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CHAPTER FOUR Pattern Synthesis 4.1.1 Fiat Plane Slot Arrays Many aircraft radars are now equipped with flat plane slot arrays instead of dish antennas, where the outline is approximatel

CHAPTER FOUR

Pattern Synthesis

4.1.1 Fiat Plane Slot Arrays

Many aircraft radars are now equipped with flat plane slot arrays instead of dish antennas, where the outline is approximately circular These are usually fixed narrow-beam antennas with controlled sidelobes Pattern synthesis typi- cally uses the product of two linear syntheses; one along the slot “sticks,” and a second across the sticks These are often designed as Taylor ii patterns, as described in Chapter 3 Since the sticks near the edge are shorter, with fewer slots, than those at the center, the overall pattern will deviate somewhat from the product of the two orthogonal patterns In particular, some adjustments may be necessary to produce the desired sidelobe envelopes in the principal planes As these arrays are often divided into quadrants for azimuth and elevation monopulse, a compromise may be advisable between the sum pattern sidelobes, and the difference pattern sidelobes and slope; see Chapter 3 An example of the design, using the zero adjustment methods described in Chapter

3, of a planar array of inclined slots, providing a cosecant type pattern, with a Taylor ii pattern in the cross plane, is given by Erlinger and Orlow (1984)

Design of the array elements, to include mutual coupling, is more complex

in that the entire array must be encompassed in the synthesis process That synthesis is described in Chapter 6, and it can be extended to the flat plane array by using double sums for mutual coupling to include all slots The iterative technique again converges rapidly A typical flat plane slot array is shown in Fig 4.1 See also Figs 6.30 and 6.3 1

Use of principal plane canonical patterns simplifies the synthesis process, but lower sidelobes can be obtained through synthesis of a rotationally sym- metric pattern, as discussed in the next section

106

Copyright  1998 by John Wiley & Sons, Inc ISBNs: 0-471-53076-X (Hardback); 0-471-22421-9 (Electronic)

Figure 4.1 Flat plane array (Courtesy Hughes Aircraft Co.)

A symmetric circular one-parameter distribution and pattern analogous to the Taylor one-parameter distribution was developed by Hansen (1976) Starting

as before with the uniform amplitude pattern, which is 24(nzk)/rcu, the close-in zeros are shifted to suppress the close-in sidelobes The single parameter is H, and the space factor (pattern) is

u > H; -

u < H -

J1 and II are the Bessel function and modified Bessel function of the first kind and order one In exactly the same manner as for the Taylor pattern, the expression for u > H forms the sidelobe region, and part of the main beam,

while the expression for u 5 H forms the remainder of the main beam The

ratio is

24 wf)

by absolutely convergent series and are quickly generated by common comput-

U = kD sin theta

SLR = 25 dB

TABLE 4.1 Characteristics of Hansen One-Parameter Distribution

where p is the radial coordinate, zero at the center, and unity at the edge IO

is a zero-order modified Bessel function In Fig 4.3 is given the beamwidth

dots will be discussed in the next section Figure 4.4 shows the (half-) aper- ture amplitude distribution for several values of SLR Edge values are given

in Table 4.1 The aperture excitation efficiency qt is found by integrating the distribution

35 Sidelobe ratio, dB

This result for efficiency is readily calculated, and is shown in Table 4.1 Table

Beam efficiency qb, the fraction of power in the main beam, might require

terms of aperture efficiency qt, which can be easily calculated from the aperture

while beam efficiency is

ON

s

E2 -sin6&

E2

0 0

(43

Here ON is the main beam null angle Now the beam efficiency can be expressed

by a simple integral over the main beam:

qb =

2 2

Jr D r7t

nearly independent of D/h Because of the low sidelobes at all azimuth angles, the beam efficiency is much higher than that for a linear distribution

0.98

30 sidelobe ratio, dB

- D/A = 50

4.1.3 Taylor Circular ii Pattern

The circular Taylor ii distribution offers a slightly higher efficiency, through a slightly slower decay of the first ii sidelobes, exactly analogously to the linear Taylor n distribution Again the aperture distribution is symmetric The uni-

beam are adjusted to provide roughly equal levels at the specified SLR Farther-out zeros are those of the uniform pattern As before, a dilation factor

0 provides a transition between the n zeros, and those adjacent The sidelobe ratio is related to the general parameter A:

The Taylor circular I? space factor (pattern) is given by (Taylor, 1960)

F( 1 u c - 2Jl(nu)“-’ TtU I - I 1 - u’/u;

The zeros of J,(m) are pn A typical ii pattern is shown in Fig 4.6 Beamwidth

is the ideal line source beamwidth u3 times 0; Table 4.2 gives the parameter A, half-beamwidth u3, and 0 for various SLR and n Hansen (1960) gives tables of the aperture distribution

The aperture distribution becomes (Taylor, 1960)

(4.13)

Note that the radial variable for circular Taylor ii is p = 2np/D The coeffi- cients F, are

-40

U = kD sin theta

Figure 4.6 Taylor circular ii Pattern, SLR = 25dB, fi = 5

TABLE 4.2 Taylor Circular Ti Pattern Characteristics

n=l l-4 n#m

m>O

(4.14)

Representative aperture distributions (half only) are shown in Figs 4.7 and 4.8 Too high a value of fi causes instability in the null spacings, reflected in a

nonmonotonic distribution As larger values of fi give higher efficiency, accu- rate for large apertures and constant phase, for each SLR there is an optimum trade on fi

Aperture efficiency is obtained from

2 7dPlPdP

0

(4.15)

integration of the product of the two series produces an integral of two ortho- gonal Bessel functions; only a single series remains:

(4.16)

Excitation efficiencies for several SLR and ii are given in Table 4.3, from Rudduck et al (197 1) Beam efficiency again is from an integral over the main beam Table 4.4 gives this versus SLR

TABLE 4.3 Taylor Circular ti Efficiency

4.1.4 Circular Bayliss Difference Pattern

Before discussing the Bayliss ii difference pattern, it is useful to consider a

The # integral for the pattern function reduces to

where I& is the zero-order Struve function (Hansen, 1995) The difference pattern in the 4 = 0 plane is then given by

level -14.90 dB, or 11.72 dB below the difference peak

The Bayliss difference pattern for a line source is discussed in Section 3.7.2 Bayliss (1968) developed a similar ii difference pattern for a circular aperture The starting pattern is the derivative of the “ideal” Taylor sum pattern

-4o- -90 80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 1 90

Angle, deg

Figure 4.10 Uniform circular aperture difference pattern, D = 20X

exist in usable form The derivative, with coefficients deleted, is

118 PLANAR AND CIRCULAR ARRAY PATTERN SYNTHESIS

to the results The parameter A and the position of the difference pattern peak

3.12 Figure 4.11 shows a circular Bayliss pattern for SLR = 25 dB, and ii = 5

where p = rtp, with p the aperture radius from zero to one The B, coefficients are

expected, larger values of n for a given SLR produce a rise at the aperture edge, which can be higher than the center value Figure 4.13 from a Bayliss

Figure 4.13 Circular Bayliss distribution, SLR = 35 dB, fi = IO (Courtesy Bayliss, E T.,

“Monopulse Difference Patterns with Low Sidelobes and Large Angle Sensitivity,” Tech Memo 66-4131-3, Bell Labs., Dec 1966.)

(4.27)

TABLE 4.5 Circular Bayliss Efficiency and Slope

4.1.5 Difference Pattern Optimization

The use of subarrays, where each can be fed with a different amplitude (and phase) for sum and difference patterns, was discussed in Chapter 3 for linear arrays The same principle was applied to circular planar arrays by Josefsson et

al (1977) The array is divided into subarrays, with quadrantal symmetry The shape of each subarray and the number of elements will generally be different; see Fig 4.14 Symmetric pairs of subarrays are then connected to hybrid junc- tions; these hybrids are connected to the sum and to the two difference com- biner networks; see Fig 4.15 As in the linear array case, a stair-step

Excellent results have been obtained with only a small number of subarrays per quadrant This configuration is especially attractive with printed circuit antennas, as the implementation is natural for realization in stripline

4.2.1 Two-Dimensional Optimization

When the aperture (array) is not circular, or when x and y distributions are not separable, recourse must be had to numerical techniques, such as constrained optimization Unfortunately, a relatively simple technique (dynamic program- ming) is not applicable, as it requires the function ahead of the change to be independent of the change But with all arrays, changes to one zero or to one excitation coefficient affect the entire pattern Optimization programs such as

Figure 4.15 Feed network for sum and difference optimization

conjugate gradient are suitable provided the objective function can be well specified, but slow Since sidelobe positions are not predictable, maximizing directivity subject to the requirement that all sidelobes be below a given envel- ope is difficult Either a sidelobe location routine that searches for each and every sidelobe peak at each function (array pattern) evaluation is used, or each pattern must be digitized sufficiently finely to accurately locate all sidelobe peaks Sidelobe searching algorithms only work well on canonical patterns, with regularly spaced, well-behaved sidelobes Numerically optimized patterns, and measured patterns, typically have a nonmonotonic sidelobe envelope Further, many “split” sidelobes occur that are much narrower The typical fine structure of sidelobes defeats most algorithms, even for one-dimensional patterns Fine sampling with filtering to find the envelope is feasible, but for two-dimensional patterns it can be extraordinarily slow Since an optimization may require hundreds of function evaluations, the computational cost may well

be excessive

For patterns where the array is small or modest in size, and the pattern features are coarse, optimization of directivity, slope, etc., with constraints on sidelobe envelope, bandwidth, tolerances, etc., may be feasible; see Perini and Idselis (1972), for example Several classes of optimization algorithms exist First are gradient schemes Simple gradient and conjugate gradient schemes tend to be fast, but they often stop at a local maximum, or hang up at a saddle point More sophisticated schemes avoid both these problems and will usually find the global maximum (Fletcher, 1986) Some of these are those of Fletcher

and Powell (1963), Fletcher and Reeves (1964), and Gill and Murray (1972)

values, forming a sequence of simplexes; each simplex is a set of points with

a vertex where the function value is maximum This vertex is reflected in the centroid of the other vertices, forming a new simplex The value at this new vertex is evaluated, and the process is repeated Special rules re-start the process

Because the sidelobe positions change as the excitation changes, an iterative

in a shaped beam pattern, often minimize the sum of squares of the differences

4.2.2 Ring Sidelobe Synthesis

For rectangular apertures, a sidelobe envelope that decays with 0 is often desirable, analogous to the Hansen circular one-parameter pattern, or to the Taylor circular ii pattern Such a synthesis can be produced by the brute-force computer techniques of the previous section, or by the methods of Section 3.9.2, where the zeros of a (two-dimensional) polynomial are iteratively adjusted This latter approach is far better

A more analytical approach utilizes the Baklanov (1966) transformation, which converts the two-dimensional problem to a one-dimensional problem Tseng and Cheng (1968) applied this transformation to a square array provid- ing Chebyshev type ring sidelobes Kim and Elliott (1988) extended this to an adjustable topology of ring sidelobes For a square array of N x N elements, let the pattern be

Now assume the desired ring sidelobe pattern is given by a polynomial of order

N - 1 in the new variable W:

I=1

Through use of trig expansions the array excitation coefficients A,, are found

(4.32)

determine the BI coefficients, can utilize either the zero adjusting process or the

determined so that sidelobe (ring) locations can be calculated This procedure

90

Figure 4.16 Baklanov ring sidelobe pattern (Courtesy Kim, Y U and Elliott, R S.,

“Extensions of the Tseng-Cheng Pattern Synthesis Technique,” J Electromagnetic Waves

square array with first sidelobe at -30 dB (Kim, 1990) The sidelobe rings are not circular, but fit the aperture geometry Such a synthesis provides higher gain for the same sidelobe level than a design based on the product of two linear distributions

Bandler, J W and Charalambous, C., “Theory of Generalized Least Pth

Approximation,” Trans IEEE, Vol CT-19, May 1972, pp 287-289

Bayliss, E T “Design of Monopulse Antenna Difference Patterns with Low Sidelobes,” Bell Systems Tech J., Vol 47, No 5, May-June 1968, pp 623-650

Erlinger, J J and Orlow, J R., “Waveguide Slot Array with Csc28 Cos8 Pattern,” Proceedings 1984 Antenna Applications Symposium, RADC-TR-85-14, Vol 1,

pp 83-112, AD-Al53 257

Fletcher, R., “A New Approach to Variable Metric Algorithms,” Computer J., Vol 13, Aug 1970, pp 317-322

Fletcher, R., Practical ikfethods of Optimization, Wiley, 1986

Fletcher, R and Powell, M J D., “A Rapidly Convergent Descent Method for Minimization,” Computer J., Vol 6, 1963, pp 163-168

Fletcher, R and Reeves, C M., “Function Minimization by Conjugate Gradients,” Computer J., Vol 7, 1964, pp 149-154

Gill, P E and Murray, W., “Q uasi-Newton Methods for Unconstrained

Optimization,” J Inst Math Appl., Vol 9, 1972, pp 91-108

Hansen, R C., “Tables of Taylor Distributions for Circular Aperture Antennas,” Trans IEEE, Vol AP-8, Jan 1960, pp 23-26

Hansen, R C., “A One-Parameter Circular Aperture Distribution with Narrow

Beamwidth and Low Sidelobes,” Trans IEEE, Vol AP-24, July 1976, pp 477-480 Hansen, R C., “Struve Functions for Circular Aperture Patterns,” Microwave and Opt Technol Lett., Vol 10, Sept 1995, pp 6-7

Hersey, H S Tufts, D W and Lewis, J T., “‘Interactive Minimax Design of Linear- Phase Nonrecursive Digital Filters Subject to Upper and Lower Function

Constraints,” Trans IEEE, Vol AU-20, June 1972, pp 171-173

Josefsson, L Moeschlin, L and Sohtell, V., “A Monopulse Flat Plate Antenna for Missile Seeker,” presented at the Military Electronics Defense Expo, Weisbaden, West Germany, Sept 1977

Kim, Y U “Peak Directivity Optimization under Side Lobe Level Constraints in Antenna Arrays,” Electromagnetics, Vol 8, No 1, 1988, pp 51-70

Kim, Y U “A Pattern Synthesis Technique for Planar Arrays with Elements Excited In-Phase,” J Electromagnetic Waves Appl., Vol 4, No 9, 1990, pp 829-845