on an index for array optimization and discrete prolate spheroidal functions

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antennas.” IEEE T rans. Antennas Propaga t . . vol. AP-27. pp. 72-

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Lab., Harvard Univ., Cambridge, MA, Sci. Ret. no. 12 AFCRL-

66-120, (Series 3) .

H. Nakano and J . Yamauchi, “The balanced helics radiating in the

404-407.

axial mode.” in 1 9 7 IEEE AP-SInr.Symp.Digesr, vol. 11, pp.

E . T. Kornhauser, ”Radiationieldfelicalntennaswith

sinusoidal current.” J . A p p l . P h p s . . vol. 22. pp. 887-891, 1951.

D. S . Jones, Theheory of Elecrrotnagnetisnz. New York:

Pergamon,1964, p. 175.

W. L . Stutzman and G . A . Thiele. AntennaTheory nd De -

s ign . NewYork: Wiley, 1981,p.265.

E . A. Wolff, Anrenna Analys is. New York: Wiley. 1966, p. 442.

S . Sensiper.“Electromagnetic wavepropagationonhelicalcon-

ductors,” in MIT Res.Lab.Electron.Tech. Res.Rept.no. 194.

May1951.

T. S. M. Macleanand R. G . Kouyoumjian, ”The bandwidth of

helicalantennas.” IRE Trans.AntennasPropagar. , vol . AP-7,

special supplement. pp. S379-386, 1959.

On an Index for Array Optimization and the DiscreteProlate Spheroidal Functions

SURENDRA PRASAD,MEMBER, IEEE

Abstract--A class of array optimization problems is considered in

hich we seek to optimize the array response in a specified angular

imiting case of these problems s the width of the specified angularector approaches zero. The optimum array patterns are also shown

o be related to the well-known prolate-spheroidal functions.

I. INTRODUCTION

We consider a class of array optimization problems where

e seek to maximize(orminimize) hearrayresponse na

pecified angular section. The maximization would lead t o an

rraydesign that ends to concent rate he largestpossible

fraction of the tot al radiated (or received) energy in a specific

ngular region. The minimization, on the other hand, is likely

o yield the formation of an effective response minimum n

he specified angular sector.

Themethodproposedhere essentiallygeneralizes the di-

optimization technique 1 -[ 3 ] to incorporate optimi-

ation of the array gain over an angular sector, thus yielding a

hole family of solutions. In fact it is shown here that the di-

ectivity optimum” solut ion becomes a special limiting case of

he new family when the width of the specified angular sector

pproaches zero. The resulting solutions are shown to be re-

amily of functions, for the ase of linearrrays, viz., the prolate spheroidal functions 4] ’

Manuscript received January 22,198l;revi sed August 14, 1981 and

ctober 3 1981.

The author is with the Department of Electrical Engineering, Indian

Institute of Technology, Hauz Khas, New Delhi-110016, India.1 A s pointed out by one of the reviewers, the use of these functions

ported by Rhodes [7].

to antenna pat tern synthesis is not new and haspreviously been re-

11 A CLASSOF ARRAY OPTIMIZATION PROBLEMS

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 direct ion in

the three-dimensional space, and c is the velocity of propaga-

tion. The transmitted/received signal isnarrowband with center

frequency oo ad/s. Let WT = { w l ,w 2 , , w,} e the vector

of complex weightsof the array. Thent is clear tha t the rray re-

sponse in the directiongiven 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-

rayandwhere [ 1 denotes heconjugate ransposeof he

complex matrix [ 1 .The problem to be considered here is the determination of

the weights w, so as to minimize (or maximize) the ratio

,/ F u)2du

F u)2du

U EUa =

E

n

where U denotesa specifiedconical egion n th e three-di-

mensional space about the main-beam direction whereas 2 is

the solidangle of ahemispherearound he mainbeam e.g.,

using the spherical coordinates, we may have

and

Also E denotes he otal power radiated/received by he ar-

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

E” = WTA U)W 7)

where A is the n x n matrix with elementsA k l given by

.If U)l2 exP [ i { Pk-PI)’ u}Wo/C] du. (8 )

Thus the power concentration ratio a of the specified sector

NNTENNASNDROPAGATION, VOL. AP-30, NO. 5, SEPTEMBER 1982 1021

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



IEEE TRANSACTIONS ON ANTENNASAND PROPAGATION, VOL. AP-30, NO. 5,SEPTEMBER 1 9 8 2

is given by

W T A W

W T B Wa=-

optimization problem can thus be formulated as one of

W so that the above ratioof the two quadratic forms

It is trivial to verify that A and B are bot h positive-definitematrices. According to well-known results in matrix

he ptimization of theatio of twouadratic

of positive-definite Hermitian matrices reduces o that of

lues and eigenvectors of the matr ix [51.A W = p B W . (10)

p l > p 2 2 p be he eigenvaluesof theaboveequa-

can be easily proved tha t these are all real nonnegative

To each eigenvalue. , there corresponds an eigenvec-

W = { w l ' , w2 , --, , ' } ~ , Iw, I = 1, such that

A W i = piBW . 11)

by substituting (1 1) in (9), it -follows that the value of~r corresponding to he choice of the th

ctor as the weight vector, is given by

(1 2)

the solution of the maximization (minimization) prob-o that of finding th e eigenvector of the system

igenvalue.

111. THE CASE OF LINEAR ARRAYS

Consider now a broadside linear array of n = 2 N 1 ele-

uniform spacing d and a real weight tapering withw k . The radiation pattern of the array, assuming iso-

N

= W TC( u ) 14)

u = (27rdn) sin 8 ,B is the angle measured from the nor-to the array, and

uo = 27rd/h sin B o and -80 < 8 < 80) is the angularector is azimuth wherein th e energy is to be maximized (mini-

or the special case when d = h/2 he matrixB can be seen to

e to the caled identity matrix

B = 27rI 18)

and heoptimumsolution is obta ined rom he eigenvalue

problem

or, equivalently

I = - l V , - N + l;-,N- l , N . (20)

Thus in this ase the maximizing (minimizing) weight sequence

is the eigenvector of 20) corresponding to th e maximum (or

minimum) eigenvalue.

The solution obtained above for the case of a linear array

with d = h/2 is of special significance since 20) is a discrete

version of the famous prolate-spheroidal wave funct ions given

by the eigenfucntions of the integral equation

and pioneered by Slepian,Pollak, and Landau [41.The proper-ties of these functions are well-known and an excellent treat-

ment of these is available in [41.

Using the terminology of the continuous case, we call th e

radiation patterns,{Fi(u)} of linear arrays corresponding to theeigenvectors {w , , ' } (as these wo are related by he discrete

Fourier transform), the discrete prolate functions. The weight-

ing coefficients (or the Fourie r coeffic ients of the Fz {u ) ) will

be called prolate sequences. h e following orthogonality rela-tions, similar to the corresponding continuous results 141 can

be easily proved for these discrete functions:

1, i = j

0, iS;i

and

Thus the discrete prolate functionsF A u are orthogonal in theinterval -7r,n) and in the nterval ( - u o , uo .

IV.EXAMPLES AND NUMERICAL RESULTS

For reasons of computational simplicity, the examples con-

sidered here are thoseof linear arrays withh/2 spacing. Resultsfor other lineararrays or for other array geometr ies can be

similarly obtained with some added complexi ty of computa-

tions.Table I summarizes he maximizingweightvectors for a

nine-element array AT = 4) for various values of eo . The radia-tion patterns for some of these values are plotted in Fig. l . It



TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 5, SEPTEMBER 1982

TABLE I

MAXIMIZING SOLUTION,Ar = 4

€0 Optimum Weight Vector k1

0.025 W 1 = (0.3301, 0.3325, 0.3342, 0.3353, 0.3356, 0.3353, 0.3342, 0.3325, 0.3301) 0.2219

0.06 W 1 =(0.3207, 0.3300, 0.3368, 0.3409, 0.3423, 0.3409, 0.3368, 0.3300, 0.3207) 0.4265

0.10 W 1 = (0.2848, 0.3191, 0.3452, 0.3614, 0.3669, 0.3614, 0.3452, 0.3191, 0.2848) 0.7366

0.20 W1 =(0.1787, 0.2787, 0.3566, 0.4145, 0.4351, 0.4115, 0.3566, 0.2787, 0.1787) 0.9695

0.40 W 1 =(0.0518, 0.1602, 0.3267, 0.4716, 0.5302, 0.4716, 0.3267, 0.1662, 0.0518) 0.9999

1 Radiation pattern of nine-element inear array: d = h/2, W =

W 1 s a function of €0

0 2

O Y

01 0 2 3 0 4 5

€ +

Fig. 2. Dependence of power concentration on E ~ .

has a direct bearing on the beamwidth of

beam-

parameter. The relationship is more clearly broughtt in Fig. 2 which shows the maximum power concentration

a,,, with eo.

It is interesting to observe from Fig. 1, that for small values

e o , we approa ch the well-known “optimum directivity” so-

[21 obtained in this case, by a uniform, cophasal array.

of course, as expected and clearly il lustrates that the

essentially generalizes and imbeds the opti-

directivity olution nto a broader class ofoptimum

1023

REFERENCES

[ I ] Y. T. Lo, S . W . Lee and 0 H. Lee, ”Optimization o f directivityand signal-to-noise atio of an arbitrary antenna array,’‘ P ro c .

IEEE. vol. 54, no. 8, p. 1033-1045, Aug. 1966.

[2 ] IM T. Ma, The ? and Applicariorz ofA m e n n u A rra ys . New York:Wile y , 1974.

[3] S. Prasad, ”Linear antenna arrays with broad nulls with applicationto adaptive arrays.“ IEEE T ra n s . Anrerztzas Propagar. . vol. AP-27,pp . 185-190, Mar. 1979.

[4] D. Slepian and H. 0 Pollack, “Prolate spheroidal wave functions,

pp. 43-64, Jan . 1961.

Fourier analysis, and uncertainty-I,” Bell Sysr., T ech . , J . vol. 40.

[ j ] F. R . Gantmacher, TheTheory of Matrices vol. I New York:Chelsea, chap. 10, (Translated by K . A . Hirsch) .

[6] D. W . Tufts and J . T. Francis. “Designing digital low-pass

filters-comparison of some methods and criteria,” IE E E T ra n s .

Aud io E lec t ro n . , vol. AU-18, pp. 487494. Dec. 1970.

[71 D. R . Rhodes, ”The op t im um line source or the best mean-square

approximation to a given radiat ion pattern.” I E E E T r a m . Anrenfzas

P r o p a g u t . . vol. AP-I 1 pp. 44W46, J u l y 1963.

A Geometrical Construction for Chebyshev-PlaneZeros

E. FEUERSTEIN

Abstract-Chebyshev-sense equi-rippleesponseerosor ni-

formly sampled antenna and digital-filter apertures may be obtained

through means of a simple geometrical construction. This construc-

tionaffords nsight ntothebehaviorof mappedChebyshev poly-

nomial zeros in the z-plane for both normal and oversampled, equi-

ripple stop-band, “super-resolution” responses.

I. INTRODUCTION

The zeros of appropriately scaled Chebyshev polynomials

may be mapped onto the z-plane unit circle by means of th e

geometrical onstructions llustrated n Fig.1 121, [ 3 1 . In

brief acircle with its center located on the line Im z ) = 0is nscribedwithin the unit circ le; the radius of the nterior

circle is given by

= 2/@1 X, ,

where h s henumber ofzeros equal to henumberofaperture amplesor lementsminusone), nd Pl, is the

main obe-to-peak idelobepower atio. Fo r a inglemain

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. This communication was prepared by F. N. Eddy,also of MITRE, from recollected discussions with, and incomplete notesleft, by theauthor.Thisworkwassupported in partunderUnitedStates Air Force ContractAF19628-82C-9001.

0018-926X/82/0900-1023 00.75 982 IEEE

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on an index for array optimization and discrete prolate spheroidal functions