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8/12/2019 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-
78.1979.
J D.Kraus, Anrennas. NewYork:McGraw Hill, 1950,chap 7.
I R E . , vol 39. pp. 668-675. 1951.
J . A.Marsh, “Current distributions on helical antennas,” P ro c .
C. L. Chen. “Theory of the balanced helical wire antenna,” Cruft
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
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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
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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