Features of excitation reconstruction in flat multielement phased antenna array face using dynamic directional patterns

Introduction

The problem of determining the amplitude-phase distribution (APD) in the apertures of passive phased antenna arrays (hereinafter – PAA) and active phased antenna arrays (APAA) is gaining ever more importance for the theory and practice of antennas development. This is associated with a necessity to ensure real-time serviceability check of antenna array basic systems at the development stages and during operation of final products.

The currently existing APD reconstruction methods are based on measuring signal parameters in the PAA channels, or imply APD recovery as per the data of field strength measurements in the antennas’ near-, intermediate-, or far-field regions. Among the well-developed methods are those based on resolving systems of linear equations [1][2], however, their use requires high computational resources. Many authors, including [2][3], agree in that transition to the plane waves and use of the Fourier transform is the fastest and the most convenient way for reconstructing APD of flat antenna arrays. However, the methods based on such an approach often demand high time expenditure for measurements.

In this paper, it is proposed to use for APD reconstruction dynamic radiation patterns (DRP) of PAA [4], measured under electronic scanning without mechanical rotations and movements of the antenna. It allows to significantly reduce the measurement time [4][ 5]. The DRP measurements can be performed both in the far and in the near regions, for example, in an anechoic chamber with the use of collimator, or by the focused aperture method. For APD reconstruction according to the measured DRPs, no increased computational resources are required due to the use of Fourier transform algorithms. Practical results of DRP application for reconstructing APD were first introduced in [6].

Basics of APD reconstruction as per DRP

Reconstruction of amplitude distribution (AD) and phase distribution (PD) as per a known DRP consists in calculating excitation parameters in the location points of radiating element centres with application of the inverse Fourier transform [7]. If the pattern of radiators arrangement in a flat aperture is rectangular or triangular, with DRP measurements taken on an equidistant direction grid in the (u, v) coordinate system, then

where A_mn, ϕ_mn – radiator amplitude and phase;
s, p – direction grid row number and column number;
F_ps, ψ_ps – values of DRP amplitude and phase components in direction with angular coordinates (u_p ,v_s );
u, v – angular variables that are direction cosines for selected direction to aperture axes X and Y, respectively;
u = sin θ cos j, v = sin θ sin j ;
k = 2π / λ – wave number;
x_n, y_m – coordinates of a radiator located in row with number m and column with number n.

Proceeding from expression (1), we shall take the advantage of the inverse fast Fourier transform algorithms to reconstruct APD, which will considerably reduce the time of DRP processing, especially for antenna arrays with apertures having a large electrical size and, consequently, a large number of radiators.

Due to the measurement specifics, radiation pattern of a single radiator in the array does not influence the DRP, therefore, in the first approximation, the DRP can be regarded as directivity multiplier (DM) of the PAA. It is known that the DM of a flat antenna array with equidistant arrangement of radiators along each axis is a 2D periodic function [8] in the (u, v) coordinate system. The DM periodicity is determined by the diffraction maxima structure. In a PAA with triangular arrangement pattern of radiators, whose phase reckoning in formation of oblique phase front starts from the centre, the coordinate origin lies between the radiators (Fig. 1). The diffraction maxima have a different sign of the phase component (Fig. 2). The period of complex-valued function of such PAA’s DM is a region shown in Fig. 2 with a red line.

For reconstruction of APD in the aperture, it is necessary to have the values of DM in the period region. The size of DM period region depends on spacings between array elements d_x and d_y (see Fig. 1), normalised to wavelength λ, and is defined by the expressions:

At certain values of d_x and d_y the DM period boundaries go beyond the visibility scope (see Fig. 2). Electronic scanning allows to direct the PAA beam beyond the visibility scope and measure the level of the received signal. To measure DRP in direction (u₁, v₁ )  in the aperture, it is necessary to form the phase distribution represented as ϕ (x,y) = 2π (xu₁ + yv₁), therefore phase advances at spacings d_x and d_y must be equal to

At large deflections of the beam, phase advances at spacings may exceed 2π, which is technically implementable, as the integer amount of 2π can be left out of consideration in phasing of the radiators.

In accordance with the sampling theorem (the Kotelnikov theorem), DRP should be measured with increments

where Lx and Ly – aperture electrical sizes in X and Y coordinates.

For this reason the total number of DRP measurements, e.g. within a single period region, shall be at least

However, in most practical cases, to reconstruct APD as per DRP it will be sufficient to have the values taken in a region corresponding in shape and size to DM half-period (hereinafter – cell), with parameters T_U = 1 /d_x, T_V = 1 /d_y . It makes it possible to bring the number of measurements down to N_min = [T_U / ∆u] [T_V / ∆v] ≈ NxNy, which roughly corresponds to the number of radiators in a rectangle delineating PAA aperture. Assuming that for a PAA comprising 1000 radiators the time of a single standard-mode measurement amounts to ≈10 ms, the total DRP measurement time will be just ≈10 s, and APD calculation on a medium-performance personal computer will take fractions of a second.

Notably, a complex-valued DRP carries information on the location coordinates and excitation of the radiators. This is demonstrated by an example of modelling DRP of an APAA with circular aperture (≈ 600 radiators) and reconstructing its APD. If, using the DRP data in a single-cell region (Fig. 3), we reconstruct APD for an aperture of larger size (Fig. 4; hereinafter, the values by axes X and Y correspond to the number of wavelengths) but with the same spacing of radiators, then amplitudes (Fig. 5) and phases can be reconstructed with very high accuracy. Notably, amplitudes of the radiators located beyond the limits of the initial radiating aperture have values close to zero.

Reconstruction in the presence of regular excitation errors

The effectiveness of the method of APD reconstruction as per a measured DRP can be illustrated by an example of reconstructing excitation of the APAA described above. The initial AD is tapered (AEF = 0.9), with AD random errors σ_а0 = 0.02. Added to cophased PD are errors with σ_ϕ0 = 10° . Also entered in the aperture were malfunctions in the form of disabled radiators, individual ones and groups of such (black rectangles in Fig. 6), as well as a constant phase addition of +60° in one full row and one half-row (Fig. 7). Additional random APD errors, updated at each PAA beam deflection [4], were not taken into account in this case.

The result of APD calculation as per the values of DRP amplitude and phase components lying within a single-cell region almost matches the initial APD: RMS deviation of reconstructed AD from the initial AD was σ_а ≈ 0.0015, RMS deviation of reconstructed PD from the initial PD – σ_ϕ ≈ 0.2° .

As a result of APD reconstruction, the radiators disabled in the initial aperture (with amplitudes equal to zero) had insignificant amplitude deviations from zero value. However, since phases of those radiators take virtually random values within an interval of ±180°, an additional indicator of a disabled radiator is substantial deviation of its phase value from phases of other elements (Fig. 8).

Based on the findings obtained, it can be inferred that in the presence of regular excitation errors in the PAA aperture, it should be sufficient for APD reconstruction to have values in that DRP region which corresponds to a single DM cell.

Reconstruction in the presence of unphaseable radiators

A special case is that when the antenna aperture contains elements uncontrollable in phase (the so-called unphaseable elements). If, for instance, the PAA aperture has a faulty half-row in which element phases do not change in the course of scanning, then the accuracy of APD reconstruction as per DRP region corresponding to the region of a single DM cell drops sharply (Figs. 9, 10).

In this case, the degradation of APD reconstruction accuracy is caused by lack of DRP data in a single DM cell. Acceptable accuracy of APD reconstruction can be achieved by expanding the DRP processing region to the size of DM period (Figs. 11, 12). Fig. 12 shows the DRP phase component demonstrating phase opposition of DRP diffraction lobes, and Figs. 13, 14 – results of APD reconstruction as per DRP data in the DM period region.

A specific feature of the proposed approach consists in that after reconstruction the unphaseable elements have amplitude close to zero (see Figs. 10, 14), which allows to very accurately determine their position in the aperture. Moreover, in implementation of the proposed approach difficulties may occur concerning separate identification of non-radiating elements and elements uncontrollable in phase. This peculiarity has been verified experimentally.

Conclusion

The basic possibilities for reconstructing APD as per a measured DRP are presented. The use of this method is the most effective for multielement PAA/APAA with flat apertures and equidistant arrangement of radiators, row- and column-wise. For such arrays the DM is a 2D periodic function, which makes the process of DRP measurement and processing considerably easier.

It is shown that with triangular arrangement of radiators in the aperture, in the absence of unphaseable radiators in PAA/APAA, for APD reconstruction it is sufficient to use the DRP data measured in the region of a single DM cell, and in the presence of unphaseable radiators, it is necessary to process the DRP data measured in the region of two DM cells (of the same DM period).

The most important advantages of the considered method are:
• possibility to use it for measuring DRP in standard mode of PAA beam scanning;
• taking measurements without mechanical rotations and movements;
• small time expenditure for measurement and processing;
• reliable determination of malfunctions in PAA/APAA excitation and control systems that do not change during scanning;
• reliable prediction of directional characteristics in the visibility scope.

The key aspects of the presented results were experimentally verified on samples of APAAs under development.