Fast Synthesis Method for Large Aperture Array Pattern in the Presence of Array Errors

In this paper, an improved array synthesis method with array errors is proposed for large aperture arrays. Because of the array errors such as amplitude-phase errors and positions errors, the performance of the array synthesis is reduced seriously. Firstly, the ideal fast low sidelobe synthesis method is obtained based on the discrete Fourier transform (DFT) method.)en, by using Taylor expansion to remove the coupling relationship between the position of the element and the scanning angle, the compensation matrix for the pattern function and the array weighted vector with amplitude phase and position errors are derived. At last, the conversion relationship between the array with errors and the array weight vector is corrected in the iterative process. )e theoretical simulation experiments verify the effectiveness and robustness of the proposed method for the linear array and rectangular array pattern synthesis. )en, the influence of Taylor expansion order on the pattern synthesis results is analysed.


Introduction
Array antennas have been widely utilized in radar, communication, navigation, remote sensing, telemetering, and other fields due to their unique characteristics, such as multifunction, flexible beam control, and high tracking capability [1][2][3]. In recent years, several excellent and innovative ideas have been proposed for the real application of array antennas. e configurations of the metamaterial photonic bandgap (MTM-PBG) periodic structure and reflector-slot-strip-foam-inverted patch (RSSIP) are used to improve the performance of the densely packed array antenna in synthetic aperture radar (SAR) and multiple-inputmultiple-receive (MIMO) systems [4][5][6]. A novel planar microstrip array antenna is presented based on a simplified composite right/left-handed transmission line (SCRLH-TL) for application in circularly polarized SAR systems in [7]. Meanwhile, a new wideband microstrip antenna based on the falcate-shaped patch is fabricated with the reconfigurable capability of circular polarization [8].
ese advanced methods have effectively improved the applicability of the array antennas.
In the application in different fields of the array antennas, several shapes of the array pattern are required which can be formed with different array weights [9][10][11]. By optimizing the weighted excitation of array elements, a technology that meets the expected demand is formed, which is called array pattern synthesis technology or beamforming technology. In actual engineering applications, different requirements require the array patterns with different shapes. For airborne/spaceborne early warning radar, the observed moving target is in the clutter background, which is widely distributed and with discrete strong clutter [12]. In order to reduce the influence of sidelobe clutter and improve the detection performance of moving targets, the ultralow sidelobe can be formed by the array synthesis technology. Not only like this, it is also necessary to form deep nulls in certain angular ranges to suppress strong clutter points or interference signals in many applications.
e analytical synthesis methods such as Chebyshev method, Taylor method, and Woodward-Lawson method are simple to be implemented. e methods can also achieve the smallest peak gain loss and main lobe broadening under certain constraints. However, the Chebyshev synthesis and Taylor synthesis techniques are only suitable for low sidelobe level and cannot form the expected pattern of other shapes. e Woodward-Lawson synthesis is a classic and effective synthesis method. But the synthesis results are with high sidelobe using the sinc function.
e intelligent optimization methods including genetic algorithms and simulated annealing can realize the synthesis of the pattern of various shapes by associating the cost function with the shape of the pattern and constraining the magnitude and phase of the weight vector in the iterative process. However, intelligent optimization methods often require multidimensional parameter optimization, and the calculation complexity of the intelligent optimization method increases rapidly as the array size becomes larger. Meanwhile, the local convergence is more likely to produce. e convex optimization methods are also a type of method to solve the pattern synthesis by converting the pattern synthesis problem into a constraint problem. ese methods can easily constrain the performance of the pattern, but they also face the problem of huge amount of calculation for the synthesis of large aperture arrays.
In recent years, the iterative fast Fourier transform (FFT) synthesis method has been proposed for uniformly distributed large aperture arrays by Keizer et al. creatively [29][30][31]. e advantages of this method, which is especially suitable for pattern synthesis of large aperture arrays, are that they can complete the pattern synthesis fast, efficiently, and with strong convergence. e important prerequisite for the application of this method is the property that the array elements are periodically spaced and without array errors. However, due to the influence of factors such as antenna assembly and integration, thermal load, and aging of the array elements, the amplitude-phase errors and position errors of each element of the large aperture array are unavoidable. en, the property for array elements with periodic spacing can be destroyed by array errors, and the performance may be decreased in practical application.
is paper considers the large aperture array synthesis method in the presence of amplitude-phase errors and position errors. e amplitude-phase response errors and position errors of each element can be obtained in real time by the internal calibration and the external calibration [32,33]. Based on this, for large aperture arrays in the presence of array errors, an improved iterative discrete Fourier transform (DFT) synthesis method is proposed, in which the influence of array errors is compensated. e Taylor expansion method is used to remove the coupling relationship between the position of the array element and the scanning angle. ereby, the pattern function and the array weighted compensation matrix are derived in the presence of amplitude-phase and position errors. After the iterations, the accurate pattern synthesis results can be obtained. Simulation experiments verify the effectiveness and robustness of the algorithm for the synthesis of large aperture for linear array and two-dimensional array.

Fast Array Pattern Synthesis Approach
We assume an ideal n-element linear array with halfwavelength spacing, and w � [w 0 , w 1 , . . . , w N− 1 ] T is the array excitation vector. e position vectors in X-axis and Yaxis are denoted by n � nd and y 0 n � 0 are the nominal coordinates of the nth element in X-axis and Y-axis, respectively, d � λ/2 is the element spacing of the linear array, and λ is the wavelength. e array pattern can be obtained by multiplying the array factor and the array element pattern. e array pattern that only needs to consider the array factor is given by Obviously, DFT and inverse discrete Fourier transform (IDFT) can be used to transform between the pattern function f of the uniform linear array and the excitation weight vector w of the array element as follows: , where DFT[·] is the DFT operation and IDFT[·] is the IDFT operation.
For the large rectangular arrays, literature [29] introduces the fast Fourier techniques (FFTs) into the pattern synthesis technique.
is method utilizes the Fourier transform relationship between the pattern function of the periodic array and the excitation vector of the array. en, the pattern and weighting vector are corrected iteratively according to the expected sidelobe level. Because this method is based on DFT calculation, it has the advantages of being fast and efficient. However, this method may fail with the unavoidable array errors.

Proposed Array Synthesis Methods
We assume that the real array element position vectors are which can be given by where Δx � [Δx 0 , . . . , Δx N− 1 ] T and Δy � [Δy 0 , . . . , Δy N− 1 ] T are the array position error vectors and Δx n and Δy n are the position error of the corresponding array element on X-axis and Y-axis. 2 International Journal of Antennas and Propagation e actual pattern with the amplitude-phase errors and position errors can be formulated as where e amplitude-phase deviation caused by amplitudephase errors has no dependence on the angle, while the phase error caused by the position error is different in different scanning angles. Consider using the Taylor expansion method to eliminate the coupling of position error and scanning angle in phase error.
e K-order Taylor expansion of the error term which is introduced by the position errors is obtained as follows: where d(n, u) � d x (n)u + d y (n)v, d x (n) � j(2π/λ)Δx n , and d y (n) � j2π/λΔy n . And, it is required that the array error disturbance |Δx n u + Δy n v| cannot be greater than the wavelength, which is satisfied in almost all scenes. Based on (4) and (5), we have where the polynomial expansion of d k (n, u) is where C q k � k!/q!(k − q)!. Based on (6) and (7), we have erefore, using Taylor expansion and the time-domain circular convolution, the pattern function is expressed in the form of multiple DFTs: where ⊙ denotes the dot product operation, * denotes the circular convolution operation, and T is the compensation matrix of Fourier transform as follows: where C 1 · denotes the calculation of a cyclic convolution matrix operation. For any vector b of length M, the cyclic convolution matrix of b is given by According to (9), the weighted vector of the array is given as follows: ere, the specific iterative steps of the proposed improved method are given as follows: (1) Set expected pattern parameters, Taylor expansion order, and threshold of two adjacent excitation errors ε 0 . If the purpose of the pattern synthesis is only to control the sidelobe level, the pattern parameters include the left and right sidelobe levels. (2) According to the given array parameters, the Fourier transform compensation matrix T and its inverse matrix T − 1 can be calculated using formula (10).    International Journal of Antennas and Propagation (4) Perform IDFT operation on the array weight vector, and perform error compensation on the obtained directional pattern function f l according to formula (9). (5) According to the expected pattern parameters, correct the pattern function and update f l . (6) Perform DFT operation on the updated pattern function, and use equation (12) to perform error compensation on the weighted vector obtained. Take the first n elements as the weight vector and normalize it after this iteration. (7) Calculate the root mean square error (RMS) of two adjacent weight vectors ε l � Judge whether the convergence condition is satisfied. If ε l > ε 0 , then let l � l + 1 and skip to step 4. Otherwise, terminate the iteration and output the array excitation w l .

Simulation Results
In this section, the simulations are performed for the pattern synthesis of the linear array and the planar array. e working wavelength of the arrays is 0.2308 m. e actual arrays are with the amplitude-phase errors and the position errors. Array errors are independent and identically distributed random variables. In the following simulations, the amplitude-phase error RMS is 0.1 dB/0.1°and the position error RMS is denoted by d max . Meanwhile, the measurement accuracy of the array deformation calibration equipment or method is millimeter level.

Synthesis of a Linear Array.
e uniform linear array is distributed on the X-Y plane, as shown in Figure 1, of which the number of array elements is 400. Figure 2 shows the synthesis result of the linear array pattern using the conventional DFT method and the proposed synthesis method in this paper. e position error RMS d max is 0.05λ. e number of DFT points is 2048, and the Taylor expansion order K of the proposed method is 3. e sidelobe requirements are −40 dB in the 0°-60°interval, −50 dB in the 60°-80°interval, −30 dB in the 80°-100°i nterval, −50 dB in the interval of 100°-120°, and −40 dB in the interval of 120°-180°, and the sidelobe of the pattern is symmetrical about 90°. e number of sidelobes that do not satisfy the sidelobe constraints versus the iteration number is shown in Figure 2(b). Figures 2(c) and 2(d) show the histogram of the sidelobe level distribution of the two methods, respectively. It can be found that the classical iterative DFT method has obvious elevation of the sidelobes for the array with the position error of the array elements and the improved method can obtain the accurate synthesis result of the pattern efficiently and quickly.
Next, the linear array is applied to compare the proposed method with the Chebyshev method, FFT method, and virtual array method. Suppose that the number of linear array elements is 400, the sidelobe requirement is −40 dB, and the position error RMS d max is 0.1 λ. Figure 3 and Table 1 show the pattern synthesis results of the four methods. It can be found that the sidelobe levels (SLL) in the proposed method are basically below −40 dB with the ratio of 99.38%. Figures 4 and 5 show the pattern synthesis and sidelobe level distribution histogram results versus K. It can be found     that, with the increase of the Taylor expansion order, the distribution of sidelobe levels gradually concentrates below the sidelobe level required. When the order K is 3 or 4, the synthesis result can probably meet the requirements under this simulation conditions.

Synthesis of a Planar Array.
e rectangular array consists of 5000 elements. e row number is 50, and the column number is 100. e synthesis results of the array pattern using the classical DFT method and the error compensation DFT method in this paper are shown in Figure 6. e sidelobe level required is −50 dB, and the threedimensional position error RMS d max is 0.05λ. e Taylor expansion order of the method in this section in Figures 6(b) and 6(d) is 5. ere are high-level sidelobes in the results of the conventional synthesis method. It can be noted that the improved method can still obtain better results than the conventional method for two-dimensional arrays.

Conclusion
is paper has described an improved pattern synthesis method for the large aperture array in the presence of array errors.
e compensation for the amplitude-phase errors and positions errors is derived to correct the relationship between the array pattern and weighted vector. Compared with the conventional method, the improved method is more effective and robust for the large array with amplitudephase and position errors. At the same time, the pattern synthesis of multiple sidelobes and the influence of Taylor expansion order on the results are simulated, which provides guidance for the parameter selection in subsequent actual projects.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.