Properties of Fourier Series - GATE Study Material in PDF

Srfpdrtbds fj Jfurbdr Xdrbds - OKUD Xtuey Lktdrbki ba SEJ Ba tmd prdvbfus krtbhid, wd idkrat tmd Gksbhs fj Jfurbdr Xdrbds, tmd ebjjdrdat typds kae kii kgfut tmd ebjjdrdat Jfurbdr Xdrbds spdhtruls. Afw, idt us tk`d tmd ebshussbfa jurtmdr kae idkra kgfut tmd . Tsbao tmdsd Srfpdrtbds fj Jfurbdr Xdrbds, wd hka idkra tf lkabpuiktd Jfurbdr Xdrbds, wmbhm bs wmkt wd wbii idkra ba tmdsd . Umdsd OKUD Aftds krd usdjui jfr OKUD DD, OKUD DH, OKUD LD, kae OKUD HD. Umdy krd kisf usdjui jfr ftmdr dxkls ib`d GKWH, GXAI, BDX, EWEF, BXWF dth. Rfu hka odt tmdsd OKUD stuey lktdrbki efwaifkede ba SEJ sf tmkt yfur dxkl prdpkrktbfa bs lked dksy kae yfu khd yfur pkpdr. Gdjfrd yfu odt stkrtde tmfuom, yfu smfuie dasurd tmkt yfu krd hkuomt up wbtm tmd gksbhs fj Daobaddrbao Lktmdlktbhs. Ikpikhd Urkasjfrls Iblbts, Hfatbaubty & Ebjjdrdatbkgbib Ebjjdrdatbkgbibty ty Ldka Zkiud Umdfrdls Ebjjdrdatbktbfa Skrtbki Ebjjdrdatbktb E bjjdrdatbktbfa fa Lkxblk kae Lbablk Ldtmfes fj Batdorktbfa & Xtkaekre Batdorkis Zdhtfr Hkihuius Zdhtfr Batdorktbfa Ubld Xboakis & Xboaki Urkasjfrlktb Urkasjfrlktbfa fa Xtkaekre Ubld Xboakis Xboaki Hikssbjbhktbfa Uypds fj Ubld Xystdls Batrfeuhtbfa tf Ibadkr Ubld Bavkrbkat Xystdls

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Srfpdrtbds fj IUB Xystdls Batrfeuhtbfa tf Jfurbdr Xdrbds

Srfpdrtbds fj Hfatbaufus Ubld Jfurbdr Xdrbds (HUJX) Umd vkrbfus prfpdrtbds fj Jfurbdr sdrbds mkvd gdda ibstde dxpikbade gdifw. Gdjfrd ofbao batf tmdl, idt us odt jklbibkr wbtm tmd rdprdsdatktbfa hfavdatbfa.

  Idt xxt  <∗ h d   kae yyt  <∗ E d    Umda x t ↕ h   kae yt ↕ E

Mdrd HUJX stkaes jfr Hfatbaufus tbld Jfurbdr sdrbds kae ha kae Ea krd Jfurbdr sdrbds hfdjjbhbdats. Afw wd wbii iff` batf tmd prfpdrtbds fj Jfurbdr trkasjfrl. Umd prfpdrtbds wd wbii ebshuss krd> Ibadkrbty Ubld Xmbjtbao Jrdqudahy Xmbjtbao Ubld Xhkibao Ubld Bavdrsbfa Ebjjdrdatbktbfa ba Ubld Batdorktbfa ba Ubld Hfavfiutbfa Luitbpibhktbfa Xylldtry prfpdrty Idt us iff` kt tmdsd prfpdrtbds ba edtkbi afw.

;) Ibadkrbty 

BUmda,j xxtkx↕t hgy kaetyy↕t ↕kh EgE  

b.d. Jfurbdr Xdrbds bs k ibadkr fpdrktbfa.

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7) Ubld Xmbjtbao

Bj xxt ↕ h xxtt  tt ↕↕ dd∗hh  

Umda khhfrebao tf tbld smbjtbao prfpdrty,

b.d. Lkoabtued fj Jfurbdr Xdrbds hfdjjbhbdats rdlkbas uahmkaode wmda tmd sboaki bs smbjtde ba tbld.

8) Jrdqudahy Xmbjtbao

Bj xxt ↕ h dd∗xxtt↕↕ hh∗ +

Umda khhfrebao tf jrdqudahy smbjtbao prfpdrty,

5) Ubld Xhkibao Bj x(t) bs pdrbfebh pdrbfebh wbtm pdrbfe U tmda x(kt) wbii gd pdrbfebh wbtm pdrbfe U/k U/k 0 k13

Bj xt ↕ h xkt ↕ h hflpfadats hmkaods jrfl ς  tf kς fr jrfl   tf  Umda

Umus, kjtdr tbld shkibao JX hfdjjbhbdats krd tmd skld. Gut, tmd spkhbao gdtwdda tmd

jrdqudahy

4) Ubld Bavdrsbfa Ubld bavdrsbfa prfpdrty stktds tmkt

Bj xt ↕ h xt ↕ h∟    Khhfreb Khhfre b a o t f t m b s prfpdr t y , b j x t ↕ h  Umda  xt  ↕ naςh   h Umda

:) Ebjjdrdatbktbfa ba Ubld

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?) Batdorktbfa

BUmdaj xt∠↕xteth↕  h  h    Bj xxt ↕ h  kae yyt ↕ E xt ∟yt ↕ U h E =) Hfavfiutbfa

Umda

Mdahd, tmd hfavfiutbfa ba tbld eflkba idkes tf luitbpibhktbfa fj Jfurbdr sdrbds hfdjjbhbdats ba Jfurbdr sdrbds eflkba.

2) Luitbpibhktbfa ba Ubld Eflkba Bj sboakis krd luitbpibde ba b a tmd tbld eflkba, tmda tmd jfiifwbao pmdafldafa mkppdas

Jfr xt ↕ h  kae yt ↕ E xtyt ↕ U. h ∟ E

_d mkvd

Luitbpibhktbfa ba tbld eflkba idkes tf hfavfiutbfa ba Jfurbdr sdrbds eflkba.

;3) Xylldtry

BBjj xtxt bbss rdkiblkobtmakry daakry↗tmdah ↗hh∗∟  h∗∟ Xylldtry prfpdrtbds stktd tmkt

Jbae tmd Jfurbdr sdrbds fj jfiifwbao sboaki -

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xt  ∛  d K  4, U 5, kae ς     xt  ∛<∗  d Jbae tmd Jfurbdr sdrbds fj jfiifwbao sboaki -

xxt x↕t  ;       ∗ xt  xt  ;  ↕∗⁅ d∗ß⁅   d ⁅ xt  Ψ<∗  d d  Jbae tmd Jfurbdr sdrbds fj jfiifwbao sboaki -

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Wdhkii,

DJXMdrdfj, Kpdrb 4f,ebeυbhtr5kbk,baU fj pui: sds ↗  sbah    sbah  d  ∛<∗  ß sbahß  d  x8t  ∛<∗      x8t  8 ∛<∗ sbah 8  d 

xxtt ↕ x8t sba;h  a 8 8  8   ß xt  x8t  ;↕ 8 sbah8 a.d  PJXHJXH\  8 sbah 8 a. d      . d  xt  8 ∛<∗ sb a h ad a d 8            xt  8 ∛<∗ sbah8 ada d :|Skod

Klpibtueds fj Jfurbdr hfdjjbhbdats edpdae fa tmd edrbvktbvds fj tmd sboaki. Bj tmd ;st edrbvktbvd fj tmd sboaki tdaes tf bajbabtd (b.d. ;st  edrbvktbvd fj tmd sboaki bs ka blpuisd) tmda Jfurbdr hfdjjbhbdats krd prfpfrtbfaki tf ;/a. Bj tmd 7ae edrbvktbvd fj tmd sboaki tdaes tf bajbabtd (b.d. 7ae edrbvktbvd bs ka blpuisd) tmda

Jfurbdr hfdjjbhbdats krd prfpfrtbfaki tf  .Bj ltm edrbvktbvd bs k blpuisd tmda Jfurbdr hfdjjbhbdat krd prfpfrtbfaki tf a; Jbae UJX fj tmd obvda pdrbfebh sboakis.

 .t 039t  039 t 9 xt  7K;  0  9t9U Xbahd, x(t) bs ka dvda juahtbfa Umdrdjfrd, ga < 3

  j εt  ↗ ∑ b . d ⇗Jfurb dr hfdjjbhbdats∜     k   ∠ xtet   ∠  t et   ∠⁅ 7K ; ;   et ⁅               t  ⁅ ?|Skod

         k  ß=   U   U             ß 8 U k     ß =    k  ∠ xthfsaςtet   ∠ ⁅  thfsaςtet   ∠⁅ 7K(;  t⁅U) hfsaςtet   ∠ ⁅ thfs thfsaςtetet   ∠⁅ hfshfsaςtetet   ∠⁅ t hfshfsaςtet ⁅       Phfshfsaςt  aςt sbaaςt\\   |⁅    hfsaςt  aςt.sbaaςt⁅    k   hfs(aς U⁅)hfs3³ 7 U  sba(aς U⁅) 7U    hfsaςU hfs(aς ⁅7)   sba(aς ⁅)aς UsbaaςU 7 ςU7σ k∗    hfsaσ ;aσsbaaσ}   hfs7aσ hfsaσ aσsbaaσ7aσ sba7aσ }    hfsaσ; hfsaσ;    ;hfsaσ ;hfsaσ ⇗ k   7hfsaσ7 ςU7σ   ß 7hfsaσ;      0 jfr vkifeeudvkifjuad fj a k   hfsaσ  ;   30jfKr dvda xt    ∛<  hfsaσ  ;hfsaςt =|Skod

_d wbii hfatbaud wbtm tmd Xylldtry Hfaebtbfas bas Jfurbdr sdrbds ba tmd adxt krtbhid. Ebe  yfu ib`d tmbs krtbhid fa fa Srfpdrtbds fj Jfurbdr Jfurbdr Xdrbds6 Idt us `afw `afw ba tmd hflldats. Rfu lky kisf danfy ‘ 

Xylldtry Hfaebtbfas ba Jfurbdr Xdrbds Jfurbdr Urkasjfrl Srfpdrtbds fj Jfurbdr Urkasjfrl

2|Skod