Deterministic algorithms belong to two large groups of algorithms for synthesizing radiation patterns with heuristic algorithms [9] . We can classify the deterministic algorithms for the detection of directions of arrival according to several criteria that may relate either to the approach used, to the information to which they are used, or to the implementation approach. What interests us in this section is their computation time during the synthesis of the antenna arrays.

However, we generally retain the following three main classes:

In [5] - [7] , the BARLETT method, known as the spatial Fourier transform, is among the first known techniques for estimating the direction of arrival. This relationship means that in order to increase the resolution, it is necessary to increase the number of elements of the antenna array or the inter-element distance factor. The average computing time is 0.17566 s. After using the genetic algorithm on BARLETT, the average time increases, and we get 0.265775 s.

Limited by its low resolution, more advanced techniques quickly had to be used.

The CAPON or MVDR (Minimum Variance Distorsionless Response) estimation method. This exploited method [4] - [7] consists of estimating, by the criterion of maximum likelihood, the power received in a given direction by considering the other sources as interferents. CAPON has reached an average computation time very close to 0.178447 s. After using the genetic algorithm on CAPON, the average time is 2.274417 s.

In order to minimize the prediction error on the response of any element of the network [6] [7] , PRONY has implemented a method of finding weighting coefficients of the network, which will minimize the average value of this error. [7] The latency of PRONY is 0.190271 s, and the application of the genetic algorithm on PRONY rather worsens, and they reach 0.263529 s.

This is attributable to BURG, Maximum entropy estimation, whose goal is to find the directions that maximize the directions of arrival [6] . MEM’s latency time is 0.177403 s, and the application of the genetic algorithm on MEM gets worse, reaching 0.269666 s.

The methods of subspaces are based on the decomposition of space into a noise space and a signal space by searching for the eigenvectors of the correlation matrix of the observation vector [7] - [13] . It was PISARENKO who had the idea in 1973 by showing that its lowest eigenvalues corresponded to noise, which made it possible to divide the space in two and to deduce the directions of arrival.

This method displays a calculation time per lap of 0.172794 s, and by applying this, we obtain 0.264251 s.

The goal of PISARENKO’s harmonic decomposition technique is to minimize the root mean square error of the network output under the constraint that the norm of the weight vector is equal to unity. The eigenvector of the correlation matrix, which minimizes this mean square error and minimizes the execution time, is associated with the smallest eigenvalue [2] [7] . Its latency time is 1.173082 s, and with the genetic algorithms on PISARENKO, we obtain 0.272009 s.

The Minimum Standard Method was developed by REDDI, KUMARESAN, and TUFS, which optimizes the weighting vector by solving the equations. Minimum standards run in 0.172794 s, and with Genetic Algorithms, we reach 0.264251 s.

The MUSIC approach, offered in its basic version by SCHMIDT, is one of the most popular techniques used for estimating directions of arrival. It has a high angular resolution and also makes it possible to determine, in addition, the number of sources and the power of the incident signals.

The pseudo-spectrum then gives:

On the other hand, the MUSIC algorithm does not work if the noise and the incident signals are strongly correlated. [6] [7] [11] , its latency time is similar to others, around 0.17947 s, and in combination with Genetic Algorithms, we have 0.268659 s. This confirms that MUSIC is better because it improves resolution at the same execution times.

The ESPRIT method exploits the rotational invariance of the signal subspace and the translational invariance of the structure of the array of elements by breaking it down into two identical antenna sub-arrays, one of which can be obtained by translation of the other. [7]

The algorithm uses the same signal model as the MUSIC algorithm, but it has the advantage of drastically reducing the computing power and memory required for storage.

Besides all these methods, the maximum likelihood method is considered to be asymptotically efficient and without distortion [7] - [13] .

They are often preferred over other methods when they have simple analytical solutions.

Unfortunately, the analytical resolution of this problem is cumbersome and difficult to implement [6] . Hence, there is a need to rely on other optimization methods, in particular global optimum approaches, to resolve them [7] - [13] .

Several techniques for estimating directions of arrival have been studied in this section. They all have limitations, among others: complexity and computation time, which are not always favorable to real-time applications, accuracy, and antenna size.

Two types of heuristics are mainly used: construction heuristics (for example, greedy methods), which iteratively build a solution, and descent heuristics, which from a given solution seek a local optimum.

More advanced heuristics have been developed and have given rise to a new family of algorithms: meta-heuristics.

The goal of a meta-heuristic is to succeed in finding a global optimum. To do this, the idea is both to browse the search space and to explore areas that appear promising.

In [14] - [16] artificial neural networks are characterized by their efficiency and performance in terms of the speed of convergences. Tools made it possible to test and compare analytical and neuronal methods. The learning base was produced using the analytical synthesis method. Measurements were carried out on several lobe configurations in order to prove the effectiveness of the neural network approach. The results obtained show a good agreement between the simulation and the measurements.

Genetic algorithms are programming techniques that mimic biological evolution as a resolution strategy. It is initialized by a set of probable or randomly chosen candidate solutions, and then this set is coded in a certain way. A metric called the cost or adaptation function (fitness in English) allows each candidate to be quantified. A selection criterion is then applied in order to retain some of these candidate parents, who will be used to produce other candidate sons by random mutation and crossing operations. The stop criterion allows either to resume operations from the metric calculation or to retain a final solution close to the global optimum [1] [8] .

Article [1] [8] makes it possible to notice that for the arrival directions, the times elapsed by GA are 4 times greater than those previously obtained by analytical methods. However, they hover around 1 s, which is just as acceptable.

At the end of this work, they were able to observe that for the arrival directions, the times elapsed by GA are 4 times greater than those obtained previously by analytical methods.

To meet the demands in terms of speed and quality of service, which could be triggered by big data and the Internet of Things, how can we use antenna networks and neural networks to reduce their computing time?

In order to improve the execution time of smart antenna synthesis algorithms by:

We used the powerful MATLAB R2016a programming environment with its various toolboxes:

The tic and toc functions are the tools provided by Matlab to measure the program performance. They start the timer to access the execution time of the function. The tic function starts the stopwatch, and toc reads the time elapsed since the start of the tic function.

For the observations, we used RFS sector antennas of the 1700 - 2200 MHZ frequency range with the following characteristics:

Optimizer® Dual Polarized Antenna, 1710 - 2200, 65 deg, 18.0 dBi, 1.3 m, VET, 4 - 14 deg.

The implementation of smart antennas goes through two successive stages: the determination of the directions of arrival and the consequent steering of beams.

Design:

Build a Matlab code to visualize the different algorithms and compare them.

Figure 1 summarizes the work covered by this power.

We consider a MIMO system made up of m_t antennas on transmission and m_r antenna on reception. We denote by x the vector of size m_t containing the symbols received. The relation which connects x and y is then written:

$y=Hx+n$ (7)

where H is the matrix of channel of size m_t × m_r and n is the noise vector. The capacity of the MIMO channel:

$C=\log \left(\det \left(I_{m_r}+\rho HQ{H}^{*}\right)\right)$ (8)

In this formula $I_{m_r}$ is the identity matrix, $\rho$ is the signal-to-noise ratio, and Q is the correlation matrix of the emitted symbols.

To overcome these problems, the solution is to design a system in which the diagram would be dynamic, with “radiation holes” and privileged listening directions. This is the principle of smart antenna systems: transmit or receive in the directions of interest and remain “deaf” or “mute” in others, depending on the position of users and sources of interference. It is illustrated in Figure 2 .

It promises very significant capacity gains and is the ultimate solution that will significantly increase throughput.

Array antennas come in several geometries [3] , the most common of which are linear, circular, or planar. They are called uniform if the antenna elements are evenly spaced.

Consider a uniform linear network of elements regularly spaced apart by a distance (see Figure 3 ). These sources are supplied with the same amplitude and with a phase gradient. For a point P located in the far radiation zone, the total field is the summation of the field radiated by each of the sources, namely:

$E\left(\vartheta \right)=E_0\left(\vartheta \right){\displaystyle \underset{k=0}{\overset{K-1}{\sum }}\exp \left[jk\left(2\pi \frac{d}{\lambda }\sin \vartheta +\phi \right)\right]}$ (9)

where $E_0\left(\vartheta \right)$ is the radiation of an isolated element.

The network factor, which depends on the law of excitation of the elements of the antenna and their arrangements, is defined by:

$AF\left(\vartheta \right)={\displaystyle \underset{k=0}{\overset{K-1}{\sum }}\exp \left[jk\left(2\pi \frac{d}{\lambda }\sin \vartheta +\phi \right)\right]}$ (10)

In this section, the general concept of the DoA estimation methodology using NNs is stated. Consider N uncorrelated signals with amplitudes $h_i$, $i=1,\cdots ,N$ impinge at the Ne element antenna array of a SBS like in Figure 4 and Figure 5 .

Let the angles of arrival ${\theta }_i$, $i=1,\cdots ,N$ composed a vector $\theta =\left({\theta }_1,{\theta }_2,\cdots ,{\theta }_N\right)$. M beams are used to cover the desired sector. If beam switching takes place, the m_th beam gives the main output of the system’s total received power

$P_m^{out}\left(\theta \right)$, $m=1,2,\cdots ,M$, or

$\begin{array}{l}{P}_1^{out}\left(\theta \right)=h_1^2{P}_1\left({\theta }_1\right)+h_2^2{P}_1\left({\theta }_2\right)+\cdots +h_N^2{P}_1\left({\theta }_N\right)\\ P_2^{out}\left(\theta \right)=h_1^2{P}_2\left({\theta }_1\right)+h_2^2{P}_2\left({\theta }_2\right)+\cdots +h_N^2{P}_2\left({\theta }_N\right)\\ ⋮\\ P_M^{out}\left(\theta \right)=h_1^2{P}_M\left({\theta }_1\right)+h_2^2{P}_M\left({\theta }_2\right)+\cdots +h_N^2{P}_M\left({\theta }_N\right)\end{array}$ (11)

It is assumed that the signals are subject to power control impinging on the base station at the same mean power level, which in the following [18] [19] procedure is considered to be unity.

$h_1^2=h_2^2=\cdots =h_N^2=1$

This became

$\begin{array}{l}{P}_1^{out}\left(\theta \right)=P_1\left({\theta }_1\right)+P_1\left({\theta }_2\right)+\cdots +P_1\left({\theta }_N\right)\\ P_2^{out}\left(\theta \right)=P_2\left({\theta }_1\right)+P_2\left({\theta }_2\right)+\cdots +P_2\left({\theta }_N\right)\\ ⋮\\ P_M^{out}\left(\theta \right)=P_M\left({\theta }_1\right)+P_M\left({\theta }_2\right)+\cdots +P_M\left({\theta }_N\right)\end{array}$ (12)

The system of Equation (11) shows that the contribution of each signal to the total received power depends on its angle of incidence and the power pattern of the receiving beam. A power vector mapped to the corresponding angle vector can be constructed

$P^{out}\left(\theta \right)=\left\{P_1^{out}\left(\theta \right),P_2^{out}\left(\theta \right),\cdots ,P_M^{out}\left(\theta \right)\right\}$ (13)

A NN is a structure of interconnected information-processing units called neurons, organized in the form of layers [18] [19] . NNs are trained to model complex relationships between certain inputs that produce certain outputs, determined by the weight connections of the neurons. In our problem, NNs should be trained to accept as input the measured/calculated power for each beam, and give as output the signals DoA.

For the NN to work, the number of the output nodes must be equal to the number of the angles of arrival to be estimated.

Therefore, generally, the network should know the total number of incoming signals in order to perform DoA estimation for all of them.

Let K be made up of a vector of N ${\theta }_k$ random, it is generated initially in a random way. The index k indicates the k^th vector. The elements of the vectors ${\theta }_k$ are les ${\theta }_{ki}$, $k,i$ are the integers. [18] [19]

For each ${\theta }_k$ cooresponding $P_k^{out}$ which is calculatate from (28) and (29). Randomly, verses ( $P_k^{out}$, ${\theta }_k$) are generated, which is the training set of our neural network.

The activation function of the hidden layers is the hyperbolic tangent function, and that of the output layer is the linear function.

$AF\left(\vartheta \right)=\frac{\sin \left[\frac{K}{2}\left(2\pi \frac{d}{\lambda }\sin \vartheta +\phi \right)\right]}{\frac{1}{2}\left(2\pi \frac{d}{\lambda }\sin \vartheta +\phi \right)}\exp \left[j\frac{K-1}{2}\left(2\pi \frac{d}{\lambda }\sin \vartheta +\phi \right)\right]$ (14)

To obtain a maximum of radiation in a given direction, it is necessary to find the phase gradient that maximizes the modulus of the network factor in this direction, namely:

$2\pi \frac{d}{\lambda }\sin {\vartheta }_0+\phi =0$ (15)

In other words, the pointing direction of the network will be given by the relation:

${\vartheta }_0={\sin }^{-1}\left(\frac{\phi }{2\pi }\frac{\lambda }{d}\right)=0$ (16)

We can thus adjust the orientation of the radiation of an array antenna by playing on the phase gradient between its antenna elements: this is the principle of scanning antennas.

Another very important factor to consider when determining the direction of arrival is the availability of traffic cells or channels. The parameters that can influence the availability of cells are:

The above paragraph illustrates some blocked cells and the corresponding alarms. We can see in Figure 6 and Figure 7 . All blocked cells have zero user connections.

To model this situation, we will introduce the distribution of Paul Dirac.

It is not necessary to go over the details of this theory developed in mathematics

lessons but simply to remember the few elementary results that interest us in the case of the two fundamental signals that will be used:

The distribution of Dirac δ (t).

Product of a function, regular distribution, by the Dirac distribution on

$P^{out}\left(\theta \right)=\left\{P_1^{out}\left(\theta \right),P_2^{out}\left(\theta \right),\cdots ,P_M^{out}\left(\theta \right)\right\}$

$x\left(t\right)\delta \left(t-t_0\right)=x\left(t_0\right)\delta \left(t-t_0\right)$

$P^{out}\left(\theta \right)\delta \left(\theta -t_0\right)=P^{out}\left(t_0\right)\delta \left(t-t_0\right)$

To introduce the blocked TX/RX of a BTS, we use the Dirac function:

${\delta }_x\left(\begin{array}{c}1\\ 0\end{array}\right)$

The learning base becomes

$\left(P_k^{out}\delta \left(\theta -t_0\right),{\theta }_k\right)$

${\delta }_x=1$, if the TX/RX cell is functional.

${\delta }_x=0$, if the cell is blocked.

Choose the configuration of the antenna network, which is either linear, planar, or circular. Enter the other parameters, such as:

The number of M elements:

However, the number of sources must be strictly less than the number of antenna elements (L < M for linear and circular arrays or L < (M × M) for planar arrays). In other words, an array of K elements can only properly detect at most (K − 1) sources.

The inter-element distance in terms of lambda: d is set by default to 0.5, but the user can also modify it at will and observe the influence.

The number of samples N whose default maximum value has been set to 100.

The radius of the ring a (which is only taken into account for the circular network) is set by default to 0.25.

Enter the parameters of the sources to be detected, namely:

The frequency of use f0, the default value of which is set at 1.8 GHz, but the modification of which does not significantly affect the results. So you can try it at 2.4 GHz or 5 GHz.

The signal-to-noise ratio, the default value of which is fixed at 30 dB, is modifiable.

Directions of arrival (“DDA” in French and “DOA” in English) according to the number of lobes desired. We indicate where the source (s) we want to detect are located.

The analytical methods will serve as a reference. Synthesis using PMUC and RBFNN.

The synthesis is applied to each of the analytical methods developed previously:

MLP_sur_Barlett, MLP_ON_Prony, MLP_ON_Capon, MLP_ON_MEM, MLP_ON_MMSE, MLP_ON_MUSIC, MLP_ON_MinNorm. RBF_ON_Barlett, RBF_ON_Prony, RBF_ON_Capon, RBF_ON_MEM, RBF_ON_MinNorm, RBF_ON_MMSE, RBF_ON_MUSIC.

For all these summaries, we must set:

The number of neurons in the input layer of the Artificial Neuron Network (ARN), ML for the case, by default set to 1;

The number of neurons in the output layer, which is automatically displayed equal to the number of rows of the vector/matrix P;

The learning rate, which determines how quickly the learning algorithm converges;

The learning algorithm, we have 11 choices possibles (traingdx, traingdm, traingd, trainlm, trainbfg, trainrp, trainbr, trainscg, traincgb, traincgf, traincgp). The value chosen by default is traingdx because of its acceptable results;

The momentum, or momentum constant, which is a value introduced to prevent the learning algorithm from getting stuck in a local minimum, also increases its speed of convergence;

MSE: Mean Square Error, or Mean Square Error (EQM), which is a tolerance threshold constituting one of the 2 criteria for stopping the learning phase;

Iter_max: the total number of samples in the learning base. Its value is generally set at 5000 but is modifiable to up to 10,000.

The transfer functions of the three layers of our ANN.

For each layer, we have the choice between 7 values: purelin, tansig, logsig, hardlim, hardlim satlin or satlins. A combination with purelin on all three coats gave us acceptable results. We have used it by default. However, the experimenter can modify them and observe the influence on the syntheses obtained. [20] - [23]

For the case of synthesis:

MATLAB automatically generates a small interface in the foreground that shows the evolution of the mean squared error as a function of the evolving number of examples already presented to the PMUC during the learning phase. The stop criterion can be either reaching the tolerance threshold (MSE) or reaching the maximum number of examples presented (iter_max).

To validate our approach, we simulated the classical approaches and the same methods coupled to NNs on the planar antenna under the following conditions 3 sources (50˚, 80˚, and 120˚).

The simulation gives us the following results ( Table 1 ):

The results show a clear improvement for NNs with radial function in general. We can see it in Figure 8 .

Table 2 shows directions of arrival from MUSIC and RBFNN, algorithm execution times, and errors.

Table 3 below shows the results obtained on 9 × 9 planar networks. The results are satisfactory. But the results are disappointing when it comes to a linear network.

Table 4 shows directions of arrival from MUSIC and PMUC, algorithm execution times, and errors.

These results clearly show that the measures taken with the intention of reducing the computation time do have a significant positive effect on the computation time, since they allow it to be saved around 5%.