1989 08 circular array and nonsinusoidal waves.pdf

8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf

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2 5 4

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL.

31,

NO. 3, AUGUST

1989

Circular Array and Nonsinusoidal Waves

MAHA

M. AL-HALABI

A N D

MALEK G .

M .

HUSSAIN, MEMBER, IEEE

Abstract-In this paper, the theory of circular array antenna based on

nonsinusoidal waves, with the time variation of a rectangular pulse, is

developed. D ifferent antenna patterns such as peak-amplitude, peak-

power, energy, and slope pattern are derived and plotted. The antenna

patterns yield the resolution angle for the circular army as a function o f

array radius and frequency bandwidth. The effect of additive Gaussian

noise

on

the angular resolution capability of the circular array is

analyzed. The analysis is based

on

calculating slope patterns by using

linear regression algorithm for different signal-to-noise power ratios.

K ey Words-Nonsinusoidal wave, circular array, angular resolution.

Index Code-l13c/d/g.

I. INTRODUCTION

HE PRINCIPLE of circular array beamforming based on

T he infinitely extended periodic sinusoida l waves results in

the resolution angle as a function of array radius and frequen cy

[

11-[3].

The increase of frequency for a small resolution angle

is limited in practice by atmosp heric attenuation, w hich can be

severe at high frequencies. Since the circular array has many

applications for radar, radio communications, and direction

finding, it is desirable to develop its theory based on

nonsinusoidal waves. Nonsinusoidal waves yield high-resolu-

tion, all-weather capabilities for radar and a high rate of

information for radio communication.

The objective of this paper is to investigate the angular

resolution capability of the circular array based on (nonsinu-

soidal) rectangular pulses. In Section

11,

antenna patterns such

as peak-amplitude, peak-power, en ergy, and slope pattern are

derived. In Section 111, computer plots of the derived antenna

patterns are presented, and the resolution angle is obtained as a

function of array radius and frequency bandwidth. The

dependence of the resolution angle on frequency bandwidth is

important in practice. In Section

IV,

antenna slope patterns a re

derived for different signal-to-noise power ratios to study the

effect of additive Gaussian noise on angular resolution of a

circular array . Conclusions are given in Section

V .

11. THEORYF C IR C U LA R

RRAY

ASED

N

RECTANGULAR

PULSES

A

circular array of

N

omnidirectional, equally spaced,

antenna elements is shown in Fig.

1.

The radius o f the circular

array a, and its center is at the origin of the

x ,

y, and

z

coordinate system. The position

of

the nth array element in the

Manuscript received July 26, 1988; revised January

5 ,

1989.

M. M. Al-Halabi is with the Department of Electrical and Computer

Engineering, Kuwait University, Kuwait.

M . G. M. Hussain is with the Department of Electrical and Computer

Engineering, Kuwait University, Kuwait. He is also a visiting professor at the

University of Michigan Radiation Laboratory, Department

of

Electrical

Engineering and Computer Science, Ann A rbor, M I 48109.

IEEE Log N umber 8928166.

t z

Fig. 1.

Geometry

of

an N-element circular array with radius

a.

x

-

y plane is defined by the angle

2an

N

I ,=-,

n = l ,

2, e - . , N.

The d istance R, from the

nth

array element to a point P(r, 8,

4 )

in the far field

is

given by

[ 2 ]

R,

=

r -

sin 8 cos

(4- 4,)

( 2 )

where

r

%

a

is the distance from the center of the circular

array to the point

P(r, 8,

4) .

According to

( 2 ) ,

if a planar

wavefront is arriving from a source at point P(r, 8,

4)

in the

far field, the relative time delay at sensory with respect to the

center of the array is

r, =

a /c ) in

0

cos

(4-

4,)

(3)

where c is the speed of light.

Let a planar wavefront with the time variation of a (noise-

free) rectangular pulse U ( t ) of duration AT and peak ampli-

tude A be incident at the array sensors of Fig. 1 from the

direction of P(r,

8, 4 ) ,

U ( t ) = A l i I ( t / A T )

A [ ~ t )~ t -

T ) ]

(4)

, O s t s A T

= t , elsewhere

where u ( t ) is the unit step function. The array sensors

transform the received wavefront into voltage signals V, ( t ) ,

= 1 , 2 , ,

N

V,,(t)

=

U ( t -

T,,).

The sum of the voltage

0018-9375/89/0800-0254 01 OO 989

IEEE


2/8

AL-HALABI AND HUSSAIN: CIRCULAR ARRAY AND NONSINUSOIDAL WAVES

signals yields

2 5 5

p m e =

psine= 0 . 2 5

l N 1 N

V ( t )= - V , ( t )

=

- U ( t-ATp sin 8 cos 4) ( 5 )

where the factor 1 / N is for normalization, and

p

is a design

parameter defined by the ratio

N =I

N,=I

a a

C A T c

= - = -

A f . (6)

psi.@= 0 . 5

A

In

( 6 ) ,

Af = l / A T is the nominal frequency bandwidth.

- -

For

a circular array with a large number

of

sensors N

%-

1

one may use the approximation A 4 = 2 ?r/N d4 , and 4

0.5

pslne= . 7 5

4, and replace the summation in (5 ) by integration, to obtain

pine-

1

1 2*

V ( t ) = g ( t ) = - 1 U ( t - A T p sin 8 cos 4 ) d4

t-

2a 0

Fig. 2.

Time variation of the voltage signal

g ( t )

given by

(8)

and

(9)

for

different values of

p

sin

0 =

0,

0.25, 0.5,

0.75,

and

1 .

( t - A T y p sin 8 cos 4 ) d+]

A

=

1 and different values of p sin 8 = 0,

0.25 0.5,

0.75,

function of

p

sin 8,whereas its peak am plitude is constant for

p

s

u ( t - A T - A T P

sin

e

COS

4 )

d6

7)

and

1 .

According to Fig.

2,

the duration

ofg t)

is an increasing

sin 8 0.5 and decreases for p sin 8

>

0.5.

The change of pulse ch aracteristicsas fun ctions of an gle, as

*

The time variation of g ( t ) is a function of

p

sin

8.

The two

integrals in

7)

result in the following time variations for g( t ) ' :

1)

p sin

8s

0.5

t / A T r - p sin 8

r O

2) Fbr p sin 8> 0.5

O

A A

- p

sin 8 s t / A T s p sin 8

A A ( t / A T )-

---

sin- '

(

)

,

1

- p

sin

8 s t / A T s

1

+ p

sin 82 7 r

p

sin

8

0,

t / A T r 1 + p sin 8

t / A T s - p sin 8

- p

sin 8 s t / A T s 1 p sin 8

t / AT ( t / A T )-

g ( t ) = in-'

(-)

sin 8 -:in- ' ( p sin

8

)

1 - p sin O s t / A T s p sin 8.

A A ( t / A T )- 1

p

sin 8

p sin 8 / A T c 1 + p sin 8

0 t / A T 2 1 + p sin 8

9)

The function g( t ) given by 8) and 9) is plotted in Fig. 2 for

in Fig. 2, allows one to calculate different antenna patterns

pattern and slope pattern. Due to the geometry of the circular

Detailed derivations

of

8) and (9) can

be

obtained by writing to the

such

as

peak-amplitude pattern, peak power pattern, energy

authors.


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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 31, NO. 3, AUGUST 1989

02

01 I I I

0 2 . 4 6 8 ld

P s i n e

-

(a)

:\

--

0 I I I I

0 ps in

e

6 8 10

o \

08I

- I

a 10

-I

X I 0

10.0

0.8

7

o

8.8

5

-8

4

.o

3.0

(4 = 90 )

e = 90

I

I

JO

e = 90

(4 = 45 1

(b)

Fig.

4.

Three-dimensional plots of peak-a mplitu de pattern A 0,b) for (a) p

=

3

and b) p =

5 .

(9)

yield the normalized peak-amplitude pattern

p

sin

8 5 0 . 5

p

sin

8>0.5.

(10)

array in Fig* 9 the antenna patterns are

Of

the

The peak-power pattern

p( )

s the square

of A @:

azimuth angle

4,

and their maximum is in the broadside

direction, which is along the Z-axis. Th e relationship in (8) and

p e)= A ~ ) ] Z . (1 1)


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AL-HALABI AND HUSSAIN: CIRCULAR ARRAY AND NONSINUSOIDAL WAVES

x 16)

10.0 *

0.0 -

0.0

-

?.O-

0.0 -

5.0 -

4

.0

3

a

2.8

1 .e

(0

= 90

257

No- '

- I

x

f 0

10.0,

0.0

7

5.0

4.0

3.8

2 .0

0 = 90

(4 = 45

(b)

Fig. 6. Three-dimensional plots

of

energy pattern

E(B,4)for

(a)

p

= 3 and

b)p = 5 .

Fig. 5 .

Three-dimensional plots of peak-power pattern

P(O,+)for (a) p

= 3

and b)

p

=

5.

results

in

the normalized energy pattern

The normalized energy pattern

E

(8) is defined as the ratio

1

T

p sin

8,

p sin 810.5

r

2

p sin

8 + -

(1 - p

sin

8)

s in - '

E ( 8 )=

n

W 0) A 2 A T

[ 1 - - - 1 ) 2 ]

1

112

--

1

where W 8) s the energy

of

g ( t ) for 8 > 0, and W 0) =

Evaluating the integral in

(12) for g ( t )

given in

(8)

and

(9)

n

p sin 6 7r2

'

A 2 A T

is the energy of

g(t) for

on-axis reception

8 = 0.

p

sin

6>0.5

(13)


5/8

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IEEE

TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL.

31,

NO. 3. AUGUST 1989

4

.o

3.0

2 .e 2

.e

I

e

1 0

x 16'

-10.0

-9.0

-0.0

-7.0

-6.0

- s e

-4

.o

-3.0

r

- /

/*

e

=

14.4 '

(6 = 45* 1

(b)

p = 5 .

Fig. 7. Three-dimensional plots of slope pattern S(8 ,+) for (a) p =

3

and (b)

where

A T ( p s i n 0 )

AT I

- p

sin 8)

( t / A T )- 1

sin- '

( )

d ( t / A T ) . (14)

p

sin

8

A slope pattern S(8) can be derived by plotting the slope of the

(least-square) line that best fits the rising section

of the

function

g ( t )

shown in Fig. 2 versus

p

sin 8. The slope can be

calculated numerically by using linear regression

[4].

psine- 0

1O

AT 2AT 3AT 4AT SAT

ps i .@-

0 .2 5

l30 AT 2AT 3AT 4AT 5 i T

0 . 5 s . pSin8

0 . 5

0

i10 AT 2AT 3AT 4AT 5AT

051 , ,

ps in8=o.15

0

1fJ AT 2AT 3AT 4AT 5AT

0.5)

p s i n

8

1.0

2 AT

psine-

1 . 2 5

0.5 pshe-

1 . 5

t-

Fig.

8.

The time variation

of

the voltage signal

q f)

given in (18)

for

different values of

p

sin

8.

111. ANGULARESOLUTION

Computer plots of A @ ,P @,E ( 8 ) and

S (0)

versus

p

sin

8

are shown in Fig. 3. The peak-amplitude pattern

A 8)

and the

peak-power pattern

P 0)

nclude a flat section in the v icinity of

the beam axis. Such beam patterns are not desirable for

achieving good angular resolution. The energy pattern E ( 8 )

and the slope pattern S(8) drop sharply in the vicinity of the

beam axis and rest to small value. The characteristics of these

beam p atterns are attractive for go od angular resolution.

The resolution angle

for a

circular array receiving (or

radiating) nonsinusoidal waves can be calculated from the

antenna patterns of Fig. 3. The resolution angle is defined as

the half-power beam width. Let

K

be the value of

p

sin 0 at

which

P ( 8 ) ,

or

E ( 8 )

equal

0.5,

p

sin

0 = K . (15)

Using the small angle approximation sin

8 = 8 = E

15)

yields the resolution angle

c = K / p = K c A T / a = K c / a A f . (16)

Hence, a reduction in the resolution angle can be achieved

by

either increasing the nominal frequency bandwidth

Af

or the

array radius a. An increase of

A f

yields other advantages such

as good range resolution, protection against electronic coun-

termeasures, and possible detection

of

the so called stealth

targets that are covered by radar absorbing materials.

Three-dimensional plots of the peak-amplitude pattern


6/8

AL-HALABI AND HUSSAIN: CIRCULAR ARRAY AND NONSINUSOIDAL WAVES 259

Ad, ), peak-power pattern P(8,4), energy pattern E ( 8 , ),

and slope pattern

S(8, )

are shown in Figs. 4 hrough

7.

The

base of each plot is a polar coordinate with variables 8 and 4;

the range

0

defines a circle with radius 8. The plots

are for a) p = 3 and

b)

p =

5 .

Increasing the value of the

design parameter p yields a reduction in the beamwidth and

sidelobe levels. According to 6), can be increased by either

increasing the radius a of the circular array or frequency

bandwidth

A f.

IV NOISE ONSIDERATION

In practice, when a wavefront is received by the array

sensors in Fig. 1, thermal noise will be superimposed on the

voltage signals at the output of the sensors, assuming no

interference or multipath signals are present. Thus, noise

suppression is necessary prior to forming a beam pattern. This

task can be achieved by employing a sliding correlator (SC) at

the output of each sensor [ 5 ] . In the case in which the received

signal is a noise-free rectangular pulse, as given in

(4),

he

output of each SC at sensor is a triangular pulse C, , ( t )of

peak-amplitude A and duration 2AT,

(17)

where

7,

is the propagation delay defined in

3).

In the

presence of thermal noise at the input of SC, the output

triangular pulse C, t)will be distorted, but the distortions are

minimum in the sense of least-mean-square error. In analogy

to

(7),

the sum of the triangular voltage signals from the

sliding correlators results in the voltage signal

A ( t- , , ) /A T , O i t i A T

A 2- t- , ,) /A T ), A T i t I 2 A T

n t )

=

2~ ( t - A T p sin 8 cos 4)

&

AT

( t- ATp sin

0

cos 4)

AT

The time variation of q ( t ) is shown in Fig. 8 for different

values of

p

sin 8. The duration of q ( t ) s an increasing function

of p sin 8 , although its peak amplitude is a decreasing one.

Based on (18) and Fig.

8,

one can derive, in analogy to Fig.

3,

antenna peak-amplitude pattern A( ), peak-power pattern

P ( ) ,

energy pattern E @ ) , and slope pattern

S(8) .

These

antenna patterns are show n in Fig. 9 for different values of the

design parameter p

=

2 , 3 , 5 and 10. The beamwidth and

restlobe levels of the different antenna patterns decrease as the

value of

p

is increased.

To investigate the effect of additive thermal noise on

(circular) array beamforming , we calculate numerically by

linear regression

[4]

antenna slope patterns for different

signal-to-noise power ratios

(SNR)

at the array sensors of Fig.

1. In our analysis, we consider band-limited white Gaussian

noise with zero mean superimposed on the rectangular pulses

received by the sensors in Fig.

1.

The samp les of the Gaussian

noise at the array sensors are independent, and the variance

an2 of the noise samples, which equals noise power, is

different at each of the sensors. The SNR at each sensor is

defined as the ratio of signal power S to the average noise

0.6

0.4

02

-

P. 10

01 . . . . . - 1 . I . . . . . . . . . l . . . .

00 2oo 40° 60° 80° 100°

0-

(a)

p -

5

p -

10

1. . . .

I . . . I . .

00

200

4 O 6 ° 800 1000

e -

(C)

I

I 1

00

2 400

600

800 1000

01

e - -

(d)

Fig.

9.

(a) Peak-amplitude pattern A @,

(b)

peak-power pattern f i e , c )

energy pattern

E ( @ ,

and (d) slope pattern S(@. The beam patterns are

derived based on (18) for p

=

2, 3 , 5, and 10.


7/8

2 6 0

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 NO.

3,

AUGUST

I989

21

I

psme=

0.5

0

2

psine=

0.75

1

0

2

I

I

0

AT 2 AT 3AT

t

(a)

1

I pine- o

0.4“R

m = 3 d B

s ( w

0.4

SNR. 3 d B

0.2

0

0.2

0.4 0.6 0.8 1

0

. ,

. .

. . .

.

.

.

, .

s m e - 0 . 2 5

0

psinQ=

0.75

psine-

1

0

I I I

AT 2AT 3 T

t -

(b)

The sum of rectangular pulses of duration A T with additive

Gaussian noise received by

the

circular array in Fig.

1

with = 16

sensors: (a) SNR = - 2 dB and b) SNR =

10 dB.

Fig. 10.

power

P

[6], [7]:

( l / A T ) I A T A 2

t

NAz

(19)

-

S

P

S N R = - =

l / N )i

’,

i

’,

n =

n = l

where A is the peak amplitude of the received rectangular

pulses, and AT is the duration.

Fig.

10

shows the time variation of the sum of rectangular

ps in

e

-

(b)

Normalized slope patterns S( @ ) or a circular array with (a ) N = 16

sensors and

b)

=

32

sensors, receiving rectangular

pulses of

duration

ATw ith additive Gaussian noise. The plots are derived for SNR

= 3

dB, 4

dB, 8 dB, 10dB, and the noise-free case for which SNR

a

Fig. 11.

pulses of duration AT, with additive Gaussian noise received

by the circular array in Fig. 1 with

=

16sensors; a) SNR

=

-

2 dB and b) SNR =

10

dB. The characteristics of the pulses

in Fig. 10(b) are mo re distinguishable than Fig. 10(a) because

of the larger

SNR.

Linear regression algorithm

[4]

s used to

calculate antenna slope patterns for two cases: 1) the received

rectangular pulses with ad ditive noise are directly summed to

form a beam pattern, and 2) the received pulses are passed

through sliding correlators for noise suppression prior to

forming a beam pattern. Slope patterns for case 1) above are

shown in Fig. 11 for a) = 16 sensors, b) N =

32

sensors,

and different values of SNR =

3

dB, 4 B,

8

dB, 10 dB, and

the noise -free case in which SNR

03.

Slope patterns for case

2)

are shown in Fig. 12 for the same values of nd

SNR

as

in

Fig. 11. According to the plots

of

Fig. 11, increasingSNR or

the number N of sensors yields a reduction in the restlobe

levels without any significant narrowing of the bearnwidth.

The plots

in

Fig. 12 are not affected by the ch ange

in

SNR; the

slight variations in the restlobe levels are due to the signal

distortions associated with correlation processing.


8/8

AL-HALABI AND

HUSSAIN: CIRCULAR

ARRAY

A N D NONSINUSOIDAL WAVES

26

1

0.4

0.2

‘Rm-lodB

I Rm- -

01

I I 1 I

0 1

2

3 4

p s i n e-

(a)

m = 2 d B

0

2 3 4

m = 2 d B

0

2 3 4

psine-

(b)

Fig. 12.

Normalized slope patterns S(0) for triangular pulses of duration 2ATp roduc ed by sliding correlators at the output of the

sensors in Fig. : (a)

N =

16 sensors @) N = 32 sensors. The plots are derived for SNR =

-

2 dB.4 dB, 8 dB,

10

dB, and the

noise-free case for w hich SNR

W .

The values of SNR are before correlation processing.

V. CONCLUSIONS

The principle of circular array beamforming based on

nonsinusoidal waves with the time variation of a rectangular

pulse is developed. Antenna peak amplitude pattern, peak-

power pattern, energy pattern, and slope pattern are derived

and plotted. The antenna patterns yield a resolution angle that

can be reduced by either increasing the array radius or the

nominal frequency bandwidth. The slope pattern is the most

attractive for achieving good angular resolution. In the

presence of additive thermal (Gaussian) noise, the sidelobe

levels of the slope pattern can be reduced by increasing either

the signal-to-noise power ratio or the number of array senso rs.

r11

r21

r31

r41

PI

161

171

REFERENCES

M . T. Ma, Theory and Application of Antenna Arrays. New

York:

Wiley, 1974.

C. A. Balanis,

Antenna Theory Analysis and Design.

New

York:

Harper & Row, 1982.

E. A. Wolfe, Antenna Analysis.

M.

G . M .

Hussain, “Line-array beam forming and monopulse

techniques based on

slope

patterns of nonsinusoidal waves,” IEEE

Trans.

Electromagn. Compat., vol. EMC-27, no. 3, pp. 143-151,

Aug. 1985.

H.

F.

Harmuth, “Synthetic aperture radar based on nonsinusoidal

functions: IX. Array beam forming,” IEEE Trans. Electromagn.

Compat., vol. EMC-23, no. 2, pp. 20-27, Feb. 1981.

B.

D.

Steinberg, Principle

of

Aperture and Array System Design.

New

York:

Wiley, 1976.

L.

W. Couch, Digital and Analog Communication System. New

York:

McMillan, 1983.

New

York:

Wiley, 1966.

1989 08 circular array and nonsinusoidal waves.pdf

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