328 A New Regressor for Bandwdth Cacuaton of a Rectanguar Mcrostrp Antenna Had Sadogh Yazd 1, Mehr Sadogh Yazd 2, Abedn Vahedan 3 1-Computer Department, Ferdows Unversty of Mashhad, IRAN, h-sadogh@um.ac.r 2-Eectrca and Computer Engneerng Department, Shahd Behesht Unversty of Tehran, IRAN me.sadoogh@ma.sbu.ac.r 3-Computer Department, Ferdows Unversty of Mashhad, IRAN, vahedan@um.ac.r Abstract- Mcrostrp antennas (MSAs) offer a number of unque advantages over other types of antennas. In MSA desgn, t s mportant to determne the bandwdth of the antenna accuratey because t s a crtca parameter of a MSA. To cacuate the bandwdth of the rectanguar mcrostrp antennas wth thn and thck substrates, we present a new method based on the support vector regresson (SVR) and Fuzzy C-Mean (FCM). The support vector regresson (SVR) s a statstca earnng method that generates nput-output mappng functons from a set of tranng data. The bandwdth resuts obtaned usng SVR and new proposed SVR are n exceent compance wth the expermenta resuts avaabe n the terature. Index Terms- Mcrostrp antennas; support vector regresson; FCM. I. INTRODUCTION In recent years, deveopng ow cost, mnma weght, ow profe panar confguraton mcrostrp (patch) antennas, capabe of mantanng hgh performance over a wde spectrum of frequences has been a maor trend [1, 2 and 3]. A mcrostrp devce n ts smpest form s a sandwch of two parae conductng ayers separated by a snge thn deectrc substrate. The patch can assume any shape. They are used where compatbty wth mcrowave and mmeter wave ntegrated crcuts (MMICs), robustness, abty to conform to panar and nonpanar surfaces are requred [4, 5 and 6]. In MSA desgn, t s mportant to accuratey determne the bandwdth of the antenna as a crtca parameter. Severa technques varyng n accuracy and computatona effort have been proposed [7, 8 and 9]. Anaytca and mathematca soutons are used to understand the physca aspects and for computer-aded desgn, but they suffer from mtatons. Desgn of MSA eements havng wder bandwdth usng a smpe method to cacuate the bandwdth of eectrcay thn and thck rectanguar MSAs are then requred provded that the theoretca resuts are n far agreement wth the expermenta resuts. In ths work, a new method based on the support vector regresson (SVR) and FCM s presented whch effcenty addresses ths probem. Once the antenna parameters are determned, the bandwdth s cacuated usng the proposed SVR. Ths paper s organzed as foows: In secton II, the bandwdth of a MSA s descrbed foowed by SVR expaned n secton III. Secton IV ncudes the appcaton of SVR to the computaton of the bandwdth of rectanguar MSAs and ts smuaton resuts for exstng data. In secton V new verson of SVR s ntroduced foowed by the resuts of appyng the method n secton VI. Fnay, secton VII draws concuson of ths work. II. BANDWIDTH OF RECTANGULAR MICROSTRIP ANTENNAS The rectanguar mcrostrp antennas are made of a rectanguar patch wth dmensons wdth, W
329 and ength, L, over a ground pane wth a substrate thckness h and reatve deectrc constants εr, as ndcated n Fg. 1. Fg. 1 A rectanguar mcrostrp antenna. The bandwdth of ths MSA can be determned from the frequency response of ts equvaent crcut. For a parae-type resonance, the bandwdth s expressed as [10]: 2G BW = db (1) ωr ωr dω Where Y=G+B s the nput admttance at the anguar resonant frequencyω r. For a seres-type resonance, G and B are repaced by R and X, respectvey, where Z=R+X s the nput mpedance at resonance. The bandwdth of a MSA can aso be expressed as [11]: s 1 BW = (2) QT s Where s s the votage standng wave rato (VSWR) and QT s the tota quaty factor whch can be wrtten as: 1 Pd + Pc + Pr + Ps = (3) QT ωrwt Pd s the power ost n the ossy deectrc substrate, Pc s the power ost n the mperfect conductor, Pr s the power radated n the space waves, Ps s the power radated n the surface waves, and WT s the tota energy stored n the patch at resonance. It can be shown that ony three parameters, h / λd, W, and the deectrc oss tangent, tan δ, are requred to descrbe the bandwdth. The waveength n the deectrc substrate, λ d, s then gven as: λ0 c λ d = = (4) ε r f r ε r λ 0 s the free space waveength at the resonant frequency, f r and c s the veocty of eectromagnetc waves n the free space. The method ntroduced n ths paper cacuates the bandwdth of rectanguar MSAs based on ony these three parameters,.e. h / λd, W, and tan δ. III. SUPPORT VECTOR REGRESSION Support vector machnes (SVMs) were orgnay ntroduced by Vapnk wthn the area of statstca earnng theory and structura rsk mnmzaton amng at creatng a cassfer wth mnmzed VC dmenson [12]. SVR s consdered as a supervsed earnng method whch generates nput-output mappng functons from a set of abeed tranng data. The mappng functon can be ether a cassfcaton functon,.e., the category of the nput data, or a regresson functon. Intay deveoped for sovng cassfcaton probems, support vector technques can be successfuy apped to regresson. Suppose the tranng data {( X 1, Y1 ), ( X 2, Y2 ),...,( X, Y )} X R, s gven where X denotes the space of the nput D patterns (e.g. X = R ). In ε-sv regresson, the goa s to fnd a functon f(x) that at most has a devaton of ε from the actuay obtaned targets y for a the tranng data [12]. The regressor must not ony ft the gven data we, but aso make mnma errors n predctng vaues at any
330 other arbtrary pont n R D. Nonnear regresson s accompshed by fttng a near regressor n a hgher dmensona feature space. A nonnear transformatonφ s used to transform data ponts from the nput space (wth dmenson D) nto a feature space havng a hgher dmenson L ( L > D). The nonnear mappng s denoted D L byφ : R R. Ths probem can be stated as a convex optmzaton probem; hence, we arrve at the formua stated n [12]: 1 2 Mn W + C ( ξ + ξ ) 2 = 1 T s. t. y W φ( Χ ) b ε + ξ (5) T y + W φ( X ) + b ε + ξ ξ, ξ 0 Where C > 0 s a constant, ξ, ξ are sack varabes for soft margn SVR, whch aow some devaton arger than ε as precson. It turns out that n most cases the optmzaton probem n (5) can be soved more easy n ts dua formuaton: Max s. t 1 2 ε, = 1 = 1 ( α α )( α α ) K ( X, X ) ( α + α ) + y ( α α ) ( α α ) = 0, α, α [ 0, C] = 1, α = 1 (6) Where α are Lagrange coeffcents and matrx K s termed as a kerne matrx such that ts eements are gven by: T K ( X, X ) = φ ( X ) φ( X ),, = 1,2,... M. By sovng (6), we can fnd Lagrange coeffcents and by repacng them, we have: W = ( α α ) φ( X ), thus = 1 (7) f ( x) = ( α α ) K ( X, X ) + b =1 IV. SVR BASED BANDWDTH CALCULATION For the SVR, the nputs are h / λd, W and tan δ, whe the output s the measured bandwdths BWme. The tranng and test data sets used n ths work have been obtaned from prevous expermenta works [13, 14], and are gven n Tabe 1 were used to tran the SVR. The 6 data sets, marked wth an astersk n Tabe 1, were used for testng. The tranng and test data sets used are aso the same as those used for ANNs [15, 7] and FISs [16]. The eectrca thckness of antennas gven n Tabe 1 vary from 0.0065 to 0.2284, n physca thckness from 0.17 to 12.81 mm, and operate over the frequency range 2.980 8.000 GHz. Some kerne functons have been aso used for SVR ke poynoma wth dfferent degrees, rada bass functon and near functons. Three evauaton methods were used, namey apparent, hod out and eave-one-out to compare average error of the proposed method wth ANFIS (Adaptve Neuro Fuzzy Inference System appeared n the appendx).
331 Tabe 1. The measured bandwdths for eectrcay thn and thck rectanguar mcrostrp antennas [13, 14]. Patch no h (mm) F r (GHZ) h / λd W (mm) tan δ Measured [13, 14] BWme (%) 1 0.17 7.740 0.0065 8.50 0.001 1.070 2 0.79 3.970 0.0155 20.00 0.001 2.200 3 0.79 7.730 0.0326 10.63 0.001 3.850 4 0.79 3.545 0.0149 20.74 0.002 1.950 5 1.27 4.600 0.0622 9.10 0.001 2.050 6 1.57 5.060 0.0404 17.20 0.001 5.100 7 1.57 4.805 0.0384 18.10 0.001 4.900 8 1.63 6.560 0.0569 12.70 0.002 6.800 9 1.63 5.600 0.0486 15.00 0.002 5.700 10 2.00 6.200 0.0660 13.37 0.002 7.700 11 2.42 7.050 0.0908 11.20 0.002 10.900 12 2.52 5.800 0.0778 14.03 0.002 9.300 13 3.00 5.270 0.0833 15.30 0.002 10.000 14 3.00 7.990 0.1263 9.05 0.002 16.000 15 3.00 6.570 0.1039 11.70 0.002 13.600 16 4.76 5.100 0.1292 13.75 0.002 15.900 17 3.30 8.000 0.1405 7.76 0.002 17.500 18 4.00 7.134 0.1519 7.90 0.002 18.200 19 4.50 6.070 0.1454 9.87 0.002 17.900 20 4.76 5.820 0.1475 10.00 0.002 18.000 21 4.76 6.380 0.1617 8.14 0.002 19.000 22 5.50 5.990 0.1754 7.90 0.002 20.000 23 6.26 4.660 0.1553 12.00 0.002 18.700 24 8.54 4.600 0.2091 7.83 0.002 20.900 25 9.52 3.580 0.1814 12.56 0.002 20.000 26 9.52 3.980 0.2017 9.74 0.002 20.600 27 9.52 3.900 0.1976 10.20 0.002 20.300 28 10.00 3.980 0.2119 8.83 0.002 20.900 29 11.00 3.900 0.2284 7.77 0.002 21.960 30 12.00 3.470 0.2216 9.20 0.002 21.500 31 12.81 3.200 0.2182 10.30 0.002 21.600 32 12.81 2.980 0.2032 12.65 0.002 20.400 33 12.81 3.150 0.2148 10.80 0.002 21.200 Test data set Startng wth eave-one-out method for computng error of SVR aganst ANFIS, average error for both methods s shown n Tabe 2, usng dfferent kerne functons for SVR. Tabe 1. Average error of SVR and ANFIS for bandwdths for eectrcay thn and thck rectanguar mcrostrp antennas wth eave-one-out method. kerne Average Error SVR Average Error ANFIS Poy, p=2 37.8586 37.2162 Poy, p=3 19.6580 37.2162 Poy, p=4 37.5095 37.2162 erbf 28.8429 37.2162 rbf 44.8668 37.2162 Lnear 20.4241 37.2162 Usng separate tran and test data sets (hod out method) determned n Tabe 1 resuts n measured error ndcated n Tabe 3 for both ANFIS and SVR. Tabe 2. Error of SVR and ANFIS for bandwdths for eectrcay thn and thck rectanguar mcrostrp antennas wth hod out method kerne Error of SVR Error of ANFIS Poy, p=2 0.0787 0.1757 Poy, p=3 0.1431 0.1757 Poy, p=4 0.1998 0.1757 erbf 0.0833 0.1757 rbf 0.2530 0.1757 Lnear 0.0612 0.1757 Fnay, same tran and test data sets were used n apparent method wth a data represented n Tabe 1. The measured error of our proposed method aganst ANFIS s shown n Tabe 4Tabe 3.
332 Tabe 3. Error of SVR and ANFIS for bandwdths of eectrcay thn and thck rectanguar mcrostrp antennas wth apparent method. kerne Error of SVR Error of ANFIS Poy, p=2 0.1113 5.4071e-016 Poy, p=3 0.1282 5.4071e-016 Poy, p=4 0.1142 5.4071e-016 erbf 6.5168e-004 5.4071e-016 rbf 0.2741 5.4071e-016 Lnear 0.0744 5.4071e-016 As we can see from Tabe 2 to Tabe 4, when tranng data set s used to test ANFIS, t can compute bandwdth of MSAs wth mnmum error; however when test data set s dfferent from tranng data set, ANFIS cannot outperform SVR for some kerne functons. Some of the key notes about the proposed SVR as opposed to ANFIS are: 1- ANFIS uses a near pecewse technque, the proposed SVR, however, s a nonnear pecewse approach. 2- Swappng kernes to obtan better resuts s more possbe n the proposed SVR. 3- A propertes of SVR can be ncorporated n the proposed SVR such as kerne trcks, hgh dmensona space, and empoyng margn n the regresson. 4- SVR s more genera due to utzng margn and permeate aspects. V. NEW SUPPORT VECTOR REGRESSION As mentoned earer, the support vector machne s an approxmate mpementaton of the method of structura rsk mnmzaton. Ths s based on the fact that the error rate of a earnng machne on test data s bounded by the sum of the tranng-error and a term whch depends on the Vapnk-Chervonenks (VC) dmenson. In ths method, optma hyper pane s determned to guarantee the mnmum error for test sampes, whereas, neura networks fa to guarantee to fnd optmum hyper pane for test sampes. Therefore, SVR yeds better resuts compared to ANFIS as ndcated n secton 4. There are, however, the foowng probems n the SVR: a) Snce each sampe appears as one constrant n support vector, ncreasng tranng sampes s equvaent to ncreasng the number of constrants. Sovng equatons to fnd optma hyper pane then becomes fary hard. b) Fndng sutabe kerne for modeng of nonnear space s not straghtforward. We, therefore, propose a new verson of SVR whch works based on dvde and conquer prncpe whch can sove two aforementoned probems. Input space s dvded to severa subspaces so that n each subspace, a SVR modes the data. Ths causes that the new generated space ncorporate the propertes of hgh dmensona space. A weghtng procedure s then performed usng probabty densty functon of each subspace and gves the porton of each SVR accordng to generated rues. Resuts of weghted SVRs are, then, combned to perform the fttng task. The proposed SVR method s revewed n deta n the foowng steps. Step 1: Input tranng data s dvded nto subsets usng a custerng agorthm such as fuzzy c-means (FCM) whch assgns weghts to any nput data. Fg. 2 ndcates as exampe 3 parttons custered by FCM. The PDF (Probabty Densty Functon) of each custer s obtaned whch are shown. The correspondng weghts of an nput data are then cacuated based on membershp vaues to each partton (.e. custers)
333 Fg. 2. Assgnng weghts to nput data Step 2: Each avaabe subset for any partton s apped for tranng of each Support Vector Regressor (SVR) as depcted n Fg. 3. Therefore, for tranng sampes of partton 1 n Fg. 2, SVR1 s traned (as shown n Fg.3). Fg. 3. Appyng SVR n each partton Some kerne functons have been aso used n ths work for SVR such as poynoma wth dfferent degrees, rada bass and near functons. Resuts were examned for best state. Step 3: Ths step nvoves testng procedure. We used eave-one-out method for computng average error for our proposed method and compared t wth ANFIS. In order to cacuate the output of the proposed system, membershp vaues were computed for each test sampe ( w ). 1 ~ 1 T w = ~ exp( ( x ) ( ) ) 0.5 t μ Σ xt μ (8) 2π Σ ~ Where Σ = ασ and Σ s covarance of tranng sampes and μ s the mean of tranng data. xt s a test sampe and denotes the determnant. In equaton (8), α s a varabe to contro spreadng of the Gaussan dstrbuton consdered for sampes of each tranng set. By appyng equaton (8) to test sampes, correspondng weghts are obtaned so one can normaze these weghts by dvdng any weght to the sum of them. Fnay, these normazed weghts are mutped by each test sampe to generate fna vaues. Fg. 4 depcts ths procedure.
334 In Fg. 4 output vaue s the computed vaue for bandwdth of MSA of test sampe. In order to compute the error of eave-one-out method for evauaton of performance, we can compute dfference between computed vaue and measured vaue of bandwdth of MSA of any test sampe and then obtan average vaue for a of data, as gven n Tabe 1, for nstance. We now compute the bandwdth for eectrcay thn and thck rectanguar mcrostrp patch antennas by the proposed SVR. Frst we used eave-one-out method for testng our method aganst ANFIS wth dfferent kerne functons for SVR and dfferent vaues forα. In ths experment we consdered vaues 0.1, 0.2,.., 1 for α wth the number of custers set to be 2. Average error for both methods s shown n Tabe 5. As t can be seen from Tabe 5, wth α=0.1 or α=0.2 our proposed method has the mnmum error for the gven kerne functons. Fg. 4 Testng procedure VI. EXPERIMENTAL RESULTS The proposed method was aso compared wth some conventona methods presented n [11, 17, 18 and 19]. Fg. 5 represents comparson between computed error n cacuatng the bandwdth of MSAs wth a number of conventona methods, ANFIS and our proposed SVR method. We used smpe crtera for computng error as absoute vaue of dfference between measured BW [13 and 14] and computed vaue usng each method. Snce the proposed methods n [11, 17, 18 and 19] have used the data set n Tabe 1 for tranng and testng smar to our method, we too used the above mentoned crtera to perform the comparson. For each case (each row n Tabe 1), the obtaned error s cacuated and shown n Fg.5 for each method. It can be seen that the resuted error n our proposed method s ess than the other methods. Mean vaue of computed error for each method appeared n Fg. 5 s aso represented n Tabe 6.
335 Tabe 4. Average error of proposed SVR and ANFIS for bandwdths of eectrcay thn and thck rectanguar mcrostrp antenna kerne Average Error New SVR α=0.1 α=0.2 α=0.3 α=0.4 α=0.5 α=0.6 α=0.7 α=0.8 α=0.9 α=1 Average error for a α Average Error ANFIS erbf 1.6950 1.6950 2.9466 2.9466 2.9428 2.9466 2.9470 4.156 4.213 4.217 3.0707 37.2162 rbf 56.308 56.207 57.979 58.079 57.979 58.079 58.079 58.475 58.495 58.495 57.817 37.2162 poy (p=2) 15.803 15.803 16.641 16.641 16.641 16.641 16.535 18.105 18.181 18.181 16.917 37.2162 poy (p=3) 7.856 7.856 8.314 8.314 8.314 8.314 8.314 8.594 8.6089 8.599 8.3089 37.2162 poy (p=4) 16.232 16.232 17.328 17.416 17.416 17.416 17.329 20.731 20.898 20.898 18.190 37.2162 Lnear 17.290 17.290 17.637 17.637 17.637 17.636 17.637 19.4233 19.513 19.514 18.121 37.2162 Fg. 5. Comparng error of some conventona methods and ANFIS wth the proposed SVR-x-axs:sampes, y- axs:resuted error Tabe 5. Mean vaue of error of computed bandwdths obtaned from conventona methods presented n [11, 17, 18 and 19], ANFIS and the proposed method for MSAs Method Error [11] 0.7241 [17] 0.1891 [18] 0.4337 [19] 0.1359 ANFIS 0.1984 Proposed SVR 0.0117 VII. CONCLUSION In ths paper we used Support Vector Regresson (SVR) method to cacuate bandwdth of eectrcay thn and thck rectanguar mcrostrp patch antennas. Resuted error from our method was compared to ANFIS whch showed that the method resuts n ower error than ANFIS and other conventona methods. A sutabe method for cacuaton of optmum nput parameters ( h / λd, W, tan δ ) s suggested to be carred out as the future work wth desred
336 constrants over each parameters to obtan desred bandwdth. An artfca search method, therefore, s to be proposed for searchng nput parameters to converge to desre bandwdth as shown n Fgure 6. Fg.6. Searchng probem of optmum nput parameters for obtanng desred bandwdth whch s appeared n the future work. Rea-tme processng of nstantaneous system nput and output data. Ths property heps usng ths technque for many operatona research probems. Offne adaptaton nstead of onne systemerror mnmzaton, thus easer to manage wth no teratve agorthms nvoved. System performance s not mted by the order of the functon snce t s not represented n poynoma format. Fast earnng tme. System performance tunng s fexbe as the number of membershp functons and tranng epochs can be atered easy. The smpe f then rue decaraton and the ANFIS structure are easy to understand and mpement. APPENDIX ADAPTIVE NEURO FUZZY INFERENCE SYSTEM (ANFIS) Recenty, there has been a growng nterest n combnng neura network and fuzzy nference system. As a resut, neuro-fuzzy computng technques have been evoved. Neuro-fuzzy systems are fuzzy systems whch use neura networks theory n order to determne ther propertes (fuzzy sets and fuzzy rues) by processng data sampes. Neuro-fuzzy ntegrates to synthesze the merts of both neura networks and fuzzy systems n a compementary way to overcome ther dsadvantages. ANFIS has been proved to have sgnfcant resuts n modeng nonnear functons [20]. In an ANFIS, the membershp functons (MFs) are extracted from a data set that descrbes the system behavor. The ANFIS earns features n the data set and adusts the system parameters accordng to gven error crteron. In the ANFIS archtecture, NN earnng agorthms are used to determne the parameters of fuzzy nference system. Beow, the advantages of the ANFIS technque are summarzed. REFERENCES [1]. P. Bartha, K.V.S. Rao, R.S. Tomar Mmeter wave mcrostrp and prnted crcut antenna Artech House - Boston 1991. [2]. Kyun Han, Frances J. Harackewez and Seokchoo Han," Mnaturzaton of mcrostrp patch antenna usng the Serspnsk Fracta geometry, Department of Eectrca & Computer Engneerng, Southern Inos. [3]. A. K. Skrvervk, J. F. Zurcher, O. Staub and J. R. Mosg, PCS antennas desgn: the chaenge of mnaturzaton, IEEE AP Magazne, vo. 43, Aug 2001. [4]. C. A. Baanes, Antenna Theory: Anayss & Desgn, John Wey & Sons, Inc., 1997. [5]. Pozar and Schaubert, Mcrostrp Antennas, Proceedngs of the IEEE, vo. 80, 1992. [6]. Waterhouse, R. B., Targonsk, S. D., and Kokotoff, D. M., Desgn and Performance of sma Prnted Antennas, Trans. Antennas and Propagaton, 1998, vo. 46, pp. 1629-1633. [7]. Gutekn S, Guney K, Sagrogu S, Neura networks for the cacuaton of bandwdth of rectanguar mcrostrp antennas, Apped Computer Eectromagnet Soc J 18:46 56, 2003. [8]. Guney K, A smpe and accurate expresson for the bandwdth of eectrcay thck rectanguar mcrostrp antennas. Mcrowave Opt. Technoogy Letter 36:225 228, 2003.
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