for depends onthe distance between groups. If we call this distance , thecriterion that we have chosen is to ensure for each frequency

point that the scattered field corresponding to the highest ordermode vanishes from one group to another below somethreshold value so that

(24)

Therefore, must be

(25)

where has been proven to provide a good tradeoff

between accuracy and efficiency.

F. Computational Cost

Below we are going to analyze which is the number of operations required for the computation of the matrix product

using the new grouping strategy, i.e., (22). Thenumber of operations needed for filling all the matrices

is not considered sincethis is done only once, while the matrix product is performedseveral times by the iterative technique (biconjugate gradientsstabilized) that solves the MoM matrix system so these matrixproducts are the most time-consuming process. The number of

operations needed to evaluate the matrix product for eachstep of the grouping strategy is analyzed separately.


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Fig. 4. H -plane filters (measures in millimeters). (a) Four-cavity filter with rounded corners at coupling windows r = 1 mm. (b) Six-cavity filter with roundedcorners at coupling windows r = 2 mm. (c) Six-cavity filter with rounded corners at resonant cavities r = 3 mm. (d) Round rod bandpass filter from [49].(e) Wedge bandpass filter from [50].

Step 1) Aggregation step: the products andmust be computed for all the groups (there aregroups). For all these products we need

operations since , and is thenumber of MoM cells.

Step 2) Translation step: we need to compute the productsfor and

for . Since and arediagonal matrices of size , those products suppose

operations. This must be repeated for allthe groups with . In the end, we need

operations for this step.Step 3) Disaggregation step: the products and

must be computed for all groups. Thatmeans operations for this step.

Step 4) The computational cost of computing the field scat-tered by cells of the same group over themselves

supposes operations.Finally, the total number of operations needed with the new

grouping strategy to compute the vector is of the orderof

(26)

If the number of MoM cells is much bigger than thenumber of groups , and groups are placed far away so thatis small compared with , we expect a number of operationsof the order of .

If we do not use the grouping strategy, and compute the entirematrix , and then multiply by vector , this matrix productneeds a number of operations of the order of

(27)

The order in number of operations can, therefore, be reducedupto a factor of provided that ismuchbigger than and .However, the relation between , , and depends on the par-ticular geometry that we are analyzing since we can not freelychose the number of subgroups as in the FMM, and de-pends on the separation along the propagation axis of the dif-ferent groups. The same reduction of the computational time

applies to the computer memory needed to store the matricesinvolved in the calculations when the grouping strategy is used.

This result is especially important since the MoM matrixsystem of (1) is solved using an iterative technique that doesnot directly solve the matrix system of (1), but instead approx-imates the solution iteratively through multiple matrix-vectorproducts of the form . With the new grouping strategy, thecost of these products and the memory needed is reduced by ,the number of groups used in the grouping strategy.

Besides, with the new grouping strategy it is not necessaryto compute the entire matrix since only the smaller matrices

with must be computed.

III. RESULTS

The accuracy and efficiency of the new grouping strategy istested with the analysis of five -plane filters in rectangularwaveguide. These filters, whose top view is shown in Fig. 4, areas follows.

1) Four-cavity -plane coupled cavity filters. The couplingwindows present rounded corners (radii mm) due tolow cost manufacturing techniques [47].

2) Six-cavity -plane coupled cavity filters. The couplingwindows present rounded corners (radii mm) dueto low cost manufacturing techniques [48].

3) Six-cavity -plane coupled cavity filters. The resonant

cavities present rounded corners (radii mm) due tolow-cost manufacturing techniques [48].


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Fig. 5. Frequency response of the H -plane filters of Fig. 4 compared with results from the bibliography.

4) Bandpass filter with seven round rods [49].5) Nine-pole bandpass filter with wedges [50].The frequency response (reflection coefficient) of these five

filters has been computed using the MoM method of [18] ac-celerated with the new grouping strategy presented here. Thebiconjugate gradients stabilized method has been used to ac-celerate the solution of the MoM matrix system of (1). Sincethe MoM method uses the Green’s function of a line source be-tween two parallel plates, only the portions of the metallic sur-face of the structure that are outside of the parallel-plate wave-guide must be segmented using the MoM. This means that onlythe coupling windows for the first three filters, or the round rodsand wedges for the last two filters, must be segmented into MoMcells. The new grouping strategy groups the MoM cells intogroups so that those MoM cells that are close together in the

propagation direction must be enclosed in the same group forthe grouping strategy to be efficient. This means, for instance,that the first filter is analyzed with all MoM cells split intogroups, one for each coupling window, and similarly the MoMcells of the fourth filter are grouped into groups, one foreach round rod.

The results of the analysis of the five filters are presented inFig. 5, where they are compared with other analysis methods[47]–[50]. Fig. 5 shows a very good agreement with the resultsfrom the literature for the five filters analyzed.

Once the accuracy of the new method is fully proven, we willtest the efficiency improvement of the new grouping strategy.For that purpose, we present in Table I the CPU time (in sec-

onds per frequency point) required to analyze each one of thefive -plane filters of Fig. 4. Table I shows both the CPU time

TABLE ICOMPUTATIONAL COST FOR THE ANALYSIS OF THE FIVE FILTERS OF Fig. 4.

COMPARISON BETWEEN THE MoM METHOD OF [18] AND THE SAME METHOD

ACCELERATED WITH THE NEW GROUPING STRATEGY PROPOSED IN THIS

PAPER. CPU TIME IS IN SECONDS PER FREQUENCY POINT

required by the MoM with the Green’s function of a line sourcebetween parallel plates [18] and the CPU time of the same anal-ysis method accelerated with the new grouping strategy.

Results from Table I show that the time is reduced by a sig-

nificant factor of 2 or 3, depending on the filter. However, thenumber of groups varies from five in the first filter to ten inthelast filter. As mentioned before, a time reduction of could beexpected when the number of MoM cells is much bigger than thenumber of groups and the number of guided modes . Since

and are not selectable, we cannot ensure a time reductionof . In the geometry of the last two filters, there is a muchshorter metallic surface for each group so the ratio ,i.e., the number of MoM cells per group, is much smaller, andso the last term in (26) is more important than in the case of the three first filters. This results in a smaller time reduction.However, this time reduction becomes especially significant indesign processes where many simulations of the structure must

be performed in order to obtain the optimal values of the designparameters.


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There are other factors that could explain that the time reduc-tion is lower than such as the fact that we have not consideredother processes that consume time, such as the computation of the elements needed to fill the matrices , , , ,

, , and , or the time spent in reading and writing inmemory and disk.

A similar time reduction could have been obtained with aclassical segmentation approach [51], where a generalized ma-trix of circuit parameters is obtained for each group and then allthe matrices are cascaded in order to obtain the global circuitparameters of the problem. However, the segmentation strategypresented here provides much more accurately the field insidethe structure since all the currents in all the structure are ob-tained at the same time, and the field is directly computed fromthe amplitudes of these currents. With a classical segmentationstructure, a complex process must be followed in order to findthe modal amplitudes in each intermmediate point of the struc-ture. Next these modal amplitudes are used to expand the fieldinside each segment of the filter, and small discontinuities usu-ally appear between adjacent segments due to loss of accuracyinherent to the segmentation process.

IV. CONCLUSIONS

A new grouping strategy has been presented in this paper.This new strategy accelerates the MoM method with the Green’sfunction of a line source between two parallel plates presentedin[18]. The efficiency improvement of the new grouping strategyis especially important when the MoM matrix system is solvedusing an iterative technique such as conjugate gradients, bicon-

jugate gradients, biconjugate gradients stabilized, and otherssince these techniques require multiple matrix-vector products

of the MoM coefficients matrix by a vector that iteratively con-verges to the solution. This matrix product is not solved directlywith the new grouping strategy, but is instead solved withoutloss of accuracy using a more efficient three-step procedure.

The accuracy and efficiency of the new grouping strategy hasbeen tested with the analysis of five -plane filters. The reflec-tion coefficient of these filters has been successfully comparedwith results from the technical literature, and a significant timereduction of 40%, 50%, or 60% has been obtained dependingon the type of filter and the number of MoM cells per group.

ACKNOWLEDGMENT

The authors would like to thank the R&D&I Linguistic As-sistance Office, Universidad Politécnica de Valencia, Valencia,Spain, for the linguistic revision of this paper.

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Ángel Belenguer Martínez received the Telecom-munications Engineering degree from the Univer-sidad Politécnica de Valencia (UPV), Valencia,Spain, in 2000.

In 2000, he joined the Universidad de Castilla-LaMancha, Castilla-La Mancha, Spain. His researchinterests include methods for the full-wave anal-ysis of open-space and guided multiple scatteringproblems, acceleration of electromagnetic analysismethods using the wavelets, FMM, multilevel fastmultipole algorithm, and use of specific basis and

Green’s functions for the MoM.

Héctor Esteban González (S’93–M’99) received

the Telecommunications Engineering degree andPh.D. degree from the Universidad Politécnica deValencia (UPV), Valencia, Spain, in 1996 and 2002,respectively.

He then joined the Joint Research Centre, Euro-pean Commission, Ispra, Italy. In 1997, he was withthe European Topic Centre on Soil (European Envi-ronment Agency). In 1998, he rejoined the UPV. Hisresearch interests include methods for the full-waveanalysis of open-space and guided multiplescattering

problems, computer-aided design (CAD) design of microwave devices, electro-magnetic characterization of dielectric and magnetic bodies, and the accelera-tion of electromagnetic analysis methods using the wavelets and FMM.

Vicente E. Boria Esbert (S’91–A’99–SM’02)received the Ingeniero de Telecomunicación and theDoctor Ingeniero de Telecomunicación degrees fromthe Universidad Politécnica de Valencia, Valencia,Spain, in 1993 and 1997.

In 1993, he joined the Universidad Politécnicade Valencia, where he has been a Full Professorsince 2003. In 1995 and 1996, he held a SpanishTrainee position with the European Space Researchand Technology Centre (ESTEC)–European SpaceAgency (ESA). His current research interests include

numerical methods for the analysis of waveguide and scattering structures,automated design of waveguide components, radiating systems, measurementtechniques, and power effects in passive waveguide systems.

Dr. Boria Esbert has served on the Editorial Board of the IEEETRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.


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Carmen Bachiller received the Communication En-gineeringdegree from the Universidad Politécnica deValencia (UPV), Valencia, Spain, in 1996, and is cur-rently working toward the Ph.D. degree in electro-magnetism and RF circuits at the UPV.

From 1997 to 2001, she was a Project Engineerwith the ETRA I + D Company, where she was in-volved with research and development on automatic

traffic control, public transport management, andpublic information systems using telecommunicationtechnology. In 2001, she joined the Departamento de

Comunicaciones, UPV, as an Assistant Lecturer, where she currently teachessignal and systems theory and microwaves. She has participated in severalteaching innovation projects.

José Vicente Morro received the Telecommuni-cations Engineering degree from the UniversidadPolitécnica de Valencia (UPV), Valencia, Spain, in2001, and is currently working toward the Ph.D.degree at UPV.

In 2001, he became a Research Fellow with theDepartamento de Comunicaciones, UPV. In 2003, he

joined the Signal Theory and Communications Divi-

sion, Universidad Miguel Hernández, where he wasa Lecturer. In 2005, he rejoined the Departamento deComunicaciones, UPV, as an Assistant Lecturer. His

current interestsincludecomputer-aideddesign (CAD)design of microwave de-vices and electromagnetic optimization methods.

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