# Adaptive Antenna Array Control Algorithm in Radiocommunication Systems
## 4 Model and Experimental Tests of an Adaptive Microstrip Antenna Array
The key problem of ensuring the proper operation of the adaptive antenna is the algorithm for generating \(w_{m}\) weights. This algorithm can be implemented in software or hardware. The algorithms used can be divided into two groups:  
* With feedback;
* No coupling. The basic feature of systems using uncoupled algorithms is the generation of \(w_{n}\) scales only on the basis of input data and control of weighing systems with a small but finite error. The operation of such a system is not corrected by output signals, as is the case in systems with feedback. All algorithms are based on two basic elements:  
* Inverted covariance matrix of received signals \(s_{n}\);
* On the control vector. The signal at the input of each element of the antenna array is a composite of thermal noise, signals of interfering sources and the desired signal in the antenna illumination zone FOV (field of view). Figure 6: Adaptive antenna system. In a simplified form, we can write it as the following equation:  
\[s_{n}=s_{noise}+s_{disruption}+s_{signal}\]  
The goal of each algorithm is to generate weights that will result in the maximum reduction in the influence of interfering signals and the domination of the output signal by the desired signal. The basic operations of the algorithm are presented in Figure 7. The designed algorithm model is a mathematical model that graphically reflects the operation of the adaptive antenna array process in the base station of the mobile telephony system. It ensures the adaptation of the radiation characteristics of the antenna array to the changing situation in the sector covered by the antenna. The model was created based on formulas and expressions describing the LMS adaptive algorithm [24]. The useful signal used in the model is described by the following relationship:  
\[d_{ni}=A_{di}e^{-j(-n\varphi_{di})} \tag{4}\]  
where:  
\(d_{ni}\)_i-th_ useful signal received in the _n-th_ array element;  
\(A_{di}\) amplitude of the _i-th_ useful signal;  
\(\varphi_{di}\) difference between the phases of the _i-th_ signal received in two consecutive antenna components. The phase shift \(\varphi_{di}\) is determined based on the given angle of arrival of the signal using the following relationship:  
\[\varphi_{di}=\pi sin(\theta_{di}) \tag{5}\]  
where:  
\(\theta_{di}\) is the angle of incidence of the _i-th_ useful signal. Interfering signals are described similarly. \[z_{ni}=A_{zi}e^{-j(-n\varphi_{zi})} \tag{6}\]  
Figure 7: Basic operations of the adaptive algorithm. The useful signal vector can be represented as:  
\[\mathbf{X}_{x}= \begin{bmatrix}x_{0i}\\ x_{1i}\\ \cdot\\ x_{ni}\end{bmatrix}  \tag{7}\]  
and the interference signal vector as:  
\[\mathbf{X}_{z}= \begin{bmatrix}z_{0i}\\ z_{1i}\\ \cdot\\ z_{ni}\end{bmatrix}  \tag{8}\]  
For the case of many useful and interfering signals, the signal vectors are as follows:  
\[\mathbf{X}_{x}= \begin{bmatrix}x_{00}&x_{01}&\cdot&x_{0i}\\ x_{10}&x_{01}&\cdot&x_{0i}\\ \cdot&\cdot&\cdot&\cdot\\ x_{n0}&x_{01}&\cdot&x_{ni}\end{bmatrix}  \tag{9}\]  
\[\mathbf{X}_{z}= \begin{bmatrix}z_{00}&z_{01}&\cdot&z_{0i}\\ z_{10}&z_{01}&\cdot&z_{0i}\\ \cdot&\cdot&\cdot&\cdot\\ z_{n0}&z_{01}&\cdot&z_{ni}\end{bmatrix}  \tag{10}\]  
The covariance matrix of the useful and interfering signals is obtained from the following relationship:  
\[\mathbf{\varphi}_{signal}=\mathbf{X}_{x}\mathbf{X}_{x}^{T} \tag{11}\]  
\[\mathbf{\varphi}_{disruption}=\mathbf{X}_{z}\mathbf{X}_{z}^{T} \tag{12}\]  
Total covariance matrix:  
\[\mathbf{\Phi}=\mathbf{\Phi}_{signal}+\mathbf{\Phi}_{disruption}+\mathbf{\Phi}_{noise} \tag{13}\]  
where:  
\(\mathbf{\Phi}_{noise}\) is the \(n\) by n noise covariance matrix. \[\mathbf{\Phi}_{noise}= \begin{bmatrix}\sigma^{2}&0&\cdot&0\\ 0&\sigma^{2}&\cdot&0\\ \cdot&\cdot&\cdot&0\\ 0&0&\cdot&\sigma^{2}\end{bmatrix}  \tag{14}\]  
where:  
\(\sigma\) is the noise power.