limits of compensation in a failed antenna array

11
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. V C 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). V C 2014 Wiley Periodicals, Inc. 635

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Page 1: Limits of compensation in a failed antenna array

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

Page 2: Limits of compensation in a failed antenna array

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

Page 3: Limits of compensation in a failed antenna array

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

Page 4: Limits of compensation in a failed antenna array

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

Page 5: Limits of compensation in a failed antenna array

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

Page 6: Limits of compensation in a failed antenna array

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

Page 7: Limits of compensation in a failed antenna array

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

Page 8: Limits of compensation in a failed antenna array

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

Page 9: Limits of compensation in a failed antenna array

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

Page 10: Limits of compensation in a failed antenna array

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

2 0.2856 0 0 0 0 0

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19 0.2856 0.2743 0.2500 0.3917 0.2985 0.3548

20 0.3256 0.0928 0.2087 0.2969 0.1649 0.1842

644 Acharya et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 24, No. 6, November 2014

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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