where * = 2n -sin 8. ug = cosh (& In [q +..-=i]) If we define w by the following expression, w E exp (j+)

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1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP32, NO. 10, OCTOBER 1984 Taylor Patterns for Discrete Arrays 1089 ALFRED T. VILLENEUVE, SENIORMEMBER,IEEE AbstractThe equivalent of a Taylor pattern for continuous apertuies is The variable u is given by the following expression: developed for discrete arrays. This array factor can be realized exactly, * ', since it belongs to the appropriate class of patterns. A method for u = uo cosdetermining the element excitations is also presented. As an example, the 2 pattern and excitations are calculated for a 41element array with d = 6 where and a 25 db design sidelobe level. Comparison is made with the patterns obtained by applying root matching and sampling of a continnous Taylor d aperture. * = 2n sin 8. INTRODUCTION In this expression, 8 is the angle measured from the array broadside, d is the element spacing, and h is the wavelength.' The value OLPH developed a pattern for a uniformly spaced array of uo is determined by the desired sidelobe level according to D that has the minimum possible beamwidth for a given maxithe following expression: mum sidelobe level. The pattern is obtained from the Chebyshev polynomial, and for an array of 2N + 1 elements, it has 2N nulls. All sidelobes are of equal height. The array excitations ug = cosh (& In [q +..=i]) that are required for low sidelobe excitations may have large values for the end elements. where q = 10sL/zo and SL is the design sidelobe level in db. The Taylor devised an analogous pattern for continuous apertures roots or zeros of the patterns are given by the following expres [I]. This pattern, called the "ideal" pattern, also has sidelobes sion: that are all equal and requires an aperture distribution that is singular at the ends of the aperture. Taylor modified the ideal pattern by making only the first Fi sidelobes approximately equal and by making the far out sidelobes decay as sin (@/X. This decay results in finite values of the aperture distribution at the aperture edges. The Taylor distribution is widely used and is often applied to discrete arrays. One method of determining the required aperture distributions for discrete arrays is to sample If we define w by the following expression, w E exp (j+) the pattern can be rewritten in product form: the continuous distribution at the locations of the elements. 2N Elliott describes an alternate method for determining the element excitations (21. The excitations are made equal to the coeffi p= 1 cients of the polynomial obtained by multiplying out the 2N The peak occurs when w = 1, that is, when $ = 0. The value of factors containing the N zeros of the Taylor pattern that lie on A is determined by the selected normalization criterion and for each side of the main beam. The resulting patterns are discrete convenience may be set equal to unity in the remaining developarray approximations to the Taylor pattern. In this paper a pattern analogous to the Taylor pattern is those for 1 < =p < =N so that the pattern may be rewritten as developed directly for discrete arrays. The approach is similar fol!lows. to that used by Taylor except that it is specifically developed N for discrete arrays. The starting point is the Chebyshev polynomial. TZN(U) = F(w) = n (w wp)(w WB) The method is described in the following section. p=1 PATTERN ANALYSIS The Chebyshev pattern of a linear array of 2N + 1 elements is a polynomial of degree 2N in the variable u and, consequently, has 2N zeros. The polynomial is given by the following expression [3]:,v (N + n)! T ~ N n=o N + n (2n)!(N n)! = (1)"'" ( ) ( 2uy = cos (2Ncos1 u), 1 <u < 1 = cosh (2N cosh' u), 1 < I u I. x ment. The roots for N < p < =2N are the complex conjugates of This expression is the Chebyshev pattern. Manuscript received January 12, 1984; revised May 29, The author is with the Antenna Department, Radar Systems Group, Hughes Aircraft Company, El Segundo, CA As for any discrete, uniformly spaced array the pattern is periodic in $, with period 2r. Thus, as with any such array, if the pattern is not to repeat in real space the element spacing must not exceed one haifwavelength X/84/ $ IEEE

2 1090 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP32, NO. 10, OCTOBER 1984, \ If we follow the technique of Taylor, and move the roots so that beyond a certain root they coincide with those of a sin [(LV + 1)$/2]/ sin ($/2) pattern, the far out sidelobes will decay while the near in sidelobes will stay relatively constant at the design sidelobe level. The nth zero of the sin [(ZN + 1)$/2] / sin ($/2) pattern is given by 2n $(jn=n. 2N+ 1 The modified pattern is then Fig. 1. Pattern of 41 element array, n' = 6, SL = 25 db. This last expression is a sin [(UV + 1)$/2]/ sin ($12) pattern with the first Ti 1 zeros replaced by those of the Chebyshev pattern. In addition to being replaced, the zeros are also shifted progressively so that the nth zero coincides with the nth zero of the sin [(UV + 1)$/2]/ sin ($/2) pattern. This shift is accomplished by multiplying each Chebyshev zero by a factor CJ, where n2n U= (2N + 1)$Ti a function of $ is illustrated for a 41element array with a design sidelobe level of 25 db, and Ti = 6, in Fig. 1. It exhibits the characteristics of a continuous Taylor pattern. However, it is inherently the pattern of a discrete array. Therefore, the element excitations can be determined to produce this pattern exactly. ARRAY EXCITATION The array excitation can be derived in the following manner. The pattern is expressed in terms of the element excitations, an, as follows: Its value at $ = (m27r/(zn + 1)) is The pattern may then be rewritten as sin F,(W)SE($) =ejn+12 (2N2+ $) N<p<N. m=l This expression is valid for any uniformaly spaced array of 72N+ 1 where = a$n. This pattern is the discrete array equivalent elements. For the case of interest here, E(2mn/(2N + 1)) is zero of the Taylor pattern for a continuous aperture. The pattern as exceot I when ~~~ 4ii \ 1) < m < (ii 1). Thus, for the discrete

3 VILLENEUVE: TAYLOR PATTERNS FOR DISCRETE ARRAYS 1091 TABLE I EXCITATIONS OF 41 ELEMENT ARRAY WITH A = 6 AND SL = 25 db array equivalent of the Taylor pattern, the array excitation tions, the array pattern was calculated from these excitations coefficients are given by the following expression: and was found to be identical to the pattern in Fig. 1. The theory can also be worked out for an even number of 1 ap= 5 E(=) elements, say 2N. the finds one Then following relations: 2Ar+ lm=(zi) 2N+ 1 1 uo = cosh ( In [v +1). ei(~mza/(zn+l)), N<p=GN I 2N+ 1 [ : E(0)+2c E (El) 2(2Ar 1) p = 1,..., 2N 1 where El I = and 1 n1 ap = E Et!?) ej([p1/2 1 m n/(2n)) 2N m=(n1) The excitation coefficients for the 4lelement, 25 db. E = 6 = L! [E(O) + 2 x array are listed in Table I. As a check on thexcitation calcula 2N (A G) Os I)<p<N ([. :] y)]

4 1092 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP32, NO. 10, OCTOBER 1984 Fig. 2. Comparison of patterns calculated by the method of this paper and by sampling of continuous aperture distribution, A = 6, SL = 20 db, 2N + 1 = 21. (2) 2 N x sin2 q= 1 E(0) = n 1 q=1 Comparison with Root Matching Techniques It is of interest to compare the results of this method with those obtained by root matching of the pattern of a continuous aperture of length (2N I 1)d. Consider a pattern with E equal to 6 and a 30 db design sidelobe level. The root locations of the pattern of a 21element array as calculated from the method derived in this paper are compared with those of the pattern of the continuous aperture in the top part of Table 11. A similar comparison for patterns with Z equal to 3, a 30 db sidelobe level and an array with 11 elements is made in the bottom part of Table 11. From these tables it is evident that the 2Nroots of the patterns of the discrete arrays and the continuous apertures essentially coincide. Consequently, the excitation and resulting pattern of a discrete array as obtained by root matching of the continuousaperture pattern must be essentially the same as that obtained by the method of this paper. Camparison with Sampling Method An additional comparison was made with the results obtained by exciting the array elements with the values of the continuous aperture distribution evaluated at the locations of the elements. The length of the continuous aperture was taken as (UV + 1)d. The element excitations for a 2lelement array with 20 db sidelobes and ii = 6 are compared in Table 111 for the method of this paper and for the sampling method. There is very little difference in the excitations. The patterns are shown in Fig. 2. The patterns are essentially indistinguishable. For larger arrays the agreement is even better. CONCLUSION From the foregoing development, it is seen that for a discrete, uniformly spaced array, the equivalent of the Taylor pat

5 VILLENEUVE: TAYLOR PATTERNS FOR DISCRETE ARRAYS TABLE II COMPARISON OF ROOTS OF PATTERNS BY THEMETHOD OF THIS PAPER AND ROOTS OF PATTERNS OF THE CORRESPONDING comuous APERTURES 2N + 1 = 21, A = 6, 30 db (This Hethod) *A (Continuous Aperture) radians radians t t t4 t * * t t t t *P *2r& (p = 6,... 10) *2r & (p = 6,..., 10) 2N + 1 = 11, A = 3, 30 db SIDELOBES n *A (This Method) $A (Continuous Aperture) r * TABLE III COMPARISON OF ELEMENT EXCITATIONS BY THE METHOD OF THIS PAPER AND BY SAMPLING OF CONTINUOUS TAYLOR DISTRIBUTION, SL = 20 d ~, A = 6,2N + 1 = 21 Element Excitations (This Method) Element Excitations (Aperture Sampl ing) oo0Oooo tern for a continuous aperture can be developed directly. The REFERENCES corresponding excitations can be found directly, without having [I] T. T. ~ ~ hign ~ of l ~ ~ for, ow beamwidth to make any approximations. and low sidelobes, IRE Truns. Antennus Propagut., vol. AP7, pp. 1628, [2] R. S. Elliott, Antenna Theory und Design. Englewood Cliffs, NJ: ACKNOWLEDGMENT F renticehall, pp. 1981, 144, The author would like to thank Mr. Cliff Williams who edited A,fred T. Villeneove (S.52A353M.58), for a photograph the manuscript. please see page 462 of the May 1982 issue of this TRANSACTIONS.