The Chaotic beamforming adaptive algorithm is new adaptive method for antenna array’s radiation pattern synthesis. This adaptive method based on the optimization of the Least Mean Square algorithm using Chaos theory enables fast adaptation of antenna array radiation pattern, reduction of the noisy reference signal’s impact, and the improvement of the tracking capabilities. We performed simulations for linear and circular antenna arrays. We also compared the performances of the used and existing algorithms in terms of the radiation pattern comparison.

The successful design of the adaptive antenna system depends on the selection and performance of the beamforming algorithm used for radiation diagram adaptation and adjustment to the specific scenario of incoming signals. Spatial filtering of desired signals and minimizing the impact of interfering signals are necessary in order to achieve improvements of the transmission quality and the wireless communication systems capacity.

The simplest and most widely used adaptive algorithm is the Least Mean Square (LMS) algorithm and its modifications [

The LMS algorithm and its modifications achieve good results only in cases of antenna arrays with a large number of elements. Also, it is necessary to specify a large number of algorithm parameters with the exactly defined reference signal, which makes these algorithms very complex. This paper uses the Chaotic beamforming adaptive (CBA) algorithm [

The paper is organized as follows: in Section

Figure

The antenna array geometry: (a) linear and (b) circular.

Total signal received by the antenna array

Taking the first antenna in the antenna array as a reference, the matrices are given by the following equations:

In this paper, we used the Chaotic beamforming algorithm [

Figure

Block diagram of the Chaotic beamforming algorithm.

The task of the chaotic optimization is to determine

Description of variables is as follows:

Chaotic optimization is based on the chaotic search. The search procedure which is composed of two parts, global and local search, is shown as follows.

Chaotic search consists of the following:

Global search is as follows:

Choosing the parameters for Chua’s equations.

Initialization of the initial conditions

The normalization of variables

The determination of the maximum number of iterations

Forming variables

In the

The coordinates of the vector

Local search is as follows:

Determine the number of iterations for local search

In the

Coordinates of vector

By using the Chaotic beamforming adaptive algorithm, we obtained linear and circular antenna arrays radiation patterns for different scenarios of incoming signal. We analyzed the impact of noisy reference signal on the algorithm performance in Section

Here we considered the impact of noisy reference signal on the Chaotic beamforming algorithm performance. White Gaussian noise is added to the reference signal. Signal to noise ratio (SNR) values used in the simulations were 3, 5, and 7 dB. The obtained radiation patterns for the different SNR values and the combination of the desired and interfering signals’ incident angles are shown in a normalized form. The normalization is performed with respect to the diagram with the highest gain in the direction of the desired signal.

Figure

Radiation pattern of normalized array factor for different values of SNR: the angle of arrival of the desired signal is

Figure

Radiation pattern of normalized array factor for different values of SNR: the angle of arrival of the desired signal is

Based on the results shown in Figures

Figure

Comparison of the normalized radiation patterns obtained by the Chaotic beamforming algorithm and the LMS algorithm: the angle of arrival of the desired signal is −20° and the angle of arrival of the interfering signal is 45°, (a) in the (

Figure

Comparison of the circular array radiation patterns obtained by the Chaotic beamforming algorithm and LMS algorithm, shown in plane of azimuthal angle 60°.

Based on the results shown in Figures

Here we considered the effect of interfering signal’s angles of arrival on the algorithm performance in cases of accurate and noisy reference signal. To verify the tracking capabilities of the algorithm, the interfering signal’s angles of arrival are selected to be very close to the desired signal’s angle of arrival.

The obtained radiation patterns for the different combinations of desired and interfering signal’s angles of arrival are shown in a normalized form. In all cases, the number of antenna array elements is

Figure

Normalized radiation pattern of the linear antenna array: (a) the angle of arrival of the desired signal is −10° and the angle of arrival of the interfering signal is −20°; (b) the angle of arrival of the desired signal is 0° and the angles of arrival of the interfering signals are −10° and 20°.

Figure

Normalized radiation pattern of the linear antenna array: the angle of arrival of the desired signal is 20° and the angle of arrival of the interfering signal is 5°; SNR = 7 dB.

Based on the results shown in Figures

Chaotic beamforming algorithm is applied on antenna arrays with a different number of antenna elements, special emphasis being on the algorithm performance in the case of antenna arrays with a small number of elements. The obtained comparative radiation patterns for different combinations of angles of the desired and interfering signals are shown in a normalized form. Linear antenna arrays with

Figure

Comparison of the antenna arrays radiation pattern with

Figure

Comparison of the antenna arrays radiation pattern with

Based on the results shown in Figures

Based on the foregoing, it can be concluded that the applied algorithm performs a very good adaptation of the radiation pattern and suppression of interfering signals in the case of antenna arrays with a small number of elements.

In this paper, we used the Chaotic beamforming adaptive algorithm, based on the optimization of the LMS algorithm using Chaos theory. The obtained radiation patterns are shown in normalized form, for different values of desired and interfering signals’ arrival angles, different levels of noise in the reference signal, and a different number of elements in the array.

We analyzed the influence of the noisy reference signal on the performance of the Chaotic beamforming algorithm. In all analyzed cases, the Chaotic beamforming algorithm accurately directs the main lobe in the desired signal’s direction of arrival. It sets deep nulls in the interfering signal’s direction of arrival regardless of the reference signal noise level. Based on these results, it can be concluded that the Chaotic beamforming algorithm shows robustness to the presence of noise in the reference signal, which justifies the use of chaotic parameter optimization of LMS algorithms.

To verify the tracking capabilities of the algorithm, interfering signals’ angles of arrival are selected to be very close to the desired signal’s angles of arrival. The Chaotic beamforming algorithm precisely estimates the interfering signals’ direction of arrival and sets deep nulls on the diagram in these directions whether the reference signal is noisy or noiseless. It can be concluded that the applied algorithm has very good tracking capabilities and suppression of interfering signals.

In the cases of antenna array with small number of elements, precise estimation of the desired and interfering signal’s angle of arrival is achieved whether noisy or noiseless reference signal was used. In the case of antenna array with a small number of elements, the Chaotic beamforming algorithm sets deep nulls in direction of interfering signals. It can be concluded that the applied algorithm makes very good adjustment of the radiation pattern and suppression of interfering signals in the case of antenna arrays with a small number of elements.

Based on the aforementioned, the Chaotic beamforming algorithm proved to be very precise and relatively simple to use, which makes it suitable it for analysis and adjustment of radiation diagram for complex antenna systems, such as planar and conformal antenna arrays.

The authors declare that they have no competing interests.