Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 415069 13 pageshttpdxdoiorg1011552013415069

Research ArticleAnalysis of Arbitrary Reflector Antennas Applyingthe Geometrical Theory of Diffraction Together withthe Master Points Technique

Mariacutea Jesuacutes Algar Jose-Ramoacuten Almagro Javier MorenoLorena Lozano and Felipe Caacutetedra

Departamento de Ciencias de la Computacion Universidad de Alcala Alcala de Henares 28871 Madrid Spain

Correspondence should be addressed to Marıa Jesus Algar chusalgaruahes

Received 2 November 2012 Accepted 4 January 2013

Academic Editor Miguel Ferrando

Copyright copy 2013 Marıa Jesus Algar et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An efficient approach for the analysis of surface conformed reflector antennas fed arbitrarily is presented The near field in a largenumber of sampling points in the aperture of the reflector is obtained applying theGeometricalTheory ofDiffraction (GTD) A newtechnique namedMaster Points has been developed to reduce the complexity of the ray-tracing computations The combination ofboth GTD andMaster Points reduces the time requirements of this kind of analysis To validate the new approach several reflectorsand the effects on the radiation pattern caused by shifting the feed and introducing different obstacles have been consideredconcerning both simple and complex geometries The results of these analyses have been compared with the Method of Moments(MoM) results

1 Introduction

Reflector antennas have been used since 1888 when electro-magnetic waves were discovered by Hertz Although theseantennas are very common in radar systems they were not inwidespread use until the SecondWorldWar [1 2]Thereafterthe demand of reflector antennas has grown in severalfields such us radio astronomy microwave communicationspace communications and radio links working atmillimeterfrequencies [3]

This increasing interest is due to the pencil-beam-shapedpattern and the possibility to configure the radiation patternof a reflector by changing the shape andor its feed arrayIn this way the characteristics of its radiation pattern forexample the effective area or the relationship between themain and the secondary lobes can be adjusted to achievespecific requirements For that reason these antennas havea wide range of applications like satellite communicationsand radar systems which demand an accurate and efficientanalysis of complex reflectors

The shape and directivity of the radiation pattern of areflector antenna depend on the electric and magnetic fielddistribution on its apertureTherefore the configuration of itsradiation pattern is closely linked with its geometrical modelConsequently according to the geometry of the reflectorthese antennas can be classified into dihedral cylindricalparabolic elliptical spherical offset and hyperbolic reflec-tors The most commonly used is the parabolic reflector dueto its highly directive radiation pattern

High frequency techniques such us physical optics (PO)[4] or the GeometricalTheory of Diffraction (GTD) [5] havebeen used extensively in the literature [6ndash11] to evaluate theradiation pattern of reflector antennas For instance in [7] itis remarked that the GTD analysis provides good accuracy inthe results for the prediction of the main beam and sidelobesof a parabolic reflector Furthermore it has been demon-strated that the physical optics radiation integral provides anaccurate solution for predicting the far field radiated by reflec-tor antennas However due to the large electrical dimensionsof these structures the PO integral accounts for most of the

2 International Journal of Antennas and Propagation

computation time For that reason [11] presents a domaindecomposition algorithm to compute the radiation pattern ofa finite aperture with a high computational efficiency In thisregard a new technique to compute the PO integral in amoreefficient way is presented in [6] The procedure presented in[8] is based on a series of Fourier transforms of an aperturedistribution which takes into account the curvature of thesurface Additionally the aperture blockage effect on the pat-tern of a quadruped supported primary-feed parabolic reflec-tor is evaluated in [9] where the scattering process is obtainedas an interaction between the reflector structure and theaperture-blocking feed together with its supporting struts

This paper is focused on the problem of analyzing theradiation pattern of arbitrarily shaped reflector antennasapplying GTDThe electrical field in a large number of pointssampled on the aperture of the reflector is obtained Thatis the ray tracing algorithm is calculated for each observa-tion point and this process spends too much CPU timeTherefore GTD has been combined with a new techniquenamed Master Points to accelerate the calculation of theelectromagnetic field in this kind of analysis The speed-upis possible since this method reduces the number of times inwhich it is necessary to calculate the ray tracing to obtain theelectromagnetic field in a given number of observation pointsor directions

In the ray tracing algorithm the Conjugate GradientMethod (CGM) [12] is applied to obtain the reflection pointson the reflector surfaces This method minimizes a costfunction which depends on the distance between the sourceand any given observation point or direction Usually thegeometric model of a complex body is composed of flat andconvex surfaces However the geometrical model of a para-bolic reflector is composed of concave instead of convex sur-facesTherefore a new distance function to analyze a concavesurface is proposed

This paper is organized as follows Section 2 reviewsthe geometric and electromagnetic analysis of a canoni-cal parabolic reflector The ray tracing algorithm used tocompute the radiation pattern of these antennas and theimprovements done in this process are presented in Sections3 and 4 respectively Section 5 is focused on the application ofthe Master Points technique to perform this analysis FinallySection 6 shows several cases of study to validate this newapproach including the analysis of the blockage in a parabolicreflector caused by an obstacle

2 Application of the Ray-Tracing Algorithm toAnalyze an Arbitrarily Shaped Reflector

As it has been outlined in the Introduction section when theradiation pattern of a reflector antenna is computed the firststep is to calculate the electromagnetic field distribution on avery dense mesh of points sampled on its aperture The mainpurpose of this step is to perform it as fast as possible To com-pute the ray tracing on a complex structure two situations aredistinguished depending on the geometry of the object

(i) geometry composed of flat surfaces in this casethe search for reflection points on the geometry is

Figure 1 Simple reflection on a convex surface

performed analytically using the Image Theory (IT)[13]

(ii) geometry composed of curved surfaces regardingarbitrary surfaces like NURBS [14 15] the searchfor reflection points becomes more complicated andminimizationmethodsmust be applied in particularthe CGM [12] which minimizes a cost function isused in this paper additionally the minimizationprocess for a convex surface and for a concavesurface is different the following sections describe thedifferences for both surfaces

21 Reflection Point on Convex Surfaces The search forreflection point on a geometry composed of convex surfaceis complicated as the Image Theory cannot be applied Asimple schema describing this situation for a given sourceand observation point is shown in Figure 1 To determinethe point within the area on which the reflection will takeplace the CGM minimizes a cost function This function isdifferent depending on whether the near field or the far fieldis analyzed For instance in the analysis of the near field thiscost function is expressed as follows

119889 (119906 119907) = 119889119894 (119906 119907) + 119889119903 (119906 119907) =1003816100381610038161003816119894 (119906 119907)

1003816100381610038161003816 +1003816100381610038161003816119903 (119906 119907)

1003816100381610038161003816 (1)

where 119889119894(119906 119907) is the distance between the source and the

candidate reflection point and119889119903(119906 119907) is the distance between

the candidate reflection point and the observation pointHence

119894is the vector from the source to the candidate

reflection point and 119903is the vector from the candidate

reflection point to the observation pointThe minimization process of the CGM method imple-

ments the following steps

(1) First of all a seed point over the convex surface isselected

(2) Then the cost function and its partial derivatives areevaluated in this point

International Journal of Antennas and Propagation 3

(3) If the partial derivatives are null the minimum ofthe function has been reached Consequently thesearch process finisheswith that seed point as the finalsolution the reflection point

(4) Otherwise the seed point is shifted over the curveaccording to the direction given by the partial deriva-tives

(5) This last seed point is evaluated repeating step (2)

This iterative process will be completed when the mini-mum of the function is found or when it exceeds a certainnumber of iterations

This shift from one seed point to another is done accord-ing to the partial derivatives of (1) which are given by thefollowing expressions

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) + 119903 (119906 119907)] sdot 119903

119906 (119906 119907)

120597119889 (119906 119907)

120597119907=120597119889119894 (119906 119907)

120597119907+120597119889119903 (119906 119907)

120597119907

= [119894 (119906 119907) + 119903 (119906 119907)] sdot 119903

119907 (119906 119907)

(2)

where 119903119906(119906 119907) and 119903

119907(119906 119907) are the partial derivatives with

respect to the parametric coordinates 119906 and 119907 of the reflectionpoint

When the far field is analysed the CGM minimizes thefollowing cost function

119889 (119906 119907) = 119889119894 (119906 119907) + 119889119903 (119906 119907)

=1003816100381610038161003816119894 (119906 119907)

1003816100381610038161003816 + (119863 minus 119907 sdot 119903 (119906 119907))

(3)

where 119889119894(119906 119907) is the distance between the source and the

candidate reflection point119889119903(119906 119907) is the distance between the

candidate reflection point and a perpendicular plane to theobservation direction

119894(119906 119907) is the vector from the source

to the candidate reflection point 119907 is a unitary vector whichdetermines the observation direction 119863 is the independentterm of the equation that defines the perpendicular planeto the observation direction and 119903(119906 119907) is the candidatereflection point on the surface

The minimization process and the associated shift of theseed point are performed as in the previous case The partialderivatives of (3) are given by the following expressions

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) minus 119907] sdot 119903

119906 (119906 119907)

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) minus 119907] sdot 119903

119907 (119906 119907)

(4)

Once the reflection point has been found on the surfacea shadowing test is conducted to affirm that neither reflected

Figure 2 Simple reflection on a concave surface

nor incident paths are hidden by any surface Likewise thereflection point must be located on the curve and Snellrsquos lawmust be satisfied at that point [16] as follows

119894sen 120579119894= 119903sen 120579119903 (5)

If any of these conditions are not satisfied the reflectionpoint found by the CGM is not the right solution and thealgorithm will need to find another point

22 Reflection Point on Concave Surfaces The cost functionsconsidered in this case for either the near or the far fieldanalysis are the same as in the case of a convex curve andthey are described by expressions (1) and (3) respectivelyFigure 2 shows this case of study schematically

Instead of the search process for the reflection point ona convex surface in which the CGM looks for a minimumof the cost function in this case the CGM looks for botha minimum and a maximum of the function presented pre-viously

Before this process starts a sampling of the curve onwhich the reflection point has to be found along the paramet-ric coordinates 119906 and 119907 is performed Each of these samplesis considered as a seed point for the search of a minimum ora maximum As in the minimization process over a convexsurface the shifted over the curve is done according tothe sign of the derivative functions expressions (2) or (4)depending on the case (the near field or the far field resp)Once the solution point is found if it belongs to the curvesatisfies Snellrsquos law and none of the incident and reflectedpaths are shadowed by any obstacle it can be concluded thatthe reflection point has been found on the concave surface

In order to better understand the minimization processimplemented by CGM Figure 3 shows a schema applicableto convex or concave curves

3 Improvements Done overthe Ray-Tracing Algorithm

The geometrical model of reflector systems is mainly com-posed of curved surfaces in particular concave surfacesHowever in the analysis of the radiation pattern of antennason board complex structures the geometrical models used

4 International Journal of Antennas and Propagation

Selection of the seed point

Move over the surface

End

No

Yes

Number ofiterations

No

Yes

Point belongsto parametric

space

Yes

No

Calculation 119889( )119906 119907

120597119889(119906 119907)

120597119906= 0

120597119889(119906 119907)

120597119907= 0

Figure 3 Flow chart for the minimization process

0

05

1

002040608113

135

14

145

15

Figure 4 Representation of function (1) over a concave surface

correspond to bodies such as ships tanks or aircrafts Most ofthese objects are composed of flat surfaces or convex surfaces

As it is mentioned earlier the process followed to searchreflection points is different for concave and convex surfacesThe CGM minimizes a cost function which depends on thesum of the distances between the source and the candidatereflection point and between this point and the observationpoint or direction expressions (1) or (3) respectively [13]

Nevertheless the analysis of concave surfaces seeks either amaximum or a minimum of the same function

To ensure that the minimization or maximization of thedistance function is adequate this function must have anabsolute minimum or maximum That is the function mustnot present a smooth variation between their values as inthis case the algorithm would fail to converge towards aninexistent solution point

If the distance functions of expressions (1) and (3) arerepresented on a concave surface the obtained plot is verysimilar to the one shown in Figure 4

As it can be observed in Figure 6 the function hasvery smooth variation and it does not present any absoluteminimum or maximum Therefore both functions (1) and(3) on the concave curves are not adequate to ensure thatthe CGMalgorithmwill converge towards the right reflectionpoint Even if the seed point is very close to the real reflectionpoint the CGM is not able to find that point minimizing ormaximizing the function in Figure 4

In order to clarify that the minimization process withthis function is not possible the parabolic reflector with 1mdiameter shown in Figures 5 and 6 is analyzed The hornfeed is located on the focus (0 0 04) and a set of observationpoints are arranged in a straight line The reflection pointsobtained are those that appear on the surface of the reflector

As was demonstrated in Section 2 the parabolic reflectortransforms a spherical wave into a plane wave This meansthat the rays from the reflection points must be parallel to theradiation axis of the reflector This affirmation implies that

International Journal of Antennas and Propagation 5

Observation points

Reflection points

Perspective

119909

119910

119911

Figure 5 Perspective view of the simple reflection points obtainedwith function (1) or (3)

Top

119909

119910

Figure 6 Top view of the simple reflection points obtained withfunction (1) or (3)

the reflection points will be the projection of the observationpoints on the reflector surface However the reflection pointsshown in Figures 5 and 6 are not the projection of theobservation points on the surface of the reflector

For that reason the CGM is not able to find theminimumor maximum absolute of the cost function in the appropriatesurface Instead it finds a minimum or a maximum in anearby surface on which the reflection does not take placeAs a result it is necessary to establish another cost functionwhich presents an absolute maximum orminimum to ensurethat the algorithm can find the suitable solution

In the first step it could be thought that as any reflectionon any type of surface concave or convex must satisfy Snellrsquoslaw this condition can be set as the function to be explored byCGMHence this functionwill take the form of the followingexpression

10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (6)

where 119894is the unitary incident vector and

119903is the unitary

reflection vectorFigure 7 shows the plot of expression (6) on a concave

surfaceComparing Figures 4 and 7 it can be concluded that the

new function shown in Figure 7 presents a higher degree ofvariation between their values than Figure 4

012345678

05

1

0020406081

05

1

Figure 7 Graphical representation of the Snell law

However despite this growth in the variation the ade-quate convergence of the algorithm is not guaranteed becausethe function does not present either an absolute maximumor minimum yet It presents multiple local maximum orminimum points This is not enough to establish Snellrsquos lawas the new cost function to look for reflection points in thistype of surfaces

In order to guarantee that a point on a surface is areflection point one more condition must be satisfied theincident vector the observation vector and the normal in thatpoint must be coplanar that is all of them should belong tothe same plane If this condition is added to expression (6)the cost function is transformed into this new expression

10038161003816100381610038161003816119899 sdot 119894minus 119899 sdot

119903

10038161003816100381610038161003816+10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (7)

where 119899 is the normal to the surface in the reflection pointIf the values taken from expression (7) are represented on

a concave surface the graph shown in Figure 8 is obtainedComparing Figure 8 with Figures 4 and 7 it is clear that

the last one exhibits the best features for its minimizationor maximization with an absolute minimum or maximumTherefore it can be deduced that this function will make itpossible to carry out the search of reflection points on concavesurfaces in a satisfactory way

Considering again the example of the parabolic reflectorof 1mdiameter shown in Figures 5 and 6 now the same test isdone with function (7) If the CGMworks with this functionit can be asserted that it can find the points of the simplereflection on the concave curves as shown in Figure 9

Figure 10 shows how in this case the reflection points arethe projection of the observation points on the curves of thereflector The rays reflected on the reflector are parallel to the119911-axis and then the spherical wave impacting on its surface istransformed into a plane wave confirming the good behaviorof the reflector

In this way the correct performance of the CGM hasbeen demonstrated working with the new cost function forreflection points search on concave surfaces

6 International Journal of Antennas and Propagation

002040608105

1

0

0123456789

Figure 8 Graphical representation of function (7) over a concavesurface

119909

119910

119911

Figure 9 Perspective view of the simple reflection points obtainedwith function (7)

119909

119910

Figure 10 Top view of the simple reflection points obtained withfunction (7)

4 Master Points Algorithm toAnalyze Reflector Antennas

As it has beenmentioned previously the analysis of the radia-tion pattern of reflector structures can be done calculating theelectromagnetic fields at the aperture and then transformingthe near field to the far field [7] To perform this analysis

119909

119910

119911

Figure 11 Geometrical model of a single reflector with its observa-tion surface

119909

119910

119911

Figure 12 Observation surface sampled at 1205823

a fictitious surface like the one shown in Figure 11 must beplaced on the aperture of the reflector to cover it completelyThis surface must be perpendicular to the radiation axisof the reflector It is sampled obtaining a huge amount ofobservation points in which the near field will be obtained(Figure 12)

Once the near field has been calculated on the set ofobservation points applying the high frequency techniqueGTDmost of the CPU time is spent obtaining the ray tracingfor each observation point Thus the new algorithm MasterPoints has been applied to speed up this process Finallythe transformation of the near field to far field is appliedobtaining the radiation pattern of the antenna

To obtain the near field on a sampled plane of points theMaster Points techniquemakes a compartmentalization of allpoints depending on the existence of the ray tracing Figure13 shows an example of a plane of observation points This

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

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DistributedSensor Networks

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2 International Journal of Antennas and Propagation

computation time For that reason [11] presents a domaindecomposition algorithm to compute the radiation pattern ofa finite aperture with a high computational efficiency In thisregard a new technique to compute the PO integral in amoreefficient way is presented in [6] The procedure presented in[8] is based on a series of Fourier transforms of an aperturedistribution which takes into account the curvature of thesurface Additionally the aperture blockage effect on the pat-tern of a quadruped supported primary-feed parabolic reflec-tor is evaluated in [9] where the scattering process is obtainedas an interaction between the reflector structure and theaperture-blocking feed together with its supporting struts

This paper is focused on the problem of analyzing theradiation pattern of arbitrarily shaped reflector antennasapplying GTDThe electrical field in a large number of pointssampled on the aperture of the reflector is obtained Thatis the ray tracing algorithm is calculated for each observa-tion point and this process spends too much CPU timeTherefore GTD has been combined with a new techniquenamed Master Points to accelerate the calculation of theelectromagnetic field in this kind of analysis The speed-upis possible since this method reduces the number of times inwhich it is necessary to calculate the ray tracing to obtain theelectromagnetic field in a given number of observation pointsor directions

In the ray tracing algorithm the Conjugate GradientMethod (CGM) [12] is applied to obtain the reflection pointson the reflector surfaces This method minimizes a costfunction which depends on the distance between the sourceand any given observation point or direction Usually thegeometric model of a complex body is composed of flat andconvex surfaces However the geometrical model of a para-bolic reflector is composed of concave instead of convex sur-facesTherefore a new distance function to analyze a concavesurface is proposed

This paper is organized as follows Section 2 reviewsthe geometric and electromagnetic analysis of a canoni-cal parabolic reflector The ray tracing algorithm used tocompute the radiation pattern of these antennas and theimprovements done in this process are presented in Sections3 and 4 respectively Section 5 is focused on the application ofthe Master Points technique to perform this analysis FinallySection 6 shows several cases of study to validate this newapproach including the analysis of the blockage in a parabolicreflector caused by an obstacle

2 Application of the Ray-Tracing Algorithm toAnalyze an Arbitrarily Shaped Reflector

As it has been outlined in the Introduction section when theradiation pattern of a reflector antenna is computed the firststep is to calculate the electromagnetic field distribution on avery dense mesh of points sampled on its aperture The mainpurpose of this step is to perform it as fast as possible To com-pute the ray tracing on a complex structure two situations aredistinguished depending on the geometry of the object

(i) geometry composed of flat surfaces in this casethe search for reflection points on the geometry is

Figure 1 Simple reflection on a convex surface

performed analytically using the Image Theory (IT)[13]

(ii) geometry composed of curved surfaces regardingarbitrary surfaces like NURBS [14 15] the searchfor reflection points becomes more complicated andminimizationmethodsmust be applied in particularthe CGM [12] which minimizes a cost function isused in this paper additionally the minimizationprocess for a convex surface and for a concavesurface is different the following sections describe thedifferences for both surfaces

21 Reflection Point on Convex Surfaces The search forreflection point on a geometry composed of convex surfaceis complicated as the Image Theory cannot be applied Asimple schema describing this situation for a given sourceand observation point is shown in Figure 1 To determinethe point within the area on which the reflection will takeplace the CGM minimizes a cost function This function isdifferent depending on whether the near field or the far fieldis analyzed For instance in the analysis of the near field thiscost function is expressed as follows

119889 (119906 119907) = 119889119894 (119906 119907) + 119889119903 (119906 119907) =1003816100381610038161003816119894 (119906 119907)

1003816100381610038161003816 +1003816100381610038161003816119903 (119906 119907)

1003816100381610038161003816 (1)

where 119889119894(119906 119907) is the distance between the source and the

candidate reflection point and119889119903(119906 119907) is the distance between

the candidate reflection point and the observation pointHence

119894is the vector from the source to the candidate

reflection point and 119903is the vector from the candidate

reflection point to the observation pointThe minimization process of the CGM method imple-

ments the following steps

(1) First of all a seed point over the convex surface isselected

(2) Then the cost function and its partial derivatives areevaluated in this point

International Journal of Antennas and Propagation 3

(3) If the partial derivatives are null the minimum ofthe function has been reached Consequently thesearch process finisheswith that seed point as the finalsolution the reflection point

(4) Otherwise the seed point is shifted over the curveaccording to the direction given by the partial deriva-tives

(5) This last seed point is evaluated repeating step (2)

This iterative process will be completed when the mini-mum of the function is found or when it exceeds a certainnumber of iterations

This shift from one seed point to another is done accord-ing to the partial derivatives of (1) which are given by thefollowing expressions

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) + 119903 (119906 119907)] sdot 119903

119906 (119906 119907)

120597119889 (119906 119907)

120597119907=120597119889119894 (119906 119907)

120597119907+120597119889119903 (119906 119907)

120597119907

= [119894 (119906 119907) + 119903 (119906 119907)] sdot 119903

119907 (119906 119907)

(2)

where 119903119906(119906 119907) and 119903

119907(119906 119907) are the partial derivatives with

respect to the parametric coordinates 119906 and 119907 of the reflectionpoint

When the far field is analysed the CGM minimizes thefollowing cost function

119889 (119906 119907) = 119889119894 (119906 119907) + 119889119903 (119906 119907)

=1003816100381610038161003816119894 (119906 119907)

1003816100381610038161003816 + (119863 minus 119907 sdot 119903 (119906 119907))

(3)

where 119889119894(119906 119907) is the distance between the source and the

candidate reflection point119889119903(119906 119907) is the distance between the

candidate reflection point and a perpendicular plane to theobservation direction

119894(119906 119907) is the vector from the source

to the candidate reflection point 119907 is a unitary vector whichdetermines the observation direction 119863 is the independentterm of the equation that defines the perpendicular planeto the observation direction and 119903(119906 119907) is the candidatereflection point on the surface

The minimization process and the associated shift of theseed point are performed as in the previous case The partialderivatives of (3) are given by the following expressions

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) minus 119907] sdot 119903

119906 (119906 119907)

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) minus 119907] sdot 119903

119907 (119906 119907)

(4)

Once the reflection point has been found on the surfacea shadowing test is conducted to affirm that neither reflected

Figure 2 Simple reflection on a concave surface

nor incident paths are hidden by any surface Likewise thereflection point must be located on the curve and Snellrsquos lawmust be satisfied at that point [16] as follows

119894sen 120579119894= 119903sen 120579119903 (5)

If any of these conditions are not satisfied the reflectionpoint found by the CGM is not the right solution and thealgorithm will need to find another point

22 Reflection Point on Concave Surfaces The cost functionsconsidered in this case for either the near or the far fieldanalysis are the same as in the case of a convex curve andthey are described by expressions (1) and (3) respectivelyFigure 2 shows this case of study schematically

Instead of the search process for the reflection point ona convex surface in which the CGM looks for a minimumof the cost function in this case the CGM looks for botha minimum and a maximum of the function presented pre-viously

Before this process starts a sampling of the curve onwhich the reflection point has to be found along the paramet-ric coordinates 119906 and 119907 is performed Each of these samplesis considered as a seed point for the search of a minimum ora maximum As in the minimization process over a convexsurface the shifted over the curve is done according tothe sign of the derivative functions expressions (2) or (4)depending on the case (the near field or the far field resp)Once the solution point is found if it belongs to the curvesatisfies Snellrsquos law and none of the incident and reflectedpaths are shadowed by any obstacle it can be concluded thatthe reflection point has been found on the concave surface

In order to better understand the minimization processimplemented by CGM Figure 3 shows a schema applicableto convex or concave curves

3 Improvements Done overthe Ray-Tracing Algorithm

The geometrical model of reflector systems is mainly com-posed of curved surfaces in particular concave surfacesHowever in the analysis of the radiation pattern of antennason board complex structures the geometrical models used

4 International Journal of Antennas and Propagation

Selection of the seed point

Move over the surface

End

No

Yes

Number ofiterations

No

Yes

Point belongsto parametric

space

Yes

No

Calculation 119889( )119906 119907

120597119889(119906 119907)

120597119906= 0

120597119889(119906 119907)

120597119907= 0

Figure 3 Flow chart for the minimization process

0

05

1

002040608113

135

14

145

15

Figure 4 Representation of function (1) over a concave surface

correspond to bodies such as ships tanks or aircrafts Most ofthese objects are composed of flat surfaces or convex surfaces

As it is mentioned earlier the process followed to searchreflection points is different for concave and convex surfacesThe CGM minimizes a cost function which depends on thesum of the distances between the source and the candidatereflection point and between this point and the observationpoint or direction expressions (1) or (3) respectively [13]

Nevertheless the analysis of concave surfaces seeks either amaximum or a minimum of the same function

To ensure that the minimization or maximization of thedistance function is adequate this function must have anabsolute minimum or maximum That is the function mustnot present a smooth variation between their values as inthis case the algorithm would fail to converge towards aninexistent solution point

If the distance functions of expressions (1) and (3) arerepresented on a concave surface the obtained plot is verysimilar to the one shown in Figure 4

As it can be observed in Figure 6 the function hasvery smooth variation and it does not present any absoluteminimum or maximum Therefore both functions (1) and(3) on the concave curves are not adequate to ensure thatthe CGMalgorithmwill converge towards the right reflectionpoint Even if the seed point is very close to the real reflectionpoint the CGM is not able to find that point minimizing ormaximizing the function in Figure 4

In order to clarify that the minimization process withthis function is not possible the parabolic reflector with 1mdiameter shown in Figures 5 and 6 is analyzed The hornfeed is located on the focus (0 0 04) and a set of observationpoints are arranged in a straight line The reflection pointsobtained are those that appear on the surface of the reflector

As was demonstrated in Section 2 the parabolic reflectortransforms a spherical wave into a plane wave This meansthat the rays from the reflection points must be parallel to theradiation axis of the reflector This affirmation implies that

International Journal of Antennas and Propagation 5

Observation points

Reflection points

Perspective

119909

119910

119911

Figure 5 Perspective view of the simple reflection points obtainedwith function (1) or (3)

Top

119909

119910

Figure 6 Top view of the simple reflection points obtained withfunction (1) or (3)

the reflection points will be the projection of the observationpoints on the reflector surface However the reflection pointsshown in Figures 5 and 6 are not the projection of theobservation points on the surface of the reflector

For that reason the CGM is not able to find theminimumor maximum absolute of the cost function in the appropriatesurface Instead it finds a minimum or a maximum in anearby surface on which the reflection does not take placeAs a result it is necessary to establish another cost functionwhich presents an absolute maximum orminimum to ensurethat the algorithm can find the suitable solution

In the first step it could be thought that as any reflectionon any type of surface concave or convex must satisfy Snellrsquoslaw this condition can be set as the function to be explored byCGMHence this functionwill take the form of the followingexpression

10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (6)

where 119894is the unitary incident vector and

119903is the unitary

reflection vectorFigure 7 shows the plot of expression (6) on a concave

surfaceComparing Figures 4 and 7 it can be concluded that the

new function shown in Figure 7 presents a higher degree ofvariation between their values than Figure 4

012345678

05

1

0020406081

05

1

Figure 7 Graphical representation of the Snell law

However despite this growth in the variation the ade-quate convergence of the algorithm is not guaranteed becausethe function does not present either an absolute maximumor minimum yet It presents multiple local maximum orminimum points This is not enough to establish Snellrsquos lawas the new cost function to look for reflection points in thistype of surfaces

In order to guarantee that a point on a surface is areflection point one more condition must be satisfied theincident vector the observation vector and the normal in thatpoint must be coplanar that is all of them should belong tothe same plane If this condition is added to expression (6)the cost function is transformed into this new expression

10038161003816100381610038161003816119899 sdot 119894minus 119899 sdot

119903

10038161003816100381610038161003816+10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (7)

where 119899 is the normal to the surface in the reflection pointIf the values taken from expression (7) are represented on

a concave surface the graph shown in Figure 8 is obtainedComparing Figure 8 with Figures 4 and 7 it is clear that

the last one exhibits the best features for its minimizationor maximization with an absolute minimum or maximumTherefore it can be deduced that this function will make itpossible to carry out the search of reflection points on concavesurfaces in a satisfactory way

Considering again the example of the parabolic reflectorof 1mdiameter shown in Figures 5 and 6 now the same test isdone with function (7) If the CGMworks with this functionit can be asserted that it can find the points of the simplereflection on the concave curves as shown in Figure 9

Figure 10 shows how in this case the reflection points arethe projection of the observation points on the curves of thereflector The rays reflected on the reflector are parallel to the119911-axis and then the spherical wave impacting on its surface istransformed into a plane wave confirming the good behaviorof the reflector

In this way the correct performance of the CGM hasbeen demonstrated working with the new cost function forreflection points search on concave surfaces

6 International Journal of Antennas and Propagation

002040608105

1

0

0123456789

Figure 8 Graphical representation of function (7) over a concavesurface

119909

119910

119911

Figure 9 Perspective view of the simple reflection points obtainedwith function (7)

119909

119910

Figure 10 Top view of the simple reflection points obtained withfunction (7)

4 Master Points Algorithm toAnalyze Reflector Antennas

As it has beenmentioned previously the analysis of the radia-tion pattern of reflector structures can be done calculating theelectromagnetic fields at the aperture and then transformingthe near field to the far field [7] To perform this analysis

119909

119910

119911

Figure 11 Geometrical model of a single reflector with its observa-tion surface

119909

119910

119911

Figure 12 Observation surface sampled at 1205823

a fictitious surface like the one shown in Figure 11 must beplaced on the aperture of the reflector to cover it completelyThis surface must be perpendicular to the radiation axisof the reflector It is sampled obtaining a huge amount ofobservation points in which the near field will be obtained(Figure 12)

Once the near field has been calculated on the set ofobservation points applying the high frequency techniqueGTDmost of the CPU time is spent obtaining the ray tracingfor each observation point Thus the new algorithm MasterPoints has been applied to speed up this process Finallythe transformation of the near field to far field is appliedobtaining the radiation pattern of the antenna

To obtain the near field on a sampled plane of points theMaster Points techniquemakes a compartmentalization of allpoints depending on the existence of the ray tracing Figure13 shows an example of a plane of observation points This

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

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International Journal of

International Journal of Antennas and Propagation 3

(3) If the partial derivatives are null the minimum ofthe function has been reached Consequently thesearch process finisheswith that seed point as the finalsolution the reflection point

(4) Otherwise the seed point is shifted over the curveaccording to the direction given by the partial deriva-tives

(5) This last seed point is evaluated repeating step (2)

This iterative process will be completed when the mini-mum of the function is found or when it exceeds a certainnumber of iterations

This shift from one seed point to another is done accord-ing to the partial derivatives of (1) which are given by thefollowing expressions

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) + 119903 (119906 119907)] sdot 119903

119906 (119906 119907)

120597119889 (119906 119907)

120597119907=120597119889119894 (119906 119907)

120597119907+120597119889119903 (119906 119907)

120597119907

= [119894 (119906 119907) + 119903 (119906 119907)] sdot 119903

119907 (119906 119907)

(2)

where 119903119906(119906 119907) and 119903

119907(119906 119907) are the partial derivatives with

respect to the parametric coordinates 119906 and 119907 of the reflectionpoint

When the far field is analysed the CGM minimizes thefollowing cost function

119889 (119906 119907) = 119889119894 (119906 119907) + 119889119903 (119906 119907)

=1003816100381610038161003816119894 (119906 119907)

1003816100381610038161003816 + (119863 minus 119907 sdot 119903 (119906 119907))

(3)

where 119889119894(119906 119907) is the distance between the source and the

candidate reflection point119889119903(119906 119907) is the distance between the

candidate reflection point and a perpendicular plane to theobservation direction

119894(119906 119907) is the vector from the source

to the candidate reflection point 119907 is a unitary vector whichdetermines the observation direction 119863 is the independentterm of the equation that defines the perpendicular planeto the observation direction and 119903(119906 119907) is the candidatereflection point on the surface

The minimization process and the associated shift of theseed point are performed as in the previous case The partialderivatives of (3) are given by the following expressions

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) minus 119907] sdot 119903

119906 (119906 119907)

120597119889 (119906 119907)

120597119906=120597119889119894 (119906 119907)

120597119906+120597119889119903 (119906 119907)

120597119906

= [119894 (119906 119907) minus 119907] sdot 119903

119907 (119906 119907)

(4)

Once the reflection point has been found on the surfacea shadowing test is conducted to affirm that neither reflected

Figure 2 Simple reflection on a concave surface

nor incident paths are hidden by any surface Likewise thereflection point must be located on the curve and Snellrsquos lawmust be satisfied at that point [16] as follows

119894sen 120579119894= 119903sen 120579119903 (5)

If any of these conditions are not satisfied the reflectionpoint found by the CGM is not the right solution and thealgorithm will need to find another point

22 Reflection Point on Concave Surfaces The cost functionsconsidered in this case for either the near or the far fieldanalysis are the same as in the case of a convex curve andthey are described by expressions (1) and (3) respectivelyFigure 2 shows this case of study schematically

Instead of the search process for the reflection point ona convex surface in which the CGM looks for a minimumof the cost function in this case the CGM looks for botha minimum and a maximum of the function presented pre-viously

Before this process starts a sampling of the curve onwhich the reflection point has to be found along the paramet-ric coordinates 119906 and 119907 is performed Each of these samplesis considered as a seed point for the search of a minimum ora maximum As in the minimization process over a convexsurface the shifted over the curve is done according tothe sign of the derivative functions expressions (2) or (4)depending on the case (the near field or the far field resp)Once the solution point is found if it belongs to the curvesatisfies Snellrsquos law and none of the incident and reflectedpaths are shadowed by any obstacle it can be concluded thatthe reflection point has been found on the concave surface

In order to better understand the minimization processimplemented by CGM Figure 3 shows a schema applicableto convex or concave curves

3 Improvements Done overthe Ray-Tracing Algorithm

The geometrical model of reflector systems is mainly com-posed of curved surfaces in particular concave surfacesHowever in the analysis of the radiation pattern of antennason board complex structures the geometrical models used

4 International Journal of Antennas and Propagation

Selection of the seed point

Move over the surface

End

No

Yes

Number ofiterations

No

Yes

Point belongsto parametric

space

Yes

No

Calculation 119889( )119906 119907

120597119889(119906 119907)

120597119906= 0

120597119889(119906 119907)

120597119907= 0

Figure 3 Flow chart for the minimization process

0

05

1

002040608113

135

14

145

15

Figure 4 Representation of function (1) over a concave surface

correspond to bodies such as ships tanks or aircrafts Most ofthese objects are composed of flat surfaces or convex surfaces

As it is mentioned earlier the process followed to searchreflection points is different for concave and convex surfacesThe CGM minimizes a cost function which depends on thesum of the distances between the source and the candidatereflection point and between this point and the observationpoint or direction expressions (1) or (3) respectively [13]

Nevertheless the analysis of concave surfaces seeks either amaximum or a minimum of the same function

To ensure that the minimization or maximization of thedistance function is adequate this function must have anabsolute minimum or maximum That is the function mustnot present a smooth variation between their values as inthis case the algorithm would fail to converge towards aninexistent solution point

If the distance functions of expressions (1) and (3) arerepresented on a concave surface the obtained plot is verysimilar to the one shown in Figure 4

As it can be observed in Figure 6 the function hasvery smooth variation and it does not present any absoluteminimum or maximum Therefore both functions (1) and(3) on the concave curves are not adequate to ensure thatthe CGMalgorithmwill converge towards the right reflectionpoint Even if the seed point is very close to the real reflectionpoint the CGM is not able to find that point minimizing ormaximizing the function in Figure 4

In order to clarify that the minimization process withthis function is not possible the parabolic reflector with 1mdiameter shown in Figures 5 and 6 is analyzed The hornfeed is located on the focus (0 0 04) and a set of observationpoints are arranged in a straight line The reflection pointsobtained are those that appear on the surface of the reflector

As was demonstrated in Section 2 the parabolic reflectortransforms a spherical wave into a plane wave This meansthat the rays from the reflection points must be parallel to theradiation axis of the reflector This affirmation implies that

International Journal of Antennas and Propagation 5

Observation points

Reflection points

Perspective

119909

119910

119911

Figure 5 Perspective view of the simple reflection points obtainedwith function (1) or (3)

Top

119909

119910

Figure 6 Top view of the simple reflection points obtained withfunction (1) or (3)

the reflection points will be the projection of the observationpoints on the reflector surface However the reflection pointsshown in Figures 5 and 6 are not the projection of theobservation points on the surface of the reflector

For that reason the CGM is not able to find theminimumor maximum absolute of the cost function in the appropriatesurface Instead it finds a minimum or a maximum in anearby surface on which the reflection does not take placeAs a result it is necessary to establish another cost functionwhich presents an absolute maximum orminimum to ensurethat the algorithm can find the suitable solution

In the first step it could be thought that as any reflectionon any type of surface concave or convex must satisfy Snellrsquoslaw this condition can be set as the function to be explored byCGMHence this functionwill take the form of the followingexpression

10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (6)

where 119894is the unitary incident vector and

119903is the unitary

reflection vectorFigure 7 shows the plot of expression (6) on a concave

surfaceComparing Figures 4 and 7 it can be concluded that the

new function shown in Figure 7 presents a higher degree ofvariation between their values than Figure 4

012345678

05

1

0020406081

05

1

Figure 7 Graphical representation of the Snell law

However despite this growth in the variation the ade-quate convergence of the algorithm is not guaranteed becausethe function does not present either an absolute maximumor minimum yet It presents multiple local maximum orminimum points This is not enough to establish Snellrsquos lawas the new cost function to look for reflection points in thistype of surfaces

In order to guarantee that a point on a surface is areflection point one more condition must be satisfied theincident vector the observation vector and the normal in thatpoint must be coplanar that is all of them should belong tothe same plane If this condition is added to expression (6)the cost function is transformed into this new expression

10038161003816100381610038161003816119899 sdot 119894minus 119899 sdot

119903

10038161003816100381610038161003816+10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (7)

where 119899 is the normal to the surface in the reflection pointIf the values taken from expression (7) are represented on

a concave surface the graph shown in Figure 8 is obtainedComparing Figure 8 with Figures 4 and 7 it is clear that

the last one exhibits the best features for its minimizationor maximization with an absolute minimum or maximumTherefore it can be deduced that this function will make itpossible to carry out the search of reflection points on concavesurfaces in a satisfactory way

Considering again the example of the parabolic reflectorof 1mdiameter shown in Figures 5 and 6 now the same test isdone with function (7) If the CGMworks with this functionit can be asserted that it can find the points of the simplereflection on the concave curves as shown in Figure 9

Figure 10 shows how in this case the reflection points arethe projection of the observation points on the curves of thereflector The rays reflected on the reflector are parallel to the119911-axis and then the spherical wave impacting on its surface istransformed into a plane wave confirming the good behaviorof the reflector

In this way the correct performance of the CGM hasbeen demonstrated working with the new cost function forreflection points search on concave surfaces

6 International Journal of Antennas and Propagation

002040608105

1

0

0123456789

Figure 8 Graphical representation of function (7) over a concavesurface

119909

119910

119911

Figure 9 Perspective view of the simple reflection points obtainedwith function (7)

119909

119910

Figure 10 Top view of the simple reflection points obtained withfunction (7)

4 Master Points Algorithm toAnalyze Reflector Antennas

As it has beenmentioned previously the analysis of the radia-tion pattern of reflector structures can be done calculating theelectromagnetic fields at the aperture and then transformingthe near field to the far field [7] To perform this analysis

119909

119910

119911

Figure 11 Geometrical model of a single reflector with its observa-tion surface

119909

119910

119911

Figure 12 Observation surface sampled at 1205823

a fictitious surface like the one shown in Figure 11 must beplaced on the aperture of the reflector to cover it completelyThis surface must be perpendicular to the radiation axisof the reflector It is sampled obtaining a huge amount ofobservation points in which the near field will be obtained(Figure 12)

Once the near field has been calculated on the set ofobservation points applying the high frequency techniqueGTDmost of the CPU time is spent obtaining the ray tracingfor each observation point Thus the new algorithm MasterPoints has been applied to speed up this process Finallythe transformation of the near field to far field is appliedobtaining the radiation pattern of the antenna

To obtain the near field on a sampled plane of points theMaster Points techniquemakes a compartmentalization of allpoints depending on the existence of the ray tracing Figure13 shows an example of a plane of observation points This

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

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VLSI Design

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Shock and Vibration

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

4 International Journal of Antennas and Propagation

Selection of the seed point

Move over the surface

End

No

Yes

Number ofiterations

No

Yes

Point belongsto parametric

space

Yes

No

Calculation 119889( )119906 119907

120597119889(119906 119907)

120597119906= 0

120597119889(119906 119907)

120597119907= 0

Figure 3 Flow chart for the minimization process

0

05

1

002040608113

135

14

145

15

Figure 4 Representation of function (1) over a concave surface

correspond to bodies such as ships tanks or aircrafts Most ofthese objects are composed of flat surfaces or convex surfaces

As it is mentioned earlier the process followed to searchreflection points is different for concave and convex surfacesThe CGM minimizes a cost function which depends on thesum of the distances between the source and the candidatereflection point and between this point and the observationpoint or direction expressions (1) or (3) respectively [13]

Nevertheless the analysis of concave surfaces seeks either amaximum or a minimum of the same function

To ensure that the minimization or maximization of thedistance function is adequate this function must have anabsolute minimum or maximum That is the function mustnot present a smooth variation between their values as inthis case the algorithm would fail to converge towards aninexistent solution point

If the distance functions of expressions (1) and (3) arerepresented on a concave surface the obtained plot is verysimilar to the one shown in Figure 4

As it can be observed in Figure 6 the function hasvery smooth variation and it does not present any absoluteminimum or maximum Therefore both functions (1) and(3) on the concave curves are not adequate to ensure thatthe CGMalgorithmwill converge towards the right reflectionpoint Even if the seed point is very close to the real reflectionpoint the CGM is not able to find that point minimizing ormaximizing the function in Figure 4

In order to clarify that the minimization process withthis function is not possible the parabolic reflector with 1mdiameter shown in Figures 5 and 6 is analyzed The hornfeed is located on the focus (0 0 04) and a set of observationpoints are arranged in a straight line The reflection pointsobtained are those that appear on the surface of the reflector

As was demonstrated in Section 2 the parabolic reflectortransforms a spherical wave into a plane wave This meansthat the rays from the reflection points must be parallel to theradiation axis of the reflector This affirmation implies that

International Journal of Antennas and Propagation 5

Observation points

Reflection points

Perspective

119909

119910

119911

Figure 5 Perspective view of the simple reflection points obtainedwith function (1) or (3)

Top

119909

119910

Figure 6 Top view of the simple reflection points obtained withfunction (1) or (3)

the reflection points will be the projection of the observationpoints on the reflector surface However the reflection pointsshown in Figures 5 and 6 are not the projection of theobservation points on the surface of the reflector

For that reason the CGM is not able to find theminimumor maximum absolute of the cost function in the appropriatesurface Instead it finds a minimum or a maximum in anearby surface on which the reflection does not take placeAs a result it is necessary to establish another cost functionwhich presents an absolute maximum orminimum to ensurethat the algorithm can find the suitable solution

In the first step it could be thought that as any reflectionon any type of surface concave or convex must satisfy Snellrsquoslaw this condition can be set as the function to be explored byCGMHence this functionwill take the form of the followingexpression

10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (6)

where 119894is the unitary incident vector and

119903is the unitary

reflection vectorFigure 7 shows the plot of expression (6) on a concave

surfaceComparing Figures 4 and 7 it can be concluded that the

new function shown in Figure 7 presents a higher degree ofvariation between their values than Figure 4

012345678

05

1

0020406081

05

1

Figure 7 Graphical representation of the Snell law

However despite this growth in the variation the ade-quate convergence of the algorithm is not guaranteed becausethe function does not present either an absolute maximumor minimum yet It presents multiple local maximum orminimum points This is not enough to establish Snellrsquos lawas the new cost function to look for reflection points in thistype of surfaces

In order to guarantee that a point on a surface is areflection point one more condition must be satisfied theincident vector the observation vector and the normal in thatpoint must be coplanar that is all of them should belong tothe same plane If this condition is added to expression (6)the cost function is transformed into this new expression

10038161003816100381610038161003816119899 sdot 119894minus 119899 sdot

119903

10038161003816100381610038161003816+10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (7)

where 119899 is the normal to the surface in the reflection pointIf the values taken from expression (7) are represented on

a concave surface the graph shown in Figure 8 is obtainedComparing Figure 8 with Figures 4 and 7 it is clear that

the last one exhibits the best features for its minimizationor maximization with an absolute minimum or maximumTherefore it can be deduced that this function will make itpossible to carry out the search of reflection points on concavesurfaces in a satisfactory way

Considering again the example of the parabolic reflectorof 1mdiameter shown in Figures 5 and 6 now the same test isdone with function (7) If the CGMworks with this functionit can be asserted that it can find the points of the simplereflection on the concave curves as shown in Figure 9

Figure 10 shows how in this case the reflection points arethe projection of the observation points on the curves of thereflector The rays reflected on the reflector are parallel to the119911-axis and then the spherical wave impacting on its surface istransformed into a plane wave confirming the good behaviorof the reflector

In this way the correct performance of the CGM hasbeen demonstrated working with the new cost function forreflection points search on concave surfaces

6 International Journal of Antennas and Propagation

002040608105

1

0

0123456789

Figure 8 Graphical representation of function (7) over a concavesurface

119909

119910

119911

Figure 9 Perspective view of the simple reflection points obtainedwith function (7)

119909

119910

Figure 10 Top view of the simple reflection points obtained withfunction (7)

4 Master Points Algorithm toAnalyze Reflector Antennas

As it has beenmentioned previously the analysis of the radia-tion pattern of reflector structures can be done calculating theelectromagnetic fields at the aperture and then transformingthe near field to the far field [7] To perform this analysis

119909

119910

119911

Figure 11 Geometrical model of a single reflector with its observa-tion surface

119909

119910

119911

Figure 12 Observation surface sampled at 1205823

a fictitious surface like the one shown in Figure 11 must beplaced on the aperture of the reflector to cover it completelyThis surface must be perpendicular to the radiation axisof the reflector It is sampled obtaining a huge amount ofobservation points in which the near field will be obtained(Figure 12)

Once the near field has been calculated on the set ofobservation points applying the high frequency techniqueGTDmost of the CPU time is spent obtaining the ray tracingfor each observation point Thus the new algorithm MasterPoints has been applied to speed up this process Finallythe transformation of the near field to far field is appliedobtaining the radiation pattern of the antenna

To obtain the near field on a sampled plane of points theMaster Points techniquemakes a compartmentalization of allpoints depending on the existence of the ray tracing Figure13 shows an example of a plane of observation points This

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

International Journal of Antennas and Propagation 5

Observation points

Reflection points

Perspective

119909

119910

119911

Figure 5 Perspective view of the simple reflection points obtainedwith function (1) or (3)

Top

119909

119910

Figure 6 Top view of the simple reflection points obtained withfunction (1) or (3)

the reflection points will be the projection of the observationpoints on the reflector surface However the reflection pointsshown in Figures 5 and 6 are not the projection of theobservation points on the surface of the reflector

For that reason the CGM is not able to find theminimumor maximum absolute of the cost function in the appropriatesurface Instead it finds a minimum or a maximum in anearby surface on which the reflection does not take placeAs a result it is necessary to establish another cost functionwhich presents an absolute maximum orminimum to ensurethat the algorithm can find the suitable solution

In the first step it could be thought that as any reflectionon any type of surface concave or convex must satisfy Snellrsquoslaw this condition can be set as the function to be explored byCGMHence this functionwill take the form of the followingexpression

10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (6)

where 119894is the unitary incident vector and

119903is the unitary

reflection vectorFigure 7 shows the plot of expression (6) on a concave

surfaceComparing Figures 4 and 7 it can be concluded that the

new function shown in Figure 7 presents a higher degree ofvariation between their values than Figure 4

012345678

05

1

0020406081

05

1

Figure 7 Graphical representation of the Snell law

However despite this growth in the variation the ade-quate convergence of the algorithm is not guaranteed becausethe function does not present either an absolute maximumor minimum yet It presents multiple local maximum orminimum points This is not enough to establish Snellrsquos lawas the new cost function to look for reflection points in thistype of surfaces

In order to guarantee that a point on a surface is areflection point one more condition must be satisfied theincident vector the observation vector and the normal in thatpoint must be coplanar that is all of them should belong tothe same plane If this condition is added to expression (6)the cost function is transformed into this new expression

10038161003816100381610038161003816119899 sdot 119894minus 119899 sdot

119903

10038161003816100381610038161003816+10038161003816100381610038161003816119894otimes 119903

10038161003816100381610038161003816 (7)

where 119899 is the normal to the surface in the reflection pointIf the values taken from expression (7) are represented on

a concave surface the graph shown in Figure 8 is obtainedComparing Figure 8 with Figures 4 and 7 it is clear that

the last one exhibits the best features for its minimizationor maximization with an absolute minimum or maximumTherefore it can be deduced that this function will make itpossible to carry out the search of reflection points on concavesurfaces in a satisfactory way

Considering again the example of the parabolic reflectorof 1mdiameter shown in Figures 5 and 6 now the same test isdone with function (7) If the CGMworks with this functionit can be asserted that it can find the points of the simplereflection on the concave curves as shown in Figure 9

Figure 10 shows how in this case the reflection points arethe projection of the observation points on the curves of thereflector The rays reflected on the reflector are parallel to the119911-axis and then the spherical wave impacting on its surface istransformed into a plane wave confirming the good behaviorof the reflector

In this way the correct performance of the CGM hasbeen demonstrated working with the new cost function forreflection points search on concave surfaces

6 International Journal of Antennas and Propagation

002040608105

1

0

0123456789

Figure 8 Graphical representation of function (7) over a concavesurface

119909

119910

119911

Figure 9 Perspective view of the simple reflection points obtainedwith function (7)

119909

119910

Figure 10 Top view of the simple reflection points obtained withfunction (7)

4 Master Points Algorithm toAnalyze Reflector Antennas

As it has beenmentioned previously the analysis of the radia-tion pattern of reflector structures can be done calculating theelectromagnetic fields at the aperture and then transformingthe near field to the far field [7] To perform this analysis

119909

119910

119911

Figure 11 Geometrical model of a single reflector with its observa-tion surface

119909

119910

119911

Figure 12 Observation surface sampled at 1205823

a fictitious surface like the one shown in Figure 11 must beplaced on the aperture of the reflector to cover it completelyThis surface must be perpendicular to the radiation axisof the reflector It is sampled obtaining a huge amount ofobservation points in which the near field will be obtained(Figure 12)

Once the near field has been calculated on the set ofobservation points applying the high frequency techniqueGTDmost of the CPU time is spent obtaining the ray tracingfor each observation point Thus the new algorithm MasterPoints has been applied to speed up this process Finallythe transformation of the near field to far field is appliedobtaining the radiation pattern of the antenna

To obtain the near field on a sampled plane of points theMaster Points techniquemakes a compartmentalization of allpoints depending on the existence of the ray tracing Figure13 shows an example of a plane of observation points This

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

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VLSI Design

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Shock and Vibration

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

002040608105

1

0

0123456789

Figure 8 Graphical representation of function (7) over a concavesurface

119909

119910

119911

Figure 9 Perspective view of the simple reflection points obtainedwith function (7)

119909

119910

Figure 10 Top view of the simple reflection points obtained withfunction (7)

4 Master Points Algorithm toAnalyze Reflector Antennas

As it has beenmentioned previously the analysis of the radia-tion pattern of reflector structures can be done calculating theelectromagnetic fields at the aperture and then transformingthe near field to the far field [7] To perform this analysis

119909

119910

119911

Figure 11 Geometrical model of a single reflector with its observa-tion surface

119909

119910

119911

Figure 12 Observation surface sampled at 1205823

a fictitious surface like the one shown in Figure 11 must beplaced on the aperture of the reflector to cover it completelyThis surface must be perpendicular to the radiation axisof the reflector It is sampled obtaining a huge amount ofobservation points in which the near field will be obtained(Figure 12)

Once the near field has been calculated on the set ofobservation points applying the high frequency techniqueGTDmost of the CPU time is spent obtaining the ray tracingfor each observation point Thus the new algorithm MasterPoints has been applied to speed up this process Finallythe transformation of the near field to far field is appliedobtaining the radiation pattern of the antenna

To obtain the near field on a sampled plane of points theMaster Points techniquemakes a compartmentalization of allpoints depending on the existence of the ray tracing Figure13 shows an example of a plane of observation points This

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

Figure 13 Observation plane divided into 4 quadrants

Figure 14 Division process

plane is divided into 4 quadrants and the algorithm beginsto analyze the quadrant located at the left button corner

This analysis tests if there is ray tracing for both externalpoints (red points) If so a group containing all the pointsin this quadrant is formed and the next quadrant located atthe right button corner is analyzed in the sameway Howeverif this is not the case a new division is done as shown inFigure 14 This iterative process continues until the wholeplane has been evaluated or the limit number of divisions hasbeen reached As a result of our experience in the applicationof this technique to compute the radiation pattern in a vastnumber of observation points or directions 4 is a good valuefor the depth limit that can let us obtain accuracy in the resultsdiminishing the CPU time

Once a group is formed several sampled points areselected to obtain the near field only in these points It isimportant to know that the accuracy of results depends on theway this selection is done For example if the group of 7 times 7points shown in Figure 15 has been formed the results will bebetter if 4 samples instead of 3 are selected in each directionbecause the ray tracing is obtained in more samples

4 samples 3 samples

Figure 15 Taking samples in a group

Finally to obtain the near field in all observation pointsan interpolation method is applied That is for several points(samples) the ray tracing and the near field have beencalculated Applying an interpolation method to these valuesof near field the near field in all the observation points ofthe group is obtained Instead of an interpolation methodan approximation method is applied since it reduces thetotal error fitting better the samplersquos values In particularthe approximationmethod used is 2D least square minimiza-tion

5 Results

In order to validate the improvement developed in this paperin the analysis of reflection on concave surfaces an extensivestudy on the calculation of the radiation pattern of reflectorstructures in multiple situations is presented in the sequelThe results of this analysis have been compared with MoMresults

In the first section the analysis of a single reflectorhas been performed to study the effects introduced in theradiation pattern by the shift of the feed In the secondsection the radiation pattern of a single reflector consideringa feed array is shown To conclude several obstacles havebeen placed on the antenna directivity to determine the effectproduced in the radiation pattern of the antenna

51 Feedrsquos Shift This section presents the analysis of thevariations that experiment the radiation pattern of a reflectorantenna as the position of feeds is modified by applying thecombination ofGTDwith theMaster Pointmethod discussedearlier

Figure 16 shows the geometric model of the antennaconsidered in this study It is a parabolic reflector with1m diameter and its focus at (00 00 04) The observationsurface located over the reflector aperture has been sampledat 1205823 This means that at 10GHz 14400 observation pointsare obtained

This reflector has been fed with a rectangular horn thatpresents the radiation pattern shown in Figure 17

First of all to analyze the effect introduced by the shift ofthe feed the radiation pattern of the single reflector shownin Figure 16 is obtained locating the horn at its focus point(00 00 04) The results for the polar component applyingGTD combined with the Master Points method have been

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

119909

119910

119911

Figure 16 Single reflector with observation surface

0

0

20 40 60 80 100 120 140 160 180

(dB)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

120579 (∘)

Radiation pattern cut 120593 = 0∘

∣119864120579∣

∣119864120593∣

Figure 17 Radiation pattern of the rectangular horn

compared with the rigorous techniqueMoM showing a goodagreement as depicted in Figure 18

The main lobe is located at 120593 = 0∘ and 120579 = 0∘ because thehorn is located at the focus of the reflector

511 x-Axis For this case of study the horn has been movedfrom the focus point to (002 00 04) over the 119909-axis Thenew schema is represented in Figure 19

Figure 20 shows the comparison between GTD-MasterPoints andMoMat a frequency of 10GHz A cut in120593 = 0∘ andsweep from 120579 = minus70∘ to 120579 = 70∘ are represented The graphshows good accuracy for the GTD-Master Point method

In the first case the main lobe was located at 120593 = 0∘ and120579 = 0∘ However when the horn is moved over the 119909-axis the

main lobe experienced a slight offset As shown in Figure 20the main lobe is approximately located at 120593 = 0∘ and 120579 = minus2∘

512 xy-Axis It is also interesting to know what happenswith the radiation pattern of the reflector when the horn ismoved over the 119909-axis and the 119910-axisThe geometrical model

120579 (∘)

0

10

20

30

40

0 10 20 30 40 50 60 70

(dBi

)

Polar component MoMPolar component GTD

minus70 minus60 minus50 minus40 minus30 minus20

minus40

minus10

minus30

minus20

minus10

Directivity cut 120593 = 0∘

Figure 18 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909

119910

119911

119909 119910

119911

Figure 19 Single reflector with the horn shifted

considered in this case is shown in Figure 21 where the hornhas been placed at the point (002 002 04)

Results for the polar component are presented in Figure22 In this case the main lobe has been shifted to 120593 = 45∘ and120579 = minus3

∘ approximately The frequency of the simulation is10GHz

52 Feed a Single Reflector with an Array Once the effectcaused by the shift of the feed of a reflector from its focalposition has been studied it is interesting to seewhat happenswith the directivity of the antenna when the reflector is fed byan array of hornsThis is analyzed for the following two cases

(i) In the first one a linear array consisting of three hornslocated over the 119909-axis is considered

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 20 Polar component obtained with MoM and GTD cut inphi = 0

119909

119910

119911

119909 119910

119911

Figure 21 Single reflector with the horn shifted

(ii) In the second one it is considered a 2D array consist-ing of nine horns

The horns used in these situations are the same as inprevious study whose radiation pattern was shown in Figure17 and the frequency of the simulation is also the same10GHz

521 Linear Array over 119909-Axis The reflector of Figure 23 isilluminated by an array of three horns that are separated 2120582so their positions are

(minus002 00 04) (00 00 04) (002 00 04)

(8)

The radiation pattern of this reflector has been obtainedapplying GTD and MoM Figure 24 shows the results for thecut inΦ = 0∘The second andmain lobes are located at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively due to the three horns

010203040

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Directivity cut 120593 = 45∘

Figure 22 Polar component obtained with MoM and GTD cut inphi = 45

119909

119910

119911

119910119910119910

119911119911119911

119909119909119909

Figure 23 Single reflector feed with a linear array of horns locatedover the 119909-axis

522 2DArray Thereflector presented in Figure 25 has beenfed through an array of two dimensions inwhich the antennasare separated 2120582 so their positions are

(minus002 002 04) (00 002 04) (002 002 04)

(minus002 00 04) (00 00 04) (002 00 04)

(minus002 minus002 04) (0 minus002 04) (002 minus002 04)

(9)

The radiation pattern of the antennas array is the oneshown in Figure 17The cut in 120593 = 0∘ is represented in Figure26 The side and the main lobes are identified at 120579 = 2∘120579 = minus2

∘ and 120579 = 0∘ respectively The three horns locatedon the 119909-axis account for these lobes

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

010203040

0 10 20 30 40 50 60 70

Polar component MoMPolar component GTD

120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Figure 24 Polar component obtained with MoM and GTD cut inphi = 0

119909119909119909119909119909119909119909119909119909

119910119910119910 119910119910119910 119910119910119910

119911119911119911119911119911119911119911119911119911

119909

119910

119911

Figure 25 Single reflector feed with a 2D array of horns

Figure 27 depicts the results for the cut in 120593 = 90∘ Theside and main lobes are seen at 120579 = 2∘ 120579 = minus2∘ and 120579 = 0∘respectively due to the horns located on the 119910-axis

The simple cases shown in this section cannot be analysedapplying GTD without Master Points technique and the newdistance function shown in Section 4 All of them have beenrun in a PCwith an Intel Core 2Duo (only one core was used)at 187GHz

53 Blocking Produced by an Obstacle Another interestingeffect to study is the hiding part of the radiation patterncaused by an obstacle placed over the aperture of the antennaTwo different scenarios have been consideredThe first one isa simple case composed of a single reflector with an obstaclelocated over its apertureThe secondone ismore complicated

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 26 Polar component obtained with MoM and GTD cut inphi = 0

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40

minus50

minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 27 Polar component obtained with MoM and GTD cut inphi = 90∘

the geometrical model of a reinforced has been analyzedshifting the reflector on its roof

531 Calculating theDirectivity of a Reflectorwith anObstacleThe scenario shown in Figure 28 is considered The reflectoris fed by a single rectangular horn whose radiation patternremains the same as in Figure 17 and placed in the focus ofthe reflector As it is presented in Figure 28 the obstacle ishiding approximately 34 of the aperture of the reflector

Figure 29 shows the effects produced by the obstacleSo the graph obtained for the single reflector without theobstacle scenario shown in Figure 16 has been comparedwith the graph of the reflector with the obstacle scenarioshown in Figure 28 The results of both scenarios have beenobtained with the high frequency technique GTD-MasterPoints and the rigorous method MoM

The main consequence generated by the obstacle is thegrowth of the level of the secondary lobes

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

119909119910

119911

Figure 28 Single reflector with an obstacle and the observationsurface

0

10

20

30

40

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 29 Polar component obtained with MoM and GTD cut inphi = 0

532 Calculating the Directivity of a Reflector Located on aVehicle Figure 30 shows the geometric model of an armoredvehicle built with both flat and curved surfaces A reflector ison the roof of the vehicle whose radiation axis is the 119911-axis

The study of this scenario has been done at 12GHzand the reflector is fed with the horn shown in Figure 17located on its focus (6758 21 304)The observation surfaceis placed over the aperture of the reflector It has been sampledat a frequency of 1205823 which means getting the near field in atotal of 3660 observation points The results for the modelof Figure 30 have been obtained applying GTD and MoMFigure 31 shows the graph for a cut in 120593 = 0∘ and sweep from120579 = minus70

∘ to 120579 = 70∘This simulation has been done in an Intel Xeon at

213 GHz The Table 1 compared the CPU time consumed inthe analysis when GTD-Master Points and MoM techniquesare applied

If a turn of 90∘ in 120593 and 145∘ in 120579 is applied to thereflector of the armored vehicle the new geometrical modelis shown in Figures 32 and 33 This represents an interesting

119909119910

119911

Figure 30 Geometrical model of an armored vehicle

0

10

20

30

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 31 Polar component obtained with MoM and GTD cut in120593 = 0

∘

119909119910

119911

Figure 32 Geometricalmodel of the reinforced car with its reflectorshifted

case because of the blockade that will cause one of the partsof the roof Now the 119911-axis is not the radiation axis of thereflector

The simulation has been done feeding the reflector witha rectangular horn placed at the focus of the reflector(693 2083 32) and at the same frequency as in the previous

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 International Journal of Antennas and Propagation

Top

Front Right

Perspective

119909

119910

119910119909

119909119910

119911

119911 119911

Figure 33 Different views of the reinforced vehicle

0

10

20

0 10 20 30 40 50 60 70120579 (∘)

(dBi

)

Directivity cut 120593 = 0∘

minus10

minus20

minus30

minus40minus10minus20minus30minus40minus50minus60minus70

Polar component MoMPolar component GTD

Figure 34 Polar component obtained with GTD and MoM cut in120593 = 0

∘

Table 1 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 47min 7 h 14min

case 12GHz The sampling frequency of the observationsurface is 1205823 The results obtained with the shifted reflectorhave been calculated applying MoM and GTD (Figure 34)Because of the cancellation of the directivity caused bysome part of the roof the level of the second lobes has

Table 2 Comparison of the CPU time need for the analysis whenGTD-Master Points and MoM techniques are applied

GTD-Master Points MoMCPU time 53min 9 h 41min

been increased This is the main consequence of locating anobstacle in the directivity of the antenna

This simulation has been done in an Intel Xeon at213 GHz As shown in Table 2 the CPU time consumed byMoM is higher than theCPU time consumed byGTD-MasterPoints

6 Conclusion

This paper presents the improvements developed to analyzeof the radiation pattern arbitrarily shaped and fed reflectorantennas Different techniques can be applied to perform thisanalysis In particular the GeometricalTheory of Diffractionis considered in this paper Although this technique is veryuseful to compute the far field radiated by these structuresit has the drawback of being very time consumingThereforethe new techniqueMaster Points has been developed to speedup this process since it reduces the number of times in whichthe ray tracing is calculated

A complete study of the radiation pattern of a parabolicreflector fed with a horn or an array of horns and a study ofthe far field pattern degradation caused by an obstacle havebeen presented The results obtained with the new methodhave been compared with MoM showing good agreementThus it can be concluded that the new function for the CGMand the combination of GTD with Master Points represent a

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 13

good alternative to obtain an accurate and fast evaluation ofthe radiation pattern of reflector antennas

Acknowledgments

This work has been supported in part by the Comunidadde Madrid Project S-2009TIC1485 the Castilla-La Man-cha Project PPII10-0192-0083 and the Spanish Depart-ment of Science Technology Projects TEC2010-15706 andCONSOLIDER-INGENIO no CSD-2008-0068

References

[1] J S Brown ldquoUnidirectional antennas for 450 to 460mcrdquo Trans-actions of the IRE Professional Group on Vehicular Communica-tions vol 1 no 1 pp 134ndash140 1952

[2] C C Cutler ldquoParabolic-antenna design for microwavesrdquo Pro-ceedings of the IRE vol 35 no 1 pp 1284ndash1294 1947

[3] S Seely ldquoMicrowave antenna analysisrdquo Proceedings of the IREvol 35 no 10 pp 1092ndash1095 1947

[4] O M Conde J Perez and M F Catedra ldquoStationary phasemethod application for the analysis of radiation of complex3D conducting structuresrdquo IEEE Transactions on Antennas ampPropagation vol 49 no 5 pp 724ndash731 2001

[5] R G Kouyoumjiam ldquoAsymptotic high-frequency methodsrdquoProceedings of the IEEE vol 53 pp 864ndash876 1965

[6] F Vico-Bondia M Ferrando-Bataller and A Valero-NogueiraldquoA new fast physical optics for smooth surfaces by means of anumerical theory of diffractionrdquo IEEETransactions onAntennasand Propagation vol 58 no 3 pp 773ndash789 2010

[7] M S Narashimhan and K M Prasad ldquoGTD analysis of thenear-field patterns of a prime-focus symmetric paraboloidalreflector antennardquo IEEE Transactions on Antennas and Propa-gation vol 29 no 6 pp 959ndash961 1981

[8] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the Jacobi-Bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[9] W L Ko R Mittra and S W Lee ldquoAperture blockage in reflec-tor antennasrdquo IEEE Transactions on Antennas and Propagationvol 32 no 3 pp 282ndash287 1984

[10] A Boag and C Letrou ldquoFast radiation pattern evaluation forlens and reflector antennasrdquo IEEETransactions onAntennas andPropagation vol 51 no 5 pp 1063ndash1068 2003

[11] SW Lee P Cramer KWoo andY Rahmat-Samii ldquoDiffractionby an arbitrary subreflector GTD solutionrdquo IEEE Transactionson Antennas and Propagation vol 27 no 3 pp 305ndash316 1979

[12] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 1987

[13] S Y Tan and H S Tan ldquoA microcellular communicationspropagation model based on the uniform theory of diffractionandmultiple image theoryrdquo IEEE Transactions on Antennas andPropagation vol 44 no 10 pp 1317ndash1326 1996

[14] G Farin Curves and Surfaces for Computer Aided GeometricDesign Academic Press 1988

[15] W Dahmen M Gasca and C A Micchelli Eds Computationof Curves and Surfaces Kluwer Academic Publishers 1990

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionThe Artech House Microwave Norwood Mass USA 1989

International Journal of

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Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of