317
1 INTRODUCTION
The modern radars are high-powered technical means
of navigation and take important part to ensure
maritime safety. The distinctive peculiarity of marine
radar functioning is the necessity of reflected from
ships signals separation with essentially different
radar cross section (RCS) in condition of close situated
targets. In its tern, that is demanded low side-lobs
level and high directivity property (narrow enough
main beam) of antenna diagram. These properties
may be realized on the base of simple enough
technique using antenna arrays with limited number
of tuneable weight coefficients of spatial filter
(antenna array elements), for example, when there are
only two of such tuneable weight coefficients [1].
In this case the entire antenna array (spatial filter)
weights coefficients
i
W
of the processing, except
two (first and last:
1
,
N
WW
), are fixed (selected under
the condition of providing the required antenna
pattern average side lobe’s level) (
23 1
;;;
N
WW W
−
…
).
Value of the two tuneable weights coefficients are
selected for carried out the condition of providing
zero values in two points (
12
,
θθ
) of the reception
pattern [1],[2],[3],[4]. Different algorithms for
calculation fixed weights coefficients for providing
the required average antenna pattern side lobes level
are presented: on the base of weighting functions sinx
and
( )
2
sinx
, which give the possibility of changing
deep of modulation of weighing function in wide
interval (from equal values maximum and minimum
weight coefficients till the biggest possible difference
between them) and correspondingly different levels
of the average side lobe suppression and losses in
antenna’s directivity. Numerical examples were
considered which demonstrated efficiency of
weighing functions suggested for fix coefficients.
Partial diagram were calculated for this purpose.
Modified weighing functions are also considered
which provieded inccreasing level of suppression
side-lobes nearby main lobe. As all numerical
examples in this work were considered for 10th array
and for 26th array, so partial diagram were calculated
for N-2=8 and for N-2=24 element array with equal
weight coefficients values and other weighing
functions with unequal weights coefficients values.
Because the algorithms of reception diagram side
Antenna for Marine Radar with Supperdirectivity
P
roperties
V.M. Koshevyy
& A.A. Shevchenko
National University “Odessa Maritime Academy”, Odessa, Ukraine
ABSTRACT: The reception antenna diagram side lobe’s level suppression algorithm for marine radar by means
of antenna array with only a few tuning elements of antenna array is considered. The others no tuning elements
of array are choosing for obtain a given value of average side lobe level suppression and with given value of
antenna directivity without using of numerical optimization procedures. Special algorithm of interaction of
these no tuning elements for realizing Supperdirectivity properties is used. The structural diagram of array is
presented. The efficiency of suggested design has been investigated.
http://www.transnav.eu
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 14
Number 2
June 2020
DOI:
10.12716/1001.14.02.06
318
lobes suppression in given points by means tuned
weights and decreasing average level of side lobes in
other points by means weighing correction of fix
element leads to increasing the width of main lobe of
antenna diagram, the possibilities of decreasing of
main lobe width are investigated.. The algorithm was
considered for providing decreasing main lobe width
on the base of approach suggested in [13]. This
algorithm includs the forming of two partial diagrams
and using special interaction between them [5, 6].
Investigations of full diagrams for N-element antenna
array including tuned elements with different
situation suppresed points of diagram are also
provided.
2 MATHEMETICAL DISCRIPTION
The expressions, which are describing the reception
pattern of the antenna array
( )
G
θ
, may be written in
the following form:
G (
ϑ
) =
( )
21
1
d
N
j i sin
i
i
We
πθ
λ
−−
=
∑
(1)
The expressions for tuneable weight coefficients
may be presented in the form [1]:
12
21
dd
-j2 (N-1) sin -j2 (N-1) sin
N-2 2 N-2 1
1
dd
-j2 (N-1) sin -j2 (N-1) sin
G() e - G()e
W
e -e
πθ πθ
λλ
πθπθ
λλ
θθ
=
(2)
21
N-2 1 N-2 2
N
dd
-j2 (N-1) sin - j2 (N-1) sin
G()-G()
W
e -e
πθπθ
λλ
θθ
=
(3)
where G
N-2(θ) – partial diagram for untuneable (fixed)
weight coefficients, which has the following form:
1
2 ( 1) sin
2
2
()
d
N
ji
Ni
i
G We
πθ
λ
θ
−
−−
−
=
= ⋅
∑
(4)
Function
( )( )
1G
θ
(with consideration (2), (3))
may be represented in the next form:
( ) ( ) ( ) ( ) ( ) ( )
2 1 21 2 22NN N
GG G G
θ θγθθγθθ
−− −
=−−
(5)
were
2
21
2 ( / )( 1) sin
2 ( 1)( / ) sin
1
2 ( / )( 1) sin 2 ( 1)( / )sin
()
jd N
jNd
jd N jNd
ee
ee
πλ θ
π λθ
πλ θ π λθ
γθ
−−
−−
− − −−
−
=
−
(6)
1
21
2 ( 1)( / ) sin
2 ( / )( 1) sin
2
2 ( / )( 1) sin 2 ( 1)( / )sin
()
jNd
jd N
jd N jNd
ee
ee
π λθ
πλ θ
πλ θ π λθ
γθ
−−
−−
− − −−
−
=
−
(7)
2 sin /
d
φ π θλ
=
—signal phase;
λ
— wave’s length; d
— distance between antenna’s array elements;
θ
—
angle between the normal to the axis of the array
antenna and direction of the coming signal.
In this paper we consider the different weight
functions effect on reception diagram, which allow
transforming the reception diagram properties.
The expressions for the untuneable weight
coefficient are describing by the next:
(1)
2
sin
12
n
yn
W
N Nz
π
−
= +
− +−
(8)
2
(2)
2
sin
12
n
yn
W
N Nz
π
−
= +
− +−
(9)
where: n=2:N-1 ;
1
1:
2
N
y
−
=
;
2 ( 2) ( 1)
12
yN N
z
Ny
−− −
=
−−
.
The main lobe widening may be regulated by
parameter ‘y’. The bigger value of parameter ‘y’
corresponds to the less main lobe widening. For
example, the reception diagrams calculated with (8)
are shown in Figure 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2 y=3
Figure 1. Partial reception diagram
2N
G
−
(
)
θ
(N=10), y=1,
y=2, y=3 (
( )
1
n
W
)
The reception diagram
(
)
2
N
G
θ
−
with equal
weight coefficients
i
W
(equable correction), which
corresponds the condition with absence of main lobe
widening is practically coincide with the case
( )
( )
1
3
n
Wy=
shown in Figure 1.
Calculation reception diagram
( )
2
N
G
ϑ
−
according to (9) are shown in Figure 2.
319
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
Figure 2. Partial reception diagram
(
)
2N
G
θ
−
(N=10), y=1,
y=2, y=3 (
(
)
2
n
W
)
So, we can see that the weight function, described
by (8) and (9), allows correcting the reception diagram
properties widely.
Modified weighing functions may be suggested
which provided the increasing level of side-lobes
suppression nearby main lobe. Modified weighing
functions are described by the next expressions:
( )
(
)
( )
(
)
1
1
2
, 3 ;
1
4
2
k
n
kM k
nn
k
n
W
n
W W if n
N
W
n
+
−
=
= =
+
=÷
(10)
where [x] - is integer part of x; k=1; 2;
( )
k
n
W −
is
determined by (8) and (9), under k=1and k=2
correspondingly,
( )
( )
( )
1
kM kM
n
Nn
WW
−−
=
, if n =
1
11
2
N
N
+
+÷ −
.
The reception diagrams, calculated with weight
coefficients (10) are shown in Figures 3-4.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
Figure 3. Partial reception diagram
( )
2N
G
ϑ
−
(N=10), y=1,
y=2, y=3 (
( )
1 M
n
W
)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
Figure 4. Partial reception diagram
( )
2N
G
θ
−
(N=10), y=1,
y=2, y=3 (
( )
2 M
n
W
)
It follows from figures 3 and 4 that modified
weighing function decreasing diagram side-lobes
level nearby main lobe area. The using weighing
functions lead to the widening of the main lobe and
decreases directional properties of antenna array.
Decreasing the main lobe width and increasing
antenna directivity may be gotten by means
increasing the number of antenna array elements. For
example in figures 5 – 8 represented the results of
calculations for N=26 antenna array diagram (partial
diagram is formed by means N-2=24 elements).
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
W[1]
Figure 5. Partial reception diagram
( )
2N
G
θ
−
(N=26), y=1,
y=2, y=3 (
( )
1
n
W
); W [1] (equal correction)
320
-2
-1.5
-1 -0.5
0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
W[1]
Figure 6. Partial reception diagram
( )
2N
G
θ
−
(N=26), y=1,
y=2, y=3 (
( )
2
n
W
); W [1] (equal correction)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
Figure 7. Partial reception diagram
( )
2N
G
θ
−
(N=26), y=1,
y=2, y=3 (
( )
1 M
n
W
)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad
G,dB
y=1
y=2
y=3
Figure 8. Partial reception diagram
( )
2N
G
θ
−
(N=26), y=1,
y=2, y=3 (
( )
2 M
n
W
)
For comparison in figures 5 and 6 in addition are
shown the reception diagrams for equal weighing W
[1] (all weights coefficients are equal 1). The results of
calculations reception diagrams represented in the
figures shows the high efficiency of using weighing
functions, especially modified weighing functions for
the area nearby main lobe.
Losses in antenna’s directivity due to weighing
functions described by the next expression [3]:
( )
2
2
1
2
2
N
N
n
n
G
NW
ϑ
ρ
−
−
=
=
∑
(11)
So, we can correct the regulated part of reception
diagram by different weights functions. This
approach does not require the implementation of
numerical optimization procedures as were described
in [2]. Choosing of such kind weighting functions we
can get additional average side-lobe suppression, but
with widening main lobe. This widening is decreased
with increasing number of the array antenna
elements. The method of increasing angle selectivity
without losses in average side-lobe level [5], [6] may
be considered, which based on approach suggested in
[13].
In considered case at the output of N-2 antenna
array elements (no tunable part of reception antenna
(1) (see (4)) we have complex signals
i
X
[1]. By the
first N-2 signals are created the sum:
2
11
1
N
ii
i
Z WX
−
+
=
=
∑
(12)
Beside (12) the second sum is created:
21
3
N
ii
i
Z WX
−
=
=
∑
, (13)
where:
( )
1
1
,
ji
i i Ii i
X S N S Se
ϕ
−
=+=
,
sin
2,
d
θ
φπ
λ
=
Ii
N −
thermal noise. If root-mean-square value of
thermal noise is negligible small and
1
1,S
=
(12)
coincides with (4) and
2
21
j
Z Ze
ϕ
=
.
Then the sum is created [5]
1
( 2)
2
() 1
Nj
R
Z
Ge
Z
ψ
φ
−−
=
( )
1
jΨ
R
2
Z
G φ 1e
Z
−
=
, (14)
Ψ
1
1
j
j
j
e
e
e
ϕ
ϕ
µ
µ
−
−
−
=
−
, (15)
where:
µ
- coefficient (
0 1)
µ
≤≤
,
**
12 12
12 12
ˆˆ
sin2 Im ,cos2 Re
ZZ ZZ
ZZ ZZ
φφ
= =
.
From (14), using (15), after some transformations
we can get:
( )
( )
( )
2
2
2
2
12
N
N
R
G Cos
G
Cos
ϕµ ϕ
ϕ
µ ϕµ
−
−
−
=
−+
(16)
321
Considering (16) may be represented in the form:
( 2)
2
() () ()
N
R NS
G GG
θ θθ
−
−
= ⋅
,
( ) ( ) ( )
R N2 S
G θ G θ G θ,
−
= ⋅
(17)
( )
2
2
12
S
Cos
G
Cos
µϕ
ϕ
µ ϕµ
−
=
−+
(18)
where:
( )
2
N
G
θ
−
−
is determined by (4),
2
d
Sin
ϕπ θ
λ
=
.
As we can see from (17) angle selectivity of
antenna may be essentially increased by means
proper choose the value of
µ
.
The width of the main beam of (18) on the level
0,5
( )
0
S
G
is:
2
0.5
2
Cos[ ]
22
arc
d
µ
µ
θ
π
λ
−
+
∆=
(19)
From (16) it follows that if
1,
µ
→
so
0,5
θ
∆→
0.
So Supperdirectivity may be provided by means
(18).
Consider some peculiarities of antenna array
working with diagram (17). Due to functional
transform (14) linearity of processing and Principe of
superposition are breaking under affecting a few
signals from different direction. If two interfering
signals have close angels of arrival, may be provided
good enough suppression of both signals. If the
difference of arrival angels is big and intensity one is
bigger enough than another, we get the suppression
of the bigger signal. These considerations are stay in
force for the case of more, then two signals. Thus for
providing functionality of proposed principia of
selection for multitarget situation special condition
should be provided. Which suppose that signals with
approximately equal intensity would have small
difference of arrival angels, and for signals with
essential different angels of arrival would be
provided corresponding difference in their intensities.
It may be realise by means antenna with diagram
( )
( )
2N
R
G
θ
−
(17). So, approximately equal intensities
will be took place only in narrow angels interval,
determined by main lobe bean width of
diagram
(
)
2
N
G
θ
−
. For the signals which have
essential difference of arrival directions, weighting of
their intensities would be provided by the same
diagram (
( )
2
).
N
G
θ
−
Thermal noise and errors of practical realization
are limited the maximal value of
µ
and thus limited
the minimal value of main lobe beam width for real
antenna design. If
η
= noise/interference ratio
(supposed equivalent noise, which included thermal
noise and technology errors), so
1/1
µη
≤+
[5].
Using this value in (16) we can get restriction on main
lobe beam width.
Calculations
( )
( )
2N
R
G
θ
−
for different
µ
and
different weighing functions for
(
)
2N
G
−
(
)
θ
(N=10)
are presented at figures 9 – 11. For comparison are
shown the reception diagrams for equal weighing W
[1] (all weights coefficients are equal 1 for both sums
(12), (13) and
0
µ
=
)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
G,dB
teta,rad.
W[1]
m=0.9
m=0.97
m=0.95
Figure 9. Partial reception diagram
( )
( )
2N
R
G
θ
−
(N=10),
y=1, (
( )
1
n
W
),
0.9
µ
=
,
0.95
µ
=
,
0.97
µ
=
; W[1] (equal
correction with
0
µ
=
)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
G,dB
teta,rad.
W[1]
m=0.95
m=0.9
m=0.97
Figure 10. Partial reception diagram
( )
2N
R
G
−
(
)
θ
(N=10),
y=1, (
( )
2
n
W
),
0.9
µ
=
,
0.95
µ
=
,
0.97
µ
=
; W[1]
(equal correction
0
µ
=
)
322
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
G,dB
teta,rad.
W[1]
m=0.97
m=0.9
m=0.95
Figure 11. Partial reception diagram
( )
( )
2N
R
G
θ
−
(N=10),
y=2, (
(
)
2
M
n
W
),
0.9
µ
=
,
0.95
µ
=
,
0.97
µ
=
; W[1]
(equal correction
0
µ
=
)
The results of calculations partial diagrams are
represented in Figures 9-11 show the possibility of
narrowing the width of main lobe by means proper
choosing of parameter
µ
for different kinds of
weighing functions. Modified weighing functions in
this case additionally decrease the level of side-lobs
nearby the main lobe area. For N=26 the result of such
calculations are represented in Figures 12-14.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
G,dB
teta,rad.
W[1]
m=0.9
m=0.97
m=0.95
Figure 12. Partial reception diagram
2N
G
−
(
)
θ
(N=26),
y=3, (
( )
1
n
W
),
0.9
µ
=
,
0.95
µ
=
,
0.97
µ
=
; W[1]
(equal correction
0
µ
=
)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
G,dB
teta,rad.
W[1]
m=0.9
m=0.97
m=0.95
Figure 13. Partial reception diagram
2N
G
−
(
)
θ
(N=26),
y=1, (
( )
2
n
W
),
0.9
µ
=
,
0.95
µ
=
,
0.97
µ
=
; W[1] (equal
correction)
Under increasing number of antenna elements for
forming partial diagram, increasing the efficiency of
using the different kinds of weighing functions
together with approach for increasing the spatial
directivity of antenna array (see Figures 12, 13). The
special role under this plays using of modified
weighing functions. That is demonstrated in Figure
14.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
G,dB
teta,rad.
W[1]
m=0.9
m=0.97
m=0.95
Figure 14. Partial reception diagram
2N
G
−
(
)
θ
(N=26),
y=1, (
( )
2 M
n
W
),
0.9
µ
=
,
0.95
µ
=
,
0.97
µ
=
; W[1]
(equal correction
0
µ
=
)
The reception diagrams with suppressions in
given points, calculated with weights coefficients
1
W
and
2
W
according to (2) and (3) for N=10 are
shown in Figures 15-17.
323
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad.
G,dB
m=0.95;W
2
;N=10;t
1
=-1.2001;t
2
=-1.1718
y=1
y=2
y=3
Figure 15. Reception diagram
( )
N
G
θ
(N=10),
( )
2
n
W
,
0,95,
µ
=
12
1,2001, 1,1718
θθ
=−=−
.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad.
G,dB
m=0.95;W
2
;N=10;t
1
=-0.8137;t
2
=-0.7854
y=1
y=2
y=3
Figure 16. Reception diagram
( )
N
G
θ
(N=10),
( )
2
n
W
,
0,95,
µ
=
12
0,8137, 0,7864.
θθ
=−=−
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad.
G,dB
m=0.95;W
2
;N=10;t
1
=-0.2576;t
2
=-0.2293
y=1
y=2
y=3
Figure 17. Reception diagram
(
)
N
G
θ
(N=10),
( )
2
, 0,95,
n
W
µ
=
12
0,2576, 0,2293.
θθ
=−=−
As we can see, the suppression in given points for
these cases is high enough, and the side lobes level
between the suppressed points is also small enough
(about – 80 dB). The possibilities of suppression Cross
Ambiguity Function in discrete points were
considered also in [8],[9],[11],[12],[14]. Estimates the
level of side-lobs between suppression points had
been gotten analytically in [14].
The results of calculations for N=26 are
represented in Figures 18-20.
-2 -1.5
-1 -0.5
0 0.5
1 1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad.
y=1
y=2
y=3
Figure 18. Reception diagram G(θ) (N=26) with suppression
in points
1
0.2576,
θ
= −
2
θ
=-0.2293; (
( )
2
M
n
W
),
y=1,
0.95
µ
=
;
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad.
y=1
y=2 y=3
Figure 19. Reception diagram G(θ) (N=26) with suppression
in points
1
θ
=-0.8497,
2
θ
=-0.7854; (
( )
2 M
n
W
),
y=1,
0.95
µ
=
;
324
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
teta,rad.
y=1
y=2
y=3
Figure 20. Reception diagram G(θ) (N=26) with suppression
in points
1
θ
=-1.2001,
2
θ
=-1.1718; (
( )
2 M
n
W
),
y=1
0.95
µ
=
;
So, we have not only suppression in given points
of reception diagram, but and given value for main
lobe width and high enough average side-lobes
suppression.
The block diagram of the proposed N element
radar antenna array algorithm is shown in Figure 21.
Figure 21. Block diagram of the proposed algorithm
3 CONCLUSINS
In this paper antenna array design capable of obtain
the given side-lobe suppression with controlled value
of directivity coefficient is suggested. The antenna
with super selectivity properties is also suggested.
The approach is simple enough for calculations and
does not require the implementation of numerical
optimization procedures, such as [10]. It’s very useful
for practical implementation, when it’s necessary to
get the given side lobes suppression with given main
lobe properties.
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