MATEMATIK
Hjiilpmedel: Beta, Standard math. tabels, typgodkiind rd'knedosa Telefon : Erik Broma n, 07 40 -45902 2, Tobias Gebiik, 0 740-350646
Chalmers tekniska hdgskola
.1
2004-0r-t7 k]. 8.45-1 3.45
TMA132 Fourieranalys F2/Kf2, 5 poiing
OBSI Ange namn, personnummer samt linje och inskrivningsAr
l. Eit linjiirt tidsinvariant kausalt system har stegsvaret /(t) : e-tq(t), (dvs. /(t) [r utsignalen dA, insignalen iir d(t))'
a) Bertikna utsignalen dA insignalen * t?(t)'
b) En sinusformad insignal ger en utsignal, vars amplitud iir hiilften av insignalens. Beriikna vinkelfrekvensen'
2 .
J .
Besld,m clet polynom P(r) av hogst andra graden som minimerar roo
J"- uo - P(")l'"-" d'''
a) Los Laplaces differentialekvation
( u r r * u r r : 0 , 0 < r < 2 , U ) 0 '
| " ( 0 , y ) : u r ( Z , A ) : 0 , l i m s - + " o u ( r , Y ) : 0 ,
l r r . o ) : { o ' o ( r < 1 '
[ * , * , " r _ \ 1 , I < r < 2 .
b) Ge nflgon fysikalisk tolkning av problemet i uppgift (a)'
Bestiim en losning till Problemet,
( u t : ' l - L a a * ' t 1 ' a 1 - o o ( r 1 @ ' ' t ) 0 '
t ' ( t ' 0 ) : ' 2 " - " ' - o o ( r ( oo' Ledning: Fouriertransformera i r-led'
5. Let / vara en funktion i ,1(R'). Definiera F(n) : tP-* f (r +2kn)' a) Visa att F iir 2n Periodisk'
b) Hiirled ett samband mellan F:s Fourierkoefftcienter och /:s Fourier- transform.
c) Bevisa (under I[mpliga fcirutslitningar) Poissons summationsfotme]:
i , r r * ^ ) : * i i @ ) .
k = - o O n = - 6
6, Bestd,m temperatur en'tL: u(r,t) i klotet ' : 1/F + y'TZ < 1' da ( * : y 2 u : * * ( r , * ) , o < r < 1 , l ) 0 ,
t , l f t, t ) + u , ( l , t i : o , - ' u ( r , 0 ) : f (r), u begrdnsad' Formulera och bevisa samplingsteoremet'
Jrr(r) iir Besselfunktion an orning n' Visa genercrande funktionsfomel:
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