# Directional pattern of an antenna array with non-equidistant arrangement of receiving elements

### Abstract

This article presents an analysis of the directional pattern of an antenna array with a non-equidistant arrangement of receiving elements. This problem is relevant for the synthesis of antenna arrays with the required location of zeros and maxima in the radiation pattern. The analysis is suitable for obtaining an analytical expression describing the directional pattern of a flat phased antenna array, the receiving elements of which are located on the antenna curtain.

#### For citation:

Zimina S.V., Frantsev M.E. Directional pattern of an antenna array with non-equidistant arrangement of receiving elements. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2020;(4):15-21.
https://doi.org/10.38013/2542-0542-2020-4-15-21

The directional pattern (DP) of is one of the basic characteristics of an antenna array. Making an analytical description of the directional pattern is relevant in development of novel radars, since it makes it possible to synthesise antenna systems with the previously known arrangement of maxima and nulls determined by the character of problems to be solved. It is considered of practical importance to develop a multi-channel selfinterference canceller (MIC) with the use of compensating channels (CC) formed from randomly selected receiving elements of a phased antenna array (PAA). Such configuration of the antenna array (AA) allows to ensure AA curtain multifunctionality and form CC with spatially distributed phase centres, which improves adaptation of such self-interference canceller to different spatial distribution of interference signals and facilitates their more effective suppression.

To solve the set task, an option was selected where antenna elements are arranged on a certain conventional rectangular AA curtain. The objective of this paper is to derive a general formula for the directional pattern of a flat antenna array with non-equidistant arrangement of its elements.

## 1. Problem statement

Let us consider a phased antenna array with the element layout diagram given in Fig. 1. On account of practical implementation of subarrays, the arrangement of AA receiving elements across the antenna curtain is non-equidistant. For each one of the four element blocks being considered, arranged along the four sides of the antenna curtain, there is its particular initial distance from the antenna phase centre ({y_{0H}, y_{0B}}, {z_{0Л}, z_{0П}}). The distances between the horizontally and vertically arranged elements are equal to {y_{1H}, y_{1B}} and {z_{1Л}, z_{1П}}, respectively. In this way we obtain an analytical formula of the antenna directional pattern for the compensating channels.

**Fig. 1**. Layout diagram of phased antenna array receiving elements

## 2. Selection of coordinate systems and signal incidence angles

To calculate directional patterns, we shall assume that axis z is directed along the antenna curtain vertical side and axis y is directed along its lower horizontal side. Axis x is positioned square to the

antenna curtain. In selecting directions of the axes it was assumed that they form a right-hand threesome. The origin of the coordinates is located in the lower left corner of the antenna curtain. We shall assume that the antenna array phase centre is located in the origin of the coordinates.

Next, we select signal incidence angles. Let azimuth angle of signal incidence φ be reckoned from axis Х and be changing in the horizontal plane within the interval of [–π/2; π/2]. Let elevation angle θ be reckoned from axis Х and be changing in the vertical plane within the interval of [–π/2; π/2]. Negative angle values define the left-hand and lower semiplanes, and positive ones – the right-hand and upper semiplanes, respectively. Direction φ = 0; θ = 0 coincides with axis Ox, positioned along the normal to the antenna array plane.

## 3. Power directional pattern of an antenna array with non-equidistant arrangement of receiving elements

Directional pattern (array factor) of the AA CC in the horizontal direction has the view [1, 2]:

Here, N_{Г}^{ААР} = N_{ГН} + N_{ГВ} – number of horizontally arranged AA CC elements, N_{ГН} – number of elements in the lower element line, NГВ – number of elements in the upper element line), ψ_{Гi} – phase progression on the i-th horizontal element.

Since there are two horizontal element lines, then row (1) also consists of two parts, which define phase progression on those element lines:

– signal phase progression on the lower and upper horizontal element lines, respectively.

Substituting (3) in (2) and making transformations, we have:

and calculating sum values in (4), we obtain a formula of the azimuth angle array factor for a non-equidistant antenna array:

Making similar calculations, we can obtain an expression for the elevation angle AA factor:

Here:

Expressions (9)–(10) define phase progressions on the left- and right-hand vertical element lines.

Multiplying formula (7) by (8) and calculating modulus of the obtained complex expression, we obtain a view of the power directional pattern for an antenna array with non-equidistant arrangement of elements:

It can be seen from the structure of formula (11) that the directional pattern consists of the product of directional patterns of the horizontally and vertically positioned elements (two curly brackets).

Let us consider some special cases.

1. The antenna array is equidistant and symmetrical, with equal distances between the elements and equal number of elements in the upper and lower blocks of horizontally arranged elements and the left-hand and right-hand blocks of vertically arranged elements, respectively:

Under these assumptions, we have from formula (11):

Expression (12) defines DP of an equidistant flat antenna array [1], which is a special case of a non-equidistant PAA. In this way, general expression (11) for the directional pattern of a non-equidistant antenna array, under the assumptions defining an equidistant antenna, comes down to the classical formula of directional pattern described in text books.

2. Let us consider special cases of a nonequidistant PAA. The antenna array is symmetrical and non-equidistant, with equal number of elements and equal distances between all horizontally arranged and vertically arranged elements, respectively. The distances between the element lines and the phase centre are different.

In this case antenna power DP has the view:

It follows from expression (13) that the directional pattern of a non-equidistant symmetrical antenna array is similar to the DP of an equidistant antenna, but there is an extra number of local minima appearing in it. Depending on the tasks to be solved by AA in a radar system, it is possible to shape a DP with the required number of either maxima or nulls.

Application of this theory is especially relevant if in the process of signal processing, depending on the current jamming environment a respective loop is selected from the elements of a flat non-equidistant AA to enable formation of nulls in the directions towards currently active interference signals.

## 4. Use of directional pattern formulas for forming antenna array DP nulls

In principle, there exist quite a few methods to form nulls in selected directions using theoretical formulas (11) and (13) for directional patterns; however, in each case the selected solutions will feature ‘side effects’ in the form of remaining ‘excess’ nulls or maxima. In order to accurately solve the task set, it is necessary to arrange antenna receiving elements in a non-equidistant way not only along the array curtain perimeter but also across the entire curtain. Then, due to an increased number of adjustable distances between the elements, a theoretical possibility appears to form the DP maxima and nulls in any directions.

The obtained formulas will be illustrated by two examples.

1. Suppose, it is necessary to synthesise an AA with DP maxima in elevation in directions from 10º to 60º with a 10º interval. In construction, a 16-element antenna array was considered, containing four elements in each one of the two horizontal and two vertical element lines (N_{Г} = N_{В} = 4). The difference in distances from phase null to the left-hand and right-hand vertical element lines was z_{0Л} – z_{0П} = k_{1λ} – k_{2λ} = 5,737 · λ. The distances between the receiving elements of the antenna vertical element lines were z_{1} = 11.474 · λ. Wavelength λ = 104 mm. Given in Fig. 2 is a chart of antenna array directional pattern F(dB) vs. elevation angle θ. It can be seen from the figure that in the interval of elevation angles θ ∈ [0; π/2] the directional pattern has 5 main maxima θ_{max} ≈ 10°; 20°; 31°; 44°; 60°, which coincide, with an accuracy from several thousandth of degree to several degrees, with the maxima that had to be formed from the outset.

**. Antenna array normalised directional pattern F (dB) vs. elevation angle θ (synthesis case of a DP having maxima in directions from 10 to 60 degrees with a 10-degree interval)**

Fig. 2

Fig. 2

2. Suppose, it is necessary to synthesise an AA with DP nulls in directions from 6º to 70º with a 12º interval and with the maximum of 60º in elevation. In construction, a 16-element antenna array was considered, containing four elements in each one of the two horizontal and two vertical element lines (N_{Г} = N_{В} = 4). The difference in distances from phase null to the left-hand and right-hand vertical element lines was z_{0Л} – z_{0П} = = 4.783 · λ. The distances between the receiving

elements of the antenna vertical element lines were z_{1} = 1.155 · λ. Wavelength λ = 104 mm. Given in Fig. 3 is a chart of antenna array directional pattern F(dB) vs. elevation angle θ. It can be seen from the figure that in the interval of elevation angles θ ∈ [0; π/2] the directional pattern has 5 required nulls and 2 ‘excess’ nulls in directions θ_{min} ≈ 6°; 18°; 26°; 31°; 40°; 47°; 70° and a maximum in the direction of 60º.

**Fig. 3**. Antenna array normalised directional pattern F (dB) vs. elevation angle θ (synthesis case of a DP having nulls in directions from 6 to 70 degrees with a 12-degree and maximum in direction of 60 degrees)

In this way, by using different loops with elements of a flat PAA for forming nulls depending on the current jamming environment, it can be possible to obtain antenna array directional patterns of different shape, thus enabling suppression of jamming signals in a non-adaptive way.

## Conclusion

The presented paper offers a general formula of the directional pattern of an antenna array with non-equidistant arrangement of receiving elements. This formula allows to analytically construct a directional pattern with the necessary number of maxima and nulls without resorting to computer simulation. The non-equidistance feature makes it possible to form additional nulls of the directional pattern. An analysis performed allows to draw the following summary:

1. The formula for directional pattern of a non-equidistant antenna array is a more general one as compared to the classical expression defining DP of an equidistant flat antenna array and provides a possibility to consider special cases of non-equidistant antenna arrays with different number of receiving elements and different distances between them.

2. In a non-equidistant symmetrical antenna array with equal number of receiving elements at the opposite sides of the antenna perimeter and different initial phase progression for each perimeter side, the directional pattern differs from the DP of an equidistant antenna by the presence of additional nulls, which makes it possible to arrange them in additional directions where suppression of jamming signals is required.

3. For better accuracy in the synthesis of antenna array directional pattern extrema, it is necessary to select finer spacing for their arrangement, since it reduces distortions introduced by the inverse sine function.

4. This approach and the formulas given in the paper allow to simulate any antenna array with arbitrary spatial arrangement of its elements.

## References

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2. Монзинго Р. А., Миллер Т. У. Адаптивные антенные решетки: Пер. с англ. Под ред. В. А. Лексаченко. М.: Радио и связь, 1986. 448 с.

3. Горелик Г.С. Колебания и волны. Введение в акустику, радиофизику и оптику. 3-е изд. Под ред. С.М. Рытова. М.: Физматлит, 2007. 656 с.

### About the Authors

**S. V. Zimina**Russian Federation

Zimina Svetlana Valerievna - Cand. Sci. (Phys.-Math.), Assistant Professor, Senior Researcher. Research interests: phased antenna arrays, signal processing, adaptive antenna arrays.

Nizhny Novgorod

**M. E. Frantsev**Russian Federation

Frantsev Michael Evtifeevich - Deputy Department Head. Research interests: phased antenna arrays, active interference protection, primary signal processing, radar.

Nizhny Novgorod#### For citation:

Zimina S.V., Frantsev M.E. Directional pattern of an antenna array with non-equidistant arrangement of receiving elements. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2020;(4):15-21.
https://doi.org/10.38013/2542-0542-2020-4-15-21