Analytical scannable‐shaped beam pattern synthesis via superposition principle

National Natural Science Foundation for Young Scholars of China under, Grant/Award Number: 61801090 Abstract A shaped beam pattern (SBP) synthesis algorithm, which obtains the desired SBP in an analytical way, is proposed. The new algorithm describes the SBP with the weighted superposition of a set of pencil beam patterns (PBPs). First, the array weight of the PBP is obtained via analytical algorithms, such as; Chebychev and Taylor, etc. Then, by computing the weights of the PBPs with the least‐square method, the synthesized SBP is expressed explicitly. Different from the state‐of‐the‐art algorithms, the proposed algorithm can obtain the desired SBP analytically, avoiding the time‐consuming iteration processes. Moreover, the proposed algorithm enables the array antenna to scan the beam in different directions. Simulations with linear and planar array antennas are provided to assess the superiority and effectiveness of the proposed algorithm.

SBPS problem is solved with an analytical algorithm that is super simple and effective. One obvious disadvantage of this existing algorithm is that its performance would degenerate when the center of the SBP beam is far away from the broadside axis ( Figure 1). To make the algorithm scannable, this article determines the weighted superposition weight of the PBPs via the least-square method (LSM) * . With the LSM, the proposed algorithm would minimize the total power difference between the desired SBP and the obtained SBP. In such a way, the proposed algorithm is expected to scan in different directions with a low sidelobe level (SLL). Two advantages of the proposed algorithm are (1) its computation complexity is decided by the required angular resolution of the beam pattern which, therefore, makes the proposed algorithm work much faster than the existing algorithms. Its advantage would be more obvious when the array size becomes large because the computation complexity of the most existing synthesis algorithms is polynomially proportional to the array size and (2) the array weight can be analytically obtained by giving the array layout and does not require any iteration process as the existing algorithms do.
The remainder of the article is organized as follows: the SBPS problem expressed by the weighted superposition of a set of PBPs is introduced in Section 2; the analytical proposed algorithm is proposed in Section 3; simulations and discussion are carried out in Section 4 and the conclusions are drawn in Section 5.

| PROBLEM FORMULATION
Without loss of generality, a linear array antenna consisting of N elements with the position being r n , n = 1, …, N is considered. Denote E n (θ), a n (θ) and w n the far-field electric field, the array factor and the element weight of the n th (n = 1, …, N) element, respectively. The total far-field electric field of the array antenna can be expressed as the sum of those of the elements: where a n (θ, θ c ) = exp(jκr n ( sin θ − sin θ c )) with κ ¼ 2π λ being the spatial wavenumber, 0 λ 0 being the electromagnetic wavelength and 0 θ c 0 being the center direction of the beam. By vectorizing, the above electric field can be rewritten as where Restricted by the limited resolutions of kinds of hardware, the beam angle is usually discretized. Denote the mainlobe and the sidelobe of the SBP Θ ML and Θ SL , respectively. Denote the angular-step Δθ, the discretized angles in the mainlobe are θ l m ∈ Θ ML ; l m ¼ 1; …; L M with L M ¼ ⌊ Θ ML Δθ ⌋, and those in the sidelobe are θ l s ∈ Θ SL ; l s ¼ 1; …; L S with L S ¼ ⌊ Θ SL Δθ ⌋. To obtain the desired SBP, the PBPs pointing to different directions, i.e. f PBP ðθ; θ l m Þ, are combined as in [9,21]: with w l m ;SBP being the weight of the l th m PBP and w PBP ¼½w 1;PBP ; …; w N;PBP � H .
For uniformly distributed array antenna scenarios, the array weight of the PBP (5) in this article, that is w PBP , can easily be obtained via Chebychev or Taylor algorithms. In [9] and [21], w l m ;SBP is determined by the amplitude of the desired SBP.
Marked the existing algorithm as LQ Alg . The LQ Alg is clear in theory and simple in applications. However, one of its obvious disadvantages is that the synthesis performance would deteriorate seriously when the SBP scans in different directions, especially when its center is far from the broadside axis. -601 scans from 0 o to 60 o , the beam performance worsens along with the centre angle. Specifically, the LQ Alg have low SLL and low mainlobe ripple level (MRL) when θ c = 0 o and 20 o ; however, the SLL and MRL would become larger when θ c increases. Worse is that when θ c = 60 o , the SLL and MRL are almost inapplicable. In this article, a simple scheme based on the LSM for computing w l m ;SBP is proposed. The new algorithm is capable of scanning in different directions with slight performance deterioration.

| THE COMPUTATION OF w l n ;SBP
Vectorizing (4), it yields with In the LQ Alg , the selection of w SBP (b n , n = 1, …, N in [9]) is only related to the pattern shape which does not consider the influence of the beam centre direction, that is, θ c . For example, the LQ Alg set b n = 1 for a normalized flat-top beam pattern. To make the array antenna scannable, an intuition is to select w SBP so that w SBP would vary along with different centre angle θ c of f SBP (θ). The natural idea is to obtain f SBP (θ) as similar as possible to the desired SBP, that is, f desired,SBP (θ), which, therefore, can be expressed as the following optimization problem: and l ∈{l s , l m }. The symbol 0 [x l ] 0 denotes the matrix consisting of the vector x l . In this problem, the sidelobe of the SBP is also controllable. Hence, F PBP (θ ) in (7) should be replaced by (8), it yields kF desired; SBP − F PBP w SBP k 2 2 , representing the sum of the squared differences of L = L M + L S linear equations. To minimize this difference, the LSM is used to estimate w SBP with the following equation F H PBP F desired; SBP ¼ F H PBP F PBP w SBP . Hence, the weight w SBP can be obtained as: where the symbol 0 † 0 represents the pseudo-inverse of matrix. Algorithm 1 described in the following table gives out the specific details on how to compute the total array weight via the proposed algorithm to obtain scannable SBP.

| SIMULATIONS AND DISCUSSION
This section carries out simulations with both linear array and planar array to validate the performance of the proposed algorithm. Since the proposed algorithm is the improvement of LQ Alg in [9], only the LQ Alg is compared (the other existing algorithms for SBP are not analytical. Hence, the array weight cannot be explicitly expressed). All simulations are operated on a 64-bit windows personal computer with 3.5 GHz, Intel Core (TM) i3-4150.  (4) is − 20 dB. Taylor algorithm is used to estimate w PBP in (5). The angular step Δθ = 0.5 o . Figure 2 shows the performance difference between the proposed algorithm and the LQ Alg :

| linear array scenarios
� The proposed algorithm has lower MRLs than the LQ Alg in all cases, while it has higher SLL in the first two cases. The MRL of the proposed algorithm becomes worse from θ c = 20 o to θ c = 60 o . However, the variation is inner −2 dB, which makes it acceptable in most applications. Comparatively, the MRL of the LQ Alg deteriorates very fast. Especially, the obtained beam via the LQ Alg is not a flat-top when θ c = 60 o because its MRL is too large. � In the first two cases, the SLL for the proposed algorithm is lower than −20 dB, satisfying the desired − 20 dB requirement. In the last case, i.e. θ c = 60 o , the SLLs for the proposed algorithm and the LQ Alg are −18.2 and −8.3 dB, respectively. It shows when the beam is far away from the broadside axis, the proposed algorithm could obtain a lower SLL than the LQ Alg . � One advantage of the proposed algorithm is that it can remain the MRL (varies inner −2 dB) when the SBP scans in different directions while the MRL would become large via the LQ Alg .

| Cosecant square beam pattern
The related parameters are set similar to those in the previous sub-part. Figure 3 shows the synthesized cosecant square beam pattern via the proposed algorithm and the LQ Alg . The desired normalized cosecant square beam pattern is defined by where [θ 0 , θ 2 ] ∈ Θ ML and norm( csc 2 (θ)) represent the normalized form of csc(θ). The figure shows that the obtained cosecant square beam pattern matches the desired one better than the LQ Alg . The advantage of the proposed algorithm is more obvious when the centre of the mainlobe scans to a bigger angle. It is worth mentioning that the SLL via the proposed algorithm is higher than that via the LQ Alg . However, the SLLs obtained via both algorithms are below −20 dB, which satisfies the designed requirement. When the MRL is the more important factor, the proposed algorithm would be more applicable; otherwise, the LQ Alg is preferred.

| Discussion
In the two previous parts, both linear array antenna and planar array antenna are simulated to assess the performance of the proposed algorithm. The advantage of these two algorithms is that both are analytical. Their array weights can be computed directly given the array layouts. Since the performance of the LQ Alg would deteriorate when the mainlobe scans in different directions, the proposed algorithm tries to make an improvement. By updating the superposition weight of a set of PBPs with the LSM, the proposed algorithm enables the beam to scan in different directions with tiny mainlobe degeneration. Meanwhile, the SLL obtained via the proposed algorithm is higher than the LQ Alg . Gratifying is that the SLL via both algorithms satisfies the designed requirement for most scenarios.
To further validate the performance in the real applications, data including the element pattern and the mutual coupling are simulated with a full-wave simulator. The mutual coupling among elements is considered using the active element pattern (AEP) [23][24]. The AEP of an element can be obtained by exciting it, while the others are connected to matching loads. Figure 5 shows the synthesized flat-top beam pattern and cosecant square beam pattern via the proposed algorithm. Both results are similar to those in Figures 2 and 3. These results further validate the effectiveness of the proposed algorithm for its robust scanning ability.

| CONCLUSIONS
The article proposes an analytical SBPS algorithm for uniformly distributed array antenna, which can obtain the array weight analytically. It is an improvement algorithm for the LQ Alg in [9]. Instead of determining the superposition weight of a set of PBPs by the amplitude of the SBP as in [9], the proposed algorithm computes the superposition weight with the LSM. In such a way, the beam direction is also considered updating the superposition weight, which makes the proposed algorithm to be able to scan its beam with tiny MRL degeneration. It is important to note that the SLL via the proposed algorithm is worse than the LQ Alg when the mainlobe is close to the broadside direction even it still satisfies the desired value. Besides, compared with the other existing algorithm for the SBP, which solves the SBPS problem with iterative processes, both the proposed algorithm and the LQ Alg would operate faster when the array size is large.  Figure 2. and (b) the cosecant square beam pattern with the same parameter settings in Figure 3