Optimization of Antenna Arrays for SLL Reduction Towards Pareto Objectivity Using GA Variants

Transcription

1 5 IEEE Syposiu Series on Coputational Intelligence Optiization of Antenna Arrays for SLL Reduction Towards Pareto Objectivity Using GA Variants Sudipta Das, Gopi Ra, Pragnan Chakravorty, Durbadal Mandal, Rajib Kar, Sakti Prasad Ghoshal National Institute of Technology-Durgapur Durgapur, India Abstract One of the ost striking aspects of nature inspired algoriths is their capability of reaching a pareto front for a set nuber of objectives with uch lesser coputational cost than the classical ones; this is priarily due to the intrinsic intelligence that they inherit fro nature. In this paper, as a first exaple, optiization of linear antenna arrays with dipole eleent pattern is exeplified for side lobe (SLL) reduction with fixed ain-lobe bea width using real coded genetic algorith (RGA). Though ulti goal optiization sees to be possible with proper cost/fitness function forulation in linear arrays, such a task becoes extreely difficult when it coes to planar arrays. This calls for the use of ulti objective variants of an algorith to reach the pareto-objectivity. As a second exaple, nondoinated sorting genetic algorith II (NSGA) is considered as the optiization algorith for SLL reduction and fixed ainlobe directivity for concentric regular hexagonal antenna arrays (CRHAA). The results show good outcoe with respect to side lobe reduction and directivity. I. INTRODUCTION Nature inspired algoriths are those coputer algoriths which are inspired by the interactions aong the species or living entities as they search for the requireents (like food, roo etc.). Evolutionary and Swar intelligent algoriths are exaples of nature inspired algoriths. Aong the evolutionary algoriths, genetic algorith is perhaps the ost widespread and robust optiization procedures used in the doain of counication technology. Genetic Algoriths are inspired by natural phenoenon of evolution where the properties of inheritance, accidental changes and survival of the fittest take place in the crossover tool, utation tool and the selection tool respectively. The ter generational eans that the solutions (resebling the species) are evolved in iterations. In this paper a broadside linear arrays of dipole eleents is considered for optiization as the first exaple. The phase difference between any two eleents is kept zero. The excitation and inter-eleent spacing of each eleent are optiized using RGA [6-8]. A cost function is defined, which keeps the SLL at low levels. The dipole antenna or dipole aerial is one of the ost iportant and coonly used types of RF antenna. It is widely used on its own, and it is also incorporated into any other RF antenna designs where it fors the driven eleent for the antenna. Dipole antenna is constructed with two thin dipole eleents that are syetrically fed at the centre by a balanced twowire transission line []. There are several types of dipole antennas such as hertzian dipole, half-wave dipole, sall dipole [] etc. Radiation resistance of the half-wave dipole was 73 Oh which atched with the line ipedance [3]. In this paper linear array of dipole eleent has been taken. There are several paraeters by varying which the radiation pattern can be odified. These paraeters are geoetrical configurations (e.g. linear, circular, planar, spherical etc.), inter eleent spacing, individual excitations (aplitude and phase) and relative pattern of individual eleents [-9].The non-unifor current excitation and optial unifor intereleent spacing allows for increased degrees of freedo in design. All these procedures control both peak and average SLL [5]. If the array eleents are place syetry along with the z- axis about the centre of the array, the nuber of attenuators required and the coputational tie are halved. Aplitude and inter-eleent only control is also easy to ipleent and less sensitive to quantization error [9]. As we shift fro linear to planar arrays the need for ore coplex algoriths is strengthened; in view of this, a ulti-objective variant of GA naed NSGA is used for optiization of concentric hexagonal arrays. Multiple objective syste designing has been the ajor concern for last few decades. While dealing with syste design probles with ultiple trade-off paraeters, basic odification to be done in the optiization algoriths is in their selection tools, ore precisely, the fitness assignent procedures []. Other issues those coe with this odification are the selection pressure and the preature convergence. Consequently, ultiple objective optiization algoriths focus on aintaining the diversity while conducting the search for a set of good solutions. Several approaches of ultiple objective optiization algoriths are ade since in the past decade. Developents of algoriths have been based on the fraeworks [-], archiving strategies [5] fitness assignent procedure [6] etc. While dealing with ultiple objective probles it is desired that the species having the qualities of perforing well in all the circustances (ore-or less, in a trade-off) be focused. Hence, a concept of doinance is developed which ay be viewed as a odified selection tool. The next section gives a brief outline of RGA and NSGA algorith; Section III and IV take the optiization exaples of linear and hexagonal arrays respectively. This work is part of a project funded by SERB, Departent of Science & Technology- Governent of India under the grant SB/EMEQ-39/ /5 $3. 5 IEEE DOI.9/SSCI

2 II. THE USE OF GENETIC ALGORITHM AND ITS MULTIOBJECTIVE VARIANT Genetic algoriths have been very effectively used in optiization of Antenna designs and areas of Electroagnetics in general [7, 8]. The use of GA has soewhat effectuated the classification optiization solutions as local or global. Algorithic steps of real coded genetic algoriths (RGA) can be found fro [7] which is an excellent treatise on the subject; hence there is no point in repeating those steps here; therefore, only a brief description on NSGA- is given here. Goldberg [8] realized the basic objective to carry out successful search for acceptable solutions for MOP. Elitist non doinated sorting based GA (NSGA II) follows not only the sae steps of genetic algorith for single objective, but soe additional steps to tackle ulti-objective probles. This work utilizes real coded NSGA II [] for the considered proble and the whole search procedure as applied in this work is listed step by step below: i. A population of randoized individuals is created aintaining liits of every variable. Each population eber is a string of probable current aplitudes (each string is called a chroosoe ) one for each ring. Stopping criteria ( generations), tournaent size, crossover and utation types with individual internal paraeters. For this work a siulated binary crossover [9] operator with index to control the spread factor as is opted for crossover operation, and for utation, polynoial utation [] operator with external paraeter to control utation of is opted. ii. iii. iv. Then non doinated sorting is carried out to record non-doination rank of every chroosoe in the population. Until the stopping criteria satisfied the following steps are repeated; For ating selection crowded distance based tournaent selection operator is called for filling the ating pool. This operator operates on a pair of chroosoes, and favors ore potential one (less non-doination rank calculated at step ii or v if found, otherwise (if both the chroosoes are at sae front) it selects the one which is fro relatively less crowded region. Solutions are cobined in a successive pair-wise order to create offspring solutions, which are then individually utated. v. The old and new populations are erged, and a nondoination rank is re calculated. Out of these solutions worst solutions based on nondoination rank are discarded. This process by default aintains the elite chroosoes fro the past III. LINEAR ARRAYS OF DIPOLE ELEMENT Consider a broadside linear array of M equally spaced isotropic eleents as shown in Figure. The array is syetric in both geoetry and excitation with respect to the array center. Fig.. Geoetry of a N-eleent linear dipole array along the z-axis. For broadside beas, the array factor is given by M n AF ( θ) = In cos[( ) kd cos( θ)] () n= where θ = Angle of radiation of electroagnetic plane wave; d = Spacing between eleents; k = Propagation constant; M = Total nuber of eleents in the array; The cost function (CF) for reducing the side lobe level is given below CF = AF θ, I ) / AF( θ, I ) () ( sl n n where θ is the angle where the highest axiu of central angle is attained in θ [, π ]. θ sl is the angle where axiu side lobe AF( θ sl, I n ) is attained on either side of ain bea. Miniization of CF eans axiu reduction of SLL. RGA technique is eployed for optiizing nonunifor current excitation weights and optial unifor intereleent spacing. As dipole antenna, one of the ost coonly used antennas is the half-wavelength ( L = λ /) dipole. Because of its radiation resistance as 73 ohs, which is very near to 5- oh or 75-oh characteristic ipedances of soe transission lines, its atching to the line is siplified especially at resonance. The far-field radiation pattern fro a dipole antenna of length L is given by []. For the half wave dipole antenna, where L = λ /, the far field radiation pattern of dipole antenna can be given by 65

3 π cos cosθ EP( θ) = (3) sinθ where θ denotes the angle easured fro the axis of the dipole to the line of site. Fro () and (3) the overall total radiation pattern is given by eleent, 6-eleent and -eleent, linear antenna arrays, SLL reduces to -37. db, db, db, respectively, against -3 db, -3. db, -3. db, respectively, for corresponding unifor linear arrays TABLE II. OPTIMAL CURRENT EXITATION COFFICIENS, OPTIMAL INTER-ELEMENT SPACING, SLL AND BWFN FOR THREE LINEAR ARRAY OF DIPOLE ELEMENT SETS π cos cosθ M n AFTotal ( θ) = In cos[( ) kd cos( θ)] sin () θ n= Here, EP( θ ) is the radiation pattern of individual array eleents, AFTotal ( θ ) is the array factor of linear array of dipole eleent. The nuerical results for three sets of linear arrays of dipole eleent designs are obtained by RGA technique. For each antenna array, constraints of the optiization variable are aintained. The best paraeters for the RGA are set after any trial runs. It is found that the best results are obtained for the initial population size ( n ) of chroosoes; and p axiu nuber of generations, N as. For selection operation, the ethod of natural selection is chosen with selection probability of.3. Crossover is randoly selected as dual point. Crossover ratio is.8. Mutation probability is.5. RGA generates a set of optial noralized non-unifor current excitation weights and optial unifor inter-eleent spacing ( d [ λ /,λ] ) for each set of linear array of dipole eleent. Sets of arrays considered are of -, 6- and - eleents. Table II shows the optial results. Table I depicts SLL values and BWFN values for all corresponding uniforly exited linear array of dipole eleent antenna. A. Analysis of Radiation Patterns TABLE I. INITIAL VALUES OF SLL AND BWFN FOR UNIFORMLY EXCITED ARRAYS HAVING ( I n =) AND λ / INTER- ELEMENT SPACING OF LINEAR ARRAY OF ISOTROPIC AND DIPOLE ELEMENTS Set No. M SLL (db) isotropic eleents SLL (db) dipole eleents BWFN ( deg.) (isotropic) BWFN ( deg.) (dipole) I II III Figs. 3, and 5 depict the optial radiation patterns of -, 6- and -eleent tie odulated linear antenna array sets, respectively, with optial non-unifor excitation weights and optial unifor inter-eleent spacing using RGA. Fro each figure, it is clearly visible that beside noticeable reduction of SLL, BWFN is also well restricted upon optiizing. As seen fro the Table II, for optial non-uniforly exited and optial uniforly spaced syetric tie odulated - Set No. ( I I... I M ) I II III Side lobe level relative to the ain bea (db) Inter-eleent spacing (λ) SLL (db) BWFN (deg.) Unifor Exitation(lada/) Eleent spacing Optiized isotropic antenna array by RGA Optiized dipole antenna array by RGA (MATLAB) Angle of arival (degrees) Fig. Optial array pattern obtained by the RGA in case of -eleent linear array of dipole eleent. Side lobe level relative to the ain bea (db) Unifor Exitation(lada/) Eleent spacing Optiized isotropic antenna array by RGA Optiized dipole antenna array by RGA (MATLAB) Angle of arival (degrees) Fig 3. Optial array pattern obtained by the RGA in case of 6-eleent linear array of dipole eleent. 66

4 Side lobe level relative to the ain bea (db) Unifor Exitation(lada/) Eleent spacing Optiized isotropic antenna array by RGA Optiized dipole antenna array by RGA (MATLAB) Angle of arival (degrees) Fig. Optial array pattern obtained by the RGA in case of -eleent linear array of dipole eleent. IV. CONCENTRIC REGULAR HEXAGONAL ANTENNA ARRAYS: (CRHAA) A structure of a CRHAA resting on x-y plane is shown in a Y Fig 5 Structure of a CRHAA resting on the x-y plane. Let M is the nuber of the rings on this structure; N is the th nuber of eleents on the q sector {,, 6} ring fro the centre; { nq, nq} th n -eleent on the q th sector on the q = of the r φ be the location of the th ring according to the cylindrical co-ordinate syste; a be the inter-eleent separation between the rings ( ) and ( a represents the axiu separation of an eleent of the first ring), and d be the inter-eleent separation on the ring. The array factor of this array is given as []: = M N 6 j rnq sinθ cos( φ φnq ) + nq (5) = n= q= AF I I e π λ where j = ; λ is the wavelength of operating signal frequency; { θ, φ } represents the angular co-ordinate of any far-field point surrounding the array geoetry; I and I nq d X represent the coplex excitation currents of the centre eleent and the n th -eleent on the q th sector on the th ring, respectively. The paraeters of the above expression are interrelated as follows: N a = d rnq = a + ( n ) d ad ( n ) (6) 3 d π φnq = sin (( n ) r ) + ( q ) nq 3 Total nuber of eleents on the aperture is given by M N = + 6 N. This work considers a eight-ring CRHAA = λ with a = d =. Thus, N for this work is 7. For this optiization proble, I = I n nq (, N ), q (, 6). The optiization proble is considered as { SLL, D} CF = (7) where SLL represents the relative peak sidelobe level of the antenna pattern and D [7] corresponds to the axiu directivity value. Usually, these two paraeters are in tradeoff and both are very sensitive to the current excitations. Tapering the current distribution will favorably cause suppression of SLL, but will equally cause the fall of D. Hence, this is a ultiple objective proble. Both of the paraeters are expressed in db. Usually, a high negative value of SLL and high positive value of D is desired. Using a positive D and negative SLL coplicates the design of selection paraeter. Rather iniizing D will serve the purpose and will ake the design of selection tool easy, the objective vector is forulated as (). The entire siulation was carried out using MATLAB software using MATLAB (R3a) on a Intel Core i5-5 CPU with processor speed 3.3 GHz and 8. GB RAM and operating syste Microsoft Windows 7 Version 6. (Build 76). For this work, NSGA is run independently for twenty five ties and the non-doinated solutions of the cobined populations are extracted. Then the ost evenly spread twenty five Pareto front representative solutions are reported. The obtained Pareto set for the considered array design proble is depicted in Fig.6 below. Two extree solutions of the obtained Pareto front are tabulated in Table III. Figures 7(a) through 7(d) depict the excitation profiles and the corresponding radiation patterns of the array geoetries corresponding to the extree points in the obtained pareto-front. Figs 7(b) and 7(d) show upper heisphere patterns only. Table III denotes that uniforly excited (isophoric) CRHAA design has an SLL value of -6.9 db and the corresponding D value is 7.56 db; Set corresponds to an excitation profile for the considered CRHAA geoetry has a dynaic 67

5 range ratio (DRR) of 3 and which results in an SLL value of B and the corresponding D value is 6.9 db; Set corresponds to an excitation profile for the considered CRHAA geoetry has a DRR of. and which results in an SLL value of -6.9 db and the corresponding D value of 7.59 db. Fig 7(a) and 7(c) depict the excitation corresponding to Set and Set of Table III; Fig 7(c) and 7(d) depict the array factors respective to the excitation profiles of Set and Set, respectively, for the CRHAA geoetry considered in this work. This table denotes that with ore tapering of currents (large DRR value), the SLL value drops ore, but the directivity also drops. This phenoenon is expected and is siilar to the coon practice of iniizing the contributions of the far-end eleents while iniizing the SLL value. Again D is proportional to the squared su of the current aplitudes [7], hence, gradual decay in the current aplitudes for far-end eleents also causes drop of D -value. It is also interesting to note that the directivity of set- in Table III is higher than that of an uniforly excited array; this ay be attributed to the increase in peak SLL and overall sidelobe levels in set- with respect to the uniforly excited array. D (db) SLL (db) Fig. 6: Obtained pareto front for the pattern optiization proble () for eight ring CRHAA design. The increase in directivity of the ainlobe at the cost of increase in the power of sidelobes is a well-established phenoenon in all fors of antenna arrays including those of unifor spacing and variable aplitudes. It ay be further observed that how an inverse aplitude taper is applied, on = of set- unlike set-, as a resultant of the paretooptiization process. Table III: Current excitation aplitudes and the corresponding pattern paraeters for eight-ring CRHAA geoetries I Centre eleent I Pattern Paraeters Unifor Set- (Lowest SLL) SLL D Set- (Highest D) I.6.. y x (a) (b) 68

6 .8.6 I.. y x (c) (d) Fig. 7. Excitation profile and the corresponding array factor of CRHAA geoetries; 7(a) excitation profile of set () of Table III; 7(b) array factor corresponding to the current excitations as in 7(a); 7(c) excitation profile of set () of Table III; 7(d) array factor corresponding to the current excitations as in 7(c). In this figure u= sinθcosφ and v = sinθsinφ where π θ (, ) and φ (, π). V. CONCLUSIONS In this paper the optial design of non-uniforly excited linear arrays of dipole eleents with unifor inter-eleent spacing has been described using the technique of real coded genetic algorith. Siulation results reveal that the optial design of non-uniforly excited linear antenna arrays with optial inter-eleent spacing offers a considerable SLL reduction with respect to corresponding tie odulated unifor linear arrays with unifor inter-eleent spacing of λ /. For the array sets having, 6 and eleents, SLLs have reduced corresponding unifor tie odulated linear arrays with a very little change in BWFN. The BWFNs of the initial and final radiation pattern reain approxiately the sae. A bi-objective current-only sidelobe suppression proble for eight ring CRHAA geoetries is dealt with. Nondoinated sorting genetic algorith II (NSGA) is chosen as the optiizing algorith. NSGA is found to successfully produce a set of well spread trade-off pareto optial solutions. Future research will be aied at dealing with other geoetries and constraints of any different areas of antenna design and analysis requires a feasible and versatile procedure, being able to perfor array synthesis by tuning antenna characteristics and paraeters. REFERENCES [] C. A. Ballanis, Antenna theory analysis and design, 3rd edition, John Willey and Son's Inc., New York, 5. [] Rao, Sion, Whinnery, John R., Duzer, Theodore Van, Fields and Waves in Counication Electronics, John Wiley & Sons, 3rd Edition, Canada (99). [3] Warren Stutzan and Gary Thiele, "Antenna Theory and Design, nd. Ed.", 998, John Wiley and Sons [] John D. Krous Antenna, Mc GRAW-HILL, Newyork, 95. [5] Stephen Jon Blank, Michael F. Hutt, On the Epirical Optiization of Antenna Arrays EEE Antennas and Propagation Magazine, 7,, April 5, pp [6] R. L. Haupt, Phase-only adaptive nulling with a genetic algorith, IEEE Trans. Antennas Propagat., vol. 5, no. 6, pp. 9-5, Jun [7] R. L. Haupt, and D. H. Werner, Genetic Algoriths in Electroagnetics, John Wiley & Sons Inc., 7. [8] G. R. Hardel, N. T. Yalapragada, D. Mandal, A. K. Bhattacharjee, Introducing Dipper Nulls in Tie odulated Linear syetric Antenna Array Using Real Coded Genetic Alorith, IEEE Syposiu on coputers and inforatics, pp. 9-5, March,. [9] K. G uney and a. Akda gl null steering of linear antenna arrays using a odified tabu search algorith, Progress in electroagnetics research, pier33,67 8,. [] K. Deb, Multiobjective optiization using evolutionary algoriths, JohnWiley & Sons, New York, NY, USA,. [] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist ultiobjective genetic algorith: NSGA-II, IEEE Transactions on Evolutionary Coputation, vol. 6, no., pp. 8 97,. [] Q. Zhang and H. Li, MOEA/D: A ultiobjective evolutionary algorith based on decoposition, IEEE Transactions on Evolutionary Coputation, vol., no. 6, pp. 7 73, 7. [3] N. A. Moubayed, A. Petrovski, and J. McCall, D MOPSO: ultiobjective particle swar optiizer based on decoposition and doinance, J.-K. Hao and M. Middendorf (Eds.): Evolutionary Coputation, Lecture Notes in Coputer Science 75, pp ,. [] K. Li, S. Kwong and K. Deb, A dual-population paradig for evolutionary ultiobjective optiization, Inforation Sciences, vol. 39, pp. 5 7, 5. [5] M. Lauanns, L. Thiele, K. Deb and E. Zitzler, Cobining convergence and diversity in evolutionary ulti-objective optiization, Evolutionary Coputation, vol., no. 3, pp. 63 8,. [6] J. D. Knowles and D. W. Corne, The Pareto Archived Evolution Strategy: A New Baseline Algorith for Multiobjective Optiisation, In 999 Congress on Evolutionary Coputation, pp. 98-5, Washington, D.C., July 999. IEEE Service Center. [7] S. Das, D. Mandal, R. Kar and S. P. Ghoshal, A generalized closed for expression of directivity of arbitrary planar antenna arrays, IEEE transactions on Antennas and Propagation, Vol. 6, no. 7, pp , 3. [8] D. E. Goldberg, Genetic Algoriths for Search, optiization, and Machine Learning. Reading, M. A: Addision-Wesley, 989. [9] K Deb, R. B. Agrawal, Siulated binary crossover for continuous search space, Coplex Systes, vol. 9, -3, 9. [] Dong Feng Li and Zhong Lin Gong, Design of Hexagonal, Planar Arrays Using Genetic Algoriths For Perforance Iproveent, In the Proceedings of nd Inteational Conference on Microwave and Millieter Wave Technology, 55-6,. 69