Khan et al. / Front Inform Technol Electron Eng in press
1
Frontiers of Information Technology & Electronic Engineering
www.zju.edu.cn/jzus; engineering.cae.cn; www.springerlink.com
ISSN 2095-9184 (print); ISSN 2095-9230 (online)
E-mail: jzus@zju.edu.cn
Detection of faulty sensor in array using symmetrical structure and
cultural algorithm hybridized with differential evolution*
d
Shafqat Ullah KHAN†1, Ijaz Mansoor QURESHI2, Fawad ZAMAN3, Wasim KHAN4
(1School of Engineering & Applied Sciences, ISRA University, Islamabad 44000, Pakistan)
(2Electrical Department, Air University, Islamabad 44000, Pakistan)
3
( Electrical Department, COMSAT Institute of information Technology Attock 44000, Pakistan)
ite
(4Electronic Engineering Department, IIU, H-10, Islamabad 44000, Pakistan)
†
E-mail: shafqatphy@yahoo.com
Received Sept. 26, 2015; Revision accepted Jan. 9, 2016; Crosschecked
ed
Abstract: In this work, the detection of fully as well as partially defective sensors in a linear array composed of N sensors is
addressed. In the first step, the symmetrical structure of a linear array is proposed, while in the second step a hybrid technique
based on cultural algorithm with differential evolution is developed. The symmetrical structure has two advantages: instead of
finding all damaged patterns, only (N – 1)/2 patterns are needed, and we are required to scan the region from 0 to 90 instead of 0
to 180, which obviously reduces the computational complexity. Monte Carlo simulations are carried out to validate the performance of proposed scheme, which is compared with already available methods in terms of computational time and mean square
error.
Key words: Cultural algorithm, Differential evolution, Linear symmetrical sensor array
http://dx.doi.org/10.1631/FITEE.1500315
CLC number:
1 Introduction
un
In the literature, several techniques are available
that address the issue of defective sensors in an array
antenna. Rodríguez et al. (2000, 2009) diagnosed the
defective sensor using the genetic algorithm (GA),
where the fitness function compares the measured
radiation pattern with the given configuration of
failed/unfailed sensors. Patnaik et al. (2007) used a
neural network (NN) approach to detect a maximum
of three defective sensors in a small array of 16 sensors, while Bucci et al. (2000) considered the ambiguity of the result in the continuous and discrete
on–off cases. Xu et al. (2007) used the support vector
*
Project supported by the Higher Education Commission of Pakistan.
ORCID: Shafqat Ullah KHAN, http://orcid.org/0000-00031969-1289
© Zhejiang University and Springer-Verlag Berlin Heidelberg 2016
machine (SVM) to diagnose the defective sensors in a
small array of four sensors, but this technique is not
applicable to large arrays because of the possible
number of combination boosts. Moreover, the available techniques are computationally expensive, as
they not only require storing the patterns of all defective sensors in the array but also need to scan the
entire region from 0 to 180.
Today, biologically inspired techniques, especially differential evolution (DE) and cultural algorithm (CA), are considered efficient and reliable tools
for optimization in the field of science and engineering ( Zaman et al., 2012).
CA and DE are optimization techniques that include domain knowledge obtained during the evolutionary process. In many optimization problems, both
CA and DE have successfully overcome the shortcomings of conventional optimization techniques due
to their suppleness and effectiveness (Reynolds and
2
Khan et al. / Front Inform Technol Electron Eng in press
where wn is the nonuniform weight of the nth sensor,
d is the spacing between the adjacent sensors, and 
is the angle from broadside. Similarly, k  2 /  is
the wave number with  as the wavelength, and
  kd cos  s is the progressive phase shift,
where  s is the steering angle for the main beam. For a
nonhealthy setup, as shown in Fig. 1, AF can be
written as
M
AF i    wn exp  jn  kd cos i    
(2)
d
n  M
nm
If either wm or w m is damaged, i.e., by putting
wm or w m equal to zero, the array factor for the mth
ite
Peng, 2005; Reynolds and Chung, 1996; Becerra,
2006; Jin X and Reynolds 1999). DE and CA are
stochastic-based search algorithms, in which the
function parameters are programmed as floating-point
variables. They are simple in structure, converge fast,
and are robust against noise. Fonollosa et al. (2013)
developed a more reliable e-nose and robust system in
which machine learning based on multiple kernels
was generated to overcome sensor failures. The outcome confirmed that multi-kernel models are more
robust to sensor failures when the sub-kernels are
trained with small sets of sensors.
In this paper, the detection of fully as well as
partially defective sensors in a linear array composed
of N sensors is addressed. In the first step, the symmetrical structure of a linear array is proposed, while
in the second step a hybrid technique based on CA
with DE is developed. In this hybrid process, the
results achieved through CA are given to DE for
further tuning. Mean-squared error (MSE) is used as
an objective evaluation function that defines the error
between the responses of desired and estimated ones.
The symmetrical structure has two advantages: instead of finding all damaged patterns, only patterns
are needed, and we are required to scan the region
from 0 to 90 instead of 0 to 180, which obviously
reduces the computational complexity. The proposed
method outperforms the conventional method of
Choudhury et al. (2013) in terms of computational
time and MSE. Monte Carlo simulations are carried
out to validate the performance of proposed scheme,
which is compared with the already available methods in terms of computational time and MSE.
damaged sensor in a noisy environment is given by
M
AF i    wn exp  jn  kd cos i      i (3)
n  M
nm
where i is the additive zero mean complex Gaussian
noise with  the variance at the nth sensor. AFm (i )
un
ed
is the pattern when wm or w m is fully faulty, as
depicted in Fig. 1. Mathematically, the measurement
noise (in dB) can be expressed as
K
2

AF i  
i 1

SNR 
K
  2 
i 1 i 
We assume that the sensor w4 fails in the array.
The method of locating a faulty element in a linear
2 Problem formulation
Consider a uniform linear array (ULA) of
N  2M  1 sensors along the x-axis with respect to
the origin. The far-field array factor (AF) for a healthy
setup of equally spaced sensors of nonuniform amplitude and progressive phase excitation can be given
as (Wolf, 1937)
M
AF i    wn exp  jn  kd cos i    
n  M
(4)
(1)
Y

w M
w2 d w1
w0
w1
×
w2
X
wM
Fig. 1 Nonuniform amplitude array of 2 M  1 sensors
with w2 sensor defective
3
Khan et al. / Front Inform Technol Electron Eng in press
array starts with the measurement of several samples
of the faulty pattern. The damaged array pattern for
the w4 sensor is shown in Fig. 2, where one can
whether the sensor is fully or partially defective. The
value of the threshold has been found on the basis of
the MSE. If the lowest error is < 0.5 = Eth , the weight
clearly observe that the pattern is symmetrical
about   90 .
wm or w m is fully faulty. If the lowest error is >0.5
CADE technique to find the weights for a partial
detective sensor. The fitness function is given by
0
Original
w4 element faulty
-10
= Eth , then the weight is partially faulty. We use the
90
G   PF i   PCADE i 
-30
-40
(6)
i 1
-50
-60
where
-70
PF i 
is the desired
response, and
PCADE i  is the value of the pattern obtained by
-80
0
Fig. 2
20
40
60
80
100
Angle (Degrees)
120
140
160
180
N 1
Original Chebyshev array and the w4 sensor
damage pattern
3 Proposed methodology
using the CADE technique. The proposed method
starts with tabulating both the faulty patterns
ite
-90
-100
2
d
Far-Field Pattern(dB)
-20
FF i F 21
and the single defective pattern Pm i 
evaluated in the range 0 to 90 . Then the computation of Cm in Equation (5) is performed by finding
the faulty sensor that minimizes Cm
faulty patterns and the single defective pattern when
wm or w m is fully faulty.
ed
In this section, we develop the proposed
method based on faulty patterns that are symmetric
about   90 . Due to the symmetric structure, the
patterns are the same if either wm or w m is dam-
0
-20
90
Cm   PF i   Pm i 
i 1
2
(5)
Equation (5) compares the faulty patterns with a
given configuration of a fully faulty sensor, and the
minimum result will give us the location of a faulty
sensor. In Equation (5), PF (i ) is the faulty pattern,
Pm (i ) is the pattern when wm or w m is fully faulty,
and m is given by 1  m  ( N  1) / 2 . Then, on the
basis of another fitness function, we will decide
Far-Field Pattern(dB)
un
patterns. The other advantage is that we have only to
scan the damage pattern from 0 to 90 , as the damaged pattern is symmetrical about   90 . The location of the faulty sensor can be found as
Original
w-4 element failure
-10
aged. The failure of w4 or w4 gives the same pattern as shown in Figs. 2 and 3. For the detection of
the faulty sensor, we have to tabulate half the number
of faulty patterns; that is, we require ( N  1) / 2 faulty
between the
-30
-40
-50
-60
-70
-80
-90
-100
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 3 Original Chebyshev array and w4 sensor failure
pattern
The proposed method is computationally efficient, as we require half the number of samples
90
i i 0 ; also we have to tabulate
( N  1) / 2 faulty
patterns for the detection of the faulty sensors. The
method of finding the faulty sensor in an array starts
with the measurement of faulty patterns. If the defective sensor fails but radiates some power, i.e., the
defective array pattern can be obtained from Equa-
4
Khan et al. / Front Inform Technol Electron Eng in press
tion (2), the defective weight is a fraction of the
original one.
A. Differential evolution
 f hq , ge if rand ()  CR or h  hrand 
rkq , ge   q , g

e
d h otherwise

where 0.5  CR  1 and h
rand
(8)
is chosen randomly.
DE was developed by Storn and Price and is
used to solve real-valued optimization problems
(Storn and Price, 1995). DE is a stochastic-based
search algorithm that has a simple structure, fast
convergence, and robustness against noise. It has
shown good results for multimodel and
non-differential fitness functions. DE is based on a
mutation operator, which adds an amount obtained
by the difference of two randomly chosen individuals
of the present population. Thus, it has got tremendous
applications in the field of science and engineering
( Das and Konar, 2006; Rogalsky, 1999). The basic
steps are given in the form of a pseudo-code as follows:
Step 1. Initialization: First we initialize Q
c) Selection operation: The next-generation chromosome is generated by the following expression:
chromosomes randomly, where the length of each
chromosome is 1 P . The P genes in each chromosome represent the weights of the array antenna,
given as
B: Cultural algorithm




d
rq, ge if error r q, ge  error d q, ge CRor j  jrand 
d q, ge 1  

q, g
d e otherwise

where error  r q , ge  and error  d q , ge  are defined in
ite
matrix S .
Step 3. Stopping criterion: The stopping criterion is based on the following condition:
iteration has reached.
CA was developed by Reynolds (1994) to
model the evolution of the cultural component of an
evolutionary computational system over time. The
main idea behind CA is to clearly attain problem
solving knowledge from the growing population, and
apply this knowledge to guide the search space
(Robert and Chung, 1996). CA utilizes culture as a
van for storing information that is available to the
entire population over many generations. The parameter setting is given in Table 1, while the flow
diagram for CA is shown in Fig. 4. CA consists of
three components: a populated space, a belief space,
and a communication protocol. The first one contains
the population to be evolved and the mechanisms for
its estimate. The population space consists of a set of
possible solutions to the problem. In this work, the
population space is DE. The second one is a belief
space that represents the bias that has been acquired
ed
S
If error  d q , ge 1    , the maximum number of
lb  wi , k  ub







i  1, 2, 3,...Q
k  1, 2, 3,...P
un
wi , k  R
 w1,1 w1,2 , ...., w1, p

 w2,1 w2,2 ,...., w2, p




 w w ,...., w
Q, p
 Q,1 Q ,2
Step 2. Updating: All the chromosomes form 1
to Q of the current generation are updated. Choose
d hq, ge from the matrix in which ge and h represent
the particular generation and length of chromosome,
respectively. Our main task is to find the chromosome of the next generation, i.e., eq , ge  1 , by using
mutation, crossover, and selection operations.
a) Mutation: To perform the mutation process, we
select randomly three different chromosomes from
matrix S : 1  c1 , c2 , c3  Q with c1  c2  c3  j
f q , ge  d c1 , ge  F (d c2 , ge  d c3 , ge ) 0.5  F  1
(7)
b) Crossover: Crossover is performed on the expression
Table 1 Parameters used in the CADE algorithm
Parameter
Setting
Population size
500
No of generation
500
Value of F
0.5
Value of CR
0.5 < CR < 1
5
Khan et al. / Front Inform Technol Electron Eng in press
knowledge. CA is used as an optimization algorithm,
its belief space is stored and updated with the population space during each generation, and the
knowledge in the belief space is used to guide the
search space toward the required solution.
The differential evolution variation operators
are influenced in the following way:

n ,g
n ,g
y i , g  Qi  F w 2  w 3

(9)
where Qi is the ith component of the individual stored
d
by the population during its problem-solving process.
The belief space is the information depository in
which the individuals can store their experiences for
other individuals to learn them ultimately. These two
spaces are connected to each other through the
communication protocol composed of two functions,
i.e., the acceptance function and the influence function. The acceptance function is used to accept the
experience of the best individuals from the population space and store them in the belief space, and then
the knowledge in the belief space can be updated
through the update function. The influence function
can guide the search space. In the present work, the
belief space is divided into two knowledge components, i.e., situational knowledge and normative
in the situational knowledge.
The normative knowledge includes a scaling
factor dsi to influence the mutation operator adopted
ite
in DE. The following expression shows the influence
of the normative knowledge of the variation operators:





un
ed
zij, g


wn1 , g  F wn2 , g  wn3 , g if wn1 , g  l

i


 n1 , g

n3 , g
n2 , g
n1 , g
if w  ui
 w  F w  w
 (10)


 n1 , g  ui  li 

n3 , g
n2 , g
w
F
w
w
otherwise
*






ds
 i 



where li and ui are the lower and upper bounds for the
n ,g
ith decision variable, respectively. w 1 represents
the jth component of the ith individual that is selected
from the gth generation by the acceptance function i,
i  1, 2,...naccepted . naccepted is the number of best individuals at the gth generation. The values dsi are updated with the difference
w
n2 , g
w
n3 , g
 found of
the variation operators of the prior generation. Normative knowledge leads individuals to jump into the
good range if they are not already there. The normative knowledge is updated as follows: Let us consider
xa1 , xa 2 , xa 3 ,..., xan
to be the accepted individuaccepted
als in the current generation. Here wmini , wmaxi
NO
YES
 a , a , a ,..., n
1
2
3
accepted

belong to and the accepted
individuals with minimum and maximum values for
the parameter i.
Fig. 4 Generic flow diagram of the cultural algorithm
6
Khan et al. / Front Inform Technol Electron Eng in press
i
and
i

(11)

(12)
otherwise
 wi ,min
i
li  
l
 i
If li

if wi,max  ui or f wmax  Ui

if wi ,mini  li or f wmini  Li
otherwise
ui are updated, the values of Li
and
U i will be updated in the same way. dsi are updated
with the greatest difference of
wi , r1  wi , r 2 found
during the variation operators at the previous generation.
faulty, and for partial fault we will use the CADE
technique to find the weights using Equation (6).
Simulation results for completeness as well as for
partial fault have been checked, which show the validity of the proposed method.
Figure 5 shows the behavior of the diagnostic
error for different values of the signal-to-noise ratio
(SNR). We examined different values of SNR versus
the MSE; the smaller the value of SNR, the greater the
MSE.
As one can clearly see from Fig. 5, the MSE
decreases as the value of SNR increases.
ite
4 Simulation results and discussion
faulty. If Cm > 0.5 = Eth , then weight is partially
d
w
 i,maxi
ui  
ui
Fig. 5 Mean-squared error versus signal-to-noise ratio
un
ed
In this section we discuss several cases based on
different numbers of defective sensors in array.
Case 1: Consider that a Chebyshev linear array
of 51 sensors with  / 2 intersensor spacing is used as
the test antenna. The array sensors are placed symmetrically along the x-axis and excited around the
center of the array. An analytical technique is used to
find the nonuniform weights for a –30 dB constant
side lobes level (SLL) in the Chebyshev array. To
diagnose the faulty sensor in the linear symmetrical
array, the radiation patterns for the fully and partial
faulty sensors were generated. The samples were
taken from the patterns in the region from 0 to 90 .
In the interval of 6, 15 samples were taken from each
pattern to scan the region from 0 to 90 for the detection of fully and partially faulty sensors. It is clear
from Figs. 2 and 3 that the failure of either w4 or w4
Case 2: In this case we consider that a sensor
fails partially and radiates some power, i.e., its weight
is not zero but a fraction of the original one. First we
consider that w2 sensor is 50% damaged and this
damage pattern is created by making the weight of the
sensor to half of the original weight in the original
Chebyshev array. Now the CADE technique is used
to locate its position. This array factor is obtained by
making the weight of the w2 sensor equal to half of
gives the same pattern, which is symmetrical about
  90 if either w4 or w4 fails. For the detection of
the original weight in Equation (1) and is represented
by PF i  in the fitness function of Equation (6).
the faulty sensor, we have to tabulate both the faulty
Then CADE technique is used to minimize the fitness
function, which in turn gives the weight of the defected sensor. MSE is used as a fitness function,
which is given by Equation (6). To check the performance of the CADE technique, these weights are
compared to the weights obtained by the Chebyshev
method in which the weight of the sensor equals half
of original weight. The weight obtained by CADE is
given in Table 2, and the recovered pattern by CADE
technique is shown in Fig. 6.
patterns
FF i F 1 and the single fault pattern
25
Pm i  if either wm or w m is damaged. The cost
function in Equation (5) is minimized for a given
90
sample i i  0 . The decision for the detection of
faulty sensor will be made on the basis of the cost
function. The minimization of cost functions in
Equation (5) will give us the location of a faulty
sensor. If Cm < 0.5 = Eth , the weight wm is fully
7
Khan et al. / Front Inform Technol Electron Eng in press
1
1
Original
w2 50% faulty
Corrected by CADE
0.9
Original
w10 25% faulty
Corrected by CADE
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
-20
-15
-10
-5
0
5
10
15
20
25
Fig. 6 Weight distribution of the original , w2 defected,
and obtained by CADE
0.1
-25
-20
-15
-10
-5
0
5
10
15
20
25
Fig. 8 Weight distribution of the original, w10 defected,
and corrected by CADE
Now we suppose that the sensor is 50% faulty by
making w5 weights half of the original value. The
of the partial fault. The weights for the original
Chebyshev array and the obtained weights for different cases by CADE technique are given in Table 2.
Now we assume that the w sensor has 25% failed.
ite
CADE technique is used to locate its position. The
array factor is obtained by making the weight of the
w5 sensor equal to half of the original value in Equa-
d
0.2
-25
10
The CADE algorithm is run to locate the partial fault
of the sensor. The weights obtained by CADE for the
detection of partial faults (25%) for sensor are given
Table 2 and shown in Fig. 8.
Table 2
CADE
Sensor
position
w25
w24
w23
w22
w21
w20
w19
w18
w17
w16
w15
w14
w13
w12
w11
w10
w9
w8
w7
w6
w5
w4
w3
w2
w1
w0
Chebyshev and normalized weights obtained from
un
ed
tion (1). Then the CADE technique is used to minimize the fitness function in Equation (6), which in
turn gives the weight of the faulty array. The weights
obtained by using the CADE technique for partial
failure are given in Table 2. To check the validity, the
weights obtained by CADE technique are compared
to the weights obtained from the Chebyshev method
of damage patterns for 50% fault. The comparison of
the weights amplitude of the defective array with the
weights obtained by CADE technique is given in
Table 2. The weight distributions of original, partially
faulty, and that obtained by CADE are depicted in
Fig. 7. From this comparison, the partial fault can be
clearly identified. Comparison of the obtained
weights by CADE with that of the defected array
shows the level
1
Original
w5 50% faulty
Corrected by CADE
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-25
-20
-15
-10
-5
0
5
10
15
20
25
Fig. 7 Weight distribution of original, w5 defected, and
obtained by CADE
Chebyshev w2 sensor
weights
50% failed
0.6426
0.6475
0.2200
0.2234
0.2554
0.2593
0.2928
0.2959
0.3321
0.3402
0.373
0.3754
0.4152
0.4197
0.4584
0.461
0.5023
0.5076
0.5465
0.5485
0.5907
0.5934
0.6344
0.6431
0.6773
0.6796
0.719
0.7203
0.7591
0.7612
0.7973
0.7995
0.833
0.8413
0.8661
0.8676
0.8961
0.8987
0.9228
0.9264
0.9459
0.9476
0.9651
0.9685
0.9802
0.9843
0.9912
0.4879
0.9978
0.9986
1.0000
0.9994
w5 sensor
50% failed
0.6437
0.2252
0.2584
0.2969
0.3451
0.3758
0.4247
0.4635
0.5085
0.5496
0.5952
0.6417
0.6815
0.7351
0.7652
0.7987
0.8418
0.8673
0.8984
0.9378
0.4857
0.9721
0.9873
0.9992
0.9983
0.9987
w10 sensor
25% failed
0.6453
0.2317
0.2643
0.2975
0.3429
0.3813
0.4341
0.4618
0.5163
0.5538
0.5916
0.6384
0.6794
0.7265
0.7631
0.1987
0.8491
0.8752
0.8987
0.9376
0.9518
0.9687
0.9885
0.9932
0.9986
0.9975
8
0.9995
0.9945
0.9835
0.9676
0.9513
0.9301
0.8978
0.8751
0.8421
0.7985
0.7584
0.7542
0.6833
0.6461
0.5932
0.5496
0.5159
0.4613
0.4271
0.3742
0.3364
0.2953
0.2571
0.2262
0.6432
0.9993
0.9928
0.9834
0.9676
0.9473
0.9362
0.8988
0.8676
0.8414
0.7993
0.7632
0.7531
0.6795
0.6384
0.5951
0.5485
0.5046
0.4597
0.4174
0.3819
0.3357
0.2961
0.2589
0.2317
0.6478
0.9996
0.9934
0.9827
0.9675
0.9486
0.9264
0.8975
0.8676
0.8451
0.7998
0.7685
0.7326
0.6891
0.6454
0.5937
0.5579
0.5163
0.4627
0.4256
0.3828
0.3482
0.2952
0.2617
0.2356
0.6513
proposed method. We assume a maximum of three
defective sensors, which yields a total of
3
N!
 2324 different patterns by the

f 1  f !( N  f )!


conventional method but our proposed method requires 1162 (i.e., half the number of patterns of the
conventional method) pattern, which requires fewer
computations and less cost. To locate the faulty sensors in an array antenna, the weight of each sensor
was considered as the optimizing parameter for the
BFO and CADE algorithm. CADE will converge to
the optimum solution. In an array diagnosis, 35 samples were taken in the range 0 to 180 in intervals of
5. We suppose that the 4th and 10th sensors are partially faulty and the 17th sensor is fully faulty. We run
the BFO and the proposed method to diagnose the
faulty sensors. Figure 11 shows the faulty pattern with
the position of 35 samples. The fault diagnosed by the
conventional method is shown in Fig. 12. Now the
same process is repeated using the proposed method,
which
d
0.9978
0.9912
0.9802
0.9651
0.9459
0.9228
0.8961
0.8661
0.833
0.7973
0.7591
0.719
0.6773
0.6344
0.5907
0.5465
0.5023
0.4584
0.4152
0.373
0.3321
0.2928
0.2554
0.2200
0.6426
ite
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10
w11
w12
w13
w14
w15
w16
w17
w18
w19
w20
w21
w22
w23
w24
w25
Khan et al. / Front Inform Technol Electron Eng in press
0
Original array
Faulty array
-10
random samples to validate the performance of the
Far-Field Pattern(dB)
-20
-30
-40
-50
-60
-70
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 9 Chebyshev array and the 4th, 10th (50%), and 17th
(100%) sensor faulty pattern
0
Original array
Faulty array
-10
Far-Field Pattern(dB)
un
ed
Case 3: In this section we consider a linear
Chebyshev array of 24 sensors taken as the reference
antenna to execute the method of fault diagnosis developed by Choudhury et al. (2013). The proposed
method is compared with the conventional method.
The fully and partial faulty patterns were generated by
making their weights either equal to zero or some
fraction of the original weights. We assume that the
4th, 10th (50%), and 17th (100%) sensor in the array
have become faulty. The faulty pattern of the 4th,
10th (50%), and 17th (100%) sensor is shown in Fig.
9. Now we consider the symmetrical counterpart
failure of 4th, 10th (50%), and 17th (100%), which is
depicted in Fig. 10. It is clear from Figs. 9 and 10 that
both failures give the same power pattern. Therefore
we have to tabulate half of the faulty patterns. The
second advantage of using a symmetrical linear array
is that the faulty pattern is symmetrical about
  90 , i.e., we need half the number of samples to
scan the pattern. First we simulated the pattern with
1–3 elements faulty. We have used a set of 1162
patterns. In each case, the faulty pattern contains the
M samples as input to check the diagnosis of fault.
In this case, we take samples of M  18, M  35 and
-20
-30
-40
-50
-60
-70
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 10 Chebyshev array and the symmetrical counterpart of (4th, 10th (50%), and 17th (100%)) sensor faulty
pattern
9
Khan et al. / Front Inform Technol Electron Eng in press
Original weights
Defected weights
1
Normalized weight excitation
is shown in Figs. 13 and 14. By the proposed method,
we diagnosed the fault by half the number of samples.
Similarly, the same fault is repeated for 18 and some
random samples for the conventional and proposed
method. The results obtained by conventional and
proposed method for 18 and random numbers of
samples are shown in Figs. 15–18. By the proposed
method even with fewer sample points, one can detect
the faulty sensor accurately. Figure 19 shows the
MSE plots for conventional and the proposed method.
0.8
0.6
0.4
0.2
5
10
15
20
Element Number
Fig. 14 Performance of the proposed method with 19
sample points
d
0
0
-10
-20
Far-Field Pattern(dB)
-30
-30
ite
Far-Field Pattern(dB)
-10
-20
-40
-50
-60
-40
-50
-60
-70
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 11 Defective array pattern with fault at 4th, 10th
(50%), and 17th (100%) sensor with 35 sample points
-70
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 15 Defective array pattern with fault at 4th, 10th
(50%), and 17th (100%) sensor with 10 sample points
ed
1
Original weights
Defected weights
Normalized weight excitation
Normalized weight excitation
1
0.8
0.6
0.4
0.2
5
10
15
0.8
0.6
0.4
0.2
20
un
Element Number
5
0
-10
-10
-20
-20
-40
-50
-60
-70
15
20
Fig. 16 Performance of proposed method with 10 sample
points
0
-30
10
Element Number
Far-Field Pattern(dB)
Far-Field Pattern(dB)
Fig. 12
Performance of the conventional method
(Choudhury et al. 2013) with 35 sample points
-30
-40
-50
-60
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 13 Defective pattern with fault at 4th, 10th (50%),
and 17th (100%) sensor with 19 points
-70
0
20
40
60
80
100
Angle (Degrees)
120
140
160
180
Fig. 17 Defective array pattern with fault at 4th, 10th
(50%), and 17th (100%) sensor with random sample
points
10
Khan et al. / Front Inform Technol Electron Eng in press
16000
Original weights
Defected by BFO
MSE by conventional method
MSE by proposed method
14000
0.8
12000
Cost function
Normalized weight excitation
1
0.6
0.4
10000
8000
6000
4000
0.2
2000
5
10
15
0
20
Element Number
500
1000
Number of iteration
1500
Fig. 19 Mean square error performance of the conventional (Choudhury et al., 2013) and the proposed method
Table 3 Time comparison analysis for six configurations of one defective sensor
Proposed method
Fault
place
Conventional method
18 samples
56.72
58.53
59.65
60.87
54.47
52.89
35 samples
117.46
121.69
115.78
109.95
108.6
107.95
Average
time
57.18
18 samples
Average
time
113.57
108.42
113.73
114.56
117.35
107.5
105.45
35 samples
Average
time
111.16
ed
1
2
5
10
20
23
Time (s)
ite
Time (s)
201.71
206.5
213.31
205.37
203.95
204.87
Average
time
205.95
Table 4 Time comparison analysis for six configurations of two defective sensors
Fault
place
Proposed method
Conventional method
Time (s)
Time (s)
18 samples
67.7
66.4
72.3
74.8
76.2
78.9
35 samples
Average
time
72.71
139.7
141.4
153.6
155.1
161.9
165.2
18 samples
Average
time152
.81
un
5, 7
20, 23
18, 20
11, 21
7, 10
8, 12
123.5
121.7
137.1
139.3
143.6
144.1
35 samples
Average
time
134.88
254.3
237.4
238.75
241.5
231.94
235.5
Average
time
240.48
Table 5 Time comparison analysis for six random configurations of three defective sensors
Proposed method
Conventional method
Time (s)
Time (s)
Fault
place
18 samples
5, 7, 12
1, 7, 12
1, 9, 11
13, 17, 24
15, 19, 22
14, 21, 24
81.7
85.3
79.3
82.6
85.5
77.3
Average
time
81.95
35 samples
169.4
173.7
167.3
169.9
175.5
164.8
2000
d
Fig. 18 Performance of the proposed method with random sample points
0
Average
time
170.1
18 samples
144.1
157.3
142.9
153.5
151.3
145.6
Average
time
149.11
35 samples
231.4
274.1
230.9
227.2
237.6
241.7
Average
time
240.48
11
Khan et al. / Front Inform Technol Electron Eng in press
Table 6 Time comparison for six random configurations of fully and partially faulty sensors
Proposed method
Conventional method
Time (s)
Time (s)
18 samples
85.7
84.4
82.8
Average
time 84.3
18 samples
35 samples
182.
6
171.
6
141.5
241.2
152.9
251.7
175.
80.9
Average
time
177.9
179.
86.7
181.
85.1
176.
147.7
Average
time 148.2
143.8
249.1
Average
time 250.01
252.4
151.3
246.8
152.3
258.9
ite
8, 12 (100%)
2(50%)
13, 18(50%)
24(100%)
14, 20 (100%)
15(50%)
1, 8 (100)
2(50%)
15, 23 (100%)
24(50%)
4, 10 (50%)
17 (100%)
35 samples
d
Fault place
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un
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