New Research Article Efficient DoA Tracking of Variable Number of...

Research ArticleEfficient DoA Tracking of Variable Number of MovingStochastic EM Sources in Far-Field Using PNN-MLP Model

Zoran StankoviT,1 Nebojša DonIov,1 Bratislav MilovanoviT,2 and Ivan MilovanoviT2

1Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18 000 Niš, Serbia2Singidunum University, Danijelova 32, 11000 Belgrade, Serbia

Correspondence should be addressed to Zoran Stanković; [email protected]

Received 9 August 2015; Accepted 1 December 2015

Academic Editor: Ahmed T. Mobashsher

Copyright © 2015 Zoran Stanković et al.This is an open access article distributed under theCreativeCommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An efficient neural network-based approach for tracking of variable number of moving electromagnetic (EM) sources in far-fieldis proposed in the paper. Electromagnetic sources considered here are of stochastic radiation nature, mutually uncorrelated, andat arbitrary angular distance. The neural network model is based on combination of probabilistic neural network (PNN) and theMultilayer Perceptron (MLP) networks and it performs real-time calculations in two stages, determining at first the number ofmoving sources present in an observed space sector in specific moments in time and then calculating their angular positions inazimuth plane. Once successfully trained, the neural network model is capable of performing an accurate and efficient direction ofarrival (DoA) estimation within the training boundaries which is illustrated on the appropriate example.

1. Introduction

Signal source localization by employing passive antennaarrays is widely used technique in different areas such as com-munications, radars, acoustics, andmedicine. Important stepin this spatial determination of source location is to performan angular direction of arrival (DoA) estimation of a signalradiated from the source. Among other things, the purposeand nature of the signal have to be taken into account whileperforming the DoA estimation, as signals can be consideredeither desired and deterministic or interfering both deter-ministic (unintentional interference) and stochastic (ran-dom function in time). In wireless communications, oncethe angular positions of desired/interfering electromagnetic(EM) source are found by usingDoA estimation, the adaptivebeam-forming algorithm can be employed to optimize theradiation pattern of antenna array so that it allocates themainbeam towards the user of interest and generates deep nullsin the directions of interfering signals from mobile users inadjacent cells.

A number of DoA estimation algorithms have been pro-posed in the literature taking into account the statistical

properties of source signals, geometry of the antenna arraysat the receiver end, multiplexing schemes, and so forth.Majority of these algorithms rely on the processing of aspatial covariance matrix of received signals at antenna arrayelements. Multiple Signal Classification (MUSIC) [1] is oneof these techniques, widely used due to its superresolutioncapabilities. However, it is of high computational complexityas it requires a demanding spectrum search procedure, result-ing in some cases in a longer run time not suitable for real-time applications. Artificial neural networks (ANNs) [2–4]represent an alternative faster approach to the MUSIC andother intensive superresolution DoA algorithms. ANNs arevery convenient as a modeling tool since they have the abilityto learn from the presented data and therefore they are espe-cially useful in solving complex problems or those not fullymathematically described. In other words, ANNs are able tomap dependence between two datasets. The learning processis an optimization procedure through which parameters ofthe ANN are optimized to have the ANN outputs as closeas possible to the target values. This ability qualifies ANNsas very suitable tool for estimating the angular positions ofsource signals [4, 5].

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015, Article ID 542614, 11 pageshttp://dx.doi.org/10.1155/2015/542614

2 International Journal of Antennas and Propagation

Stochastic source

S

s

d

y

Stochastic source

Stochastic source

1

x

x(S)N

x(S)2x(S)1

x(s)N

x(s)2

x(s)1

x(1)N

x(1)2x(1)1

𝜃Sr(S)1,1

r(s)1,1

r(S)N,1

r(1)N,M r(1)N,2 r(1)1,2

r(1)1,1

r(s)1,M

r(s)N,Mr(s)N,1

r(S)N,M

𝜃s

𝜃1

r0

...

...

Far-field scan array

yM

y2

y1

sd

Figure 1:The position of stochastic source in azimuth plane with respect to the location of EM field sampling points in the far-field scan area.

In [6] a new approach based on combination of the Mul-tilayer Perceptron (MLP) [3, 4] and the Radial Basis Function(RBF) ANNs [3, 4] is developed for two-dimensional, inazimuth and elevation planes, DoA estimation of deter-ministic signals radiated from narrowband EM sources. In[7, 8] and in [9], which was extended version of [8], anANN approach, realized by the MLP neural model, has beenpresented to provide a high-resolution DoA estimation ofstochastic signals. Since no amplitudes can be defined for thenumerical values of stochastic signals, the characterizationof stochastic signals differs from the characterization ofdeterministic signals. It requires considering the correlationbetween any two spatial points of the stochastic source inorder to provide an estimation of spatial covariancematrix. Anetwork-based methodology for the numerical computationof stochastic electromagnetic (EM) fields excited by spatiallydistributed noise sources with arbitrary spatial correlationwas presented in [10, 11]. Based on stochastic source radiationmodel developed from [10], the MPL models from [7–9]were able to efficiently perform mapping from the spaceof stochastic signals described by the correlation matrix tothe space of DoA in angular azimuth coordinates. How-ever, their application was limited to the cases of only fewstochastic narrowbandEMsources in the far-field, at the fixedmutual distance. In [12–14], the developed MLPmodels wereextended to allow an efficient DoA estimation of a numberof mutually arbitrary positioned uncorrelated stochastic EMsources in far-field.

Both the superresolution algorithms and previouslymen-tioned neural models have one limitation when performingthe DoA estimation. The number of EM sources presentedin the observed sector has to be known in advance in orderto preserve model validity and its sufficient accuracy forangular positions determination. If the model is developedfor particular number of sources assumed to be present inthe observed sector during the model operation, in caseswhen the actual number of present sources is smaller or

higher than assumed number, it is possible that model willincorrectly identify sources angular positions. Therefore inthis paper, two-stage neural model, based on combining theprobabilistic neural network (PNN) [15, 16] and the MLPnetwork, is proposed in order to overcome this limitation.The PNN-MLP model is capable of performing an efficientand accurate DoA estimation of stochastic EM sources whosenumber is changing in time and sources are also moving fastin the observed sector. The example presented in the paperdemonstrates the accuracy and suitability of the proposedneural network model for real-time applications.

2. Stochastic Source Radiation Model

Stochastic source radiation model, presented in [7–9, 12–14]and also used in this paper, starts from the assumption thateach source radiation in far-field can be represented by linearuniform antenna array with 𝑁 elements mutually separatedby 𝑑 = 𝜆/2, 𝜆 = 𝑐/𝑓, where 𝑓 is observed frequency in far-field (Figure 1). In general, the degree of correlation betweenantenna elements feed currents, described by vector I =[𝐼1, 𝐼2, . . . , 𝐼

𝑁], is arbitrary and it can be expressed by the

correlation matrix c𝐼(𝜔) [10, 11]:

c𝐼 (𝜔) = lim𝑇→∞

1

2𝑇[𝐼 (𝜔) 𝐼 (𝜔)

𝐻] . (1)

By employing the Green function marked with vectorM(𝜃, 𝜑) = 𝑗𝑧

0(𝐹(𝜃, 𝜑)/2𝜋𝑟

𝑐) [𝑒𝑗𝑘𝑟1 𝑒𝑗𝑘𝑟2 ⋅ ⋅ ⋅ 𝑒

𝑗𝑘𝑟𝑁], where 𝜃

and 𝜑 are azimuth and elevation angles determined withrespect to the first antenna element, the level of electric fieldradiated from the antenna array representation of stochasticsource, at some sampling point in the far-field, can be calcu-lated as

𝐸 (𝜃, 𝜑) = M (𝜃, 𝜑) I. (2)

𝐹(𝜃, 𝜑) is the radiation pattern of antenna array, 𝑟𝑐is the dis-

tance of far-field point to the centre of array, 𝑧0is free-space

International Journal of Antennas and Propagation 3

impedance, 𝑘 is the phase constant (𝑘 = 2𝜋/𝜆), and 𝑟1,

𝑟2, . . . , 𝑟

𝑁are the distances of considered far-field point

from the first to the 𝑁th element of antenna array. For 𝑀observation points in the far-field, we use a more generalnotation in order to describe the antenna array elementsdistance from particular points in far-field. For example, inFigure 1 𝑟

𝑖,𝑚represents the distance between 𝑖th element (1 ≤

𝑖 ≤ 𝑁) in the antenna array and 𝑚th point in the far-field(1 ≤ 𝑚 ≤ 𝑀).

For 𝑀 sampling points (𝑦1, 𝑦2, . . . , 𝑦

𝑀) in far-field scan

area, determined by the azimuth and elevation plane angles(𝜃1, 𝜑1), (𝜃1, 𝜑1), . . . , (𝜃

𝑀, 𝜑𝑀

), the correlation matrix of sig-nals received in these sampling points can be obtained as [10]

C̃𝐸[𝑖, 𝑗] = M (𝜃

𝑖, 𝜑𝑖) c𝐼M (𝜃

𝑗, 𝜑𝑗)𝐻

,

𝑖 = 1, . . . ,𝑀, 𝑗 = 1, . . . ,𝑀.

(3)

For more than one stochastic source, the EM field level infar-field sampling point, as well as the elements of correlationmatrix, can be determined by the superposition of radiationfrom all sources. If the number of stochastic sources is 𝑆, thenthe vectorM has a form

M (𝜃, 𝜑) = 𝑗𝑧0

𝐹 (𝜃, 𝜑)

2𝜋𝑟𝑐

⋅ [𝑒𝑗𝑘𝑟(1)

1 ⋅ ⋅ ⋅ 𝑒𝑗𝑘𝑟(1)

𝑁 𝑒𝑗𝑘𝑟(2)

1 ⋅ ⋅ ⋅ 𝑒𝑗𝑘𝑟(2)

𝑁 ⋅ ⋅ ⋅ 𝑒𝑗𝑘𝑟(𝑆)

1 ⋅ ⋅ ⋅ 𝑒𝑗𝑘𝑟(𝑆)

𝑁 ] ,

(4)

where 𝑟(𝑗)𝑖

is the distance between 𝑖th element in antennaarray, representing 𝑗th stochastic source, and the samplingpoint in far-field, while the feed currents vector is

I = [𝐼(1)1

⋅ ⋅ ⋅ 𝐼(1)

𝑁𝐼(2)

1⋅ ⋅ ⋅ 𝐼(2)

𝑁⋅ ⋅ ⋅ 𝐼(𝑆)

1⋅ ⋅ ⋅ 𝐼(𝑆)

𝑁] , (5)

where 𝐼(𝑗)𝑖

is the feed current of 𝑖th element in antenna arrayrepresenting 𝑗th stochastic source. Incorporating (4) and (5)into (3) it is possible to determine the elements of corre-lation matrix C̃

𝐸. When the degree of correlation between

antenna elements feed currents is unknown, its correlationmatrix c𝐼(𝜔) can be obtained by near-field measurementsas described in [10, 11]. If two radiation sources that areunder monitoring have the same angular position (𝜃, 𝜑), butat the different distances 𝑟

𝑐1and 𝑟

𝑐2, and are represented

with antenna arrays with 𝑁1, 𝑁2elements for the first and

the second source, respectively, in case 𝑟𝑐1, 𝑟𝑐2

≫ 𝑑, thenC̃𝐸2

≈ (𝑁2𝑟𝑐1/𝑁1𝑟𝑐2) ⋅ C̃𝐸2. By normalization of elements of

matrix C̃𝐸with respect to the first element 𝐶

𝐸11, the matrix

C𝐸

= (1/𝐶𝐸11

) ⋅ C̃𝐸is obtained and its elements do not

depend on the values of 𝑟𝑐and 𝑁. During the neural model

development, only the first row of spatial correlation matrixC𝐸([𝐶𝐸11

, 𝐶𝐸12

, . . . , 𝐶𝐸1𝑀

]) has to be used, because it wasshown that it contains sufficient information to be extractedby the neural model in order to estimate the source angularposition [6, 7].

3. PNN-MLP Model

Themain purpose of the PNN-MLP model presented in thispaper is to determine in real time, based on sampled values

𝜃1 𝜃1𝜃1 𝜃2 𝜃2 𝜃s

MLP-DoAsMLP-DoA1 MLP-DoA2

EEE

S

s12

MLP-DoA stage

PNN stage

PNN-SND

Switch

· · ·

· · ·

x = [Re{CE11}Im{CE11} · · ·Re{CE1M}Im{CE1M}]1×2M

Figure 2: Architecture of PNN-MLP neural model for DoA estima-tion of signals of the stochastic EM sources in azimuth plane.

of spatial correlationmatrixC𝐸, angular azimuth positions of

stochastic EM sources, which can move fast in the observedspatial sector and also their number can vary in time. Thearchitecture of this model is chosen so that calculations areperformed in two stages: at the first stage (PNN stage) thenumber of stochastic EM sources that are currently present inthe observed sector is determined, while at the second stage(MLP-DoA stage), based on information obtained from thefirst stage, the angular azimuth positions of sources in thesector are estimated (Figure 2).

The PNN stage consists of PNN with one output thatgives the estimated number of EM sources in the observedsector 𝑠, 𝑠 ∈ {1, 2, . . . , 𝑆}, where 𝑆 is the maximal number ofsources whose positions can be simultaneously determinedin the azimuth plane. PNN intended for such classificationwill be marked in further text as PNN-SND (PNN for SourceNumber Determination). The second, so-called MLP-DoAstage, consists of a bank of MLP networks intended forDoA estimation of EM sources and a switch for selection(activation) of appropriate MLP network from the bank.Bank of MLP networks contains 𝑆 networks: MLP-DoA

1,

MLP-DoA2, . . . ,MLP-DoA

𝑠, . . . ,MLP-DoA

𝑆−1,MLP-DoA

𝑆,

where network MLP-DoA𝑠has in total 𝑠 outputs and as such

it performs calculation of angular positions of sources in thecase when in the observed sector and in considered momentin time there are precisely 𝑠 stochastic sources. Switch hasa task to select in chosen moment in time, for obtainednumber of sources 𝑠, an appropriate MLP network for DoAestimation, that is, MLP-DoA

𝑠.

The training of PNN-MLP neural model is conducted insuch way so that PNN-SDN network and each MLP-DoA

𝑠

network are trained independently by using their own train-ing set. The same applies for the testing phase; however it isuseful to perform a testing of the integral PNN-MLP modelas well with the goal to evaluate the model performances inreal operating mode.


Re{CE11}Im{CE11} · · ·Re{CE1M}Im{CE1M}

· · ·· · ·· · ·· · ·· · ·

· · ·

p1(x)

PNN-SND

S

Output (decision)layer

Class layer

Class 1group

Class sgroup

Class Sgroup

Input layer

Hidden layer

ps(x) pS(x)

w(1)1 w(1)H1 w

(s)1

w(s)Hs

w(S)1 w(S)HS

x1 x2 x2M−1 x2M

Figure 3: PNN for determination of the number of stochastic EM sources, 𝑠, in the observed spatial sector.

3.1. PNN for Source Number Determination (PNN-SND).The architecture of neural network for determination of thenumber of stochastic EM sources in the observed spatialsector (PNN-SND) is based on PNN [15, 16] and it is shownin Figure 3. It consists of one input layer, one hidden layer,one class layer, and one output layer, that is, decision layer.The task of this neural network is to perform the classificationof samples of the first row of spatial correlation matrix, thatis, to determine which class among the predefined classesof this problem the sample at the network input belongs to.Regarding this problem, the classes are predefined in thefollowing way:

(1) There are in total 𝑆 classes of samples where 𝑆 is themaximal number of stochastic EM sources that can befound in the observed sector.

(2) A sample of the first row of matrix C𝐸([𝐶𝐸11

, 𝐶𝐸12

,. . . , 𝐶

𝐸1𝑀]) belongs to the class 𝑠 where 𝑠 ∈ {1, 2, . . .,

𝑆} when it is sampled for the case when there were 𝑠stochastic sources present in the sector.

According to this, PNN-SND performs mapping of thesampled values of the first row of the matrix C

𝐸into the set

of discrete values of notations of predefined classes

𝑠 = 𝑓PNN-SND ([𝐶𝐸11, 𝐶𝐸12, . . . , 𝐶𝐸1𝑀]) ,

𝑠 ∈ {1, 2, . . . , 𝑆} .

(6)

The input layer of PNN-SND is the buffer layer, and ithas the task to forward the values of the first row of thecorrelation matrixC

𝐸to each neuron in the hidden layer. For

each element from thematrix row there are two neurons from

the input layer of neural network that correspond, one for realand the other for imaginary part of element:

𝑥2𝑖−1

= Re {𝐶𝐸1𝑖

} ,

𝑥2𝑖

= Im {𝐶𝐸1𝑖

} ,

𝑖 = 1, 2, . . . ,𝑀,

(7)

so that the vector x of dimension 2𝑀, which represents thevector of buffered values of network input, is forwarded tothe input of each neuron in the hidden layer.

The hidden layer of PNN-SND is carrier informationabout the classes. Neurons in the hidden layer are divided intogroup of neurons so that each class 𝑠, where 𝑠 = 1, 2, . . . , 𝑆, hasits own group of neurons.Thenumber of neuronswithin eachgroup 𝐻

𝑠, 𝑠 = 1, 2, . . . , 𝑆, is determined during the training

phase of neural network and it is equal to the number ofsamples in the training set that belongs to class 𝑠. Activationfunction of neuron in the hidden layer that belongs to theclass 𝑠 is based on the Gaussian function [15, 17] so that theoutputs of neurons of class 𝑠 are given as

ℎ(𝑠)

𝑖(x) = 1

(2𝜋)𝑀

𝜎2𝑀⋅ 𝑒−‖x−w(𝑠)

𝑖‖/2𝜎2

, 𝑖 = 1, 2, . . . , 𝐻𝑠, (8)

where vector w(𝑠)𝑖

of dimension 2𝑀 represents the vectorof weights or vector of center of activation function of 𝑖thneuron that belongs to class 𝑠, while 𝜎 is the spread parameter(standard deviations) of activation function.

The task of neurons in the class layer is to sum the outputsof neurons in the hidden layer within each class separatelyand based on this summation to estimate the probability thatsample x belongs to considered class. Each group of neurons


of class 𝑠, 𝑠 = 1, 2, . . . , 𝑆, from the hidden layer correspondsto one neuron in the class layer (𝑠th neuron) so that thetotal number of neurons in the class layer is equal to 𝑆. Theestimation of probability that the sample x belongs to class𝑠, 𝑝𝑠(x), is performed by the 𝑠th neuron in the class layer

through its activation function based on Parzen windowtechnique [17, 18] so that the outputs of this layer are given as

𝑝𝑠(x) = 1

(2𝜋)𝑀

𝜎2𝑀⋅

1

𝐻𝑠

𝐻𝑠

∑

𝑖=1

𝑒−‖x−w(𝑠)

𝑖‖/2𝜎2

,

𝑠 = 1, 2, . . . , 𝑆.

(9)

Output layer or decision layer has one neuron that hasto decide, based on estimated probabilities in the class layer,to which class the sample at the network input is the closest,that is, where this sample has to be correctly classified. Thisneuron performs this task according to Bayes’s decision rule[19] based on the output of all the class layer neurons so thatthe activation function has a competitive nature; that is, as afinal decision, the class for which the estimated probability isthe highest is selected:

𝑠 = comp {𝑝𝑠(x)} , 𝑠 = 1, 2, . . . , 𝑆

comp: 𝑠 = 𝑠max ∴ 𝑝𝑠max (x) = max {𝑝𝑠 (x)} ,

𝑠 = 1, 2, . . . , 𝑆.

(10)

During the training of PNN-SND network, the numbersof neurons in the hidden and class layers are determinedbased on the training set, and also theweighting vectors in thehidden layer are adjusted so that the neural network performscorrect classification of all samples from the training set.Spread parameter is not determined during the networktraining as its value is set before the training. The value ofthis parameter has an impact on generalization capabilitiesof PNN-SND and this impact can be quantified throughthe number of incorrectly classified samples by the networkduring the testing phase on the set of samples not used forthe training. By multiplying repetition of network trainingfor different values of spread parameter (typically in therange [0 1]) and by evaluating the network performancesduring the testing phase, the value of spread parameter can beadjusted so that the network during the testing has as smalleras possible the number of incorrectly classified samples.

3.2. MLP-DoA Network. The main task of MLP-DoA𝑠net-

work is to perform the mapping from the space of signalsdescribed by first row of the correlation matrix C

𝐸to the

space of DoA in azimuth; that is,

𝜃𝑠= 𝑓MLP-DoA

𝑠

([𝐶𝐸11

, 𝐶𝐸12

, . . . , 𝐶𝐸1𝑀

]) ,

𝑠 ∈ {1, 2, . . . , 𝑆} ,

(11)

where 𝜃𝑠is azimuth angles vector of stochastic sources 𝜃

𝑠=

[𝜃1, 𝜃2, . . . , 𝜃

𝑠]. In the observed case elevation coordinates of

radiation sources are neglected.The architecture of developedneural model is shown in Figure 4.

𝜃1 𝜃2 𝜃s

MLP-DoAsOutput layer

Hidden layer H

Input layer

Hidden layer 1

Re{CE11}Im{CE11} · · ·

· · ·

· · ·

· · ·

· · ·

Re{CE1M}Im{CE1M}

Figure 4: MLP-DoA network for estimation of angular position of𝑠 stochastic sources in the azimuth plane.

Its MLP-DoA network can be described by the followingfunction:

y𝑙= 𝐹 (w

𝑙y𝑙−1

+ b𝑙) , 𝑙 = 1, 2, (12)

where y𝑙−1

vector represents the output of (𝑙 − 1)th hiddenlayer, w

𝑙is a connection weight matrix among (𝑙 − 1)th and

𝑙th hidden layer neurons, and b𝑙is a vector containing biases

of 𝑙th hidden layer neurons. 𝐹 is the activation function ofneurons in hidden layers and in this case it is a hyperbolictangent sigmoid transfer function:

𝐹 (𝑢) =𝑒𝑢− 𝑒−𝑢

𝑒𝑢 + 𝑒−𝑢. (13)

Following the previously used notation, y0represents

the input layer of MLP-DoA network so that y0

= x =[Re{C

𝐸11}Im{C

𝐸11} . . .Re{C

𝐸1𝑀}Im{C

𝐸1𝑀}]. Also, 𝜃

𝑠is given

as 𝜃𝑠= w3y2, wherew

3is a connectionweightmatrix between

neurons of last hidden layer and neurons in output layer. Theoptimization of weight matrices and biases values during thetraining allows ANN to approximate the mapping with thedesired accuracy. General notation for architecture of MLP-DoA network is MLP𝐻-𝑁

1−𝑁2− ⋅ ⋅ ⋅ −𝑁

𝑖− ⋅ ⋅ ⋅ −𝑁

𝐻, where

𝐻 represents the total number of the hidden layers, while 𝑁𝑖

represents the number of neurons in 𝑖th hidden layer.

4. Modeling Results

In order to verify the proposed approach, the PNN-MLPneural model is applied for determination of azimuth posi-tions of variable number of stochastic EM sources thatarbitrary changes their positions. The model is realized


Table 1: The parameters used for training and testing.

Frequency 𝑓 = 7.5GHz

Maximal number of sources 𝑆 = 3Number of antenna array elements per onesource

𝑁 = 4

Mutual distance of the antenna array elements 𝑑 = 𝜆/2 (0.02m)Sampling points distance from sourcetrajectory

𝑟0= 1000m

Number of sampling points 𝑀 = 6

Mutual distance of the sampling points 𝑠𝑑 = 𝜆/2 (0.02m)

within the MATLAB software environment. The observedsector is defined within the limits [𝜃min 𝜃max] = [−60∘ 60∘]and in this sector at any time, the maximal number ofradiation sources is assumed to be three (𝑆 = 3). Accordingto this, for construction of PNN-MLP model, PNN-SNDnetwork is first used in order to classify samples within thethree classes (𝑠 = 1, 2, 3). In addition, MLP-DoA

1, MLP-

DoA2, and MLP-DoA

3networks are used to determine the

angular positions of one, two, and three sources, respectively,in azimuth plane. The training and testing of these networksare conducted independently; each network has its owntraining and testing sets. The common conditions underwhich the training and testing sets are generated are given inTable 1.

For generating the training and testing sets for all models,(3) and (4) are used as they perform the inversemapping fromthe mapping done by the PNN-MLP model

C𝑡𝐸

= 𝑓−1

DoA (𝜃𝑡) , (14)

and then, because the neural networks do not support oper-ation with complex numbers, the first row of the correlationmatrix is, according to (6) and (11), reconverted into the inputvector of PNN and MLP networks, whose values are thenused during the training

x𝑡 (𝜃𝑡) = [Re {𝐶𝑡𝐸11

} Im {𝐶𝑡𝐸11

}Re {𝐶𝑡𝐸12

}

⋅ Im {𝐶𝑡𝐸12

} ⋅ ⋅ ⋅Re {𝐶𝑡𝐸1𝑀

} Im {𝐶𝑡𝐸1𝑀

}] .

(15)

For this mapping, it was assumed that the all feed cur-rents of antenna array elements, representing the stochasticsources, are mutually uncorrelated, so that c𝐼 is the unitdiagonal matrix. In order to generate the training and testingsets, for each element of vector 𝜃𝑡 uniform distribution ofsamples for azimuth angles of radiation source location ofthe form [𝜃𝑡min : 𝜃

𝑡

step : 𝜃𝑡

max] is used, where 𝜃𝑡

min[∘] and

𝜃𝑡

max[∘] represent the lowest and highest limit of distribution,

while 𝜃𝑡step is uniform sampling step.The sampling step 𝜃𝑡

step =

4/𝑘[∘] is used for all the training and testing sets where, by

adjusting the parameter 𝑘, the size of sampling set can bedetermined as well as the level of overlapping between thetraining and testing sets. The main criteria for choosing the

value of parameter 𝑘 for training and testing sets generationwere to minimize the overlapping between these two setsin order to obtain the real estimation of the achieved levelof generalization of trained network. During the generationof training and testing sets, in cases when there are morethan one source in the observed sector, the samples, whereangular positions of two and more sources are overlapped,are removed from the sets. Having in mind the trainingprocedure and accuracy of neural networks it is best to treatthese sources of the same angular positions as a uniquesource and this special case is considered as a case of smallernumber of sources whose angular azimuth positions are notoverlapped.

4.1. Training and Testing of PNN-SND. For training andtesting of PNN-SND network, the sets of forms {(x𝑡(𝜃𝑡), 𝑠𝑡)}are used, where 𝑠𝑡 represents the number of present sourcesin the sector during the generation of values x𝑡(𝜃𝑡). Thesampling sets for training and testing are obtained as

{(x𝑡 (𝜃𝑡1) , 1) | 𝜃

𝑡

1∈ [−60 : (

4

𝑘1

) : 60]}

∪ {(x𝑡 (𝜃𝑡1, 𝜃𝑡

2) , 2) | 𝜃

𝑡

1∈ [−60 : (

4

𝑘2

) : 60] , 𝜃𝑡

2

∈ [−60 : (4

𝑘2

) : 60] , 𝜃𝑡

1̸= 𝜃𝑡

2}

∪ {(x𝑡 (𝜃𝑡1, 𝜃𝑡

2, 𝜃𝑡

3) , 3) | 𝜃

𝑡

1

∈ [−60 : (4

𝑘3

) : 60] , 𝜃𝑡

2

∈ [−60 : (4

𝑘3

) : 60] , 𝜃𝑡

3

∈ [−60 : (4

𝑘3

) : 60] , 𝜃𝑡

1̸= 𝜃𝑡

2̸= 𝜃𝑡

3} .

(16)

The set for network training is generated for values 𝑘1

=

11, 𝑘2= 3, and 𝑘

3= 1 so that the training set of 8921 samples

is obtained. Testing set is generated for values 𝑘1= 7, 𝑘2= 2.5,

and 𝑘3

= 0.9 giving in total 7556 test samples. The trainingof PNN-SDN is conducted for different values of spreadparameter in the range [0 1] with 0.05 steps. Criteria forthe best trained networks were the percentage of incorrectlyclassified samples.The testing results for the four best trainednetworks are shown in Table 2. Notation for trained PNN-SDN is PNN-spreadwhere instead of word spread the value ofspread parameter, used during the network training, is given.Based on these results (Table 2), the PNN-0.10 network isused to realize the PNN-MLP model.

From Table 2, it can be seen that some networks haverelatively low percentage of incorrectly classified samples(below 4%) which illustrates potentially high performances


Table 2: Testing results for four 𝑖 PNN with lowest percentage ofincorrectly classified samples.

PNN modelThe number of

correctlyclassifiedsamples

The number ofincorrectlyclassifiedsamples

The percentageof incorrectlyclassified

samples [%]PNN-0.10 6130 207 3.37PNN-0.15 6120 217 3.54PNN-0.20 6084 253 4.16PNN-0.05 6080 257 4.23

of PNN and justifies their selection for realization of PNN-MLP model.

4.2. Training and Testing of MLP-DoAs Networks. For train-ing and testing of all three types of MLP-DoA networks,MLP-DoA

𝑠, 𝑠 = 1, 2, 3, the sets of forms {(x𝑡(𝜃𝑡), 𝜃𝑡

1, . . . , 𝜃

𝑡

𝑠)}

are used, where 𝑠 represents the number of sources in theobserved sector for which the MLP-DoA network performsDoA estimation. During the training phase, MLP networkswith two hidden layers are used. In order to obtain thetraining network of highest possible accuracy, for each typeof MLP-DoA network, the training of higher number ofMLP-DoA networks with different number of neurons in thehidden layers is conducted. The training of all MLP-DoAnetworks is performed by using the quasi-Newton trainingmethod with given accuracy of 10−4. For the selection ofbest trained networks, the following statistical parameters areconsidered during the testing phase: maximal error duringthe testing phase (Worst Case Error, WCE), the average test-ing error (ATE), and Pearson Product-Moment correlationcoefficient 𝑟PPM [3, 4].

4.2.1. Case 𝑠 = 1. The sets for training and testing of MLP-DoA1network are obtained as

{(x𝑡 (𝜃𝑡1) , 𝜃𝑡

1) | 𝜃𝑡

1∈ [−60 : (

4

𝑘1

) : 60]} . (17)

The set for network training is generated for 𝑘1

= 4 givingin total 121 samples. Testing set is generated for 𝑘

1= 3.2 and

therefore it has 69 samples.The testing results for six trained MLP-DoA

1networks

with the best test statistics are shown in Table 3. MLP2-23-23network which shows the lowest Worst Case Error is chosenas representative MLP-DoA

1neural network. The scattering

diagram of MLP2-23-23 network output and output of theMUSIC model obtained by model simulation on the sametesting set is shown in Figure 5. A good agreement can beobserved between the output values of neural network andreferent azimuth values. It can be noticed that the MUSICmodel has slightly better agreement with referent valueswhich is expected; however it requires significantly longerrun time than neural network (which can be seen later in thepaper in Table 6).

Table 3: Testing results for six MLP-DoA neural network modelswith the best test statistics.

MLP model WCE [%] ATE [%] 𝑟PPM

MLP2-23-23 0.9006 0.3584 0.9999MLP2-22-22 0.9682 0.3991 0.9999MLP2-12-12 0.9806 0.3805 0.9999MLP2-16-16 1.0401 0.4000 0.9999MLP2-10-10 1.0428 0.4031 0.9999MLP2-12-5 1.0700 0.3793 0.9999

MUSICMLP2-23-23

0

−60

−40

−20

20

40

60

0 20 40 60−40 −20−60𝜃 ∘)(one source)—referent values (

𝜃∘ )

(one

sour

ce)—

mod

el o

utpu

t (

Figure 5: Scattering diagram of chosen MLP-DoA1network

(MLP2-23-23) and theMUSICmodel obtained by simulation on thetest set (Case 𝑠 = 1).


{(x𝑡 (𝜃𝑡1, 𝜃𝑡

2) , 𝜃𝑡

1, 𝜃𝑡

2) | 𝜃𝑡

1∈ [−60 : (

4

𝑘2

) : 60] , 𝜃𝑡

2

∈ [−60 : (4

𝑘2

) : 60] , 𝜃𝑡

1̸= 𝜃𝑡

2} .

(18)

In order to generate the training and testing sets, 𝑘2= 2 and

𝑘2= 1.6 are used, respectively. Therefore, the training set has

1830 samples, while the number of testing samples is 1176.The testing results for six trained MLP-DoA

2net-

works with the best test statistics are shown in Table 4.MLP2-13-13 network which shows the lowest Worst CaseError is chosen as representative MLP-DoA

2neural net-

work.The scattering diagram ofMLP2-13-13 network outputsand outputs of MUSIC model obtained by model simula-tion on the same testing set are shown in Figures 6 and7. A very good agreement between the output values ofneural network and referent azimuth values can be observed.As in the previous case, the MUSIC model has a betteragreement with the referent values; however, it requires


Table 4: Testing results for six MLP-DoA2neural network models

with the best test statistics.



MUSICMLP2-13-13

−60

−40

−20

0

20

40

60

−40 −20 4020 60−60 0

𝜃1

∘ )

𝜃1∘)(source 1)—referent values (

(sou

rce 1

)—m

odel

out

put (


(MLP2-13-13) and theMUSICmodel for output 1 (source 1) obtainedby simulation on the test set (Case 𝑠 = 2).

MUSICMLP2-13-13

−60

−40

−20

0

20

40

60

−40 −20 4020 60−60 0𝜃2

∘)(source 2)—referent values (

𝜃2

∘ )(s

ourc

e 2)—

mod

el o

utpu

t (


(MLP2-13-13) and the MUSIC model for output 2 (source 2)obtained by simulation on the test set (Case 𝑠 = 2).

much longer run time (which can be seen later in the paperin Table 6).

Table 5: Testing results for six MLP-DoA3neural network models

with the best test statistics.



Table 6: Comparison between MLP-DoA𝑠and MUSIC models run

times.

Case (number of samples) Simulation run time on test set𝑠 = 1

69 samplesMLP2-23-23 MUSIC

0.03 s 5 s𝑠 = 2


0.07 s 106 s𝑠 = 3


0.05 s 63 s


{(x𝑡 (𝜃𝑡1, 𝜃𝑡

2, 𝜃𝑡

3) , 𝜃𝑡

1, 𝜃𝑡

2, 𝜃𝑡

3) | 𝜃𝑡

1

∈ [−60 : (4

𝑘3

) : 60] , 𝜃𝑡

2

∈ [−60 : (4

𝑘3

) : 60] , 𝜃𝑡

3

∈ [−60 : (4

𝑘3

) : 60] , 𝜃𝑡

1̸= 𝜃𝑡

2̸= 𝜃𝑡

3} .

(19)

The training and testing sets are generated for 𝑘3

= 1 and𝑘3

= 4/7, respectively, giving in total 4495 training and 816testing samples.

The testing results for six MLP-DoA3trained networks

with the best test statistics are shown in Table 5. MLP2-22-22 network which, among the group of models with thehighest 𝑟PPM values, shows the lowest Worst Case Error ischosen as representative MLP-DoA

3neural network. The

scattering diagram of MLP2-22-22 network outputs andoutputs of MUSIC model obtained by model simulation onthe same testing set are shown in Figures 8, 9, and 10. Again,the MUSIC model produces the best agreement with thereferent values. Neural model has slightly worse agreementin comparison to the MUSIC model; however, its accuracy isstill very good and it is achievedwith significantly shorter runtime than the MUSIC model.

4.3. Comparison betweenMLP-DoAs andMUSICModels RunTimes. As an illustration of MLP-DoA

𝑠networks efficiency

for DoA estimation, comparison of simulation run times


MUSICMLP2-22-22

−60

−40

−20

0

20

40

60

−40 −20 4020 60−60 0𝜃1


𝜃1

∘ )(s

ourc

e 1)—

mod

el o

utpu

t (


(MLP2-22-22) and MUSIC model for output 1 (source 1) obtainedby simulation on the test set(Case 𝑠 = 3).

MUSICMLP2-22-22

−40 −20 4020 60−60 0−60

−40

−20

0

20

40

60

𝜃2∘)(source 2)—referent values (

𝜃2

∘ )(s

ourc

e 2)—

mod

el o

utpu

t (



required for these networks to calculate the angular positionsof radiating sources in points defined by the testing setwith the MUSIC model simulation run time is shown inTable 6. In this comparison, the MUSIC model with spacescan resolution of 0.1∘ is used. The run times given in Table 6are rounded and they are measured for simulations runningon referent hardware platform Intel Pentium M processor1.73GHz, 512MB RAM, within the MATLAB software pack-age.

It can be seen that the neural network performs DoAestimation at much higher speed than the MUSIC model.

MUSICMLP2-22-22

−60

−40

−20

0

20

40

60

−40 −20 4020 60−60 0𝜃3


𝜃3

∘ )(s

ourc

e 3)—

mod

el o

utpu

t (



This neural network ability is very important for choosing anappropriate model for fast real-time DoA applications.

4.4. PNN-MLP Model Simulation of DoA Estimation ofThree Mobile Sources. After the training and selectionof PNN and appropriate MLP-DoA

𝑠networks according

to the testing results, the PNN-MLP model is realized.Within the MATLAB software environment, the simula-tion of movement in real time of three independent stochas-tic sources in the observed sector [−60∘ 60∘ ] is done andin real time determination of angular azimuth positionsof these sources is performed by using the realized PNN-MLP model. The first, second, and third sources are movedon the trajectories 𝜃

1= 0.8𝑡 − 50, 𝜃

2= 2.4𝑡 − 108, and

𝜃3

= 0.016 ⋅ (𝑡 − 100)2− 30, respectively, where 𝑡 represents

the time interval in seconds passed after the time when PNN-MLP model has started a sector monitoring. Sector moni-toring and DoA estimation performed by this neural modellasted 100 s. The movement of sources was selected in suchway so that during the monitoring there were time intervalsof one, two, or three sources present in the observed sector.The results of DoA estimation are shown in Figure 11. Greatreliability of the neural model to determine the number ofsources in the observed sector and in different moments intime can be observed. In addition to that, the determinationof angular positions in azimuth plane of sources presentin the observed sector in different moments in time isachieved with a high accuracy.

5. Conclusions

The neural network-based approach for DoA estimationof EM radiation of variable number of moving stochasticsources is presented in the paper. Two different neural


Source 3 Source 2

Source trajectoryPNN-MLP

Source 1

20 40 60 80 1000t (s)

−60

−40

−20

0

20

40

60

𝜃-P

NN

-MLP

mod

el o

utpu

t (∘ )

Figure 11: The time domain results of DoA estimation of variablenumber of mobile sources that are present in appropriate momentsin time in the sector [−60∘ 60∘] in azimuth plane, obtained by PNN-MLP model for the case 𝑆 = 3.

networks, PNN andMLP, are used to create the neural modelcapable of accurately and efficiently determining the angularpositions of sources in case when its number is changingin time due to their movement in the observed sector.Considered example verifies that proposed neural modelavoids intensive and time-consuming numerical calculationsin comparison to the conventional approaches and thereforeit is more suitable than conventional approaches for real-timeapplications.

By analysing the results presented in the paper, a potentialproblem while using the PNN-MLP model could appear.Increase in maximal number of sources for which the DoAestimation can be performed might lead to the higher com-plexity of PNNnetwork and the greater total number ofMLP-DoA networks in the neural model. Therefore, the overallarchitecture of the neural model becomes more complexmaking it difficult to train and reducing its accuracy for thecase of greater number of radiating sources in the observedsector. Future research will be focused to solve this potentialproblem.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This research work has been supported by the Ministryfor Education, Science and Technological Development ofSerbia. Also, it has been done within the framework of COSTAction IC1407 (COST ACCREDIT).

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