Goras, Liviu - Semnale circuite si sisteme

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lnversarea x[ -k]
x[ -nl xx'I'I-kk]l
conJ ugarea
x-[n] xI k] =x- [ -k]
aecvenge reale <-> pare x* [ -nl xI k] =-x'[ -kl
secvenf,e inag. <-> inpare xIk] =a- 111
secvenie pare <-> reale xIn]=;"1tt1 xlkl=-x-1L1
secvente lnpare <-> lnag. (xtkl+x*[-k] )/2
xIn] =-;'1tt1 (xtkl -x.'t-kl') /2
Re xlnl <-> Par xlkl (xtk1+x*[k] )/2
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Par xlnj <-> Re xlkl (xtkl -x'lkll /2
Inp x[n] <-> In xli(l xlnl =-x*[-n]
convolulia clrcularX (xlnl+x*lnll/2 xtkl Ytkl

in timp (xlnl -x-tnll /2 x'["k]Ytkl
corelatia circular5 xt-klYlkI
(xlnl +x*l-nll /z x-[k]YlkI
in tlnp (xlnl -x*f-nll /2 xlkl/N*Ylkl
produsul clrcular x[-k]/N*vlkl
xlnl *"*yylnlnJl x*[-k]/N*Ylkl
in tlmp xlnl x-tkl*Ylkl

f'diferenflerea" in tlnp x[-n]*vln1 ( 1-w*-k1x111
ttdlferentlerea" in x'[ -n] "ylnl
xlkl-xtk-11
frecvenfE x I n] vlnl
sunarea Ln tlmp x[-n]ylnl
x*[nlyI n]
sumarea in frecven$H x* [ -n]y[nl

Parceval xlnl-xln-11

( 1-w*-)xIn]

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tlnllr, gIlcBIIg $I $I$T[nE 21^-e $?ilIUI r[ $r;iltI cr

srtalele dlgcrete avind redla sunel pEtratelor nodulelor fj.nLt6, adlci
I4i-nprr=zr*t-re=r
[ lx[n] lzq- {2}

rfig
Arbale apatll sunt spatli Hllbert tiplce (Ln car.e se definegc
produse ecalare eare gen€reazX netricl gl norne)
operator reearcabil tn C' "
Un spatJ.ul este *e deplasare
operatnrrrS.
gllnl,lalrnEpaarct&lcn-uulianra,nfeovurarlreer,se poate face utlllz&rd transforuatele d, Z

1. $PITIII Cz

1.1 Definitii, structuri, subapafii rerareablle
ndtllngiScnlpeaaastalnuerl neCne'nrpeaolLoaertleoirnfldtrlepegcrirlv.e,lttecocpad€oEnnetarnuneiurcalallfrzielanrdedomae(snlnjp"crael&lulsuse1tuenlxC)tl*nC6dJes. acIouan,sftZXu,
partlcular, R. Astfel, elaeotel"e spagiuLul Cz sunt vectori cu o
lnflnltate de conponente:
x[n]=[...,x[*1],xlol,x[],1,",.1r xljleC
(3]

Rqarcln eI eu este spa&lul {vectorlal ltniar} tuturor semnalelor
discrete (unldlnenslonale). ssmeleLe dln spa$lul c' se vor nota,
prectu toate ssunalele dlscrete. er$ fel ca gl
forma xx[n[Jn]" f,a ssnlflca pdni
acun, i.n fmctie de context, notatil precun vor at&t

semnalul generfc, varlablla lndependentX fLlnd notatd cu n, cdt sl
valoarea sdtnalulul pentru argunentul n.

1. &Ooftt Spallu1 Cz este inzestrat cu operalla lnternd de adunare

pe componente a semnalelor i.n raport cu care se conportE ca grup:

x[n] +y[n] - [. . "x[-1] +]n[-1] ,x[0] +y[0] ,x[1] +y[1] , . . .1" (4]

gi operatia externi de lnnultlre c-u scatrarl din c: {5)

ax[n]=[...ax[-l],axloJ,ax[r1,...]x' a€c

(8pre deoseblre de spatiul ilz, opera!1lle de adunare gl inmullire nu

na1 sunt modulo M).

*;,friiL*4{,ffieffidCifi&l3i*t,::,. . :

'(TnInTtrEde
tulsT€traecdEenc;qlclondeE3r rolTrnJTg raoxJsnalnInl TtreEIdaslTnucTTI p)v
utr lelduoc RuJou

2

'aTBTtrTuT aTTTlpdsqns uTp aleXuauata a?.Eocs

lod aJpssTdep ep Ter Tl JB uno TrolBJedo TlT[nuV r

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f-{r) zt|.
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I t.rl :p!,rolfun eltnlsfp pzpereue6 erec yB

(el Z3u

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:ep F?TuTlep pilrolqun eurou nc qtadet
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letwe 'H 'aFlfzod rg TeeJ :fetu f,atTce W) Tnidry ep elszTJsleeree
Lm g?slxs
p
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:efeuoTsuatqp lFuTIuT tq{edsqng'g

'r{lez!,erd llu(t nrBewp lnrm eJeJe uI arnu Ttms eluauodwos
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(p dll $eqTTH eTTr{ds n3 a,raezT) ereuolsusuTp :truw rrledsqns 'v

'rpo{w14p" Tv Tg upTlprga. Te aJepaA ap Tn1ctmd
tW e7e s ry rop4ce.nd e t8 .zofefsuaas e aJezTf,e1cervJ ep apoleu
pzeezTumt e?Fl[rozaJd sTefnuJat' TzuoTsueefpTl] uelpTtong Tnlfr.ds

ut (?aT4*) atetmd ffiNTp toTetueTsyp A|ledser (evrou) auTbTao
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W eata1wss docs ldo4n
s.re ayemras ep aTTqeJf,eale.J ff{edsqns

r(}un e 'Fzeerjll. eJ irTa} W 'sarpluezard ll^ffifuo.tru ff.tuUest "S

'n1{eds 1pu
Ednco JE 'BnTd uE ';6gereo1.:edns sJTlpGtlpu oldacuoo ro(m e l^Tsuslxa
E€rEzTTT?n €3TTtuI r€ smTqord Telserp e llpTTpfap BIB?EJI o Bcerpoop
elrmrrEf,p u! Er?uT rFoA ns 'exETpcs esnpord 'TJTJlatr 'auJou lelcads
qg TJ rE um3 rEluaE[1dns asnpolluT aITm?cIlJ?s ap eTlcunJ u! ne8/TS
roleTpuffirs InTuss)poo 16 trntruauop nc grnlebel uJ cseuTJap as llJedsqns
a?serH 'afTqpc:ue: ap JPuxu eJP! un aulfuoc "3
rolalElnas Joxn?n? T1{edeqns |lIIglrJU{tEu 'e1a:cslp 'Z
e3 ?J XI IIIUdSSnS
p n1$ods

:t tltltts t{ 10ilv,{s r-Tz [HtIgtS IS lIIoCut'tlilIl$

$ruIAln, CIloil{?S SI sl$Tf,ill ZL-o $PATIUI. SE $BHTI[[ {]'

- Subspat,lul Xo al semnlel,or care tlnd la zero pentru lnl->*.

* Subspattul" Xa al s€Bnalelor care adnlt Ilnlt6 flnltX pantru

ln l*>*.

- subspaglul s al seunaLelor rapld descrescitoare l-a zaro: {B}
ntxln] -' 0i Vk>g

ln!'e

(r.m exenpl"u esto -sannalul 9-o').

" subspagiul I' al sennalelor de modul euunbil pentrtr carel

f, lxtnl i<* {e}
{ 10}
!t -F
(1rl
adlci:

-1

f, lxtnl l<*; f, lxtnl l<*;

a.-6 a.0

$pa$luJ. 1* are o structuri ds spatlu Banach {complex} eu nonna

!d,* * lxtnl I

gl dlstanba

d*=lx-fl*f, lxtnt -rtnl I 4Lz',t

- Subepat1ul" l" este spattrul llllbert {separabt}} aL semnal"e}on €
d
discrete (sacvenis) xz 7, -> c care satisfac retralia:

4i'i=E lx[n] l3q* { 13}

lurde E-r-, se nruregte energia sennalului {rela$ia (13} repre::lntd o
restrfcsle lmpus5 unei noxma la pXtrat) " "qstfe}" ln spatj.ul cz

'r>lfu]xls Fu3m.Tepl€Tgs.lu<ll-xlluuF]lrclpnRurlBTcn.lzOleu<r-l,r[orl!>]rlr[p<u=qJr.l.foz>at>rqA":1zr[Tu{se]rlrtlluFglxcTlt6RlB,lcolur>yull.Tuelrtlleget

0<{ P}sTx€

:ETSETeT cpIBTlEs erEc C <- Z tx elercsTp z
roTeTEqFs Tu (TTqprdesau) xreqTTH InTlBds e18e "r rnTledsqns _ o
(
-.El --.tr\ r-rrl
(
-l tslxl ;l tntrtslxt
o
3 3 3{oz}
el tutxl lt ,l lrl

eun pxuTzoJds: rapTgtt :T(Ec.TTrTeuloBTl)lETEar?u1orroeButqrTterTpH,za=elTdlusnTelrdleuerpenudcg.,1ree1nlyn11c{{TelpdpuueodFc.ruan3r6tuqppg

t-.{a
f,t"l'i 3 ltl---tt I
(6rl ol 1u1rtu1xl
' - ld 6

:repfgg Tnt €e1p1TTp6euT coT erp (6>b>I)

(8r) r-bi*dT

.T FoTpe '{I-d)/d=b 16 o>d>1 gcep ,eeuamee eq

(r,t ) ol r"t4 + al [u]xl : a{ [ulif+ [u]rl

(er)
:Ttsnotut1 TnT EelplTTEDsut cpIsTSEs elprilss pnop erEcTrO

3tsll wy6'l [r]/- trlxl -.l' l\=;rf-x;=&p

:qrd (TaluereJTp p pcrou Ec) ElTqJap exsa E{uElsTp rpT

T(rr ) -y6{ 1u1rl lt =l [u]xl
---:-la
:ulrd llTutJap else 'Psrou a:Dc
nr1uod 'dT rloeueq Tltpde ep erpfncTlred lrnznc luns zT Tg tI aTTTlpdS

'RlTrrtr o16:eua ep .rorsrBuros 'g:)sl3rTr8 :8aIT[mTZnIcq Pl8TXg
rnrtEds(gns) lTEtu Tr EA ar rnTlpds(qns)

rl.Vlll$ Il l0lltrs s-Tz tfft$I8 IS lU0ClIt 'nvttls

$N[TA[B, CIBCUITB $I SI$?8HE 2l--o $tarIU[ !E $xllIALB c'

t#tfo,F,lxln) l3<* (zr1

sdulsbcsrpeatelludleLe:nseerEnlaeltnnuftunelts'tee,gplrdlne analogie cu 12, epatiul semnaleLor
Putere fln1t6.

nloocdau- llegsluoburenrpaeabsglpllil€e(c1strGl-voma=na)plerdleotprrarleotazcelaonlrtsamltlelnobddutelleesl(oe1fr,rnc-n-oa-mL')epropensoepnnettorcluotlnvcdpaeerspoiatruricraneat
lnterval fl.nlt est€ flnlti.

C. Alte subspatfl. (sub)spatllle inflnlt d1lrenslonale de nal sus se
tn orLeare dlntre
pot puns ln evldanti alte subspatil cun ar fl;

Hllbert) esilnal.elor p€ntru care x[n]=0 pentru n<0.

- Subspaglul eetrtalelor discrete "antlcauzalen: subspatiul

(eventual Hllbert) sonnalelor pentru care x[n]=O pentru n>0"

- subgpatllt€ semlal"el.or reale respectiv fumgi-nare: xIn]€R

respectlv xlnlsJR Vneu.

satXsDfaeeg&lgunra.trsme uplotot,rdreeflanttrri"eeubtlelp(ad$el1ecx€amreplusirecuaplerj"gndi icsaeumzantarel,eectca,re). st

1. hza camald !n ez Dafinln sennnalul 6[n-1]" seuralul care are

toate agantloanele nulo cu excep$ta egantonulul 1. Orj.ce element
x[n] ee xp[ona]te*.e.er"l+exi[n-1f]o0rm[an:--r] +x[0]& [n] +xlll0 [n-r] +" '.
E221

llulllnea {6[n*i]), LeZ forneaz[ un set llnlar indepandent. Acest s€t
este ortonoffsat |n raport cu produaul
scalar deflnlt pentru spaflul
Hllbert 13, peu*m care sunt adavdrate relatill"e de ontogonalltate:
(0 [n-*{] ,6 [n-J] ;=6rJ
{zs1

unde 6r, este slfibslul Lul Rronecker"

dln u'rmR6epmreacrci.4.entfinddeoasrgeubmireenatud"l-Lnsetrme nl"al(ufltuxla' tm)€8g. t n, acegta

EtrnaC'J'e(lHp'cqTscneTIunapsugnTpnlorETJlddEPdapsf,rIEPcs3nnTBppCu€T:1XPSauTIdJ1ounulo:lsnccTlEr'lollplJuoEluJlTontulTtTe{lplsgdacs un alsa

TnT{Eds
1n1de3 )
aladld'uooucndInnsJaa:xdJEHJ€?3JaTqSITg'JEnITElpJSdsTanoTTnJsonpPoOJdlllBTaluJTTuuTpJaaup e 96u91

'aTenzn olerasTp aTpuu3s e?TnE P?sTxS '.rEIeJs lnsnpo:d ap RlPJauab
ptTgerrBurar ern13nrls ep a?Trego ro11{glTTTqTsod E?Tro?Ep alTqesoep

efeluene pluTzard '"T InTtpds u3 'n:1sou InzP3 uI 'l:aqTTH n1{eds

m-rlu! roTelEuuas Earprpeeul zT IfrInILtdS CUEII g1O X;'xilmg 'Z

("r< [uJrf , [n]x>)€x'u-'rJ [u]r{l+'rzl [u]xlf

, ,,r =rt< tutx r tulrf> lglxjy'=rrl [u]rf- [u]xl= ( tulrf , [u]x1rrp
""='r< [u]rf- lulxllgu-

reclrlau 15

(ez) ,l 1_rl-*ll f |\=< trrt4' trrl{)l=rrl [u]xl

: EuJou pzeaJaua6 JeTEOS fnsnpord "PreTuTITqPsas
EuroJ o e15e l.TuTIop T€Jlss rET€cs fnsnpo.Id'a1euo6o1:o luns

tsz) 0=<[u]/t'Iu]x>

ETt?Ter cEIsTlEs erec [u]A Tg [u]x eTPulllcts enoc
'TelTutr P erEnT ap elferedo gsTJTuu€s .o+ erTrnToq4s erPf, uI

t'e) iul r iul .* 3= ; lul tfrrt - * "C;,tt= rr 4 [u].d' lff ] x>

[ro]uoJ elsauTJop a$ zT UTp eluaurols gnop 9 ireTess : TsT{elar
Tnsnpord

'rlIeqITH nltede alse zr rnrtpds(qns) =T nI &iltuJs tfxstwoud 'r

lTu[]uT rsTTTtrpds eTE Xz€q ap '=f TS zT eTPuoTsue[ITP

aTT]T?sTra13sr"c alenuTluoc uI

rElu8zard'eTEuoTsueugp lTuT] lreqTTH eITTlBds erpoTreltrE aIaTolTdPJ
uU luzTTpue uv 'olTqasoop tztP3lTTTnTI qelpuo1d1s{tcrpdePqrnesJoaluPemlrolrdlwlrglsTaP:UeaeTanarluad
UaqITH eTTTipds(qns) luns

.l T$ .I FaqITt aIIII?ds z'l

zt nllffis It l0llt{$ L-TZ Ilflt$ls Is tllncln 'ltYlnls

$BtIt[[, cncutTB $t st$tBilE 2L-r $nilul EB $B;il[[ Ci

deflnlte pe zt care nu satlsfac condltllle de apartenenttr la 12;
seanalele dl"screte constante, perlodlce etc., in general sennalele care
nu scad (suflclent de repede) la zero pentru argrunente care tlnd 1a
pl"us gilsau ninus lnfinit. o parte dlntre semtalele rrexterioare'l
gpatlulul Htlbert 1= apartin spatiulul f= care va fl prezentat in

contlnuare.

3. S,slRvnflr (1) Desl un sennal dlscret y[n]cc" nu aparllne
{sub)spatiulul L", nprodusul scalarn dlntre acesta gi anunlte
sennale x[nl dln 12 poate avea valori flnite. An utlllzat ghlli-nele
pentrt a marca faptul ci nu este vorba de un produs scalar dlntre

elenente ale acelulagl spa9lu Hilbert, Rlgnrros, se poate vorbl de
actiunea unor functionale asupra anunitor elemente din 1=, actlune
definltf, de o relatie ldentlcd cu cea de deflnlre a produsului

scalar gl in care apar€ o functie (sennal) dln afara spatlulul
llllbert. Cu alte suvlnte, fornallsctul proprlu spallulul Hllbert
poate fl extlns gl asupra unor perechl de s€mna1e dintre care unnrl
este exterlor spaSiulul Hllbert, dar face pereche crr altul, din
spa!,lul Hllbert, care este "nal cunlnte decAt I se cere ca elenent

d1n spallul Hllbert".

+. Pnmmn $cnran tff spArnn I2 (sub)spatlul I" este spatiu

Hllbert. Produsul scalar a doui elenente din tr= se deflnegte

conforn reLetiel:

<xlnl ,yln|rr.=l#2fo f r- tnlytnl (28)

Produgul" scalar genereazf, norna: (2e)

t#rforl,i,tl'lx I ni f;r' =f? x ia-JlJ&fF[=

gi netrlca:

d7, &1n7, ylnl 1 =f,x[n] -y lni lr,=@=

_ = (.3--0.)

,jl-t-*T*:"8 l*r"t l'.tgn#*rktni l'-zne(xfnl ,rf.nltv

'z nes p STeTTqBTJSA uf, ?ueJnErI
sTTTrIasEJsaurPnlEpJuonudsTaEJUo3TTtcErPs 'lzUanJensElpalaeTlpaulElIuoJtosIusPuPJJl &€?TLunuap

'Esnroxa exsa

uBJaoXIeTTuTrTrTaTgJztTnlnlpuuonX3lAsauETTJdeloEJlptoeElTcduITEs':qnaJT?6l s1'oa!s.d:raE!pnuz3o'1upTPuraJnoerspaeTlpTrsJuI opacTTx' nTa(lTTlndl6ETuecfJoeT1rdcTugulEuJBfaToesqIJaePTJTua.TE:€lan^pu)

?TTqETTE^ ep

eaJEqrsq3s lnrl} p-s ErPc uI p El€eroJsuPr? a18e zc UTP TEures Tnun
E tETsrelETTq) z .ElsEroJstrsxl 'aTtrTuTJep uTrd Z gtlufiOfffivu& c

.E{eralPTTq

pT$nuTETnllrsq€rou[:utToanstEsoputPltr(rtlrlToudsTruBpTlred(olTPftqxtT)zrrpT^tTeEPslnpartrsxJusoeTls)euTsesAxr?eTlPlTTTuzTnduunaP1pn1peralPs1un(udpo)JlxesuuBErTur:Jasp'[uug]nxl

t,+rB-,, " '+p ItJx+oP [0Jr+r-P lT-]x+ " '= (p)X<->
:3 p*rgc ul TTlueTcTtrFoc puB^e (orpuopro ep gITqeTrE^) p BTTqeT.rpi\

u3 eTTTres Tg re TnTnTfeds a1a1euna6 ar1rtr Feoaltmrq g{uepuodeeroc

o TTTqqs alEod o8 TnTnTtrPds TnzEr uI T9 BD o t&tlflorsmuf 'I
"n

,l up Olgms Jom 0p JeTJnoJ I$ u 'p alalunro}suu$ ['l

"rI W aXTufJop a1euo1{ctm1 Tetm s eunTtoe
?o l-errne6p1Tsseugo"cua=T[urq]lsr1l"neTluBaWo€FtsTe'n"ldpneTxaaJxse€pro'1lm1 rTeTremuntcdu'o"c1ln1lne1lTn1n{zeeder

gc

euTfipdtu nu elRzTTTln sTefeuues arluTp Tnun Prep JpTI{c 81TuT} TToIEA
uTtqo oE zTnI TunTTpterdpeTlpnozsecTnuTn! s1npsourdJp'T=nIelodJ pugzTlTln 'pugo TT{€nlTs
FlTuTJsp FTPTUTT(Tt{tsas)
luns '=I
llpuoTlanr'Jcleo '1ez1cerd TEU us mrc P€E 'e1ss JPTPcc
Tnsnpord
u=[u]x t Er'tu/l=fu'lx luns elduexg 'lTuTJuT
pT puTl eJps

tl€8 @T <- u pug" orcrz ET luel pufl erPo gleTpuuos luns f",1rq1nr1un1u{eqdss "g
qN-redc nu e.rBJ eTeTEtreS
oI rffrrll [ds tsucds

'Tenlrou (TnTn1E.rlpd) gsndu'g

eTtsTrls€J o s?se 8I TnpTlpde eg{ptrgap uTp eT{cTrtsar Pr ugcjrsweg

,t flilft$ It l0IIYas 6-TZ lIxIst$ Ic [II0cIIe'tlYlxts

sB[[lt[, ctacff!TB 5i 5!$!846 23--to $PITIUI, $[tITAI,I CZ 5I

dz *ts-\ 'E {

{32} pr
dt
Analog se defj-naste transformata z unilateralX. (r

3, &tSERVaYfT {1} fn cazul semnatrel-or 'cauzaleo', {x[n]=O pentnr tr
n<0), Lransfornat-eJe d ei Z unllaterale Si bllaterale sunt ,
respeetlv l-dentlce. Evldent, ln astfei de cazurl, transfornatele tr
d contin nr:nal puteni pozJ"tlve ale variabiLei d, lar transformatele
z, nuuaJ" puterl negative a1e varlabilel z. Conslderalll similare .SL
S{2e1poRtafnaacrecdpnecnitrusesermii"rlaele*lfeol'f:tannatlLcaeu"znauleuf'u" rplic6 nici un fet de
conditti prlvj"nd convergenga. Hanevrarea serlllor f,orrsale (in forni cc
deechisX) este cel putln lncomod6 dacd nu, in marea naJoritate a
caz"*rfloro 1-npoe1b1L6. Conslderarea unor condlfili de convergentd, VA
adic& a condlglllor ln care seriile se pot exprina prln sumele lor
ln form& lnchj"s6 este esen$1a15. Pentru aceasta este avantajos de se
consideraf, varlabllele d gl z ca varlabile apartlndnd p}"anulul nu
conpl^ex, transfsmatele d respectiv z putAnd fi prtvlte ca
tr
operatorl cane duc semnale dlscrete Ln func[11 complexe"
{3} C&teva propri-et59i" ale serJ"ilor de pute::l sunt: re

- r&aa de convergantE a serlei de puterl 5

x{z) =X xln} z-" (33) 6.

hrn coa

adl a (34) L1 C

r=1,,* supfiX64 exi

- eerla r']nv€rge absolut pentru orlce z cu proprletatea Nzl>r gl dup
unlforq g:*.rir;ru orice z astfeL inc&t lzl>R>r; pen
- x(z\ e*'*.e functle anaLitlcX de z pentru
- Z(zl esg*: derl.vabllX de orlc6te ori. glzll>dre; rivata se obtlne
d(el)rlvc6&ntedvtaerpmroepn:j--rlestEergtef XaleineceorlnlldolrlllIlaeuir.ennctasruenltzr l>r.
- dacl serl"a

xtsl =Yxlnl a-" (3s)

!t -6

converge pesltrtr r< !z icF., atu:lcl (361

r*li-rn sup?TxTalT,. &*i-inr ir:firFf-nfl
:,.,s lF

nlduaxe ap 'alsrxa nu Es z nes p p?plrrJoJsu€Jl pc 'iI<J Pc llJluad
't.=U nps/rs 0-J pc fTqTsod a?sa r:ETncr?JEd 'fUT>gTr spoJd€palTspeurn'uunpclsFTdxnap
uI

Ts.+<-U nJluad JOTATBUUTaS e€JP?JOdurOs Ap e?euTuLralap a?sa EUPoJOJ

'z €?euLroJsuPrl nrxuad u Ts J alaz€J T6 p p?euJoJsupJ?i

arairplsluaTandxrauJlad/TuuooIpgo'(uTZ.n,/uT)€aTaTd1"fz1uBuIJ1zgaepp^ErIlaTcUpaTdTJsanapT:u€epJu€oJTponIC6aeJl€r:c1uuIp:r1lropJaos.druoeceJlnOl ufapepu1FTplu?raoubfa€JzaAaAJudoacJ

puEAE roTaTpuures 1e 1e:aua6 Inzec uJ S[tSIgUSAffOJ Xq IrFrftOq '9

'JPfnpou 1nuou11od
a:ede treu nu Ec Tn?del uI F?suoo EaJTqasoac '=t{ TS *1 a111{eds
eT aJeoXTralaf, aTaTolldec ug a?p?uazaJd BTao nc au:og 16ee1ece
ne aupourTod pnop p uru[oc fozrlTp aJsu Teu TnTao €
arpuTlrrf,a?ap ap
pTTcns rnl Tnur?TJoFTe TS JoTaurpouTTod TTJTlrFdurJ eTauraroal (Z)
'a?T:alTp p{uabranuo3 ap rTuauop ':n61sap 'eane :on z S p
sTalpuroJsue:1 'exalduoc pITqeTJpA ap 11{cun3 ec a1e1a-:d-ra1uI 'z
Tts?euuo]suef,? TnutsTlpurro] saTB Teur Fz€ezTTTln as 'lTuTJuT l.rodns

no o?aJcsTp alaTpuuras nc em1g6a1 uf 'JoTatrpuuas ETJoal uI 'al-red
F?r€ ap ad 'araTzrglur ap TIIJoxeJado nJ e:n1e6e1 u1 '.ror.ralue
ssnpo:luT afpuIras ap a111{eds uTp p Jola?puuolisue-rl roTrTlTuTJap
paJapurlxa uT-rd Elernleu e:ede p elpuuolsupJ& 'aleuuoJ TrTqasoap

?grap p?sTxa nu z TS p afalEurJoJsrlBrl arlul tI) IIf;EAusstr0 "s E
T
'€sTqauI FrrJol u3 'e1dw.rs ATlpTaJ
aaTlTTpup TTsaJdxa rorm e ecTpp a]polezundsaJoJ z nps p JofalpuuoJsupJl J

€ 'tTnbar allunue gdnp '€eJe:Aaueu nc aluauoduoc ap lrurluT Jpunu r
tm nc TT-ras:o'.m Bd-:ernaupur Tncofug ealnd eA as 'afrqeuns ?uns aITT"ras

pugc Tounle 'pr rcT€ ap plTnzau 'xaTdwoc TnTnueTd puguTJnde a1fqerJeA
e3 sleJepTstroc 'z nes p aTaTTqeTJeA nc Tsodet ug 'an71cptlset rtJrpuoc
alTumue uf 'alemts TJ lod eJealBzundsaJoc afrrJas pugJ TJnzec TmS
'Tuaera? ap ?TuTJuT rgnrnu un pug^p TTJaF TJ .roA z TS p alalpurroJsueJl

'bfnuau alueuoduro3 ap lTuTJuI JE{rnu un pugnE f,o-[eTsuuas Tnzpe uI
'z

n"s p JoTal€uJo3srr€rl earr{1nuug eprmdsaroc €A T! (alaJcsTp rofefpuuas)
:o1a{uancas e oJETUTT TaTfnfonuoo 'nTdusxa
aq 'p aleutJo1supJl
xolslTralTp aJBo?gzundseroc (arpTnoT?rpd Trnzpc u3'xoTeupouTfod)
:oITTJas paJTfTnuu1 19 a1{eleroa l9 ey{nloauoc 'aleseldap
sp TTrol?rado arlug ITTqpls lod os sxpc aTaluapuodseroc ap '1ed1cu1.rd

u! exe6€T luns z r3 a uoEruroroJsilrtu.t ltuSflpuJnl tTtftlxt v 't

2t tIllgts t[ I0Itt{s 'I-TZ tffilsts Is tIlficil0'tlvttl$

$tHIil,B, CIItUItX $t $t$IBttt 2L-r, sPtfit[ $E $$EBAIB e3

Pentru semrale nule perrtru fi{$ {"crruza1efo}, p=m gf dcmenlul de
convergenttr al transfornatei z este exteriorul cerculul de razd r.
Pentru eennale nule pentru n>0 ('ranticauzale"), r=0 gl doneniul de
eonvergen!"i al tranefornatel z este interiorul cercului de razi'R. tn
legEturi cu Flg.1, dacE pentru sennaluL 'cauzal'r x[n], x[z] are
daanlul de razI r
de convergentl reprezentat de exterlorul cerculul
{Flg.1.a), gl seulalul y[n], "anticauzaln, cu transfornata z Y(z) are
dmeniul de convergengS reprezentat de interlorul cerculul de razX R
(Fig.1.b), semralul xln]+yfnl are transfornata z deflnit6 (convergent5)
in coroana reprezentatE in rig.l,c. Mal renarcin faptul ci pot exlsta
s€mnele dlferlte, dar avd-nd aceeagi expresie a transfornatelor z (sau
d); ca€a ce dlfer6 este donenlul de convergengi. spre exenplu,
ssnalole o[n] Si -o[-n-l] au aceeagii expresle a transfornateL ?,: z/(z-
1) dar donenll de convergenti diferite: lzl>1 respectiv lzl<1.

a-

Flg,l" Dmenli posibile de eonvergentl pentru sermaLe a. nuLe pentrtr
n<0; b" nule pentru n>0; c" caz general.

7 " Trursronuata founrm A n[ul snnEl D scngr tn cazu.L in ca::e

domenlul de convergenid al transfomatei Z a unui- senne',l diecret
include cercul unitate, seria

-' Designir r. Si R depind de eomport.area
serunale" or respective.

Tll'no(g6dtm€Ti)lTrerea.l[eT?J6€{aBlepT?Inopezrlepae.rXpo:sdP?a?Ip€EJTpTEnppTouunTToTrTTtspi}lcduTusEanETJEpxauTsfol,npETrlJ.3ureorTmTpJJJa,:?pl1eueoalupqnoeJTT'adeleJlTeapTuluT'Et=gJuzuTuTTTTJEuTaTpa?pprne[rTITmeI1]Tml3ue?uoBlPoudTsfatluuJaeIr.rmElTlaOlJ
felonetropzluToJaTlsSaa'r1E:aTqu1tIprion11e{1eedusoa13{3Tuxon;p{Jro'"mf ?Tnnl)TT{dnsTtpsdluE[Annce (rapTculoc*)

ellp nJ

{5€) z 13 [u]J t lulx [aj -J 3="< lujx, [ujJ>=*J

Efpreue6 eu:o; 'zsaTu TneItsTeauBaIJotefr1e1;1fnfttr{rl1oTdnT,lppdcE1na1ddreogTuetIXeuuTouTpla3urm} p
olPuoTlcunJ I'e

ant;r:prncp:T1uaT5rlmlu"ra?aarcJapluaT?raas?eafplulTrEreFfTlTesAqTeaTes;Xx>oaJauoJrpdparslraaadpfJTuus,e=oulorAIcodT'puuns-TTt TpSTulanT.T.IrronpuTloeT{TpppJdeas,plTqxzenaplsruaea'upzalxTusnTalraUauaTrBqEPfeITaTIfreelHqlo.drJoanunpB-TIprotpcraesud=r$dsJ

zc lllllYds ru ilffiuuld0 Is xTtt{oll?llff 'z

'e EITqETxedl ur ,(D)x EuLroJ ep aclporrad aa16o1eue elaTBuuas
T6 [u]x a?arosTp dTaTputras er?uf pxnlp6aT o glrrtzerda-r aJpJ pTnEroJ

{8€} ,p-,re(D)x/f= tn1*

:ptrf,oJ uI (Dlx ap a1{armg rg eulrdra e?eod as [u]x
sT''axnnxosfpnTr?IsrE€auulpfranacpslpFeTe?lulpperTJazaaTa-Irorrndluo!1dT9utrrpPulEaper^[l€erEJaros3JuJp€ss.cupuBloEe:cT1?,TTqaanellzgTseara€JmAEurusnuuTpaTuusZsgrdBp(T-p3eoeeTpo)AllrXrrTat=zdn.(q[aaupp1]yxeglcdellapeTrotlJroclsrutaTnUdpo
a1{cmg o a16eur;;ap 1€ [x,u-)aD Tnf aITJoTEA el€ol n:1ued e6.ranuoc

{rs) ',s-€ lulsc3= 1 o1t= Ge€ly=,rc'"| Vt x

zt tltlffi$ [g I0Iivd$ 'T-TZ [txl$ts I$ xII0clIt 'ttvltts

./! r.if@'

s[il[[[, $Itcuu[ $I $t$Ttxfl 21-r * $FtYIUt Eg $EHHIIts C"

1. &smvntT3 Precj.zin faptul eE, daei xlnl=g {sr:bpaglul funeslilor

uHrglnite), adoptarea unul f[n] in formula (39] dln 1' furnizaaaX.
o tuncglonali llnlarE gi" eontlnud pe B (E cirel" netr&S, se
denonetreazd a fl norsri dln 11 a [n]: sum& rcdul-el*r
lul f
clfluonnn1ncaptolroennenagtlel{loIcrl)on.nlaFtlonrriuneucloapne(t3lnB9u)"{nAcuareripteHuni$zuecaezisdt'tenndulestimpfoaertan'ltafrul:dnBi.s::ete,iunxtnalastldetliogrol

anune:

I'(x) =11**1n1 (4*)

x-)r

lntr-adevEro presupunind ei exlstE f[n] asNfet f.nc6t s6 ave$;

.L (x) =qslol , xlnl , =8 [n] xfn] =]];g*lot {4x)
".

pentru x*[nl=6[n-kl, k=0r1,2,... ave$ 1{x*}=0 pentru orlce. 3r
natural ce€a ee i-npllc6 f [k]=0 p€ntru tot,l keK. $5 conslder5n acr:m
xlnl=olnl, sennalul treaptl discretX unitate, a c5rui
lfurl.tH cind
n tlnde la inflnlt este L, Conform foruulel ln
care toate
componentele lul f corespunz6toare valorllor pozlt!"ve ale lul n
gunt nule, se ob!,lne cl aceeagl llnit6 tnebute e6 f1e u€ro e€sa ce
este absurd. concluzla: exlst6 funeflonale ps B e c$ror expresle
nu este de forna dlssutatl,

2. Irnfffg cnffSHnLIfrZ&tA Exfst& in ts s€&ne-le car€, degl sunt

n6rglnite, nu sunt convergente (nu tlnd la o llnlti pentru n->*pn).
Dorlm s& asoelem el"ementelor dlntr-un subspaliu inelij"s gdllnnEotaot
funcglonali (deci
un num6r) numlt ltnitA ganera]"lzatd
r,l"m(xfnl) care s5 satlsfac6 ur"nrfltoarele condtlli:
Lim(6,x{pJ nFylni }=crJn{x[n] ) +p"[,-z.m(yin] )
{42}

Lin{x [n] ] =lj.m(xlnl ] {43}

gi, pentru crice gir convergent dln subspaglul B" a3" J"ui E:

e Pe de altf, parte, formula i,n eauz[ reprezlntX forma general& a
B reprezentat, de nulsinea secmaleron
funetl,onatr-eror pe subspatiul lul
lXtfauln.n"d(cx$[lPnfaol-rn>zlanOelreouclo6rprmnedlanlnnrtme-i>as*nrope-)>;a*gsnt"llourcmloalnaatiRceuesestoptre€fuelnzpcoalmlto,inuoalrlfXee,eudstesspcehamLgariarulnet olotrumrtaucrdaorlnsr

.zI TnTlpds uI esnTcuT alsa aTBuures

ap gspTr F?spacv 'axaJ3sTp roraTpuuras Tp A JauaTA Tnrleds RzEeuJoJ

(os) [:1+u]xl"r"5ryfiit= tri**

:p3rpe ialdela:ocolne ap TnsnpoJd) 6a:1u1 X arrJo nc glesuldap
9s P €irrTri.?J aJpsTJo nc lPums Tnun TnsnpoJd m?uad oJPsaC suas u!
"+T!rpe aspo JoTafeuurcs ps€Tc "llurbreuau 1:odns nc ?aJcsTp Tputnas
TElnTuunr.rTp
TTpaH TTJoTSA € aTlTuTJep o orqse3 plTt['.rT uI ualgeouncag

's?ElTUn PscTrlEU alsa

e:eolgzr"mdsaJoo pa3TJlEu e?Tnu6Tqo TalTu'tT fnzps uT go qA-rasq6

|It'i{t:il i+I(6t ) 'l

el
T l_ ,.., ,
'*'"
Ltolx,... o g_ l=

o tl

o
u1:d q?TuTJap [uJ1{ Telua^cas P '6<-u 'nJ:1ueel 'Ta?Tur.rT apundsaroa 1S

(Br) lujx sE?t'ut r"T-n= ( [u]x)ur?

rax$aur;ap u* or*"Jfr?Trqr 'ereznpo JoTar*r.ruras rnz*c qr

{{.t} Z T+NZ t4rT= ( lff]x){ur?
i=
r*/rr

esarpoap

'oJysal suas uI p?Tu1rT aJp f,pp eT<-u n:1uad glTuITI 'luaprna 'eJp nu

{ef } zau = ,n57-l7, !)= r"r*

:TnTeuuas nlduaxa ag

Sffi{="f(5*) i"1 " t [u] x) urF?

: o"TPSaS

plTu-[T aTnxTlsuoo o a?EzrTerauab 1a1pu11 p arTuTJap ep a?elTTEpou o

(?F) ra adFtrardzWf 6 WuF?uoo?feuaTJcunE= ([u]x)ur7

rS XTllflfl$ t( lnllYdS SI-TZ fllu$ts I$ [IIfltut'!'tttrf,ts

$uillB, cllcutTl $t $t$IHl 2L-tu s?lll0t Dt silntx ca

2,2 @eratori
Un op€rator llniar deflnlt pe spatiul 12 este reprezentat prlntr-o
natrlce pitrat{ cu un nrmir infi-nit de llnii gi de coloane avdnd suna
pitratelor elenentelor de pe llnli flniti, adlc6
iF-b"ur*o{'.*, v c*, k=1 ,2,...a,
lc*l'<ou. (5r)

t.--

unde o, sunt cmponentele orlc6rui elenent din la, lar drx sunt
elenentele natricel operatorului. Inaglnea prin acest operator a unul-
s€nnal dln 1= nu e6te ngcoasin dln 1=.
Extlnderea acestor consideratil Ia spatii mai. anple dac6t 1= este
dlficlli. Precizin ci forna general5 a unui operator pe cz nu egte
cunoEcrrtl" Intuitlv, ne puten inchj"pui o parte a operatorllor ca fiind
descrlgi. de nratrice cu un nunir infinit de llnii si de coloane.
Coloanele gi Un11le natricelor corespunzdtoare unul astfel de operator
pot fl eonsiderate eLenente tvectori sau semnale) din spafiul .c".

Elenentele sennalulul y[n] rezultat in urua acllunii unui operator
A asupra unul sennaL x[n] sunt ttprodusele scalaren ale elenentelor care
constl-tuLe llnlile matricei A cu sennalul x[n]. Astfel, actiunea
operatorulul A reprezentat de o nnatrice avAnd liniile constltulte dln
vectorll (sennalele) ... a-r[nl, ao[n], arInl,... asupra sennalului
x[n] se gcrie:t......lrtf
a_,[_rl a_,[0t a_.[1] ......1111.-,t:or],'lI
ao[1]
ao[-xl ao[0]
ar[-11 aat0]
r....:.':::Jt:J[:ylni=lurlnl=11...... = llI.<<uat_o'[,ltnnn]lj',,xxx[lt'
I
arlrl...ll*tlJl

(s2)

Aga ctrm arn preclzat nai sus, slnbolul produsului gcalar trebuie
in sens general adlci irnpllcind gi elenente d;ln afara
lnterpretat
spatiului l=. o alte posibllltate ar fi utllizarea formulel produsului
scalar din I= (introducerea unei medleri intre fonnulele (52), (58)).
Subliniem ci i.naglnea oferiti de relalla (52) sau de varjante ale
sale este doar orientativE; ea este rlguroasi nr:nai ln eazul spattului
13, atuncl c6nd gunt lndeplinite conditiile (51).

ga-rBsp';dao 'aT'TQrTlrnuI 11. ir! * Tnrnuuio,s apundss-:ef iEguEls) e--id"a.rp !'tr
rB+j 6rI?iiia$FIA ap qfeJ atPTg)a[: 'fT(aluqHtaTs

puEnE pJuaAsae r':ric,.t i ', ;.]:j.J+r 'i.C€lel)€ nx ?J6d€J u3 ',aq {qo ?a Q=tl
TnT '[p :l1rJT ro3*:';::.]rri:ls'.: ,. r i:{.'l; } Trr-rada,l i1J lJcxipr uI TTzuec tsa.IETFUBI:}
'ilrl;s!.-{ 3F Tzrj.r.i L+i:n lnhu:r.1 E*ir!} F?puasap [u]x Plua^cr,s EJepTEuoc
rrT,Td slsa F,rpTrrTT €a,;r'1.$Eir1ij!] ?iiulrq.?rr! a{, E ap AT?Tnlq poui un or

p eF

'rg?,ld ap {n-":. (snil np aTp6a TJa?nd pI alecTpTr s€d m6uTs
tpmJ€n];]r-,ilss$l-iB;f;E:r l'e"ii$ite:,d.1-in{BIr*u?tgt ?e{e1}nfldsn?eJ(s)r6auT93?"sTpJ?I€Enpgsp€T4sdTPrQSJgPepB1IUanr8esAeJPTSd.eTApAUapI
113'1',ip-:edp 'Nlel€:!]r"{!i rt.} +TEOe :otaTeuobsTp p:dnsPep ap aTalusuofe pUEAE
ABT:rr?..J[.€t€xqlusa.1J{:'}+1r.::.ri.Hi!4i}sT!p?sB'-€rl?d:LeraTJapcrs-TI ?op"ureas€pTudnadpsaaJpoJ3opT6acugTl4s3upnf[ppear.€IosTPTTJdoalgPxado
"spd un 3'f, tr,lds$.Ip B'f a..IPsJaAuT m TTJHSETdS' ATXJadsar 1trryse1dap
eJpolEzrff,:1*e?:r:; ,1oT83T-x?9s qn:c; u€-teons gJpnuTluoo uI "palE?Tun nc

aT€nfi s,r{?prir};r}r{ ?rFri aludg;rr,m.rd TeTHltonPTp Tnlqnsapap ep nEs PxdnsPgp

T nr eletuegsrp pTeluasaTa pug^e apTrlp'ut ap TBTJJsep ?una ( [0]x
lequafiocXm{}e gatrtlxed Ba.rcffftslJ$a nci artssJa/\uT TF ErpTuTt erpsETdap

ap AT1:-iarJFaJ (oaIt+liJx=[u]r{} qJe?u}T eJsspfdap ep TTrolprado
JoeR:dundsaJoc aTaJTFUT rlc lpcrBm lsoJ u gg e1{gzod

ad ap Tr;r.3rrau?4T* '1ar1.r1em :oT"luaurata prlTzod ez;card B rr:luad

[::;;;:l
: : ? :iri I o o!, \l i, oor "'l=rtr

(rE) o ri't il
E f-) fl
l: ::l

prlsqusarzeasJrde'an-uu T:tterfiBs iu*lx=[u]A :[0]x Tn?u: [aIu]aT^aTmpedXsroedrE[Inu]3e:loalraTfurlaPnucaasp

TilrolpJ?do AT?)dsar alelTun fnxolpJado
:1uns dr1 ?se*€ ap TTTqI?:)x€war Trolerad6 'ezueuolfce pTnrpc Erdnss
T$TriTEr-Ittiij*i -t*1a3,::eUOc?UC!$ €a.rslIOpIOaJ ?eaJa 1da:p ne lfOle.radO ap TBIXSE

rHf!$cx{ gsrHEsro ru rfczrsv xslrKrnrJ llzus rE luolfrr\

Jrqs {s,wst}"r*r ov"r} {.${rT }rfiuv} tr[v gsru&{F{ so rsmf,)sso luo,luwt "I

JgffirrJ rm cT!:padi! ilpl Ttss as '1ez1cerd
Ea Tn?{$rp$ u{ tBlmxtr{ p?srloo
pnsurTaqrolsfloneuqu'g2118pelfreedmsrs ap aTTTtpredo
pT alpluaza*d

FTaS i1a,?*T;lr)lg!.iv1€es? F'3.r8{]3 ?ults gzeeum arPc afrTdPxepTsuoC

{lTqsl}Jptral rJBInrI$8d Trolsrad0 ['u

Ba Vlq!. Cg '{iir"g?{$ LT*T& tr[il$t$ r$ !ilnmtc 'il?tltg

${rillff, st&eulrg 3I $1$TxHf, 4#'la"x.-i3*!tJ sHtlfi, B[ SililllB Cz

tt"' o o s . .1 0 0 ...

...i ooo a

tli-n"n,t a u^. u "",t 1 0 .." t54)
I I Ar./.1 = 0 0 .."

it..Joc oi o:.laj=1l "" ^ :' ...i..1

?ransformata d a ului el"s$ent x[n] deplasat Ia dreapta c-u un pas
nodulo a[nl este produsu]" eu d aI transformatej. d a el"ementuLul x[n].

Constat&r cX dsplas,area l"a dreapta cu un pas a unui element x[n]

corespunde, pe de c parts ac&iurrrl a$upra Lul x[n] a operatoruLul

doscrls de rnatricea nespectlvH, i-ar pe de aLt6 parte orlgLnaluLul
trasf,srnatetr e! nepreuentete de p odusul eu d a 1ul x[d], transfonnata

d a Luj. x[nJ (z:aspe*"ttv *rlglnatrul a' produsului intre z-t Sl
transfoeata u a a lut x[n]].

z. &eeqtmtE ps c{ffiox*rr:E ctJ i.rm s&InA& "s[e*n]nalulcyo[nnv]o*xtu[lnia]
semnalul"ui y[n] *tr sennal&l x[n] condrice la
definj-t de ae$I"unea u:ruL operator descri.s de matrlcea clrculantd
generatH de repl-iej"le inversate sl deplasate ale lui, y[n]:
i { ltylnl
*Ii. . . vyi[el.J! vyl-t101] vl-dtr . ..fl*l-1r I
y{23 .Ii [1i . I l<y[-1-r,],*tol rl
*xlnl At".. ryi-txo]] . " .il xx[fat'] l=l t<ttyt--t-nn]l_',,xxltnnl-l>)
II I
".f |
I I : .|t : J
{55)

tn transfonnatS d aceasth acftune coreepunde produsul"uL dlntre
(z-]aler[dlu]l x[n] *Srlley[lnJ.X. l.z}
transforsatel"e d respectlv
.x{il
y[n] *x[n] {55}

3. &smvey5l {1.} Accept&m o inconsleten!5 de notalie de forna

X{d)=X{z}. Aceast& notatle eemnj.ficd transfornatele d respectiv z

a3,e sesnalulul x[n] care s€ noteaxi cu Llteri nare. ln sens
matematic rlgurns x(z)=X{d*a). Prln ufiBara, ln contlnuare, desl
notalij-Le *uuit identtcen cele douX transfonnate dlferi, ;:.9a cus an

ar6tat,.
(21 Remancinn faptul ci fi*care dlntre componentele sgonaltrlul
douX seenaie x[$] gi y[nJ est) sga] ql
repn'*zentfuri eonvclutla a

Trls8rdap uTrd a3n$-rfqo ETn?fi'3€ sTTaTTdeJ B?eo? ap T6 er TnI prElp
tf[p nps ,:T usp ?€au6Te ep TfTrsssp
rtduls un ap €?aro{ro6 asTr?eu

il6l€ eTluTexos T€ aTtrnTo^uor ep rT.:o?erdo (I) Illuruusq .9

'^TlPlmoc

elre grETuTT sTtpTa-roore?uT €p lnrnpo:d FO ?pTpost pzoerlsuoilep sg

(8sl

l'''::. rir;l;,;f ,ir;rc-i;l,r, rlo;-t;,,II l' . irt,rRe6{uf[LlrrI<.<le€p^,T!/.r[ci.unuet,ETgl.n]tpT,lxxl-us{:,'e{1,€1fei&uTIr,1r+e-d}t-{endarrus1.:plg].3e+dm,{.eru,r>lyo1.el1iJall[a-uclci[eiT]ta-pc1r,teorrdTTiteln-_g*sJxs{9w5nr.upnr.F:s$oT.r-.p"x.}ode€'s(n?BBTaTn?gs€?}s]TauB'?(JTsz[eu{ups?o]Tu)a,B{r'r8rfoTnTeJsT€[nTTr'p[U-a[1ouT]}.+1,zT](BduxeV]aA."ro?..:p{[s:,.uull{eI=Ta]AnTgn[cucTTa(]lAnnTxsTulpreponTsTllnpeTulouulnd-uepIJeesfepc'sef

Tn?)un[p€ Tnsn ?ufis og$e1e:oa 1S elfnlonuoo ep llroleledo) psndsuerl

-gs]Tnol{nBTmloTAn?mczTJopre€p'e?€J€aEsoBJTerJea?Apuuql T[*u?cjsAeeTg[un{Jlluu-{gfingnaseTpTaauTn1?c:€dTTaaerTor{rcnJfauo:oneucooleeoztrram{pndulonsnedusrosoc-aropomazcrtTp{ xl.B[Eu
uTp [uJ,{ lue{EeTa un ne etrfe1e:o3 [ujl 1gm1r sfi m vrtflrgo3 .l

AprTfTt1gzeef0tJepaa3u?pEgATInrnTfuTqln*rRrenut?oore{gnTaslp:oeozn*pnuBeona,c:Esowxal?eptlroru€nm)rT:*narasaTfnspp;ntoEeuJ?dTn'ppETeBfuTlldptaqEsnFJTlJ:ooepoJn*rsTuauuaoXoE?a3€"B:fXT.euzupr,noRca;lE1spfsuaueppezcprpl(1;onlnaT€3Jzseepnr-eaarppe??TopppJT:lrudT?sTmuoneoO)st,

roTalBe.rolsm"T4 Tnsuwrd *pundsaroc TI TaTfnToA{roe ,zpc ?sacp uI TS

L l l:i[::{c.s} l. rnt*,[u-rl,r)i = irtr*i l'" rool'iuo],,(,,t]rl,,f]l = lulx* [u]/
[<-[u']x'-[u-ja>j [fojrj 1... 0 0 tgj,f]
: g"reTmT?rBd FEtoJ ,os&ETppzenoBT-3rs?B-urlo;TeaufenuardpsrmfdnEzaprJo3ulIpxTt
{Tszne3} ?ua@BTa un na pTiEtToArIoc

aTTcTTdar e?Es? n3 €T€uess ilTp Tnun 'alss:sAuT ITPTTnTaJ
ar?uTp uJeluT lnsnpo:d

r3 11!lll$ ls 'tBIi?es 6T-Tff rflflr$ts r$ ilrnilrc 'tlYrffis

"i::i:iii ,1.1r.:?

SfiT]UI, BE $BHFi[,I C"

irii'.rr s. 'ir, i:.;:.rltr:ul.a.r astfel sle cp€fatori
ti : ,,1 t'r i ;'.lr-clei;tiil.r.t.i- cane i..-a genere.t,

:- : ..: ! -.il.i ;l_!.',F -Ji.{,!{ e] at5,e:i_, f eZr-lltat.uf f j"l.nd

,,t rL,l,egt.U:i.;,1 "
;.,. rrr.:.ri. q4i-"l:enf-e
,. xIn]*i:- 5ii vInj este
'-rt

l _,,
.::., i't':ft i!q gJ+!.:",.:?!ttsj {:1-f e*la}tfi. neC,"refit6,

:i - Tq,'r!&l rf :farr.r. rlllrl'\I ( 5e3
{ "J

g,1;y r1.-1 1 .1s I ,:rlir*f:Erunziit*are corel"aLiei sl
r,,il;rir;i:!t, +r-!jlrl iryti*. t,r.zulspu*a cel_ei,l.alte
j .r ':r.: {cel
r,ti ::.r ii1 {':r la .ti j. ta"f.t i , x[n] Cl y[n]
pnttstX seeuenge
i|i . rre*

,,1:.!.1qi-i". ;.i i':i-e il-t]-fi; adi-ch nlci un semna]
I,t,..,tt,'i r,it.n'i: F* [!n fi] t se*srnaJ- y[--n] r6t gt
::.,-. .'ict:i'+q:.t.n .:u orlce }i, 1'[-.1+,[] dec.3t

i i:il.!- i,q.l.,.&

a{nli /_ ,f,{.rJj€i,rrt A {feneastrX} este
ljri,:: ,.::j i-rt +l $ Fg dr.lgcnala pri-nci,pe.l..X
l.l! ll l
6 1.:+|li i

i {

rRezul l. tt :. , ' :tr !
- tII{-aaa[ -ll:rr.']llxxrxfl'cr[-ll1]
fit 1' " Y [ ."1 ] {'64}
rblelc r.:i..++4';-:
hqle, i;r: tili I I -x i tll
Y[j j

I

t

,,,' r,:;.r^,i iiii'r:-i.,1l Ia,*ft[ini!pnl-'+'"arL]el}"ne.sa.ic.iE{rrtuj:-lr.rlJ0l xlSIn[Ir,e]+sti'r: {61"}

.r. i.,.,,.q. :31:.r,1.46- it':'{t !:.r!.}!:ni.qte regttil_ el"ementelof fi.lpd
.' , r.' .: : ,1 :n?r'.rr.tili.l+ i:rr*,i. i:f5;f .qgt"rr. di:eptqffgbl-1!-1Af€rr,

'TTtp?ou TelseaB stuelsTguoJrr[ sp?deJce Ec TnldpJ rtlurtgeu rr

'rr rolaluures eTp (z) p rolelcrrotsuer? eTp

glua6:aauoc ep .roTTrueEop e1{cesraluT elss qfueo:eauoc ep TnTuemq

t t{€e} Vllxrpt= *, txtr (-> tul rxra

rltililt:Irnl 'r

'a-(z)!{=(p)x <-> [u]x eqcered o uprepTsuoc

.E?.Ef,rolsuer? ro?Fsc€ p p?uelsTre ap eg1{1puoo pf ,e1led
orTlnteTepueaerpodpeFudre!f'eTrJo€1e1(zsro)1{zpe):a(pdeoxTeeela&liltormocJ)lfscTueeeTrT?agrEleTufTTrelpEAlevIE{TuraTdpoourrodd1e6,garuluoce1sepo1f1rq{eet 1lrrterda

'TTrss rolsgce eTawns nr ezerJnT as gs sofelueau

slsa '(are1&oo afTq€TrpA ec a11a1rd z nes p rofeTTqplrea aendrg
E1T8Tlt?tpT'pTeurleas?JlenerdrlToeqtpfrlurepET.re1un!r)o'nf6lTdJTelrcGrurpeoropdcsuT'aT!lTu'grtocms tarpErJorgJrpcTurTaoUcnu':nzXlrue{ef'dn1Teef1lETgdrrrsetleuTpTpprTeJuTpXscdmntazt

arpolgug@sp Xurs rrT TB ,J TnTtrBds u! p Tal€urotreuprl allfglapdora

z I$ p u0lffirltu0Jsily[f flI$wxludoud 't

rIpEorrlnpMprseosdrmod ap'prqHITeTJJnlsfanElTEtETlllndcrsumoInEgzetB:romcdrTo!lrrvd13'[aeupc]/Tt'arcTungop'?[o[ucured]]zxJ[u'aT[]ulgza]f[ExTupr]runruu(ce€arss1:1eeTupun1uogpn11{Ererpo:radldpunoaasr6sdp

tze)

:EacTr?Pa ep lslueze:da: lnrolBredo

elfprplap ugrd alsa rrrptf,s rrnq r (ercrrc! nrxrlor m"fro",t .L

r3 lllffts il I0III{$ TZ-TZ Itil$IS IS ilIn0tIC 'tlvlus

$[[T&I,8, fI8f,*ITB $i $ISIBilE ?L-r, $ra?I.,t tr $rttIALr o'

z. $ffnmance sm{Hrr,ur vAara'rr,Er rmpmmffE {*u*****} {5{}

x[-n] (-) x(*) =*t+i

DonenLul da convergenla se transform5 dln l,/Rcldl<l/r respectlv

r<lzlcR pentru aecven$a inlsiati. in r<ldl<R respeetiv LlR<lzl<Llt

pentru secv€n$a lnvereatd"

3" fnensrCI,&ie*Gn sEcvErTEr coxJrffiatr (65)

x" [n] <-> xn (d*) =x* (s*)

Pffienfi"l€ de esnvergen$X cofuicld.

rg" $rtmsene mrmnrn (o t"d*ril u*oo *o***o|

Flacfud de la s€mnalul xlnl se poate defint semnalul ylnl
c.€nverg€n$X (al varlabilel z) r<lzlcR"
prln relatla:

rr"r "{6t*l ft;Xi i>o (66!

fransf ornata s€trlarulul y[n ] este Y ( d ] =y { z ) :3 ( fl* } =19 { za } eu domeniul
de gonvergen$[; rr<lzlcRt.

5. Ibuasanxa s rntP ""{k<o sau >oi; (671

xla-kl <-> d&x(dl *z-kx(zl

Relagta este vaLabild ntmal in cazul transforuatei d (z)

bllaterale. ?n cazu]. transformatelor unllaterale, pentru deplaoarea la
stfinga, trebuta elislnate egantioanele care trec in domeniul" valorilor
negatlve al-e variabilei independente n.
Einusseaansdenalnfn, indctEeLnfvlnacllt-t,ioariinttedraifcn,eosrrfeeosnrpmtuadnatzecdl6tousanerenllanlatuelulrlanele<s0tes$enJ|unalt sau nu pentru n<0,

de la n=0 ceea ce

conslderate nule'

Fornulele devin:

/2 A fogt !:ratatd in dete"l"tu 1n subsectiunea consacratd

oparatorilor.

' [ul$*[u]] €,undnse.rd eTlgTaJoso?n€'rPTIllT?Jad uS

{€r} tzf frro (rlu* if-ln Fl-r {*> lff*rr]5 t{lJ 3

; ([x-],{

<- tXIfi ss:€rrJolsuE*1 u;:rI genuTtQei g?esrsAuT E{uanpas €ITgTTBT€O € p
ElesroJsup:1 X€ ealsae€ srluTp €Tn{rn E p PXBEIJoIsUeJ? arltrgp Tnsnpo:d
stsa [uJ$ TS [t ]x aTstffis Hnop e pe1{eleroe :oleztmdseJoJ TnTnltsulllBs €
z E?EaJO$susJ$ 'gsps'Jsilu.r rsrarrcas ne egfnloaneis e3se o*i**0
"L

tee) { [u] ry* [ff]d) * [u]x= [u] qu { [u]c* fffi.tf ]

: s&"f ?pT3os€-

tri.l [n]q* [uJi+.[tr]qu [uix* [cr]q* ( [t*],i+ [er]x)
:Blsunpe ap g{el HAT?nqTrlsTp-

{si} [a]x* [ir]"f= [n].d* [rr]x

:xnTl€?nso3*

B?se sT{,nTo*,uo3 'g$r.ia*ras FXTETU€'J e1 (a{uarlces ajr?uTp

EsT$raBsaTHs6rJ?appTsrFl?\Tv[Tt"aFtnTdf"x:1rsJy?o3e#f"rBa..uJ4t:FT.e€.or3sgdi nSdfuaFeel"?Tertd?rtCmtIudlg$w*eT?rg.auT{slE€oenTpcg,reg€ErnB!Wr€OgTafP{f*enTrTTdtt?oal.$A:t{suudoffeoisa?as"6dslTsnssrxafsTaspnrTsuy{pnn: TIsrooe?rTrf'uf8ePsoTTufD€Jnmtuldwtf$sfaiielssf:

Enop seta* JoTs?eur$J]E{Jst? aTe gtuasra*uc3 sp JtrT'FTuetmrp YT{easreluT

otse TpT?n1oaur:J T.:e6J$3susr1 Tts gdua$:enuoi 6p TnTuGuloG

3e*&e

{6s} {a}s ir15'* {F}"{ tp}x {*} []f.$l d laix = [u].{* [rr]:r

srs{ffGeKn "e

'gzearqsqd ex gtua{ireo'uui}r ap e"LTTu*w*p 'aITJnzP3 al€o? lry

(89) , rlx3n&[*d\nz FO4*f F- ip)f{.-p { } [:y+r: j x

r-q {siX{-a*,.p}.ff,,$"0

.T-.ry

13 !'!T{SF$ f;f, 1ftIl?d* ,"2-Lffi [H$5$I$ [$ pIIitSSic'ntvnwns

$Bfiil,B, CIAC0IIE SI SISTIilE 2L-zq $11il0[ 0r $BhlAil cz
174l
a. Tnersgonuarn snunnurr,nr ISJLrIPLTCAT cu A'

a "x[rz] <-> x(ad) *xl3)

a

9. Tnnrero*uoral,t DTFEaE!frfEr.oR.

Diferengele {de diferite ordine) ale unui sennal discret sunt
sqmlal-e atagate acestuia confor$ urnEtoarelor defintrtii:
- pri-na dl"f,erenld a sennalulul f [n] este
Af [n] =f [n+1] -f [n] (75]

Traneformatele d gi z bilaterale al-e sennnalului diferentd sunt:
z{Af lnl } = (z-1't F(zl; d{Af tnl } = (d-x-r) r(d)
(75)

lar transfornatele unilaterale:
z{Atln} = (e-1) FlzJ -ztl01;
1 (d-1*r) (7?)
DtAf [n] ] r(d) -d-1f [0]
=

Dmenllle de convergen!,d ale transfornatelor semnaluluj- diferenti
de ordinul intii sunt cel pugin aceleagi cu doneniile transfornatelor
semralulul initi-al.
- diferenta de ordtrnul doi a semnaLultti f[n] este, prin definitie,
diferenla prlurei diferenfe adic6:

Aaf [n] =A-f,[n+1] -Af [m] (78)

- dlferenta de otrdinul k este prina dif,erenta a diferentei de ordin
{k-l}:
Aef [n] =At-lf [n+11 -6*-tf [n]
{79)

Transfornatele bj-l"aterale aLe diferenfelor de ordlnul k sunt

z{A^f tn} 1= {z-1) w(zi i D{Af tnl } = (d-'-1)tr(o) (8o)

lar transfornatele unilaterale : (81)

z{A.f [n] ] = (z-L]kF(z) *rEk-1 t z-ryx-t-tgtto,

D{A"f t::l } = (d-1-1) }r(d -d''o-ta[-, (d-1-1) t-r-lArfo
J.0

unde A'fo este diferenf,a de ordinul" i ln n=Or iar Aofo=1191

'{tt spd nr s?GlTUn Te?das?
PellElulrtoprsuTn€Tsnllcn{Ip*o,.r2d}/a,.p2u=n(d*pcB*IJ}o/oT TTnnTTnuraEctrouPEsusIlsr'leroJlsuepaeJJl
T€ ro?cresaE

?Pu:olsusr?

0-,ll

(r"8) [ry]f-ul g '-(
i,pTpoTJedTffias fnTuuu6s ,,ro**ruua6ll T€unss un e.rXuTp

€TtrnfoAuoa a?Ea sTBoT.redlms Temffis Tnun stpuJoJsue.rX Pc uPJrstmu

tggl r-rp' [E*AI]x+' ' '+p' [1]x+1' [O]x= (p)xrp* tp)x

tssl Tr -nTxr. "-;tT*Tnizxr;rr +TbJ67 *
*;,eTr-n[rrfr+P'.-et {TTffTglx {*) [ar]x
ll
(o<n<u 'Ix*u]x=1u1x1

Htffis[ffid ilc tsr(xlrrutrrEs ufftrrwu8 Es${eHsmu& "II

"lTETTnTaa Tns.IsfiuT Tnun ?unc

I*u BT gtryd TffirE 6 Tg S#tusJaSTp E sJpnt sp 11ro1e"rado pr up^rasqo
'Tnu e?Ba z ne e18a{1nuug ss,€.r€'J Tnu6ura? 'Tsms s I-u Rc ale6e

ereo;.redns TslTm:gT ?lTro?sp'q1cx*3e11un E?.puJoJisrr€xl rr! ATSnTJUI

{}s} o"e-#d ft-z't-$')i r ([Irls Hi v= i*]s

:Truw s s-xTrrTJep ep Trnp#u Hnop eTas eJ?$Tp u.:ir'1g6a1 ag;>Junm6

t€s) ffi=##.'' l"lrfi
i€TarsaJoJsn*.r? s*,s {*u Br gugd ennwt"[ *r*o*o rnrpuursg

T*z fpf*iT (->' "'+ [r' u]t( lr,lx' [0lxi+ til]gtolx = ttltr03'q

fzffia=

'{7-zl7z*11t-I)/T axss 1e1d:aeu41{€qeo€wBsJ*'3sssu$p:Tl puungJze'[eutr]?osuecleEplpTuunu ,*oo*S*

p1dea.:1

TnTsuu6s TS [u]x €r?LrT$] e1{n1o*uoo 6l8a [{]x uTp u sT 0 BT ap sutg

xrums vr$*ror flru& "$T

rX XIllIl$ fiN tflIlY{$ -;z*TZ d*8x.$1$ t$ [Lt&[U$ '[TyflHtS

;t$BfrrALE, fi&[irlT$ $!$Y&St d.,F-*i{, $t&Ixt}t tB $Hl{FALB C'

HaL obse:rv$sr efi, da:-:& sseai*l-seielurLJ-su"rn;etnt*oraft:*rqinr e6otreigll"ninej",talat rlnptoimlipi

(n), poltL t-ra:rrfoxxat*1
transforeatet x e semirai-uitil sesitrp*rLodfe sunt, rddEei-n11e eclta$j-el
; -Jfi
T tr- t *.j i ,i.- ; " i =r. ). - " M*I (88)
e'-F

1?. J'l {}a:i.t**.9*K s$rlL a<-rei-* seq}nn-:.e +are sattrsfac retra&ia

SEHdA&fftE

rrl :':1 = T= :, i tt - ;wl 'r*l t-frfl (8e)
1F*a-, s

unde r*[r:i e$"t € rog€,illr:&lul gen*rai*v:'r a,i. gemttla.!.ll]-a:.{' peri*dj..c x[n].

X[nl*Fi.,si-.d.s .,:!1.'iry--i:.' .. ] t,(tl .-q, +;i.' nrl -1:e+J?rili.,,)X^{dJ*

s {9O}

.,r,-"{d; .1r"*.{o- '*,-{,-"8! * u '*

"1TT;

?ransfoxmaf,$ { ic: t:i. rra}}tr}ta:s-:.r}:i: p:'*:rJl*d"'t*e e$Le # gerie formal& care
converge trg *i*j" r*fier:.;m;{}r}siri*fpii*.}::*i*Le-i"'rs-iti;l1.:":aqiz"'c'iiief,l*dpl {rxi.i*"
nu Rcm,a'refu eS un ia

convoJ.uf dlntre

semnaluL S*[n.i ,..SSf;r*3rii i*-Er+ *ei:::*:!::l.l-*', rJe.r s*rnnna]ui semlper'lodlc, stma
ge f,ef€ olci, 1.4 -,,u,.1.;.u.u,",j ,..-dj-lr€triti*;i{i.!= E*ne*'r:-f*r, 1:l transfsa-mnt5 cl (z}
ace&6i* ar re'fu&fii- i-a ti:;-i:.rr:-i.L;r:1*a aJ:arlsf,*xKatel-r::" aelcr deui s€$ineleo una

dlntr* g{:F:sri..,!: i:€:rn:;.iiri:.i.':1i!i ds:ri,:,dt f;x l;e,r':a f*im*a][.

fS. S[Ts$Rffi4e%K \Jr,i.:.' ni:?,-i.s-4J'i3si.T-ip1'&.ffi trRs&E H

Acesg€ib #*r i.'efr': i 1 : ;:*..'r'sr^,t,e ":+r:aa.l..B ,bi ds*i" 1a Ernnsf+r.mats
unilateraSe. i)";:: :i*f trl:f.i-* :'*zu,T-i:H, evklent,, teorema valorit fnj"ti"aLe
f l{)1 '.*'r'1*,
{Sf 1

Alte :iezu.i.';"4.:...:r :tfer;i.'it$&]'rg l& rii$tiFrcFt&rs& aslmptottc5 suntI
- DaeE Fic) el;si.ii Freiltis: ir:i>r Sj., pentrr: un S.ntreq n, pozttlv sau

neglativ,

{ e2}

atunci fIm]*& F3. fi,;'i ,.r' li<rilt':ii l:{.{iq"
- DaeX F{, i sjt.-3..':1',li il{*!-ltrl:L: izi>:; *:L i--i:;:.,. {:.'Lu!}giL fin}*>0 pentru n-}@'
- Utitrir-;Snri ii*i:i::"i.'r.:i..: t.l'€i:sf<-.j.nai*,; :* rjr,'jiLate]:41e, pentru z->1

aven

: eITTlEre-r

JeJsTlps aluaAsBs snop -rcTec aTE elu€braAuoc ap alTTueuop €c€c

(oor ) no,-',-"{})r,, r, $ t'r"}ff=

="-'{no,-,. r"l"$ff}rn)r 3= { rirt6 tutt}z

z Ta?PulroJsueJl erlTuTlap
plEFaao-lcTTTr"leeupJalIneTpnebu-ppupre€fguuFTsgsanser6aaFl:?saTluasndsralustEquraa:isn,3?€pldesa6?uJpa?lpoJescuouETmnlosonds{?sll.naI:aa€TpeT?rud1J€aaeusas€oXpasTpsTn'E:lrS€no?1es:1npo?3:uTrTaEa!n,1JqrzeeTJxa?lnqauu€3rT5'nuJIonuopIrrTIeoofnlT,ERpeJxTweTBpa?Js1Uuo(€unTJT'Ts)tdluu,zr{v/.eIo/TTJz1?ld:a'g1)aTgeJaTluluaJTToJunuaTnToToJba{Aca1lu1Upa"rnsTuo1JpEnz

(ee FP:-n (n),9 t;ni,a!1r,o" -nPr-n i7n)e (n)"!*'ft'ui=, ,-s [u)6lu]1

€c urElgrv uorrsJrsgAJgs TJisnaof,d .}I

c

(86) --(:-z- \x \/-' \ l4[fr4r'lrr'./ ]']

acaJEoap

(t6) 0-svI -

(z)rlu?;= iuli.t

*au1{qo'aTpT{.rpd
burns ap a1a{uanoas nrlued sns r€m ap e11rde1a; puezrJpTnoTlJ€d

{e6} . +0-I:(r-rsy r-[,
(z)X tr-c) luTT= [r:];m11= [c]x

NPB

{86) [(o)x".- tz]x (t*z) hI r<-lz= lolx- [*]x

:uaulJqo

{?6} [0lxu - irf)r{r-u } {-} lulxV

unc ,a1;ed plTp ap ad .re1

[u]xq[0]3;i@"xr*--T,L.([T-ro=o],x,= T<-z
[ff]xgi7urrT=
{€6} =*_z { fies-t.. -
z ({ulx- [r+ujxl 3- *f

:l I[n1?lliXS l0ItYdS L'-T7 llllsls Is flIIncltc'tltxtals

. u,qW4ffiFffi.* i,i.:ll=:ffi*

$sf;fl&Lr, {1flt:lllYl $l $l$T*H$ ir* * $F{llii[ $g $$H$&iff ij*

_{ "au

r,,r<'! iuLi"{r-riri 1-. 3-l2(ti:i(E ( t0r_ l

prln fuenultir:ea eelor doufi rluble i:neEa:"1te[:!. se ai:llne c6 dameniu"i de
csnvergentE a transformatel prcdusului este:
rrrs{ lu l{er.k* {1oz)

Integrarea se face pc un cere aflat lil lxt.ei:secbl"a d*menillor:
!ri6-! -r!.i*t-tl"" x,.p. ttf-(.'
flt(i.(**.*; r { 1CI3}
YP

tn caaul seqvenie-lor i;auzale. R" gr R, sr:nt inflnit"e 5i raza

c€rculuj. de co::verEenlx satisfe,ee autowat eertr*i1,ia a tloua.

se regnil5, integral^a cit'cr"lleE& se e al-c:uieaaJl f ulosi.rid t.eor€ma

neufduur-llsr ear€ c*nduce la

zttlnlsln] & $'t u) ;;'{jj'€- i { x04}
J*$fr"i
Ree..--' "

'"u'
unde K eete num&rul de poll diterl$i. *' {i.=}-,?u.,.,H} ai fune$iei
f{u}lu" fn eazuL unui pol mul.tiptu de '.:rd:i"n m avem
R;; *aF"-'(-*uw--l's-gr:**{-(i,m-1,.*1,)!*rui**,Jfu,-.**"rri[,I{u rr' ,}r'{*r'-:r'u):ct";t'r-'ili (105}

tra-r i-n cazqrl mrui- pol sX"mplu,

Ru*'r.td.P'tuUli;!{i"-Ii-#{.i-{*tJ uliirrgrlrv rr*rJ "x-.:r1,,-, { 1CI6i
U

Fart ieul"ariz*nd relafia {30?}

* rl":s Inl a "-e#;$"ti;r)dt*) c; ltru

pe$tru eazul f [n]=g*[r:i *i z-10 olrt.S.nem for'muXa ]"uj. Fars*va].:
ss
if [ni :*" e* f : si*] l'** {lw}
X

a edy:el txlterpretare gcr€spunde invai:j"anNei rrormej. S:,1 qfomenille n 9i
sOttb{nmEeu, npbsrutlrdarteupl tuanl€irenolarmtieei). l-ul Farsevatr repreej"ntd, ea ISt i rue.nbrul"
Fentr:u sawnalele din lu
$i cercrll
lenff,ate se af I"d {n Lnterlorul dmeniui.tli de converEenf,X a trans forqatei
r,

eltelar 3oT ere Tg

(CII } np tul^ri
s TaTpJbelut s z PlPu-ro]suprl

p?sTxa TJun?P '[q'e]aa nxluBd uuofTun abJa^uoJ [u]^; ruleue:ed ap

eXuapuadap TaluaAcos e z Ta?suJo;str€"r? sreolgzundsaroc pTJas EJeC

{uJflrftrwd Hn m .lxodw trl [xttu[ffirrl '8I

{ZII } Ap \/ -/r (n^)plJp

--l:-^'z)gP

TS [q'e] uTp A u?uad u:o;1un PsuamesP
ep aoraauoa {[u]^]]z eTJas T3unle '[q'e]an nrluad uJoJTun a6:aauoc
Ap '4
(III } "-'[n]^p *

ag 'U<lzl TnTusuop ug plsTxe {[u]^l]Z 'z pleEuT:roa;ssuRpplelpTRacrgEczPeB''[Ilqsu'eo]uannp
u?uad ElTuTtrap tr9 a 1n:1eue:red ap gquapuadap '[u]^] p{ueacas o a1g

'[Inf,Jrilr{nu€d nn n3 ilrodru f,[ rEuuArull

'ffiurp ap gluapuadepuT a?sa 1e1e:6a1u1 eaJ€oTBA pc InldeJ
ep €EEes puguTl 19 'g geXeu:oJsuef? e aldrulgap ap TaT{pTe: Tp Trqunut
Top JoTac € ?TuTluI pT z ET ep ea;e:6a1r,r u1:d auT{qo a6 Tnlp?Tnzag

{OII } nP ntuE <-> InlJ
(n)dJ

:u{oz1/{)eu.{{ul}a'1nTcgaaflslupaup*eopasTeBse"lrsazalpaETBTlpUluEngtzoJ'ssp?ucBeBrrtrlcJtor'60}sE=up?[sJ0Tl ]xE]€TETgscrleau?JneBl€uJTra'JaUAdsP<oEU:ldze<l?clpuze1?l;asunT1s:r1ddluoeErasdXdasTr'1pxSJe
f,lHxAJfls rexo H Z rElHrilrorsmrf,.I, utuutxefiilT '9I

Q ToTneloJ p z na 1:ode: uT EaJprrTJap "U TaleuroJsupnl P e1{1u1gap
u1:d au1{qo es TnXp?TnzeU

€k--(60r ) <-> [r:]rn
etrdeler Jof erv
r&ancts rurn ts U rrrtsluorsnvu.r wrerr.tf,xuxdll 'sr

rt tlilflxs [t I0ItHs u,"Tz IruI$tS lS tIInCilC 't1Yilt$

$g$E$i,E' 4l&el]ITE $X ${$1t$hH J) tq 1{-i $PiYII}L BB $BHfiAI,E O8
r;, { Lr.4}

12*

f,* s*ra{t,, a n' flo';', r:,,{l-,
JgiSsijiiJts l'raett\4v

rg" kmanle* pE cffie$& $rulrers i3rrni trailsfucmah* fi pe cercuL
sff11fialelor dlscrete" fut cazul
unitaVtaeriaart€tar3mme&erieu-l[nr:plu:irtSaln" $aRr:qqinlslete$tgurlui*i
d5o.nucXaproeli.tnrcaanmsfeor:mn avtaartUresbeilfiex;pi,rtrtrdBee f:a 0 frasgle ratiorraln {raport de

fonnar
Ezfx trz.-".;&-)t ,
{ 115}
trilaup- d.ws ] -- P, SA--'"-- - -; '--

"&f'

g1gd{fiut $.1. argum+:nt.ri} friruLr;$i.*3 f'{F-:'"} s€ sllil'i;,r i.:* rliint::9lI de & i?stfe"Li
l!r-"i.a.,fdr..l*;-l.rA-,l&S$&&l ll!"oe:-ijd*^ol':r;l.
trlbl

erqrdv{*:tel =-,*g.JA+X #rSri *ts-s-" 1 5* *;,g{*3u"=;;'rl {'}17}

e &nd *ii g)&rcusg€ ilerffi} *"rriilai e, trjU ?- FvJ i gI" aei Jtl; j
vartragl.iL{e siextru3ul.ui Si arg$m*entulltr etji
urm6toff'*!s int'*sF'rct$rile getis!*J::-i {:t* :
nu:dul,ul f tttrr::t ief €st-e Flgotlor't I (!{lal. 'Ju
rlppaeurponodcsutu:,teuis:l:l*.'&}e!*.*uj$,xrdrl'{.i€1:srnettt"n,Bex('i*lref,fl*urtl*du*erhe}i-tde-px*dex1..l*-ei talae'{lrli}clp*'stp}lf.eito-atirttrd}S-'auci
*€{-n,ru}; ergLmi€*t.rttr fur;etiel este *ga3

$fi1 stl!*& arq'iaaentelor ntmXrdt"orul.u j. slixltls

s!.rea .argumentel,E:s numit*Euluj"" Ptrltru
esmplul dln figurX avesi:

if ttr--.r !" a* *"rgiti sJs; *,# + $- y'S*e { rx8}
ffi#

3,3 Y* rtarlmlttsstf**lrseieertta* d {xi e3* iser:r $*ffiefr}* reffiercabil* din ff*

*X {z} a s*fflsirllu]"tr-i. olnJ est"e

(-_a'-?) xpu <lzl JoTTxoTpA prmdsaJoe g{ua6JaAu(c ep *IezE_r)

(8zr) T+ (e)t{EoczZ-zz (-> [u] o (ilP]r+rrE

-lA :(z) a?pErolsrrpJ? e?Ip -

'(reTmTUEd zEc un €J pzeaJppstrT ?s ersoTJalue 9?pEJotrsup-??.)

(LlI't i ,Tx',!,e i <,lz

:TnTnreuues E (z) p PlpulroJsueJl -
" I> l.-zl =lpl n:1uad EITqETerr a1{e1a-r
iSZ: {rXtTro*-uz+}*g"=3*=y;(F*u-i-=xT9vpoPlp^ow+1:=ad'T'u'+ssrrTpn+InrTp€+HTuas<p-) o'r
-
(xf-u)Q 3= (u)"9
(z} p plEurrolsu€Jl

{ser} ,-*29-y ? ne-t cEI ( [{-ff] o- lai o)sE *
p (z] p p?EeroIsuEJ?
' ,rp-,F-=ir*ET=tot n f, <->

:([{-u]o-[u]o)-" TnTnTpuEas
1r77''t11-7:t=ZT_=:lllpp>)r-t!=I'[:r{P-u- ]sWTo*=-{[ru-]{op+Tn' T'n'f+p"mpl+aspp+r(z<)->p
[)r-u] o- fff] o -

elpuJoJsupJl
'TTlTpuoc ;$eelere utr "tr>lzlpl=lpp,l Fepp E{TqeTeA
(eZI) e"-z=D--p=-prppE:1u31=oJE"*e'+T"npTtnrT*e+rmpe?s +pT(zi(-p>E?[ueuloLuroprsuprl -

lz7,rl iz6^tc[?z ,-) [u]oo 'ose=z n:?uad Bc ueuagqo

{IZI ) (t-r) z -- @p:+TT)-g--=I L -PT-T ,'-,', rLrr-.'l orn

T+Z
2Z/l trc 1e6e luBau(azl)a lnug:d pugAE e?plTun
€?dEarl) [u]g(Zll)-[u]o=[u]oo Tnlnfpuulas B €lEuroIsirerX -

'I>1,_zl=' (lpI=l(pnl-XI)ua(d'''+g"TpT+qpp+fBI ^)ap?s3arrByTAsraasrdqxoe)

{0e; I -T:Z---z---'-p--T-T-:- z-\ lLt-rrl-n

Tnrm Tal€nrro;su€r? Ig Je1ncT1:sd zeJ rm e: pto=f(aizuR)1)p{qcoprp?aoslls-mrao'"d]+Bluu[aaesr-1uT]*p$uu[rIes

-ulg+[ulg=[ujo 'alplTun g'ldpaJ? TnfnTplntes

{6rr) r (-> [r]g

al llVlllS tl IflIIYCS "-TZ tHil$ls I$ flilntu$ '[lvt[[$

'{Fi--e' ". t*" +''"#n'"'

8BlilL3, CIlCUlt[ $t $I$!E[B 2L-r, [tsHtIUr, smHLB cz

cosh (an) o [n] ( -) z (z-coeh (a) ) { rze}
z2-2zccsh{a) +t

sin (aa) o [n] <-) zs j-n (A) ( 130)

z2-2zcos(a) *f

cos (an) o [n] (-) z(z-cos (eL)_ (1311

z2-2zcos ia) +1

ein (an+.b) o [n] ( -> z tzsi:n{,p} +ain(.a-b) ) { 132)

z2-2zcas (e) *r

(fuenllle de convergentE pentru ul+*i-ureLe trei transformate sunt in

afara cerculul de razE unitate).
- TransforraLa z uniTateraLd a semnaLului a-*ho[n+k] (transformata
unul semnal deplasat Ia stAnga) este

zk-3--zk1!*aLz-1+. .. +aE-\z-k+\) =-zP k-_aa-- a(ae-at) {r1-Y3Y3),

z-a z_a

aunnllhlaBlrteaenreaallerecnEeanurteetle.seemccavereennltaelploasrrupdpreilnpfuradreseapntlteaasrlaairesspltaienvgcaailfoiccraii.renteraagpnasatlfvroeprmeanalettreulluoral

n.

3,2 $ernale eauzale, legHtnrra cu transfornata Hilbert

circularH
vom prezenta in contlnuare restriclla
ounrlteastet'rdlcetlceardaicntedruonl ecanuiuzlanl asl dunsuei 1mpus6 transfornatei z pe cercul

transforrat. ) smeasnniafel sdteiscor.reatv.a(EgsiteinnadluornaeLnicual

conslderdm un eemnal cauzal x[n]. sennahrl este invarlant. la

lnnultlrea cu treapta unltate o[n]:

xlnl o tnl =x{ni (oo [nl *.16 [rr] ] =xlnl ( 13{}

cun produsurul d1n domeniul n il corespunde convolulla specifici

transfornaLel z:

xtnly[nj ,-, -:rty(u) rr +) + t 135)

unde conturul f este un cerc de razi p astfel adoptat incAt si

satigfacE relatLa:

x,Trmpzsn.npuc 'PxpTnJirrJ lreqITH eqseJed o gzeqlJo] alo leluspudepuT
raTJnod TapTu€ms sTer?m€ut:rorJlusreredrx'pe"TPplaerrB"uTa6tPnuqt{qTgo ( rEeJ eTTtrPd
?arosTp
')TTTdETslt

(rrr) #ftnri (,;re;;i ur'f+-( [o]xi;r= [ (sre)xjur:.
tt

(otl) 0' f Z o-aiaz it.it-e61)rXr] i,urr?ijf tZ *, (/ r^l jxvi\€-tadt=- ir i/e-f5#1i'xl jd6dr'
lU
15*'' r

: e?ellnzar aTarBolPlcJn

uaulfqo 'oTlefeJ er4lfn uTp BrEuTFprry TB elue: all{-qd pugp6g

{6€r} ,'- ftur"(ors)x/f = to:"f - (sre)x

lere)x?-6p

n*?.*# I -rI Ggee+ rse) - $el z d.3- o<-e f yz
(sgee+tge+efe) (sre)X tuTT
*

(Brr ) d,*

* +H#' ry. opor"r
xI_ =( sro )

enTT('gnz3nrcTleTluxnrgaTl1mT1pdllul1enusoXaozaeroEseoe[r:reaa6o?T'ezJtrJ=s1rT:uuaplazB1csllgrlaaziLrTepErTlonT:pTTeJlllncsETurTrrupaemrnpubllepmro:dlaaTlXrnepuauaunJrpocofpcrcu"B;eesreJsstT6s"apnra1*cad:puPaausuuuz*rrgoeoeacJTiplpnFETeau:?rlpcegTuappu'a'lTgzausdprnuoanJnlJ€n6odi's16llonnungTsgtppda1uloJe'pTudd:3d:aeTIg'pelJpoPT'urFlTuzpBlpepnilroortcp6xruJ{;oanaJngals6usErlnduTleus€au?axnsaa:?dxeaps
auxrcpsBBupflpTr!n;TzepcouuTaTsnnqsarTl sslusaaJpv 'aXp?Tun uInpozxTarceTamd TPTlrlepdBgBsz
eTTqET.TEA
eg{e1ar
Elroq

({.sx } nnpl1+nnV)zrn'xlr!#-= plxl - iz)x
,z

z E?B[roJIrPJl u! 'euT^ap (tEI) ET{BIeU '(z)A
ap gNua6:aAuoJ
TF {zix Jofalpilro}suBrl ap eTaz€J luns =J 16 -r eprm

i{e€T } rdr"

st tliltx$ ff l0tlr{s ,,-TZ ftxlsls Is [IIn0IIc'tlvtffi$

sBll.&[n, etssljtT8 $t :it5Tfi$fl :"! "t $Pi?i0t $t $Etnil,B c?

,,,i! : 'jt t.

tn coneluxier 'l.i# e;.J.rjiii:r ,4 .!rii' J::i.tti' s$41ea,ee;?"i'/a-lcnfd su depeadenga 3
lff-lbert { circu-la{I },ir: -lii.*;:.,--Fil!{,',i .1 irf:,'.-a t"olaEr:narA ale
5l.{;.ri-l.ie :-eal6 gi de
transforaateJdlr.
{r
*, HETnilg $ffi IWdffie$"j'fffiH FHffiblff 5(

HTRAH$F#$${&Y& {};$:r9ffi4:ffi$H,T,4 de
ca
0aIGIHAeffiS-n3X] sr.

cduenteoNrsmceAinrneadf reetrroi-&.mnsf],a,*.::e:mm*4;ran+ls*uiba;i"r-"'"L,lL.l-:ii.,'r,r ,ti*ri" i.l:1 tI
dedomeniul
:x. Lr:i:"nd-n UI

.in!':i..:l;...J ir

*ofi 'f e}:qe'n i..,:i ;:.i. ir.--r::: ii T,. cie

4.1 i:, - ti

1. $xra*HffiAnw& r1&T{j}lg&,T.l[j*{ri: E*q"i.H

rf,sPrcTTfi

Hetoda se apll"eX in c;rs*.ri.i* i:r rc,&3:
transfonnata U slsii:e prec3-"::,n l"ii 'r:r:j.nl i:-+
serte eru un num&n f i.rr,!,t, rJ* l.*rrq:ni
astfel" de caeur""i" r..+;:,;erer:;:i:i " frr
t*,.1:;";
asiguratX pentnu j,qarrbiqi:le.az i.*it ;;i {1-1 it":,r-
f i; ,,r"iL;j-u,* r.; f l-r: ' 3 " c{)}'}ttlr rle integrare
de zero o var
interpretat E *A vrU iah:i l.,F.'i,", r:i.,:,irii ;:;.!",r'r . & -:

De exemp-!"11, e;*iqJ"naJ..":i- i,::s:t,,-,,{.,:.:;t;tr.3t..:i .F'tx } =";r+22*5+z^s est€ semna}u}
dj-scret f I rr ] *a11 s""; : +iS I r: + :t I " 11 I * i r I t{: i:l* 3 i .

?. $rmmgmnnE& flFtrffigx;+.l.i.ill,,r t$i;finru iltT.1!r,fiL-r-,*Re sElrr& fri cazurile
ln care *r-'annf*x-ri1,,9.t.+j.e*
sils:i"- *:rer:i.t*_iri f-ij:-j_n expresil alqebrlce
reprezent&.nd suluel* ge:r:-1.*:: defj$jt:^ei t,ransforaratei
;-r',1*;mi:r.rri.:.9:.,;are
Z, lrpeebuniterucugneocsqcluetn* tgel"r&iirir;i:nT,ra,rnJ;..ie-i.,eri*-ei,*_r{'if{*'iit}:vt.ei.lri"t'e;p:feX"ntr-* .rrangf or-mate z
avuCinanletrtelrffacLuPr-uaniboetlcleurln&.urlataIrlceluceuicc,eFnscrps{ceezternrex}et.:v.xcune*uji.Sn:ecef"&ei$:g.ee*le*sn*"li-rtriiit-eyi,*;8:rijr,ni';jl,:t:*,,r:cs!t!*rr:l;ife:.:rrt.r.;,r.i:!nf.:iieiL,rr,::ti+:riiie.rritqti.ijsil-ian*::.oe_1*.r.-r*:ti*'r:e*r!nuj"Ssp$ec,o*anetfvis,ile&terfgnueenicxnctil!eE,uireinoje,prsuodte1el

tn mntinu&re ne "lvffi .T'efer;. exs:3.$ixi"%* i"a $et"eral-earea inversei
transfqrsatei- ff n une"i- #e{xrer,rge "';alruza-t-* resgx*rtJ.v }a transf,ornata B
uniLateraLd, pemtrre setr6: dgsnj.rli. d€ remv*i'geng$ esta ar,ter:iorul
3l'* era-aqi-ie* #.*r$ 5,$€ir,!!aj*nr$ tot"t polii frutc&iei"
cerfitl"ui *u #es?gffirtr
r{s} "

;esJc] ?r"IR {I:u} llTddTs Tod un u?ued ,Tg

(sor) ""[ ir)r, (t s -zi;# i, .*,"-i" .:Jflgl H=,-"" tri ir1;L

:auTAap

'1-r ulpro Bp 'F6?eATrep € TsrsTEs ap zTuqTFi TnT ETn6o: prpsolo; ,erec

(srr) i#[r-oz ts]d* trz*ty, lffit-r-t,Ir4--r- ,, t,rlrI1,;&

es I<H a1q1ctr1dg3pu 6p uTp.xo pugAp ltto:delsew"rgdxasfTnrJnnFpz€TazTqnoTpc
"u<d gze: ap TnTrrcxac Tn:oTraluT uJ
ETJ€ es eJp* r_*u {strg ;e;srmnl rp T{luT?sTp:o111od Tnf,Eunu a?ss s epun
re,x'E{
(ttr) x-"8 \ElEzaE Ji*sBr_rc (r)df*f?u,= lrrjJ

Rz'€zrurnl e"r*r rar'x6a?*T soTT.xsnBu**r *Hllltorffifitffi Yi;H:l"ffi:

TTTtzsqtE)rcOeepJr6aT?d€gulnuu'{(TszI"))r(EaezTal)rxfdoJd€B(neT{T<Bmad-aru>or{sfrufmfig't{grn{j{?Joge}isdoTJyTT1e'{TTtus€Ers}aF-}nrEdTacsg'dTs{e#*"TT{}-3sTdFoaiJldss,ne*'*rTn*?Sapegdu;Szros=adl{Jp{uez$r-T)Fnr0a:dtoustT1rgaa1aJJlTTcoo1clasae?nTulersnSoz6Jene{0q[fIuTnpoauzrG3derDcaaxg[suc=pTterprdrgr8rrpoaecps]

(srr) ?= frj;$BFunf# {*re}JJil-t t!,?"
a

:auTAep eunTsJanuT ap .€Trlw.xs3 'g gelew.rogsue.r? p e{uebranuoc ap

TnTnTusrB TnioTJaX{rf uT gTS? es 63€?-T-un" Tm:as ecup '.re1ncT?rpd uI

u<d epn"m rnrnsrar J erprb?ruT 'iil1llil

$)rr 0Tg 'l
sr]{N&{tr.ssro3 "t

"sad=z

(etr) ...,eg,'T*,,^Cr,,*rJ zFr sE*.(_s,i_d.f$fu--zJf=iulJ

:flilup Tg {zi.{ <*> [ulJ e.ae; n"xluad [u]J Eorun gluaacas o
llstxa Tilrn?e 'U<lzl Trirrn€{lop nT €1sTntuJ Elsa (z}d eTtJ$nJ EJB(I

:exalduoe

XITqETmA ap rclgt$c{rnJi €TJOB? {rrF lBlTnzar Tn.Io?Earn ad pzEezpq

as EpffleH gffissff Frffi0* @.Fdr{run ${rfirrexrgruo crwcnfi8J,f,fi '€

r0 liltfltg il tsltta$ g€-Tfl lHilSlS IS ilt0t&m 'flvtffs

$€il[A{,5, {llif1]i9g 5! SJ$?f;fd &'i'ut--'36 $PAt{lt-r tfl $f;tlAl,t fa

,i'^:RE eirr,",ist r- dA"#}l (14?}
u 't r-I:m,F"{z} *" ^!!!:1, fl,{",} *
? i?t 4e lvzr.
?'l.ztt

&. furmxmamw,g snrtrtn&sJllxrr tmrl*tfrsup rAEBr,ffi Df; f;o*fispcwl]FffiTfi

Hetpda se i=.azepzh pe er,:n<:agteres unor Ferechi z r".e*.ar:'r.ahil".e r-rrn a;

{ r,48i

pl e<,rn.*e5" ir: uti"'l-izarea dezvoLtXr:i.j. ln frae!.i"l sinp).e a functlei F(z)
pen'Lru e Fxxlpe i=n evldentX astfel de Lerneni" Ar::est. Lucnu se real.izEazd
pr:in tJ*rv'o1-i*r:ea i.n fracSii. si-mple a func[i.el F\zllz g:i. inmuJ.tirea ap:i
a aw!:"l J-r:r r:r*mrb::i ei: var j"abfla z . Ilar:H gr aduJ- rlr.rm&r Stcrr:u]-u"i es'!:s mal.
man:s r,t*cr.rti. qladui numtrtcni-trui, prin impr&r"Sire se *hf,in put,er-i. al"e l.ui
z ,r*rsria *.l :l::rrsFLqn,i origi"nal-e de f,:rxa &[n'rnl

ffiomm.,r*, ileferltx:in6m nr'1.gj.nate: . trangf{rrslat*! F{:r:!

{ x4e}

Arree r.llrci$t{inl:sa clezvolt.a-:r:e 3'.a fractil si.mple: {L?:isq{lqCI}

f-j-;r'-, " i,*f-*L""ji'-itll--,-iil- -$r/,AT ?-+:r/r3{t-: 'F\'-';{z\--19Ag;*7

der;"t {15i1

f[n] =$ t-r"f "s [n] - *,n. b]e+ {n)

5. Dmxrwxmrr*a.b$ $ffirGxnArJIL{Jn mm&rzIFD nHRrus&KA na66 B'{"i r*ste

regu.t.atX 1* dsn*lrilu-tr lzi>R., erlstX o secvens$ unicF. fIni {nf$i
aetfel ip*€t f 11, 1 r"--> P{z}r

r. f nl =[i ;4l_ ^.dto"_; Fi'{r ;1,l rl"'; iThr'} {r52}
i li ::{0

Recunc.raptero ii'*,rmulsr.l-e cor*spunzlt.oane qJezvolt[r ii" in serj-e Tnvlr:r
in raport r:u rrarJ.ab:t.j"a z--\ .

.Exruetat ils:;rierlLarea:

.Fi7) =sj,n 1 *s 'r -:1J't" q 'r*{5-l ,z *'- " " . {19.3}
7

tpcTTdp TJ a?Ed fnap€3oJd 'pTpTlueuodxe Erm sJpr e:luTp aduences
?trop erXuTp TnTnsnpord TelErloJsuErl
eelu4elrdo:d ad alpzpq

(8sr ) [ {z-)ur* (z)d1 += @)n<->

:a111{e1a: alrcs 1od es arec n:1uad

([5r ] . 6'.Fr-r^ L u=r' 4 l+\l=u l.u1- l4]|= tu,,at'1'e

>{Z=u

'(arurfcsp) u 1n1n1uam6re eTE erBd TnlpuEes rFurJep
rofTJoTEA lrolgzundsaroc
[u]J TTuaere? Tpunu EzpesaJaluT sJpc u! Tnzec u5 Xfl111UeSeg

'Tepolaur

eleolezrmdsaroc gluamca: ap ETnwroJ pugluaza:de: a1{e1a.r Bury?Tn

(9€r) r-Brre trl lrrjJ, . . . , (tqop-Tp)= [T jJ l0e= [OjJ

3-un=

TRn€clTf,rTBoOT?oeTcqluoa1'pr1-Trzel1rnmedTsTncrTseplEnuTcdgJdXeuepscTTuTpgaeTuTJluTorslepTTs?ucpnTourJpsn€oaeIcsuTl''dnz1q:eooulrpeg'r6lz€E1uu:nToJuaorTJauTs{gguu{ea:pgTnrdJXceuaAgTs TegTTtcTgeeutqtacTutJurpanul

"g+Er-tg+ " '+r-nz-'Q+nz (z jo
ffi=GlA=tz)d{ssr }

'(z)d aTPuol{e: : z BIllUI:IOJsU€fl qJApTSUOC
ep PTn&roJ o ad
lEalg€zfcptqmel cly1tJquee6'1fTece1lugpeapnooec?zolacupao.IrTmapr€agTnznup?reTluz?TosoluTolun3Iaarcrlep?pJuaepazlcuaeJ(dmn/cIa]dJ
1elfcun; a1e rolradns uTp-ro ap rofaleATr€p tserpTneIec plTsacau Bpolau
E'er"lreeBd3opulTeplseep ed 'rolTrgru eT ap 1n1nuou11od JoTTuTcpppJ pe"rpuTujralap
nu
Toop (z),{ lalfcuny :o111od ee:alBeounc p?Tsaceu
niulrpuo fTin)[quupcAmBuoerxerreordeocTsrlorlltuupgdTratrpoflrauudearrrrerncrruldnug,rItE^focfUtlufinvgrtrIrIU{u{tSilUr0xAf'9,I

-fti=r"rr{tsr} ...,z,l,O=uI l+wZ=u iul
Wl=U

slTnzeu '0<lzl lBcug feJasP z s3TJo nr?ued e11qe1ea alse

rr nilrrt il r0HTrs LE-Ta t*tl8ls Is ilt00tIc 'flvllxs

$Ef;({t8. riq{{irTli $r $l$.nfiB 21.-=u sPl!!ft qE $frBtAI,B c"

'i,il Ll!.:rd t*Pei.4t PentrU a ob!1ne vaI or j- *l-e e*"-r,'eljte-t or, i"qlnal,*

g:,antrr.a n=Slt S "a,ra, d.,

5" RHNI;HAT

Spstiul de se.i&nsl,e Cz contine sesnaLeLe dlscre-?-e t:nidimensj.orra.l"e
rsBrsfleritate Bz:iii g$"ruri cle ntunere complexe" 3n rnterior:d" sgra.girl"l,ui
Cs se pot {:1un"egini evtrdentX o serie de suhspatj.i i'f i.l,t-rert remarcahi-l.e stlm
ar: tr= pentru care produ,errJ" sr.:a.l.er g*cer:eazd meLri"ca gi"
fi f-,
.nor.ma Sannnalel{-}r dln Cz 11 se pot alaga crii: t-l sau. g:r}j.l:raame tn
variabll-eie *u u sau *Jt (D=z--'). fi:arrsfor.uate.Le cor.esprrnzEtr:are (rio
z r:especttr"v Fou.rJ.er) sunt utile datorj"t.{ c$u espr:nq e!rts}].or r:![ntr:.e
operatoEii care afiloneazi in gidemeni.r:1" 1rt1-MF.l!! lei. r'li-ir r'loneni-u}-

tran.*format. c*r'espondenta d j,ntre con'r+l.u{ j-.e S i 3::multirea
a sau Fourler este ilanevrarea
tiri:re e i.t,e.l-e, f r.mdamental,6.
tr.a**f,or.-utafleri-,;rr d,
sec:tz€$&.elq,tr inf::{.nlte corespuncle in domeni.ul transf*::mat E:u ftatreqrrilrea
ser':-llcr. fn artusr:6-te cCInd$li funfi:use domeniului i.n sars pq]t ?na val-nrl
varj-ebi".LeLe $:.r{!r c, ssr:iile sunt- csnverge,nte gl se pr:;lt,e J-ucra cu
stunel* aegstora unor rfr+rr*.r:la de
rle gr:arJe f:Lutrt*" {cal'e e+respt"rnd,in -cenei:alu pv.r).i.noame

Arfilrnea operatr:r'11,:r ilniart p.e C* est,e Ce .qemenea LeqaLX de
con*eg.r'1u3. c3"e pr'l"rr.1"t,ls Lmternn eared in en*mj,t* cctnrti.f.,"i,i, p*et.e lua valori
lnflnif.e" s!.!iii prezentate o serle de pr:*prietEti ele tranEf*rmatelor
lntrcduse, m*tndel.e *le determinane a orl.glna3.ului preE:ufii Si"
trensfsrs*t*j-s rsror geenaLe remarcabi.le. Ee &seirnsnear esN:e d-lncutath
p':l{r5.:i".rlF.:ht:d*:rre:eFprce:iin.redctaert;n"leataar6{tHleajpl-c-bLaieclrarzttaElc.iis"rtecamtu,enlaacrleedlo)Ir:aedfiIpsericr'ritejXltr.-ei.nr'utrraetruoazmJae.)en"eJg-"u,il
transfr-r!:ri'rri,r,i:

arf,t&nrju*si:'
transforrnat
lnaglnare ale transfclrma.t"elor.

c

o ne ore e(TuTJRu ns rroTpwp RXrodsoc 'eTlEcTITures sp IeJlsE

es rup 'pcpzTl eTipoTJTureF
np nu eleJ3sTp roTeTpuss TnTueEop ug aT6reua Tg arslnd ap algrmllou
'm6Tsec '(dwff e1 grl{rpdru e;6:aue ne p1e6e slsa per6?nd) !1TuTJ
sTpea erelnd ap slarJsTp alpuulas alTunuep Ttr ?od .I uTp eTafEuuxE s
'fif? elId6InrTpletd+sffui TlnpreplmeTuerugmtase'dT:TallTuuTTlEaOpaTpa'TTafcp"pflpTueTreJupeTAeTaopreTunelDuenpd,l

lune
uT{ 'zT u; leg6raue nlfgqgap nc {Z} TaTleTeJ Ea:puput€,sB rlE^resqo

Tryffi(z) *>ri 1ulxl
:{iTleTsJ ee}sTlps eJsc

tr) r15 s3 [F]8 " [" " " [I]:rf' [o]x' [I*]x"' "l = [u]x
'sI Tunusp EoA TI
n"1ldepdps1u{ned?asuTel6glturuae6psTTA1s :ptero} ap TxolsaA luns
ersc ad lT€[€creute-: lJeqfTH
ug aund aleeid es uC a1eufr€s ap InTIBds uI

RJnl0ruls 'a1{1u1p0 I'l

ztr TIIIIUdS 'l

lr JVTT.}Ztr{
t,rt "apagzr|Tllr uT{) efe"n;trep.{lrirfJ,voc{lrsnne."yr1Su,6edT€aJtio'rcdnrro$J eTluTaauop

g;-wrp-renrl;r *Jelur
.ra1py$rrng a7e Ja7ilzad {g z afeteiro}sue$,
tt 3tl . :"04.ttp" g:pp,er1.:ad a-reoJgJn$Iryu! w eJeJJsTp eTeurrcg
OI €I uI aTeuotulto afPrrufiesr
I acypoy.r:ed prsuiles
, zI HI v.tlJIllt$IOJ IS vrlwssoJ
,It gf,n}Jndls 'aYtTurtea

T mll"vas

?I Tttrd,vds {fln8 }

z3 H'{Yi{p,['IIS flfi TltIJ,VdS

sI llYttl$ gt I8!lV{$ t'"Y"u ltEt$t$ is $ilftlil3 'lTvttt$

.{:, *:, -!:. 3$P,1T1lir SfilJ.diLg Ix

3." Pi.r'-'-lj1' .l '. (:i::.:: ?. '.+ i:tt:trr.eg (i jrr 1.t -*..:Tt, .?-i: ge, p:'+J.- l-emaale]:
c#.re r.{{ r-le r:epeCe F,e!}tru t}
r,:i-nc! la ?eso sufici-ent. *> + €o dar iftJ-cl.
::l: t.lnri ia J-nflni[; t1e exempii; seffinele}e constante, periodice,

s$ifaf&urrnfrjie!..!-ei1*aumi4te1rii-scrgaeg'erialrLm*n&alJ$-ei.et.te,.-'.rer.rlese*mida.l"ulL tx'*+pLd unlte.N.er d*p.!.asat

oper-api.i"l* inLer*fr ,sj" externX sLrnt, desj..guru cole i:,srespunz6t.oare

spat.i"u3-uf e', 9e$i este subspatiu al spatlului" C=, baea spegl,ul,ui I"
est,e efsea6i ru. cea a Fp).ftSir.:.l"uj" eu " *!:r$em&m c.H ellferenga esN:e -l-eqa'15

de resirlct:La ie:jlusi €:j&ei p;tra.tel-or yrcrJu]-ei,::: e]*neni..:Ler semnalelsr.

'f::ansf,o,trriatei"e <i Si z $e pot. cief "i"ni }a f el" c& ln Cs i;:s&

uajori'eatea seanai.e^ie de lr:teres, r,r.r au astfel, de tr"nnsfor-rnfrte,. R,emar'':5rrt

tobugtr c[ sqry.na.l-e]-e 'Lxea$]t6, ea 6i semnaLel-e periodice lnuJfrpJ.lra{.e cu
t{'eap*a tu;:-f.:ie fae parte di*q I" ,9j- air trarreforr}etc e {Si. d} ,*SR rl$n
am a:,:5t,*t in .:;ile*i.t+l"u:. a*te::ir+r"

?.. fRtrPJF:L ;?ilp*E*si f;H [= {sub}spaliri]. 1= este spati-u ltli-l:*ri Ln

r:eE*r."t *:;l,?-r:lrdl-lsl.1 geaLar deffu:r.l"t pr:l-n

#r', *;;.::.i:il - j"[.r...1 : r,*{-1T x* [::] -,'1'[37] {3}

-. , ". "..' -- lj' ,Y

c&re geniei:*afa gr:gsne;

i l-j-m)."r--'r!:L-i- A'
-'{ T'*L i,,1i."".
* I {4}
Ll-.':. I
,,i ', I 1)..' , t, :..!'l-l .,,.r'.., {.'l l"::rl l;r

c1 s*,!11 i.

olu i x [.i"r i,,:;'' 1 .'il ] "' $x I s] * l' I r:] $1'*'1,T-fd;:f -yT]:{*;x|ri i '.y llil-::[ *

il i;,',,rfifr i', - * t, ".' [lT i :'l$r':"" (-t"Tn i ;,'-l y7|}fr -.i,F;T ;.;- [
\.ltf-in"o--r*-'-*?ln-"rr"!l-,-;'.rn:.i"i;ljyh;lnr;];i1ff;rj,o-.i.-i-yTplI,lJ^{;I-'?-+l-tl*,r*tsT;i.,Tv-ol-"o"v-imt-r;-Tl;:T?"jfeicltx"[;.]'':'.=r'fitl]f,

RemarcEtr s,5 re.',it11-1ir,i.:1 di-ii defini.!ia spa{.j"r"i}..r.ii I2 ests lmpusd
(p.itratul-ulf ) n*r, iari. .

3" $mrnnng $-!]s ,h.F'I+Fs, spegS:{J$iilr I3 spn:raLele rars nu aparlin
It )spaf,1"ulu.:ir, sen.'. #ie;fiina.1"fi:!"e earr: tJ.n.i Leurt- l.a aeno c5.nrX !t . :i$ s6s
care tlnd 1s l-p,{J"rrJ"i.,l r1:l *:::e:np-lu HIt'l=,,1.-iqi1"?, xIrll=n eLr:, Fr*dteuJ
Fcalar ests,. mSfi iluro &fl xla-i. pree.!"z&t, o fiurrcf,i.onaSd (see#fiLi"'itnl"ar6
deflnllX pe 1". grrr;! r.rii,ri;agi:i- *fu,rd. ui-:i.l:i.:F"lld fo::et'ii-i.n prt.,dusuri-r.r.i- craLar
tnj"n trz $€ c.bt,l-;r ,':i,r).r:;i f j":tj;-l,r: ,:'h:i-*1.: C:r;:E Is:',,r1. ilJ-ntlr serry:e-l.r,'.::i-..?

' txl lfrilr; ''*i iei:r.r.d.u.!Eil:'fuI "rflbFb-gp i{ltsJ:?
ilif Et : .: lir[luEid .r

Xt:i r-_. i': i'l '' " .l 1.rrr,;ep' it$ i-6;1$€ r.j nf,
!;t:::r '
lil +;
.i,' .
;.:.-i- {ir-ilxJt/

i I i*i?!' I 16 ilp r.sT{p [s,{
i, I r ri,-irl;{ B}
ittotfJ,!t?l suFg"'g*TTAfiu"
' :1 r ;.1-r'f irtn&}.rt €* .n'gdruaXa
ii ; :1.::, $;*a g r"E,]{o*s{+i eTpgEi

-:. ;ill: i ili.!{iri!i €s€ [J $
" .."tqrii^ i;{guni Irtt

i :: . :i,ji{ ?E,Sfrt{.trf;} III]fg
I I r:::]lqi!:rb gfi{r*,} ? lETg;#

i. ., ; ;i [ {1. i .{1 ! EtlElLim €T[rAUl

.' I ..t r F: r'i i'!{t} q:}TnZa.r

.i, ; I !,,;1jrir.ift:i{,!;f! fl ts:$SA tS
, , 'n: i'vlrl .r19"15,t5., "t

' 'r 1!,1 ttirJa{* h,

fs

...it,ri:;r11r;; ,-

: :'!rj:iij:J,,i;

; rl ,.r

. i.';

, ,. . 11 :1.11-tr UU r. * !
,; . :;$ 4rli -rl Tf{Bi? ITS
-: :v, !3jTlr:rJ.l:r?rj

16<u n::}tr*d
ii ) 1r"i":{'1i1s'-1[i"\j '-

' {t;::U !t"X'}ttad
r 1:lir',{:;r;J1g
f-f rrdi:t{ri$ifn$ *'
i3 .9!-q:$.;d .i {"r.€€lsg{ts " &

[J

'u:lLiJ i
, i r*}+: iLil ti i{i;{tr1-t:}n

r,:i lr: i!illil;l.i;) 'tiVilf,flii

$EfiiL[, CtrtCUIfE 5l SI$tBiE fl f-i $PAtIUt t[ $[tti[[ l,

d,A-c

PropntetEti:

- Dacd a{x[n]] exlsttr conforn definitiei (8), atuncl exlst6 sl.
ccnf,ors deflnltiel (3) Sl cele dou6 valori coincid"
- Dacd A(x[n]) existd conforn atunci existX gI
definitiel (B),
e{x[n+kJ ), orlcare ar fl k, gl cele douE valori coincid (denronstratl] .
Prin ur*sare, daeb ne referin nunal Ia eennale pentru care

A(xlnl ) *A(xln+kl ) VkeZ {8}

formulel.e (3) St (8) dau acela$l rezultat gi, evident, est€ pref,erablli

forauLa (3).

6. &aGfrvaflE tn cele ce urneazEo von presupune cd eete indeplinit$

condltla (9). Astfet, definlrea nediel se va facE cu fornula (8),

care aeiqrorH condlila {9), lar calsulele cu forEula (3}.

7. Horn cA smnax. {o* corrrn6} tn cazu1 sennalelor xlnl
pentru eare media este lndependentE de orlginea tirnpului, se poate
consldera cH semral.ul este suna dintre un semnal constant (componenta
sontlnuf, a lui x[n]), n-[nl=ar*, €gal eu medla sennatrului x[n] si un alt
a@nal, xo[n], av&nd media nulf,, care retine celelatte proprietd]i ale c
lui xlnl r a
x[n.l=rn"ln] +xr 1a] (9] c

Desigur, ee@naLele constante fac parte din spatiul 1,". observdm ci, in z.
cazul ser&nalel"or coatplexe, media poate fi, de asemenea/ un semnal-
cmplex, Hodulul mediei unui sennal se va nota cu F*, iar valoarea

mediei }a p6trat este 'rputerea conponentei contlnue".

2, [uRfimTxe $i c0t{l'0[UTIA Iil I'z

Consideratiile care uraeaz6 se refer6 la anunist operatori
particularl definli.i pe 1'. Se pot stabill, prin analogi.e cu cele
expuse in capltolele anterloare, forma diverselor matrlce lnflnit
di,nensionale care sorespund diferltilor operatori (depJ.asare,
lnversaro etc. ).

Prezent6n principial, prin relatla (10), j,maElnea actiunli unui
operator asupra unul semnal dln 12, cu precizarea cX este necesar ca
toate produseLe scalare lmpllcate sd ex:.sfe: este posibll ca rezultatul"
acij.unil operatori)-on sX nu fie definlt.

Pe de a1t[ Erarte, este posibll ca seronalul rezultat sE facX parte
ehiar din 1"1

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