cRajler 7 PFT ~-/fer 8JJ

Cf}.,) Tr~J AM If;"~.e rf.

7.1

In S€iJert?f tiff /tcJ/""J / we hVd ~

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pf ~"" f 8~J () ;. h (A...tf.e-"., 6/e Ii: ~

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() f era II ,~ ..r t)vL.R Cd!f Pil a-., ~ f, J" ( ~

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(i?lMt'J J"I!~J. -Iv fN) S'-j"J.J, re.rt.J1ve

tJ..J" 1ot:J~ V;; t6 p//btJ"cJ T'(Jure11

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j 1./._.1 _1- 1 I~. 1

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J I . I ~ II ~ J .

th -h- ~ n. i -fIr ~ -e rv ( J':I" tPM -J - t£v,. - a., '4.1i--- )

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II r-//f~r B~

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~ Ju.v». b~ II of b tl41dJ J tJJ'tA -biLt ~ I~ OJ

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a t.. iu - 4; <t : 1 i1ff ff2 '3 Id4t "I J' fJr iI01 0 r -Iv

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~ ihl 0 ~ r h fM>J.i th Wh CRf t of

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rn 0 rk. I~ ~i..1rr . Tiu fj PcJ ,J -1-0 UiA I,'T.Jl

a /uvh." J" Ilof dl'J i-l-cJ .rId" J.r (0,' 'J~" J

UJh1 ;";J I ffo '" oIff~ /AJe...r) I), t. 8?V<

d/J VVlt€ ~ -l!'~ J/JheJ! ~ t e ~Ci)y,J In ,'II./! J ~

~ ~oI/WM , /J ~ r, e Ol; £}e r ~ J ~ tJ2..z

S"Jn J J~

f' -€f 0v1- J .e Ja d ro J :e d ~

~ UJ/t-y~'.rF () 11d,"'1 7J

l..1Je r J'

D

II

:I F. /f~r g~J

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I-IJ(t..))

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flu prob!'

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O(IJ ~IO arJ

wi~ OJrJ ~.l]

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of ~~arA~

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~~ f',,)lf

b ~ oI..It~/ .(J.Itd IIdIvA

I I1"-1(0) S'v. cR ~

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0.J.rr~- ",M

f' If-t; ba,.~f

QJ.:r~J tv

a GD"'1 h, 0 '" t ro -fo ry f-e.;; lie r

1-/ (CJ - .2.7T~)/'1 .}

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tLa f~6~d

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b;) ~h I 'f-L-'J

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;-L +t~rf~ fu.<P.o. ()f. ~CUR l' tf6.Je

f//f-erJlcti b-e .eCU'~ d.£feJ-"',vol 6;

1 . ~.".~~ \j-

I-/a. ~I~)::: H (e M ~ )~IT'G II

II

-I

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II ' ~Tr'

~.

1

II -J71} \1 - e~i - ,

TJu de~'1hll v.: I/oli/ef ~ ~<JfM.r f 110 . D,.

th. ro-fp+j?~ a., J f:h ~ tv-/: of f/lf-erJ

J2. _ 0 J.. 1'1-1T<"- /-/ /

c~ h-e 'f I~ h,.q" -t~ J 114; 'J ~ If v:~ ~II FTII

6; ~

M - F 6 I-'1

/ E~ lMM P /-el: < 'c! ".- '. .

~Lf (w) =11-1,W) ~ I--/Hp (t.)) =c Hi W~' -zH z ~ /«1

::: Hew-IT)

Q.-bt01d flfe~ baJ.

I (h t ~Ii-JLP (~ UO[.]

L \~(1--J v, cd ". h'5h P~f$ ~"'p'

for}lh

low I' oJ} C:i)h-()

TLJ ./

11 PO

II-pll (~)" H0) '" EkJ h [,,] 2-"II

H*' IJ Yl

H p:: (i) ~ (- "t) ~) (-I) h [h J t..-IJ ~-I.>Q

U.r'"3 '* Ilfo/~f~aJe d.eC-o#trf,"';"O"> We hal/€.J)o

I I ~h

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n LP(i):: 5", [21>, J i! J-,;,~ f} ['u" -I] i-

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~

2: f-I (~J)

/

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oJ II iJ<q. IA) eS; '" f /d htt II-e -h, 1-'" f I~,::I"

St.<llbft /f-er s EfIJ(z ~ ) R

~~tJr

;;-/ (? 01) S'.e f

-e.r-ft:\t hJ 4

~J .rIAb'/Y~ ~ ~tJ.r

J>o

HHf () = Tk> h [2",J Z-2:- L h [2h1-1 J .z

-2111+1

hi =-Po,

-----...r-

E (12);L

i £"-,(i )

[ HH

-/ Ep> (]2)

I -/ lC -I(2;z)

Po

E\6 (~-') : L h[n.] 2->'"»0. )

. "1 -= - bot:>o

1:--- } (~ :J) =- Z h [1 h1 -J] Z-2 J?1

h1 '::.- Po

wJ<w

c#v,J

Th/s CCM,I

Eji 0') + r f-J (~o)

€ rh-f/ r1~ J ~J' ;; II()f.1)J :

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~h1pIV~i1+C1-1/iJ 1

('

!)J

+~-~ VI Cn]

fo~ pI, ~~ I, /-f.erJ',eM= 2)

r- - .I .= X Cl>h1J) ! "" .

J LfI

E91 (~~) ::

'-'0

L h [2 hi] 2 -.2111

) .2

Ej (2 ) =-

.Jfh =-tJo

/)tJ -2h1

~ h [2n,+J] i J

"" :;- -lxJ

/-1 (?) ::1>0

h-.2h1 ~ }hJ-J

[JWlJ 2 + ~ h [.2~+1 J l:

Po

h [II J i.-fJ

h1::-t,.o

-I J

Z Ej{il)

/ -/ ,/

/-hp(Z) 1= II Hr?) = f% (22) -I- i! £, (?2)

/ I H (/.2 -I I

/iHp(OI= II -?) = EIJ (? ) - i E) (?')

~Jw. J ,/

H-Jh,~ ;I

1/ /f~r

Th roe..!

~f

fr~.J' f~

-

-I ~

E~ (t) +

-,

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IwDavl

, pll /.f!/m...em .ffA+, , () h ,

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~ [11 J

c.f2AJ / c..R. J of p7) 17/ pkJ'f2

( cv... b-e j ~ ycJ,- 2 (J d 1;, ~ 6th.)

f' ! rlA.R ,) ba-.. 01 f aJ'f {'lI ~ r f vJ 11.

(VJ

c.:P, 'i) It .r

1-4 (~):: I-J ( ~k t) ..)

~--~~--- - --

Ar~ - () I ~ . M... J,... - .,) -> -

h'rJt~1 ~~ ('Gt?, ck.G;:;h?foJ'(? /-!(~J

,' fJ pi0 !~I?h.:N u,;" f (j "...,,:h- :t11-1 l)

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/-11

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E~.e (i M) =L

m:::-lx1

~h

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.,

. AH~ (i)::: 1-1 (Wf1 i)

, ~/ )( ~.Q - ,111 il'1- L- ~ WM E:~ (~ WM )

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Ii ~Ji M

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Q. h1~/

f> lie r I-/~ (~) c~ b< w~/'-IfB/Jt i/!.;...t

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cJ)J ;;,/ f.erJJ

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F()~;'-e r Md~/- ~

Ii c~ bvh, f I~-t j aJ

Iv flow f ..

f)( [ i) II] --.

I~ -~I~II~ t>FTI

f

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t1 li ( I I. I

lr' E. (t:)? l )

Hi (ii) I =

t IN/1 -- ' ,W/'1"-/I it-I (t )

I \'

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l ,j

1-1('1-/(i) III

I 1'4-1 I,.}.({1-1) j'

L 11' 9 -I t (i J11 - -M+/

~

'>([n]

TCf2

cp

~~ (if )

Hii, (i\1)-

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IJ

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w

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DFT

..)

n::l>/ ".1'1-1)( ... ,..j ,/

..~ Ut) [II]

...

"

.) VI [n ]

~ U [h]11-1~

rnf I~ :

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~r fA) cuJf -to ck CA>'" f ~J,€ tt .J; dh J/n + M= tf- frJll'-' '(j CD J>, f iJ ~.r .

II

f\h " J C ~ be. oLv >tR b'J J<-v () S :',J

iTTot F Ifo..r w) '-tfl ft-" 1u;" ';/ \ rL"( () Me.

~ Hj. (w)" H (t.)- ~f) > :fOl-1k 6.) I.) 2./ 3 .

II, Jlo~?aP ro~-Iyf-€ ~-/fer C'a»z.

vi...t.l!r o(! r-€..? 0 " J.e ( l1.AJ " ctUt J,Ji ./

J' r 7r n..]S"t" -;;;::- fA) CII] )

7Th

h [I) J .-

wen] /.f tle '¥ 1)11) r /J 1:- tJ J--. ek <J

j

f,ILr P"~ t~aJe c()h,fiJ~J

1'- I/'IYI, /, (J tf6? -f::v f4"J f (?J- f. J, nr- 4Jl

A l4j)-~ J z

S't"fJ [~( 4-Jl-~ ) J 11 -4-P.. w[49-r<J-2

rr (4-»- ~)

~ -6 . f v<{.r ~ tIbc f~f.ed

fi Hen b~J W, 11 J"u/fqJ

U;h dif,'b 11 .¥'J-- t~fedf Nconrtrucf);~" ,'.f tlltifr

th. J'wmll olv JP 8'hi;J-r ()J ~ flfe~. b~

~ I'd. ~J ¥'I tJhffi' J. sidn J ')( [;] .

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II

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TIu. C{)"V;-/' L Iv ;- p~ f J r~ U11 s Iru ci-,IJ"

ca" b.{!.

CttOVf!

S'v. eX a

I, Ih

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M

Jd; sf; '-'CY tI0c t~ {J N CD.A. did"

6 e rJJ!.t J a h ('1- band;; 1ft?r.

i d er h IN ;tb h I/o l.J'~ g. Pro /H-'" h-~J .

J?6UJVt,(f clo rn~ :

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d. If]

II

111A~ dgh,~ :II

h dMp = ;} Sf.] .

proof' 'I II

I '

I t' f11 -I

,h f,n J ==1 - DTF T { r 1-1 ( w -I ~==()

'JIT-A--I \.1 -11

e /11 h [t1]--

-

7/)]

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~!I h

1,

[n] -= IDTFT {/-I (/.J - 2';'~) J'2rr~J-n

h:: e 1'1 en ]

s:'i It Cfl..

~

/ , f'v/-I

[h) ~ IDTFT { L..g,-=. ()

:: IDTFT{ I}

= ~ [n ]II

h II (M h J -=I I /n h en] -:

1-1 (fA) - ~~~)J

-L ten]M

3 In tI&.~ -i;dorn~ "

! XI J- I

Ep (z) = . ~ M,f[ ~ J - H~M -I M

H w:: &; (~ ) +- i 5 (r ) +II

-(M-I)

.. \ I + £: E ( 2111 )/1-/

pt()~ f : /XJ

E.J (t) ==j, [h M1-i J z- ~

0= D ,.,. /\A-I.A ~..I ./ f rf

TL.r /

(\1-1~ -..R M

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IPMJ 1'/-1..>- v i-tit bf7/h dwlJ~

tVYt (V)- b~ d ~. / f~ / b.e c tNUJ:e

[0 - /:; (?!!) ] -:: /I' fI1

Ex~/! :

c&.+ fVJ= 4.

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~OJe~ /::7R Ii /f~ fA);-t!<.- ~1Mt.r f~

f-III jl ,- 2 - :J - S!(i r::: O. l J~ 1- i'l + :f. 2 Z - 2, J' z + /.s i

-f, ~? -10-I-i -~ + r

Exle :

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W -.2Lc. - (VJ I

M-VA,

-

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1.

II

If (J) 1II

1-1 (-z)I

v'fA)~::: e

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!- 'I - .J7T. It"" Ii f II H (I,) -' f J + H (t-,) - 7T J+ H (4J - 7" )

l) II. -v "v..;Ie $' {! h v' 17~ ..; J.;!~J- , "h [~] e [,,1"+ e ~ e ~ e

- h [8:] e-,;i-J~ I - e27T~

'if'

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4- X O,~S :: I

h J] Ii I c.

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II

f:j5 (~) 11'0

E ~ II

-, ('J) II~

f~ (~) ~

d.

3

EJ (~)I

::

Z { h f 4-h ) J -: Jr!t {h «[.O~I]J :-J.+ ", ;?_I

Z [h [4_+>J) = d.l + 2>-1 +- i>

Z {J, [ </.0+n ] :- - ,<,g. - z - I

J~ 4eo (z ) :-

I

[I (~4-) = / if I,S- ;Z-4

4- II - 4- -8E1 (t ):: 3 ' :: 2 + -2 J--2

11 1-

EJ (i <F-) ":= ,[

- 211 8 - ~-, II

II

II

-' -l f -I

H(z ) ~ (), 'l i.f t if]. I. z - 2.j i- -I- / I S ZJ; I

') ,

~

- JI. M~ ~ ' [; (~ )

12 ::: '

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7.3 (YJ'

,'

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-I II () I. I~ rA / t'l I~'" I p'.&?( € y-J

.I" tR..e ~~ '-Ih~;j cb","h,aJ~J CM~ ~ -I:h C¥hJy.r;.r

C0Jd SJJ1 t.

i 1.

I"J M-fiAJ ..kJ ck Mt dt><?) ~{}V-vla1PJtff~ r-Je a J cie D/J-R J b~,.J:

II r

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II .:

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J11cux; IfJcJ2Pv; cb. ~'/11J-f J ' f-; J1~ b~ .r ~. . -tl II

FJ 1 . ~ {JFT tM.~r.:r ;,y}<.)~~1: ~O-<~"C

; Ve-,.. s s '" c1 tJ cd-

1-11 ~t.J)" f--f «J- 2;:) ~= 0> 1/ ,., M-i

t'.delUivJ~ Co !h ~~brf S'hLVh :~0,"

tJlR ~J"f~ fh Jl,'O ~ I 'J" w~J~-f.f~~s

~ a)::- £? (iM) -I-? E-t (lit) + .. + ?IIT~T'/ .,

~ L£,d~~j1 )/ I~ Vo[it]

~ ~~-} I~!!J}

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;J 1.1[it J

DFT

\

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E-}~} -J Lh.., [17)---

FJ2.

Va [ill -:

VM-I(>1J

Fcu2J.

XD1=TII

II DFTII

If+ fII

S"1 ~ f:t.r; Or vJ <J 0 I> k t.J ,. tJ. t£,\

~t{v< (0) ::

(J (fA) - ? rrl<)11 ,;

11 - 0 J'vI!-"7f- --I"""'" 4

IS II ef" f vaAl 1;.;, -th h.£.t J.,) 0 ~ k S' he, «> hII J 11- J. .~ f.Jl .~ /1

tJUt Tr~J! Jl" Fh r7i'" it I of l"JI'-,'rre,,1IJ

b

.. {lip.. _I l'1li. -(1'1

fll:z):' ~ ( ~ ) f ~ h (~ ) +,.. + ~ i?r,~f

@J -.,) ~ ~j-{~J~{~!~..

~~~f1EJ-4C~.

J [h)

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a J' S'h.;yt,JJfb~w ~0&./1} f./h Ia. ~ )

X[~JI Ii

~J~ ~

~..1 ) r:iMl!" L-4 ~

Iw

1Io [~] =- /)( [Y} N ]

~ (/1] = ~ (YJ11 + I]

II.

F~4.

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'Xc. J -;{ s/p l~~ f.q,r It? U;' ~J' t;{ ~ b./h of

I

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0.'" d

fr~,-".,Jt,

II

II

II r-;Nlu,~] H!:!J

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ff [iJ l1- ~ J:- 14 [n J f

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MO/X,' ht~ j)RA.;,-Jn J oR cI f)FT h' /f~ /l~.r

, .,? ,"vJ. "',,;

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-erAcf(, J{~ btJ 1'1/

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