Signale und Systeme 1 - FetSignale und Systeme 1 Ergänzungen zum Skriptum, Formelsammlung,...
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Transcript of Signale und Systeme 1 - FetSignale und Systeme 1 Ergänzungen zum Skriptum, Formelsammlung,...
Signale und Systeme 1Ergänzungen zum Skriptum,
Formelsammlung, Prüfungsbeispiele
vonRené Repp
Wien, 16. September 2009
Teil 1Ergänzungen zum Skriptum
, \ )
rhL I
-6
, , \ IJE,/
rr='r7o
v-v. , l .v t+v.dv
- t F("r.;)
, t 'A , . . t . . ,*u =t= + n0(rl e"'
[k ')1; jvtJ-
aT
d'rI= 1 l - r - , , ; lutu)t ' .1- l
lo l / . . , '
-Fr eou'tt,tvet s(la iebun P-\ t
. . { Y ( i , , r i l - ,or , . t , - .' r ' ' . ' t " ' =- i9 l x( t re- dt =-. I <( i )r dv r t
I. l - i ! ' f ,
= -1 l l . 1r)x( ' i )C" dT' lkTt ) ' t
.2o
,1 O, I fe{uc"1-\nleqr(" t , ,orr._,Y___
Ltnd d
otS(3
: (or)e- 61r
A
lol
4 TaLJeh^u.Q,
J:(xtor))=
(r
a- -11
, , I
1 [z( j ' , -1u") l = f ] x[ ir"-,)ej , .än -,r v '=dvt -
) l: . . . . . -
I v / n )) f , , t f ; ' -- zr J xt1v) et" dvo\dvt : cru ' t 'y yx\vt l
-Ft equ enld, (er . - , i ic l ionS.
[* e-r,.l,rr =
) *('r!-j'tdl '
I , i - ,le- a)
1 il ' i
- - l ri i l - , i r1,1, , I f , . , , .1lx t t r i le ' dv ' ]d ' r ' : l , ' ' ( r )L*- rd( i l l
ouS
;Jr f , l -X (.1u') c{ r'' =
I . Kon j r,'3iert Lomgls/.4 ?r't(""trli or
Se' 4 =_Y? d,oon g, l f j*= J^e* - i f l i jv tJ^= ej ' l {"{rf ß ()
6. Tooe ier - \,(osinus 'T-rcns{or,,'o{,,on
/ - i . , , ,
* Ix{r)e ' - 'dTI
. | -
i , ,? ,
X ( iv) = I x(r)e ' - .d.r)
ot - t , , - ,
= lxt t le o (t
\+: !
/ ule/. xh)' x(-?)
obu be, at6'5t)
Gr(3 s i /
,1 |; I X(tv) er" 'dv
o
. l
a.rvrolar = j " ( r , ie," ' dt '
a.| . -
2 J x tr l coslor )dr
'?" dv = Ä ) x{i,) u"'d,v .
' r '= - ' r , dt '= -dtot
T = - L r-- , )" j ' r 'dT'=
lxCtr)t
h6
=> E in:€Lcen i6 6bi3e G leic\,^v"
- tX( iv)= | x( , i ) ier ' r*eJ" ' )dt =
) J " ' ' ' , \ -
.q
Gl. 3.55 x ( , r ) ' := | r r . , , )c"' ' " /n l " 'JY
vt ' -v t dv ' . -dv !
, ? - ;n 'a , /7 . . - ,u ' , ' , r - .t = - i . I x( ju)e" 'dr ' 'Äl x ( ju ' )e ' - 'du' onoroS t ' 'c
@ x o ,of- Al
x(t)= n I x{ ju) (e 'ut 'e 'vr ) dv = - J
xt jv/cosl ' t lc1 '
oo
Touriuc - Si nu 5 -'f r on s{or rnot' on?"
6}odloqe Vc( 3€hc'7\sc'c'!e - 'g bcl 6 l lo: ' '* ' : -T r o"s{ orpoLicn
peq€^ xG'r)= -x(T) e 'ho\ l r4o^ ov^u
t . , , , . , x{1r)= [ r ( . ) (dr ' " -e 'v ' )dr -D ql (3-55)
i= - ] . ( t )e '" 'o ' r - '> ' "
lo zu'.
^( - - iurrT= -- IX(rv/e- dv
/uJ
; [ , - , r / ru? - j ! ' t ) J--+ x(.r) -
-1
) x tlvl 1e- - e ,l ov ->o
Fr^e{ u entbe.e, ch
L- tvT I I I II Xz(rv- ivt)e ' - d lu ldV'JJJ
?'=d,t"
r )=
1j I s\^ ' lr t )
dttJI e"r't=i I
' Ysa
^ 'L GL(b3i)
, ta I I t
*X.(1u) j= Jx, \ / )1,r1-oo
- -1r- , , . , , - t r I
f L \1\Jv - )v ' t l 6 v
" i l r 1
e""tt) l = J "lr" ' l i l ' '<r " ' )e"" dr ) dr'
q= J x.'ft'1s"'' F[",r')]dr'
? ^ 'bt s 'cte sor"r tx'( ju)= 'J x*( t )e '" 'dt e '5
a t (a ao) F[,^d
m(t c. (3 .12)
(1u)
(1u')
- ti - f
J- LX,
= lX^,t
I= lx^:> Gl
ct 0n)
k or r e\ ot i on sp, od u\xt
c((3.?s) l ' ) xz ( ' i ' ) drr - ' t
y. (T "-t)d,r' -"- rfi(
[x^t1,\ f^[x"1,sf
, r_t , . l ' . T, '_T
= ,lrt
c^{f t )
ct (3- eo)
dt'
' r . ( t .r1',, (T) = | x.1 (f +
' t
q
- I , l( r '))
o.s Gl' (1'(1)
\Jr I bbs l-h6nom€n
6r.(:.n)T". vg('r-Tt;
d'r'' - atYv,
obe"
r--r' = tn/vJ 3, n (d), , 1-F 'sf)=
$ - f6.,r;sr- Q<.lhea
oJ. LrLl 5.q /-la
-- ' t ( f ( ' - jz l (k{X' t I r r
,4 | . . . - jht f . t r t_ - l t . / L. te tuL=- J Lr(t le "" - -r^ I L7-_ Jot,
\-r . !- | )^tu-t)t"L )+ _ /-= lJLq-f^ ) t ur / - uL
k.-e o
V",3".51."c.se ger"'"8q"^s,k ^j 4
DeJen Ct.(5 j)
GJ c(.(5.4)
5.2
lL.Lincorrt i l
3
2os(r t ) - iZsi^(nL) -4- \ - - l - -
2G t) \ - y L,o, e{,s
- @s0nk) * is;(znk)--?a-
4o
c{ (s./6) {l{St
c."= fft
(2""t- / - e-i'?i!)
T li'
, l . . - j t ' t , L '^ , .1"){ ik 'd ' r=
"; ) o". x.tt)e- dr + Zi J ':}a
"^= * \ * ,oa'u 'a,
tt'an
I x( t ' ) e -)
o
= 61 6an 1 @r-C(z
Zc,it vers&iebung
4 ',,,- ^A | \ _ ju lÄ
- -
in J x( t - t " le qt - 2r l
"a- - lut ! l*( . \ä,k" ' , , l t ,- e 21 l^ ' r , ,L
e d'f
,,,e3e. Lt ?er I oliai{bL
,1,tl 1t= r--t" Är'=ä't1-t t+to
- r[,qöt^" cu - c"t(l^,r)c* - j s,.{rt"-r')c"
{ 'nr [ ' O1 l.1' ! \ r-. CV= - Cu --., CV = g
4.D( ffere"L' oLion
*r"nt= ! ,
cr(r r l )
Ct<'
l4t€S(oLror\,|| , , . I
J " tt' ) or'
f , r* . iu '=j1, r^ "*).ru"
2rt l
^l,31
1lr'r J
1
T-= l ) l c"e ' . 'd ' '
l -
,kt ao' /c l -
{e""dt"= A ät k-u '
P.
- L"k
^ | - ,GL) i ,I l >* l . " tä"-- ' dt = c i
2 . (ee{le + L'4kt ' on
r r ra oq\ . r1.1 . >. (s) <--)qr
l /AJ, , t . i
Cl (E'"9i l - c, e iLr r
t . )Cl lb -5{ I lxhl coslkT+
2. O (T-2rn l g- dt
{ G - zo-) eivt d, - ,!, ,- trtn . J
2ilnt,r e:n f,- l-P'tr rn
bcLr,scl,kkn ?e, c'cl^
5.
p,, l 1r '= 1 l "T '
drt = lkdr'
{
6-f - 6n <-> C-n' ci.
ktom,.,er s{cL,t z+f = e+ib+o-jb '2'- ReL2zl
= l i " I cos(kt)co: Yu - l iu ls in(kr)s 'nVn
= *,^r cosi<r - il sio kt
4. (o"1"5'ert lxo-gk.e -t. '"üt' on
Spieqet, . rnr
^ | - ikr
, - i1 lx\-r /€ qt -
/^I Lr 'e\odq f ' .uE'o^
/-- , \ C-v.+Ct^ -61.\s.c4 ) --T- = cu=
5 Lrn ger oJe f-"tlU""
r r , / r t ' , c*-C-*.-*
u(\ > r i '1 J 1
- ; ( - t ) t ' . ,x(c ' ) e- dt
, ;
" - j
( -L) t 'drr
- ' cr(s.cc)
l.<. - L+ m 1 m=k-L
^1,' ä I xt<-Lrl
-zil
-^ |- lzir )
* [^r.)i,*ar1 = ä | a1c*i<;iu')dt" ' t t ) o ,atGct l
e /da
lu l, { 1 ,1 L. )kt l; l ; l \ ( r /e dr+- L - , i
2\
^ | . , / - ;v<= ; : I x l r le- -. t ( J ' - \
21 -1i, . . ^
-1r . t , l l f ' - _ j l - ( t r t , I Jr ll l Lc:"e ' l lAczte - l -
zl4 { Fr
( l " - t , ) r 'dT'
- , t ct . ( ; . t^ t")
Ln l -I c,,i. c"., "jt"v
Lk
, \-,-Lk
c"1j"n I c,- u l' lc,( | . lc"l,Lcl- > l t ''L t t "
-x+X(l2: L-t
?e...dis.he lÄorre\o{.r on
Cn\ crn g''"
?r "d,^LtC{. (5.54) subsL.
?orse.,oL - G lei.h,rn5
Gi.(5 55 ) x l ,
n''it c,=
xr ,-(J lculk'a
A
V,Il3 l '= x 'l fä l
lXu l"
6 - Obe, s.v"i n3o^q eri
6.3 Leistt.,"q sqrlse"
^, /r ,, ' ' /,\ r, r u.i{r"{l l ."s(1,-,,t + Q1.) * t"i lu-tn.rs(ko,.t+{a-)qt. tb. ,46l P(t t=
\ .^oto + L_.,r v=t
@@
+ L L {un llr.t 2 cos (k"at. tt,q ) cos (Lc''^t + (ir")k:^ L=4
=^(r)r" ' l t 2cosx cosy: cos (r-7)* 6o5(x*71
e,lJ luqr lL '1\L !o> ( kr . - r , t + 9."- 1r , ,1-,g;.) * cos (t<,r^t * Yrn * Lu^t , Vi.)]t
\-r1J
YlL-\
r ( t ) =
E-
21 lu-tE'\ coskk-L)r.,t + y" - qi{L'^
((f)
{u* lll,.l cos (9"" -qi.) *
* Z X {r* tr"l.o.s [(l't) c''t * 9'* tci"-{
L LJ lvvl lLLlcosL(k-t rar . ,L 1 \ r , , - Y, i J
ktL
>ch oLter be^ rF' e\
_,4t4
= ä.6'I l r " r r vo n| : -OL' - ; t )Jr(
{ t . l ' ' l rJ '+ .u'(Gt (6 zr)
r t
J,u'lr ,l' (.1 - c"i{^)
) - u'(1, l 'coi {n
- ,frtiit,r. l'sid 9a
(*f tt
C\ , ! , l - lle\ Chfr cv\\.(, oe,9Prsi
u(t) ' UI2 s 'n lct ' l ' )= i r - !^
far l " - iLort , ,Ix =
- l r ( t )€ ' dt =
Tt)
zak
|Z-17
2^u2tl
L-e t / r
drf
6q N rc\nt I i,<o' c 57stc"'e
G\ (6 3ö y(.r ) . o"+ o^ü . j ' ;€ i ' * " rü ' ( - i '+; ' '1 '*ry1'(ej" .ä ' ' )3
^l . . \
- ?(d '+Lrä" ' )
3ar.* . )€ j .*äj t t )
Clo + ".r;) ea!j'- +..
d'*'*ä''
r" '* j tal 'at = rJ
+
o.^ü d1e' '
d'1 .., r" nB \-
O" u--T
OrU -q
'::
j"T
+ lonü '
eJ"* e
\ r/ - Loploce- | ca^slo(r 'aoLron
l .Lcr(? G) nt.,>r-r#},. .=*, -+- *f i
' - ' (sr ' t ts-Pi l" \s-P" ) sn^ sn'
ar(r ,,r) rcs). $,,f,*y.... - 1fr.,,+ -T- r.,L r.ä#, /.r,-v)<
(s,pn)n RG) = rn (s?,f"/+ re(s-pnl-z* . . + rr-1 (s-p4)1+ rn g-7)+ r< + (s-p^)k
- I -
ct (".4q) $ tr-r^t*ntr)= ra (l^-,r{s-prf-1+ r,(t^-zXs-pi)"-} +' "
+ r*'' 2-(s-Pn) + 'q-'t + ft(s-e.)u1-
. . .usrr - ' q( .h 4t
X G) = f [c^ r^lt) * c. *,{")] d'"d< - 6 l x t) ö'" h + c. l xJ"\ ö" d r = c \G),,,xls))
/ G-r)
Lß) - = ri * rn$|. + t r^ Sfi ,/ t=r'
(s-e^)a(5)l= r,s-P^
-t .a
, l - l
Teil 2Formelsammlung
?^ 5&of t€^
€x i sk"t
L, nec,i L"L ,.14) = 61v1(1)+6"x2(t) a-o oa&(ju)* otXz(iu)-
Dq"(i6L x (T ) o--o X(jv) <:> X{jt) '--" 2rrx Cu)
ZetLdeVn,,n3 x(oT) o ' - -< $, x(1*)
?c.t vasctnido,^.1 x(T -t) o-----. €,-rut' X(jvl
Fce1ue.1 ver sctniel',J x(jv-JLb) --o
gju"t*1n;
ei
A.
z.3-
q
5G.
]
) .
40 -
l lx l r . r lda
Xtjr,)
2elt dtRe,e^t i oticn
2c.t i"Le5roLi o"
x("tr) o-.- lirf xr1r,)
f ,,.t.tr.' "- i ,ti * rr x(o)5(u)Tre,lurn+Jiffe.unt ioU.i,
ff., ).--- -txft)
-trre1rre^:u ' ntegr olion
| "qn) ar,.- f x(c) + n x(o)f,(t)
-@
J.)" S y -aet,
I iee,'qen 5 q.\,i oft{ a
"4 . Xonj uSiert, tro- pk*e .4it&^kur^ x"<t) *- X.(ju)
2. lteet{e Leit(q"LLion x*('r)= x(1)4--> X(-jv) = X'(ju)
x'(v). X'(-v) 7 -x, '(y) . x ' t(-u)lx ((-v) . {x l(v) 7 Q(-u)= - Q(y) n^J(u)
3. lr.$inöl e le,tft-.lrtion x*(d . - x(f) <:> Xtjv) = -X*(jv)_x,(u). xr(,v) z X,(rr)= x-(_y)
( . XonjuSiet [xom plsls Speküotf;ul'x^ X\jv).-'-" xr(-t)
5- gpiegel r,^5 x(-c) o""e 21(-je)
6. Ge,"le Zeit€kt. c^-l ge,de gTeLtrolfut r(-r)',.(t)a:: X(ju)- XEY)
:Fq.^r ie, -l.losinus -f5 6^5 f s1.-6li o n X(jv),2Jxh)cos(ut)dt
/ ' t r ) : a I
x!u) <os(ur) duö
1- Lr^3e,oJe ?cjt(L.t u^l t^"5e,de Spe{,tlrol fl,rL' x G'i - -x(r)e''
Tourier S i ou: -'l-coa 9 fo r rn otr o n X{ju) ' -21 J x( t ) s i^ (vt)d
,ax (r). f I x( iv) si "(vr )dv
i,
x tjv).- xtju)
'l
x ' x( xr t x) f T"tt, . , . tqut ."
X.*xz+ x ' + x)J
x5[r) .
x' (r) =
Tol{,-5sproAukt<--
.F
kormrrt"tiv x^ x x! = X!* &o 59o ziouv (xn x x..) x xj =
d istr ib"{r'v Xa x (x1+x1) -
€ i^se\e-".t tx x = ><
x.r (r) x x"(r) o--. x. tjv) X.(jr)X.1(i) xz(r) o---- 7i X^tv) x Xl(ju)
q
?o, sevo[ Gte,<\,r*n5 l1rt.)l"dt .
-a<,--
5' 3ao{e"' 3is
. I lx( iv) l dv
g. le r teS,,"S i^ 3e( oäe u gnSeroäe -I-eik
x(d= xs(r') + xu(t)
ebcr.s. [:; Xtju). x3{j.,) + x'(j.,)
x3(t) o--\(1u) , x,k)*--)(ttD
x(t') redlr ge,"dc <=.. X(j,/) reeü c,. g*Jc
x(t') r<el,\ 1 r.r.rge,oJe 4-> X(jv) i..o3i;' u u^te,oÄ(
x(t) ree[ : x3{ju). X'(v) , >Q !,/) = j xtv)
3. l.(o,>"\es Signol x(r) =O F t<O,rt9 ' x3t t ) ' * 'L1' {x{ t ) .T< o , x3 (t) , -xv tr): ix(-t )
1r l : ( r ) + x(- t )J
äLx\t)- r ( - r )J
än[rtu)". 'sed.b"tc S i3 nclenegi " d ictat4
a i l , , . , rL, I<<e = t-- = l \ , ( r ' l l Cl t '
'' ,t+r , r'rn'
Xt6 ... L(<.kliv wvf
s i g{\o\\eist ^J
3. 'f ou[ te( - K€(he^
1l"--- : - |q-
T,r l
Detoae€ Form
l- pc,ioÄisd^:
2ir - pe, i "{is<t" '
u&). f cuetq"^Lu{t) e-j htdt
5 .2. 6r."alese^de E. i3e,r ScJ.roften
aL
J[xtr) ldt <F1*<-
x(i-). o,r & ({.) + or.xr(r ) o--r
x (t-T .) o.- g-'uoc.'.
T.= |'" l,
- r^ )
f - - iz ikr .J
(T le- dr
I | . - rx ' i I1lx(t le dT'La I
x(c) =
xtT).
fl 4ne;u' ----
I ao=O
>- /. , .alL1kT
l -. '-lkTarLq<:
u(L\, us xG)
C^. usc<
4. e.x i >Lqnt
2 Li nes, i tät Ck= O^Gk f oacz<
3 - ?€.it ve( s/"ieh,"J
( )tffe,enlielioo x'")(r) o- 0\^)" c(
cr." G(ik) cux
)i<c-\loom f, ff t- rnr^; --V--,-
5- lnleg.oli"nr
,f ,'(t) a., ---<,u
lü
4' konj,rgi.,t ko-gte,,.e -F,^kLion xa(1) --
6_*
2. Reette -Fu^t<Lio n x*(t) = x[11a:.; a-n-a*
x( l : co * ä R.(2. ." . 'u l c-=", "*= \ -k4
x('r)= I * L, t*"|c"{t<t. cn)V.t
{r ( t ) = | c( tL)e ' ' '
1ä
2!l
^ ! l
- rL? '&= iJ x l ' i )e- d(h
x=; l , ( t )dr
3
I
'Sp'e5slung x(- ' i )
Ger.de Fu^kl 'on
U^3e,oÄc TiJokU.o5
| . ,\ x(r)c"s (kr ) ott
qI x(t) sin (rr) dc
z
5.( t - f -e l1r ," l u.J 1, . (orr e lor- , on
,7?etiodrsc!'< -tro\{ ua3: xr ({) ä kr(t) = za ) -'
it1 - z(T- i ') d-'! 6< ca^c1k
7(r) ' gr f")x r(r) o---- ct . . Gt. , [ )c,n
?erioJiviae [orretoLicn x, (r) g x.(t)" * f".1t.,1""1.- r') d.tto--- ci* c"*
i-lr - ( odlrk t x ' t ' i - l \L(T) <>-- LC'raa-L' 'a
2i\ tI t \ r ,_,rPo,sevo\. G\e.( \ . {""q i I lx(rr l dr - - Ltal
) - t t
)
lX l . lx '* l , lxnl 'I 1. .
'.(r)= Z. i 2c" sr n(kt )
x(t) ' c" t rc* cos(t"z) ^c<. Ä
- i
z-tl
RG) =F'tt
l r rq S +b--r S +--?btt+b.
(s -p.) ( s -p.) -. ( s-p,)
e ia{."ue LJt,r1aLn,. ri = (s-Pi ) lfs) lr-r,-
€ i,a(<,a.+ Tkoajw3iort k" -plcxe
Ctws\a'. T<,r". 1"T",^
z(r )N(s)
€isf b i
(s- o- , ) t+ v; '
Red,tts:eiti3 oor{ odisdne Si 3nole4 1!- -^X(s)= 7i+, l"h)e '.dt
6eeue Lrurgf.,, ossrc(L..n/ . lo"a !r."rl irr S €i.scl.]r^ ,si,b,.b<:!,_<n
l"tel^rfo"h f rer lcLn'. " , rn lz 16
s-P' (s -P4) ' ( 5-P, )k
t,,,. = ls- Pn)n RG) l| . \ .p1
- ^, l , -t " -a=
l r . är l ts-pa*Rfr)J, o.- L d" ' r - . - " , ) (or . t?r ( -L= Ll ;J t - , "" j ! ,Pa
?.3 6 tu.J leqe^Je Liqengc\,often
, ,1 . L ineorrLot x(c)" c.x,( t ) t crxz(t) o< X(s), cnXr(s).crXz(s)
2 z.e,!de\anun5 x (or) o.....- .* X (:) , ., o
l . ZatversC'r ,g loqa5 x(t- t )e (r- t ) o- . d ' "oXG) , , r .>o( . Trqurnr ve's<!ri eL,,.3 X (s- s.) .-.--o gr.t x [t)
5 . ?eil d, ffere^LicgLi c n x'(r) o..... s X(s) - x(o-)
x(')(.r) o...-. skX(s) - s!-'x(o-) - rt'-x ^'^)(o,)
- .. - (u''h-)'[
6 - 2e^LinL.gr.t ion J xtc)dt ' "---- t x(!)
? Difq<,enl io l io^ e" ' Uq.. . - - .o. ,s(orr . io,bcn $xfs) ' - -o (T )(x( t )..\ |-t.\ lre?, e \\e >(91ze
rfo l { u^5sotz xr( t)xx,k)= J x^( ' i ' ) x"( t-c)d1r" ' --"- ' X'(s)Xr-(r)
€nJc.,ertsotz x(-) -- [- [s xA)Jtro
A n fe^Sgr,.ertsotz x(o+) = !'. [s x611
Teil 3Prüfungsbeispiele
l-c.T
trl
ß
), ,ges, DaL {i, r.,t0
g 11. Ia + !1+ 13
. ! -^ _:-- ; = U+
: .uR
(3u-rr) l r rL :>
I.1, - jc. CyU
j c"C/ (,1 r ju 9L)
\c- ,C^(4+ ic 'cL)- - - u-4 + (cr)a LC1
. . . => ( ju.) iu LCrCz+ ( j c j f u f , + ju,u (c.-c.) t u({ 'S) . o - ? r jc,)
u"tt) Lc^c, . u"(d f, + u'(t) (c"-<,) + u tr) (**g) - c
I + (j !,).L C1
{u-1-c. t r - gu= O
ges'()G t ' )(li) lorsteu*t,
-;t Integco{or r.^. Le'{ze,tg(icder
^ -s ^ -29 -3s\te + ,e - e J
(t-z) €(<-z) - (!-3) €(r-?)
G(r)= s Hc). + (/ -r €(r) - 3 ("-4) e(t-a) + 3
/ r^ ( l - le ' , lät ' - c")
t'/h (") =
H (s).
nl
y' +Ll ' - arSu'+Vu 3es, h (t)
G(s)= 0 '5t
+3 = a+- .---- t,t )- (9r_ - ci") e(")
A
\\>l_ 51+35+,(Oq€r' (i) syrl4^ -DcL ,(ii) yr.1a) {ir u(t)' 3cosG<)
( i ) Y" *3y'+ roY : 1-t
(\i) yft(r) = yp(ä = j [a<iDl cosfzc + t{6tiDl
lG(r?)l" ?ä 9<{5r)-orcfcgrl']= -{
- \r71(r\ = ft c"s (z'c - ii )
t *l_6 ,t +\G(j l )=
7I x(<) 5(er- ud) dttTeceala6s y (c) =
f o ' c 'bLx(tz) . cru
o'E- b rr--o t '= t r y (c)=
g es: ).r(-,)+ srnTl €t t )o rt f,4 + da'( 3cos ch(d=
s(t)= [h'a] +[h]t(.) [Un. U(o-)-h(o-)= or8
= ors 5(ö - dt'(.or" + sin'r) e (r)
C(ju)=
Y(d=
c,9 - Ju*z(1u+r ' )z*4 $u+z)1+4
Gtj) lcosftr orc(c(j)) l = 9f>3 cas
Ju + j
(1v +zt + A
(r+ o1 osr)
ges.XG)
, . . '
x(t)-- r e(t) - (c-a)€(t- ' { ) o---o Y(s)= +=
t:5ona
I tn e"s' l r . l ' l r ' l
r (t)
T-
lr l - IJrtc'; eiu"^tat an'2al' { r ' +
i i e.,-^r rt = j*^( d'"^'- ä--") = j# (.i'* - i") = -r
$ t.,-^--aitu^r.) -- rfl,n {aj"-a'f = " E!9
IE;
x(t) = sin(5t) .e. t(s-) ges, X tjv) u. Skiz?e
X('u)--) zi
nt ,J sr4(
-n
t ' f1L
f . .c- - . ; ' \ - r t t ,ue"_.-p.- . le ot :J\"
-I
- i(v*5)n i{v+tl ?-
( ' - I- t -
_ J rvr) , / J
a i -z i , , . ! -v-5h1'J L - ; (v-5)
5.r)dJ"dt =
^-j(u-5)n ^.J(u-5he- -e
siftv*ln.l _l
lr''fx(inl
y'1(t) *z*y ' ( t t +Y(t)= I y (o)=41 yt1e1= -ay O L',r< A ge:' y(t) n,t{elr Lcfoce
s.yG) - s.u (o)- yt{o) +2usYG)- 2| 'y(o) + YG) -- o
\ / , \ S+Zg'qTt5i= ------ - -
s '+ 29s + 4
Y(s)= s* *(s+of t ,4- uz
?,.1= - r . t z lsz- ' t '= -et i J4- $ ' '
4+ (u-a) -=-
' s'+ 2\}s +4
y (t) = e u' cos (fr---F t) e {t) t (o-a) s( s,"F:B)as.1
J -c-
^6:e.de(t)
qe9'Toq'{c(-R' 'heflrrpt det
^- obeuelc
. - l -e( - 2il
= ;; (ai!"-a) -.luliiet*
= i fn(e r f -a) -L lz.ucrf -a
^- €a)k
jznt4 2n1t*,1 ) | r r r t l."= ä j t+-ntat - f f \- f i +t)1" =
o
*r.)= i - lch.. äe* ).*k.?_
k+o
* j1-i-r l a'u"a . - !nia:'"a" -T.I *(.) ä'ntd" =)
rl
^ | - ik<,
i i l l te- dt =
. - l (.t l- iru' I I-J u '* l l
A/\
t;rtr [(-.1)"-4i
/ tn\ lZn [- i*nl= ?
.A;i"-- 2tc,l = A
yn 1.'1 = g" e (t)
u (t) = ot e(t)3e-s: )..r(t)
HG)=# 6(,)=suO-4
Y(s)= c(s)l lG)= e J6')
y* (t) = o.( ,1 - e") ero
=1; U(s) =5 +/\
t^ , { \= . \ l : - - - . - . - - I
- \ 5 9+4 I
ges. G(s)
eSa
t{ -+^ ciri\,frqiv) 1i1-1^ (v\
Y-u)l
=/ l +
( i )
tv)
v-u tli) G'(V-u) 0ii)
y= rJ + c^[u- G.(Y-u)-]
, ^ -
a ' , -
|\ rv i !^^\ t - \ . |2 \
/ t +L^, Ui- i \ t r1
4 + 6'aC^'t
u- c"(V-u)
c,(5)= fl
qes: x?, lx , l
!?4
4l , / / - !zrr - jR\ l_-:-- : l i l_. j r +__t/ , - . . t lqnl t z 2i \v " , l
I? A f, - iz<\ .-d<.--
T; . Ik l -e /at- t{x
. ti l l
,1 sint dt -- i ' t--*li =zi
i- \ x(ddt
, }= G J\e ' -
- 1ll?-
_ l - l 'n -t
4nj \z
fltaI I
qnJS*,- i * t>t^r-4.*\l - .1 '= o,r-s6 .16i' 6\i )
o{:f*11...+'"----;i urnl
P.. ^osl t
C -- ^oopF
L.) =2nf ,. {.50Ha
u(!)- loVcos(crL) t 5 Vcos (1.,t t $)
3er: ?
u^= H ut= Hei'\t1 z^-- R*# zr=z*fu. gr74
P= Re[u^fi] * r<e[u'rJl = \!o\!::t^r/a
1e:, X(s)
Z4
x (,c) -
,c €(t) + (-z<+z) €(-E-^l
= t €(t) -2 (c-al e(r-,t).+
(t-?.) e(r-L)
+ (c-z) €((-Z) o..-{ X(s) =
x(')= d;ä Ees, x(c)
x(n" GiGÄ = .-. *(r) =(zet-d") ef.)a
s+4
R/z LlL
I- !cges, Systern-DcL, berte.e (6ißga
)^I
r= t u"=-(rR-t9) L=-r j .Sclre. -I(R+ j" 'eJ'c) = - * (. l
-1.fc)!e.crj")= -(/- j.,f) rr=Su 'u=# ,
yrr)= t:ff),0,{t)'!9-tI')UD
u(',)
,, (."g
r r.r - (s + o'a)1sa + zs + too)s
ges: BoJedio3corn.n
G(ju). o a ($.ra) aoo($'+ |"; 'r))v
20 r"slc(jr)l dD. 20 ("3 I J!o/ -^ I
JB
I l \ ' \ , , .4 i , ' , \
= 6'1_:ä'o1ä'i-l.1.:-AO
I : v4o,^ . oAbu >to,4 , 20 loj IYICD+2odg
r - , . ,a I iu \d.c [ ( r (Jul l = occ\ät4J
"--\'-a
L. V<4.) t u =V >>O r^ : I IL
:oqF/
+ 2oto{ | -r2--1 + o,1 JI * ,l l dD -'40 1,
'-''.+-1-+
tr
!, yzz,tg r Od9- v>t,to | AAt%lvläg- \OAg
u = ro 1 20to5 lo12l49= -&ag
| . , , r i r r . \ . - I iy \! drc\:Ll +Otlr 'o,r-: *4 / - OrC l*{oi
20rs l jä I dE---
Itr
g'. - 2olSlvl + zoJg
-rr
v <z^o - ov>>)o'. 1l
e-
'll'-a
-11?,E'
/t otclic\
,t60dg
(oJ9
20ds
-2aÄ9n.k,
.,,/ - xÄ9,
-zot$
u ((') -- s* cos 'E ges' {d = o7 cos (z + (y )
d^(s)-- ä V(s).[uß)-v6)] G^(s) v(s)= u(s) #hc(s)-- H. /ar. u c!n. *r,.ntrk)= f (d"*ä") 6*=4,,-.. = 9+ cu<.o q.)- \.f!4
cy"= G(l)ä= ää=ä 6", t" ' cy.a.c,)= ädA'u" ct , , .o €ak+!A
y,l(T)= cr^ei' ,cr-^d;' = ä cos(t+or,.eq)
See{}X.= t(rDt&l
t lJ x =
\ | l I Xa
;^ l l " -T '* l= f^rzn-t- i ) = i" T 'ä='ä 1","
'r" =,"+[ I""r"- i'n"r']=,"X(,
i(r) i(t)= 2Ft + 60 cos(c.rt)+ qR cos( 3-'t - 1s")
r . ,=Zn{ / f=FoH} ges, ?*R. s05I
ro.LA l. oe," q= Si,n,u
l,^.=R16 u^=*.i*. t^ u"=rfr;r,f{.=I" ro^= 1{;;rr r*:--;ft;r:Orrscechneq ,..^"J g,i.r:ehte,,., i a
Pn-= Rr* * n"[u.ro)l + rts[u.IöI = 4 66r hJ
3e5. x(s)
x(t) = €(t) + (-'c+a) €(c-a) + (t-z) e((-z)
= €(r) - (t-4) e(t-r) +(c-1) €(r-z)
x(s) = ä- .'1. * €"+ "t' - 9l'€"
$e5 " Ge1 Gq
W -
t t , - \ . \I = qq\u\49-r J
0 ,1 +c')
Gs=GnGt C,- 4-
-
x(t)= cos(zt) cos((,r) ges, X(ju), X(s)
Y=y=
Gr (u- YGz)G,a
4 +6aGr
x(") = { (eit' .e.it') (ejqt * Erqt) =
{ ( ee. * ett . e5" * ä6)
f - l -+I s'+36 ;i-:7 I
X(iu)=
UEgestlG G) / TB= Rc
(iÖ o,tsL..,r"e ".^ Ctj,r)
u) u.t
tt)r.. u.- un I,re= Ua= U€
U^ = U€'
.4 r ieRc + (io)z(Rc)a
, - .$r /nz\ lz f l lJbßr+!e/ - \*- i
^. t -Rm;t L^. L+7:. , Lt= j-t--l lJ** / l l l (J l (L
?-
+Lt-
! . .
Iz(i)
c{jc").
6.t1v) =
C(Jo) =
( v)! +ju + 4
!u)t + 3jv+ 4J u-5 s
/ q(j,) - i . , c(j). ^
io RC,ri!l[c" 15,t'1r.c1'
,-,;t y(r)= qF) , q(r)= 8J1}) .b.ai< v"3B
' ' \ ^ st+s+{
vrr>/ 'Z S.+3S*, t
jlil6tjt
lnß
-) t
*[ctj,rl
Y(rrl3es' Y{jv)
'. t t'r = .e.t i- ::-('::-tr*'il = -"* [=f#L'l' T'-t^
- - 'vk^rrJ/L . , lX(jr. ,) = (t .-r^) e- s,t i l
. - iv(r" ' rJ/z . , , , \Y(jv) = G(iv) x(jü = gr"ryIll u-____:f' )
u(t) = ti sin3(trrt) 5es, tl(jc,)
+' -rS *("), utä-"' = sin'(t) . t ro'f c"s=.,n v=lg
x('r). { (arin" - sin(rt)) s'r's tt i5ono-ettieche' To '."c136'-(' 'n5
x (jr) = -+ [:rfr-n) - ]6'(v+/) - 5(u-:) *5(u'r)]
. t ' '^ ' .
u(j"') = *^ ",j*^)
= j* [:5(.,--a) - 36(5"r4) -t(#,-e)
xk1= 6g".€(r) / a>O ges' x3ft) , xrk)
*5(ä.d
- : * [. f ( c., -u^) - 35(r-., *.-,..) - ö-(. -:,.^) .ö-(* t11)]
a>o r xs(r) = x,,(t)= f x(t)T<O: xs(r) = - xuk). ä "(*)
x. (t) = -äeffi+ (a"en+ 4e-) ecO
e=-: .
c€o yp(t) = e+ b<
b+ 2o +Zbt =
yek)= - l+r
cdn - lrr )(o+)- z --
ä(ra'--: +z.t) e(<)
( r ) y '+2y
(i ') yt (c) .
Y(T)=
y(t)=
Ye'(t)' b
-2- +z< b.A4+7.''= -7
ufö= co(n") e(t) ge:' y."{t)
3(r)=
y"r (c)
#t' u,-l - €(r-z) r €(t-t)""" ?
= Ju(t,) g(t-et) dcr = I cos(r,{ e(Ö 5(c-?) dt'-
II cos{nll e(r') e(t-T '-z)dr'tJ
* .[ cos1,r.') e(T ') 6(c-tr-?)dtl --ot_a t^'3
= cos(nt) e(t) - € (t-1) Jcos(nr')dtl
; e(t-r)lcos(nC)dt'
= cosht)e(c) - f sinfn{t-r! ett+) + { srn[z(t-r)J ert-r)
l (^ t
I\uE\!
.L
ges' Syste. DGL / beu3e^e 9€cnAerÄfottr
^)u.r
l r l ( t \ -L
e,= R^.# ä= Rr*#.= &+äg
c(i,r)= F.- + = - ,(,c^ (,1 f j(.,Racl) --- c^.1-: j :&c.J rr. 24
iurcz('{ + j ', ' nn c^') cL 4+jd1^c^
unl R,tc.r + u. = - uJ ua' u. t- u1.'1= u-i!-9 , y{t1'
, r R^C. . , - - , , rßr9-, , " .) ä* y: -u 6- -u:c l - rD = Qrc.{
o- a.y'*)---uä-uä
q4.ru) z-ft dt=rvar
I (.\ = s- I
sz +isrt-u(a -- 3+ 1si^z ges , ) (c)
I t - - ) , lpr--2 y"( t )= C"d"+Ceizt
u(t)= 3ea+ zl.fej' l )e(z), 3d(o)t2r-fc9lei"l =-3,2r^1ffi ei"\
= -2rzl"LEd^'^*ut\
- -ä-ö s in( ' t + arrol)
y(t)- y*k)+ yp{c) = Ct€" +C,öe - f, + ftq sia(t,+z,,ro})
q(5) --Sa+,tor ' (S + 4 Ses: 2015 tc(j,,)l dS r"J ti,(ia{
rv-46(iu)= -r- ---:--
' -U-+loJV +4
Ir u<<,1 ' 639D=A 5db
U>>,(:2oloj \uJdB
:zcü9/Drk
::-_--+
f (..\ -- -Lc 4+5 rr,k elR qes; Ortsk,,rvcA+9
tct"g lctyl l dD. 2ol"s\ iu-r ldB - 2olol l -uL + 'ro'^ jv +41 dB
-'-
*1.-.='.'.'..
n". v\<4' 6ö9V.4 : -2odbU>r, , t , - to lo3iuldß
'2d9
-1,08
n2otylct)t|läg
Cty)= -e 1JJ93 C(jo).-e C(j t )=- '*k
G(i"r= -,r {ll! -- -..e 1iJbr5 =
=_.s4- i" '$
Jht6(jdl
ß4!"tr
a
ges, !, xa (c)
.4=J n*, = ". ji a"a"'- *" [j" a'" -,
"-t'] l, = *. [io, -j.: 1x= Ä n=ä
x^(ö. l*^\ cos(c*9) = * -s(""1)
= - * ::y
ges' G(s)
Y (4 + u^- q^qr )= L^"\
- v G,r 4. - - . - - - - . - ._
-' \.r ,{ r G^ - G^G, Ä .ä-c"
Y- cn (u-v+azV)
)u *y, = q, l - ao,* 1"t' 4,9,6,)
n f o 41" Lo -aJ /
^ t ( r t . - -a\b=LrJ, c- 'Cr 'e 3l / , - \ r
s
r ll R"()-
-u.\-1- "
Ret. O S ge9d'1to95e^
tzo 5 oRen
Xet'. Slst-en\ -EL/ A.f€^5s beÄ i03e4344(.r.
l.lc (o-1 = tJq (o+) = u.
Uc +?si + n. i t L { i - -o i= . *
#'** ff' -t'ru''=" v.")' ,t#)
''"4 /". *; \! y't"). S )(t)- o
y"(t) . -rs ß::& y'r") * $ v {t). o
+ l -- ! t , .-
r 9- , /L!
y"(t)+ 2eu.yr(t)* u"z y(t)= o rv.=A r y@+)= t l
ses: xq
-lrl- -4 4
- l , r l r ,3 . \1 ' t ' r^ l l +!_- ( , . . / \ = l -4{ ; +l - ' - - : - - I; ; - t L . r - , I ' l
a" = "- l
E+l i -++el=<' t ' r 'J a11.Li.
lx{,r)l
Y (t) = x k) c"s(v"<) 5e', Y(1v)
.r,)a, ]
--
= ä (xÜr,-,,11* Xt1(v-u)l::-=:*'#:::=l::::
G (j.,) =
6{1d-
LlnUc
A / ;,uC
12,+ Se.',+ 4/jo"C
^
tj-)'LC * j,,o tc '
a
4ro ctrlUu)'+ Jv RF + 4 t jv) '+ jv or,1? + 4
Xol oS\G(j")\ dE= - 20 lo5 1 -ua t ju o ra? +z1l
9 <<,1 : -261511 ,11 = O dB
v-- 4 1 - 2ol03lo \^11- 45 tJ öD
V>>,41 -(olo3 lUl
o
-Lolb
-q0d8-(od9/D€k
u (r)' ( cos (e<) ges'yJt)
e(d - e(t'a)v| 5(t-t') r.^{t'))
= u(<) - u(t-^) = ( fcosp'r)- cos (2t -2!
les' 9chuinsu^Jsseholt
Schu,ing
I t -
, ,L
u^SsgeUoLt =
nl1, I; I cos G)d'c
_I l"nß
4l
fr- J co i tt)dt-n 11
r-----;-
x(r) .
t.ry:14-t""T= { (sin(r)-si^( +)) " +
= { iit' 4 co,(zt)ld'r= .+^ [" - + -,t{1 =tdr'- tar
t= Ot \ j5. . .
q
ge: 'X(s) Hinuei: : 9. 'b:Li tut ion u=GA , JA"a" 'R
a' Y d ' t 2u>
- : _;
dule duX(s) e ar= a*+ r.,,=AfIs-'
I ':-JU
l4- ) lT '
2 I-i, fn '= F?= ' l ;
l.l,'l"u-20d9
- lodB
ges' 6(s)
O '((n
-1qc\! , . ao\Slo, \ d0, x=,t6"ru x o,oL
(-rcr"3lctjv)\ dß= 2ofloSlo,orl ' t"5l$. 'r\-
c!,r) -- o'o1'(ä*' t)
, - lts ly+4[- ts l f+r lJdß
s+ or4 _(sr4)(e+'{o)( lvra ) ( j$r a )
c,(s)- 2
G(s) = 4 rs 3es , h (t)
A- s la aL
H(s)= cG) + = (4+s)s s+4
Hc)= + - * . -"ut t l=( / -ze")erc)
)"-7t*y=o y(o)=l ,yr(o)"O ges' y (r)
?(r) -- st- s +,{ = 6 p.,"= i tff -i =
)(t) ' 6,.f ' ' .o'(f;"-v) i).4 - s.os{
y'(t) = 9^ f ' tcos 1f;t*{) - od' 'sln(Er'{)
[ )$.q=ä e=orcton6=?
1) c--- '\-^-- 2-cos Y
y(T). rd '"cos(€' i -ä)
I l )o= f ,<"sf -$siny
= -.t
5es. Sys lco )GL
in Origino( vor iob\eA
a+ 12
g.= -eß"I" + R.Ia. i".= -jgrRzCzI.r
T.) u"- -Ra(2+ j -RrC)r l
rr) u6= R.r^ i- (e+1ä)r".
uE = ll( jG'RRna - R-Rn) Ia. 1.1e
l la=cc;, . \= - [s. 2+. iur?zcruE x^ (E**.r) r j - t tcn
U.. ra(R+R,i + I_cr RTcn = 9E:!!&jß! .+ in ti)
Rr"a.r + jc.RRaCl
nc,.^ uitü +(ä-0u"(t) = -p^ [zrre tt) + Rzc. u:(Ol
G(s)= s(sa+(s+(oo) oa<' T4oolc-Dioqcomms+,(
6(iu)= aoo ju(*J * äj '* ' t)iu+ 4
10 toslc(i!)l dB = qod6 + 2olosl,ul & + 2o$l *i - * i'. ,r. I dD - ro t5 I ju+a ldB
-_-T- --x-odg l l : v((^: od9
- ?r 55dD u-- l ; - }dg ,eo t"rrri- loao ut'{' -2oloilul
\. + orc ( ;rfoo .fiju *a) - orc (3u+/)
-_}--T--
I ' valo.v- to.U >>ro r
orc[cgu!.
Tt vt<^0,vz, lo 1
u t5t0;
l I : V4(I ' oU: l : -11(V>: l : -n l t
ltfL
n
20loJl69,)ldt
9es' t / x^(t).v
+r[.-ä(dr"-d)l= ä ri,r 'ä q' -1
- ,1 , ^i=f Jsintdc= { x^( . )= l* ,1
*. ' f , fsi, t u"o.- {nsJ1,t-ai*)a"=
cos (t, + e")
, l \ ,x^ (t) - ä cDs (c- 1) = { s;n(c)
C =,lOFp i(!) = /rf 6 <ps(<-rrt) + <r 6 c-os (eo"Q c.r.r= ^oOnia Ses'U
r ,=fa rr '#A 7^=+ zr.* :" u, . 1^r4 s l=rr l3t.- Jqi J rs4 e
tJ ---hunl'.lui\" -- rF V
