~

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 2, FEBRUARY 1987

L-

J -30 c - \

-60

1 2 1 3 14

%a

Fig. 2. The cross-polar level versus koa with E , and d / a as parameters. Solid curves correspond to interrupted lining, kod is chosen from Table I with ko = 15/a. The dotted curves correspond to continuous lining, k ~ d is obtained from (14).

have two different values for transverse electric (TE) and transverse magnetic (TM) waves, say u I and u?. For TE waves (e; = 0) only E [

will appear, so u1 = EO - p2)L’2a, while for the TM part of the modes both E~ and E ! play roles. It can be shown (e.g. [ 2 ] ) that u2 = u1 ( E J E ~ ) Since X is affected by the TE part and Y by the TM part of the mode, (12) and (13) are modified in this case to

X = (koQ/U[) tall (Uld/O) (18)

Y= - (EI/Eo)(koa/u*) cot (vzd/a). (19)

Now since the arguments of the tan and cot functions above are different, there is a possibility that the condition X = Y be satisfied at real koa and E [ . This, however depends on the amount of anisotropy or, more specifically the deviation of the ratio € , / E , from unity. Using (16) and (17) it is easy to show that the maximum of E [ /

E~ occurs when t / l = 1/2. Thus, we fix t / l and use (18) and (19) to minimize I X - Y I numerically as E , and k0d vary. This results in a relation between E , and kod for minimum cross-polar radiation which is given in Table I. It is to be noted that exact nulling of I X - Y ( occurs for E , > 8.0 while for E, < 8.0, only a finite minimum is obtainable. At the critical value E, = 8.0, kod is seen to attain its minimum value I . 17 which is equal to - 1.5a/4.

Next, we compute the cross-polar level C’ in (1 1) versus koa for given E, and /rod, and the results are shown in Fig. 2. The solid curves correspond to slotted dielectric lining with t / l = 1/2 and the values of d / a are chosen such that the cross-polar level is zero or minimum at koa = 15 (by using Table I). For comparison, cases of continuous lining with E , = 2.5 and 10 are shown by dotted curves. It is clear that significant improvement is obtained by slotting the dielectric lining if E, is greater than at least five. Actually for E, less than five, the anisotropy is small, and almost no improvement over the continuous lining case is achieved. It is worth noting that the cross-polar levels in Fig. 2 can be reduced by increasing koa at the rate of 6 dB per octave, provided that kod is kept constant at the center frequency. e.g., by doubling the abscissa and reducing d /a to half their values. the values of C’ reduce by 6 dB all over. Thus, in order to get the same cross- polar levels as the waveguide with interrupted lining, the koa of the continuously lined guide will have to increase by more than a factor of six.

CONCLUSION

One concludes that by slotting the dielectric lining of a circular waveguide, reduction in the cross-polar radiation fields of 15-30 dB

TABLE I E , VERSUS d / a FOR MINIMUM CROSS-POLAR RADIATION AT ku = 15,

t / l = 0.5

6, 10 8 7 5 d /a 0.0827 0.078 0.08 0.1

over a wide band of frequency is achieved. This improvement in performance requires that the interrupted dielectric lining should have E, > 5 in order that the cross-polar level be small at the center frequency and E , < 10 to have this improvement over a wide band. An optimum value E , = 8.0 at t / l = 112 is thus found. It is believed that this new version of the dielectric lined waveguide presents a very good improvement at the expense of only modest complication in manufacturing. A sequel to this communication will present a more parametric study of the dominant mode, the higher order modes, and their excitation.

REFERENCES

[ I ] C. Dragone, “High frequency behavior of waveguides with finite surface impedance.” BellSyst. Tech. J . , vol. 60, pp. 80-116, 1981.

[2] J. R. Wait, Electromagnetic Waves in Srrutr;fed Media. Elmsford, N Y : Pergamon, 1970, ch. 8.

Synthesis of Near-Field Patterns of Arrays

M. S. NARASIMHAN, SENIOR MEMBER, IEEE, AND BOBBY PHILIPS

Abstract-A new- technique of synthesis of near-field (NF) amplitude and phase patterns of linear, planar, or volume arrays of finite size or arrays located on a planar contour of finite size is presented. The array could consist of point dipoles or directive elements. The criterion for prescribing the NF (amplitude and phase) pattern information in the synthesis problem for unique determination of array excitation currents is also stated. The proposed near-field synthesis technique is based on the potential integral solution of source currents, Nyquist sampling of the near-field data and the technique of linear least square approximation (LLSA). The NF pattern synthesis technique is illustrated to synthesize a variety of NF patterns with a number of array configurations. Applica- tion of the proposed NF pattern synthesis technique to minimise distortion in far-field patterns of arrays mounted on a conducting platform and to realize array antennas with low sidelobes in the near and far field is also presented.

I. INTRODUCTION

Synthesis of near-field (NF) patterns of an array of point dipoles or directive elements plays a very important role in a number of applications. Minimizing illumination of a conducting platform on which a radiating array is mounted and minimizing the NF coupling between two array antennas placed side by side will be typical examples to emphasize the need for NF pattern synthesis. Until now no systematic attempt has been made to formulate an accurate NF synthesis procedure for array antennas. The only available technique in the literature for NF pattern synthesis is based on Green’s dyadic involving spherical vector wave functions (SVWF) [ 11. In this

Manuscript received July 30, 1985; revised Februaly 4, 1986. The authors are with the Electromagnetics and Antennas Group, Centre for

Systems and Devices. Indian Institute of Technology, Madras 600 036, India. IEEE Log Number 8612226.

0018-926X/87/0200-0212$01.00 0 1987 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. AP-35, NO. 2, FEBRUARY 1987 213

method the spherical mode coefficients a,, and b,, are required to be computed from the prescribed volumetric NF pattern of the array. Out of a large number of significant spherical mode coefficients obtained, only the first few coefficients which equal the unknown complex excitation currents of the array are considered and the rest are discarded. This leads to loss of NF pattern information, thereby resulting in less satisfactory pattern fit in the sidelobe region. Moreover, the earlier method becomes extremely involved, if the linear array consists of open-ended waveguides as array elements of a lineadplanar array.

In this communication, given the array configuration, we enunciate a criterion for prescribing the NF pattern information in our synthesis problem for unique determination of the array excitation currents. Subsequently, we propose a new method of synthesis of the near field of an array which employs potential integral (PI) solution of the source currents, sampling the NF at Nyquist rate and the technique of linear least squares approximation (LLSA) [2]-[4]. The proposed technique overcomes almost all the limitations of the earlier method [I] and offers impressive fit of the synthesized NF pattern with the prescribed NF pattern for a number of array configurations. Application of the proposed NF synthesis technique in minimising distortion of far-field patterns of arrays mounted on conducting platforms and in realizing array antennas with low sidelobes in the near and far field is also presented.

11. NEAR-FIELD PAITERN SYNTHESIS

By near-field pattern we mean both amplitude and phase patterns throughout this communication. We assume the array to be finite in size so that for each array configuration we can define a sphere with a minimum diameter D = 2a, which completely encloses the array. For the sake of notational simplicity in our discussions to follow, without any loss of generality, we consider only point dipoles as array elements.

The following rules given without proof should be adhered to while synthesizing the NF patterns of an array.

1) With a linear array of point dipoles with direction of current flow perpendicular to or along the array axis, we can synthesize the NF pattern only in a single azimuthal plane. Similarly for an array of point dipoles located on a contour of arbitrary shape lying on a plane with direction of current flow of the dipole perpendicular to the array plane, the NF pattern can be synthesized only on a single 8 = constant surface.

2) With a planar array of point dipoles having their directions of current flow confined to the array plane, it is possible to synthesize a volumetric NF pattern prescribed over the forward hemispherical surface (0 I 9 I 27i and 0 5 0 5 n/2).

3) Synthesis of a volumetric NF pattern prescribed over a complete spherical surface (0 I 4 5 27r and 0 I 8 I n) is possible with a spherical array of point dipoles with their directions of current flow confined to the surface of the sphere.

Now we proceed to illustrate a new NF pattern synthesis technique which could be applied to any one of the four typical array configurations mentioned in the previous paragraphs of this section irrespective of whether the array is uniformly spaced or nonuni- formly spaced.

A . Criterion f o r Sampling the Prescribed NF Pattern

In order to accurately reconstruct the NF from a discrete set of samples selected at Nyquist interval over a spherical surface (which encloses the source antenna) yith a radius equal to the prescribed NF distance in the synthesis problem, we have to sample the near field at angular intervals A8 I h/D and Ab 5 X/D [SI. We define 40max = A&,,,, = h/D. Selection of appropriate values of A8 and Aq5 depends

1 2

(e )

Fig. 1 . Configurations of different type of arrays considered.

on the nature of the prescribed NF pattern. If the number of spherical modes required to reconstruct the NF pattern is greater than ka, then A8 and Aq5 should be considerably less than X/D [6]. However, without loss of generality, we will illustrate the NF synthesis procedure with a NF sampling interval of and

For the sake of notational simplicity and without loss of generality, the NF synthesis technique will be illustrated considering a uniformly spaced linear array of point dipoles located along the z-axis. The direction of current flow of each dipole is along the z-axis (Fig. l(a)). The array possesses M discrete elements. It is supposed that the H, component of this linear array over a sphere of radius r in the near field is prescribed and we sample this H6 at 8 = 6" for i = 1, 2, . . - , N. Sampling is done at the Nyquist rate. Based on the potential integral solution, the unknown complex excitation currents of the linear array are related to H6 through the following matrix equation:

EX= F (1)

where E is a 2N X 2 M real matrix which involves the position coordinates of all the array elements and the sampling points, and F is a known vector of length 2N obtained from the prescribed near field. In (1) the real vector X of length 2M represents the unknown excitation currents (where the real and imaginary parts have been separated). We wish to determine the real vector X of length 2M in (1) that minimizes the Euclidean norm llEX - FII . If the rank of E is less than 2M, we require simultaneously that llXll should be a minimum [2].

' If 3 dB beamwidth of the NF pattern to be synthesised is broader than the 3 dB beamwidth of the same array, excited with currents of uniform amplitude and phase at the same radial distance, it is possible to select AB 2 M,,, and Ad 2 Admai.

c

214 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. Ap-35, NO. 2, EEBRUARY 1987

20 40 60 B O .

-RECONSTRUCTP PATTERN

+ ~30. -40; SYNTHESIZED + : 0' Plane .9 A8.1.915' --40

- -50

-- 60

I

-10 - m - 9-20 - s

.g

E,

-

m PATTERN - -30- v

SYNTHESIZED

a-40-

=-50 -

-60- ( C ) I

0 20 40 60 80 0 20 40 60 BO e .deg. 9.deg.

Fig. 2. Comparison of the prescribed and synthesized NF patterns at a Fig. 3. comparison of prescribed and synthesized NF patterns at a radial radial distance of 4.24 h in the ,#I = O", 30" and 90" planes for a planar distance Of 64 h in the principal planes and reconstructed patterns in the array of point dipoles with 3 A X 3 X size. ,#I = 30" plane of a 16 X x 16 h planar array of point dipoles.

m. NUMERICAL RESULTS

Unless otherwise stated, in all the numerical examples to follow the array is uniformly spaced with an interelement spacing d = 0.5 X. Further, the array element is an electric dipole. The pattern normalization constant is denoted by C.

Volumetric NF pattern synthesis will be the first example considered. Here, the NF pattern synthesis is performed on a planar array of point dipoles located in the x - y plane with dipole currents directed along the negative y-axis (Fig. l(c)). The planar array is located symmetrically with respect to the x-axis as well as the y-axis. The prescribed NF pattern, which is symmetric about the z = 0 plane, is given by

where r = rl = (9 &$, = 0.707 (D2/X); p = 8. In all, there are 49 unknown complex excitation currents and there are 101 constraints. The prescribed and synthesized NF patterns over the azimuthal planes 4 = 0, 30" and 90" are computed and are compared in Fig. 2.

Volumetric NF pattern synthesis will be greatly simplified if NF pattern synthesis for a planar array is performed only in the two principal planes. Such a procedure is adopted to synthesize the principal plane NF patterns of a 16 X X 16 X planar array of y- dirkcted point dipoles. Computed results on the synthesized and prescribed NF patterns of this array are illustrated in Fig. 3. These computed results are generated with 33 ( = M ) unknown complex excitation currents and 95 ( = N ) constraints.

The next numerical example considers synthesis of a Gaussian

amplitude pattern (with a constant phase) by an elliptic array of point diples (with direction of current flow along the z axis) in the 8 = 90" plane (Fig. l(d)). The prescribed and synthesized NF patterns of the elliptic array are shown in Fig. 4(a).

As an example of NF pattern synthesis for a linear a.rray with directive elements, we consider 25 open-ended rectangular wave- guides each excited in its dominant mode with a uniform interelement spacing of 0.5 X as measured from the center of one rectangular waveguide to the adjacent waveguide and located along the z-axis (Fig. l(e)) so that the E-field of the waveguide aperture is parallel to the z-axis. Mutual coupling effects are ignored. The synthesized and prescribed NF patterns for the Eo component of the array are shown in Fig. 4@).

A comparison is made in Fig. 5 on the pattern fit realized, when the same NF pattern is synthesized with 1) d = h / r , and 2) d = 0.5 X. When d = Vsr, since the number of sampling points and the number of unknown complex excitation currents are identical, a standard matrix inversion (MI) algorithm (algorithm 1) has been used while computing array excitation currents. When d = 0.5 X, an algorithm based on LLSA (algorithm 2) is used for obtaining the array excitation currents. The computation time required to obtain the complex-excitation currents is approximately the same for both algorithms. In algorithm 1, there is no control of interelement spacing, whereas algorithm 2 does not have this restriction. Further, since d < V2 whenever algorithm 1 could be made use of, a better pattern fit is realized with reduced radiation efficiency for the array.

A study is also made on a uniformly spaced linear array with point dipoles, with directions of current flow directed along the array axis, for ascertaining how the pattern fit of the synthesized NF patterns with the prescribed pattern can possibly be made more accurate particularly in the sidelobe region. In this study the same NF pattern

IEEE TRANSACnONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 2. FEBRUARY 1987 215

I O I D -12X; d =0.5X : & =90" P l a n e _ _ Prescr ibed

' r l :0.25D2/X

I AQ=Aemax ~4.74' Oer a-0.023rn ;

Synthesised csc2e Pattern on onc side ( 0 660') a n d

3 0 . 6 c W I

/ 1 GausGan pattern ( 9 5 60')

. - * 0 :E

L W

I I I

0 20 40 6 0 80 100 120 110 160 I -10 I

9,deg.

(b)

Fig. 4. (a) Comparison of the prescribed and synthesized NF patterns at a radial distance of 60 X in the 6' = 90" plane for an elliptic array of point dipoles (with direction of current flow along the z-axis). (b) Comparison of the prescribed and synthesized NF patterns at a radial distance of 36 X in the 4 = 90" plane for a linear array of waveguides excited in the dominant mode (Fig. I(e)).

1 . c

0 .E

- - 0.6 I e

9c d

-1 0'

20 40 6 0 80 %deglO0

120 1.40 160 18( I I I

D=8X ; d=X / r r t1=0.35DZ / X

(Matr ix inversior ( A l g o r i t h m l )

D = 8 X ; d = X / 2 rI= 0.35D2/X

n e = A8,,~=6.92~ _ _ Prescribed -Synthesized

( A l g o r i t h m 2 )

IO

Fig. 5. Comparison of prescribed and synthesized NF patterns using algorithms 1 and 2 for a linear array of point dipoles.

.~ .

216 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 2, FEBRUARY 1987

Fig. 6 . Comparison of prescribed and synthesized NF patterns of a linear array of point dipoles obtained by varying (b) Ab'. (c) d, and (d) D.

is synthesized by varying 1) AO, 2) d, and 3) D/X (array lengh). Results of this study are illustrated in Fig. 6.

N. APPLICATIONS

A. Synthesis of NF/FF Patterns with Low Sidelobes In order to synthesize NF patterns with low sidelobes at a

prescribed NF distance r = r1 and also to ensure that the low sidelobes realized at r = rl are maintained for all values of r > r l , it i s found that the 3 dB BW of the NF pattern to be synthesized should be equal to or greater than 1.33 times 3 dB BWU, which is defined as 3 dB BW of the array when all its elements are excited with uniform amplitude and phase. Fig. 7 illustrates this. Further, given the dimensions of the array (D/X X D/X for a square array or D/h for a linear array), the broader the 3 dB BW of the prescribed NF pattern to be synthesized in comparison with 3 dB BWU, a better fit between the synthesized and prescribed pattern is realized particularly in the sidelobe region while attempting to realize a low sidelobe array with an interelement spacing of 0.5 X. This is vividly illustrated in Fig. 8. While synthesizing a pencil beam NF as well as FF patterns with low sidelobes (Fig. 7), it is mandatory that in addition to near-field amplitude, its phase (held constant for the case considered in Fig. 7) is also synthesized. If the phase pattern (in addition to amplitude pattern) is not synthesized in the near field, a pencil beam for the array will not be formed at various far-field distances (Fig. 7). This explains the importance of phase in NF pattern synthesis.

B. Minimization of Pattern Distortion of Array Antennas Mounted on a Conducting Piatform

When an array or aperture type of antenna designed to radiate a desired far-field pattern in a clear site environment in free space is mounted on any platform, there will be a significant distortion in its radiation patterns, if the height of the antenna from the mounting

oO I 20 4 0 e*de$O 80 0 20 LO 60 80

9.deg.

9 =90*Plane r l = D 2 / X

pattern m v. -20

I*-3 0 - I e --30

Z-40 0 - --LO

m --Reconstructed 7J-

m pattern - - - - 7J

-

? -50 -

-60 - (a) - 60 0

-10 - 0.90' Plane r l i 2 0 2 1 A

I$ I 90. Plane r l = m

- -10

9 .deg .

Pattern

- 30

- 40

- 50

- 60

20 40 60 80 90 9 .deg.

Fig. 7. Reconstructed Hd patterns with currents obtained by synthesizing the NF patterns of Fig. 3 for several values of r, .

-lz0t \ Fig. 8. Comparison of the prescribed and synthesized NF patterns and

reconstructed far-field patterns of a 30 X linear array of point dipoles with platform is not very large [7]. However, by synthesizing appropri- low sidelobes.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 2, FEBRUARY 1987 217

0 10 20 30 40 50 6 0 70 80 90 8.deg

16 I I I I I D=l?.h

1.4- Dimenstons ~n metres

1.2 - _ - Pattern without obstacle present

1.0 - a n d n o NF Pattern synthesis -Pattern with obstacle present

per formed.

I 1 5 8 n I I 1 7 L I II I I

V Pattern wlthout obstacle present ._ = n6 and no NF pottern synthesis E

F P a t t e r n with obstacle present

E - - I performed L

0.41- O I Jv"

Pattern wlth obstacle present Pattern without obstacle present

and NF pattern synthesis performed a

/

Fig. 9. (a) comparison of the clear-site (viz., free space with no obstacle present) far-field (csc? 8 + Gaussian) pattern of a linear array of point dipoles with the distorted far-field patterns of the same array when it is mounted on a conducting platform. (b) Comparison of the far-field (cscL 8 + Gaussian) clearsite patterns of a linear array of point dipoles (D = 12 X d = 0.5 X) with the far-field patterns of the same array when located over a conducting platform after an appropriate NF synthesis is performed on the array.

ately the near-field pattern of the array one can almost avoid illumination of the mounting platform by the array and hence reduce significantly pattern distortion in the far field. In order to illustrate this, as a first example we consider a linear array designated array-1 with a length of 12 h and a uniform interelement spacing of 0.5 h. This array-1 is excited with currents obtained from a synthesis of NF pattern, which greatly minimizes illumination of the conducting platform. Subsequently this array-1 is made to radiate in the presence of the platform,2 whose dimensions are shown in the inset of Fig. 9(a), and we observe that the far-field pattern distortion of this array is significantly reduced as illustrated in Fig. 9. The geometrical theory of diffraction technique detailed in [7] is employed to compute the pattern of the array radiating in the presence of the conducting platform. Fig. 10 illustrates a second example for minimization of pattern distortion of a linear array ( a r r a ~ - 2 ) ~ mounted on a conducting platform by a careful synthesis of the array NF pattern.

C. Other Possible Applications of the Technique of Linear Least Squares Fit [3J in NF Pattern Synthesis

The least squares problem posed by

EX=F (31

The linear array is mounted parallel to the z'-axis of the platform coordinate system (Fig. 9(a)) and the center 0 of the linear array has the coordinates x' = 0; y' = 0 and z' = 1.08 m (36 X).

The linear array is mounted parallel to the z'-axis of the platform coordinate system (Fig. 10(a)) and the center 0 of the linear array has the coordinates x' = 0, y ' = 0 and z ' = 1.08 m (36 X).

i

i I I I I I I I I I _ _ Pattern without obstacle present

l - Pattern with obstacle present

performed and NFPatiern synthesis

oa

" L 2 6 ; -

Fig. 10. (a) Comparison of the clearsite (viz. , freespace with no obstacle present) far-field (pencil beam) patterns of a linear array of point dipoles with the distorted far-field patterns of the same array when it is mounted on a conducting platform. (b) Comparison of the far-field (pencil beam) clear- site patterns of a linear array of point dipoles (D = 12 X, d = 0.5 X) with the far-field patterns of the same array when located over a conducting platform after an appropriate NF synthesis is performed on the array.

where E is an N x IM matrix and F is an N-vector can be solved with a variety of additional constraints. Given an arbitrary K X M matrix C of rank K I ~ and a K-vector D, we can seek the solution M-vector X that minimizes (IEX - FII subject to

C X = D. (4)

This requires the consistency of (4) which is assured if M > K = K1. This is the least squares problem with equality constraints, referred to as problem LSE in [3], [ 8 ] . Performing NF pattern synthesis subject to the requirement that the far-field pattern should follow more or less a desired pattern can be formulated as a problem LSE.

If we have a linear least squares problem in which we would like some of the equations to be exactly satisfied, this can be accomplished approximately by weighing these constraining equations heavily and solving the resulting least squares system [3]. Employing this procedure, the pattern fit in the sidelobe region in the NF pattern synthesis problem can be improved at the cost of pattern fit in the main beam and vice versa.

We can also solve (3) subject to

G X r D ( 5 )

where G is an arbitrary K by M matrix. This is the linear least squares problem with linear inequality constraints, referred to as problem LSI in [3], [9]. A special case of the above is to minimize [ [EX - FII subject to X 2 0. This can be used to impose such constraints as nonnegativity, or that each variable have an indepen- dent upper and lower bound, or that the sum of all the variables be

218 E E E TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 2, FEBRUARY 1987

less than a specified value. Applications of this will be the following. Suppose NF synthesis is done only with current amplitudes, with phase prescribed in advance, nonnegativity can be imposed on current magnitudes. In such an event, G is the complex identity matrix and D is the null matrix. Similarly upper and lower bounds can be imposed on each current magnitude.

V. CONCLUSION The new synthesis technique proposed in this communication,

based on PI solution of the source currents, considerably simplifies the NF pattern synthesis problem and is applicable to any type of array antenna of finite size. Impressive NF pattern fit obtained in several numerical examples considered demonstrate the efficiency and computational accuracy of the algorithms proposed and the sampling criterion employed to sample the NF patterns. The proposed NF pattern synthesis procedure enables one to realize very low sidelobes for the array both in the near and far fields. Further, this technique enables one to synthesize appropriately the NF patterns of an array so that the distortion in the far-field pattern of the array usually observed on mounting it on a conducting platform without taking resort to any NF pattern synthesis, is significantly reduced.

r21

[31

r41

r51

r71

REFERENCES M. S. Narasimhan, K. Varadarangan, and S. Christopher, "A new technique of synthesis of the near or far field patterns of arrays," ZEEE Trans. Antennas Propagat., vol. AP-34, pp. 773-778, June 1986. G. Golub, "Numerical methods for solving linear least squares problems," Num. Math., vol. 7, pp. 206216, 1965. C. L. Lawson and R. J. Hanson, Solving Least Squares Problem. Englewood Cliffs, NJ: Prentice-Hall, 1974. IMSL Library, Problem-Solving Software System for Mathematical and Statistical FORTRAN Programming, 9th ed., vol. 2, Houston. TX: IMSL, June 1982. J. J. Tavormina and D. W. Has, "Spherical near-field antenna measurements with SA Model 2022," presented at Antenna Measure- ment Assoc. Conf., Oct. 1980. M. S. Narasimhan, S . Christopher, and K. Varadarangan, "Model behavior of spherical waves from a source of EM radiation with application to spherical scanning," ZEEE Trans. Antennas Propa- gat., vol. AP-33, pp. 350-354, Mar. 1985. M. S. Narasimhan and B. Philips, "Pattern distortion of aperture antennas radiating in the presence of conducting platforms," ZEEE Trans. Antennas Propagat., vol. AP-32, pp. 887-890, 1984. R. J. Hanson and C. L. Lawson, "Extensions and applications of the householder algorithm for solving linear least squares problems," Math. Comp., vol. 23, pp. 787-812, 1969. J. Stoer, "On the numerical solution of constrained least squares problems," SZAMJ. Num. Anal., vol. 8, pp. 382-411, 1971.

A Generalization of Van den Berg's Integral-Square Error Iterative Computational Technique

for Scattering

A. J . MACKAY AND A. M c ~ ~ ~ ~ ~

Abstract-Van den Berg shows a way of using spectral iteration (SI) within a scheme that shows good convergence. His method can be reinterpreted as a global expansion of the field quantities in terms of a set

Manuscript received April 4, 1986; revised July 8, 1986. This work was supported in part by SERC.

The authors are with the Department of Electrical and Electronic Engineering, University College of Swansea, Singleton Park, Swansea SA2 8PP, UK.

IEEE Log Number 8612265.

of basis functions. It is shown here that the orthogonalization of these basis functions can considerably improve convergence rates. This involves extra storage requirements, but only a negligible decrease in speed of computation.

INTRODUCTION

There are considerable merits in using a method of spectral iteration (SI) as a computational technique in electromagnetic scattering. Proposed first by Bojarski [l] and later refined by the same author [2], [3], this method has had much recent attention due to the potential increase in speed over conventional moment method techniques. One of the problems with SI however, is that of convergence. If the method is used as first proposed [I] , or, for example, as in papers by Tsao and Mittra [4] and Kastner and Mittra [5], there are many regimes where convergence is either very slow or nonexistent. Ad hoc relaxation factors, e.g., Kastner and Mittra [5] have been used to improve the situation but with only marginal success. Recently, however, Van den Berg [6] showed how SI can be put onto a much firmer footing where convergence is achieved in a way guaranteed to reduce an error norm on each iteration.

In Van den Berg's notation, his method hinges upon recognizing the pair of equations

Xc")(X)=X("-l)(X)+rl(")gc")(x) (la)

F(")(x)=F("-')(x)-tl'")f'"'(x). (1b)

X(")(x) is the nth estimate of the current density on a conducting body. F(")(x) is the nth estimate of the field difference between the known field on the object (known as a consequence of the boundary condition) and the estimated scattered field. f(") and g(")(x) may be regarded as field-like and current-like basis functions related by the Green's function.

Van den Berg chooses g(")(x) on the basis of the methods employed in spectral iteration, and calculates q(") in such a way as to minimize F(")(x) in the mean. Specifically, by minimization of

err(")= 1, I ~ ( " ) ( x ) l 2 dx

where D is the surface of the conducting body.

A NEW INTERPRETATION It may be recognized from (la) and (lb) that

X("'(x) -X'O)(x) = q'"g"'(x) + q"'g'2'(x)

+ * - * + q(")g(")(x) (2a)

i.e., the current density and field difference are represented explicitly as an expansion in terms of their respective basis functions. The relationship betweenf') and g(') is rigid, but the choice of g(') can be modified in any desired way.

Now, for err'") + 0 for increasing n it is required that err(") should be made as small as possible for a given n. This will mean that F(O)(x) should be as "close" as possible to the sum in square brackets in (2b) for any given n. Essentially, F(O)(x) is being expanded as

F(O)(X) = q ( ' ) f q X ) + ?pf'2'(X) + . . ' + q(")f(")(x). ( 3 )

Strictly speaking, err(") tends to zero for increasing n only if the sequence f ( ' ) (x) is a complete sequence of functions. However, if the

0018-926X/8870020--0218$01.00 0 1987 IEEE