Abstract: This paper presents the application of Radial Basis Function neural network in antenna array systems and

in the estimation of polarization rotation estimation in the ionosphere. Radial Basis Function neural network is used

as it satisfies both universal and best approximation property. We present the architecture of the network, as part of

the total system. Presented results show low mean error values and very good match between the referent values and

gained one, which shows the successfulness of the particular neural network.

Keywords: Antenna Array, Array Factor, Neural Network, Radial Basis Function Neural Network

Received: May 19, 2021. Revised: July 2, 2022. Accepted: August 5, 2022. Published: September 14, 2022.

1. Introduction

HE artificial neural networks theory and design advanced

significantly in the last decades. Also, signal processing is

the field where much progress has been achieved. Neural

Networks (NN) are highly suited for solving difficult signal

processing problems, mostly because of their non-linear

nature, their universal approximation property and their ability

to learn from environment in both supervised and unsupervised

ways..

A network of many simple processors (units, nodes, or

neurons) presents the NN, where all these units are

interconnected with unidirectional communication channels

(connections) labeled with numerical data, also having a small

amount of local memory [1]. For the NN we can think as a

black box that has certain input and produces certain outputs.

The NN structure and neurons models determine the

functionality of this black box.

It is interesting to note that NN resembles the brain in the

following two aspects:

- A learning process provides the NN to acquire a

knowledge

- Synaptic weights presented by interneuron connection

are storing that knowledge

The one of most exciting properties of NNs is their

functional approximation capability that makes them suitable

for applications in signal processing, control, communication

channel equalization, pattern recognition and system

identification. The approximation capabilities of various types

of multilayered feedforward architectures have been

investigated since 1990’s with much interest. Actually a

feedforward NN may be seen as a rule of computing the output

values of the neurons in the ith layer from the values of the

outputs of the (i-1)th layer, that actually present a mapping

from the input space Rn to an output space Rn.

The Stone-Weierstrass theorem is effective analytical tool

for function approximation analyses [2]. The relationship

between the Kolmogorov’s theorem and approximation

principle of the feedforward networks was found in 1980’s.

This theorem states that a continuous multivariable function

may be expressed, on a compact domain, in terms of sums of

compositions on single variable functions. It was shown that

NNs with at least one hidden layer are capable of

approximating continuous function if the activation functions

of the hidden neurons are differentiable.

A suitable candidates for NNs application are antenna

systems because of their associated nonlinearities. A

multilayer perceptron with single hidden layer is capable of

approximating any smooth non-linear input-output mapping,

provided that there are sufficient number of neurons in the

hidden layer, to an arbitrary degree of accuracy. This is

referred as universal approximation property. Radial Basis

Function NNs (RBFNs) poses the universal

approximation capability, which was proven by Park and

Sandberg [3], [4]. As an extension to this property, a property

of best approximation is defined. The model that most

closely approximates the generating function, by some

defined distance measure, from given set of models, is said

that poses the best approximating property. RBFNs poses

this property [5], and that’s why they were applied to

antenna array direction of arrival estimation and

beamforming [6], [7], [8].

In this paper we present an examples of RBFNs application

for intelligent antenna array synthesis and also for

Faraday Polarization Rotation (FPR) estimation in the

ionosphere. Next section describes the RBFNs architecture,

and in Section 3 some results are presented.

Overview of RBF NN and Antenna Systems

MAJA SAREVSKA

AUE-FON, Kiro Gligorov 5, 1000 Skopje, NORTH MACEDONIA

WSEAS TRANSACTIONS on ELECTRONICS

DOI: 10.37394/232017.2022.13.11

Maja Sarevska

E-ISSN: 2415-1513

Volume 13, 2022

Fig.1 architecture of RBF NN

2. RBFN NN Architecture

Fig.1 [9] presents the architecture of RBF NN with two

layers, one hidden and one output layer. The number of

neurons in the hidden layer is S1 and the number of neurons in

the output layer is S2. The weighted input of the neuron is the

weighted distance between the input vector and weight vector,

and each net input of the neuron is dot product between that

distance and bias vector. After passing through the radbas

function the output of the hidden layer is generated, and that is

actually the input vector for the second layer. The transfer

function of the output layer is linear.

Fig. 2 RBF neuron architecture

Fig.2 [9] presents the structure of RBF neuron. Input vector

has R elements, and output is presented with value a. The

block ||dist|| presents some distance measure between the input

vector and connection weights vector, and b presents the bias.

At the end on the figure we can notice the block of the radial

basis function as the activation function of the neuron. Radial

basis transfer function block produce 1 whenever the input

vector is matched with the weight vector which means it

behaves as detector.

The input vector for antenna array synthesis is presented

with limited number of values of array factor or radiation

pattern, and the output presents the phase and amplitude step

for the linear antenna array [10], [11]. For faraday polarization

rotation estimation, the input of RBF NN is latitude of the

monitored place and day time value, and the output is the total

electron content [12]. NN has been used for different types of

antennas [13], [14], [15], and here we discuss antenna

array synthesis and faraday polarization rotation

estimation with RBF NN.

3. Results

Array of elements placed along one direction are presenting

the irregular linear antenna array, with different mutual

distance among the neighbouring elements and with different

amplitudes and phases among the excitations. The array factor

in this case has the form:

(1)

(2)

Where M+1 is the number of antenna elements, An are the

amplitudes of excitations, β is the phase constant (β=2π/λ, λ is

the wavelength), dn are the mutual distances,  is the space

angle and αn are the phase differences of the excitations. Even

for modest number of antenna elements the array factor

determination requires complex calculations for what we need

to use computer calculations. Different distributions of mutual

distances or mutual excitations among the elements produce

variety of radiation patterns (array factors). Antenna array

synthesis presents the determination of the parameters of the

antenna array for already required radiation pattern or array

factor.

In our analyses first we assume unit and uniform amplitude

distribution where the inter-element distances is constant, and

where the synthesis will be consisted of the phase difference

determination between the neighbouring antenna elements. For

this antenna array synthesis we assume regular linear antenna

array where the radiation pattern is determined by the number

of antenna elements M, mutual distance of the elements d and

the phase difference α between the two neighbouring elements.

This simple structure will be a strong basis for further analyze

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DOI: 10.37394/232017.2022.13.11

Maja Sarevska

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of irregular antenna arrays. The input of the NN is the

radiation pattern vector and the output is the corresponding

value of α for given number of antenna elements and mutual

distance.

The designing procedure for the RBE (Radial Basis with

Exact solution) is outlined below.

The matrix P could be organized in R element vectors F,

which are presenting the radiation pattern in R points (Q

vectors in total).

We have two steps:

1) Network designing

1) Form the input vectors {Fq, q=1,2, . . .,Q}.

2) Generate input/output pairs {Fq,αq}, where q=1,2, . . . ,Q.

3) Design the RBE.

2) Network testing (Generalization)

1) Form the vectors F’ for the testing input samples.

2) Present input vectors F’ to the RBE.

3) Get the output of the network.

In the next step we assume linear amplitude distribution

with parameter A. Again the phase difference between two

neighbouring antenna elements is α. It this case the array factor

will be:

(3)

where

(4)

This time the output NN layer will have two neurons, one

giving the value of α and the other giving the value of A.

The designing procedure for the RBE is outlined below.

The matrix P could be organized in R element vectors F,

which are presenting the radiation pattern in R points (Q

vectors in total).

We have two steps:

1) Network designing

1) Form the input vectors {Fq, q=1,2, . . .,Q}.

2) Generate input/output pairs {Fq,tq}, where q=1,2, . . . ,Q,

and t is two element vector containing the α and A

value.

3) Design the RBE.

2) Network testing (Generalization)

1) Form the vectors F’ for the testing input samples.

2) Present input vectors F’ to the RBE .

3) Get the output of the network, α and A.

The main goal is to observe the influence of additional

parameter in the NN and that is parameter A, to the

performance of the network. That will give us information for

further inclusion of other parameters or different amplitude

distributions, which finally would lead us to irregular array

synthesis.

First we may consider regular linear array with uniform

amplitude distribution. Fig.3 presents the antenna array factor

for 9 antenna elements and with excitation phase difference of

-44 and 44. The mutual distance between the elements is half

wavelength. The training samples were picked with angle step

(Alfa Step) of α of 0.5, 1, 1.5, and 2, in the range (-

4545). The input samples for the array factor were for

angle step of 0.25, 0.5, 1, and 2. The testing samples were

picked with step of 0.05. The results for the mean error rate at

the output of the NN are presented in Fig.4.

020 40 60 80 100 120 140 160 180

Teta

Array Factor

-44deg

+44deg

Fig.3 Array factor for M+1=9 and uniform amplitude distribution

Now we may consider linear amplitude distribution antenna

array. Fig.5 presents the array factor for six antenna elements.

For the experiment we fixed α=45 and we used training set

for A=(01) with steps 0.005; 0.01; and 0.02. The array

factor input samples were chosen for angle step of 1. The

mutual element distance was one half wavelength. The mean

error for A at the output of the RBE is presented on Fig. 6.

0.5 1 1.5 2

10-5

10-4

10-3

10-2

10-1

100

101

Alfa Step

Mean error

0.25

0.5

Fig.4 Mean error for M+1=9 and d=0.5λ

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Maja Sarevska

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020 40 60 80 100 120 140 160 180

Teta

Array Factor

-45, uniform

45, linear with step 0.5

Fig.5 Array factor for M+1=6, unıform and linear amplitude distribution

4 5 6 7 8 9 10 11 12 13

10-10

10-8

10-6

10-4

10-2

100

M+1

Mean Error

0.005

0.01

0.02

Fig.6 Mean error for A=(01) and α=45

Some of the results using RBF NN for Faraday Polarization

Rotation (FPR) are presented in Fig.7 and Fig.8. We my notice

a low mean error values and good match between the referent

values and RBF NN values.

Fig. 7. Ionospheric FR angle of the P-SAR signal (f = 440 MHz) and L-SAR

signal (f = 1250 MHz) obtained by FPR-RBF model

Fig. 8. Ionospheric FR angle of the P-SAR signal (f = 440 MHz) and L-SAR

signal (f = 1250 MHz) obtained by FPR-RBF model

The FPR-RBF model was applied for FR angle estimation

of the P-band SAR (f = 440 MHz) and L-band SAR (f = 1250

MHz) signals which propagate through the ionosphere above

the Mediterranean area. Ionosphere FR angle of the P-SAR

and L-SAR signals in 24h winter period for latitude la (N) =

35.5 and 39.5 obtained by FPR-RBF model are shown in

Fig. 7 and Fig. 8 respectively. For each latitude with this

model we generated the FR angle values for daytime period

with resolution of 6 min which means 241 samples. The results

gained with FPR-RBF model are compared to referent values

on TEC (Total Electronic Content) values read from existing

maps. We may observe a good match of the results gained with

neural model with the referent values can be observed. The FR

angle values of the P-SAR and L-SAR signals in 24 h summer

period, obtained by FPR-RBF model, are shown in Fig. 9 and

Fig. 10 for latitude la (N) = 35.5 and 39.5, respectively. The

same resolution of 6 min which means 241 points is used.

Again we compare the results gained with neural model with

referent values gained from existing maps. We may observe a

good match with the referent values for the summer as in the

case of winter. All results are generated for B||avg = 26 T

belonging to the field change range valid for the

Mediterranean area. Bavg is the average Earth’s magnetic field

intensity in T

The effect that influence the work on satellite

communications systems especially for work of SAR systems

in L and P band is Faraday Polarization Rotation of the EM

wave in the ionosphere. That’s why of great importance is the

estimation of the angle of Faraday rotation in ionosphere for

prediction and error correction in work of these systems

caused by FPR. That’s why the unavoidable task is to estimate

the concentration of free electrons on a propagating path in

ionosphere, or TEC value. Mainly we use empirical models of

the vertical profile of the ionosphere to estimate TEC values

today, that are gained based on large number of measured

results gained with satellite systems or vertical sounding in

longer time period. These models unavailable to ordinary users

and are very complex. Alternative to these models based on

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Maja Sarevska

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available measured data is to form neural network model for

TEC values. It is proven that this model is easy to be realized

and it can determine TEC values quickly based on space and

time information of the signal propagation in the ionosphere.

Previously shown results for used FPR_RBF neural model for

FR angle estimation for winter and summer period in the

Mediterrean region provide good evidence that the use of NNs

to solve this problem is a good choice. The results also prove

that neural network models are a good alternative for the

expensive and hardware demanding numerical models but also

for software for description of the ionosphere influence on EM

propagating waves. This models are also good alternative for

slow and rough estimation of manual reading of TEC values.

Fig. 9. Ionospheric FR angle of the P-SAR signal (f = 440 MHz) and L-SAR

signal (f = 1250 MHz) obtained by FPR-RBF model

Fig. 10. Ionospheric FR angle of the P-SAR signal (f = 440 MHz) and L-SAR

signal (f = 1250 MHz) obtained by FPR-RBF model

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WSEAS TRANSACTIONS on ELECTRONICS

DOI: 10.37394/232017.2022.13.11

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Conflicts of Interest

The author(s) declare no potential conflicts of

interest concerning the research, authorship, or

publication of this article.

Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

The author(s) contributed in the present

research, at all stages from the formulation

of the problem to the final findings

and solution.

Sources of funding for research

presented in a scientific article or

scientific article itself

No funding was received for conducting this

study.

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