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This paper presents an array pattern synthesis algorithm for arbitrary arrays based on coordinate descent method (CDM). With this algorithm, the complex element weights are found to minimize a weighted
L
_{2} norm of the difference between desired and achieved pattern. Compared with traditional optimization techniques, CDM is easy to implement and efficient to reach the optimum solutions. Main advantage is the flexibility. CDM is suitable for linear and planar array with arbitrary array elements on arbitrary positions. With this method, we can configure arbitrary beam pattern, which gives it the ability to solve variety of beam forming problem, e.g. focused beam, shaped beam, nulls at arbitrary direction and with arbitrary beam width. CDM is applicable for phase-only and amplitude-only arrays as well, and furthermore, it is a suitable method to treat the problem of array with element failures.

Comparing conventional antenna, antenna array provides advantages in flexibility and versatility. By changing the complex weights (including amplitude and phase) on each array element, the radiation pattern can be controlled and reconfigured. The antenna array pattern synthesis problem consists of finding weights that satisfy a set of specifications on the beam pattern [

As an advantage of convex optimization, once an array pattern synthesis problem is convex, the global optimum must exist [

While the usage of convex optimization is restricted to the class of problems that can be transformed to be convex, the metaheuristic algorithms, e.g. GA, DE and PSO, can provide much more flexibility. They are not restricted in terms of types of array pattern synthesis problem. Furthermore, these evolutionary algorithms are able to deal with a huge parameter set without getting trapped in the local minima [

In this article, an array pattern synthesis algorithm for arbitrary arrays based on coordinate descent method (CDM) is presented. CDM has been first mentioned in [

This paper is an illustration of the utility of CDM for array pattern synthesis problems. In Section 2, a briefly analytical description of CDM is presented, along with the formulation of the array pattern synthesis problems and corresponded fitness function. It is shown in Section 2.4, how the applications of adaptive weights function accelerate the optimization process and help achieving better results. Example of the applications of the CDM in different array pattern synthesis problems with comparisons to known algorithms is shown in Section 3. Then there is the conclusion in Section 4.

Coordinate descent methods were among the first optimization schemes suggested for solving smooth unconstrained minimization problems. The main advantage of these methods is the simplicity of each iteration, both in generating the search direction and in performing the update of variables [

We have

The optimization problem for

As the optimization in (1) has no closed-form solution, we resort to an iterative method based on coordinate descent. In coordinate descent optimization, each parameter is optimized in turn, while holding all other parameters fixed. Because the parameters are interrelated, optimization over the entire set of parameters must be

performed a number iterations. We start with arbitrary

denote the optimum value of nth coordinate (the nth element) of the ith iteration; then, the coordinate descent estimate at the next iteration

Now compute the

For the sake of clarity, our array pattern synthesis problem is described for a linear array. Consider an antenna array composed of N isotropic elements placed at arbitrary and known locations

where

and

polar angle.

The goal of array pattern synthesis problem is to find optimal parameters including array weights so that the designed beam pattern satisfies a set of specifications [

For a power pattern we define

For the n-th element the fitness function is given by

with

where

Due to following important characters of

it is easy to prove

as shown in

with

which do not change

A crucial advantage of CDM is that a weight function

Because of the mathematical property of CDM, an arbitrary function

It should be emphasized that the effect of the usage of the weight function will be different according to the type of optimization problems. For problems which can obtain a perfect matching, e.g. Dolph-Chebyshev pattern (

A simple example is given here: when synthesizing the Dolph-Chebyshev pattern with CDM (shown in

feedback to the fitness function. Here weight functions like

be employed, in order to enlarge the weights of the SLL in the fitness function. Thus the convergence is observably accelerated, which can be seen in

Another example for the optimization problems without perfect matching is given here with the desired function being a rectangular function (

Examples of the array pattern synthesis problem using CDM are presented and compared with PSO, GA and DE. All simulations are carried out on Intel^{®} Xeon^{®} CPU E5-2687W v2 @ 3.40 GHz with 256 GB RAM computer. All simulations use the fitness function (4) with different weights. In the following we use

In this example the desired pattern is a Dolph-Chebyshev pattern with 32 elements and SLL by

The synthesis of shaped beams is a classical problem since it has many applications ranging from radar and remote sensing to communication systems [

The second example (

Beam pattern reconfiguration of an antenna array with faulty elements has practical importance for antennas which cannot be repaired immediately. The possibility of reconfiguration of the antenna array with faulty elements has been considered by several authors over the years [

are failed, while the faulty elements do not radiate at all (“on-off” fault [

In this example the minimization of the SLL of a

An iterative method, which is based on the coordinate descent method to synthesize most of the current beam

patterns from linear to planar array, is presented above. CDM allows changing the fitness function with a multiplication of a weight function

We thank the Editor and the referee for their comments.