Limits of Compensation in a Failed Antenna Array
Om Prakash Acharya, Amalendu Patnaik, Sachendra Nath Sinha
Department of Electronics and Communication Engineering, Indian Institute of Technology Roorkee,Roorkee 247667, Uttrakhand, India
Received 22 August 2013; accepted 6 February 2014
ABSTRACT: In large antenna arrays, the possibility of occurrence of faults in some of the
radiating elements cannot be precluded at all times. In such situations, the radiation pattern
of the array gets distorted, mostly with an increase in sidelobe level and decrease in gain.
Although it is not possible to restore the pattern fully by rearranging the excitations of the
functioning elements, compensation methods have been reported in the literature for restor-
ing one performance parameter of the array and making a trade-off on some other parame-
ter. In this article, we have made a study on the tolerance level of this compensation
process. One part of the study deals with the thinning in the failed array, that is, to find a
limit on the minimum number of functioning elements of the array that can restore the dig-
ital beamforming of the failed array. The second part of study deals with finding the maxi-
mum number of element failures that can be compensated. The study was carried out by
optimizing the amplitude excitations of the failed array. Instead of classical optimization
techniques, particle swarm optimization was used for the compensation process. VC 2014
Wiley Periodicals, Inc. Int J RF and Microwave CAE 24:635–645, 2014.
Keywords: antenna arrays; failed antenna arrays; sidelobe level; particle swarm optimization
I. INTRODUCTION
In modern wireless communication systems, the antenna
array is an integral part to achieve the spatial diversity,
which is known as array beamforming. The demand for
large aperture antennas exhibiting increased capabilities
and reduced cost and complexity is growing day by day.
Because of large size of these arrays, possibility of failure
of one or more elements is always there due to the degra-
dation of performance of the associated circuitries such as
transmitter/receiver modules and power supplies as they
have finite life time. The failure in array elements may
also be due to some unforeseen reasons like vagaries of
weather or natural calamities. The effect of such failures
usually manifests itself in terms of a degradation in the
antenna pattern resulting in sharp variations in the field
intensity, increased sidelobe levels (SLL), and decrease in
the gain and directivity of the antenna. Thus, entire sys-
tem performance is affected due to element failure. The
replacement of the defective element is not possible on
every occasion, and it is a tremendous challenge for the
engineers to establish an uninterrupted communication by
restoring the main beam in the direction of interest and
also reducing the SLL to maximize the signal to noise
ratio. As in an active antenna array, it is possible to con-
trol the excitations of the array elements remotely, the res-
toration of the beam pattern, close to the original pattern,
can be achieved by reconfiguring the excitations of the
remaining functional elements. This provides a cost effec-
tive alternative to hardware replacement, thereby increas-
ing the array life.
In view of increasing demands of antenna arrays in
radar and communication systems, the development of
healing systems for failed arrays has received considerable
attention in recent years. A number of methods have been
investigated to improve the radiation pattern of the array
in the presence of the failed elements by reoptimizing the
weights applied to the remaining elements. Peters [1] pro-
posed a method to reconfigure the amplitude and phase
distribution of the remaining elements for minimizing the
average SLL via a conjugate gradient method. Bu and
Daryoush [2] used the biquadratic programming method
in reconfiguring the array by changing the phase of each
of the remaining active elements. Mailloux [3] used the
method of replacing the signals from failed elements in a
digital beamforming receiving array. A practical failure
Correspondence to: A. Patnaik; e-mail: [email protected].
DOI: 10.1002/mmce.20807
Published online 24 March 2014 in Wiley Online Library
(wileyonlinelibrary.com).
VC 2014 Wiley Periodicals, Inc.
635
compensation technique for active phased arrays was intro-
duced by Levitas et al.[4]. The orthogonal method has also
been used for the same problem [5]. In addition to the clas-
sical optimization methods, several heuristic search evolu-
tionary techniques have been successfully implemented to
handle the problem of failed antenna array. Array failure
correction for linear array antennas based on genetic algo-
rithms [6, 7], simulated annealing technique [8, 9], and par-
ticle swarm optimization (PSO) [10, 11] have been
successfully implemented for optimizing the performance
of the antenna array with the failed elements. Hybrid opti-
mization techniques [12, 13] have also been applied to
improve the array pattern in the presence of failed elements.
In another approach, the correction procedure following the
failure of some elements of the planar antenna array was
introduced by means of a neural network [14]. On keen
observation of the above referred literature, it can be found
that, in all compensation techniques, there is always a trade-
off between various performance parameters of the array.
However, so far as the practical application of the array is
concerned, there are always some tolerance levels attached
to every parameter of the antenna. This work is a step to
quantify some of these tolerance levels while compensating
failed antenna arrays.
The limit of compensation is studied from two different
aspects. The first investigation is carried out to determine
the minimum number of operational elements whose excita-
tions need to be adjusted to restore the radiation pattern of
the array, whereas the second one is to determine the maxi-
mum number of element failures in an array that can be
compensated. Although the investigations are made on a
specific array, the overall conclusions drawn are equally
applicable to other arrays also. In this work, we have used
the amplitude-only weights synthesis to restore the damaged
array pattern as the variation of the phase weights has very
little effect on SLL. In practice, methods for changing these
complex weights for suppressing the sidelobes are slow and
ineffective for large antenna arrays because of hardware
complexity in its implementation.
To implement the compensation process, any of the
methods mentioned previously can be used. As far as this
work is concerned, we have approached the compensation
task as an optimization problem and resorted to the use of
PSO instead of analytical techniques. The simple reason of
using the stochastic evolutionary computational technique
is its robustness and easy application procedure. At this
point, it may be emphasized that any other classical opti-
mizer or evolutionary optimizer can yield same compensa-
tion result and each method has its own complexity level.
So far as this work is concerned, our aim is not to compare
results obtained from various optimization techniques but to
quantify the extent up to which the compensation is possi-
ble. The comparison of different optimization techniques
for array failure compensation can be studied as a separate
problem.
The rest of the article is organized as follows. In Sec-
tion II, a brief description of PSO algorithm is given. The
formulation of the optimization problem for the array fail-
ure correction is discussed in Section III. In Section IV,
investigations and observations on the limits of the com-
pensation are presented. Conclusion is given in Section V.
II. PARTICLE SWARM OPTIMIZATION
Although the PSO algorithm is described elsewhere in the
literature [15–18], for the sake of completeness, we dis-
cuss the procedure in brief. This optimizer was introduced
by Kenedy and Eberhart in 1995, while investigating on
the idea of collective intelligence in biological popula-
tions. PSO is inspired by the ability of flocks of birds,
schools of fish to adapt to their environment, find rich
sources of food, and avoid predators by implementing an
information sharing approach [15]. The concept behind
this optimization method is based on how and when the
information concerning new promising regions discovered
by any of the individuals, particles, or agents throughout
the optimization process is transmitted to rest of the
swarm, that is, new discoveries or improvements achieved
by any particle is available to rest of the population. It is
simple to apply, easy to code, and is a high performance
evolutionary computational technique. This algorithm is
capable of solving difficult multidimensional optimization
problems and has already been applied successfully for
solving many electromagnetic problems [16].
According to PSO terminology, every individual
swarm is called a particle. Initially, PSO starts its opera-
tion by randomly generating the particles and locating
them within a space with dimensions equal to the number
of design parameters used in the optimization process.
The initial random position and velocity are represented
as xi and vi, respectively. At each time step, a function fiis calculated using the particle’s positional coordinate as
input, which represents a quality measure or goodness of
the position in terms of the parameters to be optimized.
The position of each particle is updated according to the
best position attained by the particle so far, that is, perso-
nal best (pbest), depending on the fitness function at that
position and the best position of the whole group, that is,
Figure 1 Chebyshev array pattern and the mask used for corre-
sponding damaged pattern.
636 Acharya et al.
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 24, No. 6, November 2014
global best (gbest) until that instant. The new position is
achieved by the particles by updating their velocities
according to:
vi ðs 1 1Þ5 wviðsÞ1 c1r1 ðpbest ðsÞ2 xi ðsÞÞ1 c2r2 ðgbest ðsÞ 2 xi ðsÞÞ
(1)
where vi(s) is the velocity of particle i at iteration s, xi(s)
is corresponding position, pbest,i represents the best posi-
tion by particle i, and gbest is best position discovered by
the swarm. r1 and r2 are two uniformly distributed random
functions in the range [0, 1]. c1 and c2 are two scaling
factors which determine the relative pull of pbest and gbest.
The parameter w is a number called inertial weight which
specifies the weight by which the particle’s current veloc-
ity depends on its previous velocity and how far the parti-
cle is from its personal and global best positions. The
convergence of PSO depends on the values of w in each
iteration and is set according to [17],
w5wmax2wmax2wmin
T3s (2)
where wmax and wmin are maximum and minimum value
of weighting factor and are chosen as 0.9 and 0.4, respec-
tively. T is the maximum number of iterations and s is
value of the current iteration. Once the velocity has been
determined, it is easy to move the particle to its next loca-
tion. The velocity is applied for a given time-step Dt and
the new coordinate xn is computed as
xiðs11Þ5xiðsÞ1Dt 3 viðs11Þ (3)
During this iterative process, the particles gradually
settle down to an optimum position in the space and the
fitness at that position satisfies the requirement of the
optimization problem.
III. PROBLEM FORMULATION
The far field pattern of an N-element linear array with
equally spaced elements having nonuniform amplitude
and progressive phase excitations is given by [19],
FðhÞ5 EPðhÞFmax
XN
n51
wnejðn21Þkd cosh (4)
where wn accounts for the nonuniform current excitation
of each element. The spacing between the elements is d, his the angle measured from broad side, EP(h) is element
pattern, and Fmax is peak value of far field pattern. EP(h)
5 1 for isotropic source. Element failure in an antenna
array causes sharp variations in the field intensity, increas-
ing both sidelobe, and ripple levels of power pattern.
Assuming no radiation from the failed elements, PSO is
applied to recover the SLL close to the desired level. It
minimizes a cost function, and returns optimum current
excitations for working elements that will lead to the
desired radiation pattern with suppressed SLL. In this sce-
nario, the goal is to find the minimum number of array
elements whose amplitude weights have to be determined
to restore the peak SLL of the array to that of the original
array. The following cost function was used for the opti-
mization process:
C51
Ntot
XNI
i51
ðjFPSOðhiÞj2MaskðhiÞÞ2 (5)
where Mask is an upper bound on the array factor enforc-
ing a peak sidelobe power of 230 dB in the region jhj �BWFN (beamwidth between first nulls). The main lobe
region is defined as the region jhj � BWFN. In this
region, the Mask is valued as 0 dB. The FPSO(h) is com-
puted using the samples hi of size NI which exceed Mask
and is normalized by the total number of samples Ntot
used to sample the entire range. Figure 1 shows both the
original Chebyshev array pattern and the Mask used in
the cost function to restore the pattern.
IV. INVESTIGATIONS AND OBSERVATIONS
As mentioned earlier, extensive investigations were made
on a 32-element linear Chebyshev array with interelement
spacing of k/2. The reason for choosing this array is to
mark clearly the SLL disturbances in the presence of
faulty elements in the array. The Chebyshev distribution
was applied for determining the amplitude excitation of
the elements for the array to have maximum SLL of 230
dB. The element failure in the antenna array was consid-
ered as complete and implemented by making the applied
current excitation to the element as zero. The aim of
investigations was to determine (i) maximum possible
array thinning that can be achieved in a failed array and
(ii) maximum number of element failures that can be cor-
rected. Array thinning in a failed array, that we have
Figure 2 Compensation for element failure at fifth position by
adjusting six of the remaining elements (solid line shows the cor-
rected pattern, dotted line shows the defected pattern).
Limits of Compensation 637
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
attempted here, is different from the concept of array thin-
ning in a normal array where the elements are removed
systematically without a substantial degradation in its per-
formance. In a failed array, the aim of thinning is to per-
turb the excitations of minimum number of functional
elements to restore the system performances. The presence
of failed elements and unexcited elements in the array
makes the thinning process more complex in a failed array
than in a normal array.
A. Investigation-IThe performance of the PSO-based compensation tech-
nique was tested on the Chebyshev array with failure of
an element located at different positions in an antenna
array. The distorted pattern due to the presence of failed
element increased the SLL from 230 dB to higher levels
and also increased the half-power beamwidth (HPBW)
depending on the failed element position. In this case, the
goal was to restore the peak sidelobe power of the failed
array near to its original value by reoptimizing the ampli-
tude excitations of minimum number of the remaining
working elements, which were obtained by PSO. In the
first attempt, the element failure was considered at fifth
position, and the compensation process was applied to
recover the performance in the failed array. The amplitude
excitations of six working elements were considered as
the optimization parameters for PSO optimizer and new
amplitude excitations were calculated for those working
elements to restore the pattern. The performance of PSO
for the element failure at fifth position is demonstrated in
TABLE I Compensation for Single Element Failure at Different Positions in the Array Carried Out by Optimizing Mini-mum Number of Working Elements
Failed
Position
Maximum SLL
of Damaged
Pattern (dB)
HPBW of
Damaged
Pattern
No. of Elements
Excitation
Adjusted
Position of the
Compensating
Element
Recovered
SLL (dB)
HPBW of
Recovered
Pattern DRR
5 225.52 3.95� 6 1st, 6th, 7th, 8th,
9th, 32nd
229.99 4.14� 5.94
6 225.1 3.94� 8 1st, 2nd, 7th,
8th, 9th, 10th,
11th, 32nd
229.8 4.2� 8.62
7 224.38 3.93� 12 1st, 2nd, 3rd,
4th, 5th, 6th, 8th,
9th, 10th, 11th,
12th, 32nd
229.85 4.35� 8.01
8 223.9 3.91� 14 1st, 2nd, 3rd,
4th, 5th, 6th, 7th,
9th, 10th, 11th,
12th, 13th, 14th,
32nd
229.27 4.5� 16.25
9 223.43 3.89� 16 1st, 2nd, 3rd,
4th, 5th, 6th, 7th,
8th, 10th, 11th,
12th, 13th, 14th,
15th, 16th, 32nd
229.6 4.8� 26.31
Figure 3 Compensation for element failure at sixth position by
adjusting eight of the remaining elements.
Figure 4 Compensation for element failure at seventh position
by adjusting 12 of the remaining elements.
638 Acharya et al.
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 24, No. 6, November 2014
Figure 2 for a broadside array. The obtained corrected
pattern has SLL of 229.99 dB and HPBW of 4.14�. So
for this array with element failure at fifth position, pattern
recovery can be possible by perturbing a minimum of six
elements, that is, 19% of the total elements. In each sub-
sequent attempt, investigations were carried out to deter-
mine the minimum number of elements whose excitations
need to be changed for pattern recovery, when the loca-
tion and number of the failed elements are varied in the
array. First, we observed the compensation process by
Figure 5 Compensation for element failure at eighth position
by adjusting 14 of the remaining elements.
Figure 6 Compensation for element failure at ninth position by
adjusting 16 of the remaining elements.
TABLE II Compensation for Multiple Element Failure in the Array Carried out by Optimizing Minimum Number ofWorking Elements
Failed
Position
Maximum SLL
of Damaged
Pattern (dB)
HPBW of
Damaged
Pattern
No of Elements
Excitation
Adjusted
Position of the
Compensating
Element
Recovered
SLL (dB)
HPBW of
Recovered
Pattern DRR
Two Element Failure
2, 7 222.93 4.0� 12 1st, 3rd, 4th, 5th,
6th, 8th, 9th,
10th, 11th, 12th,
13th, 32nd
229.83 4.4� 9.44
2, 8 222.76 3.98� 14 1st, 3rd, 4th, 5th,
6th, 7th, 9th,
10th, 11th, 12th,
13th, 14th, 15th
32nd
229.79 4.64� 9.76
Three Element Failure
2,3,7 222.44 4.08� 12 1st, 4th, 5th, 6th,
8th, 9th, 10th,
11th, 12th,
13th,14th, 32nd
229.95 4.66� 8.11
2,5,8 220.4 4.06� 14 1st, 3rd, 4th, 6th,
7th, 9th, 10th,
11th, 12th, 13th,
14th, 15th, 16th,
32nd
229.82 4.82� 14.51
Four Element Failure
2,3,5,7 221.6 4.17� 12 1st, 4th, 6th, 8th,
9th, 10th, 11th,
12th, 13th, 14th,
15th, 32nd
229.87 4.78� 7.49
3,4,6,8 220.55 4.16� 14 1st, 2nd, 5th,
7th, 9th, 10th,
11th, 12th, 13th,
14th, 15th, 16th,
17th, 32nd
229.8 4.96� 8.33
Limits of Compensation 639
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
changing the position of the failed element and then by
varying the number of failed elements in the array. After
recovering the pattern successfully for a failed element
at fifth position, the compensation procedure was applied
on an array having faults located at sixth, seventh,
eighth, and ninth positions. The aim in each case of pat-
tern recovery is to obtain the minimum number of func-
tional elements whose excitations have to be adjusted. It
was found that the excitations of minimum 8, 12, 14,
and 16 number of the elements have to be adjusted
when the element failure occurred at sixth, seventh,
eighth, and ninth positions, respectively, to produce a
pattern with minimal loss of quality in the system per-
formance. It was observed that when the position of the
failed element is nearer to the center of the array, the
excitations of more number of functional elements need
to be adjusted for the performance restoration. The
results of this compensation process obtained by PSO for
failure of elements in different positions are shown in
Table I. The corresponding distorted and recovered pat-
terns are shown in Figures 2–6.
It can be seen from the data, that the pattern compen-
sation requires that in each case, the amplitudes of the
first and last element of the array need to be changed.
Further, all the compensated elements lie in that half of
the array where the faulty element is located. It can also
be seen from the data that as the location of the fault
shifts toward the center of the array, the recovered SLL
and HPBW become poorer. This shows that array failure
compensation can be possible by perturbing the excitation
values of some of the functional elements near to the
defective one and this limiting value is different for faults
at different positions. If the designer has a priory informa-
tion about these limitations, the compensation process will
be easier and the array will be operational within no time.
In the next investigation, the same compensation
approach was applied to an array having multiple failed
elements with an aim to produce a pattern with minimal
loss of quality in the system performance. Initially, a two
element failure was considered at second and seventh
positions in the array and for the recovery of radiation
pattern the amplitude excitations of minimum 12 elements
TABLE III Optimized Amplitude Weights Computed for Single Element Failure at Different Positions (as Given in TableI) in a 32-Element Array
Element Initial Chebyshev
Optimized Weights Computed for Single Element Failure at Different Positions as Given
in Table I in the Antenna Array
Position Pattern 5 6 7 8 9
1 0.44388 0.168256 0.116 0.01346 0.00005 0.0001
2 0.24331 0.24331 0.1174 0.12482 0.061533 0.089
3 0.30354 0.30354 0.30354 0.16248 0.1112 0.038
4 0.36838 0.36838 0.36838 0.24568 0.144267 0.0403
5 0.43670 0 0.43670 0.33968 0.2323 0.0559
6 0.50723 0.560467 0 0.44534 0.329383 0.125
7 0.57851 0.515578 0.624775 0 0.449783 0.198
8 0.64897 0.55978 0.587075 0.61244 0 0.2971
9 0.71699 0.6324 0.60885 0.61578 0.615267 0
10 0.78094 0.78094 0.739525 0.67658 0.646867 0.529
11 0.83923 0.83923 0.7662 0.74736 0.70785 0.5009
12 0.89036 0.89036 0.89036 0.79814 0.765767 0.5374
13 0.93301 0.93301 0.93301 0.93301 0.832683 0.6252
14 0.96604 0.96604 0.96604 0.96604 0.893117 0.67
15 0.98858 0.98858 0.98858 0.98858 0.98858 0.7507
16 1.00000 1.00000 1.00000 1.00000 1.00000 0.803
17 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
18 0.98858 0.98858 0.98858 0.98858 0.98858 0.98858
19 0.96604 0.96604 0.96604 0.96604 0.96604 0.96604
20 0.93301 0.93301 0.93301 0.93301 0.93301 0.93301
21 0.89036 0.89036 0.89036 0.89036 0.89036 0.89036
22 0.83923 0.83923 0.83923 0.83923 0.83923 0.83923
23 0.78094 0.78094 0.78094 0.78094 0.78094 0.78094
24 0.71699 0.71699 0.71699 0.71699 0.71699 0.71699
25 0.64897 0.64897 0.64897 0.64897 0.64897 0.64897
26 0.57851 0.57851 0.57851 0.57851 0.57851 0.57851
27 0.50723 0.50723 0.50723 0.50723 0.50723 0.50723
28 0.43670 0.43670 0.43670 0.43670 0.43670 0.43670
29 0.36838 0.36838 0.36838 0.36838 0.36838 0.36838
30 0.30354 0.30354 0.30354 0.30354 0.30354 0.30354
31 0.24331 0.24331 0.24331 0.24331 0.24331 0.24331
32 0.44388 0.174067 0.119 0.13028 0.121433 0.1095
640 Acharya et al.
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 24, No. 6, November 2014
TABLE IV Optimized Amplitude Weights Computed for Multiple Element Failure at Different Positions (as Given inTable II) in a 32-Element Array
Optimized Weights Computed for Multiple Element Failure at Different Positions as
Given in Table I in the Antenna Array
Initial ChebyshevTwo Element Failure Three Element Failure Four Element Failure
Element Positions Pattern 2, 7 2, 8 2, 3, 7 2, 5, 8 2, 3, 5, 7 3, 4, 6, 8
1 0.44388 0.061217 0.01865 0.0059 0.00055 0.016025 0.000133
2 0.24331 0 0 0 0 0 0.000933
3 0.30354 0.1059 0.0524 0 0.023 0 0
4 0.36838 0.209567 0.102433 0.1502 0.068875 0.13335 0
5 0.43670 0.3044 0.167817 0.1556 0 0 0.154633
6 0.50723 0.399433 0.269483 0.2929 0.220288 0.325075 0
7 0.57851 0 0.376433 0 0.344688 0 0.2855
8 0.64897 0.582333 0 0.4955 0 0.2899 0
9 0.71699 0.546 0.5545 0.5053 0.466813 0.5131 0.3316
10 0.78094 0.58775 0.55045 0.5193 0.462563 0.477275 0.553433
11 0.83923 0.680517 0.62165 0.6179 0.64225 0.50665 0.4735
12 0.89036 0.805717 0.681067 0.6896 0.6086 0.6356 0.507333
13 0.93301 0.817417 0.759 0.7207 0.697375 0.7006 0.639167
14 0.96604 0.96604 0.852733 0.7898 0.73485 0.748625 0.7296
15 0.98858 0.98858 0.889717 0.98858 0.8467 0.877925 0.808633
16 1.00000 1.00000 1.00000 1.00000 0.927863 1.00000 0.887667
17 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.954
18 0.98858 0.98858 0.98858 0.98858 0.98858 0.98858 0.98858
19 0.96604 0.96604 0.96604 0.96604 0.96604 0.96604 0.96604
20 0.93301 0.93301 0.93301 0.93301 0.93301 0.93301 0.93301
21 0.89036 0.89036 0.89036 0.89036 0.89036 0.89036 0.89036
22 0.83923 0.83923 0.83923 0.83923 0.83923 0.83923 0.83923
23 0.78094 0.78094 0.78094 0.78094 0.78094 0.78094 0.78094
24 0.71699 0.71699 0.71699 0.71699 0.71699 0.71699 0.71699
25 0.64897 0.64897 0.64897 0.64897 0.64897 0.64897 0.64897
26 0.57851 0.57851 0.57851 0.57851 0.57851 0.57851 0.57851
27 0.50723 0.50723 0.50723 0.50723 0.50723 0.50723 0.50723
28 0.43670 0.43670 0.43670 0.43670 0.43670 0.43670 0.43670
29 0.36838 0.36838 0.36838 0.36838 0.36838 0.36838 0.36838
30 0.30354 0.30354 0.30354 0.30354 0.30354 0.30354 0.30354
31 0.24331 0.24331 0.24331 0.24331 0.24331 0.24331 0.24331
32 0.44388 0.1247 0.11405 0.1233 0.116363 0.1367 0.120033
TABLE V Pattern Recovery for Maximum Number of Element Failure in a 32-Element Linear Array
Position of
Failed Antenna Array Pattern
Characteristics Recovered Array Pattern Characteristics
No. of Failed
Elements
Failed
Elements
% Age of
Failure
Maxm SLL
(dB)
HPBW
(�)BWFN
(�)Maxm
SLL (dB)
HPBW
(�)BWFN
(�) DRR
2 2, 3 6 226.48 4.0 16.0 230.0 4.1 10.6 5.61
3 2, 3, 4 9 225.19 4.08 18.0 230.0 4.3 11.2 4.41
4 2, 3, 4, 5 12 223.81 4.16 18.0 230.02 4.58 11.8 4.51
5 2, 3, 4, 5, 6 16 222.26 4.24 18.0 230.01 4.75 12.4 4.02
6 2, 3, 4, 5, 6,
7
19 220.85 4.34 18.0 230.04 5.0 13.2 4.93
7 2, 3, 4, 5, 6,
7, 8
22 219.91 4.45 20.0 230.01 5.2 13.8 4.73
8 2, 3, 4, 5, 6,
7, 8, 9
25 219.63 4.58 20.0 230.01 5.44 14.0 4.51
9 2, 3, 4, 5, 6,
7, 8, 9, 10
28 221.6 4.72 32.0 230.0 5.74 14.8 4.40
10 1, 2, 3, 4, 5,
6, 7, 8, 9, 10
31 220.53 5.1 22.0 230.0 5.76 14.8 4.57
Limits of Compensation 641
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
were adjusted. Another two element failure at second and
eighth positions was also considered and it was found that
in this case, the excitations of the 14 elements need to be
adjusted. Similarly, two different cases of three element
failure and four element failure were taken into considera-
tion, and the minimum number of functional elements
required for the compensation was obtained. The results
for the compensation process are shown in Table II. The
results reveal that the minimum number of functional ele-
ments required for compensation mostly depends on the
position of the faulty elements. It was observed that when
there is a single element failure at seventh position, a
minimum of 12 elements took part in the compensation
and the same number of elements were able to recover
the radiation pattern when four elements at different posi-
tions before the seventh positions became nonoperational.
Another important parameter of antenna array design is
dynamic range ratio (DRR), which is also obtained for the
corrected radiation pattern and the values are given in
Tables I and II. It is observed that when the element fail-
ure occurred more toward the center of the array, and the
number of faulty elements are more, then the corner ele-
ments of the array located in that half where the faults are
located have a negligible effect on the radiation pattern.
Furthermore, it was observed that the failure of the
element affect much the SLL and the effect on HPBW is
less. The variations depend both on faulty element posi-
tions and the number of faults. For example, a single ele-
ment failure at seventh position increases the maximum
SLL to 18.73% of its original value whereas the variation in
HPBW is only 1%. For the case of multiple element (e.g., 2,
3, 5, 7) failure, the maximum SLL of the damaged pattern
reached to a value which is 29.3% more than the initial SLL
and the HPBW of distorted pattern increased by 7.4% of its
original value. But the recovered radiation pattern have the
maximum SLL value for a single element (seventh) failure
is 229.85 dB and for multiple element (e.g., 2, 3, 5, 7) fail-
ure is 229.87 dB. To get this value of maximum SLL, we
have to compromise with the value of HPBW. The HPBW
of the corrected radiation pattern is 4.35� in a single fault at
seventh position in response to 3.88� of original pattern and
this value is 4.78� for multiple failures at second, third,
fifth, and seventh positions. There is a trade-off between the
maximum SLL and the HPBW. The performance of one
parameter cannot be improved significantly without sacri-
ficing the other parameter. Similar results were obtained in
each case of the compensation. The optimized excitations
of the functional elements which were involved in the pro-
cess of compensation were obtained by the PSO optimizer
and their normalized values are reported in Tables III
and IV.
B. Investigation-IIPresence of failed elements in antenna arrays damages the
radiation pattern and gives rise to a greater SLL. The pat-
tern can be restored with the reduced SLL by reoptimizing
the weights of the remaining elements at the cost of gain
of the array. As the number of failed element increases,
the gain of the antenna array reduces. The compensation
techniques applied to a failed array can recover the pat-
tern only for a certain maximum number of element fail-
ures, beyond which the performance of compensated array
falls below a specified level of acceptability. In this sec-
tion of the article, the performance of the compensation
technique based on PSO optimization is tested on the
same 32-element linear array having maximum SLL at
230 dB, BWFN of 10.4�, and HPBW of 3.88�. Different
number of element failures were considered in the array
with a goal for determining the maximum number of ele-
ment failures for which the pattern can be recovered. Spe-
cifically, the criterion was to achieve a SLL close to 230
dB and the HPBW within a specified limit, which was
Figure 7 Compensation for 10 element failure in 32-element
array with main beam at 30�.
TABLE VI Pattern Recovery for Maximum Number of Element Failure in a 20-Element Linear Array
Position of
Failed Antenna Array Pattern
Characteristics Recovered Array Pattern Characteristics
No. of Failed
Elements
Failed
Elements
% Age of
Failure
Maxm SLL
(dB) HPBW (�) BWFN (�)Maxm SLL
(dB) HPBW (�) BWFN (�) DRR
2 2,3 10% 223.62 6.6 18 230.09 7.8 22 10.77
3 2,3,4 15% 221.32 6.8 20 230.04 8.0 22 4.8
4 2,3,4,5 20% 218.27 7.1 20 230.01 8.6 24 4.95
5 2,3,4,5,6 25% 215.99 7.6 20 230.01 9.2 26 6.06
6 1,2,3,4,5,6 30% 214.44 8.0 22 230.03 9.4 26 5.42
642 Acharya et al.
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 24, No. 6, November 2014
considered to be 50% greater than the original value. PSO
was used to optimize the amplitude excitation of the
remaining functional elements in the failed array to
recover the pattern close to the original one. Table V
shows the SLL, HPBW, and BWFN of the recovered pat-
tern for different number of element failures in the array
and the DRR of the recovered pattern. It was observed
that the compensation technique applied over the failed
antenna array effectively performed the job of sidelobe
suppression. The SLL can be successfully recovered even
for larger number of failures but each time the HPBW
increases, thereby reducing the gain. In the process of
compensation, the BWFN of the recovered pattern was
also improved compared to the damaged one. The results
presented in Table V shows that in a 32-element array, a
maximum of 10-element failures can be compensated.
The recovered pattern for the 10-element failure has SLL
of 230 dB and HPBW 5.76� which is 48% larger and
BWFN is 14.8� which is 43% larger than their respective
original values.
It was found that if the number of failed element is
increased from 10, the recovered pattern has an SLL of
229.97 dB with HPBW of 6.0�. So, the beamwidth crosses
the limiting value and hence the gain falls below the level
of acceptability. Therefore, for the present array, the com-
pensation technique enables the recovery of reasonable
antenna performance when as many as 10 elements, that is,
nearly 30% of elements are not operational. This approach
of finding the limiting value of the number of element fail-
ure that can be compensated was extended for the patterns
with main beam directed at any angle. It was found that in a
32-element array with SLL at 230 dB and having main
beam directed at 30� can able to recover the pattern for
maximum 10 element failure within the limiting value. The
pattern of the original 32-element linear array has main
beam directed at 230�, SLL at 230 dB, and HPBW of
TABLE VII Optimized Amplitude Weights Computed for Multiple Element Failure at Different Positions (as Given inTable V) in a 32-Element Array
Two
Element
Failure
Three
Element
Failure
Four
Element
Failure
Five
Element
Failure
Six
Element
Failure
Seven
Element
Failure
Eight
Element
Failure
Nine
Element
Failure
Ten
Element
Failure
Element
Positions
Initial
Chebyshev
Pattern 2, 3 2, 3, 4 2, 3, 4, 5 2, 3, 4, 5, 6
2, 3, 4,
5, 6, 7
2, 3, 4,
5, 6, 7, 8
2, 3, 4, 5,
6, 7, 8, 9
2, 3, 4, 5,
6, 7, 8, 9,
10
1, 2, 3, 4, 5,
6, 7, 8, 9,
10
1 0.44388 0.178 0.0489 0.0289 0.0074 0.0002 0.0048 0.0011 0.0283 0
2 0.24331 0 0 0 0 0 0 0 0 0
3 0.30354 0 0 0 0 0 0 0 0 0
4 0.36838 0.2923 0 0 0 0 0 0 0 0
5 0.43670 0.386 0.3436 0 0 0 0 0 0 0
6 0.50723 0.3556 0.2266 0.2981 0 0 0 0 0 0
7 0.57851 0.4232 0.333 0.3076 0.3161 0 0 0 0 0
8 0.64897 0.423 0.3944 0.3641 0.261 0.3571 0 0 0 0
9 0.71699 0.5294 0.4943 0.2847 0.3437 0.2347 0.2373 0 0 0
10 0.78094 0.6101 0.4905 0.5894 0.4427 0.2845 0.2863 0.27 0 0
11 0.83923 0.7248 0.6275 0.5678 0.4457 0.4299 0.294 0.3583 0.2272 0.2186
12 0.89036 0.7499 0.6421 0.5868 0.6168 0.5157 0.4153 0.3406 0.2616 0.2328
13 0.93301 0.7608 0.8319 0.8354 0.6322 0.6022 0.4652 0.3953 0.3181 0.2817
14 0.96604 0.9084 0.8279 0.7832 0.7293 0.6786 0.582 0.5298 0.4288 0.3801
15 0.98858 0.8719 0.8495 0.8022 0.8246 0.7316 0.6712 0.6768 0.4819 0.5494
16 1.00000 0.9567 1.0000 0.8939 0.9498 0.8186 0.7529 0.6995 0.6241 0.5605
17 1.00000 0.8656 0.9712 0.9595 0.8995 0.8996 0.7473 0.7741 0.7056 0.6688
18 0.98858 1.0000 0.9763 1.0000 0.9586 0.9313 0.8643 0.9212 0.771 0.7919
19 0.96604 0.9191 0.9902 0.9835 0.9456 0.9652 0.9131 0.8926 0.9353 0.856
20 0.93301 0.935 0.9946 0.8808 0.9864 1 1 0.9009 0.8667 0.9347
21 0.89036 0.8965 0.9223 0.92 1 0.9044 0.7982 1 0.9866 0.8935
22 0.83923 0.7593 0.9191 0.9526 0.9346 0.9308 0.9251 0.8924 0.9412 1
23 0.78094 0.8743 0.8761 0.8375 0.8085 0.9126 0.8959 0.9562 1 0.9801
24 0.71699 0.6992 0.7671 0.7926 0.8292 0.8057 0.7649 0.8042 0.8654 0.8523
25 0.64897 0.6463 0.7379 0.6383 0.743 0.7632 0.8034 0.7808 0.8515 0.8713
26 0.57851 0.6093 0.6711 0.6036 0.6692 0.657 0.6105 0.6867 0.755 0.8293
27 0.50723 0.4347 0.6249 0.5384 0.5491 0.5332 0.5593 0.6165 0.6696 0.6877
28 0.43670 0.5084 0.4248 0.4393 0.5525 0.506 0.4954 0.4753 0.6108 0.571
29 0.36838 0.3926 0.4339 0.4075 0.3397 0.3576 0.3443 0.4374 0.4672 0.4687
30 0.30354 0.3513 0.3322 0.2831 0.2802 0.3958 0.3471 0.3439 0.315 0.4271
31 0.24331 0.2563 0.3245 0.2834 0.3569 0.2706 0.275 0.2216 0.3436 0.3173
32 0.44388 0.2606 0.365 0.2215 0.2486 0.2028 0.211 0.2744 0.2966 0.3502
Limits of Compensation 643
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
4.42�. The recovered pattern for the 10-element failure has
SLL of 229.95 dB and HPBW 6.63� which is exactly 50%
larger than the original value. Figure 7 shows the original,
distorted, and the recovered patterns for this case.
The same procedure was also implemented on a 20-
element linear broadside array having SLL 230 dB, BWFN
of 17�, and HPBW of 6.3�. The compensation method
applied on the array for different number of failures and the
obtained results are presented in Table VI. It can be seen
that in a 20-element array, when a maximum of six ele-
ments become inoperable, it is possible to recover the pat-
tern having SLL close to 230 dB and the HPBW below the
limit, that is, 50% larger than the original pattern. When
one more element fails in the array the beamwidth goes
beyond this limit. So, in this case also, pattern restoration is
possible for 30% of element failure in array.
The results in Tables V and VI show that the maxi-
mum SLL of the recovered antenna is achieved by sacri-
ficing some other parameters within the limit of
acceptability. The optimized amplitude excitations
obtained for the cases considered in Tables V and VI are
given in Tables VII and VIII, respectively.
V. CONCLUSIONS
In this article, an attempt has been made to quantify the
tolerance level of thinning in a failed antenna array. The
minimum number of elements for which amplitude pertur-
bation is required to get an acceptable recovered pattern
was determined. It is found that this number depends on
the positions of the failed elements. Another investigation
was made to determine the maximum number of element
failure that can be compensated for the pattern recovery
with a specified acceptable limit. It was found that for a
50% relaxation in the HPBW, pattern restoration is possi-
ble for fault in around 30% of the elements. Again, this
value varies with position of the faults. Although the
results are mentioned for a 32-element and 20-element
Chebyshev arrays, the overall conclusions drawn from the
study are equally applicable for other arrays as well, but
the quantitative values may vary. As there is a growing
demand to add flexibility in the large arrays, these results
can be used while developing self-healing arrays because
of reduced computation time and cost.
ACKNOWLEDGMENT
The authors thank the anonymous reviewers whose con-
structive comments helped to improve the manuscript.
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TABLE VIII Optimized Amplitude Weights Computed for Multiple Element Failure at Different Positions (as Given inTable VI) in a 20-Element Array
Two Element
Failure
Three Element
Failure
Four Element
Failure
Five Element
Failure
Six Element
Failure
Element
Positions
Initial
Chebyshev
Pattern 2, 3 2, 3, 4 2, 3, 4, 5 2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6
1 0.3256 0.0226 0.0296 0.0002 0.0373 0
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BIOGRAPHIES
Om Prakash Acharya received his
undergraduate degree in Engineering from
Marathwada University, Nanded, Mahara-
shtra, India in 2001, and obtained his Mas-
ters from Biju Patnaik University of
Technology, Odisha, India in 2005–06. At
present, he is working toward his PhD
degree in the department of Electronics
and Communication Engineering, Indian
Institute of Technology, Roorkee, India. Prior to this, he was work-
ing as a Senior Research Fellow in a DRDO, India sponsored pro-
ject in the same department from 2009–2010. During 2006–2009,
he served as a Lecturer at Silicon Institute of Technology, Bhuba-
neswar, India. His research interests include antenna arrays, array
signal processing, and application of biologically inspired computa-
tional techniques. He has published more than 10 papers in journals
and conferences. He is a student member of IEEE.
Amalendu Patnaik received his Ph.D., in
Electronics from Berhampur University in
2003. Prior to joining as an Assistant Pro-
fessor in Indian Institute of Technology,
Roorkee, India, in 2007, he served as a
Lecturer in National Institute of Science
and Technology, Berhampur, India. Dur-
ing 2004–2005, he has been to University
of New Mexico, Albuquerque, USA as a
Visiting Scientist. He has published more than 50 papers in journal
and conferences; coauthored one book on Engineering Electroman-
getics, and one book chapter on Neural Network for Antennas in
Modern Antenna Handbook from Wiley. Besides this, he has pre-
sented his research work as short courses/tutorials in many national
and international conferences. His current research interests include
array signal processing, application of soft-computing techniques in
Electromangetics, CAD for patch antennas, EMI, and EMC. He
was awarded the IETE Sir J. C. Bose Award in 1998 and BOY-
SCAST Fellowship in 2004–2005 from Department of Science and
Technology, Government of India. Dr. Patnaik is a life member of
Indian Society for Technical Education (ISTE) and Senior Member
of IEEE.
Sachendra N Sinha received his Bache-
lor’s, Master’s, and Ph.D., degrees, all
from IIT Roorkee (erstwhile University of
Roorkee), in 1972, 1974, and 1984,
respectively. In 1974, he joined the
Department of Electronics and Communi-
cation Engineering at IIT Roorkee, from
where he retired as a Professor in 2013.
During his tenure at IIT Roorkee, he
served as Head of the Department and Dean of Students’ welfare.
Currently, he is an Emeritus Fellow in the department. His current
research interests deal with numerical solution of operator equa-
tions arising in electromagnetics, fractal antennas and apertures,
active antennas, smart antennas, artificial dielectrics, UWB commu-
nication technology, and soft computing techniques. On these
topics, he has published more than 80 technical papers in national
and international journals and conferences. Several papers authored
or coauthored by him have won best paper awards. He has success-
fully carried out a number of research and consultancy projects for
industry and various agencies of Indian Government which include
the Department of Space, Department of Science and Technology,
Ministry of Information and Communication Technology, and
Defence Research and Development Organization. He is a life
member of the Indian Institute of Public Administration (India),
Fellow of the Institute of Electronics and Telecommunication Engi-
neers (India), Fellow of the Institute of Engineers (India), and Sen-
ior Member of IEEE (USA).
Limits of Compensation 645
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce