Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna
The maximum number of elements of a nonredundant configuration on a square array antenna is estimated empirically employing the investigated structure of differences between the elements of the configuration mapping onto the scan. Отримано емпіричну оцінку максимальної кількості елементів безнадлишк...
Gespeichert in:
| Datum: | 2009 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine,
2009
|
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/59641 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine, / L. E. Kopilovich // Радиофизика и радиоастрономия. — 2009. — Т. 14, № 3. — С. 183–187. — Бібліогр.: 10 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860055337371959296 |
|---|---|
| author | Kopilovich, L.E. |
| author_facet | Kopilovich, L.E. |
| citation_txt | O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine, / L. E. Kopilovich // Радиофизика и радиоастрономия. — 2009. — Т. 14, № 3. — С. 183–187. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| description | The maximum number of elements of a nonredundant configuration on a square array antenna is estimated empirically employing the investigated structure of differences between the elements of the configuration mapping onto the scan.
Отримано емпіричну оцінку максимальної кількості елементів безнадлишкової конфігурації на квадратній антенній решітці, що грунтується на вивченні структури різниць між елементами її відображення на розгортці.
Получена эмпирическая оценка максимального числа элементов безызбыточной конфигурации на квадратной антенной решетке, основанная на изучении структуры разностей между элементами ее отображения на развертке.
|
| first_indexed | 2025-12-07T17:00:42Z |
| format | Article |
| fulltext |
Радиофизика и радиоастрономия, 2009, т. 14, №2, с. 183-187
© L. E. Kopilovich, 2009
Empirical Estimate for the Maximum Element Number
of a Nonredundant Configuration on Square Array Antenna
L. E. Kopilovich
O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine,
12, Acad. Proskury St., Kharkiv, 61085, Ukraine
E-mail: kopil@ire.kharkov.ua
Received August 18, 2008
The maximum number of elements of a nonredundant configuration on a square array antenna is
estimated empirically employing the investigated structure of differences between the elements of the
configuration mapping onto the scan.
Introduction
The problem of constructing a nonredundant
configuration (NRC) of elements is of urgent
necessity for radio interferometry. Here, a mo-
mentous problem is how to build the NRC with
the maximum number of elements to obtain the
maximum number of interferometer baselines.
Such NRCs on square n n× grids (so called
“Golomb squares” [1]) with the maximum possi-
ble elements are found at 22n ≤ in a number
of papers [1-7]. At the same time, the size of
modern array antenna, as well as the number
of its elements, can substantially exceed these
values [8] that requires the elaboration of meth-
ods for building NRCs on large grids.
Such NRCs were obtained in [4] with using
combinatorial constructions – cyclic difference
sets (see also [9]). However, it remains unclari-
fied to what degree the element number of the
NRC thus found is close to a maximum.
The rigorous estimates for the NRC element
number on a square grid were obtained in [1, 3],
however, the former is too overrated, while for
applying the latter the information on NRCs
on linear grids of large lengths, unavailable to date,
is required. In this connection, the obtainment
of the upper estimate based on available empirical
data is of interest.
Analysis of the mapping structure
of the NRC on the square grid scan
Consider an arbitrary k-element NRC on the
n n× grid and analyze the system of vector differ-
ences between its elements (see Fig. 1). Number
the cells of the first grid row from 1 to n,
of the second one – from 1n + to 2n, and so on.
Fig. 1. An example of a nonredundant configuration
(NRC) on the square grid. The NRC elements are
denoted with bold dots. The vectors connecting
the elements are pointed rigthwards or strictly
downwards (type I) or leftwards (type II)
L. E. Kopilovich
184 Радиофизика и радиоастрономия, 2009, т. 14, №2
The vector connecting the cells can be pointed
rightwards or strictly downwards (type I), or left-
wards (type II). When scanning the grid, the NRC
on it passes to a point system on the segment, and
the differences between these points can be either
one-fold (i. e. non-repeated) or two-fold ones.
Obviously, vector differences on a 2-D grid be-
longing to the same type cannot pass to equal dif-
ferences on the scan, while only some part of these
belonging to the opposite types passes to the equal
ones. Thus, the share of the one-fold differences
on the scan can be expected to exceed half of their
total number ( 1) 2.K k k= − Besides, it should
be taken into account that the NRC placed
on a square grid, the latter being rotated through
90 ,° would give an alternative point system on the
scan (see Fig. 2). In addition, for most of n values,
at least, several different NRCs with a maximum
number of elements can be built [7]. This allows
suggestion that for almost all values of n a maxi-
mum-element NRC on the n n× grid can be built
for which the share (α) of the non-repeated differ-
ences on the scan exceeds 50 % of their total
number,
50 %.α > (1)
These qualitative considerations were verified
by using the available data. The n n× grids
at 3 22n≤ ≤ with the NRCs found in [4, 6, 7]
placed on them were scanned in two variants (as
in Fig. 2), and it was found out that for all n in
this range, except for 7,n = this suggestion was
validated at least in one of such variants (see
Table 1).
Now we will seek the upper estimate for the
NRC element number under the assumption that
for 7n > a maximum-element NRC can be found
for which condition (1) is fulfilled.
Upper estimate
of the NRC element number
Consider a k-element NRC placed on the n n×
grid. When scanning the grid, the NRC elements
pass to a point sequence
1 2 ... ka a a< < < (2)
on the scan.
Here, we apply the approach suggested for
the linear case [10]. The order of differences
Fig. 2. A grid with an NRC placed on it, and its scan (a). The same grid rotated through 90°, and its scan (b).
The share of non-repeated differences between the points on the scan is denoted with α (%)
Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna
185Радиофизика и радиоастрономия, 2009, т. 14, №2
between elements ja and ia of sequence (2),
,j i> is the difference ,j i− at this point, the
sum of differences of the first order is
2
2 1 1( ) ... ( ) ,k ka a a a n−− + + − <
and similarly, the sum of differences of order ν is
2
1 1( ) ... ( ) .k ka a a a nν+ −ν− + + − < ν
It follows that the sum of differences of orders up
to t is
2 ( 1) 2.tS n t t< + (3)
On the other hand, the number of these differen-
ces is
( 1) ... ( ) ( 1) 2 ,k k t kt t t ts− + + − = − + = (4)
where ( 1) 2, 1 .s k t t k= − + ≤ ≤
Should the differences between the elements
on the scan be all distinct then one might write
1 ... ,tS ts≥ + +
and comparing the last inequality with (3) find the
estimate of the maximum element number. Vir-
tually, in such a way the estimate for the maxi-
mum element number of an NRC on a linear grid
can be obtained [10].
In our case, some part of differences on the
scan are two-fold, but using condition (1) and with
the Eq. (4) considered, we may write
( ) ( )2 1 ... [ / 4] [ / 4] 1 ... [3 / 4],tS ts ts ts≥ + + + + + +
(5)
where [c] means integer part of c. If one writes
4 ,ts r= + β with the r integer and 0, 1, 2β = or 3,
inequality (5) can be rewritten in the form
2(1 ... ) ( 1) ... (4 ).tS r r r r≥ + + + + + + + β −
When performing here the summation opera-
tions and returning again to the ts one obtains
2 2 25 1 1 5( ) ( ) .
16 8 2 16 16tS ts ts tsβ⎛ ⎞≥ + + + β >⎜ ⎟⎝ ⎠
Further, one correlates the last inequality with
(3) and obtains:
2
25 ( 1)( ) ,
16 2
t t nts +<
whence
n k m K a (%)
3 5 6 10 60.0
4 6 11 15 73.3
5 8 16 28 57.4
6 9 24 36 66.7
7 11 27 55 49.1
8 12 34 66 51.5
9 13 45 78 56.7
10 15 53 105 50.5
11 16 64 120 53.3
12 17 82 136 60.3
13 18 87 153 56.9
14 19 101 171 59.1
15 21 112 210 53.3
16 22 125 231 54.1
17 23 149 253 58.9
18 24 168 276 60.9
19 25 184 300 61.3
20 26 198 325 60.9
22 29 244 406 60.1
Table 1. The characteristics of the maximum-element
NRCs and point sequences obtained from them when
scanning the location grids. Here, n is the grid side-
length, k is the number of the NRC elements, m is the
number of non-repeated differences between points
on the scan, and α is their share in the total number
of differences equal to K k( k 1) 2−=
L. E. Kopilovich
186 Радиофизика и радиоастрономия, 2009, т. 14, №2
( )1 1 1 1 (2 ) ,s nb t nb t< + ≅ +
with 8 5 1.265,b = =
and
( )( 1) 2 1 1 (2 ) ( ).k t nb t f t< + + + =
Function ( )f t has its minimum at ,t nb=
thus
1.265 1.124 0.5k n n< + + (6)
or ,ek k≤ where ek is the integer part of the
expression in the right-hand side of (6).
Results
Table 2 shows the maximal found element
numbers of NRCs placed on n n× grids at 7n >
taken from [7], in comparison with their estimates
as obtained by (6) and those known from litera-
ture. It can be seen that the values of ek are
smaller than the estimates given in [1]; as for the
estimates in [3], obtained only for 11,n ≤ in this
range they roughly equal to ours.
As is seen from Table 2, there is a “reserve”
kept in the discrepancy between ek and the cor-
responding value of k. Apparently, owing to this
estimate (6) is valid also for the case of a possible
value of n when condition (1) is not fulfilled.
The comparison of the estimates for large
arrays given by (6) with the results obtained in [4]
(see Table 3) shows that the number of NRC
elements given by the suggested method, though
not maximum, is nevertheless wholly acceptable
for feasible purposes.
Conclusion
The upper estimate obtained for the element
number of an NRC on a square grid is more ef-
ficient than those earlier known. It can serve a
guiding line when building a large size array an-
tenna with the maximum NRC element number.
References
1. J. P. Robinson, “Golomb rectangles”, IEEE Trans,
Vol. IT-31, No. 6, pp. 781-787, 1985.
2. J. B. Shearer, “Some new Golomb rectangles”, Elect-
ron. J. Comb., Vol. 2, No. R12, 1995.
3. J. P. Robinson, “Golomb rectangles as folded rulers”,
IEEE Trans, Vol. IT-43, No. 1, pp. 290-293, 1997.
4. L. E. Kopilovich, “Nonredundant apertures for opti-
cal interferometric systems: maximization of the num-
Table 3. The element numbers in the NRCs on large
grids [6], and their estimates with (6)
n k ek
25 30 37
30 35 44
40 47 58
50 58 71
Table 2. The estimates of the maximum number
of the NRC elements on square grids. Here, n and k
are the same as in Fig. 1, ek is estimated with (6), rk
and r1k are the estimates of k taken from [1] and
[3], respectively
n k ek rk 1rk
8 12 13 13 13
9 13 15 15 14
10 15 16 17 16
11 16 18 19 18
12 17 19 21 –
13 18 21 23 –
14 19 22 25 –
15 21 23 27 –
16 22 25 28 –
17 23 26 29 –
18 24 28 31 –
19 25 29 33 –
20 26 30 34 –
21 27 32 – –
22 29 33 – –
Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna
187Радиофизика и радиоастрономия, 2009, т. 14, №2
ber of elements”, J. Mod. Opt., Vol. 45, No. 11,
pp. 2417-2424, 1998.
5. J. P. Robinson, “Genetic search for Golomb ar-
rays”, IEEE Trans, Vol. IT-46, No. 6, pp. 781-787,
2000.
6. Yu. V. Kornienko, “Construction of nonredundant
antenna configurations on square grids by a ran-
dom search technique”, Telecommunications and
Radio Engineering, Vol. 57, Nos. 2&3, pp. 23-30,
2002.
7. J. B. Shearer “Symmetric Golomb squares”, IEEE
Trans, Vol. IT-50, No. 8, pp. 1846-1847, 2004.
8. R. Schultz, “Radio astronomy antennas by the Thou-
sands”, Experimental Astronomy, Vol. 17, pp. 119-139,
2004.
9. L. E. Kopilovich and L. G. Sodin, “Multielement
System Design in Astronomy and Radio Science”,
vol. 268, Dordrecht/Boston/London: Kluwer Acade-
mic Publishers, Astrophysics and Space Science
Library, 2001, 190 p.
10. B. Lindström, “On inequality for B-sequences”,
J. Combin. Theory, Vol. A6, No. 2, pp. 211-212,
1969.
Эмпирическая оценка максимального
числа элементов безызбыточной
конфигурации на квадратной антенной
решетке
Л. Е. Копилович
Получена эмпирическая оценка максималь-
ного числа элементов безызбыточной конфигу-
рации на квадратной антенной решетке, осно-
ванная на изучении структуры разностей между
элементами ее отображения на развертке.
Емпірична оцінка максимальної
кількості елементів безнадлишкової
конфігурації на квадратній антенній
решітці
Л. Ю. Копилович
Отримано емпіричну оцінку максимальної
кількості елементів безнадлишкової конфігу-
рації на квадратній антенній решітці, що грун-
тується на вивченні структури різниць між
елементами її відображення на розгортці.
|
| id | nasplib_isofts_kiev_ua-123456789-59641 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| language | English |
| last_indexed | 2025-12-07T17:00:42Z |
| publishDate | 2009 |
| publisher | O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine, |
| record_format | dspace |
| spelling | Kopilovich, L.E. 2014-04-09T15:34:37Z 2014-04-09T15:34:37Z 2009 O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine, / L. E. Kopilovich // Радиофизика и радиоастрономия. — 2009. — Т. 14, № 3. — С. 183–187. — Бібліогр.: 10 назв. — англ. https://nasplib.isofts.kiev.ua/handle/123456789/59641 The maximum number of elements of a nonredundant configuration on a square array antenna is estimated empirically employing the investigated structure of differences between the elements of the configuration mapping onto the scan. Отримано емпіричну оцінку максимальної кількості елементів безнадлишкової конфігурації на квадратній антенній решітці, що грунтується на вивченні структури різниць між елементами її відображення на розгортці. Получена эмпирическая оценка максимального числа элементов безызбыточной конфигурации на квадратной антенной решетке, основанная на изучении структуры разностей между элементами ее отображения на развертке. en O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine, Антенны, волноводная и квазиоптическая техника Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna Емпірична оцінка максимальної кількості елементів безнадлишкової конфігурації на квадратній антенній решітці Эмпирическая оценка максимального числа элементов безызбыточной конфигурации на квадратной антенной решетке Article published earlier |
| spellingShingle | Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna Kopilovich, L.E. Антенны, волноводная и квазиоптическая техника |
| title | Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna |
| title_alt | Емпірична оцінка максимальної кількості елементів безнадлишкової конфігурації на квадратній антенній решітці Эмпирическая оценка максимального числа элементов безызбыточной конфигурации на квадратной антенной решетке |
| title_full | Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna |
| title_fullStr | Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna |
| title_full_unstemmed | Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna |
| title_short | Empirical Estimate for the Maximum Element Number of a Nonredundant Configuration on Square Array Antenna |
| title_sort | empirical estimate for the maximum element number of a nonredundant configuration on square array antenna |
| topic | Антенны, волноводная и квазиоптическая техника |
| topic_facet | Антенны, волноводная и квазиоптическая техника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/59641 |
| work_keys_str_mv | AT kopilovichle empiricalestimateforthemaximumelementnumberofanonredundantconfigurationonsquarearrayantenna AT kopilovichle empíričnaocínkamaksimalʹnoíkílʹkostíelementívbeznadliškovoíkonfíguracíínakvadratníiantenníirešítcí AT kopilovichle émpiričeskaâocenkamaksimalʹnogočislaélementovbezyzbytočnoikonfiguraciinakvadratnoiantennoirešetke |