1991 08 sidelobe performance in quadratic phase conformal arrays.pdf

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IEEE TRANSACTIONS ON ANT E NNAS AND PROPAGATION VOL. 39, NO.

8 ,

AUGUST

1991

important for the case of targets having low innate backscattering,

such as very small objects or larger targets employing broadband

low-observable characteristics. Additional improvements in pro-

cessed signal fidelity will be investigated by tests involving alternate

deconvolution procedures [131.

ACKNOWLEDGMENT

The authors would l ike to thank LT Soonpuen Sompaee of the

Royal Thai Navy for his development of acquisition and signal

processing computer programs that were used in this effort.

REFERENCES

[ l ]

J .

D. Young, “Radar imaging from ramp response signatures,”

IEEE Trans. Antennas Propagat., vol. AP-24, pp. 276-282, May

1976.

J.

D. DeLorenzo, “A range for measuring the impulse response of

scattering objects,” in

1967

NEREM Rec. , vol. 9, Nov. 1967, pp.

80-81.

C. W. Hammond, “The development of a bistatic electromagnetic

scattering laboratory,” M.S. thesis, Elect. Eng. Dept., Naval Post-

graduate School, Dec. 1980.

[4] M. A. Morgan, “Time-domain scattering measurem ents,” ZEEE

Antennas Propagat., Newsletter,

vol. 6, Aug. 1984.

[5] M. A. Morgan and

B .

W. McDaniel, “Transient electromagnetic

scattering: Data acquisition and signal processing,”

ZEEE Trans.

Instrum. Meas., vol. 37, pp. 263-267, June 1988.

[6] S . Sompaee, “Computer algorithms for measurement control and

signal processing of transient scattering algorithms,” M.S. thesis,

Elect. Comput. Eng. Dept., Naval Postgraduate School, Sept. 1988.

M. A. Morgan , “Scatterer discrimination based upon natural reso-

nance annihilation,” J

Electromagn. Waves App l .,

vol. 2, no. 516,

P. D. Larison, “Evaluation of system identification algorithms for

aspect-independ ent radar target identification,” M .S. thesis, E lec.

Comput. Eng. Dept., Naval Postgraduate School, Dec. 1989.

[9] N. J. Walsh, “B andwidth and signal-to-noise ratio enhancement

of

the NPS trans ient electromagnetic scattering laboratory ” M.S. thesis,

Elec. Comput. Eng. Dept., Naval Postgraduate School, Dec. 1989.

S. M. Riad, “Impulse response evaluation using frequency-domain

optimal compensation devolution,” presented at 23rd Midwest Symp.

Circuits Syst., Aug. 1981, Univ. Toledo, Toledo, OH.

[ l l ] E . P. Sayre and

R.

F. Harrington, “Time-domain radiation and

scattering by thin-wires,” Appl. Sci . Res., vol. 26, pp. 413-444,

1972.

[12]

R. S. Elliott, Antenna Theory and Design. Englewood Cliffs, NJ:

Prentice-Hall, 1981.

[13] F. I. Tseng and T. K. Sarkar, “Deconvolution of the impulse

response

of

a conducting sphere by the conjugate gradient method,”

IEEE Trans. Antennas Propagat .,

vol. AP-35, pp. 105-109, Jan.

1987.

[2]

[3]

[7]

pp. 481-502, 1988.

[8]

[ lo]

Sidelobe Performance in Q uadratic Phase

Conform al Arrays

E. J Holder

Abstract-Sidelobe performance in an array with a quadratic phase

distribution is related to the array curvature and conditions are derived

that insure well-behaved sidelobes for quadratic phase conformal arrays.

A similar condition is derived for parabolic shaped arrays and results

Manuscript received June 15, 1990; revised March 18, 1991. This

work

was suppo rted by the S enior Technology Guidance Council at the Georgia

Institute

of

Technology.

The author is with the Georgia Institute of Technology, Georgia Tech

Research Institute, Atlanta, GA 30332.

IEEE Log Number 9101241.

are given to illustrate the validity of the derived requirement for well-

behaved sidelobes for both parabolic and circular arrays.

I. INTRODUCTION

It is well known that the behavior of sidelobes in a conformal

array is dependent upon the array geometry. For certain canonical

array geometries such as linear, circular, spherical, and cylindrical

arrays, the far-field patterns can be formulated in analytically

tractable expressions

[11

However, for other nonlinear array ge-

ometries, the far-field patterns are not conveniently calculated in a

tractable form and the sidelobe performance for these arrays must

be completed numerically

[ 2 ] .

n this communication we show that

sidelobe performance in a quadratic phase conformal (QPC A) array

(an array with a quadratic phase distribution across the aperture) is

related to the array curvature and derive conditions that will insure

well-behaved sidelobes for quadratic phase conformal arrays. Also,

the sidelobe performance of a linear array with nonuniform spacing

is related to the curvature of an equivalent conformal array.

The primary reason for analyzing QPCA’s is that the analysis

provides immediate insights into the relationship between array

curvature and sidelobe performance. Using the expression for phase

as a function of arclength derived in ( 2 ) below, an explicit expres-

sion for a QPCA geometry can be derived. This expression is

somewhat complicated (magnitude of the tangent vectors lie on an

arc of sin-’) and, as such, conformal arrays have not been designed

with quadratic phase geometry.

Howev er, the results derived from Q PCA’s apply to more general

array configurations. By expanding phase as a function of arclength

in a Taylor series, all array geometries are seen to exhibit near

quadratic phase ch aracteristics fo r sufficiently limited aperture size

(i.e., when the third-order series terms become insignificant). In

particular, the parabolic array is sho wn to have quadratic phase

when phase is expressed in Cartesian coordinates rather than as a

function of arclength and the circular array has near quadratic phase

for limited aperture size. In both the circular and parabolic cases,

the results derived from QPCA’s are shown to successfully predict

sidelobe degradation as a function of array curvature.

Consider a conformal array defined on the curve (x , y x)) in the

two-dimensional plane whe re is a continuously differentiable

function of

x.

For simplicity we will assume that

y ( 0 )

= 0 and that

the antenna elements are omnidirectional. Assume that a plane wave

is impinging on the array at an angle

t9

with respect to the positive

y-axis. The phase of the plane wave at ( x , y (x )) relat ive to the

phase at 0,O) s given by

p S ( x , y )

=

T ( x s i n 6

a

+ y c o s e )

where h is the wavelength of the propagation signal. In order to

analyze sidelobe degradation, we will assume that the plane wave

signal impinges at

t = 0”.

The phase of the electrical pattern at

( x , y ( x ) for an incident wave impinging at an angle 8 given that we

have a signal wave impinging at 0” is given as the difference of the

respective phases (i.e., the beam is formed to maximize the pattern

responses at 0”).

The curve y can be expressed independent of any coordinate

frame by defining

y

in terms of its tangent as a function of the

arclength parameter.

a s )

=

tan-’ ( y ’ )

0018-926X/91 01.00 991 IEEE



IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,

VOL.

39, NO. 8 UGUST

I 9 9 1

sin

( F O V / 2 )

1

cos ( F O V / 2 )

a‘ s)SI <

1235

- s - 5 s s s + 7)

and a’(s) s recognized to

be

the curvature of a plane curve.

If

sin ( F O V / 2 )

1 cos ( F O V / 2 )

a‘(s) =

Y

1

+

y y ) 3 ‘ 2

’

Then (1) can be written in terms of the arclength parameter s:

The far-field pattern is expressed as

F ( 0 )

= J p d s .

(3)

Y

The integral in ( 2 ) is tractible when

a”

=

0.

However, this

condition insures that the array aperture has constant curvature, in

other words the aperture is a circle or a line. The sidelobe structure

for linear and circular apertures is well behaved and well under-

stood. If we place the additional symmetry constraint on the array

geometry a s)= -a( -s), then the far-field pattern in

(3)

can be

expressed as

F ( 0 ) = 2 / s ’ 2 e ~ ’ ( ‘ - C 0 s ~ B ’ P ~ ( s ’

os ( p , ( s ) sin

0 ) ds 4)

0

where S is the total arclength of the array aperture and

P, S)

= / s c o s a ( l ) d l

P, s) = L s s i n a ( ( ) l .

It is apparent from

4)

that F ( 0 )

= F )

and for cos

0)

= 1 this

expression simplifies to the standard zero-order Bessel function for

s =

2 r .

F ( 0 ) =

/ = c o s ( s i n

( 0 )

sin

s)) d s .

II SIDELOBEERFORMANCE

OR QUADRATIC

HASE

When a’ is not constant, the far-field pattern

(3)

is generally not

tractable. However, for a conformal array that produces a quadratic

phase distribution across the array, an analysis of sidelobe behavior

provides insights into the relationship between array curvature and

sidelobe performance. Let

p s ) = as + bs2,

hen

F

becomes

where

Applying the change of variables

F

is written as

Sidelobe performance is degraded because si n( u2 ) is an even

function with respect to U. When zero is included in the interval of

integration ( d l , , ) then the areas under the curve for u <

0

begin

to add constructively with the areas for

U > 0

causing sidelobe

degradation. The requirement that zero not lie in the interval

( d , , , ) yields the following requirement:

a

2 b

-s -+

0

if

ab

>

0

a

2 6

+ + 0,

if ab <

0. 5 )

Since

p’ (

s) = a

+

2 bs , it follows that a sufficient condition fo r

well-behaved sidelobes for quadratic phase conformal arrays is

p ‘ s )

>

Oor p ’ ( s )

<

0

- S K <

s 5 S + .

Relating the requirement in (5) to the array geometry we have

=

( s i n e ) s + ( c o s 8

l )a ’ (o ) -

2

and with a = s in(0) and b = ( co s ( 0 ) l ) a ’ ( 0 ) / 2 he inequalities

in

(5)

imply

Since

s = 0

is

an

arbitrary point on the aperture, it follows from

(6)

that a sufficient condition for well-behaved sidelobe is

A . Parabolic Array

L / 2 . Substituting 1 ) in

3),

the far-field pattern is expressed by

Consider a parabolic array defined by

y = cx2

for

L 12

x

F ( 0 ) =

S L ’

j 2 r / ~ ) ) s s l n 8 ) + c x* c os 8 ) K)) J dx

Note that for a parabolic array the electrical phase is quadratic in x .

Applying the change of variables

- L / 2

u =

E ( c + )

where a = s in(0) and b = c ( co s ( 0 ) 1 ) the limits of integration

1 become

F ( 8 ) = e ’-

I b l

where the

k

sign ambiguity depends upon the sign of

b ,

and

d ,

=

m (

s -+ )



1236

IEEE TRANSACTIONS ON A NTENNAS A ND PROPAGATION, VOL. 39, NO. 8, AUGUST 1991

s in (FOV /2)

I t a n a l

=

I Y ( X ) I <

Angle

of

Arrival

Degrees)

Fig. 1. Parabolic apertu re antenna pattern ( C L = 0.6)

and where 0

4

a/2.

Then

- L / 2 L / 2 .

9)

Angel

of

Arr~val Degrees)

Circulararray

(or s

= 1).

ig. 4.

Angle of Arrival Degrees)

Fig. 2.

Parabolic aperture antenna pattern ( C L=

1).

Angel of Arrwal Degrees)

Fig. 5 .

Circular array (or s= 1.5).

Figs. 1-3 illustrate the sidelobe performance of a parabolic array

for various values of cL. Notice the degradation in performance for

Figs. 4-6 show the patterns from ideal circular aperture arrays for

a s = 1, 1

S

and

2.

Observe the degradation in sidelobe perfor-



IEEE

TRANSACTIONS

ON ANTENNAS AND PROPAGATION, VOL.

39,

NO. 8 , AUGUST

1991

1231

mance for

a’s

> 1 For

a

circular array to satisfy

7),

it is seen

from (9) that the circular arc must not exceed on e radian.

III CONCLUSIONS

In this communication we show that sidelobe performance in an

array with a quadratic phase distribution is related to the array

curvature and derive conditions that will insure well behaved side-

lobes for quadratic phase conformal arrays. A similar condition is

derived fo r parabolic shaped arrays and results are given to illustrate

the validity of the derived requirement for well-behaved sidelobes

for both parabolic and circular array geometries.

REFERENCES

S. W .

Lee and

Y.

T. Lo, “On the pattern function of circular arc

arrays,”

IEEE Trans. Antennas Propagat.

vol. AP-13, pp 649-650,

July 1965.

J .

K. Hsiao and A.

G .

Cha, “Patterns and p olarizations of simultane-

ously excited planar arrays on a conformal surface,”

IEEE Trans.

Antennas Propagat .,

vol. AP-22, pp. 81-84, Jan. 1974.

End-Loaded C rossed-Slot Radiating Elements

F. Manshadi

Abstruct-Three cavity-backedcrossed-slot antenna con figurationsare

described that offer simple design, easy frequency tuning, and are

lightweight, low loss, and

low

cost. These antennas are designed for

mobile satellite MSA T) vehicle phased-array applications. The slots in

these antennas are end-loaded. The end loading makes the

slots

effec-

tively longer, and hence reduces their resonant frequency. Therefore,

relatively small radiating elements can be achieved for large-angle-scan-

ning phased-array antennas. These antennas have g ood RF characteris-

tics and provide a relatively wide bandwidth without needing external

tuning circuits for impedance matching . Measurements for the return

loss

and the far-field pattern of these an tenna s are presented.

I.

INTRODUCTION

Crossed-slot antennas are important array elements due to their

planar construction and wide beamwidth, which is desirable for

large angle scanning. Shallow-cavity uniform crossed slots were

used as single radiators by Lindberg [l]. He showed that these

antennas produce near hemispherical circularly polarized fields and

are excellent radiators for use on high-speed aircraft due to their low

profile construction. King and Wong

[2]

have shown that making

the slots nonuniform tends to flatten the voltage standing-wave ratio

(VSWR) response of those antennas. They discussed several nonuni-

form stepped slot configurations; however, the length of the slots in

those antennas were on the order of one wavelength, which is not

suitable for large-angle-scanning phased-array applications. Tap ered

crossed-slot antennas are also used as elements for MSAT vehicle

phased-array applications [3]. These tapered crossed slots are small

Manuscript received July 23, 1990; revised April 16, 1 991. This work

was carried out at the Jet Propulsion Laboratory, California Institute of

Technology , under contract with the National Aero nautics and Space Admin-

istration.

The author is with the Jet Propulsion Laboratory, California Institute of

Technology, Pasadena, CA 91 109.

IEEE

Log umber 9101592.

in size but are narrow band and therefore require external tuning to

increase their bandwidth. Moreov er, these antennas are backed by

cavities filled with dielectric materials that are relatively heavy,

expensive, and reduce the antenna efficiency.

In this communication, three cavity-backed, crossed-slot antenna

configurations are described that offer simple design, easy frequency

tuning, and are lightweight, low loss, and low cost. These antennas

have good RF characteristics and provide a bandwidth of about

12 , for a VSWR of

2

: 1, without needing any external tuning

circuits for impedance matching. Unlike the conventional crossed-

slot antennas, these antennas are end-loaded as show n in Fig. 1. Use

of dumbbell-shaped end-loaded slots was first proposed by S ilver for

a waveguide slot array [4]. He suggested that dumbbell-shaped slots

offer lower resonant frequency and are easier to fabricate than

ordinary rectangular slots. In the end-loaded slots, the field coupled

into the slots is more uniformly distributed across the length of the

slots, making their effective length longer. The longer effective

length reduces the resonant frequency and makes possible the use of

an air-filled cavity. The air-filled cavity provides wider bandwidth

and is many times lighter and less expensive than a dielectric-filled

cavity.

The radiating elements described here were primarily designed

for MSAT vehicle phased-array antennas. However, these antennas

also have important applications in airborne systems such as a

high-speed aircraft with a constantly changing look angle to a

satellite or a ground station

[

11. These applications req uire antennas

that are light, physically small, and have wide upper hemispherical

coverage.

CHARACTERISTICS

F

CROSSED-SLOTNTENNAS

Slot antennas are desirable for applications where wide-

beamwidth, compact, low-profile, and low-cost antennas are re-

quired. For circular polarization, a pair of slot antennas in a cross

configuration should be used. The input signal can be applied either

by a section of microstrip transmission line (coupled to the slots) or

by straight coaxial probes. Use of straight coaxial probes keeps the

antenna element simple and minimizes the losses introduced by

microstrip lines. To avoid generation of higher order modes inside

the cavity, four symmetric probes should be employed. Each pair of

probes is placed in the null field of the other pair to minimize cross

polarization. The center conductor of the probes is soldered to the

top plate of the cavity.

Due to the high

Q

of their cavity, crossed slots are inherently

narrowband. The bandwidth for large crossed slots is generally

reported to be about 10% or less [l], [5]. However, the bandwidth

is dependent on the shape of the slots and the dielectric constant of

the substrate. The bandwidth is larger for wider and nonuniform

slots or for lower dielectric constant substrates such as air. Rela-

tively wide bandwidth can be obtained if wide slots are used that are

on the ord er of o ne wavelength long.

The resonant frequency of the cavity-backed crossed slots is a

function of the physical dimensions of the cavity and the slots, the

shape of the slots, the dielectric constant of the substrate, and the

location of probes. For a cavity-backed narrow uniform slot, the

length of the slot is about half the resonant wavelength. Making the

slot longer, the cavity deeper, or the slot narrower lowers the

resonant frequency. The input impedance of the antenna and the

resonant frequency of the cavity are also dependent on the location

of the probes. The resonant frequency of the cavity is increased as

the probes are moved closer to the center of the cavity. One factor

that affects the resonant frequency is the shape of the slots. By

0018-926X/91 01.00 01991

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1991 08 sidelobe performance in quadratic phase conformal arrays.pdf

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