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Progress In Electromagnetics Research, Vol. 127, 139–154, 2012

ADAPTIVE BEAMFORMING WITH LOW SIDE LOBE
LEVEL USING NEURAL NETWORKS TRAINED BY
MUTATED BOOLEAN PSO
Z. D. Zaharis* , K. A. Gotsis, and J. N. Sahalos
Radiocommunications Laboratory, Department of Physics, Aristotle
University of Thessaloniki, GR-54124 Thessaloniki, Greece
Abstract—A new adaptive beamforming technique based on neural
networks (NNs) is proposed. The NN training is accomplished
by applying a novel optimization method called Mutated Boolean
PSO (MBPSO). In the beginning of the procedure, the MBPSO is
repeatedly applied to a set of random cases to estimate the excitation
weights of an antenna array that steer the main lobe towards a desired
signal, place nulls towards several interference signals and achieve the
lowest possible value of side lobe level. The estimated weights are
used to train efficiently a NN. Finally, the NN is applied to a new set
of random cases and the extracted radiation patterns are compared
to respective patterns extracted by the MBPSO and a well-known
robust adaptive beamforming technique called Minimum Variance
Distortionless Response (MVDR). The aforementioned comparison has
been performed considering uniform linear antenna arrays receiving
several interference signals and a desired one in the presence of additive
Gaussian noise. The comparative results show the advantages of the
proposed technique.

1. INTRODUCTION
Smart antenna technology is a very interesting and challenging
issue in modern communications [1–3]. One of the major interests
concerns the design of antenna arrays that produce radiation patterns
dynamically shaped according to certain signal directions that vary
with time. In particular, the array must form a main lobe towards a
desired incoming signal called signal-of-interest (SOI) and several nulls
Received 28 February 2012, Accepted 6 April 2012, Scheduled 11 April 2012
* Corresponding author: Zaharias D. Zaharis (zaharis@auth.gr).
140

Zaharis, Gotsis, and Sahalos

towards respective undesired or interference incoming signals. Taking
into account that the geometry of the array is time-independent, a
dynamically shaped pattern is achieved by applying on the array
elements appropriate excitation weights that vary with time. These
weights are calculated in real time by beamforming techniques [3–17].
Therefore, the beamforming algorithm must be completed as fast as
possible.
Our study presents a new adaptive beamforming (ABF) technique
suitable for antenna arrays [3–6, 9–17]. The technique is based on
neural networks (NNs) [5, 18–26], which use training sets produced by
a novel binary variant of Particle Swarm Optimization (PSO) [27–
33], called Mutated Boolean PSO (MBPSO) [10]. In the MBPSO,
the update of particle velocities and positions is performed using
exclusively Boolean expressions, while former binary PSO variants
update the particle velocities using real number expressions [34]. Since
real number expressions need more CPU time to obtain a result than
Boolean expressions, the MBPSO becomes more effective than other
binary PSO variants. Moreover, the MBPSO algorithm involves a
novel process of adaptive velocity mutation that makes the algorithm
more effective than the conventional Boolean PSO [35]. Both the
Boolean update mechanism and the adaptive mutation process make
the MBPSO a robust algorithm suitable for NN training.
The proposed technique has been applied to uniform linear arrays
(ULAs). It starts by selecting a set of random cases where a
ULA receives several interference signals and a SOI at respective
directions of arrival (DOA) in the presence of additive zero-mean
Gaussian noise. The above directions are usually calculated by DOA
estimation algorithms [1, 19, 22, 24, 36–40]. The DOA of the SOI and
the interference signals represent the input parameters for each case.
The MBPSO is applied to each case in order to extract the array
excitation weights that steer the main lobe towards the SOI, place
nulls towards the interference signals and achieve the lowest possible
side lobe level (SLL). These weights are used to train a NN. The
NN derived from the training procedure is the actual beamformer.
In order to test its effectiveness, a new set of random cases is
selected. Then, for every case, the NN is applied to extract the
excitation weights and the produced radiation pattern is compared to
corresponding patterns extracted by the MBPSO and a well-known
robust adaptive beamforming technique called Minimum Variance
Distortionless Response (MVDR) [1]. The above comparison shows
the advantages of the proposed technique.
Progress In Electromagnetics Research, Vol. 127, 2012

θ0

x1

s0
θ1

s1
sN

θN

x2

.
.
.

.
.
.
xM

141

∗
w1
∗
w2
.
.
.

y

∗
wM

Figure 1. Beamformer block diagram.
2. FORMULATION
The condition described by the beamforming theory [1] is a ULA which
is composed of M isotropic sources and receives several monochromatic
signals sn (k) (n = 0, 1, . . . , N ) from respective angles of arrival θn
(n = 0, 1, . . . , N ). An angle of arrival (AOA) is defined here as the
angle between the DOA of a signal and the reference direction which
is normal to the ULA axis. The variable k indicates the kth time
sample. The signal s0 (k) is the SOI, while sn (k) (n = 1, . . . , N )
are N interference signals (see Figure 1). The SOI is considered as
reference signal in terms of power and thus its mean power is given by:
Ps = E |s0 (k)|2 = 1

(1)

where E{·} denotes the mean value. Besides, each mth array element
receives an additive zero mean Gaussian noise signal nm (k) (m =
2
1, . . . , M ) with variance σn calculated from the signal-to-noise ratio
SNR in dB as follows:
2
σn = 10−SNR/10
(2)
The signal xm (k) at the input of the mth element can be calculated
by the following expression:
¯ ¯
¯ ¯
x(k) = a0 s0 (k) + [ a1 a2 . . . aN ] s(k) + n(k)
¯
¯
¯
(3)
where
T
x(k) = [ x1 (k) x2 (k) . . . xM (k) ]
¯
T

s(k) = [ s1 (k) s2 (k) . . . sN (k) ]
¯

(4)
(5)

T

n(k) = [ n1 (k) n2 (k) . . . nM (k) ]
¯

(6)

are, respectively, the input vector, the interference vector and the noise
vector, while
an = 1 ej
¯

2π
q sin θn
λ

2π

· · · ej(M −1) λ q sin θn

T

,

n = 0, 1, . . . , N (7)

is the array steering vector at AOA θn . Also, the superscript T
indicates the transpose operation. In (7), q is the distance between
142

Zaharis, Gotsis, and Sahalos

adjacent elements of the ULA and λ is the wavelength. Equation (3)
can be written in the following form:
¯s
x (k) = a0 s0 (k) + A¯ (k) + n (k) = xd (k) + xu (k)
¯
¯
¯
¯
¯
(8)
¯
where A = [¯1 a2 . . . aN ] is an M × N matrix called array steering
a ¯
¯
matrix. The vectors
xd (k) = a0 s0 (k)
¯
¯
¯s
xu (k) = A¯(k) + n(k)
¯
¯

(9)
(10)

are respectively the vector of the desired input signals and the vector
of the undesired (interference plus noise) input signals. According to
Figure 1, the array output is calculated as follows:
y (k) = wH x (k) = wH xd (k) + wH xu (k)
¯ ¯
¯ ¯
¯ ¯

(11)

]T

where w = [w1 w2 . . . wM
¯
is the excitation weight vector and
the superscript H indicates the Hermitian transpose operation.
Equation (11) can be written in the following form:
y (k) = yd (k) + yu (k)

(12)

yd (k) = wH xd (k)
¯ ¯

(13)

where
H

yu (k) = w xu (k)
¯ ¯

(14)

are, respectively, the desired and the undesired component of the array
output. The mean power of yd (k) is expressed as:
2
σd = E

wH xd (k)
¯ ¯

2

=E

wH a0 s0 (k)
¯ ¯

2

= wH a0 aH w
¯ ¯ ¯0 ¯

(15)

Also, the mean power of yu (k) is expressed as:
2
σu = E

wH xu (k)
¯ ¯

2

=E

¯s
wH A¯ (k) + n (k)
¯
¯

2

= wH ARi AH w + wH Rn w
¯ ¯¯ ¯ ¯ ¯ ¯ ¯
(16)
¯
¯
where Ri = E{¯(k) sH (k)} and Rn = E{¯ (k) nH (k)} are respectively
s
¯
n
¯
the interference correlation matrix and the noise correlation matrix.
Given that n (k) consists of uncorrelated zero-mean noise signals, it
¯
2
¯
results Rn = σn I. Thus, (16) can be written in the following form:
2
2
σu = wH ARi AH w + σn wH w
¯ ¯¯ ¯ ¯
¯ ¯
(17)
One of the parameters used to measure the effectiveness of a
beamformer is the signal-to-interference-plus-noise ratio (SINR). Due
to (15) and (17), SINR can be calculated by:
SINR =

2
σd
wH a0 aH w
¯ ¯ ¯ ¯
= H ¯ ¯ ¯H 0 2 H
2
σu
w ARi A w + σn w w
¯
¯
¯ ¯

(18)
Progress In Electromagnetics Research, Vol. 127, 2012

143

The basic process performed by the MBPSO is the minimization
of a fitness function F . The inverse of SINR could be used as an
expression of F . As F is minimized, SINR is maximized, meaning that
the main lobe is steered towards the SOI and nulls are formed towards
the interference signals. Our technique becomes more challenging by
setting an additional requirement which is the minimization of the SLL.
Taking into account the above considerations, F can be described by
the following expression:
2 ¯ ¯
wH ARi AH w + σn wH w
¯ ¯¯ ¯ ¯
+ γ2 SLL
(19)
F = γ1
wH a0 aH w
¯ ¯ ¯0 ¯
where coefficients γ1 and γ2 are used to balance the minimization of
the two terms given in (19).
3. MUTATED BOOLEAN PSO
The Boolean PSO (BPSO) is a binary PSO variant described in [35].
The MBPSO is a novel version of BPSO proposed by the authors [10].
In the BPSO and MBPSO, the position Xs and the velocity Vs
(s = 1, . . . , S) of every particle of the swarm are represented by J-bit
strings. The search space is defined by an upper and a lower boundary.
A large fitness value is assigned as a penalty to particles being outside
the search space. Provided that the optimization process minimizes the
fitness function, these particles are gradually moved inside the search
space.
An important novelty found only in the BPSO and MBPSO is the
exclusively Boolean update of Xs and Vs given below:
vjs = r1 · vjs + r2 · (pjs ⊕ xjs ) + r3 · (gj ⊕ xjs )
xjs = xjs ⊕ vjs

(20)
(21)

where ·, + and ⊕ are respectively the “and ”, “or ” and “xor ” operators,
xjs and vjs are respectively the jth bit of Xs and Vs , pjs is the jth bit of
the best position Ps found so far by the sth particle and gj is the jth bit
of the best position G found so far by the swarm. Moreover, r1 , r2 , and
r3 are random bits and their probabilities of being ‘1’ are respectively
defined by the parameters R1 , R2 , and R3 . The exclusively Boolean
update makes both the BPSO and MBPSO more effective than a wellknown binary PSO (binPSO) variant that uses real number update
expressions as described in [34].
Both the BPSO and MBPSO control the convergence speed of the
process by controlling the velocity length ls which is the number of ‘1’s
in Vs and is not permitted to exceed an upper limit lmax . Therefore,
144

Zaharis, Gotsis, and Sahalos

if ls > lmax then randomly chosen ‘1’s in Vs change into ‘0’s until
ls = lmax .
To increase the exploration ability of the swarm, an adaptive
mutation process applied to particle velocities has been involved in
the MBPSO. This novelty makes the MBPSO more effective than the
typical BPSO. According to this process, every ‘0’ in Vs may change to
‘1’ with mutation probability mp , which linearly decreases as follows:
itot − i
, i = 1, . . . , itot
(22)
itot − 1
where i is the current iteration number, itot is the total number of
iterations and mp0 is the initial value of mp . Usually, mp0 has relatively
small values to avoid pure random search. Comparative convergence
graphs presented below in section 6 exhibit the superiority of the
MBPSO in comparison to the typical BPSO and the binPSO proposed
in [34].
The MBPSO is an iterative technique and like every evolutionary
technique is time-consuming. Nevertheless, this problem is not crucial
because the MBPSO is used here only for NN training which is not
a real time procedure. The real time procedure is performed by the
trained NN which responds very fast.
mp (i) = mp0

4. MINIMUM VARIANCE DISTORTIONLESS
RESPONSE
The Minimum Variance Distortionless Response (MVDR) is a robust
adaptive beamforming method that aims at minimizing the mean
power of yu (k), while yd (k) is preserved [1]. Thus, the optimum
excitation weight vector is derived by minimizing the quantity wH Ru w,
¯ ¯ ¯
H a = 1, and is given by:
while w ¯0
¯
¯ ¯
R−1 a0
wmvdr = H u −1
¯
(23)
a Ru a0
¯ ¯ ¯
0

where

2
¯
¯¯ ¯
Ru = E xu (k) xH (k) = ARi AH + σn I
¯
¯u

(24)

is the correlation matrix of yu (k).
5. NN-MBPSO BASED ADAPTIVE BEAMFORMING
A NN is a structure of interconnected information processing units,
called neurons, organized in layers [18]. During the training of a NN,
the weight connections of its neurons properly change in order to model
Progress In Electromagnetics Research, Vol. 127, 2012

145

Figure 2. Block diagram illustrating the NN structure and the NNMBPSO based adaptive beamforming methodology.
the mapping between certain inputs and their respective outputs. NNs
have been broadly applied in various problems of electromagnetics and
mobile communications [5, 18–26]. Due to their fast response and easy
implementation, NNs constitute an attractive solution for real time
applications, such as beamforming and DOA estimation [5, 19, 22, 24].
In the proposed ABF technique, the NNs are trained by L
¯
randomly generated angle vectors θl = [θ0l θ1l . . . θN l ]T paired
with the respective optimal excitations weight vectors wl =
¯
[w1l w2l . . . wM l ]T . The first element of the lth angle vector, θ0l , is
the AOA of the SOI, while the other elements, θnl (n = 1, . . . , N ), are
the AOA of the interference signals. The weight vectors are optimized
by applying the MBPSO on the fitness function given in (19).
¯ ¯
The L randomly created pairs (θl , wl ) constitute a set employed
for the supervised training of a feedforward Multilayer Perceptron
(MLP) NN [18]. The training takes place in MATLAB R R2010a
environment, using a very efficient implementation of the fast
and effective Levenberg-Marquardt backpropagation algorithm [41].
Figure 2 illustrates the proposed ABF method, giving also the NN
structure. The NN is composed by (a) an input layer of N + 1 nodes
fed by the angle vectors, (b) two hidden layers and (c) an output
layer of M nodes that gives the corresponding weight vectors. The
number of nodes for each hidden layer depends on the number of
training pairs and the dimension of the angle vector. The criterion
of their choice is the better NN training performance and the accuracy
of the results. More details about NN training using the LevenbergMarquardt backpropagation algorithm in MATLAB can be found
in [19].
The introduced NN-MBPSO based adaptive beamforming
methodology is summarized in the following steps:
¯
1. Random generation of L angle vectors θl denoting the AOA of the
SOI and the interference signals.
¯
¯
2. Production of the optimal wl that correspond to θl using the
146

Zaharis, Gotsis, and Sahalos

MBPSO algorithm.
3. Creation of a MLP NN and back propagation training using the
¯ ¯
collection of the randomly created pairs (θl , wl ), l = 1, 2, . . . , L.
4. The trained NN instantly responds to any input angle vector,
giving as output the excitation weight vector that makes the
antenna array produce a radiation pattern with the desired
characteristics concerning the main lobe, the nulls and the SLL.
6. NUMERICAL RESULTS

Average Fitness Value

Three different scenarios are considered to test the performance of the
proposed technique. The first two scenarios concern a 9-element ULA
(M = 9) with q = 0.5λ and SNR = 10 dB receiving respectively three
(N = 3) and five (N = 5) interference signals, while the third scenario
concerns a 7-element ULA (M = 7) with q = 0.5λ and SNR = 10 dB
receiving three (N = 3) interference signals. The parameters used by
the MBPSO in all the scenarios were: S = 20, R1 = 0.1, R2 = 0.5,
R3 = 0.5, lmax = 4, mp0 = 0.10, and itot = 500. A set of 5000 random
cases (L = 5000) is selected for each scenario. Each case is a group of
N + 1 values randomly selected from a uniform angle distribution and
given respectively to θn (n = 0, 1, . . . , N ).
Initially, a comparison in terms of convergence among the
MBPSO, the conventional BPSO and the binary PSO (binPSO)
proposed in [34] is made. Thus, the three methods are applied to each
one of the 5000 cases of the first scenario to extract the corresponding
w. The convergence graphs of the three methods are recorded for each
¯
case. In this way, comparative graphs showing the average convergence
are constructed (see Figure 3). It is obvious that the MBPSO converges
a little slower than the BPSO, but it finally achieves better fitness
5

bin PSO
BPSO
MBPSO

4
3
2
1
0
0

100

200

300

400

500

Number of Iterations

Figure 3. Comparative graphs showing the average convergence of
the MBPSO, the conventional BPSO and the binary PSO (binPSO)
proposed in [34].
Progress In Electromagnetics Research, Vol. 127, 2012

147

values. Moreover, the MBPSO converges faster and achieves better
fitness values than the binPSO. The above comparison justifies the use
of MBPSO-based data to train a NN.
The excitation weight vectors extracted by the MBPSO for the
5000 random cases of each scenario are used to train a NN. The
trained NN is compared to the MBPSO and MVDR in terms of
performance by selecting a new set of 1000 random cases. Then, the
three algorithms are applied to each case to extract the excitation
weight vectors, respectively wNN , wMBPSO and wMVDR , as well as
¯
¯
¯
the corresponding radiation patterns produced by these vectors. The
weights of each vector are normalized with reference to the weight of
the middle element of the array. The amplitudes of all the weights
found by the above procedure range from 0.05 to 2.
The 1000 patterns derived by the NN are statistically analyzed for
each scenario regarding the absolute divergence ∆θmain of the main lobe
direction from its desired value θ0 as well as the absolute divergence
∆θnull of the null directions from their respective desired values θn
(n = 1, . . . , N ). The statistical results are illustrated in Figures 4, 5
1500

Number of AOA

Number of AOA

500

1000

500

0
0

0.5

1

1.5

2

2.5

400
300
200
100
0
0

3

0.5

1

1.5

2

∆θ °
main

∆θ °

null

Figure 4. Statistical distributions of the main lobe and null angular
divergences derived from the NN for the 1st scenario (M = 9, N = 3).
400

Number of AOA

Number of AOA

2000
1500
1000
500
0
0

0.5

1

1.5

2

∆θ °

null

2.5

3

3.5

4

300
200
100
0
0

0.5

1

1.5

2

∆θ °
main

Figure 5. Statistical distributions of the main lobe and null angular
divergences derived from the NN for the 2nd scenario (M = 9, N = 5).
148

Zaharis, Gotsis, and Sahalos

Number of AOA

Number of AOA

400
1500
1000

500
0
0

0.5

1

1.5

2

2.5

3

300
200
100
0
0

0.5

°
∆θnull

1

1.5

2

∆θ°
main

Figure 6. Statistical distributions of the main lobe and null angular
divergences derived from the NN for the 3rd scenario (M = 7, N = 3).

Figure 7. Optimal patterns for SNR = 10 dB, M = 9, a SOI
arriving from θ0 = −13◦ , and three interference signals arriving
from AOA −56◦ , 20◦ and 46◦ (SINR NN = 19.16 dB, SINR MBPSO =
19.14 dB, SINR MVDR = 19.35 dB, SLLNN = −16.73dB, SLLMBPSO =
−16.15 dB, SLLMVDR = −12.23 dB, DNN = 9.35 dB, DMBPSO =
9.35 dB, DMVDR = 9.32 dB).
Table 1. Average angular divergence and average SLL values.
Scenario
M
N
∆θmain
∆θnull
SLLNN
SLLMBPSO
SLLMVDR

1st
9
3
0.33◦
0.75◦
−13.61 dB
−13.44 dB
−12.26 dB

2nd
9
5
0.46◦
0.95◦
−13.28 dB
−13.18 dB
−11.74 dB

3rd
7
3
0.47◦
0.73◦
−12.57 dB
−12.55 dB
−11.25 dB
Progress In Electromagnetics Research, Vol. 127, 2012

149

and 6. Considering a confidence level of 5% for all the scenarios, the
main lobe divergence is less than 1◦ and the null divergence is less
than 2◦ . The above analysis as well as the average absolute divergence
values ∆θmain and ∆θnull given in Table 1 show that the NN has a high
percentage of success in steering both the main lobe and the nulls.
In addition, the above set of 1000 patterns derived by the trained
NN is used to calculate the average SLL value denoted as SLLNN .
Respective values, SLLMBPSO and SLLMVDR , are calculated from two
similar sets of 1000 patterns derived by the MBPSO and the MVDR.

Figure 8. Optimal patterns for SNR = 10 dB, M = 9, a SOI arriving
from θ0 = −28◦ , and five interference signals arriving from AOA
−44◦ , −13◦ , 3◦ , 38◦ and 59◦ (SINR NN = 18.91 dB, SINR MBPSO =
18.93 dB, SINR MVDR = 18.76 dB, SLLNN = −14.34 dB, SLLMBPSO =
−13.48 dB, SLLMVDR = −12.11 dB, DNN = 9.45 dB, DMBPSO =
9.45 dB, DMVDR = 9.43 dB).
Table 2. Normalized optimal weight values for SNR = 10 dB, M = 9,
a SOI arriving from θ0 = −13◦ , and three interference signals arriving
from AOA −56◦ , 20◦ and 46◦ .
m
1
2
3
4
5
6
7
8
9

w NN
−0.553 + j0.277
−0.332 + j0.657
0.205 + j1.059
0.930 + j0.903
1.000 + j0
0.930 − j0.903
0.205 − j1.059
−0.332 − j0.657
−0.553 − j0.277

wMBPSO
−0.525 + j0.285
−0.302 + j0.585
0.223 + j1.000
0.917 + j0.887
1.000 + j0
0.917 − j0.887
0.223 − j1.000
−0.302 − j0.585
−0.525 − j0.285

wMVDR
−1.179 + j0.479
−0.735 + j1.254
0.344 + j1.837
1.378 + j1.325
1.000 + j0
1.378 − j1.325
0.344 − j1.837
−0.735 − j1.254
−1.179 − j0.479
150

Zaharis, Gotsis, and Sahalos

Table 3. Normalized optimal weight values for SNR = 10 dB, M = 9,
a SOI arriving from θ0 = −28◦ , and five interference signals arriving
from AOA −44◦ , −13◦ , 3◦ , 38◦ and 59◦ .
m
1
2
3
4
5
6
7
8
9

w NN
0.841 − j0.229
−0.447 − j0.824
−1.023 + j0.239
0.009 + j1.137
1.000 + j0
0.009 − j1.137
−1.023 − j0.239
−0.447 + j0.824
0.841 + j0.229

wMBPSO
0.733 − j0.264
−0.469 − j0.759
−1.000 + j0.153
−0.019 + j1.000
1.000 + j0
−0.019 − j1.000
−1.000 − j0.153
−0.469 + j0.759
0.733 + j0.264

wMVDR
0.974 − j0.328
−0.495 − j1.071
−1.294 + j0.256
−0.026 + j1.231
1.000 + j0
−0.026 − j1.231
−1.294 − j0.256
−0.495 + j1.071
0.974 + j0.328

These values are given in Table 1. It seems that SLLNN approaches
SLLMBPSO but it is better than SLLMVDR . Both facts are predictable
because the NN is trained by the MBPSO, which takes into account
the SLL minimization as shown in (19), while the MVDR does not.
Thus, in many cases the NN produces notably better SLL values than
the MVDR. Such cases are shown in Figures 7 and 8. The values of
SINR, SLL and directivity D, derived from each case, are given in the
legend of the respective figure. Also, the normalized optimal weight
values are given respectively in Tables 2 and 3.
7. CONCLUSION
A new robust ABF method, that combines the optimization
capabilities of the MBPSO with the speed and efficiency of NNs,
has been developed. NNs have been trained by optimal training
sets derived by the MBPSO, in order to learn to produce the proper
excitation weight vectors that make the array steer the main lobe
towards the SOI and form nulls towards the interference signals.
Emphasis has been given to the production of radiation patterns with
lower SLL compared to the MVDR, which is a popular ABF technique.
Extensive simulation results prove the generalization capabilities of the
properly trained NNs and show that the proposed NN-MBPSO based
adaptive beamforming methodology succeeds the above mentioned
goals.
The cases studied here show that the MBPSO converges a little
Progress In Electromagnetics Research, Vol. 127, 2012

151

slower than the BPSO and faster than the binPSO, and also achieves
better fitness values than both the BPSO and binPSO. The CPU time
required by the MBPSO to converge and the NN training is not an
issue, since neither the MBPSO nor the training is involved in the real
time procedure of the actual beamformer. After its training the NN
responds instantly. Therefore, the proposed beamformer seems to be
quite promising in the smart antenna technology.
REFERENCES
1. Gross, F. B., Smart Antennas for Wireless Communications with
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09.12022806[1]

  • 1. Progress In Electromagnetics Research, Vol. 127, 139–154, 2012 ADAPTIVE BEAMFORMING WITH LOW SIDE LOBE LEVEL USING NEURAL NETWORKS TRAINED BY MUTATED BOOLEAN PSO Z. D. Zaharis* , K. A. Gotsis, and J. N. Sahalos Radiocommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece Abstract—A new adaptive beamforming technique based on neural networks (NNs) is proposed. The NN training is accomplished by applying a novel optimization method called Mutated Boolean PSO (MBPSO). In the beginning of the procedure, the MBPSO is repeatedly applied to a set of random cases to estimate the excitation weights of an antenna array that steer the main lobe towards a desired signal, place nulls towards several interference signals and achieve the lowest possible value of side lobe level. The estimated weights are used to train efficiently a NN. Finally, the NN is applied to a new set of random cases and the extracted radiation patterns are compared to respective patterns extracted by the MBPSO and a well-known robust adaptive beamforming technique called Minimum Variance Distortionless Response (MVDR). The aforementioned comparison has been performed considering uniform linear antenna arrays receiving several interference signals and a desired one in the presence of additive Gaussian noise. The comparative results show the advantages of the proposed technique. 1. INTRODUCTION Smart antenna technology is a very interesting and challenging issue in modern communications [1–3]. One of the major interests concerns the design of antenna arrays that produce radiation patterns dynamically shaped according to certain signal directions that vary with time. In particular, the array must form a main lobe towards a desired incoming signal called signal-of-interest (SOI) and several nulls Received 28 February 2012, Accepted 6 April 2012, Scheduled 11 April 2012 * Corresponding author: Zaharias D. Zaharis ([email protected]).
  • 2. 140 Zaharis, Gotsis, and Sahalos towards respective undesired or interference incoming signals. Taking into account that the geometry of the array is time-independent, a dynamically shaped pattern is achieved by applying on the array elements appropriate excitation weights that vary with time. These weights are calculated in real time by beamforming techniques [3–17]. Therefore, the beamforming algorithm must be completed as fast as possible. Our study presents a new adaptive beamforming (ABF) technique suitable for antenna arrays [3–6, 9–17]. The technique is based on neural networks (NNs) [5, 18–26], which use training sets produced by a novel binary variant of Particle Swarm Optimization (PSO) [27– 33], called Mutated Boolean PSO (MBPSO) [10]. In the MBPSO, the update of particle velocities and positions is performed using exclusively Boolean expressions, while former binary PSO variants update the particle velocities using real number expressions [34]. Since real number expressions need more CPU time to obtain a result than Boolean expressions, the MBPSO becomes more effective than other binary PSO variants. Moreover, the MBPSO algorithm involves a novel process of adaptive velocity mutation that makes the algorithm more effective than the conventional Boolean PSO [35]. Both the Boolean update mechanism and the adaptive mutation process make the MBPSO a robust algorithm suitable for NN training. The proposed technique has been applied to uniform linear arrays (ULAs). It starts by selecting a set of random cases where a ULA receives several interference signals and a SOI at respective directions of arrival (DOA) in the presence of additive zero-mean Gaussian noise. The above directions are usually calculated by DOA estimation algorithms [1, 19, 22, 24, 36–40]. The DOA of the SOI and the interference signals represent the input parameters for each case. The MBPSO is applied to each case in order to extract the array excitation weights that steer the main lobe towards the SOI, place nulls towards the interference signals and achieve the lowest possible side lobe level (SLL). These weights are used to train a NN. The NN derived from the training procedure is the actual beamformer. In order to test its effectiveness, a new set of random cases is selected. Then, for every case, the NN is applied to extract the excitation weights and the produced radiation pattern is compared to corresponding patterns extracted by the MBPSO and a well-known robust adaptive beamforming technique called Minimum Variance Distortionless Response (MVDR) [1]. The above comparison shows the advantages of the proposed technique.
  • 3. Progress In Electromagnetics Research, Vol. 127, 2012 θ0 x1 s0 θ1 s1 sN θN x2 . . . . . . xM 141 ∗ w1 ∗ w2 . . . y ∗ wM Figure 1. Beamformer block diagram. 2. FORMULATION The condition described by the beamforming theory [1] is a ULA which is composed of M isotropic sources and receives several monochromatic signals sn (k) (n = 0, 1, . . . , N ) from respective angles of arrival θn (n = 0, 1, . . . , N ). An angle of arrival (AOA) is defined here as the angle between the DOA of a signal and the reference direction which is normal to the ULA axis. The variable k indicates the kth time sample. The signal s0 (k) is the SOI, while sn (k) (n = 1, . . . , N ) are N interference signals (see Figure 1). The SOI is considered as reference signal in terms of power and thus its mean power is given by: Ps = E |s0 (k)|2 = 1 (1) where E{·} denotes the mean value. Besides, each mth array element receives an additive zero mean Gaussian noise signal nm (k) (m = 2 1, . . . , M ) with variance σn calculated from the signal-to-noise ratio SNR in dB as follows: 2 σn = 10−SNR/10 (2) The signal xm (k) at the input of the mth element can be calculated by the following expression: ¯ ¯ ¯ ¯ x(k) = a0 s0 (k) + [ a1 a2 . . . aN ] s(k) + n(k) ¯ ¯ ¯ (3) where T x(k) = [ x1 (k) x2 (k) . . . xM (k) ] ¯ T s(k) = [ s1 (k) s2 (k) . . . sN (k) ] ¯ (4) (5) T n(k) = [ n1 (k) n2 (k) . . . nM (k) ] ¯ (6) are, respectively, the input vector, the interference vector and the noise vector, while an = 1 ej ¯ 2π q sin θn λ 2π · · · ej(M −1) λ q sin θn T , n = 0, 1, . . . , N (7) is the array steering vector at AOA θn . Also, the superscript T indicates the transpose operation. In (7), q is the distance between
  • 4. 142 Zaharis, Gotsis, and Sahalos adjacent elements of the ULA and λ is the wavelength. Equation (3) can be written in the following form: ¯s x (k) = a0 s0 (k) + A¯ (k) + n (k) = xd (k) + xu (k) ¯ ¯ ¯ ¯ ¯ (8) ¯ where A = [¯1 a2 . . . aN ] is an M × N matrix called array steering a ¯ ¯ matrix. The vectors xd (k) = a0 s0 (k) ¯ ¯ ¯s xu (k) = A¯(k) + n(k) ¯ ¯ (9) (10) are respectively the vector of the desired input signals and the vector of the undesired (interference plus noise) input signals. According to Figure 1, the array output is calculated as follows: y (k) = wH x (k) = wH xd (k) + wH xu (k) ¯ ¯ ¯ ¯ ¯ ¯ (11) ]T where w = [w1 w2 . . . wM ¯ is the excitation weight vector and the superscript H indicates the Hermitian transpose operation. Equation (11) can be written in the following form: y (k) = yd (k) + yu (k) (12) yd (k) = wH xd (k) ¯ ¯ (13) where H yu (k) = w xu (k) ¯ ¯ (14) are, respectively, the desired and the undesired component of the array output. The mean power of yd (k) is expressed as: 2 σd = E wH xd (k) ¯ ¯ 2 =E wH a0 s0 (k) ¯ ¯ 2 = wH a0 aH w ¯ ¯ ¯0 ¯ (15) Also, the mean power of yu (k) is expressed as: 2 σu = E wH xu (k) ¯ ¯ 2 =E ¯s wH A¯ (k) + n (k) ¯ ¯ 2 = wH ARi AH w + wH Rn w ¯ ¯¯ ¯ ¯ ¯ ¯ ¯ (16) ¯ ¯ where Ri = E{¯(k) sH (k)} and Rn = E{¯ (k) nH (k)} are respectively s ¯ n ¯ the interference correlation matrix and the noise correlation matrix. Given that n (k) consists of uncorrelated zero-mean noise signals, it ¯ 2 ¯ results Rn = σn I. Thus, (16) can be written in the following form: 2 2 σu = wH ARi AH w + σn wH w ¯ ¯¯ ¯ ¯ ¯ ¯ (17) One of the parameters used to measure the effectiveness of a beamformer is the signal-to-interference-plus-noise ratio (SINR). Due to (15) and (17), SINR can be calculated by: SINR = 2 σd wH a0 aH w ¯ ¯ ¯ ¯ = H ¯ ¯ ¯H 0 2 H 2 σu w ARi A w + σn w w ¯ ¯ ¯ ¯ (18)
  • 5. Progress In Electromagnetics Research, Vol. 127, 2012 143 The basic process performed by the MBPSO is the minimization of a fitness function F . The inverse of SINR could be used as an expression of F . As F is minimized, SINR is maximized, meaning that the main lobe is steered towards the SOI and nulls are formed towards the interference signals. Our technique becomes more challenging by setting an additional requirement which is the minimization of the SLL. Taking into account the above considerations, F can be described by the following expression: 2 ¯ ¯ wH ARi AH w + σn wH w ¯ ¯¯ ¯ ¯ + γ2 SLL (19) F = γ1 wH a0 aH w ¯ ¯ ¯0 ¯ where coefficients γ1 and γ2 are used to balance the minimization of the two terms given in (19). 3. MUTATED BOOLEAN PSO The Boolean PSO (BPSO) is a binary PSO variant described in [35]. The MBPSO is a novel version of BPSO proposed by the authors [10]. In the BPSO and MBPSO, the position Xs and the velocity Vs (s = 1, . . . , S) of every particle of the swarm are represented by J-bit strings. The search space is defined by an upper and a lower boundary. A large fitness value is assigned as a penalty to particles being outside the search space. Provided that the optimization process minimizes the fitness function, these particles are gradually moved inside the search space. An important novelty found only in the BPSO and MBPSO is the exclusively Boolean update of Xs and Vs given below: vjs = r1 · vjs + r2 · (pjs ⊕ xjs ) + r3 · (gj ⊕ xjs ) xjs = xjs ⊕ vjs (20) (21) where ·, + and ⊕ are respectively the “and ”, “or ” and “xor ” operators, xjs and vjs are respectively the jth bit of Xs and Vs , pjs is the jth bit of the best position Ps found so far by the sth particle and gj is the jth bit of the best position G found so far by the swarm. Moreover, r1 , r2 , and r3 are random bits and their probabilities of being ‘1’ are respectively defined by the parameters R1 , R2 , and R3 . The exclusively Boolean update makes both the BPSO and MBPSO more effective than a wellknown binary PSO (binPSO) variant that uses real number update expressions as described in [34]. Both the BPSO and MBPSO control the convergence speed of the process by controlling the velocity length ls which is the number of ‘1’s in Vs and is not permitted to exceed an upper limit lmax . Therefore,
  • 6. 144 Zaharis, Gotsis, and Sahalos if ls > lmax then randomly chosen ‘1’s in Vs change into ‘0’s until ls = lmax . To increase the exploration ability of the swarm, an adaptive mutation process applied to particle velocities has been involved in the MBPSO. This novelty makes the MBPSO more effective than the typical BPSO. According to this process, every ‘0’ in Vs may change to ‘1’ with mutation probability mp , which linearly decreases as follows: itot − i , i = 1, . . . , itot (22) itot − 1 where i is the current iteration number, itot is the total number of iterations and mp0 is the initial value of mp . Usually, mp0 has relatively small values to avoid pure random search. Comparative convergence graphs presented below in section 6 exhibit the superiority of the MBPSO in comparison to the typical BPSO and the binPSO proposed in [34]. The MBPSO is an iterative technique and like every evolutionary technique is time-consuming. Nevertheless, this problem is not crucial because the MBPSO is used here only for NN training which is not a real time procedure. The real time procedure is performed by the trained NN which responds very fast. mp (i) = mp0 4. MINIMUM VARIANCE DISTORTIONLESS RESPONSE The Minimum Variance Distortionless Response (MVDR) is a robust adaptive beamforming method that aims at minimizing the mean power of yu (k), while yd (k) is preserved [1]. Thus, the optimum excitation weight vector is derived by minimizing the quantity wH Ru w, ¯ ¯ ¯ H a = 1, and is given by: while w ¯0 ¯ ¯ ¯ R−1 a0 wmvdr = H u −1 ¯ (23) a Ru a0 ¯ ¯ ¯ 0 where 2 ¯ ¯¯ ¯ Ru = E xu (k) xH (k) = ARi AH + σn I ¯ ¯u (24) is the correlation matrix of yu (k). 5. NN-MBPSO BASED ADAPTIVE BEAMFORMING A NN is a structure of interconnected information processing units, called neurons, organized in layers [18]. During the training of a NN, the weight connections of its neurons properly change in order to model
  • 7. Progress In Electromagnetics Research, Vol. 127, 2012 145 Figure 2. Block diagram illustrating the NN structure and the NNMBPSO based adaptive beamforming methodology. the mapping between certain inputs and their respective outputs. NNs have been broadly applied in various problems of electromagnetics and mobile communications [5, 18–26]. Due to their fast response and easy implementation, NNs constitute an attractive solution for real time applications, such as beamforming and DOA estimation [5, 19, 22, 24]. In the proposed ABF technique, the NNs are trained by L ¯ randomly generated angle vectors θl = [θ0l θ1l . . . θN l ]T paired with the respective optimal excitations weight vectors wl = ¯ [w1l w2l . . . wM l ]T . The first element of the lth angle vector, θ0l , is the AOA of the SOI, while the other elements, θnl (n = 1, . . . , N ), are the AOA of the interference signals. The weight vectors are optimized by applying the MBPSO on the fitness function given in (19). ¯ ¯ The L randomly created pairs (θl , wl ) constitute a set employed for the supervised training of a feedforward Multilayer Perceptron (MLP) NN [18]. The training takes place in MATLAB R R2010a environment, using a very efficient implementation of the fast and effective Levenberg-Marquardt backpropagation algorithm [41]. Figure 2 illustrates the proposed ABF method, giving also the NN structure. The NN is composed by (a) an input layer of N + 1 nodes fed by the angle vectors, (b) two hidden layers and (c) an output layer of M nodes that gives the corresponding weight vectors. The number of nodes for each hidden layer depends on the number of training pairs and the dimension of the angle vector. The criterion of their choice is the better NN training performance and the accuracy of the results. More details about NN training using the LevenbergMarquardt backpropagation algorithm in MATLAB can be found in [19]. The introduced NN-MBPSO based adaptive beamforming methodology is summarized in the following steps: ¯ 1. Random generation of L angle vectors θl denoting the AOA of the SOI and the interference signals. ¯ ¯ 2. Production of the optimal wl that correspond to θl using the
  • 8. 146 Zaharis, Gotsis, and Sahalos MBPSO algorithm. 3. Creation of a MLP NN and back propagation training using the ¯ ¯ collection of the randomly created pairs (θl , wl ), l = 1, 2, . . . , L. 4. The trained NN instantly responds to any input angle vector, giving as output the excitation weight vector that makes the antenna array produce a radiation pattern with the desired characteristics concerning the main lobe, the nulls and the SLL. 6. NUMERICAL RESULTS Average Fitness Value Three different scenarios are considered to test the performance of the proposed technique. The first two scenarios concern a 9-element ULA (M = 9) with q = 0.5λ and SNR = 10 dB receiving respectively three (N = 3) and five (N = 5) interference signals, while the third scenario concerns a 7-element ULA (M = 7) with q = 0.5λ and SNR = 10 dB receiving three (N = 3) interference signals. The parameters used by the MBPSO in all the scenarios were: S = 20, R1 = 0.1, R2 = 0.5, R3 = 0.5, lmax = 4, mp0 = 0.10, and itot = 500. A set of 5000 random cases (L = 5000) is selected for each scenario. Each case is a group of N + 1 values randomly selected from a uniform angle distribution and given respectively to θn (n = 0, 1, . . . , N ). Initially, a comparison in terms of convergence among the MBPSO, the conventional BPSO and the binary PSO (binPSO) proposed in [34] is made. Thus, the three methods are applied to each one of the 5000 cases of the first scenario to extract the corresponding w. The convergence graphs of the three methods are recorded for each ¯ case. In this way, comparative graphs showing the average convergence are constructed (see Figure 3). It is obvious that the MBPSO converges a little slower than the BPSO, but it finally achieves better fitness 5 bin PSO BPSO MBPSO 4 3 2 1 0 0 100 200 300 400 500 Number of Iterations Figure 3. Comparative graphs showing the average convergence of the MBPSO, the conventional BPSO and the binary PSO (binPSO) proposed in [34].
  • 9. Progress In Electromagnetics Research, Vol. 127, 2012 147 values. Moreover, the MBPSO converges faster and achieves better fitness values than the binPSO. The above comparison justifies the use of MBPSO-based data to train a NN. The excitation weight vectors extracted by the MBPSO for the 5000 random cases of each scenario are used to train a NN. The trained NN is compared to the MBPSO and MVDR in terms of performance by selecting a new set of 1000 random cases. Then, the three algorithms are applied to each case to extract the excitation weight vectors, respectively wNN , wMBPSO and wMVDR , as well as ¯ ¯ ¯ the corresponding radiation patterns produced by these vectors. The weights of each vector are normalized with reference to the weight of the middle element of the array. The amplitudes of all the weights found by the above procedure range from 0.05 to 2. The 1000 patterns derived by the NN are statistically analyzed for each scenario regarding the absolute divergence ∆θmain of the main lobe direction from its desired value θ0 as well as the absolute divergence ∆θnull of the null directions from their respective desired values θn (n = 1, . . . , N ). The statistical results are illustrated in Figures 4, 5 1500 Number of AOA Number of AOA 500 1000 500 0 0 0.5 1 1.5 2 2.5 400 300 200 100 0 0 3 0.5 1 1.5 2 ∆θ ° main ∆θ ° null Figure 4. Statistical distributions of the main lobe and null angular divergences derived from the NN for the 1st scenario (M = 9, N = 3). 400 Number of AOA Number of AOA 2000 1500 1000 500 0 0 0.5 1 1.5 2 ∆θ ° null 2.5 3 3.5 4 300 200 100 0 0 0.5 1 1.5 2 ∆θ ° main Figure 5. Statistical distributions of the main lobe and null angular divergences derived from the NN for the 2nd scenario (M = 9, N = 5).
  • 10. 148 Zaharis, Gotsis, and Sahalos Number of AOA Number of AOA 400 1500 1000 500 0 0 0.5 1 1.5 2 2.5 3 300 200 100 0 0 0.5 ° ∆θnull 1 1.5 2 ∆θ° main Figure 6. Statistical distributions of the main lobe and null angular divergences derived from the NN for the 3rd scenario (M = 7, N = 3). Figure 7. Optimal patterns for SNR = 10 dB, M = 9, a SOI arriving from θ0 = −13◦ , and three interference signals arriving from AOA −56◦ , 20◦ and 46◦ (SINR NN = 19.16 dB, SINR MBPSO = 19.14 dB, SINR MVDR = 19.35 dB, SLLNN = −16.73dB, SLLMBPSO = −16.15 dB, SLLMVDR = −12.23 dB, DNN = 9.35 dB, DMBPSO = 9.35 dB, DMVDR = 9.32 dB). Table 1. Average angular divergence and average SLL values. Scenario M N ∆θmain ∆θnull SLLNN SLLMBPSO SLLMVDR 1st 9 3 0.33◦ 0.75◦ −13.61 dB −13.44 dB −12.26 dB 2nd 9 5 0.46◦ 0.95◦ −13.28 dB −13.18 dB −11.74 dB 3rd 7 3 0.47◦ 0.73◦ −12.57 dB −12.55 dB −11.25 dB
  • 11. Progress In Electromagnetics Research, Vol. 127, 2012 149 and 6. Considering a confidence level of 5% for all the scenarios, the main lobe divergence is less than 1◦ and the null divergence is less than 2◦ . The above analysis as well as the average absolute divergence values ∆θmain and ∆θnull given in Table 1 show that the NN has a high percentage of success in steering both the main lobe and the nulls. In addition, the above set of 1000 patterns derived by the trained NN is used to calculate the average SLL value denoted as SLLNN . Respective values, SLLMBPSO and SLLMVDR , are calculated from two similar sets of 1000 patterns derived by the MBPSO and the MVDR. Figure 8. Optimal patterns for SNR = 10 dB, M = 9, a SOI arriving from θ0 = −28◦ , and five interference signals arriving from AOA −44◦ , −13◦ , 3◦ , 38◦ and 59◦ (SINR NN = 18.91 dB, SINR MBPSO = 18.93 dB, SINR MVDR = 18.76 dB, SLLNN = −14.34 dB, SLLMBPSO = −13.48 dB, SLLMVDR = −12.11 dB, DNN = 9.45 dB, DMBPSO = 9.45 dB, DMVDR = 9.43 dB). Table 2. Normalized optimal weight values for SNR = 10 dB, M = 9, a SOI arriving from θ0 = −13◦ , and three interference signals arriving from AOA −56◦ , 20◦ and 46◦ . m 1 2 3 4 5 6 7 8 9 w NN −0.553 + j0.277 −0.332 + j0.657 0.205 + j1.059 0.930 + j0.903 1.000 + j0 0.930 − j0.903 0.205 − j1.059 −0.332 − j0.657 −0.553 − j0.277 wMBPSO −0.525 + j0.285 −0.302 + j0.585 0.223 + j1.000 0.917 + j0.887 1.000 + j0 0.917 − j0.887 0.223 − j1.000 −0.302 − j0.585 −0.525 − j0.285 wMVDR −1.179 + j0.479 −0.735 + j1.254 0.344 + j1.837 1.378 + j1.325 1.000 + j0 1.378 − j1.325 0.344 − j1.837 −0.735 − j1.254 −1.179 − j0.479
  • 12. 150 Zaharis, Gotsis, and Sahalos Table 3. Normalized optimal weight values for SNR = 10 dB, M = 9, a SOI arriving from θ0 = −28◦ , and five interference signals arriving from AOA −44◦ , −13◦ , 3◦ , 38◦ and 59◦ . m 1 2 3 4 5 6 7 8 9 w NN 0.841 − j0.229 −0.447 − j0.824 −1.023 + j0.239 0.009 + j1.137 1.000 + j0 0.009 − j1.137 −1.023 − j0.239 −0.447 + j0.824 0.841 + j0.229 wMBPSO 0.733 − j0.264 −0.469 − j0.759 −1.000 + j0.153 −0.019 + j1.000 1.000 + j0 −0.019 − j1.000 −1.000 − j0.153 −0.469 + j0.759 0.733 + j0.264 wMVDR 0.974 − j0.328 −0.495 − j1.071 −1.294 + j0.256 −0.026 + j1.231 1.000 + j0 −0.026 − j1.231 −1.294 − j0.256 −0.495 + j1.071 0.974 + j0.328 These values are given in Table 1. It seems that SLLNN approaches SLLMBPSO but it is better than SLLMVDR . Both facts are predictable because the NN is trained by the MBPSO, which takes into account the SLL minimization as shown in (19), while the MVDR does not. Thus, in many cases the NN produces notably better SLL values than the MVDR. Such cases are shown in Figures 7 and 8. The values of SINR, SLL and directivity D, derived from each case, are given in the legend of the respective figure. Also, the normalized optimal weight values are given respectively in Tables 2 and 3. 7. CONCLUSION A new robust ABF method, that combines the optimization capabilities of the MBPSO with the speed and efficiency of NNs, has been developed. NNs have been trained by optimal training sets derived by the MBPSO, in order to learn to produce the proper excitation weight vectors that make the array steer the main lobe towards the SOI and form nulls towards the interference signals. Emphasis has been given to the production of radiation patterns with lower SLL compared to the MVDR, which is a popular ABF technique. Extensive simulation results prove the generalization capabilities of the properly trained NNs and show that the proposed NN-MBPSO based adaptive beamforming methodology succeeds the above mentioned goals. The cases studied here show that the MBPSO converges a little
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