Baixe Resolução - Haykin - Sinais e Sistemas - signals cap 3 e outras Provas em PDF para Engenharia Elétrica, somente na Docsity! CHAPIER. 3
3.1 :z 38n .32 : En
o tn x 2 “e 13 e € -J43h
Ca) x[n) = cos (“En + E )= + .e
2
Nas a,-7 cy
es oose nike [-6,-8,....6)
«Le IT Lit
x[3]=Gelê ,xlisl:q-20€ ; xlkho kata
kef-s...,eY
EA
Ne21 Choose ne frio... 104
x(-5] =x(51=5
al) 23 5xb] .-Tj
x Lo] = à k e$-1,...,1DY
+ xLkI
(e) xnJa E gin-2m]+e & En+sm] Nba ao =E
ms -do
Choose nike 1 -2,...,38)
A
mL sk EL
xEk]= | [Dare ? o q e ]
= + [2 + avs(22k)s (E ] kef-a,..,3)
2 xEkI
+ja
a
L
06) = db (ce É, pci)
s-En(Tk) kef-s,..,2)
& Ixte3]
(o) N-s Ss nça RE
choose nketo,.... uy
ZA : 47 -5SZk
- -5 47 sm
xt = 4 (02 +20 15".ae 4 EL
(o) XxUk]= E Slk-zm]-aslks2m]
me-0s
N=-6 — No =, choose nk&et-2,.. 2%
-intta nã A
x(n)-€ ê a (2) rat SEC)
x(n]=2 cs (SEn)- 1-2 (-1)9 ne f-z,.3y
(dg) N=7f — So = E
choose nl el-s,..,2]
9 e Ss
xinj=de 1 "ge dn = 2 sin(£)
ne f-3,....3%
(e) N = — So:
chose n,ke T-3,...,2)
27 2m
win]. TO Ls çtto 2 cos (258) ds
nefj-3,...,3)
(p Na — so
choose nike 1-5... ,64
-1Ek
x kJ) «e 36
« (1-5 Ela sn (EF (noi)
[E sin (Eta -t))
net-s,... 63
3.3
(Co) xt) = sin(27t) + cos (37t)
- gar º eram gsm + gismt
2% + =
Go = gcd (27,37)=7
t t
«la. 5 x [2] = ns
x[3] =x [-3) ==
x UK] 0 , kd
a]
(o) b)-3 st-t mj+S(t-Sm)
tr=- os
t-+ ae
XLk] = + (e? Sta se Er)
= 5 (1+ cos(22%))
“js Ixckal 4 X[+
eras a
-4+-3-2-4 0) 1 234 EK 3 -2 “> o 1% 3
(e) x(t)=-5 EM g(t-am)
ma-09
T-6 q
xkl=7 (1+€
x[k]= 4( L+ €
xlk]= E Sb)
Lia
e dA
Ci-k
')
10
(ds) Wo = 7
x[2]= e TE xlaJe elf
A : 7,
22º X[-4u]- 2e"*E
x [4J-
:M , :a : mm
“1 ant + -F37t EAR
x(t) « é E, 42017 ,7* s 224º,
e teme soe dE ris
xtt) = 2 co(s7t-D)ruco (unt+ 7)
(2) wo=27
X[k] = entar -u<ksu
x (6) = s q tante tto) sin (97t)
leu o sin (7€)
Ds a
(f) Wo = TT
x [kJ =|k| -3<k<a
nlt)= 2008 (nt)+ uy ws (27t) + ts (bat)
—
3.5
(ay x En] (5º u En-u]
x) = É xn)e PA E (tiny
n=
n=-c
recto |<, always
2
(8) onty one parod |
a an
Ix(e*2)|- ===
Ju-+ os nata sn?
- Sin AU
sx) o sm tont (
bol
(db) x[n] = à la l<a
n=-
ds
xtett) = É (aci), É (ae
às 4
=>" DRE Tod
x(2 ) I- 020? + rag
|- a?
? (+07). 20 cost)
Ixte 9]
|x (ei) [= ate 25)
(3) onty One period t
2 - cos LS
gx(e8)
)
59 y-n
12
(o) xInd=5+5 cs (X.n) Inl<w
À a Nº 4 IN A
x(2t%). A = (1+£ +e Je?
n=N A
a DsdiNAA)O) 1 so(BNCAN-CESA)
Xe J== sin(tt- sm * + sin (2 1
Er) (QN CRS)
x
vo sola tON Aa)
o ia (ZE 1
sita)
Note : 69) On&y one period ! ASsumina, N=7
ê 35
3
6
25
= $ 2
“45
2 1
0.5
E ' º t 2 % 2 0 2 4
Omega Omega
(didn] = £ [e-2n]
X (em) = s ele-anje dO = TM
n= to
|x ce *)] =|
axe) «com
15
(o) x (ei) cer E elnler
-E 7
; au tn) (ny
a(o o +) e to Ja )
=
a
sm (õn)
x En
] Taio) 0%
n= 4% cx (4) = E+=S) =
+ NE
="
(dj x[n 55 (4, Gin tn 4 Peq )
- 8
tis 09]
x End a (n7 +)
(o) xln] = (3 -sntaçeRO e Pan Tsinta).
e E PiREaM do)
o o atnespa ot
x[n] (1 £ ? Dt , ds
Tax ã >
A EEN Ss gta)
4 SODA dO
el da)
=
A
no s gi gts 1
ne. xbiJ= aal 3 ds +J = dnj=o
-*2 «Jan
e a (ant FT i-e
n=-3 «al (4 Ja ds +J iz daj=
no +
néctos xd (1-C00) o)
="
. 7. “
(g) mn] = (Jean - Peltda)
2
nzo « xfo]-o
não : x[n]--2 Ci cs (Zn)
dao an Lo
E:
(o) x(t) = e ult-)
X0dw) = Ê x (t) od et de = ? pts a
- Do
X (jo) - e Cgw+3)
ê ju+3
|X (ga) | = LD
VwzZso
E x (30) = CG - tan”! (L)
(b)
(e)
7
02 T T T 5 T T T
001 — +
AMO) 00 - Palo) o |
0.005
“tl
x tt) = e
X (gu) = 1 e UTI a 4 $ o btit q
X (ão) é W41
x Cia) [= x (30) + Xljw)-o
A Xen] % X (jo)
Tw Q
xt) = te Put)
. doing mtegaal
XGu)= Pro SE it qr by ports
. l
“Go = Tras
[x Cjo)| = À
V Qu eu2)7 (yu) >
gx Ga = to! (is)
20
3.8 cos (tu) lo | <=
(Gy) X(ju)- l
9 , Dlhenise
Ya au - Je
-A + e juwt
x Lt) “om J z Le deu
- a
ta Sto (t+ jts (t-
-m J (e? Poção DO) des
-7/a
- I 7/a ã
too qr do? + 1 de =
t=- x(1)= x (4) = >
SEE SH
tati xt) sim EE
j )
A. Vale yet e
Fon Eme Et
5-1) =
+ t=1
q >
+ xe) = “Als : [PASTA .
sin Fem), stelmatt ), eruisp
Ar tem) 2e(m)
mesma
(o) X Cgu) = 272% u (q)
A Po o-20 que CP Cit-2)0
xtt- [e 2 do = ode as
x (6) = !
2n(2z-3t)
2
t) =
x t+) t2 4
===
- 7 Gt. -
(a ) 14) = (Ze zu pic daJ=5 ) jute 24
t=2 1 x(2) = =
. E - sin (2te -2))
ttz x (t) EO)
a £
(e) se) md Sae dito
x) = ws(st)-Sasn (at) — tto
t-o = xo) =o
Q ,t=o
=| À eg E go(lst) ,tFo
da
(prt E(Le* “das - fe E go)
1- cos (2t)
rt ?
xt) =
E=o = x (0) =0
o » t-o
a x(t) = | |- vs (2t) tão
mt
Poa]
22
3.9) Note -
G = continvous P = periodic
D = discreto À = aperiodic
-3t
(a) x(4)= 2 oe (rt) UML) — CA + Fouder
Transform
o 0 se rémt + Fost
as PROC AR Cs -
x(go)= É x ge de
o P o: +3 (ro m)t (3 dirt
o
X Liu) = — o + — 1
3+ 3(w-7) 3 + | (+)
===
(e) xtmi. [8680 vg on(B) lnisão
O otherwise
D, A : discrete time Founer Transform
x(ui2) = E eitÊ ca
N=-19 (E 2) ma
40 a(F-snjo da - 2 *h8
Xe) =» e EST
(439) - 38 (505 -2))
(e ) - so (0-8)
==
(o) xln)] > D,P : dire time Fourier Transform
27
N=7 2 So SE Uncose n,k elo... 6Y
as
(o) XEk] : DP es PD o: discrer time Fovier fenes
N=5 > so= 55 chooce «nf ef-z2!
xIn]= 2 (sa Zon + sm En)
ea
cos (E)+ Gsn(E)= EE , lol< 7
(9 Ku]
CA «> AC o : Fourier Transform
õ , otherwise
Mm o .
x(t)= — Set au
-n
04 jo(t +)
(sd e das
x(t)= Sos (at) + at
mts) '
pet cx(-t)=
| ted
208 + | we (mt) Lad
mts z
en
(dj X (30). —s» Fourier Transform
x (t) = (de? 2 IE q + fede dwt clio )
+ Eos
xt) = od e ti es fale 1) dos )
ab
. It — (t+1)
xt) dE (1-4 )
come
(e) x (0) —» discrete time Founer Transform
A .
x En] “a / mean
x En] E CD A | não
n=0 xl0] =o
0) ; (2-e
ato) = | 3 (1) não
(f) XCk) + DA «os PC Fourier Series
x(£) = z(gnlsmt)+ sin (47t)+ sin (5 mt)+ sin (67t))
(3) x (2) =|sn(s) | —s discrete time Fourier Transform
o
| . 3 7 às
xUnJ= (fc ce qn + smae “do)
x o
2d Seul? oo)
xEnJ= a CREDO n& La
Ns=l x[-1] =o
21
Q po N= th
xtnd = | 4 Gy nai
A n2-1 e
E) X(jo) S gw +Iz
a = DD
) é (qu)? +45 ju +6
. o a 3
X (ão) o Ju +2 Ju+3
» X(t) = (247 + se ED ute)
b) XU3 .— Ao
(8) (ás) (jw)?+ 4 ju+rs
, 2 z
X Cia) ” ju +41 o qJu+3
nx tt) = ne (et er) utt)
(o) X (ju) = — 228
(gu)? + Bqw+z
; ll -2 t
X Cju) = JU + + du ++
a xt) =(2! - 2e “E ) ue)
Na o 6
(4) X (900) - Ml tdo
) (Cau)? +3 jo +) Cu sw)
. -1 1 =—1
X (90) = a Guta * Jory
a xlt) e(uot+ e2t e) lt)
30
a xtnJe SEJA (4 (A) 4 L)N) uTa
3.13
2t
(a) xtt)= so (gt)e Cult)
-2t qnt “ot - q7t
x(t)= £ -. Te q (4) O RT 2 Srt)
j.2 iz
-2t
ez ulb) Es L
2 +
2 dt s(£) o, S (tw -4))
de ] = —t— E -— A
Xlja d> ( 2+4 (w-7) 32 3055 )
ps
-3|t-2l
() xlt)je ca
-a|t |
e? ET b
9+ w*
s(t-2) es qrizo e)
-j2o
.. . . BA
X (go) wz + 9
E
(o) x (4) « [| Se tet ) [ so (2nt)]
sin (wt£) ET |
mA <——
at
3]
stt).sa(t) E > Sljw)x Sa Cu)
Stu) Salgw)
á.
A 7 -27 27
2 X (jo) 5 sitgu)* Sa lza)
lo]
s- , m ajulgan
=KGw)=9 2 lol sz
o , Otheruise
5 S (gu) x Saldw) = X(juw)
2
:
-31 -27 «mn O 7 2, 3” so
(o) x(t)
h
SE (42 sn (4) ult))
ót -jt
x (4) - É (dor ul£) +)
+ -2t + Ê
é ult) —? (+ ju)
2% sta) —s SCjlw-0)
qo Ss) es jo S (ju)
. - b
X (go = go (55) (nu “Gm )
32
t
2 X(5 q) = e ( ! —
$ (2+3 (o -1))* (2+ 3 Cum =)
K =——
(e) x(t)=- 7 SST qr
-% at Sl)
sin( mt)
Al «—
xt pa q
t x
Ssttat es Sd, ssço)s(a)
-c0
gw
a , W=o
“XGgm | |ul<7mr , vão
o , Otherwise
=="
+
(b x) = gy tos 0 Fe rtta-) (Eos)
“ut
u (+) es, Es
s(t-2) <—s, Sl gu) griao
s(=t) es 2 S (F(20))
«XCja) = et oz qrózo 2 1tiTO)
á uejza + — 2+ 30
===
(9) x(t) - So dO) 2 sh SP Qto) |
sitt) x so (4) «— Silo). Sa (ju)
d s(t) <—s jo Sl gu)
35
= s (a) — -jtx)
cos (2) Salgw) «ma Lo S,MteB + 4 s,(6-3)
4
2 sin (20)
<—s ret (2t)=
Co ( PAS 2 t
a cos(Z) Laado vs vet(20-D)
e “a + rectte(=+3))
o xtt)=-gt (rece (2(t-5))+ rect(z(t+3)))
une sp xo] "ag xtt)
A
E q t
ml
(9 XGgo - SELL
2 Sete) e, ret (t)
o es 4% ult)
3
E, (jm). Salja) Es (lt) x sat)
o | ES
« xlt)= e “ult) »* rect(t)= | [= 0 O gear
2e*snh(y . t2i
36
(PD X (ga) = Im (7 1
Ju +?
-380W
L 28 <—s g(t-s)
!
ju+2 — qr u(t)
-s(-t
Im (S(jo) e add (st) = Set st)
x(4) = LIT una) — 22 060% 4 (ema)
2 a
; DR EL)
(9) XCjo) = +
= ento «e—s ret (t)
[Sitgu)]? es st) x sy tt)
2-Mil ltl<-2
o x(€) =. [ o , o e
> x tt)
3.15
(ty xl] (5) u[n-2]
x[n] =(5)* (4) qln-2]
pç
s [n-2] es q e! (e)
day do edan a
x (e ) vt pod
a
(o) x [n]= (n-2)(uln-s]- uln-63)
x [nj « (n-2) 8 [n-8]
x Enj= 3.8 [n-5]
. x (4%) = 2 Tisa
(o) x Enj= sin (0) ()º ula-1]
x Inj. £!EN cerêmo (ANA) utn=)
d> Í
a sã,
x In] = = (q) Lone" o “o )
ana o | t
(+) ulo JOL “és
(4) x [0] - [-setEm) | x [cette |
An min-2)
s[nJ4 sa 0) ss S(2i8) Sa (e)
so(Zo) [| pInt< E
An o ,&<lnls7m
“e
(2) x(4iº) - E) a le 2
de [te te-3) fe)
jet «—s - 3.8 En-4]
2
Tr o 263)" alo)
E Sus - an s EnJ
« xUn]J =-jn(2 (4 ulnje?? Eat ulnJe 7a")
*(-3).8 [n-4]
xi) = cu cos (T(n-y) e) nos) uTn-4]
31f
(0) ylt) = x(t-1)
: YCgu) =e 2X (gu) = 27% asa te Le)
() gt) = cos (mt). x(t)
jnt “3
y &) = 2 entro «x tt)
=X (ja) = X (gt) + X (gu +)
Y (go) = sin (0-7) | sio(co+77)
qA+ 7”
* (jo) = (Eno + SA Qu)
+
(o) ylt) = x(2t+1) =x (2t-1)
x Gus 2 Lx) e x)
Y (0) = Soto) 32 sin (04)
sin? (00)
YCju)= 32 =
(9) ytt)= xl. do!) ndeté ca) tatoo)
«Y (gu) = ei x(3 cm) 4267 2Ê “xl
— rim + x(3& )
- -33 :
do zsin( 0) ,-d30- sin (60)
x Co) =2 E TT 73
EO o sa (LB)
Cu
==
(e) y lt) =x (6t
Gta) es SE X (ga)
(qu) = 4 E (22t)
r(juw)= 3 ( 2 ce (0) a galo)
Tt *
(D yo) = (2t-3) xt)
y lt) = 2tx(t)- 3x(t)
From (4): tx(t) es 3(20Lo. 2 sin Co
«Yu = 3 4 ( Ef 2 sato), 6 safe)”
42
(3) From (e) s0=) 4 (T) dt
a Y (30) = 34 sor tro ar a(s) 8 Cu)
Y (Ju) 2 sin mi2e
K +
(h) gG)= x LT) de
“XY (gu) = 280 (e) —— +n(2) 8 (to)
Y (9 = 2890) 4+2m 8(w)
Es
30 =
6) tt) = E (0)
* X (ju) = jo - Asa Cm
X(90) = 42 sino)
3.18
e x (eis) es x[n+43]
(n+a)(-1) +? u ln+3]
(b) Y(2*) = Reot Ix(e*)Y — Even [xln]y
H
(a) Y (e)
!
cg lo) + XEDA XE nb) utn)+ Cmt)" uto]
2 =
yto3 = en(C&) ulni=(-47Put-nI)
45
. NT
» yin] = 2 vos (Zn sin =6)
so (50)
(6) JL = cos (EE) xU-1E5 + (xIn-2]+X 0421)
sin (= (n-2)) sin (E (n+2))
se(E (0-5) * mn)
= y EnJ= +(
(o) ylkJ = XCk] 4 X06) ES (x007)*
: um
24 [n) = sin 2 Z6 n)
A D o
sm e)
(4) YLK] = Rear | XC6IY ES gx (x009)
2
“uma
y ln) = sa (son)
sia (55 n)
3.2) o 2
dm DTET Lya
a Pd <——s=2 > | —- ul 3
O o pp nã (6) und
S 44 ru
-212 (16)º 5
7”
co 2 a. o Ho
() se tomo E né S (a
- «2
Uva
2m * 2úo
-— na
=
4 do FT a +
(c P = -a lt]
) -Co (2 )* ad é de
o
-an(a) fe + q
k-o sin 2 (-S& k) NA(S) =
A
E
E we have So = TE ,M=5
5
So: TD. 3 (1) 2
n=-s
= 100
(2) f sin? (7t) at ET VA
-0 xt 2" .5g + dos
7
322
Ca) xit) EL, 22 u(w)
t Í
ul-t) — “223%
1
2500 z2ne “Cuco
À
o xt) = 37 + Z-3E
(b) RE E, X Cjw)
gt o 2
[+ wa
> a es ame te!
' PAi) = Io)
(e DIFS;
) [3 [kJ de pio
sin ENTE N > 20
ua
[! Inlss sn (25 6)
o s<Ini<io 20 sin( ob)
HT
- sin (o aos kiss
o sin =") ao Lo s <lk|< 1
,
3 kiss
: XLk) |? Ie
o s<lkl<ito
xTk+1.20] + x [kJ
k
323
(o) 4 X (jo) do
nu mo
4
(o) É |xGu]P ceu
= infegers
2m x(o)
27m (1)
27
2 n É Je de
"
50
(1) Furthec , if xCt)even : x(t)=-=(€)
xt) = So xao teta
ST> ku
XLk] = -+)e FE q
Td, x (-t)e
X[-k] 2, LT -JkGoT
Ter» x(T)e aT
=-t dT=-d
flip order of $
XE] = XTk]
= X*[k]
e The XLkT is rear voued or Im IXLtJ%=o
(b) xin-no] = Tx (02%) ção (00) da
+
2
t a a rn js
“am Xe MM) ednto pêdo qa
-— ma
DIET , j
a xin-no] Ss x (e) grid
(o) & ito son x ln] = Z x [n] e ifeSen , dksen
=<N>
«sz Xin] e JE Pe) son
k=<n>
- XTk-ko]
dem
+ p fodon (ny E [ko]
(a) Let E faxtt)+ by(6)Y = SC gu)
co - Jwt
S(ja) = ) (oxtt)+ bytt))e de
-to
o - o :
ca Pat eds bd ya
-% - Do
-ax(jo)+ by (ju)
a ox(t) + tytt) EL a xXCgu) + bY Co)
(e) Let E [xl ylIy = C(e 7)
cu E E Mt yin) e
n=-te L=-g
ço o : +
=> xt] 3 yln-2] gira), dot
E=-c N=-c
ll es)
-S xuctt ye)
t=-0
= x(e0) y(e Dr)
a xln)a yin] 2 xçeit) y (e)
(8) Let É [xt]. 903% =M (ef)
IA
- co
m(e Mt). Do xlolyln]. e”
N= —00
x[n] a + J xte3T JedEn al
. co |
M (e) “E y tal. = J x (e )pstn TE
52.
ar) MT) É ginle! Pnçcinr qr
h=-to
=) XT) 3 yin] er Tr am
n=-to
ll
= x(T).y (e ID jan
=, x (254) *y (23)
e xln] .yln] Er, = X(2*")4 y(ein)
27
- C[kJ N= do
(3) let É [x[1103 En3%
EJA = - -Ik ton
CltJ-= Ze Pao TE y ln e3)e ?
-T xt) sz +egtn- 43 q SSL, nitenot
A=<n> nt =<N>N
ms
No 3 xt got y rh]
L=eN>
a
-N. x [E] o. ylkJ
r
MÊS, nx [k).y ck]
- xin)B yinI
(4) Let É [xt yt)7 = MUkJ
MET = Lo xt). y(t) ereta
<T2
. |
Iwo: OC go lT
Bw=|2É
ç -W2ga*
R dos
Bw 7
(=) (27
Bus
V2 à
o Td.Bw =
If à increases
q) Td increases
(2) Bw decrases
(3) Td. Bw ctayz the same
3.27
x) = To! cms
Ta By > mg , so Bw >
la) xt) q
A E A (Er
aT
—
Td
56
gw > É
&T
(b) xt) % x (t) = (e ,HI<2T
o , otherunse.
27
a - z a
Em a (2T-I6) de = 2 / uT%? yTL3 tie
32
T is TÊ
S (GT 4? a”
->7 ) dt =2 / uT2 yTE + ETA
= + 73
32 |
Ta = as TI
br
Ta= [E T
tw 2 Vs
2N2T
3.28 e
Íf > (+ 2 am
Ta = A, tento! de *
co
É ecoa
Bw =
as [febres ras
3 |xCot) |? at
Time - Bandusdth product às invariunt to
scal ing
de ok
55] x(46)-S Xlkl1e
k=-co
(a) (1) xtt) = s X Lk) (cos (kaot) +3 sin (host)
E=-o
to
cos (04). cos (kwot) - sin (96) en(k wo t)
= cs (kw t + Op)
ta,
Tros : x(t)= às +,2 dade cos (k tuo t + 8%)
al
compare us : co
x(t) = Co + ce cos (k.wo t+ 6)
=
Gi) ck = Vap?+ bz
= 2 J(Pe XIII)? + (Tm (XEEIS)*
Ok = tan”! Le
Aja
“ton! Im [XUkII
Re ÂxTkIS
Giu) Ce =/Qu?+lop?
Ok = -ton” RE
33) x(t)= x (t-T) holfwne cynmetry
XI= í xt) e et q
Ta Shao
dk + 4 x(tJe] ea
Ta
XL = Ste
T/
XTkI + Petty eta dk + f x (T)e
2 3a dt Y ?
-J Foot
6!
Tf .
xt =| Px et (cota by
o
when k even (=0 (EB, AM, o) the integrand
vanishes,
Thus : TA
“jk wçt
+ Saxttpe beta E oda
X LkJ= º
o , E even
3.31, N-1 ea
(o) xin] = = XT 0 n=o, to Noi
=0
eguivalent to :
x[0]= X[0] eU Mt, x tn] tNT DO Co)
x] = X 00) e MD x TN] INDO
XEN -1)= XL03 ce MOMOÊNCD A a x Enc) NUM)
con be untten :
x[0] qto) AN ct) ato) X [0]
xL1] =| 239 Rot) o UNA Ly x [1]
x (NJ 3 OmolMc) q iN) ao (Na) X EN-1]
ubih ie: XaV X , Vis NX N motrix
bz
Vis defined as Vere = po te)
where r(=r00) = O,l, 0 Nt
c (= cotuma) = 0, 1,000 N-1
pon! -Ikaon
ir xinise =-0,L,-...,N-
(o) x [3 NE En] ,E-o No
n=>0
N. x [0] = xC0].e Í8Seto) xUN-1] e (OM CN)
N.x0] = x(0).ç ADMeto x EN Ito QN)
NX IN] = x(0] UN URL), 4 tn) g N) Rin)
con be wrntten :
X [0] ] [e inato, PELO EI 4
x 03 jafe IUNAÇ po Ca-A
x Inc) [MDL NDA
whihic. X=2WYX , Wis NxN matrix
W is defined as iiwrc= + gt Lo (e)
' N
where r,c gre defined as (0)
(o) W.V ie defined as
N-1 A us
WVr e - = q Amo to 4 JUS) (Ce) d
k-o N
& 3
(o) -3 x*[k) TC] -2 xl] 6*k]
k=-7 k=-g
a
.£ xe sua] É |xtef-z |se3/*
k=-7 ka-7 E-=-7
Thus, we con re-write (b) as :
à z $ 1 é 2
msE; =AS Ixco[ at + E |xtrI-609/-Z [60h]
Tem Ea-7 Fez
(d) Minimize MSEz.
MSEz is minimized uhen the sum at the middle
vanishes :
* XTkI = 6 TkI
(e) MSEZ mia = + f Nope -É [x t:9]*
Ter, k=-3
Às jd increases :
|X tr)? is always greater thon zero, «o
the som :
z 2
> |xt+1|?>0
k=-7
. , z
So is Tape dt
Às & increases E |x tg]? increases ond
="F
MSE3 mn decrences
65
bo
In foct , 4 ic expected as | o , MSE3 mino
3.33 | t> N 2
MSE = JS |xt)- 3 ce gett)| dt
ta-t, t, la
N 2 2 No
(a) leto)- = ce sete) Pelxte)) -x4) E tata)
=|
mo Na -
-x 6) 3 cLcgk (t)
E-o
N-t
N-
+32 E Cmt oelt) gole
M=o =0
Thos : ta N-t
MSE = S bes - 3 a Fut) 6 *Ci)at)
Ea Ut k=o “e
T
ta x»
S xt) gr (6) de
tati é,
E a (o
N-10 Nm
+72 2 e mt (o Pat) ém O)
m=o k-o -t &
Let : Cas P x (6). e (t) dt and use the
k “tespnta
MSE =
Ex (esfPat — 5 ce fUqk)- fe.
TE vt EE k-o fat, E
E ce. 6ÊLk] + z EE : Lew?
E Ck-6Ck)|Z
E É F eco” d+ “a k-S LE]
N-1
Tt
-— >
tati Eco
to
Analogous to 3.29 cr =0Lk) a S xt) er"
t;
sr]
ta
(b) MSE min = ! J
N=!
fe =. jon]?
tato &
(o)lZ dE -
* ) tati ko
| ta 2 Not 2
- [Sie d- 2 fe tekl
ati tt,
MSE mia = O when :
tr N-
Tso = = q cel? |
ts E-o |
(c) We con ser thot the vrthogonality relation for
the Walsh guncton is as follows :
f i pk=-e
. * E=
o Pk). te lo) d L jk+e so: fk =|
2 ct çã
6) x (8) =| 2 + +
o , O Xt<C Do qet<i
“a i | !
Co = 2 dl = Ca=2(4) ==
a
3jy cy = 2(b)=0
Cos f-zde =-&
fa €s = 2(0)=0
cof -2dt =-+
EA
de va
+S fntant)- LES a + Éfei (art) 25 (8)
a
Za
= O. 1265.
(d) gk(t)= 28! E Bm (6) El pp alt)
Go lt)= 10, & (6)=t
fat) = Det) LO) = (32.1)
ta (4) = D+ E (Bt?) Bo bo (st3 26)
du lt) = Dt T(stt at) 2 L(262.,)
=p (28tt 304743)
gs) = Std (354% a04242) É L(st23t)
= vo (ais tº -as0 +32 +45)
The orthogonatty rélofion for Legendre polynomial
às
' * o 2
4 Pk (6) df (E) de = She 2
>
E
ak +!
a
(1) xte)=[ 2 |, OS E<s
o (IS ELO, Lts
So: fe =
7
, Ya
Co = — y 2 (1) dt =
Va
3 3
C, => d 2+ dt =
Ss Va
CG: 2] 3t2 dio
f Va 2
C3= 5 S st? .2tdt = 1,024
o
9 Mo |
C4="2 Lylsstt-sot?+a)dt = 0.527
ota 5 a
Cs = Jaolaist - so t + Fst)dt=1.300
5
XU)= 5 cede)
E-o
Va
5
ME = S[ 4 de - 2 ão lee 12]
= O. 1886
(ii) x (4) = sin (mt) o , ISts<si
Co = + 4 sn(gt) dt =o
CG = & Jtsia (mt) dt = 0.985
CG = > / + (st?) )sa(7t) de =6
Ca = 1 J + (st?-3L)en(nt) dt = iss
Cy = = J + (35tt-30t7 482) sin(7E)de = 6
72
1
Cs =>) vo (est? - 280 L3 +75 t)sm(nt)t
Cs = 0.213
5
Kit) = 2 cogkit)
20
A Treo = e 2 =
ME = + [SC cr) de - 2 [cul ]
-0 2E+
Mse = 3x 104
caia
o) x(t) X(4)
M M
cc ANN A Co.
LC TA
(b) X() = É X Uh] e Stuot NO
=-to
| Tia jet
x[-= fo RW) e! CE ar a)
A
co ,
Xlgu)= 4 xlt)e Pa (3)
-to
Using : x(t) em Ft)
-» Co
Va dito dd
XUJ = Ta Xit) e
—Plot 4
.. p al
TA
Ph E
él E |
! B Y
i
N I
Ê H
of 6 —
75
ofjus!
O fast
[H | H
E
NAL
O
Um 4]
P3.35-3
0.4
-
oz
Ê
So
ê
=-02
=
-0.4
pr , "Y
i nº ll
É i L T a
Ed + LE
obetrtes!
Í
jr n A ' o '
Oferta!
jcos(omega*n)
Bp3
Je:
P335.
— Plot cof es -
Paas
2)
|
1
08
=06
E
&
*04
02
a
II
|
T 1
|| se
— 0.6
E
8
*o4
o2
10
eo
P3a.36e
— Plot 1 of a —
P3.36-1
a & 91
q
——&
o
-O = 9 +
—o Fe ”
—e
2 o lo
—s a o
—s
—e Je e
o
o — e a
+ o
o
e o o
O
a e
e— — e
l
Ss 879º 4 8B747Tº & 3 3º
8 s 8 s ç.8
1 1 1
Sep Dix > Bop lx > Bop Dix >
oa 98 o
4
a
< —— e— +
o
Gn qd
o 2
Gi q
+
pb e———ta
da T
es
vans 8 é a oº Ga q q e?
$ 3.83 q ED
Ibl xi bixo tBix
Ez
P 3.36
— Plot acf 2 -
P3.36-2
o 9
02 o 50
o. g
70.15 =
E > 4 º º º
x 01 >»
0.05 “so
3 4 6 9 2 4 8
e
04 100
a
v
8
o E
— EM x
v
04 j f -100
9
0 1 2 3 4 º 1 2 ô t
0.6 g
e 50
Es 8
= = a
Taz v
| Í É
dó 9 f ? 9
õ 2 4 E 8 o 2 4 6 8
3.36
To compute the time signals from FS coeffidents
vsing ft (.) , we look qE the expressions :
- 27
Fe. RAS Kk]efTkt
le=%
27 kn
DTES : X[n)=- 2 xXCk] ef N
<Ny
Using a finite number FS coetficients (ones
that are significantiy big) ,q cdiscrete
upproximation con be done us follow “az
approximation of FS: [niJ= 7 RlkJelsrio
ng ie the sompled time vector and can be
determined rem :
À Dk o T=a NT
T NT 4
steprze
P 3.28
- Plot 10f 2-
P3.38-1
15 T T T T T T
1 d
3
E
2 05 d
£
*
o À
“05 j 1 1 4 1 1
o 50 100 150 200 250 300 350
Time index(n)
2.5 Im T T T T F T T
2 4
s 1.5 4
a 1
E
5
2 os J
e
s q J
-0.5 +
A 1 1 1 1 1 1 1 4
º 50 100 150 200 250 300 350 409 450
Time index(n)
Eb
0.25
o
do
Xlkiinormal
o
a
o
0.05
o
0.25
o
A o
ES
-
YlKki:vent tach
005
P 3.38
— Plot a of a -
Pa.38-2
T T - T T T
0
sff recaseso paoacmassao ecoa
10 40 50
Frequency index(k)
T T T T T
Mia me
10 30 40 50
Frequency index(k)
60
Paso
—Pot30fs3 -
P3.39-3
Using Unttion
mox (.) , the peale overshoct can
be deusa
. J Peale
av 1.0891
59 1.0894
99 L.os 95
91
3.140 «F27 4
(o) XL] = xt)e Tt qo, T=4
from example. 3.26 asith T=1 ;X C6)= ye) de
TSiwo “3 oto
we hove x(t-to) 5 € Jlcoo: x (k]
A le =0
ond LTk] = [ ve o km — where Goo=
480 (CE) pão a
gt Two
+ .k=0
Yu1=[ kz
2 asim ( ) , e to
qu?am
! .
Alo 2 ES$27 cl]
= | 4
x Lk] “ã E =
= 1 (ET Mn
usa (SD) tals ). k+o
k227
(o) x (4) = Õ x [k] e Jeett
k=-c
since x(t) is real ond even, XCk] =x [-k]
, — :
*&) = E X [kJ etcest, > X [Je vt x(0]
k=1 =-Do
-Ê x [Je iDe8t ST x kit ix to]
=
Es six Le]
42.
-£ XLk] (o Jvekt, q -Jwokty rs
- BLkI ws (kwot)
k=o
where BkJ] = [ xiol. ,kE=o
2x0k] ,uto
A 3
(e) MI (6,2 B [KI cos (k ot)
0
15
P3.40-3
P3.uvo
— Plot 3 0-5 -
x10*
+
03 04 as
02
0.1
-03 -0,2 -0.1
-04
1
1
o
trom,2)soo [7]
i
19
í
-0.2H
-D4F
qu
=0.6-
-0.8-
02 23 04 0.5
0:
-0,3 -0,2 0.1
=0,4
Ps.uo
- Plot uofs -
P3.40-4
04 0.5
03
0,2
01
-0.3 -D,2 -0.1
-04
o2 03 04 os
01
-0.3 -2,2 -0.1
-0.4
x 10%
em
+ a o a J
1
(rom,62)soo, Jezla
(9%
0.4
-0.2+
-0.6
-0.8+
G7
P auvo
— Plot 5Df 5 -
P3.40-5
”
e
x
+
a o q
(rom, s6)soa,[6elg
-o2H
-0.4H
-06H
(9x
-0.8H
0.2 03 04 05
91
-03 -0.2 -0,1
-04