A Linear Antenna Array Failure Correction with Null Steering using Firefly Algorithm

The element failure of digital beam forming array antenna systems used in defence equipment increases the side lobe power level which distorts the beam pattern of the antenna array. The problem of array failure correction becomes more complex when null steering conditions are required to be added. In this paper, the problem of linear antenna array failure has been addressed with multiple wide band null steering using firefly algorithm (FA) by controlling the amplitude and phase excitation of array elements. A fitness function in the form of template has been formulated to obtain the error between original (pre-failed) side lobe pattern and measured side lobe pattern and this error function has been minimized using FA. Numerical example of element failure correction of element failure of array along with multiple nulls is presented to show the capability of this flexible approach.


Keywords:    Digital beamforming array antenna systemlinear antenna array,   side lobe level,   element failurenull steeringoptimization,   firefly algorithm

The digital beamforming antenna array is one of the most important components in wireless communication systems used in defence applications to improve the system capacity and spectral efficiency. The pattern synthesis of active antenna array is required in many defence applications like radar, satellite communication, sonar, mobile communication etc., where interference signals are considered to be a main problem. It is necessary to produce nulls in specified direction to reduce the effect of interfering signals during antenna beamforming. The desired patterns of linear antenna array with suppressed sectors in the interference directions can be easily achieved with well known analytical techniques, if all array elements radiate properly. In literature, the different conventional123  and evolutionary techniques4567891011 have been applied for producing nulls in antenna pattern. In general, the antenna array used in different applications consists of large number of radiating elements or sub-arrays. Due to ageing or excessive temperature, there is always a possibility of failure of one or more elements in the antenna array system. The failures of elements in the antenna array destroy the symmetry of the array and may cause sharp variation in field intensity across the array and distort the pattern in the form of increased side lobe level and mainbeam width. The replacement or repair of the defective element of the antenna array is not possible in systems which used for critical situation like war field, space platform etc. 

  It is possible to re-synthesize the radiation pattern of array with minimal loss of quality and suppressed sectors in the interference directions. It can be realized by controlling the excitation of the normal antenna element of the array without replacing the defective element. Many conventional techniques have been proposed to solve only the problem of element failure of antenna array by improving the array pattern in presence of defective elements like a numerical technique based algorithm12 to re-obtain the directional pattern of linear antenna array with single element failure conditions, partial compensated the degraded pattern of planar array using combination of accumulated averaging scheme and conjugate gradient algorithm13, shore’s side lobe sector nulling method14, an orthogonal method15 and conjugate gradient based method16.

Generally analytical approaches are unable to handle the antenna array failure problem, when a non uniformly spaced array is considered. This problem is also a hard problem for numerical approaches due to randomness of the geometrical layout of the remaining non defective array elements and of the desired beam shape. Population-based, stochastic search approaches can provide an effective solution for such complex problems, as they tend to explore multiple solutions simultaneously, relying only on zero order information. Many stochastic search methods have been proposed to solve the problem of antenna array failure using genetic algorithm (GA)17 18, particle swarm optimization (PSO)19, adaptive neural system20, simulated annealing (SA)21, firefly algorithm (FA)22 and hybrid methods23. Some stochastic approaches have been also proposed to solve the problem of antenna array along with null steering problem using simulated annealing (SA)24,  particle swarm optimization (PSO)25 and hybrid method26.

  In this paper, an effective method based on the firefly algorithm (FA) is proposed for array failure correction of arbitrary linear antenna arrays along with null steering problem. The FA algorithm is a swarm intelligence based algorithm272829 which can solve problems with continuous variables in multi-dimensional spaces more naturally and efficiently. The FA has been demonstrated to outperform artificial bees colony algorithm (ABC) in terms of convergence and cost minimization in a statistically meaningful way30. The performance of the firefly algorithm has been found more superior than particle swarm optimization (PSO) in terms of finding optimum solutions for the desired beam patterns of ring antenna array31. The array-failure correction along with null steering is a much more complex problem than simple side lobe reduction in antenna design for a uniformly spaced linear array. In this paper, FA has been successfully applied for linear antenna array failure problem and the antenna pattern has been corrected along with multiple wideband null steering in the desired directions using amplitude and phase control.

For linear array beamforming, one of the three approaches i.e. amplitude-only, phase-only or the amplitude-phase can be employed32. It is difficult to compensate the degradation of a damaged array pattern with amplitude-only approach, as the failed elements introduce an asymmetrical aperture distribution. The phase-only approach with constant amplitude requires a large number of array elements to produce low side lobes pattern33. Consequently, amplitude-phase approach is used for redistribution of weights of the antenna array to correct the damaged pattern.

The linear antenna array of P identical radiators with a uniform spacing of half a wavelength between different elements is shown in Fig. 1. The array factor (AF) of an arbitrary antenna array can be generally written as,

AF=W S( ϕ, ϕ m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiyqaiaacA eacqGH9aqpcaGGxbGaaiiiaiaacofadaqadaqaaiabew9aMjaacYca cqaHvpGzdaWgaaqaaiaac2gaaeqaaaGaayjkaiaawMcaaaaa@4368@        (1)

where W= { w 1 , w 2 , w 3 ....... w P } T , w p C CN ,p=1,2,...P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maacmaabaGaam4DamaaBaaaleaacaaIXaaabeaakiaadYcacaWG 3bWaaSbaaSqaaiaaikdaaeqaaOGaamilaiaadEhadaWgaaWcbaGaaG 4maaqabaGccaWGUaGaamOlaiaad6cacaWGUaGaamOlaiaad6cacaWG UaGaam4DamaaBaaaleaacaWGqbaabeaaaOGaay5Eaiaaw2haamaaCa aaleqabaGaamivaaaakiaadYcacaaMf8Uaam4DamaaBaaaleaacaWG WbaabeaakiaaysW7cqGHiiIZcaWGdbWaaWbaaSqabeaacaWGdbGaam OtaaaakiaadYcacaaMf8UaamiCaiabg2da9iaaigdacaWGSaGaaGOm aiaadYcacaWGUaGaamOlaiaad6cacaWGqbGaaGzbVdaa@607D@         (2)

is the weighting vector, ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@396A@ and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@396A@ m are the direction variable and the main beam direction respectively and S is the steering vector. CCN is a subset or the set of the all complex numbers, indicating the weights of elements of linear antenna array.

The steering vector S of linear array of P identical radiators is given as

 S=exp{ jkd( p P1 2 ).(cosϕcos ϕ m ) }p=1,2,...P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiiaiaado facqGH9aqpciGGLbGaaiiEaiaacchadaGadaqaaiaadQgacaWGRbGa amizamaabmaabaGaamiCaiabgkHiTmaalaaabaGaamiuaiabgkHiTi aaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaiaac6cacaGGOaGaci4y aiaac+gacaGGZbGaeqy1dyMaeyOeI0Iaci4yaiaac+gacaGGZbGaeq y1dy2aaSbaaeaacaWGTbaaleqaaOGaaiykaaGaay5Eaiaaw2haaiaa ywW7caWGWbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6caca GGUaGaaiOlaiaadcfaaaa@5F4C@         (3)
where k is the wave vector and d is the spacing between array elements. The pth radiator or element failure of array is done by setting the weight wp equal to zero in Eqn. (1). The element or radiator failure of antenna array causes the distortion in side lobe level (SLL) and main beam pattern, which is corrected as per original antenna pattern and multiple nulls are generated in desired interference directions by recalculating the amplitude and phase of the non failure elements or radiators using the firefly algorithm.




The objective of the work is to restore the SLL of the original pattern and optimize the directivity of the antenna array along with multiple wideband null steering in the desired interference directions. To achieve the objective, a template has been constructed on the basis of the specified SLL, the required shape of the main lobe and the desired multiple suppressed sectors in interference directions as shown in Fig. 2. The designed template is then cast over the antenna array pattern generated by individual solution provided by Firefly Algorithm to evaluate their cumulative difference. The determined cumulative difference is taken as a fitness value of the solution.



The firefly algorithm (FA) is a nature-inspired algorithm based on the social behavior of firefly swarm. The fireflies show flashing behaviors with different flashing pattern for communication with each other, to find mates and search for pray. The Firefly Algorithm was proposed by Yang29, in which three idealized rules have been used:
a) All fireflies are unisex and each firefly can be attracted by other firefly irrespective of their sex;
b) Attractiveness of each firefly is proportional to its brightness. For any two flashing fireflies, the firefly with less brightness will move towards the firefly with high brightness. Attractiveness between the two fireflies is proportional to their individual brightness or attractiveness and distance between them;
c) The cost function of the problem is determined by the brightness of a firefly and it can simply be proportional to the brightness for optimization problem.

The pseudo-code for FA is shown in Fig. 3 and the steps involved in FA are summarized as under:

Step 1 (Initialization): Initialize the location of N fireflies30 in Z dimension space within the space boundary and is given as

x nz (0)=ran d nz (0,1)( x nz U x nz L )+ x nz L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbGaamOEaaqabaGccaGGOaGaaGimaiaacMcacqGH9aqp caWGYbGaamyyaiaad6gacaWGKbWaaSbaaSqaaiaad6gacaWG6baabe aakiaacIcacaaIWaGaaiilaiaaigdacaGGPaGaaiikaiaadIhadaqh aaWcbaGaamOBaiaadQhaaeaacaWGvbaaaOGaeyOeI0IaamiEamaaDa aaleaacaWGUbGaamOEaaqaaiaadYeaaaGccaGGPaGaey4kaSIaamiE amaaDaaaleaacaWGUbGaamOEaaqaaiaadYeaaaaaaa@564B@

n=1,2,3,…N; z=1,2,3,…Z         (4)

where x nz U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacaWGUbGaamOEaaqaaiaadwfaaaaaaa@3B98@ and x nz L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacaWGUbGaamOEaaqaaiaadYeaaaaaaa@3B8F@ represents the upper and lower limits of the zth variable in the population respectively, randnZ (0,1) is a uniformly distributed random value within the limit 0 to 1.

Step 2 (Computation of light intensity of fireflies): In this step, the brightness of each firefly is computed in terms of its cost function at the present location of respective firefly in current generation of FA. For a maximization problem, the light intensity or brightness of firefly is directly proportional and for a minimization problem it is inversely proportional to its cost function.

Step 3 (Updatation of the location of the fireflies): In this step, each firefly moves toward the other firefly with more brightness or light intensity in the population and update its location for the next iteration of the algorithm. The location of each moving firefly in the population is updated based on the attractiveness between the moving firefly and firefly with more light intensity.
The variation of light intensity and formulation of the attractiveness are the two important issues used in the Firefly Algorithm. For simplicity, light intensity is used to calculate the attractiveness of a firefly, which is further used to determine the cost function of the problem. The attractiveness β is a relative term which is judged by the other fireflies and it is change with the distance rlm between firefly l and firefly m. In addition, brightness of the firefly decreases with the distance from its source, and light is also absorbed in the medium, so we should allow the attractiveness to vary with the degree of absorption. In the simplest form, the light intensity I(r) changes according to the inverse square law I(r) = Is/r2 where Is is the light intensity or brightness at the source. The light intensity I varies with the distance r i.e. I = I0e−γr, where I0 is the original light intensity for a given medium with a constant light absorption coefficient. The combined effect of both the absorption and inverse square law is used to avoid the singularity condition at r = 0 in the expression Is/r2 and it can be approximated using the Gaussian I(r)=I0 e γ r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamysaiaacI cacaWGYbGaaiykaiabg2da9iaadMealiaaicdacaaMi8UccaWGLbWa aWbaaSqabeaacqGHsislcqaHZoWzcaaMc8UaamOCamaaDaaameaaae aacaaIYaaaaaaaaaa@460B@ form as discussed in29.
We can now define the attractiveness β of a firefly by, β(r)=β0 e γ r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaai ikaiaadkhacaGGPaGaeyypa0JaeqOSdi2ccaaIWaGaaGjcVRGaamyz amaaCaaaleqabaGaeyOeI0Iaeq4SdCMaaGPaVlaadkhadaqhaaadba aabaGaaGOmaaaaaaaaaa@47B1@ where β0 is the attractiveness at r = 0, because the attractiveness of the firefly is proportional to the light intensity seen by adjacent fireflies. The attractiveness between the two fireflies in Z- dimensional search space is determined as29 31:

x l = x l + β 0 e γ r lm 2 ( x l x m )+α ε s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGSbaabeaakiabg2da9iaadIhadaWgaaWcbaGaamiBaaqa baGccqGHRaWkjugqbiabek7aIPWaaSbaaSqaaiaaicdaaeqaaOGaam yzamaaCaaaleqabaGaeyOeI0Iaeq4SdCMaaGPaVlaadkhadaqhaaad baGaamiBaiaayIW7caWGTbaabaGaaGOmaaaaaaGcdaqadaqaaiaadI hadaWgaaWcbaGaamiBaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaa d2gaaeqaaaGccaGLOaGaayzkaaGaey4kaSIaeqySdeMaeqyTdu2aaS baaSqaaiaadohaaeqaaaaa@57BC@          (5)

where γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWccqaHZoWzaa a@3954@ is the light absorption coefficient which is fixed for the given medium and its value can varies from 0.01 to 100 depending upon the characteristics of the medium; α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3941@ is a randomization parameter used to generate randomness in Eqn. (5) and its value vary from 0 to 1; ε s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadohaaeqaaaaa@3A6D@ is a vector of random numbers drawn from a Gaussian distribution or an uniform distribution31; the attractiveness between the two fireflies l and m is represented by the product of β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaicdaaeqaaaaa@3A29@ and e γ r lm 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaeyOeI0Iaeq4SdCMaaGPaVlaadkhammaaDaaabaGaamiB aiaad2gaaeaacaaIYaaaaaaaaaa@409C@ terms. The attractiveness of firefly is given by β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaicdaaeqaaaaa@3A29@ at Cartesian distance r=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaaicdaaaa@3A59@ . For simplicity, the value of β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaicdaaeqaaaaa@3A29@ is taken unity in most cases of our implementation. The Cartesian distance r lm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaMi8UaamiBaiaayIW7caWGTbaabeaaaaa@3C1E@ between any two fireflies l and m at x l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGSbaabeaaaaa@3810@ and x m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGTbaabeaaaaa@3811@ respectively is determined as29,31:

r lm = x l x m = t=1 T ( x l,t x m,t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGSbGaamyBaaqabaGccqGH9aqpdaqbdaqaaiaadIhadaWg aaWcbaGaamiBaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad2gaae qaaaGccaGLjWUaayPcSdGaeyypa0ZaaOaaaeaadaaeWbqaamaabmaa baGaamiEamaaBaaaleaacaWGSbGaaiilaiaadshaaeqaaOGaeyOeI0 IaamiEamaaBaaaleaacaWGTbGaaiilaiaadshaaeqaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaaqaaiaadshacqGH9aqpcaaIXa aabaGaamivaaqdcqGHris5aaWcbeaaaaa@55FA@             (6)

As per Firefly algorithm, the firefly with highest light intensity is not allowed to move in any direction, while the rest of fireflies change their position according to Eqn. (5) at current generation. In this way, the algorithm gradually updates the global best (gB) solution in the successive iteration.

Step 4 : Ranking of fireflies and computation of current global best- All the fireflies are ranked based on their light intensity or brightness in the current generation and position of the brightest firefly in the population is taken as current global best (gB). The brightest firefly has a best fitness value among all the fireflies at the current generation.

Step 5 : Repeat the Step 2 to Step 4 until end condition is met by the algorithm as shown in Fig. 3. The terminating condition of the algorithm is a condition under which either the total numbers of iterations are completed or desired value of cost function is achieved. The location of the best firefly (gB) provides the best solution and the corresponding brightness of the firefly provide the best fitness value of the objective function using Firefly algorithm.





A classic Dolph – Chebyshev 32 elements linear array has been considered with a side lobe level (SLL) of -35 dB. The steering vector S of the linear array is given by Eqn. (3). The above considered linear array having 4 element failure condition with the defective elements are located at 1st, 2nd, 31th and 32th positions has been simulated. Figure 4(a) depicts the original array pattern of linear array system without element failure condition with main beam and a side lobe level of -35 dB. When the elements of the array at the above mentioned locations become defective, the SLL increase to a maximum unacceptable value of -25.08 dB at 82.9° and 97.1° as shown in Fig. 4(b).








The firefly algorithm has been implemented in MatLab as per above description. The four FA parameters, i.e., the population size U, the light absorption coefficient γ, the attractiveness  β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3943@ and the randomization parameter α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3941@ are set to values 50, 1, 0.2, and 0.25 respectively. The FA has been run to correct the failed pattern and produce the nulls at the desired direction as per the objective function described in section 2. The Fig. 4(c) depicts the corrected far field array pattern obtained using FA with main beam and nulls, which corrected the SLL value from -25.08 dB to -35 dB and produced nulls at 62° and 118°. The program has been run 15 times for 500 iterations and best result registered. The convergence characteristics shown in Fig. 5 indicate that the FA algorithm converges in around 230 generations.





The beamwidth of original, damaged and corrected patterns have been observed 4.2°, 4.6°, and 4.8°, respectively. The directivity of an antenna array is a figure of merit which determines the power density of the antenna system radiates in the direction of its strongest emission. The directivity of the original array design have been observed 14.21 dB of main lobe and -20.8 dB of side lobes as shown in Fig. 6(a), which is distorted by the failed elements of the array design to 13.88 dB and -11.2 dB of main lobe and side lobes respectively as shown in Fig. 6(b). Figure 6(c) shows the recovery of directivity to 13.6 dB of main lobe and -20.13 dB of side lobe along with two null steered at 62° and 118° by FA. 






Similarly, the original, damaged array pattern of linear array with main beam shifted at 49° and 131° is shown in Figs. 7(a) - 7(d). The corrected pattern of linear array with main beam shifted at angle 49° along with nulls steered at angle 22° and 77° is shown in Fig. 7(e). Figure 7(f) depicts corrected linear array pattern with main beam shifted at 131° with nulls produced at 103° and 158°. The normalized excitation coefficients of original linear antenna array, antenna array with 4 element failure conditions without optimization and antenna array with 4 element failure condition with optimization using FA are listed in Table 1. It has to be noted that the damaged condition of antenna array element is represented by 0 in different columns of Table 1.
It is not necessary to have null steering symmetric to the main beam. In this work, the code is written in such a way that null steering remains symmetric around main lobe. Asymmetric null steering can be done by changing the logic of the program.








Table 1.Normalized excitation coefficient for corrected radiation pattern of 32 elements linear array with null steering by FA.

The field pattern of high performance digital beamforming array antenna systems can be seriously degraded with the malfunctioning of antenna elements. In such condition, it is a big challenge to steered nulls in the desired direction along with restoration of the original pattern of antenna array. In this paper, the Firefly algorithm is proposed for solving a practical problem of linear antenna array system by re-optimizes the amplitude and phase excitations of the remaining elements to recover the original pattern of the antenna array and produced nulls in the desired direction. The proposed method proved its effectiveness to suppress the side lobe level and introduction of nulls in the field pattern of antenna array in presence of antenna element failures. The work can be extended to locate the faulty elements and determine its position in the array. Also the mutual coupling of the elements can be added in the analysis process. This method can be extended to planar or conformal antenna arrays.


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Mr Narwant Singh Grewal received the BTech (Electronics and Comm. Engg.) from SLIET Longowal, Punjab, in 1997 and ME (ECE) from Guru Nanak Dev Engineering College, Ludhiana, Punjab, in 2002. Currently pursuing his PhD at Punjab Technical University, Kapurthala, Punjab. He is working as Assistant Professor, in the department of Electronics and Communication Engineering, Guru Nanak Dev Engg. College, Ludhiana. He is life member of ISTE, IE(I) and IETE.His chief research interests include evolutionary computing, electromagnetics, and antenna design.

Dr Munish Rattan obtained his BTech (Electronics and Comm. Engg.) from from BBSBEC. Fathehgarh Sahib, in 2002 and ME (Electronics Engg.) from P.E.C. Chandigarh, in 2004, and PhD in 2010. He is currently working as Assistant Professor, in the Department of Electronics and Communication Engineering, Guru Nanak Dev Engg. College, Ludhiana. He is presently guiding 6 PhD candidates.

Dr Manjeet Singh Patterh did his BTech (Electronics Engg.) from Madhav Institute of Technology and Science, Gwalior and MTech (Electronics Engg.) from Birla Institute of Technology and Science, Pilani, He did his PhD from Punjab Technical University Jalandhar. He is presently working as Professor in Department of Electronics and Communication Engineering at UCoE Punjabi University Patiala. He has published over 60 papers in international and national refereed journals and conferences. His current interests are Digital signal processing, wireless communication systems. He is member of IEEE and life member of ISTE, IE(I) and IETE.