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Using a phased antenna array with control coupling for nulls forming in the radiation pattern

https://doi.org/10.38013/2542-0542-2020-3-6-17

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Abstract

A simple nulls forming method in the radiation pattern of a phased antenna array considered. Presented the basic calculation ratios confirmed by the simulation results are. The possibility of applying the method to form extended nulls in the radiation pattern is shown.

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Egorov A.D., Yashenkov A.O. Using a phased antenna array with control coupling for nulls forming in the radiation pattern. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(3):6-17. https://doi.org/10.38013/2542-0542-2020-3-6-17

Introduction

Patent [1] offers a method for nulls forming in the radiation pattern (RP) of a receiving phased an­tenna array (PAA) of a radar system, referred to by the authors as a “PAA with control coupling”. A PAA model was built by the authors proceeding from physical considerations, and its operation algorithm is described from the same viewpoint; however, the mechanism of practical implemen­tation of the proposed method is not elaborated on. The objective of this paper is filling of the said gap.

In brief outline, the essence of method [1] is as follows. Let there be a radar PAA intended for receiving a target echo signal. At the same time, there are several radiation sources within the radar coverage area that interfere with reception of the wanted echo signal. Hereinafter, we shall refer to those sources as interference sources, or simply interference. Relative to the PAA, the interference sources and the source of a wanted echo signal are located in the far zone and can be regarded as plane waves arriving from certain directions. The problem of taking bearings to the interference sources is not considered in [1], as it is assumed pre-solved, with the directions to the interference sources known. Such an approach, when the prob­lem of interference sources suppression is split into two stages, first solving the problem of taking bearing to the interference and then the problem of RP nulls forming in the selected directions, is known from literature.

As a result of method application, the prob­lem of receiving the wanted echo signal, when scanning with the RP maximum, and suppressing interference through formation of nulls in the side lobes region is solved. The problem of interfe­rence influence in the RP main lobe region is not considered.

The idea behind the proposed method for a simple case of null formation in the direction of single interference action can be illustrated by a diagram given in Fig. 1a. Signals received by the antenna array radiators, after their amplifying and digitising, are divided into two parts. From the tapped signal part, a signal equivalent to the compensating RP of the PAA, with its maximum directed towards interference action, is generated through weighted summation with phase distri­bution φ1, ... ,φP. Then this signal is split so as

to be supplied to each PAA receiving channel, compensation of previously introduced phase shifts -φ1, ... ,-φP is done, and this signal is sub­tracted from signals arriving from each radiator. In the diagram (Fig. 1a), the subtraction operation is shown as summation with additional inversion of the signal phase by π. This operation can be inter­preted as formation of a narrow null in the partial pattern of PAA radiator (Fig. 1b). In this case null direction in the radiator’s partial RP corresponds to the direction of interference signal arrival.

 

Fig. 1. PAA RP null forming for single interference source: а - PAA diagram; b - null forming in radiator partial RP

After interference signal suppression, in each receiving channel weighted summation of their signals with complex coefficients C1, ...,СР is performed for forming a required PAA RP.

The approach with application of compen­sating RP subtraction was first proposed in [2]. The difference consists in that in the method being considered, the compensating RP is subtracted from the radiator’s partial RP, whereas in [2] the compensating RP is subtracted from RP of the PAA. The benefit of the proposed approach is a possi­bility for pre-forming partial RPs of the radiators and subsequently using them in PAA RP scanning, which is supposed to bring about certain saving of computational effort as compared with the ap­proach proposed in [2].

Under simultaneous influence of several in­terferences, it is necessary to form the minimum number of compensating RPs equal to the number of interferences, which leads to transformation of the equivalent diagram (Fig. 1a). The compensating RPs forming blocks reproduce “horizontally”, and their number is equal to that of the interference sources being compensated. Given in Fig. 2 is a diagram corresponding to the case of two inter­ferences compensation.

When subtracting several compensating RPs from partial RP of the radiator, when using linear phase distributions for forming directions of the compensating RP maxima, due to mutual non-orthogonality of the compensating RPs, it is not possible to obtain an ideal null in directions towards interference, since in the direction to null formed by one compensating RP side lobes of the other RPs will be present. To improve the depth of nulls in the selected directions, the compensating procedure described above is repeated a number of times, which results in diagram reproducing “vertically” (Fig. 2).

Fig. 2. PAA diagram for nulls forming in direction towards several interference sources

There are no precise instructions in [1] as to the necessary nulls forming iterations or, which is the same, diagram levels (Fig. 2). It should be natural to presume that this number depends on the number of interference sources and their mutual arrangement in space. As will be shown below, under Rayleigh angular resolution of in­terference sources, deep nulls forming requires a small number of iterations. If super-resolution of interference sources is necessary, process con­vergence slows down substantially.

An unmediated implementation of the nulls forming algorithm, as illustrated by Fig. 2, involves rather resource-consuming computa­tions, associated with multiple use of discrete Fourier transform. It will be shown below that the computation algorithm can be essentially simpli­fied with transition to a matrix description of the equivalent diagram and introduction of a score correlation matrix.

To clarify further discussion, we shall intro­duce a simplified equivalent PAA diagram which does not disclose its inner structure, as shown in Fig. 3.

Fig. 3. Equivalent PAA diagram

Elements of the diagram denoted by sym­bols A1, ... , AP are digital receiving modules, in which signal is amplified and digitised. Complex coefficients A1, ... , AP include random errors as­sociated with non-identity of signal reception by individual radiators, i. e. aperture errors and spreading errors of transmission coefficients of the digital receiving modules.

Elements W1, ... , WP are adaptive weighting factors. In the absence of interference sources they form a control vector with unity amplitudes, which provides PAA phasing in the selected direction.

Complex coefficients C1, ... , CP, same as in Figs. 1 and 2, are used for forming a required undisturbed PAA RP.

Given in the next section are calculation ratios for computing adaptive weighting factors.

Calculation ratios

Let us consider a receiving antenna array consisting of P isotropic radiators, arbitrarily arranged in the Cartesian coordinate system (Fig. 4). When using radiators having individual partial diagrams, the latter can be accounted for in the ratios given below.

The radiator coordinates can be represented by the vector:

Relative to the antenna array, the radiation sources are located in the far zone and can be regar­ded as plane waves arriving from certain directions. Wave vector directed into the observation point

where   - coordinate system unit vectors (Fig. 4).

 

Fig. 4. Spatial position of PAA radiators

Let the antenna array be affected by Q ra­diation sources from known angular directions, also located in the far zone. Wave vectors directed towards the radiation sources

Vector of complex amplitudes of signals with accuracy to the constant multiplier, equal to the complex amplitudes induced on the antenna array radiators by an interference source with number q

Using vectors (4), a matrix can be com­posed, whose structure is similar to the correla­tion matrix of interference [3] acting upon anten­na array

where ( )H - symbol denoting Hermitian conjuga­tion of the matrix.

The obtained matrix (5) can be regarded as an evaluation of the interference correlation matrix, in the formation of which only phase in­formation was considered, i. e. information about directions towards the interference sources. Infor­mation on the interference source amplitudes in this case is unknown, and an assumption is made about their equality.

As an initial approximation of the diagram weighting factor (Fig. 3), we shall use the control vector which provides antenna array phasing into the observation point

Having analysed the process of computing adaptive weighting factors shown in Fig. 2, and using denotations (4)-(6) introduced above, we can demonstrate that computation of said weighting factors comes down to the iteration process

where k - iteration number, I - unity matrix with dimensionality P.

The iteration algorithm (7) is fully equiva­lent to the scheme claimed in patent [1] when using equal-amplitude adders in the diagram given in Fig. 2.

From physical considerations, process (7) is a convergent one, and the necessary number of iterations, or, which is the same, of successive cascades in the diagram of Fig. 2, can be deter­mined from statement

where ε - small number determining the diffe­rence of vector W norm between respective itera­tions.

With a given number of iterations K, the ex­pression for process (7) can be written as

where W - the sought weighting vector.

Let us find the limit of expression (9) with K going to infinity. For that, we shall perform eigenvector expansion of the matrix. In so doing, we take into account that matrix H is Hermitian and non-negative definite [3]

where Λ - diagonal matrix, consisting of eigen values of matrix I - H,

U - matrix whose columns are eigenvectors of matrix I - H.

It is easy to show that

It is known [4] that eigenvalues of Hermitian non-negative definite matrix H are real and non-negative numbers. From physical conside­rations, process (7) is a convergent one, hence, eigenvalues of matrix I - H are not to exceed unity. Indeed, matrix Λ has Q non-repeated elements that are less than unity, and P - Q tuple elements, equal to unity. In this way, raising matrix Λ to the power of infinity is equivalent to nulling of its elements not equal to 1.

Hence, the sought weighting vector is

where Λ0 - matrix obtained from matrix Λ through nulling of elements less than 1.

Introducing denotation

we have

The evaluation of correlation matrix H used in the above expressions is a diagonalisable matrix [4], for which mathematical notions of kernel and span are defined as linear subspaces of eigenvec­tors, corresponding to zero (kernel) and non-zero (span) eigenvalues. In paper [3], definitions of correlation matrix noise and signal subspaces are given, equivalent to the mathematical definitions of kernel and span, respectively.

Analysing the obtained expressions (12)- (14), using definitions given in [3], we can con­clude that matrix (13) can be regarded as a projec­tor matrix, projecting control vector W0 onto the noise subspace of correlation matrix Н.

Then solution for an optimal weighting vec­tor W can be found immediately through the eva­luation of correlation matrix Н. Similarly to (10), let us perform eigenvector expansion of matrix Н

where Ψ - diagonal matrix, consisting of eigen­values of matrix H,

U - matrix whose columns are eigenvectors of matrix Н.

In making expansion (15), it is taken into account that orthonormalised bases of the eigen­vectors of matrices H and I-H coincide.

Let us compose projector matrices for the noise and signal subspaces of matrix H, denoting them Pw and Ps, respectively

where ui, uj – - columns of eigenvector matrix U, corresponding, respectively, to zero and non-zero eigenvalues of matrix H.

Then the sought weighting vector

Computations by formulas (14) and (18) yield the same results.

Analysing ratios (16), (17), we may come to the known conclusion that the maximum num­ber of suppressed interferences Q cannot exceed the value P-1.

There is another method, known from lite­rature [10], [3], for constructing a matrix project­ing onto the subspace orthogonal to interference signals. The sought weighting vector can be com­puted as

where Y - matrix whose columns are vectors (4).

Computation by formula (19) yields the same results as (14) and (18). Notably, unlike the algorithms considered above, no eigenvector ex­pansion of the matrix is required. Matrix YH · Y subject to inversion is a real one, having the order of Q equal to the number of interferences (Q < P).

Simulation results

Let us consider the results of adaptation algorithm application using the example of a linear PAA consisting of 16 radiators with spacing of 0.542λ. For illustration purposes we shall consider forma­tion, by means of this PAA, of two RPs: a narrow one, in the form of a sharp beam, and a broad one, overlapping side lobes of the narrow one. In RP calculations it is assumed that the radiators have partial RPs varying by the cosϑ law.

Given in Fig. 5 are undisturbed PAA RPs. The solid line shows the sharp-beam RP, formed by means of Taylor amplitude distribution across the aperture [5], with the side lobe level of minus 25 dB. The dashed line designates the RP intended for operation of the side lobe suppression (SLS) system, overlapping side lobes of the sharp-beam RP. The sharp-beam RP is normalised to the intrin­sic maximum. The SLS RP is normalised taking into account the aperture efficiency factors (AEF), corresponding to the RP-forming amplitude distributions.

Fig. 5. Undisturbed PAA RPs

Fig. 6a shows the same RPs when forming nulls in the direction towards 4 interference sour­ces, with their angular coordinates specified by vector

The vector of adaptive weighting factors was computed in accordance with algorithm (14).

Fig. 6. Nulls forming in directions determined in (20): а – PAA RP; b – interference suppression ratio

Fig. 6b shows power dependence of the interference suppression ratio, calculated as a normalised square of the modulus of the vector of adaptive weighting factors W, determined for the angular direction assigned by the control vec­tor (6) corresponding to scanning direction

Figs. 7a, b, respectively, show the plots of RP and interference suppression ratio for the case of two closely located interference sources

 

Fig. 7. Nulls forming in directions determined in (22): а – PAA RP; b – interference suppression ratio

The vector of adaptive weighting factors was also computed in accordance with algorithm (14).

Application of algorithms (14), (18) re­quires computing of eigenvector expansion of matrix (10), therefore, in case of a large number of PAA receiving channels, the volume of com­putations is substantial. For this reason, in cer­tain cases it may be practical to apply iteration algorithm (7), in the process of which it is simple matrix operations only that are performed.

In case of a single interference source, one iteration according to procedure (7) is sufficient. Let us consider iteration process (7) convergence for a case of several interference sources influ­ence, using the criterion of conventional con­vergence of the Euclidean norm of weighting vector (8).

Fig. 8 shows dependence of the number of iterations on the scanning angle for a case of nulls forming in the direction towards four interference sources (20). The parameter of the family of plots is convergence accuracy e from statement (8).

Given in Fig. 9 are similar data for ar­rangement of interference sources at angles (22). Presence of interference sources closely located anglewise leads to considerable slow-down of al­gorithm (7) convergence.

Based on the numerical modelling results, it can be claimed that application of the iteration algorithm (7) is only feasible in case of compli­ance with the Rayleigh resolution criterion for the angular coordinates of interference sources. Otherwise, it is necessary to use algorithm (14), which always ensures forming of deep nulls in desired directions.

Since for calculation of the vector of adap­tive weighting factors a deterministic evaluation of the correlation matrix is used, use of algorithms (14), (18) results in formation of deep nulls in the specified directions, ensuring suppression of in­terference signals to the level of intrinsic noise of the receiving channels.

The actual suppression ratio in such a sys­tem is determined by the measurement accuracy of interference source angular coordinates. Deep null formed in the interference source direction is fairly narrow, so a signal from the interference source is received, as a rule, on the RP null slope. Due to it, using a possibility for expanding RP null by angle is of practical interest. A lot of ap­proaches to solving this problem are known from literature, e. g. [6][7][8][9]. Unfortunately, all of them apply to linear PAAs with equidistant arrangement of radiators only.

Let us consider a possibility for using PAA with control coupling by drawing on the simplest and most frequently used Mailloux - Zatman null broadening method [6][7].

The principle of the method consists in sub­stituting a single radiation source for a group of equipotent incoherent signal sources arranged on a straight line. Mailloux used a group of discrete sources, and Zatman - continuously distributed dummy sources.

Then a modified score correlation matrix of interference sources can be expressed as

where ◦ - Hadamard product of matrices, i. e. product of their element-wise multiplication,

А - matrix with dimensionality ΡχΡ, where

Δ - constant defining RP null width.

Using H instead of H in (12), (15), we cal­culate RP (Fig. 10а) and interference suppression ratio (Fig. 10b) for distribution of the interference sources (20) and Δ = 0.07.

Fig. 10. RP broadened nulls forming: а - PAA RP; b - interference suppression ratio

As can be seen from Fig. 10, use of the Mail- loux - Zatman method leads to substantial broade­ning of RP nulls as compared with Fig. 6. The interference suppression ratio in the broadened null direction in this example is at least 70 dB.

Broadening of RP nulls can also be obtained through substitution of a single direction to an interference source for a discrete group of close­ly arranged directions. In so doing, the restriction implying that methods similar to (23) are appli­cable to linear equidistant arrays only is removed. With such approach, for correct discrimination of correlation matrix eigenvectors by their belonging to the signal or noise subspaces at the stage of projector matrix composing, it is necessary to use algorithm (18) or (19).

Conclusion

In summary, this paper focuses on the specific features of applying a simple method of RP nulls forming in specified directions, suitable for application in PAA with radiators arranged on an arbitrary surface, based on simple physical representations.

It has been shown that the considered null-forming method can be regarded as a special case of applying a projection algorithm of adap­tive spatial filtering, which provides control vec­tor projection onto the correlation matrix noise subspace.

It has been shown that the use of this method enables application of the methods of RP nulls broadening, which makes it possible to relieve requirements for the interference source angular coordinates measurement accuracy and the rate of correlation matrix evaluation refreshment.

References

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About the Authors

A. D. Egorov
All-Russian Scientific Research Institute of Radio Engineering, JSC
Russian Federation

Egorov Aleksey Dmitrievich - Sectoral Head.

Research interests: radar, antennas, microwave technology, antenna measurements.

Moscow



A. O. Yashenkov
All-Russian Scientific Research Institute of Radio Engineering, JSC
Russian Federation

Yashenkov Artyom Olegovich - Leading Engineer.

Research interests: digital antenna arrays, digital diagramming, radiolocation.

Moscow



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For citations:


Egorov A.D., Yashenkov A.O. Using a phased antenna array with control coupling for nulls forming in the radiation pattern. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(3):6-17. https://doi.org/10.38013/2542-0542-2020-3-6-17

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