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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
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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
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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.
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4 International Journal of Antennas and Propagation
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
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International Journal of Antennas and Propagation 5
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
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6 International Journal of Antennas and Propagation
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
-
International Journal of Antennas and Propagation 7
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).
4.2.2. Case 𝑠 = 2. The sets for training and testing of
MLP-DoA2network are obtained as
{(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
-
8 International Journal of Antennas and Propagation
Table 4: Testing results for six MLP-DoA2neural network
models
with the best test statistics.
MLP model WCE [%] ATE [%] 𝑟PPM
MLP2-13-13 2.0540 0.3832 0.9998MLP2-14-14 2.2920 0.3709
0.9998MLP2-16-16 2.3787 0.3784 0.9998MLP2-18-18 2.4203 0.3767
0.9998MLP2-22-20 2.4504 0.3807 0.9998MLP2-23-23 2.4513 0.3864
0.9998
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 (
Figure 6: Scattering diagram of chosen MLP-DoA2network
(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 (
Figure 7: Scattering diagram of chosen MLP-DoA2network
(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.
MLP model WCE [%] ATE [%] 𝑟PPM
MLP4-22-22 3.8280 0.4047 0.9997MLP4-22-20 4.1949 0.4193
0.9997MLP4-23-23 4.2612 0.3929 0.9997MLP4-20-10 5.3978 0.4735
0.9996MLP4-18-18 4.5724 0.4862 0.9996MLP4-18-14 4.6599 0.5252
0.9995
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
1176 samplesMLP2-13-13 MUSIC
0.07 s 106 s𝑠 = 3
816 samplesMLP2-22-22 MUSIC
0.05 s 63 s
4.2.3. Case 𝑠 = 3. The sets for training and testing of
MLP-DoA3network are obtained as
{(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
-
International Journal of Antennas and Propagation 9
MUSICMLP2-22-22
−60
−40
−20
0
20
40
60
−40 −20 4020 60−60 0𝜃1
∘)(source 1)—referent values (
𝜃1
∘ )(s
ourc
e 1)—
mod
el o
utpu
t (
Figure 8: Scattering diagram of chosen MLP-DoA3network
(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 (
Figure 9: Scattering diagram of chosen MLP-DoA3network
(MLP2-22-22) and the MUSIC model for output 2 (source 2)obtained
by simulation on the test set (Case 𝑠 = 3).
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
∘)(source 3)—referent values (
𝜃3
∘ )(s
ourc
e 3)—
mod
el o
utpu
t (
Figure 10: Scattering diagram of chosen MLP-DoA3network
(MLP2-22-22) and the MUSIC model for output 3 (source 3)obtained
by simulation on the test set (Case 𝑠 = 3).
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
-
10 International Journal of Antennas and Propagation
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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Chemical EngineeringInternational Journal of Antennas and
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