Claire Schmidt53

IoT de Cachan-

GEiin

DEVOIR DE PREPARATION

AU DS N-

3

SERIES DE FOURIER & TRANSFORMATION ENZ

Elements de Correction

Exercise A

① # = Z-

^

z÷,

6"

'

Ulm - r ) si I #a 6

d 'apri , lreeemple usual b

"

Ulm I,

I si Izl > IblZ -

b

et le theorem e du retard :

am - mU ( m - m ) £ , z

- mA I z )

② an = IF Ulm ) ACH -

- II.I ,

unten -- n¥Cz =

e¥ the 4't

on a utilise ' le develop pennant ex serie euhiehen.EE?=eZvalable-VzE4

Z 2-③ - = -

2-2-4 ( 2 - 2) (2-+2)

1or #tz are . A -

- ( ⇒Et et D=L⇒I I,( Z - 2) ( ztz )

=

done= tf 's - II ) -

- E. ¥.

- ¥d¥÷E -cerium

I -21 ) 2

④ mum )Z

s -z ( ¥ )

'

= - z'

-

- ¥ ,this ^

d'

apes ,d ' enemple usual Ulm ) ⇒ I si Izl > A

2- - n

et la proprietor : man #- z Alta L conservation dm domaine de cu )

116

Claire Schmidt

IOT de Cachan-

Geiin

⑤ Sm Ulm - s ) = 53×5 " - '

Ucm - I ) € , 5% E' x I si IZI > 5

Z - 5

=

1252-42--5)

2 ene method :( Celal desire )

si 1St - ' letaczi-E.os-un-nz-n-E.si SE 't =

÷!×÷±s ⇐

pyarum. 37 Eh effete : T raison Z

??? Ulm - 37=0 si m 23 et Ulm -37=1 si m ? 3 ⇒ HI> 5

itI; De plus , nI@am = ④ x # si talesp Of raison

per rerme

Eh simplifiant , on obhreut : ACH -

-

Ix E'

× ÷z .,

.

- 12¥ ÷q =zzY÷ ,

⑥ methods : Utilise bmq # ACE) et an.Uhl Es AH ± Iz , > n

s " HI > 5

] ( Z - 15h

5- '

nun = Smx mum Es At }f= ⑤ ,

i 13131

=2-

a. Iz ) > 3

( Z -3) 2

method 2 : Utilises man -2 , - z At to avec an =3" - '

Uh ) -31×5041 # Att ) = ItZ - 3

5- '

m Ulm ) = m x 5- '

Um ) Ex - Z ( Lz z⇒"

conservation de domaine si HI > 3

de CV

= - ± Z-

3 - z

> ⇒ =cz÷z 121>3

⑦ method I : Decomposer en elements simples

Z

⇒-

-

ez . is an ;s=Zzt ¥ ;) A -1¥;),:# = - ti

Z

⇒= - Fizz '- Ii #

;

D= II;) IT -

- fi

z¥ - Ii kitties 't -45 = - f- jie'

+ f ; Iii 't=

.utjzmcei e- itEm )

= - tf ; Tx Zjsinftzn )

= I - '

sihlzn )216

Claire Schmidt

IOT de Cachan-

Geiin

HI > let - let Izl > I e- im- -

Method : Reconnoitre la transformed en Z de bmsinlwm ) f ' 'n ' '

n

an-. sin Iwm ) -

- eiw~_Z , ÷f¥,-¥¥ . ÷ZiiIzt

sina.CZ-

emceeing

= ± ei -

+ e- in = 2am

Zj E- zLei th ejw-e-iw-2jsi.nu"

261W

AH = Z si HI > A

Z1 2765W th

bmsiniwni ⇐

Ai¥=⇐¥÷w+.=z?T

ZOh ideutifie ( surtout le denominator ) avec

-et b = 2z2d ' OI cow = o

⇒ on choi , it w = 't

Tsin Ikari I 2z}h!" = zfZ¥ on a an factor 2 e - plus

Noi I € , I - '

sin I En )-22+4

⑧ { an } = [ A,

0,

0,

1 ,O

, 0,1 ,o

,o

, - . . } ao = I an = O az= o

m GN

ay = 1 ay = O as = o

af = A af so ag = o

et plus generale went :

A) p=/ 93 pt

. = 0 AS pt 2=0+ - r o A

done ACH =I an Em =

TotanE 't

%Etta,

E 't

EstaE 't E

't . .

. ta,

Ert . .

MIO

= A t Z)

t EG t ES t . . . + Emt. . .

= z- SP

on recon nail eve send geometric - e

t -D= ⑥

done ACH = E ( z' IP = ① xn÷ = z¥n si IE ' I Ln ⇐ HI > 1

p -- o

X OI h Terme A

raison

3/6

Claire Schmidt

IOT de Cachan-

Geiin

⑨ fit) = ' Te = 2 done an -_ flute ) = f (2£ ) =t3= mt )

on a done am = M a 3 = I mt 3) Ulm ) Ulm ) =L

an -_ much t 30cm ) Es I +3 I tflzl > A

( Z - I )2 Z - A

④ cz÷=a÷× # ¥ an 't bn area an -

- Z- '

( ¥ ,)etbn=E' I ¥ )

← ¥ ,-

-

E '×¥y tryin- . let zt -

- E'

¥ £LEI= a bmM

dmc Ca = an 't bn = 9ohm tan bn. ,

tazba.at . . . . t an .zbz tan. ,

bn t ambo aAt 2 t . . . th -2 )

I y "un

"

un" 4 " "

a

I I "oO n oh - J A n - 2 n - I

⇒ cm = ( m - 2) 1+1--2 = Cmn ) co -

- hobo -

-o c

, -_ aosntanho = o ⇒ on -

-

' )Ulm - I )

2

Ex 2-

:

1) f et 21T- period .

que , paire et tu E Conti, flush -

I flo )=1IT

a fly flitter - ¥t= - A

^-

fl =1-1=0

µ\

It I I' f est par 're done Lf est symihn.pe/C0y )

2

- n

-

= > we =a

T= 21T

2) ao -- I . pta!!

= ¥ !"

C-⇒ du - ¥,

Ca - ÷ ]! =÷ ( it- t ) -

- o value . move

IT

bn = If f flu )

sihlmnldn= O f est pan 're done pas de sin dani la decomposition

re

- IT - eh Serre de Fourierin paire

416

Claire Schmidt

IOT de Cachan-

GEI in

a- =¥÷÷÷÷m=÷!"

u - ⇒ union I re

÷-

pan 're

a. ten .⇒ at

"

i!÷: ..

+ .

# sin Int ) = o

O

an = Ix!"

sinful du = u÷ fuming! fsihcwnidn-acII.ecIT

Cos o = I

am -

-

- find wit' -

o÷) = ¥+4 - C - it )I costume - it

C- IT

3) Finale meat,

la wie de Fourier de f est If (a) =I IF, drown tybosinlnu ) )

4

Sfm -- II ¥11 - far ) asmn

°

am = µ÷( A- C - IT ) done

azp = o et Ept ,

= ÷¥r+ -

done Sfln ) = 2- 8- cos ( Apna )p -

- o IT42ptiT

4) f est d par morceau et 21T - period .

que ,de plus fest com IR

duc If ( ul - - flu ) en tout u ETL ( pas dediscontinue)

on Cho is it u = o car cos o = A

+ -

g

If I o ) = flo ) = I - cos o avec f Co ) =A

p=o IY2ptD~

+ - t -

T2d ' oi 1 = 2- 8- ⇐ E 1-

= -

p=o IT 42ps D

'p=o ( 2pA )

' 8

5)a

tf est une sein -

e de Riemann converge - he ar 2=2 > A

+ - t - t -

2- 1- = E 1- ( a pair on impair )n -

- n m~ per app+

p? ¥2⇒ III .

-. FIEF - I ri E÷= ⇒ IE? -

.

'Ex 's,

516

Claire SchmidtEX 3 IoT de Cachan

- GE i in

-

flu ) = A- coin t sin kn ) Sinn

a) fluid ⇐ 1- coin + since , = ,

I- "

↳ u

cos 142 - u ) = sina

⇐ sin I 2mi = cos u

µSin HE -

a ) = coin

⇒ cos (TL - Lal = cos u \

useful⇐ I - La -

- a CH on Tf - 2n = - u CUT ?

⇐ 3h = Iz CUT ) ou n -- Tf CUT ]

⇐ n = I (F) on n -- E CUT S= { Tf GF ) ; I CHI }

b) f ( a 1=1 - coin t sin 12N ) est deja nee sire de Fourier T= 2T ⇒ w = A

f ( u ) = ao t II,

@cos ( nun ) t basin ( awn ) ) avec ao - 1an = o

An > 2

a,

.

-

-n

c) Seba he formula de Parsed,

be = A boo, bn -

-o

, bio th > 3

ai t EEE,

tbi ) = I Slim du

GI

ad t EEE,

ohit bi ) = ao t I ( an 't bi ) = 1+111751=2-

les autres terms sont nuts

ITCalculous tf ) I A - win tale 2n ) du

- IT

2ftn) = ( A- Cola t sic

, In ) = ( A - coin think) ( A

- Cola t min )

= A- 612 t sink - coin t with

- coin ah Zn t Sch 2n- 1h 2n cos u t Iiic Zu )

= A-

2 coin + Zsih 2n- Zola sic 2n t coin + since )

- in -

- ( sihsn-s.ua ) = ¥141= Y )

2 2

coin = Atan GET2

sink = ¥3 " Ce formula de trigonometric peruetteit delineator flu )2

sin ( at b ) ta-

h ( a - b ) = 25in a costs pour pouvoir colder une primitive

ft ( ul = A -26in t 21inch ) - sick -

sin u t A t Z cos Kul - tacos Mn )- -

-q pain impure

pair -

ITT ) f 'Ll da = IF J! @-2 win + Lz

co , 12h - zushi ) du = tf [ 2n - Kihn + f ai kn ) .

tsin Kal- IT

= It x 2T = 2 sin I 2T ) = sin I hit ) = o

oh a bien frigate .

616