fourier transform

7
 Frnu trantbrrt t'53tolu +frurrreT rnrufr'prnt r.irron {rrn&icrr cD ooftpd or allt rn nr.ints'ttat lru Sh{61/L,pl raffirnrf FCX) n a s& &rndtron€f crrwd Vo&rqto\"q 6wtr b.{ -o fn,tfu,l.trr 4 P{.Ut=ff(01 = t v v / tltfe rha unckior _trl orl*a fu\nva{9-touuor TomNuen ('0 g) qrd ,h rr:vntenal o..rl = {ttO} ft ndBn&rl rY,e"cu fqtcp)\ =kgtn-?tP)AP . : t'lo to*niqr rutrcrrufonrnl f 01ct) nruo+ &filaA ancl o-a.n&el D e-{ocil} C$s(O $c'ro l.l coft{}ponil g inUa,r[ $"r"iol girrg fcrrr$cem n ?i{nn ry, qtx) , I *hltur,o-u'ex u'nvu-tfa)r}&fl'}n -t c'tt$ ot-'r I'x L a$na.tanzl&nn6ted , po oar,.,t fl.tP) q"G.* Hy - ltero frro crry€)t^eK nV+n:o atrnter {1'utron'q{rn flrw b? , ct .r.) ,. it.colrl + \ " Z"CSv*dp  ,, J d # futahim atulerrr ,ouru'q{ ,rd Lnp\qeq rq?s tct -ltqu o / &(L) t6,(tPt't) > o t o t t o P qctct\ f ** oL6ri- = -L,:. ^(+)drr l: ::- R) r = 1|;trt_, Ak-* f nitt*{+) )tt/r

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  • Frnu trantbrrtt'53tolu

    +frurrreT trnrufr'prnt 6r.irron a{rrn&icrr FcD to toftpd tor all t rn tnr.ints'ttat 4lrut(Sh{61/L,pl traffirnrf FCX) tn a s& &rndtronf 4crrwd Vo&rqto\"q P 6wtrb.{ -o fn,tfu,l.trr"4 P{.Ut=ff(01 =t vv/

    tltfe rha funckior &_trl uorl*a fu\nva{9-touuor {TomNuen o('0 [g) qrd|.

    ,h rr:vntenal o..rl = {ttO}

    ft ndBn&rl ,rY,e"cu = fqtcp)\ =kgtn-?tP)AP. : t'lo to*niqr drutrcrrufonrnl 6f 01ct) anruo+ h&filaA ancl o-a.n&el oD

    e-{ocil} = C$s(O$c'ro tl.l coft{}ponil ng inUa,r[ $"r"iol girrg tfcrrr$cem rn ?i{nn ry,qtx)

    , I *hltur,o-u'ex cu'nvu-tfa)r}&fl'}n r-t c'tt$ ot-'r I'x L4a$na.tanzl&nn6ted d, po

    oar,.,t =fl.tP) = q"G.* Hy-ltero frro ecrry)t^eK inV+n:o jatrnter {1'utron'q{rn ' flrw b?,

    ct c.r.) = ,. it.colrl = + \" Z"CSv*dpt , , J d# futahim batulerrr f,ouru'q{ a,rd Lnp\qeq trq?s

    tct

    -ltqu

    o/

    &(L) = t6,(tPt't) t>ot o tto

    P qctct\ -f ** oL6ri-=

    -L,:. ^(+)drr . l: ::- R) dr

    = 1|;trt_, .Ak-* f nitt*{+) a)tt/r

  • = Jf a tcr- ir), *Jt= If' n stectldr= f (s)

    *'* Pt(ott\ = fCs;rrtr'en e-p

    af-tkl ecrnuolutr,cnFCx) Ldpx) te-

    1-'H-S

    t-t, x-[p= g

    xxY) : WE6sryat trqnl{.rfvl4 FQt9 CoC$ fi ftqprnJueh rt trr-fu1iqy-11nnbf"r'n9d

    P'tPo0taGt = PtFc^\ r\ c.rr]t*fu

    F{ roo )recr)}=Ft flBgeg,{).tu}=

    frn,ut 5ar ccu 6 (t(r_rg &.tx P,rt / x-u =y*: .ctlt = d.U=go

    J a, J-^*

    = f.n[rcu*g .fr (rD &(u) rlu du= t nt.u Fundo J:ofPv 66v) ,tu= F"[ eeol e..pJrc" w\

    L'+l s evo*d= FqFl q,-irlfr ere ATv, FiT-, Filr m t+rroFR;{s ytR cWnn

  • .Ln

    lr&t!x

    o

    7l

    * {d* =+la

    4

    g

    ef x. -Ldxa

    + $r.oT^"Jft,rtr'ar +flecqfar =j- [11*oatu(ryu*Mr!ne

    f MbAp= an=& {

    v ' t

    ,&\

    9n

    + find L|"t 'fturu"qlt +rr"riltoc"wr dnq *unetian %weh \ryFco 1+ _e L\Lerio ottq,lul:e.

    joil,'-rn

    \Ag."1 _ f u0$ ptrt d.l*{''JlF

    -;-;* o l nffa, psldy'+ ( "*^Fcr) dx)x P !4- -er , ' (

    , - lbL^. < .

    a afne- tPt )trh P .

    c.11\s qq Fr{IR r{t) qFrq F+{ FTt} TcqcEil q.r{,eFFRr c,nTr+ qTRTF rfurnwgl ft,1 sr* qim"HEfsts qiqfr

  • * sotuq- % = rffi {{ u.o,t) : o s u(x, ol"qo, tzo bY fuwref trar8lr,?)n '

    Io\\, -:y--il ttarJQ, # = a$*",-e 1)o, t>0,

    - AT"X> 6 u(i, +) batn&d uIM,'za

    utth ke beunl,nTy octtd-ihrruu(qU =0 U(]{,t) h uouyrdad' --O

    and th-r \nihol esn&hcrr U q10) = -X x>o [email protected]'-*__-,---,-

    a 7W(P,t)- Apu us cPoU

    0'0ztu, e?,0

    Talrrruc( nrl {uutrot dng +fqnl{nrm dn vatan\o-t crf @CIu.re ?gtn* a r ,

    "d -vJ, fifttmrdl = * L #'finPxrrd'

    + ft J'u un,tl finpl,.tr= A [fi0*#1;-(+ pcdspx 4r* Sur (?,t) = c -p uor,t) , ^ rl!.-4 f flu*,u pvhnrudl+fr Lq (p,u - c -Lc,-, Lt(c, t! -r.pv u^cp,g [*rtt* L{' t0 u =f*n'ff

    A Ln UlCP,t) =- Aflt*hc4 t{!,rE,t)J = r.et\ \-- '/

    "trft -DI - uatp\$ =?uthnq t 'o in@ Lra ?d.14(0,0) > e -@

    Notp *aurrr-fi fu'Qo-uguorr gitt'troettry$.-r,h an b6!h grdol C @ft@*fft, 0 0

    Jo u c^l(,o) finqrAr = (erfinnx 't.)t -r ol o, 1r4(P,c) = +rslpx-P6EIP)Ol = J---@; ')o ?'+.1fl e-:&, &t^offlsri) u,, Cpr+) -Oh, a ecvt [-ft"_ q *-@

    K

  • Noutayrrl6 inrara -frwoaatrnqf5omg. 1,rn cn i,alh ri&-r 6t S *ffL

    $npx41o# a{vu Q$| = .vUor ZT .eo, +> soajert t ttrqdn&h'nL u(',O =o,ffittooc) = 4t 'glt u .^,,t) a b4uh^aa'd

    We hcNQ, at * huuifi- = ffi, a>o, f >o .___0U itt t\c bnrndrry cc'n&hfu-

    U(d,S = o --_6ard thr rnih'ctl cin&'hcyf Lt(),6 \ .-- ! L o L;^L L1c K),. I --rc

    th^fr {oqizla{ nru rrq}t(cy,.r dn h.h ndEl df, .,uJ"ttL(4 a[ 6.o-r, -0, ,ora vt Srnpr dr. -- f )t'

    t fiv frnPr+x+ & fuu,+t $inpr.{x : frivrrx$li

    .(*+ ftt$(P,D = o F,t,r) r o*!i'-{J: '' hnPdti1CI,t)9tn?xx[-- ?q(U -il

    - 1/ T

    - f ' t{EU,+).e 2ur! [ p,D

    -itimD =-P'?t4 tT\ I,6 -_ -?ti + 1,1e

    "

    ' ulwlc t it ---@

    {wthrr4 t = o in Gt *o K}r Vl(?,o) = e ._@

    NUrn t"t^*6 tt9 truruot h'na- fruryl{rtpn 6n bah h1,, 6f @@ qtbtl ,r (^x,d) Ernpxdtt'o

    t r n O t l \4 W\ f r7cY F

    )(

    L_erlfy

    .dtt = \: i fryr?xdy "fo&npxd-tr#"lj ='[ry- fr=

  • 2,, c= *-W G*^dlicro 4ron(O -w"tt b

    rrrtf,t) = f.afl+ -@f.{nia rauth"f irr^rgrgz tot'dot hbt' drrqlgftmrn dI bfu dAei 4 'utrzrfl

    (r ,:9s! ait,p fr1,rtr.U)u -T:uCx ,O =h,/r'(JCD

    F _[: r'B' r-.

    rFf . 4AG-

    4-,IIr'=L: t

    ,:8.5,G.! :T,F.:1t. (;'

    ,tg,q !,:

    &t s+4q 9 I& ifrdaX

    :]' 9 4 a t6 1 Q l n a +.4 _.^r g APYx it a l ,4d!44'i&na;ae4I $ieA i 3 g@ a 4 6q Z Y

    I,'J1T A AA^ G{A axk,^ai,as "nf ;3,. " ' . . l 5 = c |

    *a 3 a,{s ++# e. fl"#;sf, + g,e$=1dl I !;B*fi:"dfi 34# 3!+ 'F +i3+'s iH' s;:At ;fr'+id F 1, I arif -^ q $,r 3q#r ffiA ?tr:{136$f

    e. i$* 6nd':t 3fr|;'i fl,;5 i_dlF'! ilg4 s{ii ;ai"E f f,d aA 4 ?i L***F l- y$:d * a# d*l*;t a!;+ A nA $$il*: {ll

    il+d'3a g##$#' pa a *nfli $a *efl {{i g; a !i11+ $is$n$ !##ryi^ae 3;;ee3 1d

    A$i; i$le t3 +&ii- $i ll-l fl,g; $14 #a";i6 g3r#g a#;*; A r

    'ff1f,46iaif,d a XiA'$3{i4"fi3**egilaJ'e{xg:#gA*si

    #' ^,4ai-?A'

    i{j$

    6^41

    a