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1342 IEEE TRANSACTIONSON ANTENNAS AN D PROPAGATION,VOL. 41,NO. 9, SEPTEMBER 1993As a limiting example of a particular isotropic panel where themeasurements show E, =E y, and 4, = $ = 20, the graphshows 4R 20 as expected.

DEPOLARIZATIONNTO THE ORTHOGONALIRCULARLYPOLARIZEDAVEFrom the polarization ellipse and its tilt angle, both thedominant circularly polarized wave as well as the orthogonal

circularly polarized waves can be calculated as discussed above.Assuming that the incident wave is dominant RCP, the amountof emerging LCP wave or isolation relative to the emerging RCPis shown as the ordinate in Fig. 3 with the LP phase differenceas the abscissa. The parametric curves are LP amplitude differ-ences in dB. For instance, the isolation is 10 dB when the LPamplitude and phase differences are 4 dB and 25, respectively.CONCLUSION

Convenient graphs are presented to determine the effect of ananisotropic planar panel intercepting a circularly polarized beam,based on measurements using orthogonal linearly polarized (LP)waves aligned to the symmetry planes. The right circular polar-ized (RCP) additional loss values are presented for the fullrange of LP amplitude ratios and LP phase differences. Exem-plary RCP phase shift values are presented for a limited rangeof LP amplitude ratios and phase shifts. The amount of CPdepolarization or isolation is illustrated in graphical form.

REFERENCES[l] A. F. Kay, Radomes and absorbers, in Antenna Engineering Hand-book, H. Jasik, Ed. New York McGraw-Hill, 1961, Chap. 32.[2] C. k Balanis, Advanced E ngine ekg Electmmagnetics. New YorkJohn Wiley, 1989.[3] S. W. Lee, Basics, in Antenna Handbook, Y. T. Lo and S . W . Lee,Eds. New York Van Nostrand, 1988, Chap. 1.

Comments on Modeling of Cylindrical Objects byCircular Dielectric and Conducting CylindersQiu Mei-De

Recently, an approach for modeling the EM scattering ofcylindrical objects by small circular cylinders (which we call acircular cylinder model) was presented in the above communica-tion [l]. I read it with great interest. A few comments are asfollowed.(1) Under Eq. (12) on page 97 of [l ] it was stated that Ni san arbitrary large number which satisfies the relation 3 I Ni I3ka + 3 in order to have a convergent solution. It seems to methat this bound of Ni is superfluous when a 2-D perfectlyconducting cylinder is modeled by a circular cylinder modelmentioned above. To illustrate this, the specific case of Ni = 0with TM polarized excitation is considered here. In this case,n = E = 0. Eq. (12) in [ l] reduces to

wire grid model circular cylinder modelFig. 1. The geometryof the two models.

0 w 120 180 240 3m 3wAngle

a. Far field xlnb. Near fieldFig. 2. The scattered fields of the two models for a perfectly conductingcircular cylinder.- xact; 0 wire grid model;Y circular cylinder model.where the elements of vector [Ai] are given by

ai = e j k p i cos(4; - 40), (2)and the elements of [Sig] re given by

(3 )This matrix equation system is just a wire grid model of aninfinite perfectly conducting transverse magnetic (TM) cylinder[2]. The behavior of a wire grid model is well known [2]-[4] andwill not be discussed here. It should be mentioned that when weemploy a wire grid model to model the scattering of a conduct-ing cylinder, we get a matrix equation of much smaller order andhence a more rapidly convergent solution. The memory storagerequired is also greatly decreased.(2) It is found that the scattered field, especially the near field,obtained by a wire grid model that satisfies the same surface-arearule of thumb, is more accurate than that obtained by a circularcylinder model compared with the series solution of the per-fectly conducting circular cylinder.The radius of the perfectly conducting circular cylinder to bemodeled here is equal to A. The data for the circular cylindermodel is the same as in [l ] and Ni = 4. For a wire grid model,the same surface-area rule of thumb is used. Fig. 2(a) shows thefar scattered pattern of the two models. Twenty wires of radius0.05A are used for the wire grid model. Fig. 2(b) shows the nearfield of the two models. The field point is a cut along y = 0,across the cylinders diameter. Thirty-two wires are used herefor the wire grid model. It should be noted that the location

(1) of the wires in the wire grid model is different from the cylindersin the circular cylinder model in [l], which is shown in Fig. 1.ForSig][Ci] = [Ai], the wire grid model, the centers of the wires are on thesurfaceof the cylinder, whereas for the circular cylinder model [ll, thecenters of the small cylinders are on a circular boundary ofManuscript received March 1,1993.The author is with Shenyang Research Institute, Shenyang,P.R. China.IEEE Lo g Number 9212499. radius 0.9h.

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IEEE TRANSACTIONSON ANTENNAS AND PROPAGATION,VOL. 41, NO . 9, SEPTEMBER1993 1343In addition, there is an error in Eq. (10) in [l]. t should rea d 2 1

-2

, pi sin qq2 pb sin c#$,, p i sin 4; < p; sin 4;.

p; cos 4; - p; cos 4;cos-1-cos- * ih

4Pp4 =

REFERENCES111 A. Z. Elsherbeni and A.A. Kishk, Modeling of cylindrical objectsby circular dielectric and conducting cylinders, IEEE Trans. Anten-nas Propagat., vol. AP-40, pp. 96-99, Jan. 1992.[2] R. J. Paknys, The near field of a wire grid model, IEEE Trans.Antennas Propagat., vol. AP-39, pp. 994-999, July 1991.[3] A.C.Ludwig, Wire grid model of surfaces, ZEEE Trans. AntennasPropagut.,vol. AP-35, pp. 1045-1048, Sept. 1987.[4] J. H. Richmond, A wire-grid model for scattering by conductingbodies, ZEEE Trans. Antennas Propagut., vol. AP-14, pp. 782-786,Nov. 1966.

Authors ReplyAtef Z. Elsherbeni and Ahmed A. Kishk

We have received comments [l]on our paper [2] that intro-duces a general method and semianalytical solution for themodeling of composite two-dimensional scatterers. In the follow-ing section we shall provide our response to the comments in [11.1. It is pointed out that we have used an unnecessarily largenumber of terms (N,) or thin conducting wires with TM excita-tion. This comment is true, for this special case. It is possible touse the expression reported by Paknys [31, that is, & = 2ka,which yields a fewer number of terms than what we have used.Fig. 1shows the near field of the scattered TM incident planewave from 20 cylinders simulating a perfectly conducting cylin-der of radius 1 . Two different values for N, were used, namely,4 and 2. Almost no differences between the two cases wereobserved, which indicates that the choice of 4. 2 for this

special case is sufficient. However, it should be clear that theexpression for the number of terms in our paper [2] is for thinand thick cylinders, TE and TM excitations, and dielectric andconducting cylinders. It is much more general and the proposedreduction in the number of terms will not yield the correctresults for all these cases. As an example, the TE excitationrequires a minimum of three terms for the very thin conductingwires. The curves in Fig. 2clearly demonstrate this fact.2. Fig. 3 shows the near scattered field from a perfectlyconducting cylinder of radius 1 A which is simulated with 32 thincylinders as in [l , Fig. l(a>l. Our results for this geometryindicate that the value of N,.= 0 does not produce accurateresults as in [l, Fig. 2(b)]. However, when N, = 2, very goodagreement with the exact solution is obtained. This again indi-cates that the choice of & = 0 is not sufficient for wires withradius equal to 0.05A.3. It is not true that the wire grid model for the scattering by aconducting cylinder yields, in general, a matrix equation ofThe authors are with the Electrical Engineering Department, Univer-IEEE Log Number 9212498.sity of Mississippi, University, MS 38677.

I J0-4 -

-6

-8 -1 - - - 20 cylinders, Ni=2, ai=.lh-4 -3 -2 -1 0 1 2 3 4

X I XFig. 1. Near field of a perfectly conducting circular cylinder with itscenter at the origin, radius = l h a nd 20 cylinders using the circularcylinder simulation. (Excitation is a TM plane wave incident from thepositive x axis.)

10 1 I0

-10y1El3 -200e4 -30

TE_ - - -II\\\\\\

II\\\\\\

-40 1 - soluti;, .-_1- . 6 2 cylinders, Ni=O, ai=.05h. .. 6 2 cylinders, Ni=3,ai=.05h

-4 -3 -2 -1 0 1 2 3 4Xlh

-50

Fig. 2. Near field of a perfectly conducting circular cylinder with itscenter at the origin, radius = l h and 20 cylinders using the circularcylinder simulation. (Excitation is a TE plane wave incident from thepositive x axis.)

smaller order relative to the matrix equation generated by thecircular cylinder model. To illustrate this point, one can refer tothe dashed curve in Fig. 2(4.2, order of the matrix is100 x 100, circular cylinder model) and the dotted curve in Fig.3 (N,= 2, order of the matrix is 160 x 160, wire grid model)where the near-field data in these two cases are in good agree-ment with the exact solution.4. The method of modeling a perfectly conducting cylinder byletting the simulating cylinders touch the cylindrical boundaryfrom the inside (circular cylinder model) is a better representa-tion than placing the cylinders such that their centers arelocated on the cylindrical boundary (wire grid model). This is

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1344 IEEE TRANSACTIONS ON ANTENNASAND PROPAGATION,VOL. 41, NO. 9, SEPTEMBER1993

- 4 --6 - - xact solution

I- - - 32 cylinders, Ni=O, ai=.053L-10 1-4 -3 -2 -1 0 1 2 3 4

x t hFig. 3. Near field of a perfectly conducting circular cylinder with itscenter at the origin, radius = l h and 32 cylinders using the wire gridsimulation. (Excitation is a Th4 plane wave incident from the positive naxis.)

very obvious when cylinders of finite diameter are used insteadof the very thin cylinders or wires, in order to maintain the sameouter physical boundary of the simulated cylinder.5. The far-field data presented in [l ,Fig. 2(a)] for our circularcylinder model is inaccurately computed by Mei-De. We haveregenerated this data and found that it is in excellent agreementwith the exact solution as was originally shown in [2 ,Fig. 41.6. In our paper [ 2 ] ,we did not present any near-field results.The submission of [l] encouraged us to compute the nearscattered field from simulated objects. We found that the nearfield of the 20 cylinders with Ni = 4, ai = O.lh is in excellentagreement with the exact solution as shown in Fig. 1.We believethat the results in [l ,Fig. 2(b)] which represent our circularcylinder model are incorrectly computed by Mei-De and thusinaccurate.7. The complete expression of (10) in [ 2 ] is correctly provided

in [ll; however, we should point out that we have used thecomplete expression in our computations for all the resultspresented in [2]. We thank Mei-De for pointing out that (10) in[2]was incomplete.We would like to take this opportunity to correct a fewtypographical errors in [ 2 ] hat were pointed out in [ 4 ] .(a) Equation (5) should be replaced by the following equationin order to describe the H+ component in both free space anddielectric media.1 dEiiqi=--

770k0 dP (5)

REFERENCES111 Q. Mei-De, Comments on Mo delin g of Cylindrical Objects byCircular Dielectric and Conducting Cylinders,ZEEE Trans. Anten-nas Propagat., vol. X, pp. 0-0, 1993, this issue.

[2] A. Z. Elsherbeni and A. A. Kishk, Modeling of Cylindrical Objectsby Circular Dielectric and Conducting Cylinders, ZEEE Trans.Antennas Propagat.,vol. AP-40, no. 1, pp. 96-99, Jan. 1992.[3] R. J. Paknys, The Near Field of a Wire Grid Model, ZEEE Trans.Antennas Propagat.,vol. AP-39, no. 7, pp. 994-999, July 1991.[4] Q. Mei-De, personal communication, Sept. 1992.

Comm ents on A N eural Network Approach toMVDR Beamforming ProblemK.W.Lo

The above paper presented an analog neural network ap-proach for the implementation of the MVDR eamformer. Ananalog circuit was described, which consists of a conductancematrix Gwhose element values are proportional to those of thecovariance matrix of the array G, nd a set of OP-AMPswhoseoutput voltages U represent the weight vector for the beam-former. Assuming an exact covariance matrix, computer simula-tions were carried out to demonstrate the performance of theproposed circuit. The paper has been read with interest, but itappears to have a few errors. We present below a list ofcorrections, followed by some thoughtful discussions as well asour extended work on some of the results.Corrections to the paper are as follows:In (34) and (35), the p should be replaced by p/2.In (441, the second + sign should be -, and consequently,N = p B T / C .Since matrix M is real symmetric (and positive definite), ithas a complete set of orthonormal eigenvectors. Thereforethe solution of (44) an be expressed as a linear combina-tion of modes e-! and not tke- l as mentioned in Ap-pendix I.Eq. (54) should read e+., = 2 apJ( p + 2 ap , ) .

It was shown in Appendix I that the convergent time to thesteady-state response is governed by the dominant time constantof the analog circuit, T, which is the reciprocal of the minimumeigenvalue of matrix M (45). However, there was no explicitaccount on the value of T in terms of the circuit parameters. Infact, the upper bound of T can easily be deduced by realizingthat matrices G / C and p B T B / C are real symmetric and so theminimum eigenvalue of M has a lower bound equal to the sumof their minimum eigenvalues, which are 2 a u 2 / C (providedthat the number of interfering signals is less than the number ofantenna elements) and zero, respectively. Therefore, the domi-nant time constant r has an upper bound given by

C2au2rmax-

Manuscript received March 22, 1993.The author is with the Microwave Radar Division, Defence Scienceand Technology Organization, PO Box 1500, Salisbury, S.A.5108, Aus-tralia.fEEE Log Number 9212509.P. R. Chang, W. H. Yang, and K. K. Chan, IEEE. Trans. AntennasPropagat., vol. AP-40, no. 3, pp. 313-322, Mar. 1992.

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