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 [31] . A coordinate descent method can be described as follow:

We have and.

The optimization problem for is

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 as initial value. Let

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

Now compute the and compare it with. While, which is a predefined value as quality criterion, repeat the procedure until.

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. The power pattern in far field is given by

where is the complex amplitude at the nth element with, where is the real amplitude

and is the phase at nth element. is the wave number for a wavelength and denotes the

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 [1] . In this paper we reach the desired beam pattern via minimization of a fitness function f, whose similar form has been introduced in [32] .

For a power pattern we define

converges to in the senses of by i passes the entire given interval. Now we rewrite into its complex form. According to (3) the fitness function (4) can be reformed to

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

with

where denotes sum of the remaining terms in, which are not depending on and and denotes complex conjugate.

Due to following important characters of

it is easy to prove

as shown in Figure 1. That means we can use a surjection

with

which do not change and, to cover the entire. Thus the new domain of is. Because and are independent of each other and f is a quadratic function, there is only one in the new and can be solved numerically.

A crucial advantage of CDM is that a weight function can be used, or more precisely, multiplied to the fitness function (4), in order to save the needed time and steps of the optimization process or achieve better results:

Because of the mathematical property of CDM, an arbitrary function can be multiplied to the fitness function, functioning as a weight function, without the need of changing any other part of the algorithm. Hence, an algorithm can be implemented to modify the fitness function during the optimization process by means of varying the weight function. With the application of different types of weight function (or the algorithm to calculate the weight function), this array pattern synthesis algorithm can be exclusively used for different optimization problems, in order to acquire maximal speed and best results.

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 (Figure 4(a)), the application of the weight function will only alter the speed of the optimization process, without changing the result, as the fitness function will always converge to zero. But when dealing with optimization problems which does not have a perfect matching, e.g. the problem of array with element failures (Figure 8), using the weight function can both change the speed of the process and the resulted beam pattern. This character of weight function can be quite useful when a improved result is required, but it also means that the weight function should be determined carefully when not expecting a huge change in the result.

A simple example is given here: when synthesizing the Dolph-Chebyshev pattern with CDM (shown in Figure 4(a)), the convergence is quite slow after the SLL started to descend. The reason can be considered as that at the side lobe area, the and are both considerable small and thus their difference gives less

feedback to the fitness function. Here weight functions like or can

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 Figure 2(a), that the SLL descend much faster when using the weight function. It is shown in Figure 2(b), how the SLL descend when matching a shaped beam using the same weight functions. In this example the optimization problem is from [33] , which is a perfect matching optimization problem. The beam pattern of this example is shown in Figure 6(a).

Another example for the optimization problems without perfect matching is given here with the desired function being a rectangular function (Figure 7(a)). The ripple of the shaped main beam should not exceed, while the SLL less than. The algorithm to calculate the adaptive weight function is described with a flow diagram in Figure 3.

In this example the desired pattern is a Dolph-Chebyshev pattern with 32 elements and SLL by. Equal distance d between two elements is half wavelength. In Figure 4(a) original Dolph-Chebyshev pattern and three other patterns from CDM, PSO and DE are shown. The desired function is an original Dolph-Chebyshev, thus it provides a perfect matching. The main lobs from all methods do not much differ from the original one. The pattern from CDM shows the best match of the SLL (perfect matching). This can be explained with the amplitude and phase distribution which is shown in Figure 5. Figure 4(b) represents the convergence of the fitness function over the time. by using CDM converges to 0.9999 in 5.8 seconds, while by using DE 1500 seconds are needed to achieve the same results. The optimization by using PSO, which converges to 0.9999 in 76 seconds, is neither get best results nor a short calculation time.

The synthesis of shaped beams is a classical problem since it has many applications ranging from radar and remote sensing to communication systems [34] . Two examples are shown, one is a perfect matching, the other one is a non-perfect matching. The first example is shown in Figure 6(a). This pattern has been represented in [33] . All three algorithms can synthesize a very similar beam pattern as the desired pattern. However, only the CDM succeed a perfect matching. The time of optimization by using CDM is 4.5 seconds for a, while the PSO 35 seconds and DE 1200 seconds needed for the same result.

The second example (Figure 7(a)) is a non-perfect matching, because the desired function is a rectangular function, which means that it is not possible to achieve a perfect matching with finite elements. For this synthesis the weighting algorithm in Figure 3 has been used to achieve the requirements which is described in subsection 3. The progress of convergence is shown in Figure 7(b). It converges at first at high speed and then become stable after 50 seconds. Since the optimization is non-perfect matching, the is not possible to reach one.

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 [35] -[39] . In this example we take the results from [36] as a reference. In this case the elements with Nr. 2, Nr. 5 and Nr. 6 of 32 elements Dolph-Chebyshev array

are failed, while the faulty elements do not radiate at all (“on-off” fault [40] ). The result is shown in Figure 8. The half-power beam width of the results from CDM is, which is with GA. The SLL from CDM is, which satisfies the original Dolph-Chebyshev condition as well.

In this example the minimization of the SLL of a array with uniform element distance half wavelength is presented. In this case the half power beam width should keep, while the SLL as low as possible. The result is shown in Figure 9(b). The SLL has sunk from to.