International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 915293, 13 pages

http://dx.doi.org/10.1155/2015/915293

## Scalable Alternating Projection and Proximal Splitting for Array Pattern Synthesis

School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China

Received 2 July 2015; Revised 15 August 2015; Accepted 17 August 2015

Academic Editor: Felipe Cátedra

Copyright © 2015 Yubing Han and Chuan Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Array synthesis with embedded element patterns is a problem of great practical importance. In this paper, an array pattern synthesis method using scalable alternating projection and proximal splitting is proposed which considers the scaling invariance property of design specifications and constraints for the amplitudes of pattern and excitation. Under the framework of alternating projection, the scalable pattern and excitation constraint sets are first defined. Then the scalable pattern projection and iterative procedure for optimum pattern scaling factor are studied in detail. For the scalable excitation projection, it is designed as the solution to a constrained weighted least mean squares optimization, which can be solved by an effective forward-backward splitting iterative process. Finally, the selection of the weighted matrix and computational complexity are discussed briefly. Several typical linear and planar synthesis examples with or without the embedded element patterns are provided to demonstrate the effectiveness and power of the proposed method.

#### 1. Introduction

The synthesis problem for array antennas can be loosely defined as the inverse of the analysis problem, that is, to determine the element excitations and possibly the lattice geometry, so that the array radiation pattern or the excitations satisfy some prescribed requirements. It is a vast subject in the antenna research, which has applications in fields, such as radar, sonar, radio astronomy, and satellite communications [1].

There are many methods to synthesize a desired pattern. The early history of the synthesis algorithm could date back to Woodward-Lawson method [2], which was firstly presented for linear arrays, and then extended to the circular and planar arrays. The synthesis methods in this stage established the basic pattern decomposition theory, using expanding characters of Fourier series and Bessel function and solving the question with some energy optimization ways. Yet, poor controls for side lobe levels (SLLs) and ripples in the desired coverage are the common problems for these early methods, so some algorithms using Taylor or Chebyshev distribution as aperture current distribution made a great effect on solving them next stage [3, 4]. However, these Taylor or Chebyshev synthesis methods have some drawbacks. The applied array should be a circle or rectangle, and this would be difficult to apply during most synthesis procedures. Today, iterative methods based on optimization techniques are very powerful tools for pattern synthesis thanks to the modern computer. There are many types of random optimization methods that have been applied to antenna synthesis problems, such as genetic algorithm, differential evolution, simulated annealing, and particle swarm optimization [5–8]. These optimization methods are computationally expensive and they are not effective in large arrays because a large number of particles must be needed in these situations.

Alternating projection (AP) is an extremely powerful and versatile procedure for synthesizing the excitation of very general antenna structures [9, 10]. The synthesis problem is formulated as the search of the intersection of sets and is solved by an iterative projection method. On the other hand, the least mean squares (LMS) or weighted least mean squares (WLMS) method is popular in array synthesis [11], which formulates pattern of array in matrix form and minimizes the difference of realized pattern and desired pattern in LMS sense. A variation of the LMS method is described in [12], where different weights for different directions are introduced and the weights are changed iteratively according to the relative errors in each direction. In [13], a novel alternating adaptive projections (AAP) algorithm for antenna synthesis is presented based on the scaling invariance property of design specifications and constraints which arises in synthesis problems of practical relevance, and an iterative procedure is proposed to compute the optimum scale parameter (or reference level) which is simple to implement and easily integrated in standard alternating projection routines. In [14], a Gram-Schmidt orthogonalization method is proposed for the radiation pattern synthesis. It applies the Gram-Schmidt procedure to obtain an orthonormal basis from the embedded element patterns, and then the pattern synthesis is viewed as finding the projection on a linear space spanned by this basis, which can be solved by using WLMS optimization. Recently, an efficient synthesis method called the weighted alternating reverse projection (WARP) was developed in [15, 16], which combined the power of WLMS and alternating projection together by iteratively modifying the target pattern and weighting values of different directions.

With the inspiration of WARP, we have proposed an array synthesis method with consideration of embedded element patterns by using weighted alternating projection and proximal splitting [17], whose salient characteristic is that the projector over excitation constraint set is designed as the solution to a constrained weighted least mean squares (CWLMS) problem which can be efficiently solved by using proximal splitting method. In this paper, our alternating projection and proximal splitting method is extended to consider the scaling invariance property of design specifications and constraints (particularly for the amplitudes of pattern and excitation).

The rest of the paper is organized as follows. In Section 2, we present the problem statement for array synthesis with embedded element patterns. The proposed array pattern synthesis method is developed in Section 3. Several synthesis examples are reported in Section 4 to demonstrate the efficiency of the proposed method. Conclusions are drawn in Section 5.

#### 2. Problem Statement

Consider an arbitrary antenna array with elements. Suppose that the embedded pattern of the th element with sampling directions is expressed as , where and are the elevation and azimuth angles. The far field pattern obtained by the superposition of those embedded element patterns can be written aswhere is the array excitations, , and . The purpose of array synthesis is to find a set of complex excitations so that the field pattern and excitations can meet the specified requirements.

Same as AAP method [13], we consider the scaling invariance property of design specifications and constraints for the amplitudes of pattern and excitation. The scalable pattern constraint sets are defined aswhere is arbitrary nonzero scaling parameter for pattern constraints and and are the upper and low masks. The scalable excitation constraint sets are represented aswhere and are the amplitude and phase of , is arbitrary nonzero scaling parameter, and , , , and limit the amplitude and phase variations for each excitation. Obviously, for the case of phase-only array synthesis, we have . It is worth noting that and may not be convex because they are the unions of some convex sets. The purpose of the array synthesis is to find the intersection of these pattern and excitation constraint sets.

#### 3. Proposed Array Pattern Synthesis Method

##### 3.1. The Method of Alternating Projections

According to the above discussion, the array synthesis problem can be formulated as the search of the intersection of constraint sets. Our solution is the method of AP [18, 19]. Here we first briefly introduce the method of AP. Given sets with , we seek a feasible solution which belongs to the intersection of constraint sets. For the AP algorithm, the most important concept is the projection operator. The projector over a subset is defined as the operator whose value of is a point such that every point in has a distance from not smaller than . Note that when is a nonconvex set, more than such a point can exist; in this case we assume that a rule of choice has been fixed. The AP algorithm can be expressed by the iterative processwhere is the iteration time and , are the projectors over the constraint sets . That is to say, the method of AP can be implemented by projecting onto the constraint sets cyclically in each iteration, with the desired point obtained in the limit. In this paper, a weighted AP algorithm is used in the application of array synthesis with scalable pattern and excitation constraints.

##### 3.2. Scalable Pattern Projection

In this subsection, the scalable pattern projection over pattern constraint sets is discussed. We first define the projector over with a fixed scaling factor bywhere . Then the projector over the scalable pattern constraints can be defined aswhere and is an arbitrary norm definition, such as , , and norms. In this paper, we select norm since the norm dynamic projector presents the most robust operation in terms of solution quality and convergence rate [13]. In each iteration, the optimum scaling factors can be determined by following Algorithm 1. For convenience, we denote and , where and represent the real and imaginary parts of a complex vector, respectively.

*Algorithm 1 (iterative procedure for optimum pattern scaling factor ). **Step 1*. Initialize the scaling factor .*Step 2*. Obtain by carrying on the projector .*Step 3*. Calculate the relative error .*Step 4*. Update the scaling factor by . *Step 5*. If convergent or equal to the maximum iteration time, then and break; else return to Step 2.

##### 3.3. Scalable Excitation Projection

Now, we address the scalable excitation projection over excitation constraint sets. After we obtain a new pattern by using projector , an appropriate way must be found to transform to a new excitation . It can be implemented by minimizing the difference between the temporary field pattern and synthesized radiation pattern under the excitation constraint , which can be solved by a constrained weighted LMS such thatwhere is a diagonal matrix with weights for each direction; it can be adjusted according to the relative importance of the different directions of the temporary pattern. The details of choosing will be given in Section 3.4.

To solve (7), a proximal splitting method is developed. We first introduce the proximity operator [20, 21]; for a point in set , the minimization problemadmits a unique solution, which is denoted by . The operator thus defined is the proximity operator of . Similar to the definition of projector , if there are multiple minima, we assume that a rule of choice has been fixed.

Obviously, (7) is equivalent to the following unconstrained problem:where is the indicator function of :It may not be convex because of the nonconvexity of . From [17, 20, 21], we can know that (9) can be effectively solved by using forward-backward splitting iterative processwhere is the forward gradient step, is the iteration time for solving (9), and is a step-size parameter which can be obtained by the normalized steepest decent [22]:where and . is the backward proximity operator which is equivalent to the projection over the excitation constraints .

Here one remark is presented for (11). In the strict sense, the forward and backward splitting iterative process can be applied for convex functions only. When is convex, the forward and backward algorithm converges to the unique solution to (9). When is nonconvex, the convergence of this method is still an open question and the optimum solution cannot be guaranteed [23]. However, despite lack of adequate theoretical justification, the forward and backward method is routinely applied to problems in the absence of convexity with good results [24, 25]. Furthermore, when we check (11) in detail, it can be seen that the forward step is well-posed because is convex. For the backward step , it is ill-posed because is nonconvex and the projection over is not unique. In this case, we should define a fixed rule for the projector over , which is described as follows.

We first define an operator , which is the projector over for a fixed scaling factor :where and force the excitation coefficients to satisfy the amplitude constraint and phase constraint , respectively:Letting , we have the projector of over such thatwhere , which can be calculated using Algorithm 2.

*Algorithm 2 (iterative procedure for optimum excitation scaling factor ). **Step 1*. Initialize the scaling factor .*Step 2*. Obtain by carrying on the projector .*Step 3*. Calculate the relative error .*Step 4*. Update the scaling factor by . *Step 5*. If convergent or equal to the maximum iteration time, then and break; else return to Step 2.

Here we also use norm to compute the optimum scaling factors and denote and . Thus the iterative process of (11) can be rewritten asand the operator can be approximated bywhere is the number of iterations for solving (9). For the initialization of the excitation in (16), we set . The way of initialization is very efficient and has good convergence.

After the operation of , a new excitation is obtained. Then we can transform to a new field pattern by applying a linear transform . Therefore, the whole alternating projections can be expressed bywhere is the iteration time of AP.

##### 3.4. Selection of the Weighted Matrix

In this subsection, the choosing of weighted matrix is discussed. Similar to [15, 16], we first calculate the overranging differences for different directions between the obtained pattern after the operation of and the upper mask or lower mask at each iteration; that is,where is the optimum scaling factor determined by Algorithm 1. Then the weights are adjusted according to the differences in each direction, which is expressed bywhere is the used weight at previous iteration, represents the adjusted weight, and is an S-shape function with the scale parameter . The purpose of is to regularize the sum of weights in different directions to be 1, which can avoid the large variation of and improve the stability of weighted alternating projection.

##### 3.5. Algorithm Summary

The proposed array synthesis method is summarized in Algorithm 3.

*Algorithm 3 (scalable array synthesis using weighted alternating projection and proximal splitting). **Step 1*. Initialize the field pattern and matrix .*Step 2*. Compute the optimum pattern scaling factor .*Step 3*. Impose the scalable pattern constraint using .*Step 4*. Calculate the error value and update .*Step 5*. Solve the constrained weighted LMS using .*Step 5.1*. Initialize excitation .*Step 5.2*. Carry on the forward iteration using .*Step 5.3*. Compute the optimum excitation scaling factor .*Step 5.4*. Impose the scalable excitation constraints using .*Step 5.5*. If convergent, then break; else return to Step 5.2.*Step 6*. Calculate field pattern using .*Step 7*. If satisfies the pattern and excitation constraints, then stop; else return to Step 2.

Now the computational complexity of Algorithm 3 is briefly discussed. According to Algorithm 3, Table 1 lists the computational complexity for each step. From Table 1, we can see that the complexity of our algorithm is aboutwhere and are the numbers of outer and inner iterations of AP. and are the iteration times of computing the optimum pattern and excitation scaling factors. and are the numbers of array elements and pattern sampling directions. is the exponential complexity which arises from the calculations of weighted matrix and operator . It is worth mentioning that the number of inner iterations does not need to be very large to ensure the convergence of inner iteration. A small value can be set to reduce the oscillations in iterative process and accelerate the convergence of AP method [26]. For the iteration times of and , from the statement in [13], we can know that few iterations should suffice in approaching the norm exact projection to find the optimum scaling factors. So they can be set to be small values which leads to negligible increase in the computational cost compared to the fixed scaling factors.