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6.1 a jeo
(a) x(s)=-E =
cr,3s2r2
— As-IDCe +)
(s+2)(s+41) -2
. SA
Ss+2
X (Ja) = X(8) =. des
s=0+Jw JW +
poles : s=-2
zeros s=
(b) X(6) = 28?
s2radzs + 4
o ast
(s + Jz)2+(J2)*
KCgo) = X (s) = oo?
s=D0+jmw ju 212 +u4-w*
potes gs=-NZ + qJz
zeros s = o (double)
= j 2
(e) XCs) = 5-5 Tee
3e-4
CHAPTER 6
" (s-3) (s+2)
x set)
x (9) = (s? + 72)
6.4
(a) x (6) = 42 e" utt)
t2ctt) céu, Sis (5)
ds?
-2t Lu !
£ ult) <——s rã
dz t
X Cs) = ( = )
2
x (s) Ê (s+2)?
(b) x lt)= ef ult) + sin (27t) alt)
St) 4 Sad) «É, 8, (5). Sa (9)
+ Sou I
€ u lt) O Gai
. Lu 37
t PR A
sin(37t) u (t) Zesnã
( , 37
“x s) = (8m)(s2 +97)
(9) xt) = E tuto)
st) &s es (8) - slot)
t utb) PE =
4X (9)=— -0
x (= +
(d)xG= tu&)-Ct-)utt-)-Eca)utt-a)
+ (tas) ult-3)
s(+-T) Me, ente (e)
: Xg = (1- e". e2s +73)
J
o
+ -2T
=[e cos (3T) ult) dt
to
t
ES MO)
J sSÚT) dt es, “3 s(T)at + s
x A s+2
(s) s (s+2)2+9
(p x) = 24 E (ef sia) utt))
+26 És SL L(s)
d
de Lt) És cLcs) - lot)
a s
"4X (= cado lmss e)
l .* 28 +2)
+ ca (A
s?+as+2 (sT+2s +2)2
2(s*-2)
(s2+ 2942)?
X (s) =
b.3
(a) X (s) = (+) e
Lis) ——, 5 Lita
s —bo
Í <——s, et utt)
s+1
"
€
Seta
o
(1 - 2") ulb)
O MD (es)
(s+2)2
S-L(s) — -tAtt)
gts Lts) es L-T)
,
(s+2)*>
xt) = ct (Ct-2)e Et ut ao)
«—s te ulb)
-a(t-2)
=a(2t-t+2)e ult-2)
(o) X(s) =
(2s+1)> + 4
x (3) <—, AX (at)
10
(d) É xt) es X(s+2)
a(ls+2)
1 “(s42)?-2
(2) a. E x (t) «—s a(s X(s)- xtot)
x(ot) = lim sX(s) = 2
Sw
ur - 2 (SE -2)
t
() jx(2td — To Ex(E) provided thot
9 x(t)=0,t<o
2
(E)
AT =
6.8 er ult) edu,
x (6) = et
we (ont) ultt)
»
SL
Ae A (e duto p-Pt) us
Using s- domain shift property :
| | 1
X(s) «+ (ros + (Er joia )
Es 2(s+4)
2 "(s+a)i+o*
«XD Sta
(s+0) + q
=
6.9
8)
(b)
(e)
Unearity
x (t) = aox(lt + by tt)
e +
Z(s) = J a(t)e q
= $ Cox(t) + bytt))e-S de
- Í(axlt)e St + bytt)erSt) dé
e o co
= aL ix tt)e-St at] + b[3 y tt) e-Stak |
Z(s) = aX(s) + lb UCs)
Senting
2 (6)= x(ot)
Z(s) = x (ab) e St ab
E jr ed” au
9) = X(ã)
Time shift
z(b)=x(E-T)
(= Jult-T)e tas DIE
É x (9) e (NH) ao
If x(t-T)ult) = x(t-T)ult-T)
Z(s)= Ê x(9)e Pe ay = e SE x(x)
1
12.
(d) s- domain shift
z(t) = et tt)
ot
Z(s) = Ff utt) e” 2a
Jr) pts St q
Z(s) = X (s-so)
(e) convotwtion
Z(6) = xt y tt)
a
= [x(T) y(E-T)dT xt) ylty
causal
co co +
- a —S
Zi) - JUS ytk T)dt)e-SE a
[o] to
= J Cf x (T).y (Mat Je A op
(Pe e «tuga 4)
Z(s) = X(s). YLs)
(f) differentiation in s- domain
Zit) = txt)
£(s) = Ss x (t) e SE dt
o
- P x(t) let) a
= a da E - st
1: (x (te) at
x Lt) =(-2e72% +22 FE) utt)
(o) X(º) = 280144
S2+28+1
..2 + .-8
s41 (s+1)?
x (6) =(220t- ate ft) ut)
(a) X(s) - BS +4
s3+357%+2s
a 24d s=E
s s+1 s+a
x (+) =(2 4et ze-28) ut)
(w) X(s) = 28lras+s
(s+2)(s7+28+)
A 2 o
ss2 Ver o * ts+9)*
x(6) =(272t4 pet) uls)
Cf) X(s) = Be+R
s2+u4s +5
. BS +46 Rs
(s+2)? +1 (s+2)2+1
x (+) =(3e72t cos (t) - 4 e 2 (4)) Ult)
X (e) = ust+as+4o
(9) Xts) (s+2)(s*+28+0)
l + B(s+1) -— 3
s+2 (e4)z2+3*> (sn)*+3*
2
x (tb) = ( 2% da +38 e cos(zt)- + sa (28) ut)
-9
(s +H)(s>+ 28 +10)
=| S+1
s+1 (s+)7 4 3%
Ch) X(s) =
x(t) = (-20t + e* cos (3t)) utt)
s+y+e "28
s2+5s+6
z =| -2s ! =
+ +% +
s+2 s+3 s+2 s+3
x(t) = (227 at -* -3+) u(t)+(2" 2(t-2) - et)
ult-a)
(1) X(s)
"
H
b.i3 4
(a) 5=— y t)+io ylt)=2x(t) ylot)=o
x(t)= ut)
éu, s(s Y(s)-2) + 10 Y(s) = E
[ss +10] yYls) = E +19
Ys = (5807) + (555)
= Uf(s) + yº (s)
Vig -s (dna) yitoA(1-*uly
YN (s) ar =» yMtt)=-22e"* ut)
f
(b) SU) + 5 Soy tt) + ey) = th x) 3 Gondt)
ulot)=-+ ,ylot)=5 ,x(t)= efute)
da lots ss seleto +8+5
+3
=(-u-as)
a
- 1. Ss
us) -(nnisos)* (tes (s+3) )
= YÍ(s) + YNCs)
a A
HO) «Era 2 mo yilt)= (pet 40% e)
S+1 Sta s+5 A
2 aa EN
UN(s) tas "=(-2 cia? Jult)
dz
(e) ay C)+uylt)= ax6) UCS!
Eye] -2,
t-o+t
x(t) = ut)
É, (st su) Uls) -2 -s 2d
8 a +85
Ye) = ( E) +( s2 +u )
a VÊ(s) + 9º (s)
8
yHs = s(s2 44)
&o
Lã àrlot) = x(0t)-iL (ot) R vc cos)
a (6*) = x (0F) — Tutor) É - Ve (to+)
L =
L
So we have 2 Initial conditions for 4 (kt)
To solve for y (t) | note thot (a):
Vls) = L(sIts -1(o?))
y ty! [UcsoY
() (LsZ + Po +L)ICS)-L( itot)+sibt))-Ri(o!)
=sX(s)- x(ot)
| EXE) = xtot) Litot)+(sL+Ri(o+)
I(s) ( Ls2+, Rs+ + Le%+ Bs + 1 )
3 = LeTH
Ig = 1Í() 4194 SP ChsTAS
YU LSIÁS -Litos
s.X (5) - x (ot)
2 I
Ls + Rs + -—
J
13 (s)
Lilot)+ GL+r) À (ot)
LsZ + Re +
J
1º (s)
=1H RAS : x (t)= eU
lot) =2 A, Ve (o+) = UV
X(s) = + , XLot)=1Y
al
i(of) -2Ã
ot) = Ac2080 A
à (ot) = DAR « es
Mf(s = sics RR
Ce tsmlsr) — St% Se
1º(s) «EE
s*+35+2
f
5
Us) «e-BELs
(s+4)(s+2)
= ts my sc .
(s+1) (s+2)
ye) --L ,-8
s+4 s+3
. yf (e) =(pe** + e Ceu
yr) =( get. Ge Hut)
(o) R=20 ,L=1H C=5F,xG=ut),
=4V
Ap (ot)=2 À , Ve (ot)
X (s) + , xtot)=4V o i(ot)=2A,
Tot) = d=4 01
1
ou A
UF (s) - Ed = Ea
T" (=) =Tf Nr Dleto) —- WD
s2 +25 +85 ee)" + ar
da
ya SE a
Ss? +25S+5
=HYe-10
(sm)? +2*
Ms) = DHE40 362)
(sa1)2+27 (s4)2422
sus (4) «(eg Uize Cosa ze” Siret)ult
a y' (6) - (e 2e "cost ie” s ret) u 3
y"(&) = (-4 cos (at) - 3 sin (at) ) et ult)
6.Ib
(a) x(t) =er2tutt) +etutt) set ut)
g
X(s) =) xt)e Cat
-Do
co co o
Se testa slot et at + fete sta
o o -
X(s) Lo,
s+2 s+ s-|
"
ROC :(Re[sy>-2)n(Rejsy>n)a (Refsy<i)
=-1< Refsy<i
*. From now on , we use tube and properties
0) x tt) = 22t ee (28) Ult) +2"* ult)+etutt)
bXF s |
ROC : Refsy >-2
(a) X(s) = 4? S+2 '
causal ( right - sided)
x(t) = ec2tt+s) ulbas)
&) X(s) = ( Do), POC: Pefsy<i
de s-1
anticovgo?. ( left-sided)
x) = -+22t ut)
-Ss “25
(9 Xty)= si -+— +)
POC:Re [sy<o
anticousal ( left - sided )
x(t) = Se letu (-t)rul-t-s) +u (-t-2))
xt)= -u(-t)-S (t+) -S (t+)
“1d -
(4) X(5) = sc! (4720), oco Re joy o
causal Crignt - sided)
e Es u(t-2)
t
xt =! -Tu(T-2) dl
É
xW=[-Tadt a (4-4)
Rb
GiB
(a) X(s) =—EI2
s2+38+2
..2 +05
S+1 Sra
() ROC : Re [sy <-à wunticausal
xt) =(-2e t+ e) ut)
Qi) ROC + Re fsy>- causal
xW)=(22et . e-2t) utt)
(i)PDC: -2< Refey<-
-2
t
tuo sided [é | FatUsat
e , anticausal.
xWU)= -2-2t ut) - 2 et ut-t)
2
(b) X(s) = —E HI
(s+1)(s?-28+4)
16
= E sm E -1
s+1 (sa)2+3 (sa)2+43 /" FT
GW ROC : Re sy <-: anticausal
x) =[-Feta etc (at) e etn(at)
ul-t)
2f
Gi) POC - Re [s7 > cqusal,
x Co)s (Ret pet cos(st) io e co (Et)utt)
(ii) ROC:-ixRefsY<1 E causal
double-sided
e* anticausal
x(6)= et ut) (E osJTt “as sio(J5t))e*
Ut)
252 +uUsA+42
(e) X(s)
ce +2s
= AA
= 2 + S+
(i) ROC - Re (s)<-2 —lepe-sided (anticausal)
sxtt)=28(t)-[i cet utt)
(1) ROC» Rets) 2o rigmt-cided ( cousmt)
« xb)= 28G+[1-2e02E] ult)
Gi) PDC + —-2 S Pe (s) <o
ax) =2€1t - ul-t)- ec2tut)
(4) X(s) = Sirzs+r
S2 +20 +1
- o 2
= V+-sã + te+Dê
VYls) = 2,02
s+1 s+3
H(s) = C2(s+3) | als+2)
Ss+1 s+3
| |
=-2 [= + sas ]
ht) = ca (et ret) uct)
6.m d
(o) By (t) + do y (bt) = 2x (t)
(5s+140) Y(s) =2X(s)
ode
2
s(s+2)
h +) = et ult)
Cb) do y + s vtt)+ Eylt)= * (+ de x 0)
(s7 + ss+6).YU(s) = (s+1).x(s)
H(s «mst
(s+2Xs+3)
2. + 2
“ s+42 s+3
ht = (- 2 + 2 2-3) utt)
30
3]
(<) dt y (t) - 2 y (6) + so y() = x (W)+2 Exit)
dtz
(s? -2s mo) ULs) =(28+1) X(s)
HOs) = 28H
(s-) 2? + 3*
- 28) (3
(s-1)2 437 (5-1) 2+37
ht) = et( a we(zt) + sin(3t) )utt)
6.22
o) H = ASA
(a) ts) sis+2)
“= 48)
H(s) XE)
e vit) anã y tt) = x (t)+ 2 x (E)
(by Hs) .— BS
S82+2s+to
. YES)
ht = SO
e 2 ylt)+ 2 vt) + o ye) = 30 xtt)
() His) = 2istD(s-2)
(sa Xc+2Xs +3)
= 2(8s?-s-2).
s3 +6C? +Yjg TE
32.
UC
X(s)
3
sb, 6 dê 404 nd E 40) + 6 y
- 2(- ie En Es do)
(A & = o A —
o 2] 6 [2] Est 4] ,v=[0]
Hg) = (<I-A)TB+D
do
= [E | [E + [0]
s+3
H(s) =—dl-s
(s+)(s+3)
W a-[ 2],oi) c=lo a o-to]
He) = (si A)! GA+D
=[o 94º Aliado
S2+5s-6
35
inverso system :
dad? d dz d
t)-Dytb)-zu (tb) =x (6) + o x(t) - 6xlt
dt? J a? )-2y 6) dt? dt ,
ba e dos
a Hts TE terre
(o) ) s2+ as +01 Ro=- dio
a £ !
“(en) 74102 4 s
[4 Ge) |-— E Heroqrête
lj +1-31o]]joo+ + 310]
|H Cáos)]
Vo ME A
1 Jão? |
|
-1o | IO Cu
(b) H(s) = SÊ+28 je
) ) e 41 Js
| Go]- [ico + 35] jco -38]
lg0+1] -3s
[Rlg00)]
as
(O) pts = 2 jos
S+42
[5e0-a]
[H Cjos)] “ Tjasal -2 > &
[H Goo]
- re
“ - ” x
” N Ê "
é a ,
. - 2 “
ao a - .- “a
-a =
bat] Mpoes - di
M zeros - cx
so
Ae +iPk
-Hpe + FPk
M
(0) Hits) =1E LES ana [NC E Etico. ote pel.
EL de) piso - ip+|
Aly (com poco
M
Ei (a-pry+oe|
M
O RE AÇo- pr
“OM
e MET Ao -P)Z
H(s) = 1
. SX . jo - x
(+) For E (s) = S+á , H (0) = Jos Fa
LUC = mto! (Sto (Lo)
SF
Co) H(s) =S Ri Cia = dE
z HC) = 7 — tonT' (mw) —- ton”! (+)
= 4 - tonTt(w) - tono! (LS)
6.29
(e)
& Hlja). EminT (Go)
Generaizotion for Hs) = Hits). (s-c)
where H'(s) 4 a minimum phase port,
Re(c)>o
Emi (=) = E'(s) (s+c)
S-t
S+€
Hap (s) =
fiminT(s) =
H'(s) (s+ 0)
HCsS), Hlmia (=) = Hap (s)
Hap (ju) = fere
FUu+e
[tap Ciu)) = à
* Hop (gw) = 7m-2too (E)
+
P6.30
Part (a)
ans «
0 + Lestazi
q Tiatazi
-2. 5268
512734 » D.5538L
0.236 — D,5A38i
Part (di
o.86s01
c.5000 - 0.8660L
.oupoi
“ocooi
“ocgoi
icoool
aa4ti
«234
«cooui
cocoL
Imag Axis
P 6.31
- Plot 10f 2-
Pe.ai(a)
1.5
os
g
á
-1.5
—
-02 o
Real Axis
26
Peszs
-Pot 1 0€3 —
Pe.3a(a)
E
40
20
10
-30 —20 -10
-40
wiradis
L L
0.5
04H
= 03H
o
01H
-50
1,5
=
S
petmb) >
-05+
++
-1.5E
20 30 “o so
10
-40 -30 -20 -10
-50
wirad/s
Peas
— Plot zofs -—
Pe.33(b)
50 T T T T T T
-50 40 -30 -20 -10 0 10 20 30 40 50
Yo
15
Ps.33
— Plot 30f8-
Pa.33(c)
“o
T T T
1H (ml
05
-50
-10 o to
wrad's
20 30 40 50
—40
—30
wirad/s
20 30 40 50