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8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf
1/8
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
8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf
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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.
8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf
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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)
8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf
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2 5 8
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
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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.
8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf
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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.
8/9/2019 1989 08 Circular Array and Nonsinusoidal Waves.pdf
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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.