antennas.” IEEE Trans. Antennas Propagat.. vol. AP-27. pp. 72- 78. 1979. J D. Kraus, Anrennas. New York:McGraw Hill, 1950, chap 7. IRE., vol 39. J. A. Marsh, “Current distributions onhelical antennas,” Proc. C. L. Chen. “Theory of the balanced helical wire antenna,” Cruft Lab., Harvard Univ., Cambri dge, 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 197 IEEE AP-S Inr. Symp. Digesr, vol. 11, pp. E. T. Kornhauser, ”Radiation ield f elical ntennas with sinusoidal current.” J. Appl. Phps.. vol. 22. pp. 887-891, 1951. D. S . Jones, The heory o f Elecrrotnagnetisnz. New York: Pergamon, 1964, p. 175. W. L. Stutzman and G. A . Thiele. Antenna Theory nd De- sign. New York: Wiley, 1981, p. 265. E . A. Wolff, Anrenna Analysis. New York: Wiley. 1966, p. 442. S . Sensiper. “Electromagnetic wave propagation on helical con- ductors,” in MIT Res. Lab. Electron. Tech. Res. Rept. no. 194. May 1951. T. S . M. Maclean and R . G . Kouyoumjian, ”The bandwidth of helical antennas.” IRE Trans. Antennas Propagar., vol. AP-7, specia l supplement. pp. S379-386, 1959. On an Index for Arra y Optimizat ion and the Discrete Prolate Spheroidal Func tions SURENDRA PRASAD, MEMBER, IEEE Abstract--A class of array optimization problems is considered i n which we seek t o 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 s the width of the specified angular sector approaches zero. The optimum a rray patterns are also sh own to be related to the well-known prolate-spheroidal functions. I. INTRODUCTION W e cons ider a class of array optimization problems where we seek to maximize (or minimize) he array response n a speci fied angular secti on. The maximization would lead t o an array des ign that ends to concent rate he largest possible fraction of the tot al radiated (or received) energy in a specific angular region. The minimization, on the ot her ha nd, is likel y t o yield the form atio n of an effectiveresponseminimum n the specified angular sector. The method proposed here essentially generalizes th e di- rectivity optimization technique 1 -[ 3 ] to incorporate optimi- zation of the array gain over an angular sector, thu s yielding a whole family of solutions. In fact it is shown here that the di- rectivity o ptimum” solution becomes a special limiting ca se of the new family when the width of th e speci fied angular sector approaches zero. The resulting solutions are shown to be re- lated to an important amily of functions, for the ase of linear arrays, viz., the prolate spheroidal functions 4 ] ’ Manuscript rec eived Janua ry 22,198l;revised August 14, 1981 and October 3 1981. The author i s with the Department of Electrical Engineering, Indian Institute of Technolo gy, 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 ant enna pattern synthesis is not new and has previously been re- 11 A CLASS OF ARRAY OPTIMIZATION PROBLEMS Let C u) denote the steering vector of an n-element array for a give n 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 th e three-dimensional space, and c is the velocity of propaga- tion. The transmitted/received signal is narrowband with center frequency o o ad/s. Let WT = {wl, w2, , w,} e the vector of complex weightsof the array. Then t is clear that the rray re- 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- ray and where [ 1 denotes he conjugate ranspose of 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 EU a= E n where U denotes a specified conical egion n th e three-di- mensional space about the main-beam direction whereas 2 is the solid angle of a hemisphere around he main beam 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 elements Akl given by . I f U)l2 exP [i{ Pk-PI)’ u}Wo/C] du. ( 8 ) Thus the power concentration ratio a of the specified sector IEEE TRANSACTIONS N NTENNAS ND ROPAGATION, VOL. AP-30, NO. 5, SEPTEMBER 1982 1021 0018-926X/82/0900-1021 00.75 1982 IEEE
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8/12/2019 on an index for array optimization and discrete prolate spheroidal functions
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
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.