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VOL. AP-30, NO. 5, SEPTEMBER 1982


antennas. IEEE Trans. Antennas Propagat.. vol. AP-27. pp. 7278.1979. J. D.Kraus, Anrennas. NewYork:McGraw Hill, 1950,chap 7. J . A.Marsh, Current distributions onhelical antennas, Proc. IRE., vol 39. pp. 668-675. 1951. C. L. Chen. Theory of the balanced helical wire antenna, Cruft Lab., Harvard Univ., Cambridge, MA, Sci. Ret. no. 12 AFCRL66-120, (Series 3 ) . H. Nakano and J . Yamauchi, The balanced helics radiating in the vol. 11, pp. axial mode. in 1 9 7 IEEE AP-S Inr. Symp. Digesr, 404-407. E. T. Kornhauser, Radiation field of helical antennas with sinusoidal current. J . Appl. Phps.. vol. 22. pp. 887-891, 1951. D. S . Jones, The Theory of Elecrrotnagnetisnz. New York: Pergamon,1964, p. 175. W. L. Stutzman and G. A . Thiele. Antenna Theory and Design. New York: Wiley, 1981, p. 265. E. A. Wolff, Anrenna Analysis. New York: Wiley. 1966, p. 442. S. Sensiper.Electromagnetic 194. ductors, in MIT Res.Lab.Electron.Tech. May1951. T. S. M. Macleanand R. G. Kouyoumjian, The bandwidth of IRE Trans. Antennas Propagar., vol. AP-7, helical antennas. special supplement.pp.S379-386,1959.


Let C(u) denote the steering vector of an n-element array for a given spatial direction u, given by

where p , is the three-dimensional vector of position coordinates of the jth element, u is a unit vector in a specific direction in the three-dimensional space, and c is the velocity of propagation. The transmitted/received signal isnarrowband with center frequency oorad/s. Let WT = {wl, w 2 ,-, w,} be the vector it is clear that the array reof complex weightsof the array. Then sponse in the direction given by u is given by

F(u) = WTC(U).f(U)

( 2)

where f(u) is the radiation pattern of each element of the ar[ -1 denotestheconjugatetransposeofthe rayandwhere complex matrix [ - 1 . The problem to be considered here is the determination of the weights w, so as to minimize (or maximize) the ratio

On an Index for Array Optimization and the Discrete Prolate SpheroidalFunctions

Abstract--A class of array optimization problems is considered i n which we seek to optimize the array response in a specified angular sector. The optimization of array directivity is shown to be a special limiting case of these problems as the width of the specified angular sector approaches zero. The optimum array patterns are also shown to be related to the well-known prolate-spheroidal functions.


- EU --


where U denotesa specifiedconicalregionin the three-di2 is mensional space about the main-beam direction whereas ! the solidangle of ahemispherearoundthe mainbeam e.g., using the spherical coordinates, we may have

I. INTRODUCTION We consider a class of array optimization problems where

we seek to maximize (or minimize) the array response in a specified angular section. The maximization would lead t o an array design thattends to concentratethe largest possible fraction of the total radiated (or received) energy in a specific angular region. The minimization, on the other hand, is likely to yield the formation of an effective response minimum in the specified angular sector. Themethodproposedhere essentiallygeneralizes the directivity optimization technique [ 1 ] -[ 3 ] to incorporate optimization of the array gain over an angular sector, thus yielding a diwhole family of solutions. In fact it is shown here that the rectivity optimum solution becomes a special limiting case of the new family when the width of the specified angular sector approaches zero. The resulting solutions are shown to be related to an important family of functions, for the case of linear [4] . arrays, viz., the prolate spheroidal functions
Manuscript received January 22,198l;revised August 14, 1981 and October 3, 1981. The author i s with the Department of Electrical Engineering, Indian 1 Institute of Technology, Hauz Khas, New Delhi-110016, India. 1 As pointed out by one of the reviewers, the use of these functions to antenna pattern synthesis is not new and haspreviously been reported by Rhodes [ 7 ] .


Also E denotes the total power radiated/received by the ray, whereas E , is the power in the sector U. Using (2), we can write


where B is an n

x n matrix with elements

Similarly we have

(7) with elementsA k l given by


where A is the n


x n matrix




(8 )

Thus the power concentration ratio

a of the specified sector

0018-926X/82/0900-1021$00.75 0 1982 IEEE

U is given by




andtheoptimumsolution problem

is obtainedfromthe


or, equivalently The optimization problem can thus be formulated as one of choosing W so that the above ratioof the two quadratic forms is maximized (or minimized). It is trivial to verify that A and B are both positive-definite I=-lV,-N+ l;-,Nl , N . (20) Hermitian matrices. According to well-known results in matrix theory, the optimization of the ratio of two quadratic Thus in thiscase the maximizing (minimizing) weight sequence to that of is the eigenvector of (20) corresponding to the maximum (or forms of positive-definite Hermitian matrices reduces finding the eigenvalues and eigenvectors of the matrix [ 51 . minimum) eigenvalue. The solution obtained above for the case of a linear array AW = pBW. (10) with d = h/2 is of special significance since (20) is a discrete Let p l > p 2 2 p, bethe eigenvalues of theaboveequaversion of the famous prolate-spheroidal wave functions given tion. It can be easily proved that these are all real nonnegative by the eigenfucntions of the integral equation values, To each eigenvalue.&, there corresponds an eigenvec,, I I 2 = 1, such that tor W' = { w l ' , w2', --,w , ' } ~ C


AWi = piBW'.

(1 1)

Also by substituting (1 1) in (9), it -follows that the value of the energy ratio ~r corresponding tothe choice of theith eigenvector as the weight vector, is given by
(1 2)

Pollak, and Landau [ 41 . The properand pioneered by Slepian, ties of these functions are well-known and an excellent treatment of these is available in [ 41. Using the terminology of the continuous case, we call the radiation patterns,{Fi(u)} of linear arrays corresponding to the eigenvectors {w,,'} (as these two are related by the discrete Fourier transform), the discrete prolate functions. The weighting coefficients (or the Fourier coefficients of the Fz{u)) will following orthogonality relabe called prolate sequences. tions, similar to the corresponding continuous results 141, can be easily proved for these discrete functions:


Thus the solution of the maximization (minimization) problem reduces to that of finding the eigenvector of the system (1 1) corresponding to the maximum (minimum) eigenvalue. 111. THE CASE OF LINEAR ARRAYS Consider now a broadside linear array of n = 2 N 1 elements with uniform spacing d and a real weight tapering with weights w k . The radiation pattern of the array, assuming isotropic elements, is given by the Fourier transform relation



0, and


= WTC(u)


where u = (27rdn) sin 8 , B is the angle measured from the normal to the array, and

It is easy to see now that in thiscase we have Thus the discrete prolate functionsFAu) are orthogonal in the interval (-uo, uo). interval (-7r, n) and in the where uo = 27rd/h sin Bo and (-80 < 8 80) is the angular sector is azimuth wherein the energy is to be maximized (minimized), and that



For reasons of computational simplicity, the examples considered here are thoseof linear arrays with h/2 spacing. Results for other lineararrays or for other array geometries can be similarly obtained with some added complexity of computations. Table I summarizesthe maximizingweightvectors for a nine-element array (AT = 4) for various values of eo. The radiation patterns for some of these values are plotted in Fig. l . It

For the special case when d = h/2,the matrix B can be seen to reduce to thescaled identity matrix B = 27rI (1 8)




Optimum Weight Vector

W1 = (0.3301, W1 =(0.3207, W1 = (0.2848, W1 =(0.1787, W1 =(0.0518,
0.3325, 0.3300, 0.3191, 0.2787, 0.1602, 0.3342, 0.3368, 0.3452, 0.3566, 0.3267, 0.3353, 0.3409, 0.3614, 0.4145, 0.4716, 0.3356, 0.3423, 0.3669, 0.4351, 0.5302, 0.3353, 0.3409, 0.3614, 0.4115, 0.4716, 0.3342, 0.3368, 0.3452, 0.3566, 0.3267, 0.3325, 0.3300, 0.3191, 0.2787, 0.1662, 0.3301) 0.3207) 0.2848) 0.1787) 0.0518)


0.025 0.06 0.10 0.20 0.40

0.2219 0.4265 0.7366 0.9695 0.9999


REFERENCES Y. T. Lo, S. W . Lee and 0. H. Lee, Optimization o f directivity and signal-to-noiseratio of an arbitrary antenna array, Proc. IEEE. vol. 54, no. 8, pp. 1033-1045, Aug. 1966.
IM. T. Ma, The#? and Applicariorz of Amennu Arrays. New York: Wiley, 1974. S. Prasad, Linear antenna arrays with broad nulls with application to adaptive arrays. IEEE Trans. Anrerztzas Propagar.. vol. AP-27, pp. 185-190, Mar. 1979. D. Slepian and H. 0. Pollack, Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I, Bell Sysr., Tech., J . vol. 40. pp. 43-64, Jan. 1961. F. R . Gantmacher, TheTheory of Matrices. vol. I , New York: Chelsea, chap. 10, (Translated by K. A. Hirsch). D. W . Tufts and J . T. Francis. Designing digital low-pass filters-comparison of some methods and criteria, IEEE Trans. Audio Electron., vol. AU-18, pp. 487494. Dec. 1970. D. R . Rhodes, The optimum line source for the best mean-square approximation to a given radiation pattern. IEEE T r a m . Anrenfzas Propagut.. vol. AP-I 1. pp. 4 4 W 4 6 , July 1963.


m n

[4] [j] [6]


Fig. 1. Radiation pattern of nine-element linear array: W 1 as a function of 0 .

d = h/2, W =

A Geometrical Construction for Chebyshev 2-Plane Zeros

Abstract-Chebyshev-sense equi-ripple response zeros for uniformly sampled antenna and digital-filter apertures may be obtained through means of a simple geometrical construction. This constructionaffordsinsightintothebehaviorof mappedChebyshev polynomial zeros in the z-plane for both normal and oversampled, equiripple stop-band, super-resolution responses.


OY 0






Fig. 2.

Dependence of power concentration on

E ~ .

The zeros of appropriately scaled Chebyshev polynomials may be mapped onto the z-plane unit circle by means of the geometrical constructions illustrated in Fig. 1 121, [ 3 1 . In (z) = 0 brief acircle with its center located on the line Im isinscribedwithin the unit circle; the radius of the interior circle is given by
= 2/@1

is clearly seen that E,, has a direct bearing on the beamwidth of the radiation pattern and, therefore, can be regarded a beamas shapingparameter. The relationship is more clearly brought out in Fig. 2 which shows the maximum power concentration ratio a,,, with eo. It is interesting to observe from Fig. 1, that for small values of e o , we approach the well-known optimum directivity solution [ 21 obtained in this case, by a uniform, cophasal array. This is, of course, as expected and clearly illustrates that the present approach essentially generalizes and imbeds the optimumdirectivitysolutioninto a broader class ofoptimum solutions.

+ X,),

where hT isthenumber ofzeros(equal tothenumberof P l , is the aperture samples or elements minus one), and main lobe-to-peak sidelobe power ratio. For a single main lobe (Le., P, = I ) , X , = 1. Super-resolution is achieved Manuscript received July30, 198l;revised January 5,1982.
E. Feurerstein,deceased, waswith the MITRE Corporation, Bed-

ford, MA 01730. Thiscommunication was preparedby F. N. Eddy,

also of MITRE, from recollected discussions with, and incomplete notes in partunderUnited left, by theauthor.Thisworkwassupported

States Air Force Contract AF19628-82C-9001.

001 8-926X/82/0900-1023$00.75 0 1982 IEEE