Goras, Liviu - Semnale circuite si sisteme

:oTJaluT €urJsJ q!1s 3Hrot3uxsToad?paordTlaosra(prp)ut{!f,JnEuTnrprdout'trcnlT.(npu)rofuutirlTeodlalTnnplnopueJn6c

per6 ap (pld

1:ode: uT (Blu rnT {aTrJnnpTzal) aT{-:n1saf, ?uns (p)l,r a1aureou11od

(E*) *'[-,,(p).ur]

,='o'*"" [1*,rr)

lg3ur

IaJAsE ailaTe ,ia;ez'FTBtrrJou* ap auleouTloO luns 1p1!1 aTsueouTTod opun

\ (F)n Dowl
(rr) {'"!ltl:tr*L |LDlr{ pee/ r- [+! \
j= I'c ,L,tv.,,,,',t",\fifii,i-,i,.i-)l= rit'+i )r'tn's
t fptr+ -Z;

a?sadsar as ps xgcu! TaJ?sE prrurre?ap as (p)ts "r"Jff;ji*Ter

(Ef ) f gf': t{ t{Dsl'f(*.r m""u*}' { 1p , re;
: {rrT g=r"((P)tar) *( ,o) r")

apun

w'ttl pw/ 1=a\
(zr) pefi r (;" I {F) %1 .pez6> 1p1rz rz rs
\| (p) (p) r/ l- trld

pe;6 ap (p)u pnrJoJ uI ar:cs aleod as (p)n TnI Inpe-r6 X€oap cilu

1uu uourlod saTJo qa gruTJp Jsezaarqs TnTnSsaJ ewajoaJ

"{p)i{ TnT aT€ efnp()l! asailrnu as 1,...,I:T ,Tr[(p)lu1 a1aureou11o6
{p}q Tg {p}p {{a'[*'TiTunancTu:r}oanu.pr?ouuarTdo,d(I=pn(po)ptrtEr(?ps)TqX+a(pE) rluF(upu)reEatsgucJuellsfaeJaXostE)

{T}} {>,f >f >r A ,T= ( (p) tut' (p,)rw)
tp:3lelt1udaoplaenpedouurlylocdu,I:eTltnupaoTucrluaoou3llondo .roTaureouTTod p1eu1
ug au'r"rd pu"tTJ Trrnu uroA 13 a:ec ed

{0} } ," [ (p) ru] t{I= pus

tfi [lff]ri$ [{ lfiuYd$ 0z-97 [tlr$t$ Is [IIfltilt'[1tiflts

$BfllrB, cttcurll sl $t$lBt{B l-6-zr $PlTIUt ltr Srnnl xI slll l

"ffirto -(f t, to r,td))*"' ', @) r,to'),"" o,,'n'' 3.:

",o,=i* unc

AnaloE teorenel referltoare la nunere, sunt adevlrate relatille: s(i
st(d) st (d) =s,
s, (d) s* ( d) =0"* (d) ,,,66 r1(-ia *;ki ; (S7) de

i '"101 , pol

fJ,'t s, (d) =1 prl

An utlllzat notatli asenlnXtoare cu cele d1n paragraful referltor car
la teorema restului chlnezesc pentru numare. 1.
po)
Dercnstratia teoremei restului chinezesc pentru polinoame este
anaLog6 cu cea referltoare la nunere. Da(

1. 0rsnnvATrf ln cazul in incaerxepr"ejsiasutenotremmaeii marl ca unltatea, att
restului
rezlduurile 11 (d) care apar chinezesc Pr:
pot fl exprin'ate eub forna: po.

cJ-l ( {8} fl(
4- J t\.qlt/ -Sr/zro t-Jz /\rul\/ fLmrr.J 1\q.1, \1I2
po:
expresia teorenel restulul chinezese devenlnd:

lf 1l-!.\d).-.lTo,or
tr(d) ',,(d)Lm j \d) I -r \ r.o rn, (al r {4ei

[J.t I J mod !d(d)

tn aceast{ expresle reztduurlle s-au exprimat prln dezvolt4rlle
rsodulo Irq (d) lz: polinoamele r.,, (d) reprezlntl "coeflcientii"

reprezentdiilor n,, (d)-are ale reiiduurllor. Aceate polinoane se
oblin prln impdrtlrl succeslve ale rezlduurj.lor la n1 (d). Aven:

ct-L

rJ (d) =E rt,(d) tny (d) I'= (50)

t'O
=r1s(d) +rtt(d)mr(d) +rte(d) [rnr(d) J'+...nt7t",.1 (d) fmr(i) ]",-t

unde rin(d) egte restul lmplrtirli lul r.,(d) Ia m1(d), r.,,(d),
restul'lnplrtirii lui (r., (d)-r1o(d) )/m1(d) la ni(d), r'.,,IOl t'estul
inpdrt.iril tui [ (rj ta)-rjo(d) )/n1 (a)-rl, (d) ]/n11d) la m, (d erc.

'TEUTTU rU TnTnUourTOd

EuTTirTduTTJTarTp{ragpdupJupEausHnauuTItlsaasrPT3gpepTBe?rsETnp€aupr,r(pa)e?OsafSET(TpE)J,'6€uaoluaauqeo. u(lploldti

poep Teumu 1S gcep TTpba ?uns (E)26 Tg (q)lg tT:otpJado erpurrn uTJd

(es) (Vlz6= (v) 16

(p){s 1nuoul1od oTnpou a1e6a lrms (p)e;"11"r(";fftffi.A;rTrJ"ill;
'tF inTlp.ds TUCuT(lpa)p'6alesTa]eTrpScEalTsnaTdn1.rnodlprrtyacdou3f€1{uTaelucTTuleToutD1nnucouo1u1eoodutrlod
Erlop (p),b

(p)t* als uoJ,vuxdo Nn xq {rxvorrroa} rrtJilru E&oc v uxJ,u.rncafl 'r

1n1nruou11od Tnpprb leeap cru Tprtr p€r' ap rrur;;Jt"ff:1il#;

ecTJleu o ap lrtcrml Taun rrrgrp:dxa EalplTTTqTsod TaJlse ueAV

(ss ) { [Y] ):= ( tYl )J
:e1nc11,red u1 'q 1ac1:1eu T€ eJ€Tnr,rc ap uoullod'cTeXlssaTr(apl)Jb€:pelcBp1n,uaopuu1n1oadp

(?s ) '(p)6 trJ,r+\,P-\ )pF\tr-rr)uA=\p)J un a?sa {p}r xsT,(p)o

E?InzaU peJ6>(p)J per6 nc lsal uouTTod
1n1nuoutr1od alrJnof,az u! p?pTn6ar gcTXTTeue a1{cung o a?sa (p)b apun

{sE} {lPpl)6: . r*D')D= (lPp)!J1

raTros a1€od aS 'uou11od

un (p)b Tg areraJ€o alJcun; o {T-n'..,,0} <- {T-n,...,0} :tr aTd

J0lpJAdo Un Ap rrlJunJ ['[

tz5 ! O_c r=1

(IE} (p)"Fr (p)"rs 3 3= tOlo
r-{3 f
rf IlVItiN$ XI llltIfdS
:BI:3S Ualnd

f,-fr"i(t t(pr))Frlw=i lP. )'ts

lurgxou ErPq

,r-97 lll[trsts Is [IIncil0 '[1YtrH[$

tltil[8, cIlcuIIB sI st$tBfr[ l-6-r= $?tTtur, t[ Sili ; Silir

z. F,*rtr DE ur opRETon sr rronmA RpnrnrfiIr cmnrrzgsc conforn senr
coml
teormei restului ehlnezesc, orlce pollnour de grad lnferlor
gradulul unui pollnon M{d) flxat se scrle Ln forma (46). tn cazul in est(
care s€ doregte exprinarea unul operator de forna p(A) utlllztnd 6.
teorena restulul. chinezesc, se pot folosl proprletXtlle pollnoanelor rep(
sr(d) (47) {se tine seana de faptul ci produsele er(d)sr(d) sunt trar
Per.
Erl'r=A Errr=8, ,* tr=t {57} valr

pollnoane de anulare). Dac{ notin sr(A)=Erobglnem, pe baza undr
proprletitllor anlntlte, relagllle : eu
Oparatoril Er sunt operatori. de proJ"ectle pe subspatllle generate
de dlvlzorll prlnclpall al pollnonulul caracteristlc al operatorulul sat
t, Daci se utlllzeazi o descmprnene fln6, pe relatj.el
pot deflni operatorli: mal baza {49}" atur
se
tnt:
Er;s11(r)
adr
care satlsfac relagJ.lle:
lp(
8s&.*-g Y j*k, OsJ<co*1, 0<mscr*l
mul
3. Prnroana unrr poIJDot se poate denonstra ci orlce potj-nom p(d)
nedivlziblL prln d cu coeficlentl lntr-un cimp flnlt G?{il) este tra.

factor al unul polinom de forma dr-1 pentru un anunit i. Cea mal rilcA s€m
valoare a lul T se nune€rLe perioada 1ui- p(d). Se demonstreazf, ci
perioada unul polinom de Erad n ireductibll, diferit de d, cu Exr
coeficlenli in cF{M} trebuie sd dividX numiruL }t^-1. perloada
pollnomului [p(d)]'este MaT unde i este cel uai riric intreE astfel
incit Ma>c lar T este perloada lut p(d)

"

c. PrnrCIenA unrr OPERATOR Pri* definltl.e, perloada u:iuj. operator

este valoarea mlnim6 a num6rului f pentm care este adev6ratl
relatia Ar=f"

5. fummn: mnporEayr sunt operatorli av&nd p:l,inonruJ.

caracteristlc de forma dk. Forna canonlcf, a unei matrlce nu lpotente

este:

paa<'l-rd?r*€_sA"x'rT-aEro.q6'-l(Hec[-eTtppppeg]*eu)lIsuogd)lTqle.]rTxsalpaugrdq.Fc9IHnSrsuBdP!r(ual*-sTTtp-BTa[l-n{xpIJpd$l)sa}PTI'd?EaPg]{c6Io=Iuauutla{r5aT1fsTTrTud3q('agfuTI=J-unzH?crT1TtsT)rPlA€n,ll<Ttd'Hb(.pTB-caT"Ep)JaErdlt.ellrluaTHTTlalenn=aTe[zTTIor(Jauq.To-rarTrnJ>zlr'pss*eT.€ru[uAaHJ(ErTp6-'pJT"&)sl,ad.p?.cl")5l{ugaa4gsTs?lPArOqp$JpuaaFnpJ'3'aTIoPP[d[G}p?PT(suspcluTliTnl]}padpcmsl:€ill

('e) " i F )dl (p i6=r-, i F )dl (p )r (r p- r ) /JFAape-lXuI
:aTf3s
a?Bd ?s zEO lse3E UT

''(rJp,lxa?aspaTA€TTpnnJuole"J*ard-o-[(npa)dlJ]odreepr uT [u]x TnTnlBuuras ep€orJad TJun?E
(p)x ae
apT^Tp *--[{B}d] Eceg

(se) (rsP-r)lrttP)dl

eJec wruTlu t;, alsa aJtc T aJPCaTJ nrlUOd a1ls €Td€TaJ eceJsT?Es

as 'a1u1nne a?r€ n3
uTp Plncsourlc ((p)d 1n1 lnpe'rb alse'oq
'Tf,Fare3uT FusrxadTnnqBaxs?,aTtIa)cIB'Il gpeollad o aJE r[ (p)d] uouTlod a:eaa1g apun

T.rH gpTATp

(ee) , I iF) dj iilE= ip)x trp-T) (=i totYdPou6= F)x (rp-Ti

'1ua1eR;qca'nes

(re) 0o)Yd Pc!.u (p )X= (p ))fr p

:e1!e1a: ao€Jsrfes aJpJ J P$r.ruTu eeJeoTPA
ua'=1lst e'(ppluvl"dTlrepA?oTuTeqnucplaoauTala?rpsTpnJn[ou3l]exJ(apled)uoxulnaTscnlT1n-uB:onadre€e:sd:TturIsd1eTl[vTun]TuxnITTnpncTTnJ'IpEouTugIlaaespuuEPoplBEJXosaTudJeaardJl
ssunTlrv '=t(pldl=(p)od eu:o; ap cT?sTJa1"€rPO 1nuou11od pugAp v

:o1e:ado un aT"{ .uofult$Ido Hn nf, &uod\ru lg TtN}uls rnNn Eauor&Id '9

'Tnu TnTPuuas alsa

Tnle?Tnzal '1:eseldap { urnuryxPu ednp 'Te]lsv 'e?d€aJp uTp JoToxuauoduqc
,,eaJapJa1d,, rr3 a3BJ es paJps€Tdap PcTp€ o=4p ornpou rnT€uuas
leuuas rnun prdnse rnTuo?€Jado €aunrlc?
eldearp eT pzeas€Tdap

IO 0 ol

Io U :l

Il0. ::
!l
(0e) lo 0" OT 0l

lllovn U 00 rl &.

0" ol I 'ir J{

s[{ 't0 nYds o.-97 il{ilsts Is xllillII3 ',fl1Y{fl[s

$lilllB, cncutn $I $I$Ililt 1-6-zs $ttTItlL lt srmnt rI $BIIT A !

3.{ Polinor priritiv 1. I
trotlunea de pollnon prfurltlv-este lagatl de poslblHtatea genertrril
tuturor elenentelor cAnpului Ht (nodulo un pollnon ireductibll) de 2. I
c{tre un slng^lr elenent.
un pollnon treductlbll de grad N este prtutltiv.dacl gi nunal dac6 3.(
\.\'
dLvtde pollnoqul dr-l pentru n cel pulln egal cu td-t (nu dlvlde d!-1
p€ntru n.d-r). 1
c[ pentru orice l{ gl orice H existe un polinom
8e d€nonstreaza I
ireductlbll de grad n.
da fleclrul element xlnl din td satlsface relatia {.(
Transforilata gi se poate exprina ca putere a unui elementl0
xldll=x161 unde K=ld )
corespunzf,tor unui pollnonr prlnitiv. Aceasta revina Ia faptul c6 toate
sdnalele dln xx pot fl generate prln convolu$il rePetate (modulo t
pollnonul nodular lreductlbll) de ctrtre elementul primitlv. circu

l. EXnpru pentru t{=2, pollnmul dl+d+t este ireductlbil. Rezulttr ct )
nulfi-nea N[=21 poate fl structurate drePt cfurp modulo pollnomu1
{
lreductibll de nal sus. Pollnonul d <-> 6[n-1] este prlnltlv..Prln F

urfiar€, d rldlcat Ia puterlle 0...14 nodulo pollnomul d*C*t seaaa
genereazI toate elerentele nenule ale cAmpului. Acest€a au
e
transformatele d umltoare:
rezul
1, d, d2, d3, dl=d+l, d5=d2+d, d6=d3+d2, d?=1+d+d3, d8=1+d2, d9=d+d3,
sesna
dI0=1+d+d2, dl1=d+d2+d3, dl2=1+d+d2+d3, dl3=1+d2+63, dll=1+d3, autoc

orlglnalele acestor elenente se determlnl imadiat: sLmbo

6[n],6[n-1], 6[n-2], 6[n-31, 6[n]+6[n-1], 6[n-1]+6[n-2]...etc. opsra

{. PR0PnIETATILE TRilt$F0ffiATnI d

Consj-der5n o pereche x[n] <-> x{d) Fi polinonul nodular p(d)
preclzat, corespunztrtor sennalului nodular a[n] (in partlcular, pentru

a[n]= 61n1, polinomul modular este l(d)=d[-11

10 s"nnulul',*' -dfjiu::t-Hs-:rimitlv

d*-ar
este ao6[n]+a,6[n-1 ]+. . .+.r-i6tn-N+1 I .

'Eluapacerd Tosgns ap B?o1., TzaA ,t

Rreri)Esuor EaunTtoesqns ur nTT€lap u! prefe.rl a". ;"ltt"r1predo
TStsp H oTnpou cEJ es ala'aslTnnpuod:TdqoTgaTaeTca?uurnJss palcpzrTrTtluTl.nl[BeafTgrnTl;oqE[s

:fpuuas 16e1ace pzeazTumJ (greTmrTc .re1ncl1red ur) erlnToAuoeo?np
"*:ffi;l"l"l:rH:iff ffii:.l*"_arPo1:Jrsrp1lon:{u6aufl[sHu]uT]i1na:TdTlEe1udu6sraluscreTrSpr3pRTupnru[oluuaT?uusrraEaa.rTianTNoA.pu1oeclurnourpaceEBl[Eul]adT:,d[ou:]dx
aTpuuas
g11nze.r

puguTt Ts (plr Ts (p)x 'p aleu:o;suEf,? pnop rolac pdhpo:a'pugcag
(69 ). +I{T'I-'+)rf,-[-r1ru-]_illt]0rg-g"r.{{<.l{-t9tItr-l.0r/-'Ill{t/rEl.,-f]nI.ir-l_lxlu-l{+rjJx.-.x>.w+).+.lf0"x,.,=++{tt.rTt'-.._a*ffn-i[u']brt]]ri/r..f6;tIi1r{x]]tx+x1+xJt-00i:=lIr-iI_,rfrili4u/11t.pn:fi.*trfd.[i0ujxt1xtx-]
r -rf T-fi

lelfnlonuoc fnzE3 nrluad e1{e:lsuouep : ( [u]9=[u]e] arBTnslTs

urpluezaJd aJenuT?uoc uT

{s9} *'uP'{p)x {P)x <-; tnlap*[:f-u]rf t)rlx3=[uJPpou [u]^{*lulx

,[i ur.fnrortnq] .t

{re}<-) rr+1,r-uixr-r6. .:':;:i;';'.",";Ju- fi*," po* rp-u3xro
1-rI

{*ysu"r'u nomroa} rultnss rn CI]rxror ruxssrdso so flurnilrr rrrumoroB .e

tee) trld pour1p)X{p <_> tlrlu po.[{_u]x

u fiLr. E Srw$fldr0 'z

t{ee} tpl txt,

f[tfi 11{IHIC 10ruVd$ ,r-gT ti k?J,uffwrmT 'r

lHlIst$ ls flII0cilJ'tTvtffis

8ltilltr, cIt0ulTt st $tsT[r{E 16-nt SnIIUI ][ SH{iltl iln

+ (x(xx[0(([xd0]t)].oy",l[yvMt.{(yyLd-tJillOoJ-+ll+{x*,xxtrx1l+[O1ltyi]i.i[y,.Ly"l[+yiolMlx.M]d-t+21-+1.Jl.".1.+."*.d+..+'+.x.y..[[./r+.vr*xf'fx--+{1lt.]xM]a{.6--Myu!r1l-]*1,',L.2)yyl)=.[)l01.d]udl -+))l)..*d.. t, -'. (7ol 9
t'i
x[nJ*x[n]-x[n], tn domeniul transformatei d aceste proprletEti se scrlu
X(d)Y(d)=0 respectlv x(d)x{d}=X(d) {modulo un pollnon nodular preclzat, u

in particular An-r). K

5. Itf'ongfTfe este eonvoluSla cu secventa inversati. Transfornata d di

a psreondnuasluulludl cinorterseputrnaznAstsfor rcnoraretaladtleai audnouu5iasedminnatrleexa[nc]esCtei ay[ns]i tI

est€ st
transformata d a ceLeilalte secvenge inversate (obtinutS prin
transforfiarea ylkj -> ytN*kl i. {,

6. OnsrnvE$n *eanlntirn cX at&t convol-u$ia cit, gi corelalia nu pot de

fl definlte ffu6 precizarea polinomulul nodular" {A nu preciza
pollnomul" modular este echivaLent cu a defini suna gl produsul
dintre nunere nodulo M fertr a-l preciza pe l.l. ) Hal observdm cd,
esenla acestel situatii deriv& din necesitatea de a defini dt,
dll+l... atc"

.i TmnsronxnrA sg{rAurrrn }n LTrpLrcAT cu A} ( 711

a"x[nl <*> X(ad)

8. TnNsroru{ATA sucfl suna de i.a 0 1a n din xlkl este convolu&ia

dlntre xtnl gl sennalul dceomnosntasntrteaoz[6n]in" Crun transfnrmata treptei este
{4.1.-) se obflne:
(1-dlr)/(l-ci) aga cufi se

n xttJ = xlol6 [n] + (x[0] +x[1] ) 0 [n-l] +

f, (721

#'=c .{Hrt&l }s [n-]r+11 (*> n(d) t
t'0

1{ DesiEuE, mocluto }.1 .

tsr.) P-L *rP FOe' r " ' '+rp+p+1 <*> h'-ul o- [p'] o

Sf 5*r-aP+

;n>{ '[]t-ulo-[uJo TnTnTeulLs " p sl€l[Jo]srrEJl -
p*r&Ll
r-F:-f =rp 91 r-wfn' " +ap+p+I <- ) lu] e
;5-=

: Il+ll-u]F]" "'+[T-uJg+[il]gElulo TnTnTpuuas e p Bleuuo]supJl -

(9L) -ffih=ro-u 3r-=irq-,*pl-'se+''' +rp"E+pp+J (-) up

l u€ TnTnr'uriles p p g?p&rolsu'"rr *

t$rl T 4*> iul0

:1 na gle6a ,1e1{luggep

uroluoo 'Xuapz*a 'ts1se IrrlE TnTnT€uuas p p elsLroJsue"r?

dl uTp aTIqpsJEusJ trIpUHa$ Joun ap p eluErr0lsusJ,[ I"f

'aT€uuas
Fnop JoTac roTe1puJoJsue":1 ea"rt{1nuuT pT auT/\al p?sEaJ€ p p?pleroJsuErl

uS ':o1e:aua6 lnleumos 1* {sot Feu TEuTJep) [u]19 ppurms aJ?uTp

PxEfnorTO €TlnToAuoe ug"rd p11nza"r ,,ttpo1-rad* Teuu€s un p3 u4?JJpurau

,#+ 'i.
{p)dx*yP @)ur=

tlr) -,I ril

l- (P)dx(- rP+"""+rap+xp+T* {p)x {-> WT-uJox - [uix

L- !f- "L

- .Hlx

re1 '1u1x ,,l1po;:ed,, rnTn{€uq$as T€ ,,Jo?eJaua$ 1n1euras' alsa lulDx eprm

(g1) W-ulx= ixp-u)"* 3 = [Ir]x
r-f,

:er{e1a: spJsT+ps eJ€} afpmrss afacp puTTJ m ,apTpoT.rad,!

eTaTtuues TuTJap uoA sA0rGOruIIds Ecnrftfltsts H.f,ut{uotgxt{ufi -6

II lltlfls ti litllf{$ *t-97 ilrul$rs Is flllllctIt 'llvtlts

$Bilt[8, (jITCUITB $t $t$TBltB L6-rs $tmr[r m scrtttt xx $iltr

- transfortrata d a sennalulul a0(o[n]-o[n-k]): c[_:

a'(o [n] -o [n-k] ) < -> fH; ,rat=f*l-ta{d' ( zel sel
(eo)
- transfornata d a semnalulul (KiN) gau,

r- ^"perlodlc" tne
0r(o)=f0tn-rr<1
cuE
este
tret
{-r _{_r
6*(n)=p 0tn-rrcl 5,
(-) l-+dr+ dzr+. .. +dit-r.- H| arr= f -9"{-" tAf i
repr
a-dr
nult
{,2 Aplicatie pentnr cazul }l=2 in teoria codHrii
o apllcat,i-e renarcablLtr a proprletdtii de anulare a produsului slc
intern a doul secvente se referd la tehnlcile de codare gi decodare in
tranenlsllle da date. se pune problema transmiterli infornatiel sub defl
forlra unor eucceslunl de sennale dlscrete, de cele mal multe ori lnte
prln
blnare {"cuvinte") aglnilate cu elemente din spallul H[. Esentra acestor
tehnlcl se bazeazd pe ideea de a transmite mai nulte efunbolurl dec6t sPat
cele strlct necesare transnlterii informaglei dln fiecare cuvdnt, fie func
ele k. Acest lucru se reallzeazi prin "lungirea, cuvintelor transnise
cArora 1i se atageazd ln fat6 1nc6 m slsrboluri de control,
de cele i gdeipcenhdiaenr taa
slnbolurl de lnfonrafie, cu scopul de a detecta
corecta erorlle ap6rute datoritd perturbaiillor de pe canalul de
transmlsi-une. Deslgur cd nunlrul de erori ce pot fl detectate respectlv
corectate este in relatle dlrect[ cu nunlrul de slnboluri suplimentare,
n (evident, cu c&t m este mal nare, cu at&t ne agteptdnr sd puteu
detecta gi corecta nu ne propunen sd lntr6m in
nai nulte erorl -
aqenunte). vom sugera in cele ce urmeazd nunai prlncipiul care std la
baza determln6rii simbolurllor de control dln cele de tnformatle. Vom
presupune cd cele m si-nbolurite de control s{nt puse }a incegutul
cuv&ntului transmls,lar cele k slnbolurl de lnforantie in contlnuare/
astfel incit m+k:M. Dacf, not6n cuvAntul transnis cu v[n], constat5u cd
V(d)=c(d)+I(d) unde clnl reprezlnt6 cuvdntul de cod av6nd ultimele k
sLoboluri nule, lar i[n] cuvintul de inforaratie avAnd prlmele m
sinbolurl nule.
.CuvAntuL transnls este:

'p elBlf,roJ:8upJl pr[gluezerdeJ elsurEouTTd ap eTlcun]
reETTTtOnTlEeTJJEdeosPrdFaor leesoc?8eczTelucTEqcrdTn'lTlxnlfs,eJuTppurEoloea?ppeulgBzrn?lEcdul
',IH TnTnTlPdB

ecTrlEr uTId
TTTqe?uazerder groleredo rollre;Tp elTunTtcp xelTtm pou ul e1e:dre1u1
1od es ETnJla ezeq ed ureluT snpoxd rm eonpoJluT axpod es q?TuTJep
TBJ1SE eTeuues ep {plTrrIl) EauttrTnu uT po pXde3 ?uelrodut alEg
'IPTJolreA nTtrpds{ ec urpngrd o gs
eIls?luesegrcesloeT{rpe'J{rfeo-r1nde"Te'€gu}uJarnfsradpr;€r{lIurn;rfru1uneTupl8pTelext{louFIec"zEpepurdgdraueagcelsTldaJn-e1Noe1ulg1lerddpeec€aTolsTpeqaeJslueaazuraIprdfpsxu

,l,vHflzffi 'g

gp oTTrnToqqs €uTurelep 1od ae 'a1e3Ble eTnqsJX
ll3lsTs m glTnzer 'ereclg11uepl
gsJ{pmc cTeorrluo€rp erpc uTp elncsouncou uI nJ
u1:d 'TTlEnlTs elaque
uI

(tB) (P)r+ (P)r= (p)r* (P)J= (P)/

'npg

{€8) @'to'(P)t- (p)l+ 11o)J= (P)/

:11{1puoc elerpolgurn erluTp eun pugundul au1{qo 1od as
forluos sp TrnToctuts N eIEc 'r1=(9)perb rg r=(p)ppx6 spun (pig'(p)H=I-'P

€[roJ ap T-lp TnTnroullod e eJczTro?JEI o eldope a?eod ee ?cec

tzsl lul F+ [u].?= [u].tt

tr rriltls rr lonv{s 0r-9r IIIISt$ IS rutflcilc'tltIt[$

slilalt, cncut?n $l $l$TDltB L7 -' sPtTIilt ltI sBilatIE ilz $3!

SPATIUT DE SEMNALE MZ PI

S:PATITJL W I or

Mtin79t7, structwi L ge

FoPmATORr tN SPArnm 5 ti

@eratorl wrttcalarl 6 tv?
Traasfamata d in spatiul lF: proprietEt,T cal
Transforwte d aLa unor sef,,nal-e raaarcabl-7e din tf "10
11 in
8ryJu de datetz.lnare a orlginalulul unel transfatwte d
(tt3) 13 at€
e1
STSI7&IE DT$CRETE LTI'IARE INVA$ANTE IE TIT{P 11 nei
,9lsfefi6 discra'TteONpTIo,AdRuElara Tinlara, neliniare, varianta gi t
invarianta in tW
14

Detatzl"narea rdspuunsuiut sLstarleTor wdul.are invarlants 17

Rispurnsul 7a axcLtaf,Te nuLd 2A

Ststeee wdul-are Tlnlare generale 21
adapunaui in stare nuld
ad-Iizarea trutsformtel d 28

fn stare nuld a wntru detemlnarea rdapunsului

sistxaTor nodn-lare llniare

l"nvarLaate 29

REZtt&T 37

ln acest capttol lntroducm spaglul de smrale lfz ale cilrui

eLsente eunt sffiBalele discrete avdnd dreniul nult,fuea Z a nunerelor
intregi 91 codcmenlul, nultilea g={0,11...,H-1}, operatia de qrup fl1nd
cura rodulo H a conponentelor de acelagi rang ale eemralelor. O parte
dintre proprietS?lle spat,lului Nz sunt asenEn6toare cu eele ale

epaliulul !fl: spatiul l,lt este Unita spa$lului lF cind X tinde la
lnfinlt. Spati"ui" lfl constituie cadml dezvolt5rll teoriel slstoelor

dlpcreta mdulare.

1. $PAIIm il'
1.1 llefinitii, structuri

O generalizare poslbild a spatlului l.{r o ctnstltule elasa

s€Dnalelor dlscrete crantizate pentru care dmEniul este Z, nnrlti-nea

nrnerelor intragl, lar codmenlul nuttlnea flniti de nunere {0n1,...,1,1-

t"). Elenentele epatiulul xr eunt vectori de forna

x[nl=[. . .x[-1],x[o],x[1], . . .J* Osrlnlct{, Vn€Z

o reprezentare i.ntultiv[ a elmentelor acegtul spatlu corespun&

BToTeo fl&SdryF s$it gB "ro'f].3'€.TF.:r serTlTffr] str{i "rxffidj 't {n)m
6Tg ps lt
OreeeeooTT3a'saet.lxlTcutgPset1frotdfrea)u€ssffST'i6BfrfpTFpT'$ir3rT*'T?g?ip;sfusaenrsFJ$sm$f3Txffsi9fol6Ffi3utsEsT@gJpucl,oss;J?d1TlsuuJnusf"$TfaTdEip$sp{eosTTeTd{plornnnsogpdTsquostnresTuauu?rJtssegls8€aFXipsE6f#wBelrs€olEoraTtTp{rl:A.€aorerpaepoaltTJ"tudErT1leTTHluTSrsultTuTTtresuTgnr"treaITrrtrso{dlf?FEneafrtaedl.oooprEsadnxmesndrumtrEl&rgmuecAss.no4tBEd,,mEwlJlrcsz1{lr{pn€etgnsDar}tfpAusnoa.{orrlr.epnrfi
1rrn4lod tTnfnTuemp TE ?slTerFT trlr"rrl3prep ap slcf,et sTaldaoooo
qcpeptg{e*ladn1?sfp,&.1r?5oxgega{'1'TcanH1lgdTpmrilrogT*lne€druqso{rluTvEplgEsTT*SSgonu1eg"SaU"sHEpsf{euIrlgrBlss{rsFffJfi,gF&eo{TrooJ}defXsfoiurlEsrD:prse"r[u{tnnlTsUt}aoresgsq "I

"3?s'..,4x r'1x ,ggT ,[T]x sJ;t wiruu

FTn'rcettT3{!tTJu'ouWTApsnaascssdEa[Tu?sro]3:TT?(S€fiEa{ruTspJspuw,'TBp?6cJlo3u?sg,1s?ExxufaarfpA1nnmr6waacule!rB?uyEpc{rpr$sa.prTEo"gTrmssTurqruo!E]:J;T?i JurTSxengAdT,'*rTTesfetdfTfJspnuTBffepiySrGIpm.r[aps!TrlJeanxTerrrmnEeroeea1Sreo1n]q3gue6g

'(aalkznEpn

Bt8tqrffiE} eTE"xfr?ffir .ff){8.c6ffi1s HFm6ttrgtm e?se lretdns J$JFJ Tc els3

trS moe.rd 6?{rreTe sp ?Tc}?S J&iffirr{t Etrr*J?uTp ?TnXT?E$o* TnAxedns pffgdlp

FTeTcffiF Suns d{ Tsr1twds {!_T ffiTTry**strcJ p$rfltres ap sseTs Eas{t

"xH TnTteds uTp ?utreTs T{rrm B aJ*?$auajdau t"F}e

"t?TuTltrF eT 8pr"5"g?

E{6?H$uofus gaiesn?T$ es s-xu* *d arss puBJ rgH 1n1fi1*{edx JoqguundssxoJ

Tn[mo? €?TWTT ne.{*p$snop T# s'}Esd TrupuTTT3} e":€rir ep lgsT.ro
€{ry6mT Bg} {T'tbTd} Tr?rrotrTJo nrprr;T;p unr sd Jotauso,Tluesa g1.rgee1d

:fl l't?tfifis 86 T0tltJ$ E* - d'.*Yi- $frn&${$ xij trt{$$s!}'tTYtnns

$JffALE, elB&BJl6 $E $t$?v.iit l-7-* lI$tAtIm tr8Bil8AN,B $tt!
unu
Este pasi-bit$ gi detlnj.rca sF*?lului ff peste extensia de con
*rdinul" a a e@u3-ul. Galols GF(l{}, cf (lF}, un
in aceat caz fllnd 9en
n*cssard prmizarsa mdnlui in care sa daflneec oparati.lle da Fro
a&rnsra pi de imd.fi.r"e in Gr{xtr }: e]"ementeLe dln tr(lP} stutt puae
ful corsepfifidengfi cu p{:}}i.no.ffi avfurd cooficlenfi din GF{l{}, adunarsh *.
coresprifiu*$d edt:a&rlS. 6rotr:f"ncmelar- iar irwrl$irea, produsului
pltr"nowner wsdrqlf) un po}$"nem *rdnetlhl.l preefr:at, $e poate arita xIn
eS si"sta€}.e mo*la},e"te Xi"ni"me {eare nor fl lntrodqee l"a sffirpltul
aeestui *;*g*lt*ll. cietinlte pa{rtru sce}artr dln mF(}Fi , sunt ratr
ec*i"v*"1enf,e {}"r siatffis]"a daf l*d.ta pentm aealaxi dln CF(F|}. Di.n de,
m:sat m*f"i"$u fn csntfnuare vm eonsld€ra m*sal satrtale rlaftrnitc
undr
Iwstre ffi{ffi} " 5"

2" l'tfumnmx $F6&j"n"[. ]dz eate fuizsstrat cu operatla tnternX de a*mara 6"
w&:3ql s a amaieS.e'r eare detsr*lnX structura de qmqr': spal
- l lW[ *] l ,
gi x I r: l :".y{ a: } = [ . . . ,x [ x f 0l $y[ r] ! , x [ 1 ] eyl 1] , " " . 1* e@rl {ur
lnma-tr?ise ecalartr.
operali"a e;r.fmnfi de mo&rl"o M su dln x[n]

tr4$(H!={ {0,x, . . .M*3},SoQ} ; t epr
7.
ax{m}x$. ".aosf*l},egx!_$l,aoxlli,. ". 1*. a€{0,1,. ".H-1}
dec
3" PWffig$, ESW Ps:od{nsq.n} "{s}Lerr a dsufi smatre sa daflne8tc
prec
exs!f,orffi re].a$i-er! r

{xir;l ,ytn! }*ffi (xtil sylll } (r}
rY-

io care ijn;i"Le3* & ecreeffc ata fo*t el"rnbllzaLe iro degtr sunela, tn
cazurile grre*{13-c*, wr mcrf.ine un nrm*r finit de termnl nenu.l.l.
x{xr3 ot yln} reprezintE, ln no*ruf stAng sumLe in ti.W
Bengc6n e& y{i.N {i$n s€shrul" drapt
reprazlnt6 conporentele acestor
ce ;[1] gt
e€nnale, Xlr"ffis1.1;:r.r&ffi eS cj.tltoru} va face di"stincti"a necesarf,, Ln
firctle de r'r+' qf,
DooE sffiie..r-l' .xLnl at y[n] care satisfac relatia <x[ntr,y[n]>oO sunt

or,togonatre.
Ca gi epat,trul. H*" apatl,uL inzestnat clr un egtf€l de pro&ls nu este
apaltu l{llbert pre*'d}bertlan eicl relatta <x,x}40 nu i,pltci
r:tr ne@itate 6i n*.e$.
s*s- Sl Is a*eet cae ee pot defi"nl retriea Flan*ing,
prodrgul l-ntarn cu gr*nd*re *au preiduoul scalar i.n care ttumarea se face
in rcd oblsnuit. !6otiv.s!1" pentsu cere s-a preferat dafi"nlrea pr*oiurului

scaLr cu s{r$** mdr:},m H *s*s }.egat pe de o parte de eoermrta ut:-nkirli

nata t''octacrga1+3s.nalucespd.trss.s!.}pue}eeti.veteerul$.ro{pss?nffic&t)gllndpJro.f&errseunlt mdulo ll se $or

de valoarea lul

lt.



"ge tt*f1il1, afr $fi.lifl'ttf, $* '.E
i. -f 'ri
l.
rur; .".i* ai:frJii :;tj,i: ;i ;riltr:t-it.i-"-';r'' [!]1j.fi.Oa{fie13:
,ji.ir,.:r.'.i i.,9.:I.;r t'r:ir-r$f{;i].i-1i.i: ,i ;.-'l-rtrl:iet, i-"r:itr.t,rl,ilttej te
,t'i,gl ,:'.:.' i_: r,iii f{j:iLr:ipl;rd+ i:illilii:.r*a. pltj-.. jtl;ilxts-i.rt:{ {c1ti:e
''.,,::,;l':,, .i.i: -'iilrrliL iti- g!; ::!. f;-i p.i ldl.ll,llli r.ri:i.$i-ll*,.i.1-r:, 5U
'-::J -:,,'li .a,i"i ::: :.,i-.iqi,rt!1 .1.f ? i.r:l:.',r,1.:r{:.,:l r.:i.:r..:,lrt}nx-;i:lr ;!T'{? {er-:;
i:e
:r:-:ii.:rr F-'. rr-" F,.:"iF: Sfift,$j !*:j: r,!*!*"i-r',eJ:iir {trj g[]*t.ii"ll_ H]!
.i.4 fl ia
Ii
;::r,, .j. .::r: nri. it:rr:i Fiii..r'€+i.* il.ili..i l,'i I ;.1 1..,':'r,:, ,-ir'f :1.:ilt I i.,";- p1"i. i_lir:rnl:1,::i.
r:i.".-'i r,,',-,,tt,:,:.:.1il.rt:iei.'31n,,'.r{.i.iir::'rf*i{*i*+i,ti.:eIi,,,ilittuet.1::'',1:,:,,].:i.:,.'::.',':rlitr;:;1].;!ai!!:tr.i:ir.:.ir6-'aisr:itE! iig:ta*l.':.*l"i.:',rj-ll-*ii, IE

iir.::::'l:il .:'.' i:..r i'.'*r*l'*;.;;r-i i'l - :ri
Je
i' .lr r. r-t :- ii" :'i:ir,:.i.Sl.]i1l'i1til r-'t-i ,.r-::: 'li.r. l ;]j,!:t: :.,:t1i€i lr,ii.!.ii.!i,i $. fXfi :l itl

"! r:- +tii :, -qTTiil Ht it
r':,- i..:,.ii' .:.,:')
si.
il;t';l:,r *.r;r,,i,i :,i i,l:. 6 t1;ln:, !:,F€i;:"ali.t]f ] tr, j.ar.' **t: ifi j"1i. pF $p.:-h j.l.tl, M?
{t
5L1i .i'-:.i.iil: i, r.rl;.r ;,;lii. j,-:,
!a
,l_,-t-:.i.ir::i." ::I i..,'i.l;i t.1 F!:]L::j{:'.j,1-r'ri- il:efg tl,qgr::.,it.i i}Fi\.r:ej'-{)tr1.11- Fti:.i fi.
{li.,j';.,j..1\jr,:.i: r: : ,.-;.;irr, i!,.{ii;::{,1;.'3 Si*}-l g!:$ij..i}i' i,.ta, jii'fq.'i i'l i'.{". Sj+t,* Citgj"
*v-i*r,,iri ,: ., :... '.,.: t:,:.r::r.i-ii.";r l-:i, i:r,f,t" :f "j e.tx5&1s i:iar.r{ u.nuj?igr.i qle senNneJ"s
Li;1, .4-:' 'r :
l.flli::itit'.i.r..,:: . r, r .,, :t ' irrij::ij!ii1:i.,;{ iifj $e1iit;rq:! {*j-+r i]..}r;t$.}-)i,liie,tij.t:'ilt:8 r:{1j.e&t.la}sr
,il
,. ::r -., ,:r.i.r-r.!, 5i Si.i:.i:i.!?+ f:i-i|egF$i'i7*t,{lsl' e .l"1ni3l"gf

rlFflr:.tiit.:i, . : ., .: :r:r"', :i.r.q,.1,4 .s€t!!t:rii."l.ill"{}: ,vlt'J i::*ft.iJ{el i.:: !."lr$,a .4-a.:t.j.!tnli. tinttj"
{3Be}:et-,;r :t ,;l'tfi:it:riir-it.it--|Li.l.si=fieg;ji"tain-Lii-xi"I.n: J:$aati:irir:lr,f*.eFi ro.&due.gUt.::tS.e€ri,nfn{,erleuri:nXeIrrejle,
.::r.-rri.:r"
ef,errl€i-t:i-,:.i,,,'t
I t:..i.

AStfel., ;,':'r.:r, ., ,;j.-ri-::.fal,ijJ:ulUl F, repf*Z{:*t.$.1 rj* * uatrje* ,ivi${i :.l"nj.ile

co$si.j"l:r1!..:;: ,'r.i'1.!tx!i. iseffiia}r*]"ei "."a._.ln;.a.,fni,e,[n1... aslipr'a
6e{0i}61'"r.i:,: :.1 '...'.

ji-; i. il I . . ."f"ffrrti"hitt".o-.it'ji,:I!"1ijj"r.:'.'se-.'LIlrn:'li,i,..xf""'ff4lirr"i'"Iri.Ijl {F}

f ri al;tl

e.i.;i:,';.l.ii.:.,":r!':i':i-i'" r:r::r;!:i:tori-,i.*r p+t fi- f{.,:,r.ti.-* t:i fitll*ljp de carac;tefU}
,iRVefSAhi._i rr.;:rii.. :-: .::_:,t".:."ii:Sar"*"i.3 ,itSSS€{:t..i',' gi.11isi:5-:,r-: EBU neslmetfiC a_J-
&atrtee.l;:' .r.: nl. I : i:,::xi: *F"at j it,ir.t.i i{}: .

'1 pT tr$ *J€J np (+) -
ar-1rnnT*l-efi:iirr?'a;$erdidr,ra:leue.rsannTacarpra?1mpi;*do.$pId€atvJ€J:oJouuaIg(e?Tbe'anufle1€qsr?{lsqrspp]erieie;ST€Tdraspe?agup.r=aepuuJe€e11Tfanseu1pa€ur1Jeg?ensreetr1{ldrTg-a{t:u1d0a:'Ao'Tec1I'ea6gTsTz{,u'p{rniuuilosgTuag1ioozaeo{ua'Ja.{ranaxdobTanaJ-ss}I
es ii$i'*i ese 'nJpuTTTs rnun 1n-:nt u1 p1e:ntiqlu1 [u]x €IueAces Eiaprsuos
€ ep sisa 9ruIuTT €s-I€ssTdeF 1nc11qau5 ?u e ap ,AT1Tn?uT potu un r

fi I I00

{ri it L 0 = t4 j 0 T il = tvl
T oDo 0
0 ooo L

I,j U 00

:3uns sed En no s?dEaJF pT a:FSJ9ALIT n rTtqEpldap

a;i;i**3*rs.: TTJgs€Tdap a:eoquzmrdsa:oo :oTa3TJ?Pu €TEulo-{ "€9?p1T{In

rc etru*a sJppunJes ms eTedTsul-rd traleuo6eTp Tn?gnsepap n€s e:dnseap
T ri; erryiuelsTp €Tanuau-aTF prrg^p asTJ?ptri sp TgTJ3sap xunF ( [0]x

Sasueur;r3r*ce TaTtTzod p€i"€.c":esuoe no) B"r€sranuT Tg Ex€TuTf a:eseldap
ap r.r?railss-x is IT+u]x=[u]A] gJ€TUTT e"reseldep ep 1'lrolexado

ii0T r' lrl
{e} n&rTn
II .
rf,!\
rfl Il.
iL
i
L
t

';oXeeundsaxo3 atBJTpuT no ?p3JpH ?scJ e OO eTlTzorl
ed +p Tll?u€qffi{i} '1a}T.r?eu roTe?uaura1€ pTiTzod ezlcerd e n:1uad'a:et

sg lrl *T3*edxr*.x [x] aTaJTJ?eru ap lJeluezaJdal ?$Its Trox€Jado ;op tr43

i i {0}x={t{)x-:{S)i{ <= [tI-N]x=[u]{)
i?$amcT" irriirE.3d pugSdecxa) f,ola{ueA}as E, a.rPsJaAuT ap 1n:oleredo

ialc?Tun TnJo?p"rado
:1uns

*1;? ?n*:;e eF "t.TlqcilJpue-; Tro?sJadg 'eeeeuolla€ BTnrps €Jdnse TnTnTeu$as

.:"*i*1n*i;e,r3w** qriJEuopJOsJ ?cala lderp np T.roXerado ap IaJIEV
i:*f!ffif-tr m&3ffisdxfl fis I.[Hutsc 83rf,offiVJ rlzug rH ruc,f,$tlA .IJIls
i:ds} Tiurr utrfFJ srq sJru.Eufi flo r.d,urffizswdcu rmruuxd3 "r

{arevro*

TJsInrTUed UOXU:ad0 I"U

?S f;lY{H$! Aa 1lll6fiC n-{T iltflsI$ {s flIlncil3 '['tYiltfl$

sIHl{18, CIBtUItE $t 5i$?Fti!, gF-lv."r $P&TIt;T FE I]E}Ig,qIg HI

ileplasarea ra st&rrga cor*spunde nratricsi*r transpuse, resgl*ctiv
operatorllor adjuncf.J" cel"cr de deplasare la dreapta, adici unor matrice
avand elementele de eleasupra diagonalelor egale cu unltatea. Remarc&m
deplasare mr:dulo rul element sau polinom, in particular
ci operatorl.i de
operatoril de deplasare clrculard, nu au sens in 1.1".
operatorii de deplasa.ne la I
cu sau firX inversare slurt dreapta sau 1a stdnga cu mal mulgi pagl,
operatiilor descrJ,gi de matrieere eorespunzEtoare E
cr: slngur pas rfulicate la puteri
pasi. un egale cu nu:mirul de i

Transformata d a unui elenent x[n] deplasat la dreapta cu un pas T
modulo a[n] este produsul cu ri atr. transformatei d a elementului x[n].
I

constatam cF deplasarea la di:eapta cu un pas a unui element x[n]
corespunde, pe de o parte ac!,iu:iii "l-ui x[n] a operatorulul
respectiv&, iar e.s$pra al,td part.e griglnalului
descris de matricea
pe de
trasformatei d reprezent,ate de pr*dusul e:u cl a lui
X[d], transfor:nata
d a lui x[n] "

Deplasarea 1a dreapta cu nrai mulfl pagl revi.ne, a$a e:un aro arEtat,
1a apli.carea op€ratorxlui ridlcat 1a * putere egaL6 cu n'tmirul
deptrasful.i, lar in lransfornratH ,J, de pagi
al ra inrmrlfiirea ci-i d la put.erea

corespunzXtoare nrmEzulul de pagi.

t; ;[o o1 o 0 .x-,
1 tt
ir x-. J.,,
; 1l
i"*.. -a- -ad xAl'3
*rv-r.iI
-aoxi}*d1xr*. " " -Urrr**-rJ

gd2ee.nfete&nraleimtt5rn*detmlneetnuravrecurp!l,,ilyuiDcn[nEiele]acc(uiufnfniv\eureoleirrsn,o[aJeprtnreesturgat#tixo#d[rrerdup]lex"casosclsaldr&tiuesmeaed{Lf*.ef]i-satmreuuLaiet[rmy*tc[ejnne]La:r:i]cdiy*"n[rnev]u**rlxau[ngntla{]
It" ] rt :i { lttLnJ *xL"l {.

r.."rlzl vlil yl01 .{"t" r ,l'
ytCIJ yt-rl yt zJ ....fl$f*i t"_f ro.rl'iil-ll*I-'(1'vs.fr]t-'"irnnnill.'xl,'o[;nrfJotijt>> lI|I
yiol
y[r

,J,

ln transfornrat6 d aceast,H aef i.une *r.rrespunde pri:dumrr].rr i dlrrt,rc
traftsforalateLe d *le Lut xlnl *i y[nJ.

iii.i :,: r. .1. i
:ir .;.,i, . r r:r I ii ,!i:: -r lt iiti
cl .t: i'l li
:", i .1, ir:an:F.li i-',3
i'.N:::'
:,r. '1.i tl ::a'1t!,i.!
:':".:
i:li.,i ,.rlllr:'::!
i.: ii
'': *-r 1 ::lj: :ti,:! !rle,F

. r_t rt!ii,nb\n44? lF
tt

.rr I :rqq';tr\ffi!-r13
f..,?,,r:i:F._?e,f +€

. iI,.r :
:r :i !:j,,i: . : i.X:, X :4...1';€ ;; 18.6*?8 Sd. e'.i

:,

' -:t!;j?, r..t I tti tt !.ifi:,{i:fi:t::: i; l} i3* ij -l TF!}S

.t::.::,a : :tti.;.:;:j ir,,;.ff8-!i,-{'l {n?1r;-) il*

r i'trents&$?'
: :' 't ''tiF!Llr. ri!
ii !.: .1. ,r : : ;.;1 lj .i | r.,-
f, rtj: r' -.1 ! i{sii,.l& .T$ft

-:: t: :.'l'ia'r r%"1?1?ry4btg

lri:,15r1 *'?3gh.q!l

$nHt$[8, ftgfljt?x $i $3$?g6s{ Xf q $tf,IIiii sfi $Btifr$ig He

r[r:] =3r[n] +y[a] ={x[*-],Jr[.r:-gj ]*H {}riel .y[n-e] i { 13}

corctrag,la {convoX-utla} u.rrefa eltrrtre servente cr.r cel.al.t6 seq:ventH

lnvsrsaf,b;

t3) Remas'ei{m ed nierrl-sele eorespunzdtaa.re ope::atorllein de
eore1aSJ"e Si eonvoluf.l-e cu aeeS"agi elesne&t a.u colsana centralS
identi.*&, iar 'eotroanei,e din dreapta gi din st&nga sunt

interschlnnbate .

{{} $pre d*osebire rle spafiul Ms, tn spertlul. Mu rrl.r ex1st6
semnale x[.*] gi y[n,] pentri..! cijire csn,rolur.le st fi{: nulR. &ceasta
insea$r"t,l ed ni"ei p;ate fI.
un eler*si:t xilnJSM= nu ortogona). pe un a1t
elament .y[*n] gi aqentuJa deprl.asate cu orice
k, y[-r:+k]. pe {toate} regrl.icile

b. fuwnxunr ux$cex$r trs H*rktr{rx s.refr*ru&T,x

- tlperatcrruS" de inmu..!"tire {rj L1rl sem"qlal afrrj ifsi:sastrX] este

reprezentat pri-ntr*o ulat,rlee diag*na16 ar.r&rrJ pei diaE*na)_a princ,i,palfr
elementele secvengei multi"pJ.Xeatoatre, adi*6

r

I
ai-lj 0
I CI d[(}] 0 ... i{ xt-rl - l,* t-rl 'l -r: i ls!
x lrl*[?1
u I xliJl

I o €[3] lxtr.J

I [*.'. . [

reeultatul- ac{lunj"l r:peratoruj-uj. asiuLrra el"e*ne:ituluj. x[nl fi:i,nd dect;
yp[ ra: rl t-"" i"'..r,:1aa[*r:t, ]x [ *:r ]6 [ n+ j
fn I +a I S ] x [ 0 ] 6 [ n ] *a I I ]x [ ] ] * [:r.-]..l.i " . .
dlag+nal-a oprl.r*t-r:l-r:tlrri pi;;r.te etrn.tri.3i€
*i"rf,rjieei iln

bi"oe e*mtrract t:u el"emente egale cu unitat*an rastr.ll *leroentel.or flind
nule' un a*f,fel- de operaton este denre$i"L "fer**st"r:E drept"urighlular5".

5" t!lt [maure] A nffi6' sffi*An-B e, , E . pe+i alorrrl

fpqm{ls$L HimERH
{:e$}reue!xtat prtn matrj.eea

x_ry_:. Jf-ryo "x-r14. ,i { rs1

xrf'-r ffr,.}-;, Xc/, I
I
xrF-r xl..tss xrl"
I

H?gir:rBSlIOs €AUnTl3eS{r "r UT nTTs.*ap {r.I g?ry?.Q"T? -f*TTJ(ri.ts:{s/fln

es}rx v *

{sr } {pF}X {-> [n]x"e

** fi3 g,ffiugtrffiT,H TfHtrmffigs cewfiom*ffilfff& "$

'{ [t-]d4*[{},q

ea.xpeTflgsljwg?

ug:d g:trelu"$'trru,.:rl B?EsJenuT a$uerrnaa s?T?TTBtBc e p B?Riff3o;s{$e;e?
TS saXssae a;?{r;p pTnun ? p p?*e?oSsue;X av+uTp 1ns$p*sd F?,ea

[{r]A TF [!{]s s{ElJRas pn)p e TaTlpTaros :eqEzilvdsaJfi3 Tx}TnTptffiBB p
p F?GeI{iF.${€.Ai, 'Ewgaantrl u{uaasss I1.$ PTlnTo$,uo'? BaFa €E.fmgUrrl '&

"t{sY } urjl (prJr (*> t{-riJ,4 [rylK Fr6-*{ * [uj.{* [ri]x

ss.{r}sffiffiJ "s

{a{ } urlzYxP a ") e.q-uJn ms ffiwrdcfi "e

Fd}ffi;,

{e{} {r} rrrr ftf -- &-

ffirs*ruffiffiT "T

'(Plx <*> [u]x
Ta3u"s4rpa.xEasdrJ$sTrTyEJk*apfTfifsfeuJs3aT"seuomTpsoTrsraEdlTl upesnnTptesdsaclsgapaoge"ruaId
oTaTcuue$
slus?sTx6
rr! JETnpow rnlmrglo'd e{ueXslxautr
ep TeSlrwrudrssoJd *ffi Tn-{aTXpds 1nzB3
?unF aTT-rTqasoae 'NH TnTdEds uTp aTea
ap ep sd a?B6aI
n* eJpO?ETJGMBG€ Alrns *g 1n"5$ads uf p T6?Be.roJsrrpJ? aTTl€x.aTJ{lord
qT$Eq.apdnrd ;uH InTl?d$ p P?BelCIJ$ll?rt 7'l

'lulz TF [u],t a:1utrp :ETers
ap TrLJo?epl '[u]s ns TpurrT{JodoJd
TrisnpoJd pUTTS e?e?TTpuclTtJ6dsJd
e: 'aenpoJd [r.!Ju {€imF Tnm BJdnFR
Tp!.!ffias {.rqtr '*Fi XnTnTleds Inzsa uI 1S
J.clXBred€ ep Ta.T?s€ Tffiin EaunTlaH "[$'lA TS [u'lx aTer?r]was ap e1e-raua$

"s E?Y$Sg$ 5* I{uI{d$ *t-d I SgEi$t$ {$ il6tn0pl3'fivFsfi$

Sllllffir el*s$tr& $! $i$?H*u 6 *b- $ttilur, l[ $ruMil ta $8Ht

{. rf " 1l gE

5. Mrc x[a] gfif.i&efefi.wEre{flt:frfiietrdeeepits-&f.ul nlaj"€natadtonfnxjl.klceru€tetraenosnfvoorrlualtlaa atr

d{ntre {ob
tr*Btef **te 1,/{3"*di a$.e q}& poa
6& 'jry{xrstrsazl in eontlnuare in (2.3.-} ap€
ee obgS.ne;
val
! xtrrj * xlil]* [.ni . {xi+J ,.n[r! I s [rr-tJ +. . " (-"> dl-igd (2o]
rel
tEr
1"
?. ?nrrenm sw*&&ffiffi sffid:s?x*Tmren {x[nJ=11;t*ql, n>l{>O), H

f1lM pr*cradn

-bx r r,t ( ry Lgl-1ry-l4i d1t:-."yx'i4ff-Ll-di-i (211

Dsmomatrag*e; .$c&u&nd dn& pecnffi{'a x{nj si*er,ren&a furtirziati cr
perloade m, s{n*al, dmt*r*.{* *tsmr*eclsl*lrqr:t}llfla{ rntw{itr.dgererz:si*rd<l;er.oAsvam obline
rerrent& pr9mtt K o
av&rrd mm*i. deci

xlnl *x[.s:*ac! *".s[g. ]&n]{l+i./-t_".!lj&&in[n-tilr++tx.l[J] S [n*z] +. . . {22',t

de trnd*, prin ap}:{"**nea t,i:asrsfrrrmntef * affit}or na*rJ., rezultl
ir{d} *d6'tr{;f} *x[S] " r+x{l] .6f+. . " +x [&f-t] . dd-x tZ3]

8. M4t Atffi"ts Ffrmrffifets sunt. a*ele sffirrt-le cer€ aatLefac nelatla:

Fx l nj " M' o- tr' q s'i *r[n-fi] (2d!
l* *

uCe q{n} aete mgffiia}nJ. genermtorn *} sffi}alulut periodic r[n] au&ld
pcrtsda h. Yrannfor.xats d a ewralelor periodd"ce nu 6€ poate srrl.e
dac*t, ca s serte forsal&. eiar eere nu {:o[]wrge pentru nlci o valoare a
varlabl}ai ei"
&smflnaMlltualnrca&rin[n:!c]c=l.sx8, us6un{nn&s-lwaxs}mf{alscppereerdi,edtr*d3s.ip.are;**srbEt"Ianaue-L+dtnes)ps6€rtlensne6cr1reruiantlur*rloa" lluEpteelanrsldoraldntoltcrre.,
fn transfo.rxet& d a*easta ar r$,'snl- ta lnsrJ"it-rea trancfonatelor celon
dou6 swsl"B, urtlg d$ntre aeaste neexist&md"

?,3 ?ransfsr&ets d a3e unCIr sffisle reEarcabile dix ll'

- transfernata d a amlal.ulul &{nl este, evl&nt, cnnfor:r

&fLni.$,iei, egel& cu 1"

" s.xPr;€n;, rE$! thl-: -,i:ifT*.{xJ{6

T.qiP]!ii:{cd B-i =!_'ii.?ds* r'i-1 ';i'|!3T-gr4g:;'ti?tri.Es3 F-*;L:iils*.r3rts3 .*5t* irrlg€{ss ET
B-.r. .!-ri''r-:8$ts.T *:,liiar;-tii::is u3 G'i€1i.Ee.Z#ii* a'{i:'"} i}:J, n+:g €? t}cEttr ?w-}:"sLqTT{}*

q"i'ttts*l 'g rdnwe;: .3{]E*{:[i1: $.SeH t* 1.5:;Ai] R e:]:{tlii{Hi;c*'€ p eg} F,Tgi::{TS FAT"

::i:r3:i?-ri:i.r!ii Lrl Tfr W3!-r+:t€ .'E+Tryffi{l{iTi {{:Yl 't t e'r1..:rFdw'f &j- +$itsr-{!e;& Fn..f.WA*f*Sq - i

l: |'i* Ipi-tt:.:1i-1trr',t'{ E! Fr.4vi'e,r; BT .r

{I{ j r.';:!'r:-i '','.r,,i}!;.$S;.'*" -.:'r*"FosI:+8 {.1 ilrd

ltst tri#;S-'ir:;!xq

sjll 1re33.86)i.-*'il ."1 1'..-:1-q,,.
3T4?$T.?ar3',i?flsr} -{i:!T3i?F!r@g 5} ry'.t.fi$EirrEFrI.Sr:j

i, ; Jli"H* r H$ 4- ). i f,ry rr] E- {rri d?i,, tr
sr.? oL' =gp'e F ;'.;
{$il }

p'lr€ { gq.=rr A+..5x1nt-"F T*TnEpffi-q8s s i} ??1r?s:!'{iTqr::{F-ra

f}* lr i*l -a.f "f.lr I 1 ilrie

{sr ! t':_T "f'r$=1pi)*
r-€
.r-i" '+il ttl+.r [tj,Ng€{{Etr# TttAET-*$res e p p?pw;},t:r:&stx!*.l}
'T.ESTptc+p Tfue1@i:q $T 'T\iare! p:w-"p q'fj.Es;.qr!

d i:.;: ) ;:' i -*S1yF g*' "'+€F*F+&1p+T .{-} [ru]#FF
:.- ._*.='

[ulo",€ TRTn{effmafi € p s?amr{:r]|Fusl'}

p 'T>fl pe :G*ffi?Tir{r ?n{ns"F,Ba Tri.T&*{}T.re?iiT pFrErJt{xs'dSE

r1r?e3*{$ T.gqer1ts 4.a$6 FTxreqa; "xaqriffi* [srffiTd {sg T.3C}TRA sr.}T aakffid
Ffi $l 4TT.r.q*r'6.TB*1 Ip..i sffia'se*;=?e tr}p.q "{t*{p-.gi{"""+fp4E}c{} Ri} ry*l.xs*{qm}

C,0" ! *j;f {-.} {r_rl n

:[xg{ z?.a sTpsTss,i$Tw8
r,i€!] w} tr3
T€ttffi}€ "i.ri,:,lYr 3r?atwi*ff,€1ffis? ?E .1q1n:ilT3-q:*43 "{i-[sq# +$
" "' "e I e..trit]-r i T', ry if i.Ir.{.1?*{H}s

qseS ur"'d.-8, E$S.q$i! is T"EitSFsF$ "{rfl*i*$
?ft F1?FryF$ K$ i€81qY4$

$Biltail, {ItsutTc 8t $tsrtrn 17-rt 8?ltr0[ it $[nil[t ['

3.{ Exenplu de deterninue e origfulalului uei transfornate

d {tr=3}

x{d) - zdz+dt+d4+Q6+l{ +2$e 1rz)

1. +?d8

Dorln eX det"ersin6n orlgina"Lu.l rln]. Utiliz6n algorltnnr]" Lui ffiuc1td.

2d8+2d7 +df6+df{+d3 +2d2- {2d8+11 "1+2d7 +6f6a6f*16a +2dz+Z

2 ds + 1 * ( ? d7 +do +d 3 + dr +2 d, +21, d+Z d1 +Z d5 +Z da +d1 +d+ j.

2d7 +d6 +rja +d3+Zd3+? *
* 1 ? dt +2 d:+?cfa +dl +d+1 ) " t +d6 +d5 +t fia +t flz +) f,+t

2 d't +2 d3 +26f{ +6f 3 +4f+ 1 =

= 6/s.r gf 5.r2 6 a + 2 da + 2 d+l ) . ? d+ d5 + d5 +2 d4 + 2 da +2 d+ 1

rjs +d5 +3 da +B fiz +ft j+ 7= ( 616 +cl5 +Z d4 +2 cf e +Z d+1 ) . 1 +0

&ezultX c6 cmdc aI polinoamelor de la numHr6tor Sj" nuni.t$reste:

d'+d5+2d*+2d2+2d+X

Stupliflc&nd, H{d} devi.ne

yq67 = *-F,,31--=
L +d+2d?

tlrlginai-uJ" lul x(dl se obline p,rjn dezvoltarea in serie de puterl
a 1uj" x{ci} {tractLe contlnu6}:

zdz= (1 +d+Zd2) .Zd2+d3 +Zd4

d3 +Zdr * { 1 +{+ad3) d3 +dd +d5

d{+d5= (t+d+ad2} d4+dG

6t = ( 1 +d+2d3) d5+?d7 +ds
?d? +clg= (t +d+2dz) .Zdx +d6 +Z#

de+2de= (t+d+2de) " d6+de+d'10
dt +d1o= { t +d+2 d3 } . de +dr'x

Rezulth cX
X{d) =2d3+d3+o?4+dr6+2d't +d664eo. . .

xlnI iS-6*j.Q[dn"&e-e6[lnj..]+x+?[.n0&.J6[en[sn-t?-e1] j++1a,0"0[n[n-g-2] ]+*r.r6.6[n[n-9-3] ]++, 1. ""0;;O[nO-4e]+ttO0,:0z1[tn-"S. l +

'T.3ET83g arlug H oTnPot Tnsnpord Td lrr8 njrruod
nqlxusT'oIsosr}'nTTTTSnnfTJ$u3n+mst,rn!&J{€p,oTosoTrTTdTTrTnnTrfTpofgiooI?o(Tuptqrer[uoFdu.rFnonYsstzsTBTTtTTT(6T?dmlTJneoTlxITlao{TTHinmlENATl'loTTTgrcEuluCtesB!€fiJuerdcIeTadcrTonlrsrTsnBEJlaoorecpTt1geuleIsE3ET1lut3tr9Ttulsprlodr'oe"eEdnl'reeotnJluEsTeErnrrnzsuBxenr?clpruudoo9ualdJdTl
t'6TJ TB g'6Td ItTp oTTrnpnr+s utrd rTToctrts lB?uazordor TJ alEod Tlt

{er} Qr= 1e1r [Hi3t[lEt tilliF]?: Hlfr*]
:Earot ap 4{unca eP
ff91 sI rpTnpfl xaJssTp mlsTs un
sTrasep e1*a ITqETJTeA rep'reTtr[T I'6TJ rJ pzee.reGns

uflufTinrndsreralreaT?6$one{!'T{a{-eJ"oJ'T?"zlsrgqT}6u3TTaSBrpp( l"ga-l"ee".frn}c}Tr?etu3rTfradpa}1?@1le1X{1uuneucze!aled'1T€nTrrqePsuTTJero6pEITTTGgnzemeJctT(uIq"I6leT'upJ
upEre*X 1tf6oT notqsTemoXleqr).t*orTyJpoureeuT9n1p9o3clfGrpIdETIasunupneunTxlequzofolrnel1trtxr p
"7
tr1rd luzTTEaJ

1[i tt'""u:1tt'"*'xfi ,.*| ,* ,['"1'l* [u]i ,lIrul rel
{rr} l= J,,1"
i ["] t (l
[tat'rJ [[;l;1

Y:ltl*H:l3:l

'erEf,rn lryrd '{tl)SC ad
ep efetnes u 'aler1utr ap
'rr*tr"*"#"Tt.ffi"t $rTao1rolT*?re$rtJeaaApr?c€u??nsusoa?pre'eerJrs8*'?lTs8EO1apSBTec]Te?:rIf;f$aas1s'eT€arBe.rrtlgusTT
sReTTToseTlsrEepuSesepoT@aatrBcp6Jeq'seg,1pn4r1oT1n3sL3emn/I1sTsneunB:n1J.urEleuled'e{pud'Ta'sfu'ruando1d=en:efswldaio:tAdE-n?sl}oTraepnTcnsRdppzlog"aoeads-Imeo'p(TtaslTci;fJettesllsTlsTePs3srPuuTnIlodu;nnaesps
eueT?fiFsa{pmtBdslBxTopaosaEse?glselseaTdpsn1Xssusan{4sx9}sdeJoB1TsenTppooTu'e€l trecd?reqluJoJTsuTanppeeapTteeaTdil9autTtrreussTT}seaPJI€."3IHIIn*l:Tnlupja'x0dded'sI

qdrTl al{IpTnAuI T$

alusTrs& 'am1u11au 'arsTuTI anlnpl alarJslp arclst$ I'g

dfiIL tffi S HYIUUASI tUUil{IT flUnfi0$il ruruC$I0 XHruSI$ '[

:fl tlvtft$ f;{ ltlIi{{$ 'T-Ltr tml$ll I$ illllctlc'fl'lvlnls

$nt$L[, cIt{ilTg ctr cr*tgtrt l-7-tu $il?r0t, l[ $tHA[r i" ilt

Bse reaill"ntla cE, pentru orlce glsta, intrarea det€rrtnd starea

g.ln internedlul ecuatiei de gtare (care, in acest caz ecte o ecaragie
cu diferen?e loglce),lar starea (91 lntrarea) detar.mf,nt legirea (pntrm
lntarmdlul unel ecuatll algebrlce)

"

qnl ilnl

{rel xln+l I {mlpt({rif.elnf"nf 4ngl"?h

ylnln{$t[{nl.n}

Dl{nl}-xln*l I rt

FJg"? Stn*tura genera.ti a rmul etsten ro*rJar discret.

b ceJ.e *a umeazd vor studla nmal ststmele nodulare tlnlare

lnvarlante ful ej.sp pentru care ecungiile care deserlu l"aqdtrlrile dlntre

lntrara, Ft&re S1 leglre sunt de forea:

r,L?;il;iflfl#?Jfr xror =& (13l

urrde ratrleele e..instante A.-. (natrlcea caractsrlstlci a sistetrului) rrt

gl B--r dcflneec ecuatla de etara lar natricele, eto aaanen€a constante, r€{

aCa*t.fe91l Dd€*,ediseftl€ner*ecaeucnutaltdiaednetlcleerlcrueic.eSletrucocrtuearlp.luanc*fa,troearreealcizaezualzuE:i. 3.
rrarlabll ln tl,rp, c-si daoaeblraa cI mtrlcele A,BrC,D sunt c'onstante.

Blocrrrlla elwntare necesare rea[ziril unui aletan dlaeret
rodqlm llniar lnvarient In t14 nuntr

- Buatoare rcdulo ll"
- rcaloare noclulor M

3 ln fumcfi"e de posi.i,la monenttilui lnltlal no in raport fir

orlglnca, starea 1ntr}ial.i aste x[0] ser: x[n.J,

tiI;.*:i u:p *?usue}lkt)}

i.!$i ri r ;r,..'d €*#yxg "'{-&{}gg '€
i.:::-:.i,;\ sxg?ctffigru

ETI; .frx!::'i-".f: r :$rl&$.!_nii riir 1.3{r+*€& g?wp:geJdBS

i li*l t',. .r€.!** 5.?t,1i A-i.9?{ris?*Jr18H e"&'sJ
}'l€li:ll:ii
Jtt* " I gtu*e
";',{
r iilrrt& F.r,'*[S

ia;tg:
lc.Fi.i

i*?'lF r*

til

aiij: 'fixYFHfi$

$EHHALfr, 4IRS$tr?B $f $i$?a$4 17-ru $P&ITSI, N,B $$HEAI,S ffE

v! )

V-! e d
t,
tt n

Fls.s ALtqrnatasr6 de re$rezentars a unui. stret.m di-ecret ssdu.Lar li-nlar
l*lvnniaslt 3.n t$&l.

3 2 ktemgnmrea nHsprmsului s istenelor moiiulfrre fnwariant*
"

t" &xa;;'r.n mwgmar.H s &c$agrBr nB $r&Rr e*it€*:

x[nl *.qn*1ol .ij A" e rFelkJ {44}

&-0

D€roft#erefl3",* form;}ei (44) se f,aec prln induelie. Dln ecua$la de fj,
stAre, part,3-.eel.L*:rie&nd n*X, rezult6,
x[1] *"tx[*] +Ae[0] e
t45) n.

adl.cl ecua$ia {44} este satief,Xeuti pentrrr n*1. 3

6 8olu91a se pcate senla gi sub fereac

fo =,{a-ft+1r[n,,] . F A' kF6[&l
ktn*rl
iJi!*.e4*lo,l *'*o
=.Aqxlel uo'
i &'pelkl
*m
^*
."--

eTE: (E8T?nuweruu6':&n6f};'eHenpo'spueTTTJs3ET'T{en?{Tftrfxfi"3}sBTTsnuTsTI'PF??sImuT!uTTnTclulnlTd"sroE?req-
{Ig) [O]xrtsJ=o*.,f
cET[u1JEeTJpElE: e{pR:faTfslnuTuatruuaEofTldTtsl3euortpaeTTro?xreneo*ptBETrtTruTTsrneupsTnlrudpmTstTEdrors}-Erl€TteTTnlllsrpnneotl]ussotl?xraporrrdgos8?JTptttplpsmpruBwl!XruuOsnT=dosurdyTTrqnnul)unees',ss

'C=[0]rl TnT EsrTrtJap g5 u uI I*u
uTp Tauns e aruo;"radns TBlTutT €eJ€qwtr{oe uTrd erurepTsuoo u! lpnf a?se
[u]eo Tnu6uJatr, 'aTnu luns TryJ-lErT e?TeTeTaJ elsol purgJ g cere4ul ep I
€srTroT g6re1 uee e:apuod TTtflmJ luns '[u]Etrl 'Te eTe?uaue1a 16 1:r
amTsuarfp s"rE F! "TnTrrelsFs u a:apuod EeoTrleu e?sa fulq peoTrlpt{

(0sl .{(& 6I1-q-ug:* f:f-ffjr1 i

{=rr a= [rf-u]T ]
: aTTr{p?on ?p",rTTT?n n€*s aptln

04f

3.tlfJ€ try-uJ ry l0Jxu ffit=
tr

{6r}

r*t ry.f [{] ear.ry.,," H* [c]xo vp=,*,"

:wu--r{qo

'e:gBeg ap e;{enec ur 6JpAs ap TeTlpni)a e1{n1cs puTmoTrrl 'JgAepp-Jlu}

0 <r.r ffr.uV$
{8r} USU {Jj l= lff]ry

:eprm

lJ E{

({.r} [:f Jd []d .ujq 3. IC,J.rr,,gJ* lu] d

;'c?ss sua*x-i ?,: rsr&vnJfl ( Ewunffis wJftsff "2,

' 'u TnT E expOTEA
eoTro trr?ned Tcap Xau nxgued H?€Jg*trps als6 (pg) pJ plTnzeJ eprm ap

{=,rn, [:f I €flr -,o H* [0] x...*v"" [rr I eg, i,o, ** t .t-,vfi*, ur r" f r."i *

lfT+ulx UXTIL"TITJ 'glerg*epe slgs {!,?} elf,enae ee ezelodl ug

,fl tIHllS t[ !0IIT{$ *E-{-1. fin[J$t3 I$ [IIn0tIC't',ttflnl$

*'$I;,)
iri, t 5411P.;,6

(l

{:;?}

ii :,:.:, :: I I +.1 ii ! 'jl ,l . .., ! ;:ei!L i'X

}e. f':'i-"t,'lt'' ir,i!:1i+ l':. ' i. I r I :.i :'.,i1; .t l,: . :ii nX.'ii #Il:&f;
€@lli:;1 i.:'. i,. ,I
de $t,t''i ',: , ;al i i jl: , tj.lll. li.:.rr,,';,i* ;i-r*, {'-lji'{.:iili6
#S-L g";.!!. i;'' , ql-.i;:, ;l 11111

i.Jfi i..il .::r: I :I I rli:1: !',':!. 'i. i "tiri !i{'.rj. ir'iffitt"
r&eiErl.iihr.
'l't, 'tii.:,.1F:r Sil ri-''. ".,1-:r' i!.i,i.l.l'..r:i'q!.\ti
.;.11ri:il,.'rI:

: ;::q:69,"i r: Jr .'i I r: .:: :.'a tra:r'ifr iti* .tli.! qj
' .: : 1;rni .,; iPr"$*"
, I:: t.:. 1 .:1.ii
I .-". ii:4:il,iili.tr,4 lj

',- i'tr+*$e:I':fs & si
gt,:";,.::1 j::e;;"i . ].

.-.;:,{ l:!'lj!l

ete('eFrlfsx*(sTro.'rrtil)luovsreTFleplepgcaefTilcflsrrraiB?o[rEdJJTTrTBenTTnnln?uTpnTsmnlosedTlTccaoTdfFu6cmopaeTEer?lof€rrrT)alrgrtdle6'TtJdu=uTUn6rrrJCfp([e'TrEeBooprUlurressF?T-eTJrSsTterooxnclIutdrSe?eET?apTTtsplntiergdlrnegmnrolrI6solpTtooltEruiolnepetp1lenplloesuoeoegTTcToIrearPsorry:mddTdTrrE'eIgancTpf)TorWBEroT'rruelrItT'ednlrredtedEcurtuJpr1x?rr'ls4Ip!

'ry 'rogscnrr rn6u1s rm ;6 x,.-q euuu 16 'rotegap€rd rnOugs
un are arels er€ceTJ eceJpoop eliFgs TrnTcTJ lumu eu1{r!oc [Elr;s ap
TE eIels ep Tnlpro 'erBlnftTssu luns E s ecTuouec
reJxlsE Tnun arI?un)Te aTTrncoTq alEol pc FlTnzql T.epcTtrrt?rEdr
TTTqTzTATp
Tsrlol srp
Bl1rcmmEaq€1nouoTdTrtrtBprclrprEpcrctDrpeTrescupT!6FrsrgrlTssTeexueoczBTt1^'E{TBOp rTteglrTlTr}I.TeFpTrEe'1TtIGnTUpoTsBroaTleUrrlEecstrsTTepaFrJclTTsllsTBTTsrra.llqccnnExrescc

HInu eT{slTrxe PI psunds$ t'g

prl3ae 'XEztTEoB ep ro0n T3r lTil rep /grgrd nc Tfp?eTrdord TgEeTece

ltrtT*\Tqce rlsts Tnon eJeolpzuudseroc nleuolfer FcTuoueo ps
Errol rtc F1TrtcoTUU Tl etcoal '1ep wgsls ur pugluezaJder 't acT.rXBt o lrl
Ec3oe ulrd e1e11r erltr! ITT1n eolBpmC desnesroTaErqpxlsTlune! zTerurdoseeecolenXseplrcdoprX6TnzaJ
TnT Tnrossagns nc
'':

'Z erftT$rg ep Eralrlq fn1frgs nTrTJ 'B 9'6id

rl tiltfls ll l0lttal OZ. LT trru$t3 If IrI0ttIt 'flftflls

8lruitB, ctn*ut?E sI sI$t$rg L7 -,, $mtI0t Dt Stltat[ ilz

$e demonetreagE'ci daci P'Pr,...,P- sunt perioadele stdrllor unul
sietes discret dular linlar, atunci peri-oada uatrlcel A este
ffic(Pr,P* ,..,nPrl . De as@€nea, dactr natrlcea A poate fi scrisi in

forsa canoniei:

ol ( s3l
,r=[4

lo

s1 daci

t?l til (54)

sunt douE st6rl partitionate conforn partilionirii matrlcei A g1 av&nd
perloadele Po gi reapectlv P2, atunci perioada stirll

^-JItxrj) (55)

este cmmc{P'P"}.

2, llulfnrr DE crcutru Ne referj-m la un slsten discret modular liniar
descris de matrlcea caracteristica A. Graful de stare aI unui

astfel- de sist€m poate fl caracterizat de o multine de perechi intregj_

de forsa:

E = iw, [P1] , JVa [Pzl ,. . . , iY], [Pk] ] (55)

unde .It{[p" ] reprezint[ nrnirtl de cicluri de Lunglne p'i=l,...,k.
t'luJ-giclea E ee ntneEte nuLf,ine de cicluri a lui A gl fiecare pereche

de intregi Hr[Prl reprezlnti un termen cicTic. Daci considerEm o
@trtce A de forna (53) pEntru care graful de stare al natrlcei Aa are
4re ta ciclunl- de lungi-ne P, gi graful natrlcel Az are N= ciclurl de
lunglne Pro reeulttr cE existd NlPa respectiv d=P, stdri de forna {54)
av&nd perioade F, respectlv P=. GrafuJ. de stare al natricel A trebule
si contind l{a$aP1Pa stEri cu perloada mc(p'p=} gi decl
fir.{rPrPr,/cnme { P, F= )=NrH=mdc( P., P= ) cicluri de lungi_ue mc(p'p, ) .
Fot exieta 91 alte cicluri cu aceastE lungirne datorltd altor perechl

PaPz .cu acelagi ciltrnc"
DAcE ee deflnesc:

- produsul a doi termeni ciclicl ca flind termenul clclic:

x, ' Dacd T aste perioada natricei A, aatlufnieclcAirtux=l xpl.m, ni=tru1,o..r.lc,ke stare
al gi deci T trabutra sd fle un nultiplu adicd
lui P=cfiworc(P'Pr,,..,P*) . Cum insi {At-I)x=0 pentru orice stare x,
rezuft6 AF=I Si decl Plt. Deoarece, pri-il definifie, T este cel nal nlc
intreg care sati$fsce AT=I, rezulti cX T=P.

T0 0CI ol
r0 00 ol

{8s} OO IT II --
00 T0
ro ll=u

00 r0 0l
fndffifi ']: (zldg ed P?Tr4Iep PT€uortsr grTu015"3

PUrcJ LI ?;rT?STJ"?"e.r€3 BacT"r?EU aTd (xpTod Ts q€paz)

nidTr:$Tsw {ira arJ P$ eo{P-I} pc TaI?sc 5a.:1u1 3T!t TBtl Te'-3[{Ap?)sda ]-.T[naTpTm€

frp'{rs}

lJ{.ta*--'
a. r... ' t'.rt t'.rt p,fu ' t'.rt Tlfu,' trl 4

o''-*t''f-"6d
--:*--_- :sp Px€p alsa 1r{a)dlH TeSTJXEil e

T.r.nTrrs @p Baff.llTnu'-[(P)d] Tnf PpeoTred alsa "1 rS r{ fnp€r6 ap 'x ap
lTr@;Tp TTqTl3np€Jr woutlod un aese {x)d Fc€p ec Erl$uouap 8?Pod aS
6TJpnTTTSieT:fJtTsc'ip;rou TTuTraeErlTsu{l€rTsnrTFrnaeTlaanJ1p€aul'TaT$leezraaanxXna€paurTesJT3d1sf'edt3nep"T'e-gJ3a"re9slu=Ha3mraa:aTrT*lsf,aJ?oPvTwqTeuanoTTnJrn:l1TeueuaTd3€
rpaT.'EJT3nJT"TdJrTTTFpaf?i{rTawT$pozsuT6GaJpw'Fartc$pnpPTmeor1ET'p1J}Tqea{{uTpgornpTeusllpu.eel:ufe€fna3uTTslfuopo:l{uTu/eg"3l'sc'IEa=uJpToJoT1lT8'r,Jo'cF'zgTpTlasATTTF.erpaa3c?PaT3JpleE?"eo'rPuulEcOzJuopTnardTTosm{u3aenfl,olpooT{td}qrt
g1euo1$e: EJTa.lrrrIRJ Ps ssro; uTrd v acT.rlen reJE3Tro TTJTncoTu!

pa?E?{TT{iTs*{3 ap EmaE pugtr.r4 'eJBoTJaluP Tau€i"roa? uiro}uo3
.
'{trz

Jge ' [8 ]8e', [9 ]gT', {7,}7T'i r l8}={ [?Z ]82' [8 ]82' [g]Zt' [Z]Z,l' [ 9] 9' I r ] 8]
[ 9 ]' ] t' I e] €' Wlz" trlz' [r]r' [r]z]:=3's
={ Ie J c.', I T ]F'. [9] 1.', [s ] z' I e I s', I r eanr,r{1nu e?se {[9]Z/ [I]]]='S

TS i[8]*'{.e}E' [I]e]*"g TxnTOT3 ap roTTulTlfnilI Tnsnpord fn&WI '€

"-g"g i6a:ggc)eirVfpy-eFfiTyi*}srfgl e TrnTJTs ap eauTiinu flnnle ''K Tg 'g alltedsat
xtlns
tcTecTJTeqqns aTe TJlr"Trr? sp aT'{WJTnE eoeo

:eue-roa4 BgrPolpuln PrleUOUap

alped as ''g T6 'g u1p TJIToTo :oTTuauJa? aTe {TauJBaiolLTraxo??tuaP:eTcaT1lTer.[zJ3a-p3
Eioluoci
sTTqleod sTasnpord e?Eo1 ?uns TOTTOTa
Botr[1Tnff puTT] BO 'g TS '3 TJnIJTo ap TUT{Tnu Fnop s TnsnpoJd

"" [ { ed d } JWEa J ( z6' r 4 ) cpwnr=ptg= J =d ] =fi ' [ " d ] *tt 1S

f{"t ['Nvflgsg T{tiI$ds ",. LT tflflI$t$ I$ lilmll]'flvtIfi$

illilII, illclltt $l lllilil L7 -r" $?ilrur, il 83iltlx r8

Foltnocele caracterlgtlce aroclate natricclor de pe diagonala I
princlpali a natriel A stnt respectiv: t

1+drd3+dr=(l+d)'; Fr(d)=1+d; hr=l; e,=3 t

l+d+da; Pz(d)=1+d+d2; h==11 E"=1 r
a
ln acegt caz ave
Tr.(t'.1i T="'=2; Trt*t=4; Tr(")=3 .t

f*-{t bi , 'i\ ttl , "i-t l"l ,"i? tll i. } r

i1 [1]" 1 [1] , tlz] , r [r] ] =i2 [1] ,1121,1 [{] p

E,{t t1l , +-t3l}-{r t1l ,1[3] ] A
].),=),.&=t2 t1l ,1 [2J ,1 [4] ] {r tr} ,1 [3]
t
{2 t1l ,tl,2l ,2t37,1t41 , r [e ) ,1[1.2] ]
n
Gaful do stare al ratrlcel A egte prezentat in flgura Plg.7
L.
sso/-o\

L**-,.J

##

*

fi1g.7 Graful eterilor pentru errylul l.

5. hlcurllt PnlotDEI sl;nrl Este porlbllE detsnlnarea perloadei
orlclrel stdri pentru un siste dlgcret nodular llnlar degsrls de

ratrlcsa caracterLsttc[ A p€ baza urrltorului algorltr:

- Se detcrrlni forsa cano'nici ratlonali Ao.P-'lP;

- se calculeazi x.=P)r* Iq ,a . . xJ'

unde s. re;rrezilti oordonatele orespunzitoare 11n111or natricel
Area<pr(eoz)€ndtalnrsaApoglllmIlalstrl*opsllnzteruaxaazci,atrcalnaccfinnrtnedreealn.erlo. tin c,r x.(d)

- Fls r,(d)'[p*(d]l'1. Pentru flecare s. nenul ee gloeptt' cea lal



$nilu, e n0ulT[ sl $l$TltB L 7-ru $?ffitut 0x $xHriLE H" $Bilt t l

ratricei A este u"-[[trao.it natr.
natr:
tor I (6,r)
l
tA,] j
nodu
unde [Aol este o natrice nutpotenti,iar [A,] este neslngrr].arX.
Dacd se parti|loneaztr nul9lnea st6rllor corespunz[ton pmrt itlondril [A,l

ratricel A: avAn

^*=[t*'qolj {55} (

se scrle Flg.
2.
;j1*l h."* t66l
J.
"r*F [r'J=lor"*, Dor:

Prin urmare,in'graful de stare al mat"rrcei A-, sultlmea tuturor sal

stdrilor de forna

tx"l (67)
tdJ

reprezlntE un arbore zu rdd6cina z€ro (numit arborele nuli care este
chlar graful de stare aI matrtcel A*.
Analog, stXrile de forna
t0t
t58l
[''l

(nunite stlri ciclice) constituie clcl"uri ln greful de stare aL
natricei A-, aceste clcluri reprezenttnd graful de stare al matricej.

Aa.

1. GnArunt DE s'rARE pnrrnu srsTg{s roprrr$ng tnrtEns cgtrEftrLE

Reanlntil faptul ci st6rile unul sistem nu sunt unic definlte: oricc

conbi.natie liniar independent6 de varl"ablle de stare este variabild de
stare. Prln urmare adoptarea variabll.elor de stare nu este unlci.
Pentru anunite noduri de adoptare a variabilelor de stare existd
anunite avantaje" Exenplul tlpie este legat de reprezentarea ln forua
canonlci rationall a unui slsten discret nrodular U-nlar. Ae:P-aAIl,
variabllele lnlfiale de stare flind inlocrrite cu varlahile izonorfe crr
pri-mele, prln relatia: x-=px. Destgnrr, nunerul varlabilelor de stare
utillzat in descrierea unui slstea este acelagl, nirineVdimengiunaa
spatiului stirilor filnd o carasteristlc6 a sist€mului lndep€ndenti de

alegerea vaniabllelor de stare.

Un rezultat i-nportant se referE la faptul cX fiecars etanr clclicd
dln graful de stare a unui slsten dlscret llnier nodular descris ,Je

esa'rrpTpeTgJBulErTurTe[!tms;reel!tfrTrie6fnfpTolognfsu,UcnlKa$teIeJlzlJs€sdTq"Itpsel fdoftBiTTt?(lnplstTorTrzdsrTeaxTxrsnPrTluo$snrlElPBumtte?r&srtBsTTzlrTFrglTraEaf'rfal€r3JuoBrTouruvprJu[s3nsJaewlztee;uuJaTmtTztJTrleET?fEnvl trfas[sIOE"C8s

r€l'roTrelu€ TnTduexe 'so[ TPu ?elnazerd e?ss luPllnzeJ TnJpro
uI P?surfxalep ?aoJ E TJnTsTJ ap Pa{ff{trnH

torl i: o: :otJ* ;[: l]=*

[' [oo.l

: ltms areo?,gzurdealoe
p'.u(BpA+BI}jrETTuf T"TP-r'epTnapco1u1e}elrIa3ls3TPpreouBTls3T3sT'urlcn|raTTEJTx?fltnreds|efTfios
eTeaTrl€!{ 'z
rTrozTATp

'arp?s ap IpJ$ 8'6TJ

fi

t'qnlc1Tacc1a-fr,1pectaaTT' TuxTnpTeJJTPJ?usTePrppJ?osTuloceTt lelJeuoc ov TacTrlpu efoJoqJ€ pUEAP

TaoTrl€E Te era?s ap 1nler6 -
i (IrnTOTc) [rv]
Tt (aroqxE) ['V] roTeeTrlEr eTE erP?E ap aoTT'lnliP:6 eartrn:letloJ -

Itt'ovtt{6e} toJ l=,s

I tovt uJ I eTelsaJE EareuoTlTlrEd -

EuLroJ
'-v TecTlxeu ea:EuT[toleP -
:fsJ?se a1Ba1nr1suo3 ea ov
goTXETral3€Jpo eeaTrlBi pUEAB r€'TuTf reTnpou

?er3sTp lna?sTs Tnun f€ aJt?6 ep InI€r6 '1r11nzer Tlr?6aae PzEq ed

'"9 1ac1r1eu
T€ TnU aTeJoqrg nt ]rolmzT aroqrP Tnun PuTCEpEJ a:IFA 'v pacT:?BU

:ll lltllll$ M0lltl$ 9'- LT tHtt$ts Is [IIftlll3'[lvlFt$

t(?.1p:'r :: i1l ri! ${ iit$ili!.t lN

i \at::: ' i;\t'i:'-ii?Pri f \\tJtr'fl
.ikr'Tdr, ;_.t i :._I WJtj:lt"Lr
\tiYi Y,''' !:
q.-r;'t . l "rr1 ii r: .i.li .i '.;r qi tIp
i:.
r,!
qG#
l: ., -i"!' .r. . _ ,t,, _dr_
'1 1 ,r--

, ..'.:}rLt,

fld.$, u" r;,;.,: .tr:iil: $: i Vt.)ift j"tliij€41
n; ir' d: iJ.[i.ri; € i.€rffifnief.
L l!i.q! r. I ,.: ' 'i
&s,|,{'.i ""ji:r, :. i'i
U-r,,raj{.i,, -,
{7il

ii,, : ,'i

*.,it-,* ll-{t'esftt"Et" in

gi!l

*\.t::1t_. *- 'F* "ril

lf

F];{.t :: ' i ,:og-<:

!,- , ::' i. ll''i '.,i i, t'.-:t,..., .: : -!-gbt;.q;11r1.t.i, r;.f afi.i.l. de
!',i:..,'r ,.r1,,,,ir1i..1 ir*, 1*i.1.l;lte* qlg rJegllaAaf:e
.tli;.{3.i i* ':i ii
.-:i, ? ': ; ," .:.if i . ..-i! f ,.iit: **S ri*' r.rt:-gg59o111id Un*l
Ff:ir:,. ;'

3:ritij.,::r-1 :l ,- ali. t:. .i

f+;:*:i . . ' : r . jr i 1,.. '.; r+ :t*i l a*qar-g il i:*-r e exfgtd
, i, ; ir,iti?J 1.';l!.:$ ."i :r c,r3.Lsl.it:;l,y f$f h renc$j"€
CSr,l iil.il .r.:'. 'I' . ir ?r rxi.! :ii,jri:1.;.1itr-€ &Itt€r:.l.i,r adlC6

{_idali t1d:;:,:, .l

ei' 4-1dL)rl:ii.J'{ i.i,l :rr',rr,!:r-i r.;l,irl .-:.t, ,l..llltiir r-,# (eq].{t!,*l.€j eij
', ..,,. :Jf ,i :t!Hg{:r:1i1r!E'.+.lart FtiIXgltgftte
I lt,*e'L -: ,'r ; q:

e;',;.'r I iilll:.;'r',

'at€uoTsueutpTlm 8rP luTT aIETnpou
EIueTEAT(pa elsa TJTgaT U Ts T'rE:luT
arrelsTs Ttr B BaJ€lro&oJ n3 IEJuI TetrXsp
T n3 .rpTuTT JPtnpor lsf,csTp E€txsTs Tntm Bal€lrodtrc3

{8a} rtq, t["I]C3r'X*e>'+' ''+.. [+{-(u[{]-"urq]"r[:rr{]'t'[:;30]'t{*e>[:+t-(u[])i'-rurl] ttrrytl {t'e[:3if')]t=-ta)lu=lT't

< t{-ill

-= [:f -u] rrcl [:t j
a'lrlcoaTdJas?eur€raP?nE,c*urT1t*Su"r'oealrara1ppBulsoeod1JTag€pc'T1Trngl1eneuuxaeeT1srex1lus*ae"ln:pTaratanp[ruurot]c9d'1qeta"n"JcT"uJ'g"l€l*oEou'[kEuTo]qcac ep'tuAn

0-{

lzLl < Il-ulq'[{]e>= [{-u]{J:i1e f= 1u1zf

:e1se:dxa are 1S

f';piepa:a1{u1:ossabl6snlnuslnnTdsBsuNBnladTlssxgaorlnaE€rueTer?TpzuTnTnresEpTa?JnTEtl?lrpnEeuuadpTuaslulrsdnadesE€uI:Tn6ulqpoaEraeas}xsTIreunTuXaTupnespeuTrn^edalsagp:udotruou6I

HInu a$ls r4 Insund$Hu $'g

u$eJ!il"u.9ira3ofdlnpdzif*onu',lnll"re"dnrsgbpaeorreogo{c*u,trreotl1glTa1lugdresupuEee/exIT:T1enuc's1ItupLePIrof6r,zloTTlTATnETETpn.rfou?u?TPTsnuTqnn{'u:qptoruluprnIdm=sZo'nT-llurdT"lllfat(l€x7'Pla=aZ1Te--s"qP'gT'-=pTpgrlzol-z(lgI<'At}/-dT'pgqp I

'n1&re u6p TnTnlelsts BarFzTIeeU II'6T; T

P rf

_T_*

PPPPPP t.9 t.P

:arezTfeeJ sar€oryl.rn are rp+cp+P?+I 'tP*Z
EcTlsTreloPrEs asTrlEl o BTIJSep
rETilpor lex3sTp Tn3elsTS
frldux 't,p eP

TB
TJEXuaGITe TTJozTATp pugAP

(€)d9 ad ?Tt4Jep IpTI4I

"t llvnl$ ll l0Ilvl$ 8Z- LT rlilsls Is lllnollc 'ltlllts

$ntat[, 0llcurrl tI $I$ntl L7 -rn s?til0t Dx $BHltI t" $Illtil

l. Inrurrrrntr tt*rttt blnl Tinend seua de renarca anterloarE, dlscre
lnlocu
Ln cele ce uneaz[ von considera ntnal slstaele discrete noduLare
Ilnlare unldlnenslonale, cazul rultldi-nenslonal rezultAnd pe baza Transt

cazurllor unidirensionale . de unc
Rlspunsul unul astfeL de sisten se calcul"eazd cu fornula: Deten

Jztnl =p etkl nln-.kl (7{} obtlrx

Functia h[n] se nune€te functie pondere a sist@ului gi reprezlnti Rl
rispunsul acegtuia Ia sesraLul- 6[n] aplicat la nouentul n=0 in condltli
ptnnlogpiarlleetilnlluolrede{sltinairaerlztaetreo)g.iFionrvmarrllaan(ld74in1 reprezintE esentra inloct
slet€r
tjlp a sisterului
rodular: cunoagterea rispunsulul La lnrpulsul unltate pernite lmd1.r
deter:rinarea rispunsului la orice excltalie. Dernonstratia acesteL
afirna9li se bazeazd pe posl"biU-tatea scrlerii excj.tatiel in forma: n

ll (7s) 8e nu
pol1n(
e [n] -=F e tl(l 6 [n-k] slgt€r
rt"0
D
Datorlte liniaritdt,li, cun rispunsul La 6[n-hl este hln-k]
{lnvarlanla in tinp), rispunsul Ia elkl6[n-k] este elklh[n-k] €i, in I
sfirgit, rispunsul Ia sunl este suna rispunsurilor. ut1ll

Prln urnare, rispunsul in stare nuli a unul sisten dlscret nodular

llnlar (Si lnvarlant in tlnp') este convolutia dj.ntre excitatie gi

functla pondere a sistemului.

d3.5 Utilizuea transfornatei pertru deteninarea

r[spunsului in stare nnli a sisterelor lodulare liniare

invariante
Deterillnarea rispunsului ln stare nuld a sistenelor dlscrete

nodulare llniare lnvariante in tilp se bazeazi pe faptul ci
transfornata d a convolutlel a doud semnal-e este produsul

trangfornatelor d ale celor doni sennale.

Considerln din nou forna generali de descrlere a unui sistes

' tn cazul slsten€lor varlante l.n thp functia pondere este de
forna hln,kl Si nu este invarianti la translatil. Rispunaul este, 1n

aceet caz:

-vlnl =nI etkl htn, kI
9s^

'.$puod TcoTrxPs Ep 0lrrlot3utrl ' (p)H Jettu"t? eP s*7FJlil ezTrTln uT
Ert es TrTgeT elrril Tss Tt TrPrrrr elTil TPr nt rol€Flsts TnzPc l{[ ep

'[u]A TnTBUTDTTo IUF:.r5ltp 88 - rn!
i(plH Ts {p}g 9lrarcJsrnll erlurp Tnsnpord llrrllrl.p es -
P3
,[u]e TDTnTein3 B p ElEaloJ3uPrl '(P)r Pqrrerep e8 - a?

:Totler 6oEJ EA es [u]e ulfc11cre e1 plnsundqr. Ber€uf:relac E
'lar.r8TP
E luJrl erapuod TnTtrcl8Ts Pa,
paop A lrodur rm TeTtJun] B p Elrf,roJsrllJX else (p)H 'enou11od
TS
alse Tg Tn$FlsTs E xeJsu?rq ap PT{JunI alfclru ee
JE
t6l #-/;6-.r-o=(PtH
:ETpuoT{EJ sT{rord uT

'cTlrTretrJeJpc Tnflxxr;fod TfrolcrJ r4p luTporf t{
rreuqV[filqqoToeroec/aTr ErBtFilr{rpTr}E)r c}(nTpT?)sTrTBTrTenTlTtTeCrTtpT9Xr raBcpoorcr:.dF$1tT1cToeJrdolgcapmlJsueerlEl(o:p.dr/nI ulrd 'TnTnra?sTs
(sl
nc p p{ryncoTrg
Fc qarEleu Ta
a1r
{6r} tarrp.*f-fl;gp,r-j. *," Tnl
:TE{rtJ ug Fr4lqo
e1o
(8r) ef e ro 1$$,frpe- - tPlEsPv tr-wr-- (p)x TT1
P?
n/rl 'u rTr€r pPr6 ep rcutTd rm eltrt lr-wl=(p)r Inlu€uTareleo
I,L
{P}f,IIP-=(P)x(r-t?)
ezg
9lTnzer ep(m ap aJe

ler.) @lsq* (lPP))xnp=v=tP(pltlx 'Fr
@lePa+ zl

:rErA€ roTTT{$ce TE Txil€r TTqe p puturolsuerJ

{er.} luleg+ [u]:61= tul.,{
lr-uJ eg+ [1-u] :rg= [u]xl

: (I-u nc u lTnJoTuT

B-E eJEls ap eTSEnc€ (rg) dr.r? rg lmTrPAq rBT{ttI reppol larJsTp

rt IlIlflIS ll lflIltt$ 0€-LT iltIsIs I$ tIIncil0 'ItilIx$

lullH, cncull[ $I flttBlB 17-n $tilt0[ tB $[;tatt tr st

l, REnllaT ve
co
fo acest capltol egte prezentat spatiul smtaLelor diacrete
cuantlzate avind damlul rulti.rea Z a nurerelor intregl 9i codoneniul f,tu
ndtinea {0,1,...,N-1} cn l{ lntreg g1 prin. Sl in acest spatiu se poate de

d€flnl un gr'odus lntern care apare lpllclt in acllunea orlcirui Hi
op€rator llnlar asupra unul gemal dln H'. operatorulul de convolutle or
a doui scrale dlD l{' il corespunde prln transfornattr d produsu}
transforratelor d ale celor doutr seunaLe. ca

tr leqEturf, cu smalele dln X- se deflneec sisten€le dlsqrete pr
pdulare. Fona gpn€rali descriere a unui glgtesr nodular dlscret llnlar ra

este +Belnl c€
+Delnl
1fx[ny+t1n]l -exfnJ x[0] =xo dl
=orlnl

8o1u91a generali a unul astfel de aletu, conforn prlnciplului"
gtprapunarll efectelor, pr@rlu llnlarltitli, contile o parte
c-sreeFrnzitoare etirll (excltatle nuli) el o altd parte cor€apunz6toare

erclta$lei (stare nuli) :

x[n] =Aar10, .E A'-t-l'Belkl

Cmceptsle & trangfonati d ei de functle de transfer gunt utlle
in detenlnarea rlspunsului.

kalizars unul astfel de slster 1qlllcl utllizarea sunatoarelor
gl acaloarelor rodulo l{ precu gl a elaantelor de Lnt6rzlere.

* lj iiE= -:
;.1.1. ),.

r: r. :, ,lill j i !;di ;;a i-iit;"J

i | .'r'r {r, .':i. i i1 Lrlfi:..i

:.' .: : ?t!!):i i 7{:l iil-t .5i
I I ri -, i-. j i'.::i*l;';

.': , t: j 1,: r 'i I t:-{.ji,irr;}-nr.:!

.;ri, I r: I ! l i. .i j+ ,itsri:i ! 'x,Fi
'il,:- ,1 .:.+."i. i;!;itiUJ:

.:,ii :' t r -:: -!il,j ?; !ii+!$!-i{l AF
r. -t :: i ;:)ig:f iili ]?elf.{.i- J'ItJ.gt

r:'r,ai' ri jif {;}trIi]1"1{j.l
- ,.. .: ..ir : it, . i :l$f,,ji"",

.,ii .; ., :1rt'ii,lij,;i

-t r4!i:ll,i:r-a: r.:

{l:

i r.,'.

;:'

;iI'ri ri&s.rt

sftilt, ctmutl[ 5t stslilf 18-' silMt Sf StitAtt c6 stil
2.
e1" SPATIIIL D[ $EHilil,E
ci:
1.1 mfhifii, mtatii, operatii
in€
Spatlul de eelrmle Cx este spagiuL semralelsr descrise prln n-uple de

ordmate de nurere conplere {secvente de }ungfue H daflnj.te p mulg,lnea re!
""....Ux}-1s}a{usa{*uNF12,o..n.u,olg,i.xe".aiz/2cn1oinrf[ccauzualceNa=sptaar}c.a
{O, de ereaplu nultinea ri
{1,
Un seunal dln C'este o functie care se poate reprezenta prlntr*un b,
vector c-u f, cqronentea. iu

oparatia lnternd este adunarea sanal_elor dupd rei"agia:

xlnl +ylnl = [x[0] +y[0] , x[1] +y[1]

(se a&rni conpanentele de aceLagi rang). (2)
Operatia externd este innultirea cu scal_ari din C:

a.x[n] = [ax[0], ax[i-], . . .,axUv-1] I r

{se inmrJ.gegte fiecare conponenti cu scalarul respectiv).

R*rrr1. m, spefnrf, Hlf Spatiul de semnale CH ae assandnd cu
ctr acesta din urmd apar urmXtoarele
spattrul Ma. tn raport

deosebiri:

- adunarea Si innultlrea nu se nai fac aodulo X;
.-- codomenl"ul este reprezentat de nulti-mea nunerelor complexe;

prcdusul lntern este produs sealar, iar spatlul CN este sBagiu
Hilbert finit dimensional, cu toate consecln[ele cara derivx de aici,
{intre catre, esenliate sunt
gen€rarea de cXtre produsul
sca-La,r a normel gi a
netricil);
- operatcrlj.
de
depLaeare pot fl deflnigl
rodulo orice pollnon de gnad
n avdnd coeficlentil dln C.
(cea nai utlllzat6 deplasare
est€ totugi deplasarea
clrcurarx, T-._i!;"i, tfi:rttt intuitivd a unui semnal
n"oodurl"a"r ndru-1i=r0iil);
pollnonului

]lnie 'S-ar pdrea cH forna naturali de reprazentare ar fi cea de vector
onizontali ar
(ordonarea pe sugera axa tfunpulutr). Vou prefera

forna de vector csloand pentru a putea acliona in nod natural eu
operatorii" reprez*iltati de natrice
pf,trate de ordin txH.

'nTlBdB uI Tg rn6Tsep )of peAp a?Ed E6JpuoTlrrpgg z

'oJ.,ft=ll lBc{rg TeJlse '1pp I g(ftt1 ep} Ted.rel{rF iln"-rlur
lup "1. annrolnur6e sp rm nr cT6oTcrre
EDd Tntnls Tunom TTJFTToT?u?go "S
Tn?qTnzar TJ eleod InTlds rrtp TermrE
gc lfllAf,l$t

'1uemr6re
'p 'c iproTod 'q
Tg Tnpa u1rd 'p 3greu16w1 eped 16 pTper elred uTild
!ep1{eds TmpoH 'Z'6TJ
:rc rtp fE(cs Tnun B eJBluezerdar ap

'p "**T-le u r ol

' ly'lvlz- usl-r

I,t#{ ,tux1rrrrl
t(luFFre

,t"--:i

(luhletr

-ilubt}eu

0su Z=rJ

sp Trnpoc 'Z'bTa 'Fu.Irpa?TcnJre46cns€lTtrlnnsTosnJuoucTp"gfe$uere1s{eT1naurnos:r aJeluszerdaJ
eupberg eu s nrluad FITIn
elTV 'plssaou sp
as unc eg€'ruprfTTJ Tnun TJi pA euTbBtrT
lpDal 'Tg PJTTJTJ EeJpspldop e{rg:egiuncrlo
ep TeJxs€
o 'f'6Tr uI FIBJe
ed eleseld (Tqtrpr acTro puenp) auEoTluBge uFp plsuoJ .3 1nt{eds
rfp (TEar) TEuEes Tnrm p p^T?TnluT eu16eq o f,Jlrtug Iug1'EZtUdtU "2,

13 lltll$ il uttta$ n-87 Iulsts ts lltnnm'twnls

5pft11$t Af S[hFAt[ Cil 5Ell$r I i

q.,. :rl..t:l ': 2. n
u
. O:' i.,:, :;1.
!"111bq
,i:r: i:];1ri- {:fr:l€ec j_n}e al*c faptulLlt c;
funcl
l1i. ! , , 'i t , l. ;* ElSf t fa!! produsul sealax3

1. \..i:: €$1-e
il"i
: t:. ii-ti3. 1 ,f- +r- 1,.-
-, :.,
{3}
w I ,.. '
VF:{ i: i-r i i..i , '-r'i^r --qfIi'lgLf*!-,i1*ir"e*r*iir'' (4) r
f-":ii F,!jLf: {5} l-fl$ill r
't;i...!:r'1,:r] i
C*i];il-,ia;': r'.lri j r,
s4t!ii. ii.f ,i i'.. ' :rrrj'v*lfrJ.}*
i'r: :l

. ,'i-- L;1-1 *ri,1 !? {e) ,

',,:.!,r:*;: xinJ ::e$pecti.v yIn] ln L

,., i'r::,i:i:.Lc:r ti"ni.ara, panlr:l orice ha,za
:tfi,f| i:"'- I
i:., ,: "ir:c',,'ii r,';-itF irieqalitatea Lui
Lrk I i1 _
r , ": liui .ft' 1 !l.y..trrti=j-: l!-? t6) s
" 1 q--'.
{:0ns i-
.$:F 'ts"r t.';'an s
tr;+.2+:
l'i: '.'r' i:t"ra:lr-* fi UOr$at Sl ln fApOrt gtt
?" L
Jgr
d
lxinl l,=f, iNin)
:, Iii:'," (?) O::TL}]
Q.t: Lc!3
I i.. -'i
1". l t-1 {lu al

i:1-)iue i:idusd de pr:odusul scal-ar, bnze'.
r:alx:u'-t r:u + fiorffi5 este ccnvergent

r,.r.,::.i..: 1'r. 'r1i,:1,ir;ii;o.!;:i-tl4pllerlanj-bl-uful.-le=zleid$fce(apnrt-e,uuVptLarmlu"eOlaszfiaEleu,
,:: t;.t 'l .:;;,1;;:;.

. ,.,:.: :., I
:: ;': :'i . i.!ir'1;A .l&3"e se f,ace sunarea.
:r:.. ::i. .i.,ri +: conittzil lntre ceXe doux
s"Lst-e

pent-r

"TnTru?3ads Ttlguitn:.!i.sp E eji€o'lia]Trt e€JE]"r]TTdt{Ts ru?u6d
rlTlETd Tnl€Jd,r e?se slsB9€ ia?nJsotllroal-I ztl nJ TTt€tr3€ uu ap we?sTs

Tnrm €grEATozaJ qITSB3SU €psxdToeJ T6Zeq JOTTJaISeA €€JSUTII.ISIAC a

(zr ) - ( '[&u]srB$'Jt*r:rag"r11={T{$erli>rpy={] ar<Sl[ra'rX][urm.Sl{gQ'J[s2tIte?]="€0F> ?ung'atpsJTp T€z€q

TTro?)eA m TTeuoTt:odo:d Sune EsoJdTs€-I Teupq TTJt)134J\ al{,ITAnc alfE nJ

{II } rrg* a{g ur$et- {r*,r0> 1 rqe*r6

:aTTTl€TsJ pug?sTxa'g1euo6o1:o

ee tr€ elsa gro:d1cer ezr:q zE> ?sal:€ uS '{g?euiou$lJo TS nui g1euo6o1:o
else Ez€q
aJer u! eTsse eprt?T!,suoJ Tg TnTrulceds e aJeuTulJelep
ep rTqe.Ion€J zer un glffio*x$uo vzts ;r[tulffids Tuanndtpo5.||0 '{;

{ITp 8?Ttl?TXsuoo eTT}nTT €]qi 'e?sSn{uc; paao:d1;>a: Tez€q
satl?sls TE]B} TJEuLro}s{l€J"+
"TTaArTmEUEIruALSm $T ?nuTlqo t€ulE€s '*3 utp Tsxlr$6s uit 1&1. ?ErapTsuot
11 aleod plep Ez€q o rtp ?:odpJ uT Tu{iffias Tnun {P [:t]x In:lradsl
'{*"g=<[u]rO'Iu]t0> E"F{ET€J efe}sT+es arus} *H}oJ<l1ro: ezuq pUTTJ Iu]'tg

(or l < [u]rr [{.rJ{g}= [-]j jtr na eugiqo a'

:€{nin:f}J [q]X
1n:1cadg ' aXeXdope TEz€q g'+€TposE "{.-$t " ' T 'i}=4 ' {"0} Pao.rdTs€J €zeq
alseouno es eJEp n1&rys TEux llntu ss€+ es 1r"1n-:1;ads pa,rpurtlula?Gfl
'1n1n:1cads '{6}
E aJsuTu[ra1Bp ap ?€p"gfdffii3 TBtil TaP trnzE] €?6€ plsaJv
lgfenca ap Tn{runa1sTs EaJeATouBJ ijTld Ht!$tu.r€?ap ss [ryix InJ?pad$

(6) T-iI''" "I00=F < [ffi{S' [fi] tS] lrf li ks { lff]x' lry] t$>

;[*
r- af

'nd"r6?)ioedla€lfe, tumIlsTstll{lh; lJii8.r)dsegElseTr€eur }JeJe€I€la;ps : eux{qo

as TS I*S""'I'0*T el$a{1nuug
es nJ?u6d
Ezpq ap TeJls€ 0

f-,dd

(81 ['r]rySiifj{;i- lrr.ix
i -$r
: pm-Isl

qns sTun pou uI aTJ3s an T€tiues 8r'rJer T3UI1f,€ e..Ie30.l(80 EZ€q 0 atsF
I-l{'""'oI'0=:t '{tO} EppU 'ezaq o aTil:iT-4.suoa G-IEJ aTeurrres ap a1{ounJ
u3 ,p utp T€uutas TnfusTJo €a.rFT"I3* ?Tufitsd e{;} T€ufiTsiuarn-"ip ITuTJ ?JtsqTTH

1np1{eds a11dg1a1:do-:4 rffirtrddv6 K:{W€ iA'rWn:UaS XUrHnd}IO3ss0'7,

*t lllffls l{ liltlvds '-*? 1" lHu{r5 I5 1l:fl:}ltIl 'llVH$]5

s[ffi][, tgH{&f9$ $; 3{$Ttg{ .Ef,t spritsr $t sf*titt c{

-&. qJ- f-r

Un eimenl xqnl a3 spatiuhli se ec.rie: silHl
g'3"
-$',f,Inl :.* a-"L #sr!al-,l 1.3
rI'n] . sl'z] :&, lril ",5- xlklj*r^";'t-irl rIk]E*[aJ (lc]
F H. bazE
8au
iar xf kj, {ipq}ci:.}.:'iJl- .i.i"r.i xJl; I in rftErcrt fi.i haza ,*" In Jse obf.ine cu uzua

forstr,:la: c*p

tF1 d l rtl (ort

,,.e] =c8*inl,:i[.rl' =n$*- S;i;:i si",1 suba
JIE iJ
eucl
Ferechstt, xigl i <*> XIk] cores;plsic6te transforn$r$i pref;1ae.te de dete
*.mnalel"* *:rtrE*:r*.1e E}*{nj. .?n lit-e:ratur& se i-rrt.*1"*esc gi al,te pnodurl
de d*fi"nire $, i.::r-rilstl*r,xF.r'i3or, in fu;:u::!i* cl* re.laLi_ile prri-l care srmt oh
leg*te b;:xa d:i"r-'e*t& d* see reci-pr.r:r::X;
gen€
xinJ"+r i_'i;xlfrr$*i;;i eete
ace]
br,# Fq! { 3"5} subs
it.1 suhs
trlkj. -LF *;t,ii x{;rj resF
V,^e-M fl.,t
norl
xinl *F.fd- i1 ririd'*[n, { 16}
sr6
.r l,kl -;f,l; $i 1;ri x trrl lar

unde XIk] l.i,.,'i:i! '1r"-iiti el:!-f *:r6 p*l;tm f;i.-q*al:e al,t.er:nat,iu6 pri.n faetorl evic
de scarh i..'*l;.,:::.lr: illilTlgrif* rli,n fasa surn*J.or sunt, in tr,ltlmrl caz"
de iLii;j gilsau de S*[n]. ixlr
abeorblte
poat
1. Consw,lrsslnr", ffi{}&}sr[ {u***o ;,ur Fnnsmvani tu ca:ur unet baze
orto.g*i:a_1";i Sflr

f;xl,rj i"-(.,rl.r] , xlnj ]=3ftfr':*. [n] .xlnl ,=$-:f. lxtn] lr= t 171 CrpeI

:: # t;Hi;'st&r!:;--td*::::Il;;,f",'*,[email protected] :*.;H";&lk"tfSrrki.,larl\I"l$l:f,/5'nn*_d'i.r$,*[.Itifrft"r'?r1J o6f",i*fi.n:ri]],,i\J*= aeI
ila{

acAi

ado
iucr

{-:! ldt{



smilt, 0ncuII st stsrttf lf5 * $Fnlt0t 0t SIHHALI flF

- Subspatli de sennale invariante Ia aetiunea unor o1)€ratori de
deplasare cl-rcnlari: semalale dtn aceste subepatii vor fi denunite
perlodlce (perloada trebtrle, evident s[ fie divizor aI lui tl).

- SubEpatil de seonale invaniante la aetlunea unor op€r,&tori de
prolectle de tlp fereastri" Sunt aubspatii contin&nd seornal"e care au
valorl nule pentrt anunlte ranguri ale varlabllet indepe:rdente n. Un
sffiral partic-ular este se$nalul pentru care val"orlle difex:^t.* de zers
sunt egale cu unltatea. Produsul aeestui sennal- cu orlcare alt semnal
din spatiul respectiv lasi lnvariant eemnalul deoarece cele doud
s€rrnle au valorl nule pentru o aceeagt subnul!.lne a donenlulul de
defhltie, iar in af&ra acestei submil"tlmi seumalul cu care se faee
produsul are valorl egale cu unitatea.

#2. BluE $I TAN$FoRltfiru RKHARCABITX ffi s
l{
Orlco baz[ din C" contine F veetorl l3ti:"er i-ndependen{.1 . Vectorll a
unel astfel de baze s€ pot nota eu S*[ni sxu q;u $[n,i]. Frin urtnarer
ei pot fl precizatl printr*o matriee Sx'll . Deterxj.-nsrea spectrului unut
aantal revine, in ulti-ne inrtanNX, Ia determlnaraa rezultatulul
rult1pllcerll vectorului smnal ru $ trel!:l"ce, r*zultatul flind un alt
vector care apartlne tot spa$truluj" fx" Frln urfiars, det"erruinarea
spectrulul reprezint5 in fond aegj-tmea unui ogr*rateir partlcular asupra
eennalulul lnlglal, rezultatul fi"i,nd un clt smrral" h coneluzieo
deterninarea coeflclentilor dezvsl"tErii unili eesmal ln raport cu o bazd
@recare egte echivalenth crr p transf*rnnare eare duee sffitalul in

epectnrJ" s6u6.

fui cele ce urneazi vcr fl prezentate hazel"e remarsabi.le dln spa&iul"
de s€mnale Cn, UneLe dlntre accste baze sunt reale,iar altele ecmplexe.
tt eazul bazelor reel€, dezvoltare* s€ffiralel$r reale va conduce la
spectre reale, iar dezvoltarea seffiralelor compJ"exs va fl eshlvalentl
cu dezvoltarea indapendentA a p6rlilor real6 gi" Lmaginar6, spectrel"e
ftlnd deslgur coryIexe. tn eazul" baaelor ccmplaxeo spectrele sunt in
general. coqrlexe :indlferent rlacE serfinaleLe sunt. reale sau eornpiexe

{spectre reale vor eorasFunde unqr sennal"e et:wiexe}.

' SugerEm cititorului revederea ehesti'ux!il{!r de mr,tematic6
referltoare la proprtret6tile spagli.J-or Hil"bert gl ac{tuu*a o[3e: aL{)r.t] jr
in agtfel de spattl.