Array Antennas of Size 8×8 Based on Hadamard Difference Sets
The problem of synthesizing the optimized equi-amplitude array antenna (AA) on the 8×8 grid based on a Hadamard difference set is considered. By using newly found sets of this type on the 8×8 grid the 28- and 36-element AAs having a low sidelobe level are obtained. A numerical experiment showed that...
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| citation_txt | Array Antennas of Size 8×8 Based on Hadamard Difference Sets / L.E. Kopilovich // Радиофизика и радиоастрономия. — 2008. — Т. 13, № 2. — С. 210-215. — Бібліогр.: 17 назв. — англ. |
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| description | The problem of synthesizing the optimized equi-amplitude array antenna (AA) on the 8×8 grid based on a Hadamard difference set is considered. By using newly found sets of this type on the 8×8 grid the 28- and 36-element AAs having a low sidelobe level are obtained. A numerical experiment showed that by a small alteration of the structure of such a set, further reduction of the AA sidelobe radiation is possible.
Рассматривается вопрос синтеза антенных решеток (АР) на основе адамаровских разностных множеств. С использованием новых найденных множеств этого типа на решетке 8×8 получены 28- и 36-элементные АР с низким уровнем боковых лепестков. Численный эксперимент показал, что путем небольших изменений структуры таких множеств можно добиться дальнейшего снижения уровня бокового излучения.
Розглядається питання щодо синтезу антенних решiток (АР) на базi адамаровських рiзницевих множин. З використанням нових знайдених множин цього типу на решiтцi 8×8 отриманi 28- та 36-елементнi АР з низьким рiвнем бокових пелюсткiв. Числовий експеримент показав, що шляхом незначних змiн у структурi таких множин можна домогтися подальшого зниження рiвня бокового випромiнювання.
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Радиофизика и радиоастрономия, 2008, т. 13, №2, с. 210-215
© L. E. Kopilovich, 2008
Array Antennas of Size 8×××××8 Based on Hadamard Difference Sets
L. E. Kopilovich
A. Ya. Usikov Institute of Radio Physics and Electronics, NAS Ukraine,
12, Acad. Proskura St., Kharkiv, 61085, Ukraine
E-mail: kopil@ire.kharkov.ua
Received March 13, 2008
The problem of synthesizing the optimized equi-amplitude array antenna (AA) on the 8 8× grid based
on a Hadamard difference set is considered. By using newly found sets of this type on the 8 8× grid
the 28- and 36-element AAs having a low sidelobe level are obtained.
A numerical experiment showed that by a small alteration of the structure of such a set, further
reduction of the AA sidelobe radiation is possible.
Introduction
The idea of using cyclic difference sets (DSs)
for synthesizing aperiodic linear array antennas
(AAs) with a low sidelobe level (SLL) was sug-
gested in [1], and it was generalized to the 2-D
case in [2]. Further investigation of this issue in
both linear and planar cases was described in
monograph [3]. In particular, there it was shown
that the basic DS class appropriate for the con-
struction of planar aperiodic AAs is that of Had-
amard difference sets (H sets [4]).
It will be mentioned that one can build the H
sets only on integer grids of definite sizes [5].
In each case there exist several inequivalent ver-
sions of such a set, where the number of the
inequivalent H sets has a tendency to grow with
the larger grid size. Each of these sets generates
an ensemble of equivalent H sets, which can be
obtained from the initial one by a definite proce-
dure (see below); and after finding in each of the
ensembles the set ensuring the minimum SLL for
the AA, the best of them is then chosen.
The difficulty lies in the fact that now the
complete collection of inequivalent H sets is built
only for the grids of sizes 4 4,× 6 6,× and 3 12;×
as for the grids of larger sizes, little is known
in this respect.
In the recent paper [6] several new inequiva-
lent H sets on square n n× grids with 8, 12, 16n =
and 24 were obtained, and the corresponding AAs
having the minimized SLL were synthesized.
To further this line, the knowledge of new
inequivalent H sets is required.
This paper continues investigation of the AAs
based on H sets on the 8 8× grid. In this case, the
number of the found inequivalent H sets is brought
up to twenty that enables obtaining an AA with
improved characteristics. Also, a numerical ex-
periment to clarify the possibility of further redu-
cing of the AA SLL when the structure of the
underlying H set undergoes small alterations
is carried out.
General Information on H Sets
By definition [7], a k-element DS { }( , )i ia b on
an integer x yν ×ν grid is such a set that any
nonzero node ( , )a b of the grid has exactly Λ
representations of the form
mod ,i j xa a a≡ − ν mod ,i j yb b b≡ − ν
where
( 1) ( 1) ,k k VΛ = − − .x yV = ν ⋅ν (1)
An H set is a DS whose parameters satisfy
the additial condition [8]
4( ).V k= − Λ (2)
Array Antennas of Size 8×8 Based on Hadamard Difference Sets
211Радиофизика и радиоастрономия, 2008, т. 13, №2
H sets exist on grids with [5]
2 , 3 2 ,n n
xν = ⋅ ,y xν = ν 1,n ≥ (3a)
and
1 12 , 3 2 ,n n
x
+ +ν = ⋅ 4,y xν = ν 1.n ≥ (3b)
(An H set on the s t× grid is equivalent to a
perfect binary array PBA ( )s t× [9] which repre-
sents a binary array of size s t× whose autocor-
relation function remains constant with all cyclic
shifts of its elements along the both grid sides [10].
Further, we make no distinction between an H set
and the corresponding PBA.)
It follows from (1)–(3) that the element num-
ber of an H set is determined by parameter V:
( )1 2.k V V= ± (4)
Note that with the “–” in (4) the fill factor of the
array 0.5.k Vβ = < The H set for which one “+”
is taken in (4) is called complementary. Its ele-
ments are placed in all grid nodes empty from
those of the first set, so its element number
is 1 .k k= ν −
For a square grid, ,x yν = ν = ν and
( 1) 2.k = ν ν ± Thus, H sets with 28 and 36 ele-
ments can be build on the 8 8× grid.
It is known [7] that if there exists a DS with
given parameters, then there also exists an en-
semble of equivalent sets having the same para-
meter values; in the 2-D case, these sets are ob-
tained from the initial one by a simultaneous
cyclic shifting all its elements along the grid sides,
and also by the grid automorphisms transforming
it by the formulas
1 11 12 ,i i ia c a c b= ⋅ + ⋅ 1 21 22 ,i i ib c a c b= ⋅ + ⋅ (5)
with 11 12 21 22, , ,c c c c integers 1 1( ,i ia b are then
taken modulo ,xν modulo ,yν respectively), pro-
vided the determinant of the equation system (5)
11 22 21 12Det c c c c= ⋅ − ⋅
is a number co-prime with the grid sidelengths [11].
In all known cases the 2-D H sets exist in a
number of inequivalent versions unobtainable one
from another by the described procedure. Each
of such sets possesses its own ensemble of the
equivalent ones, so the total number of H sets
with given parameters is proportional to that of
the inequivalent sets.
New Inequivalent H Sets
on the 8×××××8 Grid
For the 8 8× grid, two inequivalent H sets are
given in [12] and [13], and three more – in [6]. To
our knowledge, no more inequivalent H sets are
given in the literature.
Here, we found 15 new inequivalent H sets.
Several such sets are obtained by using the for-
mula [14]
2 , ,2 ,i r j i j rw u+ +=
where 0 ,i s≤ ≤ 0 ,j t≤ < 0 1,r≤ ≤ ( )ijU u= is
a PBA of size (2 ),s t× and ( )ijW w= is an array
of size (2 ) .s t× Under specific conditions, W is
also a PBA. The fulfilment of these conditions is
difficult to check, however, one can make sure by
direct verification that in our case (at 4,s = 8),t =
W is an H set on the 8 8× grid. This allows to
obtain H sets on this grid by using the known H
sets on the 4 16× grid, and choose the inequiva-
lent ones among them. As a main source for
obtaining H sets on the 8 8× grid by this method
a wealth of H sets on the 4 16× grid given in [15]
was used; so, 8 new inequivalent H sets were
found. Besides, one such set was obtained from
[12] in this way.
One more method consists in transforming an
H set { }( , )i ia b into the set { }1 1( , )i ia b on the grid
of the same size following the rule: if sum i ia b+
is an even number then 1 ,i ia a= 1 ,i ib b= whe-
reas in the opposite case its odd component de-
creases by 1 and its even component increases
by 1 (thus, in all cases, 1 1 ).i i i ia b a b+ = + As the
result, an H set is again obtained, and among
the sets obtained in such a way 6 new inequiva-
lent ones occured.
L. E. Kopilovich
212 Радиофизика и радиоастрономия, 2008, т. 13, №2
The found collection of the inequivalent H sets
is probably still incomplete, however, by using these
sets the AAs having lower SLL than those ob-
tained earlier were synthesized.
Searching Procedure for the AA
with Minimized SLL
The normalized power pattern of a planar
equiamplitude AA is
2
1
1( , ) exp ( ) ,
k
x y j x j y
j
F q q i a q b q
k =
⎡ ⎤= ⋅ + ⋅⎣ ⎦∑ (6)
where ,j ja b are now co-ordinates of an array
element, xq and yq are the generalized space coor-
dinates: 02 ( ) ,xq d l l= π − λ 02 ( ) ;yq d m m= π − λ
d is the distance between the adjacent nodes in an
array row or column; λ is the wavelength; l, m are
the cosines of the angles between the beam and
the axes x and y; and 0 0( , )l m is the direction
towards which the beam is pointed. In what fol-
lows, pattern (6) is optimized on the domain
( ,xq−π ≤ ≤ π 0 ),yq≤ ≤ π therefore, the results
obtained are valid for any values of d λ and 0 ,l
0;m without losing generality, one can take
2d = λ and 0 0 0.l m= =
When set { }( , )j ja b represents a DS, function
(6) takes the constant value 2( )cF k k= − λ over
the net of space points { }(2 , 2 ) ,s tπ ν π ν
0s t+ ≥ [4]. Therefore, we may expect that for
a square AA, F takes its maximum value near a
point of the net
{ }( , ) ;s tπ ν π ν ( 1), ..., 1, 0, 1, ..., 1;s = − ν − − ν −
(7)
under the conditions
Abs( ) 2s > or 2;t > ( )mod 4 0,s t⋅ > (8)
first of which means that only the sidelobe region
is considered, and the second one – that the points
at which cF F= are excluded.
The calculation was described in [6]; briefly, it
is as follows. An H set for which the quantity
maxFm F= (dB) over the space points net (7)
does not exceed some given M is found; then the
value of Fm over the net of double thickness is
found, and so on. The process of double thicke-
ning continues until the magnitude of Fm is sta-
bilized, coming to a certain quantity 0.F
Further, an H set giving a value of Fm over net
(7), (8) which is less than that for the initial set is
searched for, and the described procedure gi-
ving now a new value of 0F is repeated; if this
value is smaller than the preceding one, such set
is stored, and so on.
Note that a smaller value of Fm does not in-
evitably lead to a smaller value of 0;F therefore,
to diminish the probability of omitting the real mi-
nimum value of the latter over the whole consi-
dered ensemble of H sets, the described process
was periodically repeated, beginning in each case
from a new H set of the ensemble.
The calculation was carried out through the
ensembles corresponding to all known inequiva-
lent H sets on the 8 8× grid. As the result, the
28-element AA having the SLL –12.59 dB, and
the 36-element AA with the SLL –13.86 dB were
synthesized. The power patterns of these AAs,
together with the schemes of their element ar-
rangement are given in Figs. 1 and 2.
On the Possibility of Further SLL
Reduction
Recently, the so called genetic algorithms were
applied to the synthesis of linear AAs in a number
of papers, including those where cyclic DSs as
initial ones were taken [16]. In some cases, one
can pass from a linear grid to a rectangular one
having coprime sidelengths, thus obtaining a pla-
nar AA.
In this way, it is impossible to obtain a square
AA. However, in this case one can search for an
AA with a still smaller SLL by slightly altering the
underlying H set, e. g., by shifting a few of its
elements from their places to other grid nodes,
and then calculating through the ensemble of sets
obtainable from such a set by the same described
procedure.
Array Antennas of Size 8×8 Based on Hadamard Difference Sets
213Радиофизика и радиоастрономия, 2008, т. 13, №2
We made a numerical experiment using inequiv-
alent H sets having a structure of the form shown
in Fig. 3. Here, in three rows of the grid there are
two set elements, six elements in one row and four
elements in each of the rest rows. While making by
turns all possible element shifting in the rows con-
taining only two elements of the set, one obtains, in
each case, a “disturbed” H set, and when taking
this set and that complementary to it as the initial
ones, we obtain, by using the aforesaid procedure,
the ensembles of 28- and 36-element sets.
The calculation through these set ensembles
has shown that in this way the AAs with smaller
SLLs can be synthesized. As an example, a “dis-
turbed” 36-element set ensuring the SLL –14.43
dB is shown in Fig. 4.
Fig. 1. The power pattern of the optimized 28-element AA (a), and the scheme of its element arrangement (b)
Fig. 2. The power pattern of the optimized 36-element AA (a), and the scheme of its element arrangement (b)
L. E. Kopilovich
214 Радиофизика и радиоастрономия, 2008, т. 13, №2
Conclusion
The collection of the inequivalent H sets found
also allows obtaining optimized AAs of larger
sizes. Thus, by the method suggested in [14, 15]
and somewhat simplified in [3], when having an
H set on the s t× grid, one can obtain the H set
on the (2 ) (2 )s t× grid. In this way the H sets on
the 16 16× grid can be built, and on their base
the AAs with a large number (120 and 136) of
elements having a low SLL synthesized, that be-
comes pressing in the development of modern
radio telescopes [17].
References
1. D. G. Leeper, “Thinned periodic antenna arrays with
improved peak sidelobe level control”, U.S. Patent
4071848, Jan. 31, 1978.
2. L. E. Kopilovich and L. G. Sodin, “Two-dimension-
al aperiodic antenna arrays with a low sidelobe lev-
el”, IEE Proc., pt. H, vol. 138, No.3, pp. 233-237,
1991.
3. L. E. Kopilovich and L. G. Sodin, Multielement
System Design in Astronomy and Radio Science.
Dordrecht/Boston/ London: Kluwer Academic Pub-
lishers, Astrophysics and Space Science Library,
vol. 268, 2001, 190 p.
4. R. J. Turyn, “Character sums and difference sets”,
Pacific J. Math., vol. 15, No.1, pp. 319-346, 1965.
5. P. Wild, “Infinite families of perfect binary arrays”,
Electron. Lett., vol. 24, No.14, pp. 845-847, 1988.
6. L. E. Kopilovich, “Square array antennas based on
Hadamard difference sets”, IEEE Trans. Antennas
Propag., vol. AP-56, pp. 263-266, Jan. 2008.
7. M. Hall Jun., Combinatorial Theory, 2nd ed, New
York: Wiley, 1986.
8. P. K. Menon, “On difference sets whose parameters
satisfy a certain relation”, Proc. Am. Math. Soc.,
vol. 13, No.5, pp. 739-745, 1962.
9. Y. K. Chan, M. K. Siu and P. Tong, “Two-dimensional
binary arrays with good autocorrelation”, Information
and Control, vol. 42, pp. 125-130, 1979.
10. D. Calabro and J. K. Wolf, “On the synthesis of
two-dimensional arrays with desirable correlation
properties”, Information and Control, vol. 11,
pp. 537-560, 1968.
11. A. G. Kurosh, Theory of Groups, New York: Chelsea,
1958.
12. D. Jungnickel and A. Pott, “Abelian difference sets”,
in The CRC Handbook of Combinatorial Designs,
Ch. D. Colbourn and J. H. Dinitz, Eds. Roca Baton:
CRC Press, 1996, pp. 15-64.
13. L. Bömer and M. Antweiler, “Two-dimensional per-
fect binary arrays with 64 elements”, IEEE Trans. Inf.
Theory, vol. IT-36, No.2, pp. 411-414, 1990.
14. J. Jedwab and C. Mitchell, “Constructing new per-
fect binary arrays”, Electron. Lett, vol. 24, No.11, pp.
650-652, 1988.
15. K. T. Arasu and J. Reis, “On Abelian group of order
64 that have difference sets”. Tech. Rept, No. 1987.10,
Dept of Math. and Stat. of Wright State University,
Dayton, 1987, 12 p.
16. S. Caorsi, A. Lommi, A. Massa and M. Pastorino,
“Peak sidelobe level reduction with a hybrid ap-
proach based on GAs and difference sets”, IEEE
Trans. Antennas Propag., vol. 52, pp. 1116-1121,
Apr. 2004.
17. L. Kogan, “Optimizing a large array configuration to
minimize the sidelobes”, IEEE Trans. Antennas
Propag, vol. 48, pp. 1075-1078, July 2000.
Fig. 3. The scheme of the element arrangement in the
H sets used in the numerical experiment
Fig. 4. The 36-element “disturbed” H set ensuring the
reduced SLL of –14.43 dB
Array Antennas of Size 8×8 Based on Hadamard Difference Sets
215Радиофизика и радиоастрономия, 2008, т. 13, №2
Антенные решетки размера 8×××××8
на основе адамаровских разностных
множеств
Л. Е. Копилович
Рассматривается вопрос синтеза антенных
решеток (АР) на основе адамаровских разно-
стных множеств. С использованием новых
найденных множеств этого типа на решетке
8 8× получены 28- и 36-элементные АР с низ-
ким уровнем боковых лепестков.
Численный эксперимент показал, что пу-
тем небольших изменений структуры таких
множеств можно добиться дальнейшего сни-
жения уровня бокового излучения.
Антеннi решiтки розмiру 8×××××8
на базi адамаровських рiзницевих
множин
Л. Ю. Копилович
Розглядається питання щодо синтезу ан-
тенних решiток (АР) на базi адамаровських
рiзницевих множин. З використанням нових
знайдених множин цього типу на решiтцi 8 8×
отриманi 28- та 36-елементнi АР з низьким
рiвнем бокових пелюсткiв.
Числовий експеримент показав, що шляхом
незначних змiн у структурi таких множин мож-
на домогтися подальшого зниження рiвня бо-
кового випромiнювання.
|
| id | nasplib_isofts_kiev_ua-123456789-8398 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-9636 |
| language | English |
| last_indexed | 2025-12-07T16:28:38Z |
| publishDate | 2008 |
| publisher | Радіоастрономічний інститут НАН України |
| record_format | dspace |
| spelling | Kopilovich, L.E. 2010-05-28T10:07:15Z 2010-05-28T10:07:15Z 2008 Array Antennas of Size 8×8 Based on Hadamard Difference Sets / L.E. Kopilovich // Радиофизика и радиоастрономия. — 2008. — Т. 13, № 2. — С. 210-215. — Бібліогр.: 17 назв. — англ. 1027-9636 https://nasplib.isofts.kiev.ua/handle/123456789/8398 The problem of synthesizing the optimized equi-amplitude array antenna (AA) on the 8×8 grid based on a Hadamard difference set is considered. By using newly found sets of this type on the 8×8 grid the 28- and 36-element AAs having a low sidelobe level are obtained. A numerical experiment showed that by a small alteration of the structure of such a set, further reduction of the AA sidelobe radiation is possible. Рассматривается вопрос синтеза антенных решеток (АР) на основе адамаровских разностных множеств. С использованием новых найденных множеств этого типа на решетке 8×8 получены 28- и 36-элементные АР с низким уровнем боковых лепестков. Численный эксперимент показал, что путем небольших изменений структуры таких множеств можно добиться дальнейшего снижения уровня бокового излучения. Розглядається питання щодо синтезу антенних решiток (АР) на базi адамаровських рiзницевих множин. З використанням нових знайдених множин цього типу на решiтцi 8×8 отриманi 28- та 36-елементнi АР з низьким рiвнем бокових пелюсткiв. Числовий експеримент показав, що шляхом незначних змiн у структурi таких множин можна домогтися подальшого зниження рiвня бокового випромiнювання. en Радіоастрономічний інститут НАН України Антенны, волноводная и квазиоптическая техника Array Antennas of Size 8×8 Based on Hadamard Difference Sets Антенные решетки размера 8×8 на основе адамаровских разностных множеств Антеннi решiтки розмiру 8×8 на базi адамаровських рiзницевих множин Article published earlier |
| spellingShingle | Array Antennas of Size 8×8 Based on Hadamard Difference Sets Kopilovich, L.E. Антенны, волноводная и квазиоптическая техника |
| title | Array Antennas of Size 8×8 Based on Hadamard Difference Sets |
| title_alt | Антенные решетки размера 8×8 на основе адамаровских разностных множеств Антеннi решiтки розмiру 8×8 на базi адамаровських рiзницевих множин |
| title_full | Array Antennas of Size 8×8 Based on Hadamard Difference Sets |
| title_fullStr | Array Antennas of Size 8×8 Based on Hadamard Difference Sets |
| title_full_unstemmed | Array Antennas of Size 8×8 Based on Hadamard Difference Sets |
| title_short | Array Antennas of Size 8×8 Based on Hadamard Difference Sets |
| title_sort | array antennas of size 8×8 based on hadamard difference sets |
| topic | Антенны, волноводная и квазиоптическая техника |
| topic_facet | Антенны, волноводная и квазиоптическая техника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/8398 |
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