306 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-28, NO. 3, MAY 1980

Fast Analysis of Large Antennas-A New Computational Philosophy

OVIDIO M. BUCCI, GIORGIO FRANCESCHEnI, SENIOR MEMBER, m E . AND GIUSEPPE D'ELI.4

Absnacr-A new computational approach is presented which allows a fast analysis of radiation properties of large antennas. The radiated field is first computed using conventional techniques, e.g., physical optics and geometrical theory of diffraction, in prescribed sampled space di- rections, roughly one direction per lobe. Then sampling theory is used to reconstruct the complete radiation diagram. Numerical experi- ments are presented in the last part of the paper, showing the excellent performance of the method.

F I. THE IDEA

AST ANALYSIS of radiation properties of large antennas a vital problem in today's space communication tech-

nology, where single and dual reflector, symmetric and offset, single, multiple, and contoured beam antennas are considered. Although the following considerations are quite general, we will focus our attention on a particular case-the offset para- bolic dish with nonfocal illumination-for clarity of presenta- tion. The geometry of the dish is sketched in Fig. 1; the scattered electric far field E is given by

where J is the unitary 3 x 3 matrix, K the propagation con- stant, P = d m , J, the induced surface currents on the parabolic dish surface So, andS1 is the dish area projected on the focal plane.

Fast analysis of radiation properties of the dish implies fast evaluation of expression (1). Powerful asymptotic tech- niques are available, as the geometrical theory of diffraction (GTD), with its several modifications, and asymptotic physical optics (APO). However, only recently a fast code for the non- asymptotic evaluation of (1) has been developed [ 11 by Galindo-Israel and Mittra (GIM). These authors manipulate the radiation integral (2) in order to obtain the best approxi- mation to a (double) Fourier integral. Then they expand the integrand in a suitable basis-circular functions and Jacobi polynomials-obtaining a fast convergent series representation for I, whose leading term corresponds to the case of a uni- formly illuminated aperture. The use of Jacobi polynomials easily allows the inclusion of corrective terms due to the non-Fourier transform structure of ( 2 ) or to the curvature of the reflector, which is the same.

Manuscript received February 20, 1979; revised October 15, 1979. This work was supported in part by SOC. Selenia SPA, Roma, Italy.

Universitz di Napoli, 80125 Naples, Italy. . 0. M. Bucci and G. DElia are with the Istituto Elettrotecnico,

G. Franceschetti was with the Electrical Sciences and Engineering Department, University'of California, Los Angles, CA 90024, on leave from the Istituto Elettrotecnico, Universid di Napoli, 80125 Naples, Italy.

I"

Fig. 1. Geometry of offset parabolic dish.

We will present an alternative approach for the evaluation of I which is at least as fast as the GIM one and exhibits a number of extra appealing features. In particular, the evalua- tion of (2) is not limited to the PO approximation for the surface currents J, (as practically in the GIM case), and the inclusion of the reflector curvature can be handled in a much easier way (no extra series summation is needed).

First of all, let us consider the primed coordinate system ( x f , y ' , z') of unit vectors

k ' = i cos a+$ sin a

(3)

i'= -2 sin a+$ cos a

dl 2f

tan a=-

with the origin 0' = O'(x0, yo, zo), x 0 = d l , yo = 0, zo = -f -I- (dl -I- a , *)/4f in the center of the aperture area across the dish (see Fig. 2 ) .

Simple geometrical considerations show that

where y1 = d(x - d l )' -k y2 and is the radial coordinate in the projected aperture plane. Then, substituting into (21, we get -

I =exp C j ~ ( d ~ sin 0 cos @ + z cos e)] cos (Y

. exp (jzix' + jc 'y ' ) d l ' dy' (6)

where the dependence of the integral upon ( x f , y ' ) is implied, and

u'= K sin 8' cos @'

c' = K sin 0' sin @'

0018-926X/80/0500-0306500.75 0 1980 IEEE

BUCCI et al.: FAST ANALYSIS OF LARGE ANTENNAS 3 07

r

tr

Fig.. 2. Relevant to change of integration coordinates.

and are the transverse ( to z ’ ) components of the propagation vector with respect to the primed system (x ’ , y ’ , z‘).

Expression (6) can be further manipulated by introducing a convenient direction ( B o , GO), e.g., that of the main beam, and properly normalizing all spatial coordinates, hence

where (uo ’ , UO’) are the values of (u’, u ‘ ) corresponding t o t h e direction (B0,Qo). Then

where

Equation (1 0) is essentially that given by GIM in [ 1 1 . Accord- ingly, we have the remarkable result that the mathematical manipulations in [ 1 ] correspond to take an aperture plane just across the dish. The fundamental reason why this is the optimum choice from computational viewpoint will be clari- fied in the next section.

For A = 0 , ( 1 0) is an exact Fourier transform. This trans- form relationship is approximately met for 6 = B o . This is not true in the far-out regions, as claimed in [ 11. As a matter of fact, although A(s = 1 ) = 0, the asymptotic evaluation of ( 1 0) does contain terms involving the quantity [aA/as],= 1 which is different from zero.

In the approximation A = 0 , (10) is the Fourier trans- form of a (spatially) bandlimited function. Accordingly, I’ is exactly expressed in terms of its samples at the Nyquist angular rate [2 ] , [ 31 (n divided the spatial bandwidth):

The main idea of this paper is to take advantage of this circumstance and, consequently, t o reconstruct the far field from the knowledge of essentially one point per lobe. Letting IO be the same expression (1 0 ) with A = 0 , we have exactly

+ a + x IO’(lC, o)= Em l,(nn, mn)

- a - x

sin (u - nn) sin fr - ma) u - nn z-mn

The series (14) makes use of the simplest choice for the sampling functions; other more sophisticated possibilities are also possible [ 41 .

Then, using essentially the same technique of the GIM paper, we have after series expansion of exp (jA) and suc- cessive Fourier transformation:

i ,x. + a, + z I’(u, c)= x, 1, Cn1 Io’(nn, ma)

0 - a - x

.[I+- +-] [ u - n n all2 2c= c-mn 1. (15)

a C P sin (u -na) sin (o-mn)

The following comments are now in order. I

Expression (15) is a very interesting (exact) representa- tion for the far field scattered by the dish, which requires the computation of the radiation integral IO’ in the discrete (sampled) directions u, = nn, U, = mrr only. Truncation properties of (15) will be discussed in Section 111. However, it is obvious that the number of terms to be retained is pro- portional to the angular range in which the radiation diagram must be computed. Accordingly, the effort (computing time) is proportional t o t h e scope.

The integral Io’ is an exact Fourier transform, so that fast Fourier transform computational techniques can be used. For large values of rn and n, asymptotic techniques can be adopted as well, which imply further reduction in computa- tional time. From this viewpoint, (15) represents the best technique for smoothing out and joining asymptotic and nonasymptotic computed data.

The use of ( 1 5), as well as that of the GIM method, has a serious shortcoming. The evaluation of the samples Io‘(nn, nzn) is practically based on the PO approximation for the surface currents J,. This limitation will be relaxed in the next section as a bonus stemming out from the elimination of the p-series. Elimination of this series and of the restriction t o PO makes this method a very appealing alternative to the GIM technique.

11. THE IMPROVEMENT

The use of (15) implies reconstruction of the far field from the knowledge of spatial samples Io‘(nn, m x ) , which coincide with the true scattered field only for 0 Bo. The reason is that the starting point of all the analysis has been the Fourier integral IO’, instead of the true radiation integral I’, in order t o have a (spatial) bandlimited signal. We can try to reformulate all the analysis upside down from the onset, starting from the true radiation integral and then forcing, t o an assigned degree of approximation, the band limitation requirement.

308 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-28, NO. 3, MAY 1980

~~

Fig. 3. Relevant to improved analysis of radiation from the dish.

With this in mind, let us consider an arbitrary plane in between the reflector and the observation point P (Fig. 3) . The electric far field in P can be computed, using the equiva- lence theorem, as

where H is the magnetic field scattered by the dish on x. Let us consider, for the moment, the elementary magnetic field dH(Q‘) produced by the elementary current at Q on the dish, namely:

Substitution of (1 7) into (16) shows that the rapidly varying phase term of the integrand, for K + m, is given by $(e’) = K(+ -I- /); and this phase is stationary for Q‘ = Qo, i.e., at the intersection of the straight line from the source point Q to the field point P. This implies, as known, that the major contribution to the integral (16) due to the current element J,(Q)dSo arises from a neighborhood of the pro- jection of the point Q onto C along the direction to the field point.

In order to truncate the integration exhibited in (1 6) to a finite area, we must fix such a neighborhood. When the integration is extended to all X, the integral is asymptotically given by the contribution from the stationary point Qo, which obviously equals the direct field due to the source at Q. If the integration is truncated to a finite neighborhood of Q0, an additional “diffracted” contribution is introduced. This last is negligible as compared with the direct one as long as the observation point P is well outside the transition region associated with the light-shadow boundary, Le., when the detour parameter

evaluated at the relevant stationary point 3 on the neighbor- hood periphery is much larger than one. When all the source points on the reflector are considered, the conclusion is that we can safely neglect the contribution to (16) due to the region of C such that, whatever the poinLQ, 5 > 10.

The truncated integration domain C is now defined; it obviously changes as the position and orientation of the plane C changes. Since the equation = 10 describes a para-

Fig. 4. Relevant to enlarged aperture area concept.

boloid of revolution of focus Q , shifting C toward the re- flector will decrease the relevant integration domain. This suggests that the best direction-independent choice is to make C coincident with the dish aperture plane S‘, Le., closest to the reflector. This could be rigorously justified and also makes all flash points on the rim, relevant to off-axis radia- tion, to lie on Z.

The dimensions of the relevant integration domain are now obtained from (18) for the worst possible points Q on the reflector. These are the rim points, and we get (see Fig. 4)

$ = K I I B A I + I A P I - I B P I I

= K 1 6 -6 sin (CY+@ 12 10. (19)

From (1 9) we can compute the radius a1 ‘ of the enlarged projected aperture

The conclusion is that the domain of integration in (16) can be truncated to t he enlarged projected aperture defined by (20). Then, the (integration dependent) operator [J-f?] reduces to the (integration independent) operator [J - RR] and we recover a Fourier transform relationship between the far-field and a (spatially) bandlimited field distribution, at least up to an angle 0 depending on the enlargment factor x (see (20)).

Note that the (Nyquist) angular sampling rate should now be related to the enlarged aperture of radius X Q ~ . Accordingly, we should substitute Xu1 t o a1 in (9) and (1 3). Letting:

Ll,,”l = nn, c,,, = m n (21)

and assuming as the reference direction (00, $ 0 ) that of the z’ axis, we can easily solve for the sampling angular directions:

1

KXQ 1 sin e,lm’=- J(nn)2 cos2 a+(mn?,

tan Qrtnl’=- m

r1 cos CY

It follows that aperture enlargement just implies an x times larger Nyquist rate, Le., a x2 times larger number of angular samples. This is similar t o what happens in communication theory applications, when a signal is passed through a low passband filter and then sampled. The samplingrate is properly changed with respect t o t he ideal filter case, accounting for

BUCCI et ~ 1 . : FAST ANALYSIS OF LARGE ANTENNAS 309

the smooth (and not sharp) behavior of the transfer function of the filter at the cutoff frequency. In our case, the change in the (angular) Nyquist rate accounts for the nonzero value of the field in the aperture plane outside the dish area.

When the far field is known in the sampling directions (22), the radiation diagram can be reconstructed by using the sampling theorem [2 ] , [ 3 ] , as we will show in detail here- after. The far-field samples can be conveniently computed upon use of known conventional techniques, e.g., PO in the angular range close to B o and GTD elsewhere. Measured values of the samples can be used as well. This flexibility is another very appealing feature of the presented method.

In applying the sampling theorem, it should be remarked that the Fourier transform pairs are the aperture field distrib- ution and the transverse components of the spectrum

[J-RR]- 'E(p)=L E (23)

as follows from (16)' with truncated to c. Accordingly, the components (23; should be sampled; and then the far field obtained upon multiplication by the inverse matrix operator L- ' . Expressing E in its (e ' , 9') components we get

where

Practical use of (24) requires the truncation of the series t o a finite angular range of significant field samples. Then the value of x can be chosen, so that expression (24) allows the reconstruction of the field starting from the best available values of the true scattered field in the sampled directions ( enn l '? Qnnt 'I.

111. THE NUMERICAL EXPERIMENTS

Once the computation of the radiated field has been cast in terms of an aperture problem, it follows that the radiation diagram in any plane cut through z' can be expressed in terms of the Fourier integral of an equivalent line distribution (integral of the aperture field along lines orthogonal to the chosen cut). This suggests that numerical experiments can be performed on the simplest geometry of a parabolic cylinder with line source illumination; for this case (24) also becomes scalar.

Fig. 5 shows a reconstruction experiment and the improve- ment which is obtained by increasing the aperture area, i.e., by increasing the sampling rate. The parabolic cylinder has an aperture 2al = 50X, an aperture half-angle = 30' and the (electric) source line is 1 O X far from the focus in the focal plane.

5Y 10 lr . u= k3stne 151r

Fig. 5. Radiation diagram of parabolic cylinder. Solid line: surface currents integration method. Triangles: 11 samples at (spatial) sampling rate Au = TT. Dots: 11 samples at (spatial) sampling rate Au = 47115.

I

I I I I , , I I . . 0 I 6 I 71

u.ke,sine

Fig. 6. Radiation diagram of an offset parabolic cylinder. Solid line: surface currents integration method. Dots: 7 samples. Crosses: 13 samples. Squares: 19 samples. Triangles: 25 samples.

The solid line represents the ratio I EJEi 1, Ei being the field due to the source, and has been computed using the PO surface currents integration method. Then, 11 samples of this radiation diagram have been used to reconstruct it. The samples are symmetrically located with respect t o 11 = Kal sin B = 10n and spaced of Au = 77 (triangles) and Au = 4 ~ / 5 (dots), respectively. The latter corresponds to an equivalent increase of the aperture of 20 percent and results in a negligible error in the reconstructed pattern up to the third lateral lobe.

Fig. 6 shows the relation between the number of samples and the angular range in which a good reconstruction of the is considered with aperture 2al = 50h, offset angle Il/o = 45' and aperture half-angle Il/ = 30'. The cylinder is illuminated by an (electric) source line 1 O X far from the focus. The solid line represents the radiation diagram computed using the POsurface currents integration method. Then, this radiation diagram is sampled, symmetrically with respect t o u = 677, with a sampling rate Au = 477/5. The total number of samples i s 7 (dots); 13 (crosses); 19 (squares); 25 (triangles). Inspec- tion of the figure shows that the number of samples requiied for a quite satisfactory reconstruction of the radiation diagram is slightly larger than the number of required lobes. The ex- ample of Fig. 6 exhibits a rather pathological geometry. In

310 IEEE TRANSACTIONS

most of the cases we considered, we found that the number of samples should be equal to the number of lobes plus two.

IV. CONCLUSIONS

In this paper we presented a simple reliable method for fast analysis of the far field scattered by large antennas. One of the most interesting features of this method is that it is compatible with any existing program for the analysis of radiation diagrams, thus requiring no extra programming effort but a simple interpolation scheme.

It is also noted that the basic idea of the method is com- monly used in communication theory. Sampling theory is widely used in optics [ 51 and also in the antenna field; the theory has been applied to wire antenna pattern representations [ 6 ] . (For an excellent review of sampling theory applica- tions, see [7] .) However, we would like to point out that the use we made of the sampling theorem is completely new, offering what we think is a significant advantage in reflector antenna computations. This technique, and especially its possible modifications [ 81, should play a major role in the best strategy for antenna radiation pattern analysis [ 91.

REFERENCES

[ I ] V. Galindo-Israel and R . M i m , “A new series representation for the radiation integral with application to reflector antennas,’’ IEEE Trans. AnrennasPropagat., vol. AP-25, pp. 631-641, Sept. 1977.

[2] E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory.” Proc. Roy . Sor. Edimburgh. Sect. A, vol. 35,p. 181, 1915.

[3] C. E. Shannon, “Communication in the presence of noise.” in Proc. IRE, vol. 37, p. 10, 1949.

[4] D. P. Peterson and D. Middleton, “Sampling and reconstruction of wavenumber-limited functions in N-dimensional Euclidean spaces.“ Inform. Conrr., vol. 5, p. 279, 1962.

ON ANTENNAS AND PROPAGATION, VOL. AP-28, NO. 3, MAY 1980

[SI J . W. Goodman, Introduction ro Fourier Oprics. New York: McGraw-

[6] E. Roubine, Antennas. Vol. II, Masson, 1978, p. 113. [7] A. J . Jeni, “The Shannon sampling theorem. Its various extensions and

applications: a tutorial review,” Proc. IEEE, vol. 11, p. 1565, 1977. [SI 0. M. Bucci. G. Franceschetti, andR. Pieni. “Reflectorantennasfields.

An exact aperture-like approach,” to be published. [9] 0. M. Bucci, G. D’Elia, and G. Franceschetti, “Computation of

radiation from reflector antennas. An optimum strategy,” thud Report to Selenia, Istituto Elettrotecnico. University of Naples, Naples, Italy, 1979.

Hill, 1968, p. 21.

Ovidio M. Bucci, for a biography and photograph please see page 305 of this issue.

Giorgio Franceschetti (S’6&M’62SM’73),‘ for a photograph and biography please see page 595 of the September 1979 issue of this TRAiiSACTONS.

Giuseppe D’Elia was born in Salerno, Italy, in 1950. He graduated in electronic engineering from the University of Naples. Naples, Italy, in 1976.

Since 1976, he has been working with the research ,goup in electromagnetics at the Engineering De- pamnent of the University of Naples. His main scientific interests concern the transient behavior of linear antennas and the analysis of large reflector antennas.