AD-AL03 W3 NAVAL RESEARCH4 LAB WAS44INSTON DC P/O 9/5SYNTHESIS OF TAYLOR AND BAYLISS PATTERNS FOR LINEAR ANTENNA ARR-ETC (u)AUS S1 .J P SHELTON

LiCL S IPI ED NL--51

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I NRL Report 8511

Synthesis of Taylor and BaylissPatterns for Linear Antenna Arrayst

J. P. SIHELTON

Eleciromagneiks BranchRadar Division

DTIC..E CTEJ

August 3l, 1981 H

NAVAL RESEARCH LABORATORYWashington, D.C.

Approved for public releas; distribution unlimited.

C,' 81 oi 052

SECURITY CLASSIFICATION OF THIS P AGE (When, Doi& Entered)

REPOT DCUMNTATON AGEREAD JNSTRUCTIONSREPOT DCUMNTATON AGEBEFORE COMPLETING FORM

I REPORT NUMBER 2GOVT ACCESSION NO. 3 RECIPIENT'S CATALOG NUMBER

NRL Report 8511 14j)-A/oSYNTHESIS OF TAYLOR AND BAYLISS PATTERNS FOR J/n~terimrBem.o cniun

LINEAR ANTENNA ARRAYS, 16. PERFORMING ORG. REPORT NUMBER

-rXTr~3 S. CONTRACT OR GRANT NUMBER(.)

J. P. Shelton

S. PERFORMING ORGANIZATION NAME AND ADDRESS I0. PROGRAM ELEMENT. PROJECT. TASKAREA B WQ!;4~NTNy-"'S

Naval Research Laboratory 62712N; SFj21414911Washington, DC 20375 53.060 015

It. CONTROLLING OFFICE NAME AND ADDRESS I2. REPORT OATE!

Augua- 10 81

19IA. MONITORING AGENCY NAME & AOORESS(If differen.fro ontolln Sf~f SECURITY CLASS.(och.pot

I, IS..UNCLASSIFIEDIS* DECL ASSIFICATION/ DOWNdGRADINGI .SCHEDOULE

to.- -ewxMwwxW 4zA4w.NY (of 101. Report)I 7..q 1w--

Approved for public release; distribution unlimited.r

17. DISTRIBUTION STATEMENT (of ch. .b.flact ent.r.d I,, Block 20. If dtff.,.il h~om Report)

IS. SUPPLEMENTARY NOTES

IS. KEY WORDS (ContIinue on reverse old. It noc*eary ed Identify by block num..ber)

AntennasLinear arraysPattern synthesisSidelobe control

20. ZMTRACT (Conl"no. -on *, old. It necessary nd idenify by block num.ber)

11 The history of synthesis techniques for designing linear antenna arrays with low sidelobe pat-terns is reviewed briefly, and the limitations that are encountered with very low sidelobes and/orsmall arrays are pointed out. Taylor's continuous aperture synthesis procedure is outlined, and atechnique for transforming it for application to a discrete array is described. Discrete-array designequations for Taylor and Bayliss synthesis procedures are given. A set of programs for use on aprogrammable calculator are presented.

DD IA731473 EDITION OF I NOV 65 IS OBSOLETEjAS/N 0102-014- 66011 _______________________

SECURITY CLASSIFICATION OF TNI1S PAGE (69,.. Doi&. MnIrect

CONTENTS

INTRODUCTION ......................................... 1

REVIEW OF TAYLOR SYNTHESIS PROCEDURE ............... 2

ARRAY PATTERN FUNCTIONS IN TERMS OF ZEROS .......... 4

ACKNOWLEDGMENT ..................................... 6

REFERENCES ............................................ 6

APPENDIX A - Design Equations for Linear Arrayswith Taylor-Type Patterns .................... 7

APPENDIX B - Design Equations for Linear Arrays withBayliss-Type Difference Patterns ............... 9

APPENDIX C - Programs for the HP-41C CALCULATOR .......... 11

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SYNTHESIS OF TAYLOR AND BAYLISS

PATTERNS FOR LINEAR ANTENNA ARRAYS

INTRODUCTION

The requirement for low sidelobes from array-type antennas is a long-standing one. Thecontributions to this theory extend from Dolph's utilization of Chebyshev polynomials, throughTaylor's papers on linear and circular apertures, Bayliss's extension to difference-type patterns, andfinally to recently developed techniques which provide arbitrary pattern control for linear arrays[1-81.

The purpose of this report is to examine some of the more recent applications of these syn-thesis techniques in light of their limitations and also the computational capabilities which are nowavailable. For example, at the time Taylor published his synthesis procedure, engineers had onlyslide rules, mathematical tables, and mechanical desk calculators to generate the distribution func-tions. The computational capability available to today's engineer is vastly different, and we willshow how Taylor's and Bayliss's procedures can be modified to give better results.

A more careful look at the synthesis procedures previously mentoned is presented in Table 1.

Dolph's synthesis is precise and gives minimum beamwidth for given sidelobe levels, but theseconstant amplitude sidelobes are not desirable for larger arrays because it is possible to radiate mostof the energy into the sidelobes. Taylor solved this problem by allowing the far-out sidelobes to falloff as dictated by an amplitude discontinuity at the ends of the aperture. Taylor, and later Bayliss,synthesized continuous distributions and sampled these to obtain array excitations.

Table 1 - Synthesis Procedures for Linear Array Apertures

Procedure Continuous or LimitationsDate Discrete I

Dolph/47 Discrete Poor results for large arrays

Taylor/52 Continuous Inexact for low sidelobes, small arrays

Bayliss/68 Continuous Inexact for low sidelobes, small arrays

Hyneman/68 Continuous Inexact for low sidelobes, small arrays-iterative

Stutzman/72 Continuous Inexact for low sidelobes, small arrays-iterative

Elliott/76 Continuous Inexact for low sidelobes, small arrays-iterative

Elliott/77 Discrete Applies all continuous procedures to discrete arrays

Manuscript submitted June 15, 1981.

/]1

SHELTON

Some recent applications have called for lower sidelobes and smaller arrays, thereby pressingthe limitations of the Taylor and Bayliss synthesis procedures. The problem of discretizing contin-uous aperture distributions has been treated 19-101. The technique used in this report is differentfrom those of Winter and of Elliott, but it is mathematically related to Elliot's technique.

REVIEW OF TAYLOR SYNTHESIS PROCEDURE

A brief review of the Taylor synthesis procedure is given here. The key to this procedure is theequal-sidelobe pattern function which is the continuous-aperture analog to the Chebyshev polyno-mial pattern for arrays:

E(u) = cosir -/ -A 2 , (1)

where u = 7ra sin 0 /X, a is the length of the aperture and 0 is the angle measured relative to thenormal to the array. This function has a maximum value of cosh 7rA at u = 0 and unit sidelobesextending to u = ± oo. Taylor showed that the pattern of Eq. (1) is not physically realizable from acontinuous aperture distribution, just as the Dolph array excitation becomes increasingly impracti-cal in the limit of large arrays. His brilliant solution to this problem was:

1. For all zeros of the synthesized pattern functions, which we will call E,(u), from the nthfrom the origin to -, the locations will be the same as those from a uniformly illuminated apertureof the same size. That is,

E(u ) = 0 for u = n for n .

2. For the first 11 - 1 zeros, their locations will be determined by the zeros of E(u), scaled sothat the nth zero is located at u = -f.

The aperture distribution is determined by performing a Woodward synthesis of Es(u). That is,we define a set of functions of the form

F,(u) = sin (u - n)wr/(u - n)ir,

and then construct Ea(u) from the Fn(u)

00

E8(U)= E E,(n)F n(u)" (2)

Since we have defined E,(n) 0 for n ff, Eq. (3) becomes

Es(U) = E E(n)Fn(u)" (3)

n--n+l

2

NRL REPORT 8511

Fourier transformation of Eq. (3) yields the aperture distribution:

E (u) eJ2 xu~r/adu

-1

=ff E E,(n)Fn(u)eJ2 xufr/adu. (4)

- n=-fl+l

That is, A(x) is a weighted sum of integrals of the form,

00J sin (u - n)v ej 2xur/adu.

(u- n)7r

Letting u' = u - n results in

ej2nrx/a0 sin uIv e j 2xu '7r/adul.

Since the imaginary part of the integrand is odd, this becomes

00

eJ2 nfrx/a sin u'r cos 2xu'i/a du'

e I2nirxf 1 [sin u'ir(l - 2x/a)+sin u'ir(l + 2xa) du'. (5)

4. 2 - -

A standard definite integral is

00

sin bzdz = for b > 0

.0 z = 0 forb=0= -r for b <0

Application of this integral to Eq. (5) and thence to Eq. (4) yields

A(x) = i- Es(n)eJ2 1rnx la

n=- n-+l

= E8(0) + 2 3 E,(n) cos 2rnx/a for IxI < a/2 (6)

n=1

= 0 for IxI>a/2.

3

SHELTON

The continuous aperture distribution given by Eq. (6) is sampled to give the element excita-tion values for a discrete array. This last step is approximate, and the pattern function of the arrayis obviously different from E,(u). This approximation is acceptable provided that the number ofelements in the array is much greater than i-and the sidelobe level is not extremely low. Figure 1is an example of a case in which the synthesis procedure gives an unsatisfactory result. For a side-lobe level of 50 dB below mainbeam and 1h = 8, a 30-element array has the computed patternfunction shown. The near-in sidelobes are unduly low, whereas the first eight sidelobes shouldbe about the same level.

ARRAY PATTERN FUNCTIONS IN TERMS OF ZEROS

Elliott used a synthesis technique which relates the discrete array distribution directly with thearray pattern [9]. We also use this relationship, and our procedure achieves identical results withthose of Elliott. However, the actual computations are different, and it is desirable to compare thetechniques.

Elliott expresses the pattern function as a polynomial in w, where w = eJ( 21rs/X)sin 0. Thezeros of this polynomial are given by wn, which are normally located on the unit circle. Once hehas the wn properly adjusted, he completes the synthesis by multiplying out the product expression,n(w - wn ), into the polynomial. The coefficients of the polynomial are the excitations of thearray elements.

CD

CD

0

0

-90.0O0 -60.0O0 - 30.0O0 0.0O0 30.0O0 50.O0DEG

Fig. 1 - Conventional Taylor synthesis, N f 30,S8, 50-dB sidlobes

4

NRL REPORT 8511

Our procedure also uses the pattern function zeros in a product expression. Since thepatterns are symmetric, our expression can be of the form, H(cos z - cos zn), wherez = (21rs/X) sin 0. We cannot multiply this product expression out to obtain the coefficientsdirectly since we require terms of the form cos nz rather than cos nz. Rather, we carry out asynthesis exactly analogous to that used by Taylor. Uniformly spaced pattern function samplesare found by using the product expression. These pattern samples are used in a Fourier series tofind the array illumination.

The procedure relies on the equivalent location of pattern function zeros for the line sourceand for the discrete array. Whereas the zeros for the pattern of a uniform line source distribution arelocated at u = n, the analogous relationship for a discrete array is z = ntr/N, where z = 2ws sin 0/X,where s is element spacing and N is the number of elements in the uniformly excited array.

The transformation of Taylor's procedure is easily seen to consist of locating the zeros instep 1 above at z = nv/N forn > fand then scaling the first W zeros of Eq. (1) so that the ith zerois located at z = -r/N.

Appendix A lists the resulting equations for Taylor arrays of both even and odd N, andAppendix B lists the equations for Bayliss arrays (yielding monopulse difference patterns) of botheven and odd N. Figure 2 is an example of a Taylor array pattern with sidelobe levels of 50 dB withn- = 8 and N = 30. These equations can be straightforwardly programmed for automatic processingby a digital computer. Many programmable calculators now have sufficient memory to implementthese programs. Appendix C lists programs for carrying out the synthesis and evaluating thepattern functions with an HP-41C programmable calculator.

900

C3

Fig. 2 - Discretized Taylor synthesis,N 30, n i 8, 50-dB sidelobes

0

-90.00 -60.00 -30.00l 0.001 30.00l 60.00DlEG

05

SHELTON

ACKNOWLEDGMENT

I thank Dr. Robert J. Adams for his careful and constructive review of the initial draft of thisreport. His questions led to a clarification of the relationship between this synthesis and that ofElliott [9].

REFERENCES

1. C. L. Dolph, "A Current Distribution for Broadside Arrays which Optimizes the Relationship

between Beam Width and Side Lobe Level," Proc. IRE, Vol. 34, June 1946, pp. 335-348.

2. T. T. Taylor, "Design of Line-Source Antennas for Narrow Beamwidth and Low Side Lobes,"IRE Trans. Antennas Propagat., Vol. AP-3, Jan. 1955, pp. 16-28.

3. T. T. Taylor, "Design of Circular Apertures for Narrow Beamwidth and Low Sidelobes,"IRE Trans. Antennas Propagat., Vol. AP-8, Jan. 1960, pp. 17-22.

4. E. T. Bayliss, "Design of Monopulse Antenna Difference Patterns with Low Sidelobes,"BSTJ, Vol. 47, May-June 1968, pp. 623-650.

5. ' R. S. Elliott, "Design of Line Source Antennas for Sum Patterns with Sidelobes of IndividuallyArbitrary Heights," IEEE Trans. Antennas Propagat., Vol. AP-24, Jan. 1976, pp. 76-83.

6. R. S. Elliott, "Design of Line-Source Antennas for Difference Patterns with Sidelobes ofIndividually Arbitrary Heights," IEEE Trans. Antennas Propagat., Vol. AP-24, May 1976,pp. 310-316.

7. R. F. Hyneman, "A Technique for the Synthesis of Line-Source Antenna Patterns, HavingSpecified Sidelobe Behavior," IEEE Trans. Antennas Propagat., Vol. AP-16, July 1968,pp. 430-435.

8. W. L. Stutzman, "Sidelobe Control of Antenna Patterns," IEEE Trans. Antennas Propagat.,Vol. AP-20, Jan. 1972, pp. 102-104.

9. R. S. Elliott, "On Discretizing Continuous Aperture Distributions," IEEE Trans. AntennasPropagat., Vol. AP-25, Sept. 1977, pp. 617-621.

10. C. F. Winter, "Using Continuous Aperture Illuminations Discretely," IEEE Trans. Antennas

Propagat., Vol. AP-25, Sept. 1977, pp. 695-700.

6

Appendix A

DESIGN EQUATIONS FOR LINEAR ARRAYS

WITH TAYLOR-TYPE PATTERNS

These equations will determine the aperture illumination coefficients for a linear array ofN elements to produce a Taylor-type pattern function with - sidelobes on each side of the mainbeam at a level of L dB.

This design procedure involves three steps. The first W - 1 zeros of the pattern are determined.Then the appropriate pattern function samples are determined. Finally, the array element illumina-tion coefficients are determined by a harmonic analysis of the pattern function samples.

A particular advantage of this synthesis is that the knowledge of all of the pattern functionzeros allows the computation of the pattern function as a product rather than as a polynomial. Theproduct computation involves only one trigonometric function evaluation for each pattern func-tion value. All other constants need to be evaluated only once for each array.

The pattern function zeros are given by

Zn = A+(n-1/2) 2 forn= 1 to f - 1 (Ala)

N A 2 + (ff- 1/2)2

21nN for n V to M, (Alb)

where

M = int( 1

and A is given by

A cosh 1 ~[O(L/20A r = osh- (A2a)

(L + 6.02)/27.29, (A2b)

where L is the sidelobe level (positive) in dB. Equation (A2b) is an excellent approximation, espe-cially for large L.

7

SHELTON

The pattern function is given by

zM /Cos Z- COSZnE(z)=cos- H 1- Cos z~ N even

n=1 /

M /Cos z - Cos Z" d M

The pattern samples to be used to find the array element illumination coefficients are given by

The element excitation coefficients are given by

e = 1+ 2 m cos m(2p)N N even, p 1to M+ 1

E= N

ra=1

where p is an index or element number starting at the center and moving to either end of the array.

8

Appendix B

DESIGN EQUATIONS FOR LINEAR ARRAYS WITH

BAYLISS-TYPE DIFFERENCE PATTERNS

Appendix A gave the design equations for linear arrays with Taylor-type patterns, whichproduce a main beam with slightly larger beamwidth than that of the Dolph synthesis but in gen-eral with higher gain. In some applications; such as monopulse, we might require a difference pat-tern. Bayliss presented a synthesis procedure for difference patterns, analogous to that of Taylor.In this appendix we adapt the Bayliss procedure to discrete arrays.

As in the case of the Taylor synthesis, the application of discrete arrays involves three steps.The first iY - 1 off-axis zeros of the pattern are determined. Then the appropriate pattern functionsamples are determined. Finally the array element illumination coefficients are determined by aharmonic analysis of the pattern function samples.

The pattern function zeros are given by

2irq,(if +!)z for n = 1, 2, 3, 4 (Bla)NVA2 + -n

2

21r(W+! VA 2 +n 2

for n = 5toT - 1 (Blb)

,~1 (ni + 2

for n -f to M (Blc)N

where

M = int( 2)

In this case it is necessary to find both A and qn from graphs in Bayliss's paper [4]. For 50 dBsidelobes, A = 2.42, ql = 2.78, q 2 = 3.18, q3 = 3.85, and q4

= 4.65.

The pattern function is given by

E-z sn 2 EcOs - c°Zn] /sin4-nflE(z) = sin z - Cos cos- - cos z N even (B2)2n14n= 1 2 n

/ Z fi[ Z.

sin z L"os z - N oddn=lI 2 n-1 2

SHELTON

E~z) is normalized to unity at z z 1 /3, which is near the pattern maximum. If a more precisepattern maximum is desired, a better multiplying constant can easily be found.

The pattern samples to be used to find the array element illumination coefficients aregiven by

bm = E( I(2m-1)) form =1 to - . (B3)

The element excitation coefficients are given by

e. = 2 bm sin (2 m 1)2p1) for Neven, p toM+1

=s -for N odd, p 1 toM+ 1 (B4)=2 q m s N

where p is an index of the element number starting with zero at the center of the array. For N odd,the center element of the array always has zero excitation. The excitations on one side of the arrayare the negative of those on the other side.

10

Appendix C

PROGRAMS FOR TIlE IIP-41C CALCULATOR

This appendix presents programs for the HP-41C calculator for the design equations of Appen-dices A and B. The software consists of four programs, SUM, DIF, IN. and SL. "SUM" contains theequations for synthesizing Taylor-type sum patterns; "DIF" contains equations for Bayliss-typedifference patterns; "IN" contains subroutines that are used by both programs; and "SL" is aroutine for calculating the peaks of the sidelobes of the synthesized array. The number of registersused by the programs and the number of card sides required for storage are:

Program Registers Card Sides

SUM 30 2DIF 42 3IN 39 3SL 19 2

130 10 (5 cards) .

It is possible to synthesize aperture distributions using either SUM and IN or DIF and IN. Theseprograms require at least one additional memory module. Furthermore, the programs use nineregisters for variables, indices, and constants. Table C1 correlates the number of registers avail-able for synthesis parameters with the number of additional memory modules in use. The avail-able registers are used for the pattern samples an and b. and for the pattern function zeros(cosines) zp . The number of these registers is W + M. Therefore, the size of array that can besynthesized for any given configuration of Table C1 depends on -W. For a 50-dB sidelobe require-ment, IT will be about 8. Roughly speaking, an array of 55 to 65 elements for difference and 80 to90 for sum can be synthesized using one memory module by trading programs in and out of themachine, and an array of 90 to 100 elements can be synthesized with all programs loaded usingtwo modules. The maximum array size that can be handled using three modules is 310 to 320 fordifference and about 340 for sum. It appears that one or two memory modules should suffice formost requirements.

Table C1 - Registers Available after LoadingIndicated Program Complements

Number of Memory ModulesProgram Complement

1 I2 3

SUM + IN 48 112 176DIF + IN 35 99 163

SUM + DIF + IN 11 71 135SUM + DIF + IN + SL - 54 118

11

SHELTON

The procedure for running the programs is:

1. Allocate memory by XEQ SIZE (9 + M + n).

2. Load the appropriate program complement.

3. Enter either XEQ SUM or XEQ DIF.

4. The display will prompt for N, L, and NBAR. N can be even or odd. L is sidelobe level inpositive dB. DIF will also prompt for A, Q1, Q2, Q3, Q4.

5. After calculating z n &and loading cos z n into registers starting with (9 + R), the display will askwhether you want a listing of peak sidelobes (SL) or aperture distribution (EP). After the sidelobesor excitation coefficients are listed, the display will ask whether you want the other set of parameterscalculated and listed.

The routine SL computes the sidelobe level relative to the main beam level by evaluating thepattern value at a point midway between pattern zeros. This computation is admittedly approximatebecause the pattern maximum is in general not exactly midway between zeros. The main beam pat-tern value is computed for z = 0. The difference pattern maximum is computed for z = z1 /3. Thisfactor was found to be accurate for 50 dB sidelobes. The exact multiplying factor will be somewhatlarger for higher sidelobes (L < 50), and it can be found quickly by obtaining z 1 and executing PA:

RCL (9 + R) gives cos z1ACOS gives z1k new multiplying factor, such as .4

COSSTO 02XEQ PA.

Alternatively, k can be found from Fig. 4 of BaylissCz, which defines the beam maximum by p,,where k = po/§I. (§ 1 corresponds to our z I )

Once the desired value of k has been found, go to lines 110, 111 in DIF, and exchange k,* for 3,/. It is now necessary to reload the reference main beam pattern value into R08. This calcu-

lation starts at line 61 of SUM and 105 of DIF. Alternatively, you can simply rerun the program.

The pattern value, in voltage and normalized to mainbeam level, is found by keying in thevalue of z in degrees, then keying COS, STO 02, XEQ PA. 4

The registers used are:

00 N01 M02 A 2 and cos zm for PA03 n04, 05 loop indices06 multiplying constants

Ci E. T. Bayliss, B8TJ, May-Jun 1968, pp. 623-650.

12

NRL REPORT 8511

07 accumulator for E(z), e08 main beam reference value09 to (8 + f) computed values of am, bm(9 + 1i) to (8 + 1i + M) computed values of cos z,,.

Program IN contains the following subroutines:

IN Asks for input data N, L, NBARZN Completes calculation and storage of cos z nBR Asks for choice of sidelobes or aperture distribution and branches

to EP or SLPR Prints element excitations epPA Computes pattern value for am, bm, or SL routinesEP Completes calculation of ep.

The programs use flags 00 and 01 to indicate the following conditions:

Flag 00 is set for N evenclear for N odd

Flag 01 is set for DIF executionclear for SUM execution.

The use of registers by program PA precludes the use of the plot subroutines resident in theprinter.

Note that the sidelobes and pattern values obtained with these programs are all relative to themain beam level. No information concerning gain or aperture illumination efficiency is computed.The aperture distribution can be used to compute aperture efficiency or gain.

The programs and sample printouts are listed on the following pages.

13

SHELTON

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i9x27,124 $7C085 5 40136

sb RUCL, 69 31 38? ST 86 4

21. 74 U6 2 42 12E-7

22 SORT 75 S3IT 85c 127 RCL O ~ 43 4

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o 3 W41 51 I41L

32310. @4' 34 *PRP 'DIF 52, PROMPr

85 GUos 53 VX'33*1k 91 6 S10 82 0141K 1IF- 54 w34 RCL 84 67 XEQ 'PA, 82 SF 81 5 5 ARL4~ 5 AT 88 RCL 0 5 63 X2 'IlN Se WC4365 89 8 i4 R6L8610-7 9@0+ 85 2 5 l

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14

NRL REPORT 8511

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15 i

SHELTON

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131 r 96 R, 56 r:

132 17 R.L 85 £123 RCL -5 56 Z ;8*

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16

NRL REPORT 8511

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,9.E 4-,E* '. ,

45,% P 4';., 7 r

45,89 I*W44 '' - - *

A=' '. 4L

49.23 ~ ~ r I$ *.f

49.29 .**h:20 49.31 *.**

iBfeP<=..

3 - 3 - - - I

84= 4.5 03= 3 "

El e.436E. Q4= 4.65

E_2: 1.3673 El: 8.9f9.F I= 1.9 ,1 E2: 1.65 1E4= 2.244 E3= 2.1:ES: 2. t5 E4: 2.25-E6 1 . " E 5.. 2. R7Z

6: .3K .E: ,i8; U= -E8=: t. j7.,E = ' "'

218: e.",. 2?

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4-,. 17, ,

46.. 5 *i-.r:i

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17

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