Pre-Design of Multi-Band Planar Antennas by Artificial Neural Networks

Ci a ion: Lahiani, M.A.; Raida, Z.; Veselý, J.; Oli o á, J. P e-Design o Mul i-Band Plana An ennas by A i icial Neu al Ne wo ks. Elec onics 2023,12, 1345. h ps:// doi.o g/10.3390/elec onics12061345 Academic Edi o : An onio Dou ado Recei ed: 13 Feb ua y 2023 Re ised: 3 Ma ch 2023 Accep ed: 10 Ma ch 2023 Published: 12 Ma ch 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). elec onics Communica ion P e-Design o Mul i-Band Plana An ennas by A i icial Neu al Ne wo ks Mohamed Aziz Lahiani 1, Zbynˇek Raida 2,3 , Jiˇ íVeselý3and Jana Oli o á3,* 1Na ional Ins i u e o Applied Sciences and Technology, Tunis 625 00, Tunisia 2Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, B no-K álo o Pole, 616 00 B no, Czech Republic 3Facul y o Mili a y Technology, Uni e si y o De ence, B no-S ˇ ed, 662 10 B no, Czech Republic *Co espondence: [email p o ec ed] Abs ac : In his communica ion, a i icial neu al ne wo ks a e used o es ima e he ini ial s uc u e o a mul iband plana an enna. The neu al ne wo ks a e ained on a se o selec ed no malized mul iband an ennas cha ac e ized by ime-e icien modal analysis wi h limi ed accu acy. Using he Deep Lea ning Toolbox in Ma lab, se e al ypes o neu al ne wo ks ha e been c ea ed and ained on he sample plana mul iband an ennas. In he neu al ne wo k lea ning p ocess, sui able ne wo k ypes we e selec ed o he design o hese an ennas. The ained ne wo ks, depending on he desi ed ope a ing bands, will selec he app op ia e an enna geome y. This is u he op imized using New on’s me hod in HFSS. The use o he neu al p e-design concep speeds up and simpli ies he design o mul iband plana an ennas. The indings p esen ed in his pape will be used o e ine and accele a e he design o plana mul iband an ennas. Keywo ds: mul i-band an ennas; eed- o wa d neu al ne wo k; cascade- o wa d neu al ne wo k; p obabilis ic neu al ne wo k; ull-wa e analysis 1. In oduc ion When designing a con en ional mul i-band plana an enna, a p ope pa ch geome y has o be selec ed o ob ain esonan equencies in he eques ed ope a ional bands [ 1 ]. In he nex s ep, a ull-wa e nume ical model o he an enna is de eloped in an elec omag- ne ic simula o , and he model is op imized o mee he equi ed pa ame e s o he an enna as accu a ely as possible. A p ope an enna geome y is selec ed by a designe expe ienced wi h i . In his communica ion, we ain a i icial neu al ne wo ks (ANN) o ep esen he designe ’s expe ience. ANNs a e ained on a se o no malized mul i-band an ennas cha ac e ized by modal analysis, which e icien ly p oduces app oxima e aining pa e ns. The modal analysis i sel and he way o c ea ing aining se s a e desc ibed in Sec ion 2.1. When designing a plana an enna on a p esc ibed subs a e o eques ed ope a ion bands, no maliza ion ela ed o he wa eleng h in he subs a e is pe o med, and no mal- ized esonan equencies a e mapped by ANN o he op imum an enna geome y. Fo mapping, h ee ypes o ANN a e used: •Feed- o wa d back-p opaga ion ANN [2]. •Cascade- o wa d back-p opaga ion ANN [3]. •P obabilis ic ANN [4]. In Sec ion 2.2, ANN a e b ie ly in oduced, and he p ocess o hei aining is desc ibed. In Sec ion 3, we p esen a design example o illus a e he unc ionali y o he neu al p e-design. Sec ion 4concludes he pape . In he open li e a u e, se e al pape s on an ANN-enhanced an enna design ha e been published. The pape s co e he ollowing opics: Elec onics 2023,12, 1345. h ps://doi.o g/10.3390/elec onics12061345 h ps://www.mdpi.com/jou nal/elec onics Elec onics 2023,12, 1345 2 o 11 • On-line neu al syn hesis o adia ion pa e ns. In an a icle abou he design o a cogni i e an enna a ay [ 5 ], he adia ion pa e n o a con o mal pa ch a ay has been adap ed o a complex en i onmen by a p ope phasing o elemen s. A used deep ein o cemen lea ning was based on an on-line ne wo k, which upda ed pa ame e s o aining, and a a ge ne wo k, which calcula ed he loss unc ion exploi ing da a om an expe ience pool. In a pape in es iga ing he syn hesis o con o mal phased a ay an enna (PAA) pa e ns using deep syn hesis [ 6 ], he on-line ANN and he a ge ANN c ea ed a andem ne wo k s uc u e ha minimized he di e ence be ween he eques ed pa e n and he cu en one. In he desc ibed app oaches, an enna geome ies a e ixed, and inpu signals a e op imized o each eques ed adia ion pa e ns. In his communica ion, he neu al ne wo k selec s he op imum shape o he pa ch o o m he op imum cu en dis ibu ion ela ed o mul iple esonances in he eques ed ope a ional bands. • Black-box modeling o an enna s uc u es. Compu e p ocessing uni (CPU)- ime mode a e ANNs a e ained o app oxima e he esul s o CPU- ime expansi e ull- wa e analysis o e a limi ed de ini ion space. This app oach can be applied bo h o canonical s uc u es [ 7 ] and ad anced ones. A pa ch an enna wi h a g ound plane de ec ed by spli - ing esona o s was modeled by a mul i-laye pe cep on and op i- mized by a pa icle swa m algo i hm in [ 8 ]. A high-gain quasi-Yagi an enna wi h a pa abolic e lec o was modeled by a py amidal deep eg ession ne wo k in [9]. Fo mul i-physics modeling o mic owa e il e s, he au ho s used a deep hyb id neu al ne wo k [ 10 ]. The ne wo k was concei ed as a cascade o ANNs wi h di e en pa ame e s. Fo esonan s uc u es like il e s, his modeling app oach was shown o be bene icial. Neu al black-box models usually ace he p oblem o su icien gene ali y and accep - able accu acy being eached wi h easonable e o s [ 11 ]. A su icien ly gene al model should co e a su icien ly la ge de ini ion space o an op imum dimension, which is gi en by he numbe o s a e a iables. A su icien ly accu a e model usually equi es ime-expansi e simula ions o aining pa e ns. In o de o o e come hese di icul ies, we p opose an app oxima e classi ie ha maps no malized ope a ion equencies o he op imum layou o an an enna elemen . The de ini ion space is educed by wo king wi h no malized equencies and dimensions, and he CPU ime needed o c ea e aining se s is minimized by using a ime-mode a e modal analysis. I he p e-designed s uc u e is su icien ly accu a e, hen con en ional local op imiza ion can be quick and inexpensi e. • An enna design by ANN and op imize . I he e icien and accu a e black-box model is comple ed by an op imiza ion algo i hm, a simple design ool can be de eloped. In [ 12 ], a pa ch is di ided in o pixels. The shape o he pa ch is syn hesized by combining a con olu ional ANN in he ole o a o wa d model (geome y a he inpu and pe o mance a he ou pu ) and a gene ic algo i hm in he ole o he op imize . The desc ibed app oach can be used e en o a mul i-objec i e design o an ennas. In [ 13 ], a deep neu al model was combined wi h Thomson sampling o e icien mul i- objec i e op imiza ion o e eal he Pa e o on o op imal solu ions. In [ 14 ], plana ul a-wideband an ennas we e designed by combining deep s uc u es ( he cascade o an ex eme lea ning machine, a deep belie ne wo k, and a es ic ed Bol zmann machine) and a pa icle swa m op imiza ion as a global op imize . Since p ope ly ained ANNs can p o ide he esponse e y quickly (only a ew a i hme ic ope a ions a e needed o be execu ed in pa allel), neu al models a e ad an- ageous when combined wi h e olu iona y algo i hms [ 15 , 16 ], swa m-in elligence ap- p oaches [ 17 , 18 ], and o he CPU- ime expensi e global op imize s. Bu he ques ion o he de elopmen o a su icien ly gene al and su icien ly accu a e neu al model emains. The app oach p esen ed in his communica ion is based on neu al classi ie s. A pu e classi ica ion o plana mic owa e il e s was p esen ed in [ 19 ]. A he inpu o a deep Elec onics 2023,12, 1345 3 o 11 ne wo k, a bi map wi h a pho o o he il e was in oduced. ANN classi ied he il e as a low-pass o band-pass one and de e mined he o de o he il e . In [ 20 ], he classi ie conside ed he eques ed gain, impedance bandwid h, and ope a- ion equency and selec ed among a pa ch an enna, a spi al an enna, o a ho n an enna. Then, he selec ed an enna was designed by an in e se neu al model ha mapped e- ques ed an enna pa ame e s o an enna dimensions. An enna geome ies we e gi en, and dimensions we e compu ed. In he p esen ed app oach, he ope a ional equencies o a plana mul iband an enna a e mapped by he ained neu al ne wo k o he op imum no malized geome y o he pa ch. Consecu i ely, he selec ed geome y is modeled in he High F equency S uc u e Simula o (HFSS) and op imized by he New on algo i hm. Thanks o he success ul p e- design, he local op imiza ion is su icien , and a low numbe o i e a ion s eps is equi ed . This p o ides u he accele a ion o he an enna s uc u e design. Acco ding o ou knowledge, he desc ibed exploi a ion o ANN o he an enna p e-design has no been p esen ed in he open li e a u e ye . 2. Me hods We in end o c ea e an e icien ool o he design o plana mul i-band an ennas. In o de o ensu e a su icien design e iciency, we use a classi ie o he selec ion o an op imal no malized an enna geome y, which is he i s s ep. Second, he geome y is deno malized, aking in o accoun he wa eleng h o he used subs a e, and a ull-wa e HFSS model is de eloped. Finally, he model is op imized by he local New on op imize in a ew s eps. In he ollowing pa ag aphs, he indi idual design s eps a e desc ibed in de ail. 2.1. T aining Se s by Modal Analysis The no malized an enna geome ies conside an ai subs a e wi h he dielec ic cons an ε = 1, negligible loss an δ = 0, and a hickness o h= 1 mm. Since he co esponding wa eleng h is λ0 = 300 mm, we c ea e a se o slo ed pa ches wi h he undamen al dimension 150 mm ×150 mm. Geome ies o aining pa e ns a e depic ed in Figu e 1, wi h pixel dimensions o 5 mm × 5 mm. The g een pixels ep esen me alliza ion (a pe ec elec ic conduc o in an app oxima ion, a s anda d coppe oil in an implemen a ion), and he whi e pixels ep esen an unco e ed subs a e. The size o he pa ch is ixed a 30 × 30 pixels excep o he ex ended pa ch an enna (Figu e 1b). The wid h o slo s is ixed a 1 pixel, excep o he squa e slo (Figu e 1i). T aining geome ies a e going o co e he mos impo an mechanisms o exci ing mul i-band beha io : • The con en ional pa ch (Figu e 1a) plays he ole o e e ence. The mul i-band beha io co esponds o he highe modes o he pa ch. The ex ended pa ch an enna (Figu e 1b) can be unde s ood as a con en ional pa ch wi h a capaci i e p olonga ion [21]. • The L-slo an enna (Figu e 1c) ep esen s pa ches wi h a wo-segmen slo ha b eaks he pa ch edge in o wo pa s. The cu en s in bo h sub-a eas o he pa ch a e gal ani- cally connec ed [21]. • The U-slo an enna (Figu e 1d) ep esen s pa ches wi h a h ee-segmen slo ha b eaks he pa ch edge in o wo pa s. The cu en s in bo h sub-a eas o he pa ch a e sepa a ed [21]. • The T-slo an enna (Figu e 1e) can be unde s ood as a olded slo dipole ed by a coplana wa eguide (CPW) [ 22 ]. A magne ic cu en lowing in he slo loop is he sou ce o he adia ion. Elec onics 2023,12, 1345 4 o 11 Elec onics 2023, 12, x FOR PEER REVIEW 4 o 12 • The T-slo an enna (Figu e 1e) can be unde s ood as a olded slo dipole ed by a coplana wa eguide (CPW) [22]. A magne ic cu en lowing in he slo loop is he sou ce o he adia ion. The double-U an enna, he G an enna, and he H an enna a e pa ches wi h slo s in- side he an enna elemen . These slo s di ec ly in luence he cu en dis ibu ion on he pa ch and he mul iband beha io o he pa ch. The double-U an enna (Figu e 1 ) consis s o a la ge adia o ( he whole pa ch) and a small one ( he a ea inside slo s), which de ine wo ope a ional bands [21]. The G an enna (Figu e 1g) di ides he pa ch in o an in e nal a ea and wo ex e nal ones, po en ially o e ing a iple-band ope a ion [23]. In he case o a non-symme ically loca ed non-symme ical H slo (Figu e 1h), mo e han h ee bands can be c ea ed [24]. Finally, a squa e slo (Figu e 1i) c ea es he equi alen o a loop an- enna [25]. (a) (b) (c) (d) (e) ( ) (g) (h) (i) Figu e 1. T aining geome ies o no malized mul i-band pa ches: (a) con en ional pa ch, (b) ex- ended pa ch, (c) L-slo pa ch, (d) U-slo pa ch, (e) T-slo pa ch, ( ) double U-slo pa ch, (g) G-slo pa ch, (h) H-slo pa ch, and (i) squa e-slo pa ch. Pixel dimensions: 5 mm × 5 mm. G een: conduc i e su ace; whi e: unco e ed subs a e. Wi h a ying posi ions and dimensions o slo s, 60 aining layou s we e c ea ed. Each layou was modeled in he pa ial di e en ial equa ion (PDE) ool o MATLAB [26]: • Me allic pa s o he layou we e enclosed by Neumann bounda y condi ions. • The sol e was se o e alua e eigenmodes. Figu e 1. T aining geome ies o no malized mul i-band pa ches: ( a ) con en ional pa ch, ( b ) ex ended pa ch, ( c ) L-slo pa ch, ( d ) U-slo pa ch, ( e ) T-slo pa ch, ( ) double U-slo pa ch, ( g ) G-slo pa ch, ( h) H-slo pa ch, and ( i ) squa e-slo pa ch. Pixel dimensions: 5 mm × 5 mm. G een: conduc i e su ace; whi e: unco e ed subs a e. The double-U an enna, he G an enna, and he H an enna a e pa ches wi h slo s inside he an enna elemen . These slo s di ec ly in luence he cu en dis ibu ion on he pa ch and he mul iband beha io o he pa ch. The double-U an enna (Figu e 1 ) consis s o a la ge adia o ( he whole pa ch) and a small one ( he a ea inside slo s), which de ine wo ope a ional bands [ 21 ]. The G an enna (Figu e 1g) di ides he pa ch in o an in e nal a ea and wo ex e nal ones, po en ially o e ing a iple-band ope a ion [ 23 ]. In he case o a non-symme ically loca ed non-symme ical H slo (Figu e 1h), mo e han h ee bands can be c ea ed [ 24 ]. Finally, a squa e slo (Figu e 1i) c ea es he equi alen o a loop an enna [ 25 ]. Wi h a ying posi ions and dimensions o slo s, 60 aining layou s we e c ea ed. Each layou was modeled in he pa ial di e en ial equa ion (PDE) ool o MATLAB [26]: •Me allic pa s o he layou we e enclosed by Neumann bounda y condi ions. •The sol e was se o e alua e eigenmodes. •Eigen alues we e conside ed wi hin he in e al <0; 5 ×104>. Elec onics 2023,12, 1345 5 o 11 Since eigen alues a e equal o he squa ed wa e numbe , he ope a ional equencies o an ennas can be e alua ed acco ding o [27]: n= c 2π√an(1) whe e n is he n- h esonan equency o he an enna, co esponding o he n- h eigen alue ancompu ed by he PDE ool, and cis he eloci y o ligh in acuum. The nume ical model o he G-slo pa ch is depic ed in Figu e 2. Bounda ies (black lines) a e associa ed wi h he Neumann bounda y condi ion. The dimensions o he pa ch a e ixed. Posi ion and dimensions o he slo a e a ied by changing he coo dina es o polygon e ices A h ough L. Elec onics 2023, 12, x FOR PEER REVIEW 5 o 12 • Eigen alues we e conside ed wi hin he in e al <0; 5 × 10 4 >. Since eigen alues a e equal o he squa ed wa e numbe , he ope a ional equencies o an ennas can be e alua ed acco ding o [27]: 2 nn c a π =, (1) whe e n is he n- h esonan equency o he an enna, co esponding o he n- h eigen- alue a n compu ed by he PDE ool, and c is he eloci y o ligh in acuum. The nume ical model o he G-slo pa ch is depic ed in Figu e 2. Bounda ies (black lines) a e associa ed wi h he Neumann bounda y condi ion. The dimensions o he pa ch a e ixed. Posi ion and dimensions o he slo a e a ied by changing he coo dina es o polygon e ices A h ough L. Figu e 2. Modal analysis o he G-slo pa ch an enna by he ini e-elemen me hod in he PDE ool o MATLAB. The h ee lowes modes compu ed by he PDE ool (Figu e 3) we e used o compose a aining pa e n. O he aining pa e ns we e p epa ed in a simila way. (a) (b) (c) Figu e 3. The lowes 3 modes o he G-slo an enna we e compu ed by he ini e-elemen me hod in he PDE ool in MATLAB. Figu e 2. Modal analysis o he G-slo pa ch an enna by he ini e-elemen me hod in he PDE ool o MATLAB. The h ee lowes modes compu ed by he PDE ool (Figu e 3) we e used o compose a aining pa e n. O he aining pa e ns we e p epa ed in a simila way. Elec onics 2023, 12, x FOR PEER REVIEW 5 o 12 • Eigen alues we e conside ed wi hin he in e al <0; 5 × 10 4 >. Since eigen alues a e equal o he squa ed wa e numbe , he ope a ional equencies o an ennas can be e alua ed acco ding o [27]: 2 nn c a π =, (1) whe e n is he n- h esonan equency o he an enna, co esponding o he n- h eigen- alue a n compu ed by he PDE ool, and c is he eloci y o ligh in acuum. The nume ical model o he G-slo pa ch is depic ed in Figu e 2. Bounda ies (black lines) a e associa ed wi h he Neumann bounda y condi ion. The dimensions o he pa ch a e ixed. Posi ion and dimensions o he slo a e a ied by changing he coo dina es o polygon e ices A h ough L. Figu e 2. Modal analysis o he G-slo pa ch an enna by he ini e-elemen me hod in he PDE ool o MATLAB. The h ee lowes modes compu ed by he PDE ool (Figu e 3) we e used o compose a aining pa e n. O he aining pa e ns we e p epa ed in a simila way. (a) (b) (c) Figu e 3. The lowes 3 modes o he G-slo an enna we e compu ed by he ini e-elemen me hod in he PDE ool in MATLAB. Figu e 3. The lowes 3 modes o he G-slo an enna we e compu ed by he ini e-elemen me hod in he PDE ool in MATLAB. Elec onics 2023,12, 1345 6 o 11 As a esul , we ha e 60 iple s o esonan equencies o 60 pa ch layou s. Neu al ne - wo ks a e ained o map iples o equencies on he inpu o he index o a co esponding pa ch layou on he ou pu . De ails a e gi en in he nex pa ag aph. 2.2. ANN and T aining P ocess We use neu al ne wo ks o map iple s o esonan equencies compu ed by he modal analysis o he index o a co esponding pa ch layou . Hence, he ne wo ks ha e h ee neu ons in he inpu laye and a single neu on in he ou pu laye . The inpu neu ons simply dis ibu e signals o neu ons in he hidden laye . The ou pu neu on collec s signals om neu ons in he hidden laye and p ocesses he signal using he ac i a ion unc ion [ 1 ]. In o de o selec op imal pa ch layou s, we used 3 ANNs om he Deep Lea ning Toolbox o MATLAB [28]: •Feed- o wa d back-p opaga ion ne wo k. Inpu pa e ns a e sequen ially in oduced o inpu neu ons, he ANN esponse is compu ed, and he di e ence (an e o ) be ween he ou pu and he esponse om he aining se is e alua ed. The e o p opaga es back o he inpu and changes he se ings o neu ons o minimize he e o . When implemen ing he ne wo k, we used TRAINLM as a aining unc ion, LEARNGND as a lea ning unc ion, MSE as a pe o mance unc ion, and TANSIG as an ac i a ion unc ion. T aining de ails a e gi en in Figu e 4a. Elec onics 2023, 12, x FOR PEER REVIEW 7 o 12 (a) (b) Figu e 4. T aining neu al classi ie s in he Deep Lea ning Toolbox o MATLAB: (a) eed- o wa d back-p opaga ion ANN; (b) cascaded- o wa d back-p opaga ion ANN. In o de o es he quali y o aining, ou an ennas di e ing om aining pa e ns (Figu e 5) we e c ea ed and analyzed. Co esponding iple s o esonan equencies we e in oduced o he inpu s, and ANN was asked o classi y he op imum layou : • Feed- o wa d ANN succeeded wi h 61.2%/59.3%/41.2%/61.4%; • Cascaded- o wa d ANN succeeded wi h 63.2%/52.3%/40.0%/46.0%; • The p obabilis ic ANN ailed. Since he eed- o wa d back-p opaga ion ne wo k showed he bes pe o mance, we used his classi ie in u he es s. Figu e 4. T aining neu al classi ie s in he Deep Lea ning Toolbox o MATLAB: ( a ) eed- o wa d back-p opaga ion ANN; (b) cascaded- o wa d back-p opaga ion ANN. • A cascade- o wa d back-p opaga ion ne wo k is simila o a eed- o wa d ne wo k, bu includes connec ions om he inpu and e e y p e ious laye o he ollowing laye s. The ne wo k accommoda es he nonlinea ela ionship be ween he inpu and he ou pu bu does no elimina e he linea ela ionship in be ween. Elec onics 2023,12, 1345 7 o 11 When implemen ing he ne wo k, we used TRAINSCG as a aining unc ion, LEARNGND as a lea ning unc ion, MSE as a pe o mance unc ion, and LOGSIG as an ac i a ion unc ion. T aining de ails a e gi en in Figu e 4b. • A p obabilis ic ne wo k con ains adial neu ons wi h a Gaussian ac i a ion unc ion in he hidden laye . The ou pu laye sums con ibu ions o each class o inpu pa e ns, p oducing a ec o o p obabili ies as he ou pu . The ans e unc ion o he ou pu laye picks he maximum o hese p obabili ies and p oduces 1 o he co esponding class. Fo o he classes, 0 is p oduced. When implemen ing he ne wo k, we used NEWPNN o c ea e and ain he ANN. In o de o es he quali y o aining, ou an ennas di e ing om aining pa e ns (Figu e 5) we e c ea ed and analyzed. Co esponding iple s o esonan equencies we e in oduced o he inpu s, and ANN was asked o classi y he op imum layou : •Feed- o wa d ANN succeeded wi h 61.2%/59.3%/41.2%/61.4%; •Cascaded- o wa d ANN succeeded wi h 63.2%/52.3%/40.0%/46.0%; •The p obabilis ic ANN ailed. Elec onics 2023, 12, x FOR PEER REVIEW 8 o 12 (a) (b) (c) (d) Figu e 5. Tes ing he geome ies o no malized mul i-band pa ches: (a) iangle-slo pa ch, (b) ci cu- la slo pa ch, (c) single no ch pa ch, and (d) double no ch pa ch. Pixel dimensions: 5 mm × 5 mm. G een: conduc i e su ace; whi e: unco e ed subs a e. 3. Design Example In o de o demons a e he unc ionali y o he de eloped me hodology, we de- signed a h ee-band an enna co e ing wi eless local a ea ne wo k (WLAN) channels: • 802.11b/g/n/ax: 1 = 2.4 GHz; • 802.11y: 2 = 3.6 GHz; • 802.11j: 3 = 4.9 GHz. The an enna should be designed o he subs a e ARLON 25N wi h ε = 3.38, an δ = 0.0025, and a heigh h = 1.524 mm. Since he whole pa ch is assumed o be in he hal - wa eleng h esonance a 1 = 2.40 GHz, we can e alua e he wa eleng h in he dielec ics acco ding o [27]: d c λ ε = , (2) whe e = 2.4 GHz, c is he eloci y o ligh , and he dielec ic cons an equals o ε = 3.38. Nume ically, λ d = 68 mm, and he scaling ac o ela ed o he neu al model equals o n = λ d /λ 0 = 68/300 = 0.227. In oducing he iple o no malized equencies [2.4/2.4; 3.6/2.4; and 4.9/2.4] o he inpu o he neu al classi ie , a G-slo an enna (Figu e 2) is ecommended as an op imum s uc u e. The size o he pa ch W × L was 150 mm × 150 mm in he neu al model and 34.1 mm × 34.1 mm in he ecompu ed model. The coo dina es o he e ices A h ough L o he polygonal slo (Figu e 2) p oduced by he neu al classi ie a e gi en in Table 1. The dimensions we e ecompu ed using he scale n = 0.227 (Table 1), and a nume ical model in HFSS was de eloped (Figu e 6). Table 1. Coo dina es o e ices in he polygonal G slo in he pa ch depic ed in Figu e 2. Compa i- son o he neu al model, he ecompu ed one, and he op imized one. Neu al Ne wo k Recompu ed Op imized Figu e 5. Tes ing he geome ies o no malized mul i-band pa ches: ( a ) iangle-slo pa ch, ( b ) ci cula slo pa ch, ( c ) single no ch pa ch, and ( d ) double no ch pa ch. Pixel dimensions: 5 mm × 5 mm. G een: conduc i e su ace; whi e: unco e ed subs a e. Since he eed- o wa d back-p opaga ion ne wo k showed he bes pe o mance, we used his classi ie in u he es s. 3. Design Example In o de o demons a e he unc ionali y o he de eloped me hodology, we designed a h ee-band an enna co e ing wi eless local a ea ne wo k (WLAN) channels: •802.11b/g/n/ax: 1= 2.4 GHz; •802.11y: 2= 3.6 GHz; •802.11j: 3= 4.9 GHz. Elec onics 2023,12, 1345 8 o 11 The an enna should be designed o he subs a e ARLON 25N wi h ε = 3.38, an δ= 0.0025 , and a heigh h= 1.524 mm. Since he whole pa ch is assumed o be in he hal - wa eleng h esonance a 1 = 2.40 GHz, we can e alua e he wa eleng h in he dielec ics acco ding o [27]: λd= c √ε (2) whe e = 2.4 GHz, cis he eloci y o ligh , and he dielec ic cons an equals o ε = 3.38. Nume ically, λd = 68 mm, and he scaling ac o ela ed o he neu al model equals o n=λd/λ0= 68/300 = 0.227. In oducing he iple o no malized equencies [2.4/2.4; 3.6/2.4; and 4.9/2.4] o he inpu o he neu al classi ie , a G-slo an enna (Figu e 2) is ecommended as an op imum s uc u e. The size o he pa ch W × L was 150 mm × 150 mm in he neu al model and 34.1 mm ×34.1 mm in he ecompu ed model. The coo dina es o he e ices A h ough L o he polygonal slo (Figu e 2) p oduced by he neu al classi ie a e gi en in Table 1. The dimensions we e ecompu ed using he scale n= 0.227 (Table 1), and a nume ical model in HFSS was de eloped (Figu e 6). Table 1. Coo dina es o e ices in he polygonal G slo in he pa ch depic ed in Figu e 2. Compa ison o he neu al model, he ecompu ed one, and he op imized one. Neu al Ne wo k Recompu ed Op imized A−70.0 25.0 −15.9 5.7 −13.4 4.7 B−60.0 25.0 −13.6 5.7 −11.6 4.7 C−60.0 65.0 −13.6 14.8 −11.6 13.3 D 0.0 65.0 0.0 14.8 0.5 13.3 E 0.0 0.0 0.0 0.0 0.5 −0.5 F−20.0 0.0 −4.5 0.0 −4.0 −0.5 G−20.0 5.0 −4.5 1.1 −4.0 0.6 H−5.0 5.0 −1.1 1.1 −0.6 0.6 I−5.0 60.0 −1.1 13.6 −0.6 12.1 J−55.0 60.0 −12.5 13.6 −10.5 12.1 K−55.0 20.0 −12.5 4.5 −10.5 3.5 L−70.0 20.0 −15.9 4.5 −13.4 3.5 Elec onics 2023, 12, x FOR PEER REVIEW 9 o 12 A −70.0 25.0 −15.9 5.7 −13.4 4.7 B −60.0 25.0 −13.6 5.7 −11.6 4.7 C −60.0 65.0 −13.6 14.8 −11.6 13.3 D 0.0 65.0 0.0 14.8 0.5 13.3 E 0.0 0.0 0.0 0.0 0.5 −0.5 F −20.0 0.0 −4.5 0.0 −4.0 −0.5 G −20.0 5.0 −4.5 1.1 −4.0 0.6 H −5.0 5.0 −1.1 1.1 −0.6 0.6 I −5.0 60.0 −1.1 13.6 −0.6 12.1 J −55.0 60.0 −12.5 13.6 −10.5 12.1 K −55.0 20.0 −12.5 4.5 −10.5 3.5 L −70.0 20.0 −15.9 4.5 −13.4 3.5 In he nume ical model, he pa ch was comple ed by a mic os ip eede and a wa e po (Figu e 6). The an enna was simula ed in he equency ange o 1.5 GHz–5.0 GHz. The impedance cha ac e is ics (Figu e 7a) showed ha esonance equencies a e shi ed. Figu e 6. Nume ical model o he G-slo an enna in HFSS. The pa ch is comple ed by a mic os ip eede and he wa e po . In a consequen s ep, New on op imiza ion was un o shi esonances owa ds he eques ed bands. A e he op imiza ion, pa ch dimensions we e 30 mm × 30 mm. The e ices o he op imized slo polygon a e gi en in Table 1. The impedance cha ac e is ics o he op imized an enna a e depic ed in Figu e 7b. Ob iously: • 1 = 2.4 GHz is shi ed o 2.5 GHz and is no su icien ly deep. • 2 = 3.6 GHz co esponds o a shallow minimum, while he deep one is a 3.5 GHz. • 3 = 4.9 GHz is uned success ully wi h |S11| < −10 dB. Figu e 6. Nume ical model o he G-slo an enna in HFSS. The pa ch is comple ed by a mic os ip eede and he wa e po . In he nume ical model, he pa ch was comple ed by a mic os ip eede and a wa e po (Figu e 6). The an enna was simula ed in he equency ange o 1.5 GHz–5.0 GHz. The impedance cha ac e is ics (Figu e 7a) showed ha esonance equencies a e shi ed. Elec onics 2023,12, 1345 9 o 11 Elec onics 2023, 12, x FOR PEER REVIEW 9 o 11 K −55.0 20.0 −12.5 4.5 −10.5 3.5 L −70.0 20.0 −15.9 4.5 −13.4 3.5 In he nume ical model, he pa ch was comple ed by a mic os ip eede and a wa e po (Figu e 6). The an enna was simula ed in he equency ange o 1.5 GHz–5.0 GHz. The impedance cha ac e is ics (Figu e 7a) showed ha esonance equencies a e shi ed. Figu e 6. Nume ical model o he G-slo an enna in HFSS. The pa ch is comple ed by a mic os ip eede and he wa e po . In a consequen s ep, New on op imiza ion was un o shi esonances owa ds he eques ed bands. A e he op imiza ion, pa ch dimensions we e 30 mm × 30 mm. The e ices o he op imized slo polygon a e gi en in Table 1. The impedance cha ac e is ics o he op imized an enna a e depic ed in Figu e 7b. Ob iously: • 1 = 2.4 GHz is shi ed o 2.5 GHz and is no su icien ly deep. • 2 = 3.6 GHz co esponds o a shallow minimum, while he deep one is a 3.5 GHz. • 3 = 4.9 GHz is uned success ully wi h |S11| < −10 dB. (a) (b) Figu e 7. Impedance cha ac e is ics o he G-slo an enna: (a) ini ial model, (b) op imized model. In a consequen s ep, New on op imiza ion was un o shi esonances owa ds he eques ed bands. A e he op imiza ion, pa ch dimensions we e 30 mm × 30 mm. The e ices o he op imized slo polygon a e gi en in Table 1. The impedance cha ac e is ics o he op imized an enna a e depic ed in Figu e 7b. Ob iously: • 1= 2.4 GHz is shi ed o 2.5 GHz and is no su icien ly deep. • 2= 3.6 GHz co esponds o a shallow minimum, while he deep one is a 3.5 GHz. • 3= 4.9 GHz is uned success ully wi h |S11| < −10 dB. 4. Resul s In his communica ion, we ained a i icial neu al ne wo ks on he app oxima e esul s o modal analysis. Neu al models mapped a iple o he lowes esonan equencies o he co esponding layou o a pa ch. In o de o de elop su icien ly gene al neu al models, an ena s uc u es we e no malized. In o al, 60 aining pa e ns we e c ea ed using 9 an enna layou s. In MATLAB’s Deep Lea ning Toolbox, we c ea ed a eed- o wa d ANN, a cascaded- o wa d ANN, and a p obabilis ic ANN. On ou an enna layou s, which we e no included in he aining se s, he unc ionali y o neu al classi ie s was es ed. Whe eas he p oba- bilis ic ANN ailed, he eed- o wa d ANN showed ela i ely good esul s. Using he eed- o wa d ANN, we ied o design a iple-band an enna co e ing WLAN bands 2.4 GHz, 3.6 GHz, and 4.9 GHz. The neu al model e u ned a G-slo an enna as an op imum s uc u e. The no malized an enna was ecompu ed o he subs a e ARLON 25N, and a co esponding HFSS model was de eloped. Due o he shi o esonan equencies wi h espec o he eques ed ones, he an enna was op imized by he New on algo i hm. The op imized impedance cha ac e is ics a e close o he eques ed ones, bu he ma ch is no pe ec . Ob iously, he concep o mul iband an enna p edesign by neu al classi ie s can wo k, bu much mo e e o has o be de o ed o he composi ion o a la ge and mo e