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Ne . sf time & fra
domary
¥, W) yO HD Hl) yx (07
x toy oy 13) 2) x2
x109) x2 (2)
xi(o) x3(>)
x, a)
x (D x, 0) x) (0)
ere P41MITIVIYVVVVVIVVDHOVIVUVUVUD EDD UE TEESE
ns ars MA
Pans nardy prdodvon dot deer ohern pean
s uel 2
Z [amI" -JlZ | xe]
n-o N eo
( Wwe
2 2 an a*n)
no
a wl ans gine ?
: in a = xt
cr > (a fae
nev
ne) nA j2nkn
2 xe) Z ame
"N KE ae
NA 4
Lz ce) rey
N Keo
Nol
Lez | te)
NO OKO
Pavceval's solabon %(ny> (Cv 2,374)
Prove the Pay ae
Lis? x(k) = Cio-242, -2,-2-9)
S L
Z (MWe vagy att
neo
= 40
RUG
zs (wNb, 2,0) 2 4 (worery +8)
r +
= Po. Ww
4aS OT Dy
Computational wovaplerivg 2 DPT ddveut Conaputaion,
fh Npt pet
Woof, Comply raalt pli action 2 N™
No - Of woe aadinen = N(w-1)
Faby Fouvity Tramspoewn Cert)
(Atger' thaw)
atts an odgerittne wed tp seduce connpudabienra!
. nt pli cattén Nn N
Wo: Ok Complex rath pli > NL eg
no. ob 4 addition > ON dey
Bost of 2adry ~ 2 abgoritann
\. pecimahen Ww time
g. permahin in OH,
Ape DET wring DIT [ decimation in time)
mC { Alo), We, 2, x]
» ~PPPPAAPPPPPPRPPRPPRPPeeePeFrrEeFEEhRaAbtPIPSVYVVVVVVUVUVUVUVUVI SEEDER
az oly \
WU
xt2n> j > opt per *ICED
at) — Koy, U2)
odd
Ke Leone) j Rie)
HOD, 403) pe DFT
(4) a 4p eT
x(k)
rele)
3
K(ey = EB ttny wy,"
naa
) kon, \
= Zxlen) wy +t Erne) Wy
Nico no
K(antt)
' '
aa
2 BZ xeadwye ty Zazcme wien
nzo nzo
__ —_———————
¥2 Ce) r
le “tM jo,
Ryle) Wy, akan
k 2 ok
xdtie) = AIC) 4 We kp Ce)
ky
= Wy
ks 0,1
KUe) = x Cte) 4 wat kp (kK)
Kc 4)
: kee
~%y (le 42) 4 Na xp Char)
wy (42) = *y Cte)
M2 Chet) = x Ce)
WME) g%2 (IC) Ave opt DET, 2 periodic wile peried 2Kir pam(ety xj Be cin J aE 7
We => e = : ee
ko,
vy
Xl) = xi Ck) + wit Xr ley — 5 x (0d, x0)
Klee) x(k) _ Wi x Ce) —> x6) , x3)
I
File) _ > x(2n) wn”
nro
RVC) st C0) wy? 4 tir)
0
FO) = 1 (0) wy” + 462) wr) = KOO We” 4 1 (2) V2"
xt
¥1 OD = neo) wm? 4 Ada) wi = Ale) wy? - i
1 sJan
wr = 252 -
aul)
\
Xo CE) = SS games) wy
noe
ROK) = Uy we 4402) we
° oO
¥, (0) = Ker) wor 4+ 403) ues
HO) = aay we 71) Ww!
5 6
= KA) wy? - FLD 9)SSVVVVVVVVPPVVVDDVUDEEDDEDETOVESS
Paes
Prabluty MUL fey 4 pt DET Wel
ut overs
a Xo)
y «082 ama
me
4 18) > Z
ke ON
FCW) = xy CED + wat
¥(k42) = x, Ole _
wrt welt)
ce Ap = a 4
wk x
xlo) = KyloD + we! x, (0)
XCD = xt) ty! x2 U
KIM 2 XE lo) —
Wy °x 2 lo)
X03) = xy CD wy! Kt)
Nernst
y or
(0) wet" Ko)
- x0)
20%
Yel) oot” mg! ¥(2)
' .
(0) we
ete Ly! x(3)
Nn
\
Xz Ck) — 5 xle x0)
2Cke) —> x(2)_ xO)
Nov og, corp ruth
Sly t =4
Wo of, add
Lin bit reversed
Trin bit Awevsak eda
SIP in normal ordy
ALO) 4 (00)
LU yw)
XQ) Lv)
403) a)
(0) o© ) oo fo)
ay er [Lo ¢ey
@) .0 Jor iy
ao) wedi fy
too
[Pim ermal order
Xo) n loo)
xa)
* (2)
(3)ra
Compute 4 pk DET of the Sequimce wen = (L354) =
~
wring DIT Algerith, 2
HI LY (2) XE) e
MEM = (1% 3,4) 2
e
\ _— x1 Code" xfoy- 0 we
-
2 KOS 2427 &
f yploye& —KQ):-2 «
We ee -2SCSIVVVVIYVEVEVUVEIEES ss BEESEEDL
24: 0124
Spe ori wring PIT
ya n
4
ww awe
ob
UC22N) = HGH)
ne 01 ae Obed { Alo) Le) Xue)
—u2n) — .
toy 1 Ute) AU) Lea[ 2.@atiy = cltnre)
x Cn). xC2) Alb) * 20k)
4.(2W4)')
al?)
wecry HCI) 45) PAC2-2mat) =e fan at)
|__—§ guy xcs) Farle)
(2. Cent 41) 2x U4ne sy
409), HCP) Xan Ce)
ke 01 kr0,\
1
my kd = FS xlany wen
n=o
xy (0) = 1Co) + xy)
xy OO= Ato) - ACK)
x (e)= 2H alynar) win
neo
yr!) = tay atey
x2) = 402) - Xle)
|
ror HO) = 2 Uqna wi"
noo
Ko (0)
rai) =
= 10) +25)
Te) - x5)
I
X22 (k)= Zan 8) w,”
For (0) = X(3) 42X02)
Xo. 01) = L03)- (7)
wy Cee xu) + Web x1o CE)
wy llear) = race) wi ny, Ct)
XL loy= xy C0) ¢ wy,’ Ka (6)
12 Xuld + wy) «1,00
¥ ye Ku lr) Wg Xt (0)
10322 eu OD ~ ng xy, C1
x2) > x2) (tb) 4 wy mee Ck)
Ko (Rte) = Kay Ce ~ wa ¥ 25 Ce)
X20) = ¥2) (0) + Wy ° kar (0)
¥2Mexy Ct) 4 wa! x2) (1)
ale) = %y (o)- we ¥a2 (0)
| Xl) =r) ml xe ©)Kz 0b
Me) = ACK) , wrk, CKD
2
X(K+q) > x, (8) wg a Ue)
¥ (0) > 100) + WE? x, 10)
KU) =
We xe) 49! x01)
XQ) =. ei) + Wg? 200
*(3 =: x13) re we 5 (2)
PLY = -
Nem (0)@ we? x, (0)
(5) = C1) we! xo CN)
¥lo)= ¥\ 2) — wy? %2(2)
XO) = x1 (a ~wy'x, (2
=
> OORRRRRRRARARARDFIIVIVVV TIT TT GGG GY yw ie la le lets lee te te io es,
Spoiny DIT FFT BUTTERFLY STRUCTURE
40) xX lo) 7 (oy 7 ROD
xc yu ra) we)
4) we
42) x ¥ lo X nz) * (3)
“SB
1 x, ¥, U8) ¥ (4)
ee ytd ‘
4U2 ¥, (0) x(s)
/
as) m0 (6)
“s
ms) 2) ~~
x7) pls) ms)SA —
Datawune Erk DET Fer the degree § 12 34, + 42,13
hen
eo) fi
xno). 4G 25
wend) =l-4 2-3
Rn (0) FP2aydes xpt)=3-2=5
Foy (= p4a-5 Xai) = 2-S=-)
For lOV= le Gee Kars 4-123
¥1 C0) = xn (04 my °X)7 (0) = § +52
eV = eC ty xin GD = 3B ete -22)
x(2) = (0) - wy °¥i2(0) = $-S=0
x OS) = ww C0) Hwy! X12. 01) = -3 4)
“elo =m ( Wy ohe (0) = Ss Vo)
pi) = -1-3)
t
nL aoe ¥ 2 C1) Wa Fx
ep (2) = Fay (07 — MG" Ree (07 = ©
z = -\ +3)
eat?) = x21 (0 X22C0) =
2
Spr DET k= od» 3 xe = mr Ck) + wee Ck)
RIOD = Xil0) twee X2l07 > 10TID =e
wer ex C9 403d 1S ha) rai)
2 —S-B2F-J2-414
czy = x1 (2) +E (2) =O
x3) ¥) 7) + we? x9) = 34s) + (-0-101-j0-107)
C=143))
wt
= OITA) orig
ae FPHLPAPDADIBIDRDBDHPHMHOHOHOHHSHOOBRRARAATwie a le a le ee ee ee ee
x Ck +4) ey Ce) ~ Wei ee CE)
XG) = &1 (0D - we’ x, (0) >
vis) 2 1G) = wel x2 OD = (-3-J) ~ (0-707 -j 0.701) (-1-3))
= —Ot2 + fj O-4tgr
x ley 2 xa) ~ Wg HAC > Ow
KID = x, Od ~ wg? Xe (3) = (3) - (- 9797 Jo 9 NC-4 43)
SHB) Z 4
xb = (2%, -S828-J2-41y4, & HONS ake, O,
- ONILAPO MND , 0, - sport }o-uin)( Lue ald
. (ny >
Find the B ph PFT ofthe Agueme n) aS
n
Ky (O)= Coy y AEAD > 144 ent) = 46) 24)
=e 6
Kp (0) = C22 AAI) 2p 4d xpM= 0
=e
ap =e
ry (= arate) = 2p2 7%
24
x5, 00 sO
Kop lO) = UIA ACMI =24e
acy el Jenks
mee j2n
_ eit
x) (oye ¥ lO) + Mee X12. (0? > 242 2q
xa = eu (YO twa! a ae 0 + jle) =O
i = 2-1) =9°
ea (2D Sx (0) amy X12 (O
x OSD = kw C0 = wa) ee ee
—— , /
fp (0) > X24 C0) + Wy kre lO> 4F 428
xp A = Xa + wy ar D = OF@=O
yn (2) = ¥21 (0) ~ WG? Kar 0) = 4 - GeO
Ka (4) = Fay (PD ~ GP Kae (1) = 9-020
K(o) > %) 10) +We" ky (0) = ate = (2
! .
XCD = YOO ag CI = o4(b-id)(o>=0
A) = ¥, (27 VWe™ x2 (2) - 040 +0
(B= (3) 17H) 2 GO 40 = O°
at APP PPHPAPYVBUBDBDHDVDVABDWMNMHMDVMHDOVYVNNYVOESFSIVIVVVIVVIUUVUVVVVVVEVOVIVUIIIIYDY
XH) = elo) ~ we 2 lO ~ g- bo -G
FS) = xy - Wg HCD = ©
(U7 = x 2) > Wg CD = &
x7) = ¥ (3) 7 Wyre) = ©
SPECTRAL ANALYSIS OF SIGNALS USING DET
oe”)
> 26b Sampls
——«
4D sey
fs = Kur
i” & Fove saraplls [4
Spe
ignod fm = 4th
ISL semaply a(M)
Lod ko fm
L, 25upt DFTxe) Lor k fom oto 65
[xo] = [x cae)
Ppt PFT
>) 1xa7l
prot [ preerl, Peel, Deed, tera, Pl
jriay on} |
Cee rye
X Cr eMeteRs pad sy
XY (N-K) peng ce)
\x (n-¥) | {ikea phoote suaperst - edd agen
[X CN) = - Lele)
- en
2
xen) = (U2 V2,Vt,y J
<4
vey = (12,0,0,68°,%9)
(12,0, 08, 4, er 20)
IxCe) =
Ix] —, eal
fon fs =
f.= 4 yp, ~ fs = fm
4 z
26k Aowaplts 4 Apes pagal, duit Aampling pate
fs = 8000
oko Ake (2bo Soampls Dor ok gic). Sonnples
rmdagriboat lees Syrantdnny
mA AAFFHFSSSHSSPHPPHPHOHHSHHHHHEHSHGHHGHHLRHMHOMROEtii ee eee ee ee
2cb pk PFI Wu Sanrapln lav! frorn | mal 0) 0 fo
Cru olrel
xCK)| for Keo to 127 . | fs wade
| 7 Find he man Vi y stad in |
1” 128 Values abothe, 123 Vals WE
jut Mee Con jag capeanallay
ony
take only 4
Zebact indians Ck) at did |)x 9) 2 OS may, |
— eae
—— 1)
ay th Hu Ir -
asnune 5,8. Ibe au dhe indies @bvedvth aror
¥ k- [ 5,8 Iv, 126)
f, =f 2 ¥ £000 pS x00 , It xxs000
k “| Dep 256 Zsb
(2b y soce )
2b ~
“oH
fe: [1st 25 Hz, aeowr, Soome , 3437-5 x)
(tet ible procedwee togred the fry A the 10
,0)-08 24
FAST FOURIER TRANSFORM (FFT) wing
dedmation 40 Tarynn CDF
a pt DET
3 PA
XCKD= DB xn) Ny Kz o4D3
n=0
3
\ En en
2 Zax wP 4 Zoom,
neo nee
> nen-2
Ken+2)
'
)
xe Bun) wa” 4 Ee x0n42) wy
° nz0
ne
en
= = (x swe Pa cnerd Joe
wee ee -Jare2e =
e
Le ee
= Ck
\ my
Caen) + Cotas,
2
neo
x(2e)
x(o) x(2)
x) —
x (2k 41)
xD ¥(%)
VVBDBBPBBVBHV9H9VHHHOSOHO HOH HHOKHSBHOKRESBSOSHOAHEUHE=
=
5
5
Ss
S
s
Ss
=
”
ia)
”
”
”
»
»
»
”
2
Aa
2
a
2
a
a
2
A
A
A
A
A
a
~~
1
K(2E)2 SCX) 4+ coy (m429) We
nro
C1 tor inbegee 21
2en kn
We = Wy
-j 28 5 2
eo tN Aten en
'
X(2e) = Fin 2tn42)) wn
nro —————_______ 2
nen)
\
X(2et = = (xn ay xene2y) wl2kton
n=0 4
2F+)
Oy yey =e 7
2en ne en n
Wy Wy > WW
\ x
y(read = BZ ( etn) -xCn+t2) wy] w,*"
neo po
k= Oo)
U kn
x02) = = Aun) Wr
n=0
Hoy ¥le)
\ Ki
Kk4N) > 29 ON Wy ”
neo
xy x0)
mz 04>3 n> ole) kK: ote)
9) (wy —— rer)
4tn)
92. (n) ——> X(2E +1)a
spr
C x00, ear, x(2), x03)
mond
n=o-1
QiM)= HON) 4+ Unter)
ilo? = Kl0d4 %XE2)
FD = ay 44X03)
n
gut 2 (EOD 901129) og
°
ae (0) = (x00 - 1020)
42 =r ~ 039)
k= 0,)
\
xanrZ quem wae?
nee —
. to) * sow
= Gl *9
xX Co) = qo +i)
¥ (2) = grt 9
4
\ x
2 SGr(nywy
x (ze AND 24
29 +5 we
*OD= gute) + qe)
SIGN yo ~4r 0)
ert et 9O ORC ONMARRDHOOSLEBeeeE PPP EF EF FFE PRSFIDIIIID DB».
x0)
yay)
K2d
xs)
io es
“«
9) =
i) ¥ CID
A way?
: xX
wyl
qr a) =I we?
'[p- Normneb
orb
OlP - bit Awevsep
OdaSalient featwwes of DIT:
Le Noh tomaplix, Weatitipli catcon > bog JN
Ato oh Covepe adden = N kay N
o. Tip in bit pavewsal. ay dar
Dee
olp an neumnel eA
3B. gm place abgorithun 5 comp dene witlon Hot
ATAU +
tipliateon tae place byore . addi Ken and
& poverpt alti r
salir pentane op DIF:
and
a £lp un nounced od
el in bit Aeversal ordi»
afha
Puibtro hen.
eee mR DRDRAPOHOHDODOODO PR OOROOKRRGStorrpoks apt PFT of He secmenu Cn) “Ce, Lav)
¥ (0)
2 Aw) ot ye '°
4 ge _ SN — x (uD
a7 xO 4 vn
ge 0) 2-2) C1)
\7 al) qr) vary ¥ 19)
wae} ~Y
veers lw, ey ety
06-08-24
iva, DIF)
Bpt ver twang neo bl
Ann) xCqey
Wr0
raae)
gir) xCeerr)
oT
an),
gam Che +t)
L4 Cn)
4 ge) xCak +3)
ge nO) 44 M44)
2 m= ( tn) - x(44)) we
—
SVISIOOSSKOSHOVVTVIVUDEIIIIIISD Da»Oe OS a.
gu) = Qe + grcnt2)
qr = (gr (Md ~ 4,427) we ors
qa CMe Br) 4 Ge lnar)
qu Os (@- qzinte)) Wa
. \ rH
e20)) ROME AVS Z gir (now
neo
\
XCM) = 2 Sylmw,*?
nro
_
¥(2) = gute? +9 a
ap)
nib) = qn!) agu lo xb) =-gie 'O) — 9
- = ¢
xC4)= gn (09 a" ) \ vo
w
KOH B= Z Que ve
ate 870" teal
\ en
“2. 2) 6) W7'
K (acy = 2 Qa n e
¥()=- grr ld s9er
er = gn fe) +92) KEE grrl)~ Jor Od
)
xCS)= jr lor -G ic
aw AR RARHRARRBAHRHHOOHOOHROOOMROASBeReeEMSIPSVVVOVOVUDVODIVUVEDIDEIDEDIdOdDDOS:Find Bpt DET of Ane Sequente y tn = tot (200)
8
The Sey beer Arprate
ey °
AY gg =n
27 /y —
a(n) = les fonn
(37)
Aley= 1 LAY = 0-709 U0) =0
113) == 0-707) 2xtK) = =I C5)2 — 0-707
Alo) ~ 0 AC7) ~ 0-707
& no os
gi(m= xn) 4 Uta)
ploy= 2
q: lo) = Alert Al(eV= © pw of
2(0 = 3
ee TOE Dee gz (22°
PCE:
~ zlby = 0 .
Gi >= 127+ ~¢ gl2)= 14)
5 (42 403) 4 AOD = a
Y- oto)
gu (nye 91m) +g) Ente) fa = fru) +92 Intr)
qn (00 = ailora gt (20 ger ( = gala grcr= 2
pr (1) 2 qt) +91 (3) =0
ries a
gt ee (gutrnd - gered Joy Qrr lo) =2
gre (po) = Qilo) - Glows = 0 yz (l-}- ) (3)
ay Uy = 719-7) we 22 qeu ele
atm) = (0m) 244.) wg?
G21 C1 = qrtry age lo 2
Ger) = (a (ny S5rh rug"
~ yD DARAPARBBIBBABPRASCASPHOMMOMOMOBRABBeBBeeeeeeVUVPVIVEOKOOVEVUTIISVUVEVTIII IID DS
' Alo) id
0-707 XO) i
0 xXtey °
~ 0-707 23) ae
=! aly 2
— 0107 ty 1-5
okey °
© 707 27) it)
xCK)= (0, 4,0,0 609, 4)
: papa
find gpk DFT of We Aegnin xen Cy’
é?
oy ea & ©
Hess gomye TAI ATMAGY | gy (nde (HO — Ent Hg" -
qe) > |-l =0 ples Vee ee 7
wt
ays Vl =e LOS flav) fi ph
are: \-\ 20. d f be Me
, ~ .
2 ye 2), :
glare tv 0 ve vO Pate ght
$e. (2) (5-3)
Qe = rey (2 Jk)
20401 FA bhaly aj eaty,
gn (a) = 910 4g, Cntr) Qn (P= 920) tor(arr)
qn (ey. O40=0 ger (o)= 2+ (-2))= 2-2)
gr =e jn Cr) = (Ve tyfanj uy) +
gatm= (gilm-§G ("42))ivy” : = Ay dhe
= 0 woh =-) Paces.
gir Lo) 20 9226 = 6. gaa (nny) wg?
jerlol= 2-2
Biz) =e Q22Cieee wed -)
opr vse
Dee06-08-24
DICOCRETE TIME LINEAR TIME TNVARIANT SYSTEM
Dr UTI system
—- —
, oT iW)
ae uT|
dln)
yon) xeon x Den) 2 tn « 10)
“yen SF Wa) xln-b)
NIFTPPPP PHP RRR ee eeRens:YVPSFOOOODDOVOOUHEVIVEBIEBEEDDdD DDS
Shakic + Cowal;
ACA)= 0 M0 Win) = nen wro nNZd
mento nro a neo
Aen) = Ftn+l) = renee. n to
Frabeiliiy
& [new] = yr
nNr-
h(ny= 2% wtn) ~ wnstabee sgt
th (n)= bb)" er = crake rte
umn Con it heath, )
ag ert sagtern 2 comple mepenerit
Papennt
jwon ‘ton Cansiveke omputod
yin) eo” Bead ar qrer)
0 a te gives 2)
My= Zz HL) xln-b) . fe y
4 gle”
: oo Jwok
ae “> nae .
Sa oye
"
: ov" Wwe) = %tn) Hue) (input x fore, Feopred
Sytem amd ov: ie )
\ L bigen voli ‘
gen Auvnchion
fro xin) HC)inn
“Odwwine He elp umponse — ten x(n ze 0nd
- © &en- F (5 )> )
Hn} = CON 4 ECan) r(ide 2a8ta)
sony 21
Atwy= &(n) 4 Fn) sr
“jw pe 2nd)
HlwW)y > 1 +e —
> Pe cCoyw -jAinvwo f
Wo +73
qin = tmp lwo) _——_
fy
plwoy> 14 008(%) ~} sin 5)
21S jo bo
2 4 OF jr bbe
J Hewey) = 79 .
= i =-24-949 = OlWe)
utes) = tom! an) —_
14 My
~ a)
14 (wo)
(Wo) | se
TBE
538 5 ("5-90")
(n) = mn xV7g Ke ae oe _
4 i 2
(oortpt 8 ie
a vows tan ? in am): cos, 1 amd the 543
(mn) fly" vl
Vang Alay ore ohn a 6) wt)
yn/4"
ACA) = cos MY, 7 oe
dag eg A)
jem’
dadowint He ofr 4%)
Jaen
eanabhdeen meccanannnaanannppaneesceeeSO PIFPOOOdIIHOOOEVVVVVBIBIIIEIDIDYD’:
7 a
hewn ayy" ate a” alm) | (y= = '
(ae
mend = % OND 4420) | ocacl
; we A \ afutm = 1
vinye ol Cry) Ab Le ae!
5 myn wee NA
P (we 4 re) (-e
Lt
' : | __
4(™%) = ——y 2
-) Alu JSMA,
( Inape Vy, ml, +My 4
=) rs |
= 454 Ie o3sas 403535 © by bg 4)
Be 2 0. Be a5-
= \ = 8s? E£7ee
vogh L28°67
jm —j 28-67 3 (lyn ”
pin oH" nea Dee Jl lg -28- 67")
Hi Cn z
AN
gem: € t H (-%,)
wey) 2 te Ed
1p ol I= ¥ves(nig) ~Uy (Me) 4b jos
> 1 2 1.357 (28-67°
02-7367 (528 67 oO
~j lyn 328-67" “J (myn ~2h67°)
ye tn) = x haste Byesitem i
a 2pon Briony a goon)
= 1-367 elimeng - 20-67)
z
5 (Me 8-28-47)
+ +357 @
zu
ae E357 Tn - 26-47
po OP ter
57 ottlayed by 2B67*
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= ohn
nO) *@y am)
jm
04") JMO ron )
geome PERL) + i (TM) 4% -%)
Una ~j UR)”
+1 (e" hem) ~@ H(-m) )
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amg - odd Syren - _
Ho) = . -
. ~) 28-67
Wa): 1357 (- 286) = 357 &
i a) . j2e-er"
H(-Mg) = 387 (2tb1 = n3s7&
= J2bse°
# (th) : 1 »* °. SMe), 0 84-4 69?
1 yb +4 SH
j26-56"
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reer oN Ra : ;
[v= 2 + tos (my - 28.67") 4 Sin(mn ~26-56))
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hom (LY wen)
4)
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spomee Ane aysiow
pr
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Prviodst cequunte
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ue 4 »
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ke
be Syniusis Gin.
DTFS. coegigutent aa ae 4 (+25)\ py es 0 dont
Nv
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fre ¢ Goo itwhvon (1) A by
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Rusporse of DT LT Syviem ty nen periedie inpuds
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penn we x Ain)
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e 2 a
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Parseval's wlotion | 2
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nz -00 220? ~ Spy (0)
Gwin a parieaic prchen
o 7 ; vith pevied ox
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2)" Ve 5 IY(w) | daw
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Find the EDS of Stinenee (0) for the inpat xCA)-(L) 20h)
a Oo innpulre 9eatponne Ila) ay an)
vain) is pri odic for veg ste
a on we le
wlan) > poniodic (yon penuent)
(b° > nen -pericdic
2
tnd = (4)? u(r) 2 MY“ 74 ined
h
. [wt
yo)2 1 on &) y
1-,e
u (wy = Toe. ;
> \
)
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(w) ym tite co ae ('-Fote) ayes
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i
nt
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J
xu) #1
125 - 50
: a,
V4 Uy
a y-
e \ oe
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| {! : 1), * ('s sinvs)
= \ . Co
\+ yaa -2 ww + |e, 19), ~ %/, eso
pro! a
STAN ALE LRRARADADNHMHOP HEH PEMRSSORVSWSE
Sb]Sygl= Jy! =
Cras tos) ( 0/4 — 7/2 vosvo)
writ ow an 7 Athen dale
invuge & terordoren wars po
ad
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ype [wy = xlw) Les ——— es
ylw) ee xl) pale
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Bala) = (- y wu)
2
sete boys
De-OS-24 — DTLTI austen as Thequaney eee eee hae
x) oe i
A(n)
yim? atin) « Alo)
VORVIV DEVE SD DD DE
y(z) = X(Z)- H(Z) H(z) > brome}. dncion ray
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2 (v= Y(z) ag oe (rasig
Soll x(Z) we
Se
~ BPE BSF
pie
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( oundarg
Foye uum tober accourals vv a
: 1 ip
ae punonvinaly - wpe
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)
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3+ pelt coro syste
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dow pans filler
a LPE
wiles) 2) NO) |.
we?
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qn ab wera wr ety
?
i 1
Ae . 4
Y ZO gy lle =O ele Ie
N
Tm i
w
ye’: Ine
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sa LABDPHPHDHPHPHDDDPDPRAFERBRIBASSSSSSRSEEBSSESSVSEAMIVAIVVIVDVDVVDIVVEEESESEDEEEYDS:
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aul 5g see
AL w=0 jrewy|? 2 Ht! = Y Max
2 : 4
wen ] Ho] *S os "a hwo] a _
L 'h __—__—
weny, LHe = 3% vol °
Mh : a wo
pole cere plot Ima. domann af gow erent the A
tm(2) eb filtew te
Alte eeadar id
O-freg- eonprontled ,
re freq - de omphannised
tig pes pun:
-
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z 38
9, |Awl*= grin ad weo
max al wen
|
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J. Jol 15
a ", Y Ee + € an wn
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2D
[HC] = (tain) sire)i) ie & - ww
2 oe
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wen leo) = Say 24 | mh | A
+ 4 Ml
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inz domen, 7 7 bly
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bod poss fill;
fue yet
wey (1-h2")
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2-4 Za
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val He ve = ye
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Mh putea|* - manvabene
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rs e [Wl] ‘a y,te%) + (2 Sinaia)
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wWw=0 wre
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pert
win
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toe He a “ -
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lb 3 1
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nied = Cai’) > Crayz")(-y 2)
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ry yn fille
a Geet
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min a oe t Ah
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le
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+—_j__}__ ilu
n \ ° ow
iy Hlw) = & r\
KF < |
z | HOo)| =) all w \ ke(z)
from o toan2 Hansjoom done ia A wl cave 1 pitt
Ware PoLe SYSTEM;
H(z) = Ni)
iz)
N(z) =)
nz +
me) kot leu jee
An
dow po lis. (LPe)!
H(z) 2 4
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eo = 1.0%
ws 0 7 dreamy snaphonnel
At wo | Hw) “2 mo | imle)
| ale
: ab
smh
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y
)
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wenden
Pala aie teapnisey
the pred
pevecs ~ deems to's
ioe Feet cereg
jw
ye
= ¥(gestos fine)
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imaging « ysinw
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Band poss fie:
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lav
laz*s 9 foe 2A
eae . Pi
yor. ga
at yoo thy, |¥euor| — mar
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W(t) =_ 4
l-z%
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Jn j
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/ In(z)
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“mt |e eee
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giz)z NIZ)
(2)
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Wiz) 2 tae 2ev0 at z =-l
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pole at 2=\
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red Mse b ernphorated
ak weo [Heyl mar
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( ver. eopwited,
nv [7 o Rel) ie facodl —max
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H(z)
2wos of 2:4)
5 Jo
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lo
jn
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ween jan) omer
pols at z= 4)
5 Me
Jo _ Ale ele 5
ye ye” ov BF
ie KS ebhh
In(2)
Ab we JA) | Hew] - max a
xn o kelzy
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Bama neice fil:
aCe es
2
H(z) = V4 z
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dw. Jrle IM
ye =)]-e@ ey
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;nm 5
at vw: fA, |Muo) | — vbax ame
ginle
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y= eee oy
AL weo,m > | A(t)» may v
L anne ed
2-08 Ly.
one Hwo pole BPE that baste clnbe oi pars oon
nwo ero nits Chavaduviskes 20,
oe i abu: 477
{ wats Le =
ond ts may seapen x a
ok
al Ken
po
Yas age
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zy a
[nol yy \
mw -
0 ME mMOK Ws
q
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2 Zz
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H(2) 2 (27) cee)
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(rajazt) Cty? )
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,
ye
> DPHPLPLELRLPDDPDDHDABHHBBBBHAOAKaaeeeassesFVAVEVIAOD DODD VVODIVUUUSE EDD OD dL
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s
=
&
°
lever |n(a)} 2 \" (en) ct
And @& \a(re)| =) Ve
nd G&y
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wee
2 e
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14 V4 4 297 CHO
JCM |: ety . a 7
Vackiy (-r)*
2
ate tery” Q:
+
:
(41+ +
ie)" 2 4” (2-2 0-44) 24°
° leyte 210-5 u z
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[eqwr Vl corks yoy) _
a
\
a
pay te BN
7.
2 Cy?” Ceay) = varie tea
= Chavte2rt) (han) ASIN = latter”
peri timer = S561 1-1 Har gEy™
2
= 0 4ayt-2¥° 40.44 =02 —————_—_—_ a -
V2 243 V4 - 4x0-94 x0-44 tb
A044
L
Ys dt 0.6923 en eer Ts
a
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vo
sth] ,to. 837
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my ot of, Cor le
Poles z= aja bypy rt AE
7 £5 .
yew. ¥ e
§ radius
Maaco Ge ey, 22 = ols
TT z .
w(z) = ols (1-2)
L4oqet
= 2
dt ay(nye (12-129 ari 202-021) bad
1 CM) = eae .
gt ib YO" HL) AU)
pled = act) # ot (H+)
IP PPPLRPPHPPLRHPNMNYVDVVVDAAVAAVAOBGaeece
4
&
+
te
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buvet AraH at bo, wnds ak Ob ms . Gena Ht
Ae
OH) if He Aignal xO) Jo bewaplad at poe Ym |
Find He Der af tre gerrmled Sy rimee .
yb) = $m (200 7E)
> sm (geno) > Sin (En)
sore
Te 2 at 2 66m4
9 asgon =
nro od
xen) = Si (a ate fee “y
ber a [o A, Pe)
PDBPDPPPBPPRPRPPBBBBBBBEeeEESesGF
(0) /
ran} | tos?
¥(2) “ft-) t-) . =
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° e
. Vel ,
JJp oy =
a
=3
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>
>
>
>
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:
2
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20-08-24,
Dtgtal xesonatyy
Wo — Atbonank
190
224
ab We Wo
yuna ante
\H Cw) | 3 mar
a weeded abd hoe
Mane bode“
\
(ac é
ple
0 awe So hoe 9 civeahy oh) ame
(0 WAR) (qhy veg
¥w) 2 7)
daormeacedw ah eye
mE fy em2")
H(z) as walualed on unit cole
Hw) =
j-2Me
2
yt) \ = 1
\- tos (w-wo) 4palner 2)
1+ Ws (w-Weo) - 2 lostw- wo) Sin’ (we
- '
= ——
“Re digslw-We)
poe)
___—.
\
— Fo
: ——e
(reese)
A (Bia (o-) )
L
-we)A - Aw» (w-Wo)
bs ar WweW.
Cw)" =e
chal ALHONDY 45 0
Atgtal iter wbwch prodmes marin Aespor oe
ab os wenpovRulon, Progeny .
Digntal notch fillor:
Aiizevo Biter (eve _deemphosries 7)
Magwilrde wo wo at a par holon prun'g wo-
jwo ll
attpr l-e &
jwe -j)w —j(w- We)
wiwy = 17 & e eaaeae
L : spews \
\Auo) > (\- wos (i-wo)) + (sinw-we)
)
2
= a&-2v0t[we-we) [veo))
‘ -
~
At w= Wo , \
We Qe we
2
) Hal °
CLT PPPPP PPP HHS HHPABHBABSHHSKRAgggasg
ESLFNAVEVVPRVEVVHPHOVIDIVUVUVUDEDVEUUUUEDS
Nig {lal comb filer :
pilows contain Rreguemes er uhale Wocking
eles «
”
wiz) eS hte 7
kee
MM order flr
(=)
-)2" Av) = (h tn
ye 4)” SS align te
Oe!
UM lg
hh
w (2) = tah Za
huny = Senay g(n-7)
wiry =f, |
1
uk
Weld * 2 Ate 2
OL
Wlw) = Zz nck &
Yn (w)) = wen vo ofwe2n
Wy (0) = H (ew)
W(Lw) = nm wero oée
oe
a
2m
aan
c
SLES Hom paveh cond pllaris
Produas wll periods 2 ak
-_ jes Arkegen, mmubbples of 2A/_
Mal
eal 369) |2 Irodutesy new periods callay Ak vakeg rll pli» 2 aE.
Oigital sinuwridal pec tatoy /
G prodace o |p witlrond” tia
H(z) Vo
= -t
baat + aot oud aun Silo
Q,= - WeWo a>)
H(t) = Ylv \
XL) \ inikal vals =0
We) 2
x(t) t+a,r7 44,0
Yee) Chawre 4 aoe) = rte)
snwerse Z transfor
mend 4ay ginrd +42 n-t)
-2) -@
yim = ren) = Ay gine) = A244" Pealvace
Cin put given at onby neo)
=u)
[ny = Sin we Bin)
A108" 2y
wr’ O,2 LWbWo a= )
ov
209) = wSinwy, din) Input applied at onhy nz0
yore -yGrzo.
DPPPPNN PNP NL KHOK PHP EPEEAE:
©
)
ge i PPPBEPEPE-y
5 4 (o> = Ahim We 4 toe Wo o .
D = f0) = Sinwe 19 Coro CY - 4%
> 2 or a Lv ~
2 Sinwo
> yo Oo + 2wWRWo oe go")
> : ,
> a Los SinWo Sin®Wo
ch neo,
3 house Ue nen “ty
> yen) = Sin (MIN opp ay HUE for ab esse
2 ree sense 6° Oo) yond? gin nt Ye
3 Bu &)———
| )
> (2) Ye
3 [BP] we hey
2 ee anaes
>
z <
> eles
2 Av) yew)
2