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formalisnul produsului scalar dln Lz gi fornallsmul dlstribulionals,
in continuare indicl care sugereaz5
ltom folosi atdt caracterul
nunirabll c&t Si cel nenunirabil;
actlunea seurnlficalla 1or nu conteazd. Astfel,
unui operator linlar, asupra unui element descolnpus in
prolectori, se poate scrie simboU.c astfel:

y,= Ax r= al 4* !fia c r.= a f d0r{O0l9+=1oVl',;OIofgi 0r9rrloo=r4,860ig$*ii*a,,==pSof p-, ( r o 8 )
$1s!=aJ; 6i6j=0i;

x S-au pus in eviden!6 dlferitele descompunen spectrale attt pentru
pentru y. Se renarcl operatorii identltate a ciror acliune
cdt gl
poate fl elLnlnati sau lntrodusi in funclie de necesitEli.

Deslgur cE formalismul prezentat este corect numai in situatille
3n care sunele sau lntegralele care apar in fonnule exist5 gi sunt
converEente sau pot fi interpretate in sens distrlbuliona1
corespunz6tor unor vectori ttket" particularl.
Utllitatea acestui fomallsn este legatd, in
posibilltatea de a prezenta intr-o inagine unificati pri:nu1 rdnd, de
a eiror esenfX este proprietatea de ltnjarjtate" diverse
procedee

y r* Atrr= alxr=a f O{c, =a J O{O3+ =

= t)rt{.J69.x"v =t.e{a . = ofeo ^!trt{a6iJ9.rat = o{t0-i ,- (10s)
J
afO{=E{*tJ=Srr6i

aio{=g:' - f{oi+{=Vj

pJs"ucutreitemp*ocsaapFteuounrnamecp:atrciolo,udnaucacsitntisuncnvaveereacactorut.ornutrenll'tr"r5dk-roeiustt'ttftro(ierdbmsiustuetrtrai"lirbneauasntiuieepcrrt)iragnunuirnanotiaeetsl,'eifcxuudonatalrcct"ilini,tiL,tueaistsiuvtp5rsa,e

ZI tnINSCnS /373 elumuas ap InllBds '{z
apuwes ap 'tz
*Nallsr(.eilrnot ?IJos 1nIIBdS
rzriler prlpoEry '02
*r51J eleuuras 3p F{ledg
FpmJoifsuan rf szug op "6I
aruuas FIFds
"8I
*r eteuwas ep Inlleds

2w epumas ep tnlFds .LT

*lt arcuwos ap Inllsds 9I

,w aFuuas sp Boqltnl I '9I

alorcsrp etuelsls rs eleuules

*-il[][ e vgJltilvd

uostzug 3i

lsznBc eappldums q6o nquoc u1 itmrcnJ :J'

sp elclorruA g{ugul o s}so gfEiordns e1 tt
!ryinp qneT
'glql* UIP plncsptl ec4crd

no rodp *nmluoc r* alun$ 1-urr4 ?zerp'n15
fislon[ 'r9

'efwr$au rycep

gn4gst rBur epo ptry$ edumcs tcril a€J

st*ftit {, fiB{tjtTt 5i $l5Ifgf 15-' fiuiltNtt 0t stixttt t1 stiilti

M{JLTTMEA &H SEpfNALE Mr un as
va 1o
I&iltrj0ftrA Sf s€J{s*ee. F,l J
;Pe"tm:nn-logrr e d*f, inJ-{it, natafiii aenna
1 slnEu
Strurtu:-$ri pos:b-z"le aJe ffilti.nii Mlt *""r*gii'
opsrae4).rl Sl (0,y)
valoa
cirw$ s'&ss#"sya. frE TEORZA EUMEP#,["$R 4
Teat:amt* ale *rp#rflrJ-:l nnmerelo-n intregl dacd
trssrema restuju_r c&-lneeas* 7
Filnrfla llri guJE{, perJoadE, rdddcinl prinitive de se
"sPSm"fttfrrfunf crf:$rff*fvru*gns.Era!a.{lIae[AtLn' Egut-flttci-npeiirat#or-1I.n 7
J.rcrar: pe # el- ar
9
cazul genert-! 15 repre
,semnaJe ld zl , Rtt', C' 16
RESUNAT 17 dintr
18
2L valoa
22 pe to

di-n tn aeest capltol prezentdm cea nal efunpli mul[lme de semnaie utr],5 2. 11

punct de vedera conceptual: muJ"Sjmea sernnalelor rJiscreta crrantlzate Una d
dfedxsadaoveeiocnfedcullorllanreerpmdttfnuaHlpe&edn,,nronecllmlnuttluruel$e,nal.,c.nhofi$P$f'.(sinueqsrerlv$edsisanenczv&n,eeao$rrseElnnfailaifnatertgieinrttdetrssieeaatleou.gssaeLrltmttfllfdeeetaedeLlnllnlpt*n.todrrtrmrrlaeie-esvnrafperiilmsrlnrsedsmiliertrreaesgetmnsl*dpnatinent&eglMa:plnucl=eeodrein{l*rll0eofecfd,el,ereeooL"rpmhmEnritien.enenL.dantn.d,eifa,,uilMccenlni,aaeiu*ristrLnpaeete}or,cr,mrior,peLnrdopeleeaodepsmprsnotreealpufbouznembiemgicrulefnnruLurrltrlenr6-lLelldnoiostaiadenc6rI
ucsastnlrtnoueerctmtJ6u"uerglaq.l&sliit*nnu1rlfadaccepuelslcuaeoglmnicenmxeaoplpedte.r,olcpnurinepo-ate,srctguictlleeu" laaucrteiilsl*ttaesti epmaaulli!sr1rtrluniHd€i$ue,lucvalornreurlesgelgmrp1so1laltttae1 nultL

1. ffitr&TMEA $E $El{lW,E Hl multl:

3.tr Ternino3cgie, definilii, notagil domen

mcs1o1.arnetg$sdCunpureaonentllcdnceeeimnrn"ere[iaepns.Trt €t}cir{nzepeetl"{ldit,lontsausaBtellne$pisn.Erulien"{gnRpnitlttrueu-No:nseedsfrmuoelnemnngnaceuinllnrlt{eil.cnem)orleln,emcsEsceiotdunedentorsisean€ectmm,ncnndlouaaadL]l"osueuJigggLecujen.aridu,rlet.ecrtagEearvenrnieadolndeeaundJntet sing

eodoc
denrs

l.e co

Etreste

1n ca.:
este

unai-

si sli

numer

aldln
multl

(cupr
11,

gemna

prlvl

mnunletrr

i'i,'

silntl, clnct,llt sl s;slffig 15*, mtllt[t 0[ srilAtt il silt

nultixe care poate fi,de examplu,nulflnea {0,.. .,t{-1} sau orice nultlme 5.
lzomorf6 cu aceasta) reprezantAnd valoarea se$na1ulul {Fig.1.b}"
s6nnal€1s sunt dlscrete cuantlzate decl vor fl denunlte d1glt&}e I1,t i
(lndlferent de valoaroa Lui M).
I
3. CmrtsnTTr DE srxBol,rzaRF Pentru slnbollzarea unui semnal din xl
vm uttllza oricare dintre urnf,toarele notatii: tu-
f) sln]5, cu n=0, (notatrie speclfic6 semnaleLor discrete)
e.i((
2) s[0], L'11
a5{
3i {o,a), aeil nFX
Pal
4) [a],a€x
s€t
Ultlna notat.ie va fi folosit& ce1 nai rult, ea sugerAnd repnez*ntarea
t1l
unui semnal dlscret prlntr:un vector (matrice col.oand);
ffi.1
s[n] * s[o {1}
s[1 raJ

"t: Pet

av&nd, in acest caz, un slngur eLement, s[0]. Vonn putea scrie agtfel: tri
s[n]=s[Q]*[6] dasl va]oarea eemnalulut este agH" Prj"n urmare, fit
Xl={[0], It], [?],". ., fM-tr]]
d€
4. &sc***Trr (1) Hullirnea de sennale Xl este, deslgur, foarte
slmpl&: 3.n esenf,5, ea constl dln M nultere privite ca samnale. Degi
acest, gru*ct de vedere poate fi conslderat o conpl.lcatie 1nut1l6,
este lxportant s& f,acem distlnctie incd de ps acun i,ntre semnale
nu$er€. i-uAnd in conslderare "dou[ fe]-url de.nunere", von put€a
Eii punem tn avldentf,, chiar ln
mullirnea il1, concepte tlplce dln
s6
teoria sem:rrtl-e1or: ortogonaU.tate, spactre, operatori, aslgur6nd
trat[rii un punct de eedere unitar.
{2} Consfdar&rn ed cj-titorul a acceptat punctul de vedere conform
cdrula semnalele sunt .FrodeJe natsratlce ale semralelor fi-zi*e care
retin numal inforrngia uti).X pentru o aplicatle dat5; in acest caz
gunt excluse pozltll de tlpul "sema1ele nu au corespondent flzlc"

eete acelagi" numdr natural cuprlns Lntre 0 gi M-1.

5 sltnnolut n s$te consacrat pentru a reprezenta argumentul

se4nalelor dlgcrete.

'TFeJlul eJaunu ep

eBuTctrzrfT{eTstBrn3Ju3dcTrGf?&nscJa?latro?jBnpcsrs,lolsoudenaaBnJluoucn6srcgBJ?ssTsqEndseuTEErlrmlF€eFerpeTXrneoJ?nRrTEunlU:ledlppcpTpeaTp}l1Tsap€TuToTTCEf TRau?sTuupTJtlr

nTdFdE pp F?T/\T:d €TBrxFs ep EecTllnr ElTUIlep e?sa erec e?sd
Tndqs rrTp r€I"Os m T€ fp(nBs un arltrtp snp)f,d :lrlre?xe eTfprdo -

: (B-rEoqEuunds6,roc eTT-rr.lJnrls snporluT m-s erpJ uf eTTT{TpuoJ
{rg} Tmpc€ Ter? ug p?T^T:d TJ s??d sJeuroJsueJ? gspecv 'ersr
Tts Tx.Q { Bp TH{rnBs un puB?Tnzo.I ',{ E?us?suo3 o na (eTPuraB ep aJpcareo
nTlEds Tnffi prjEuTlr€,ie) [u]s TEwr€r6 Tnun Tn8npoJd aTniTlsuoc I! cTdTl
T:npoil uT ElT^Trd T] a?pod Tpures lTE ur TBuuBs
f,nldmlrd'a1T;rBsTp
g?Tmue o FJ Tnldel leTurlqns .Iep 'aITtFlTTTqTsod
sTnun a-rgs[:rosEErFr"]
1tg uFnT nll'€6[Jd g?Ttrr]ap e16a BrEc ue?sad$
81eo? e.rlr:epTsl,ts3
aBuCT{T'rJuOfrTH?lTTesuoEfafsi
E:nlgBeT ug eurexxe 11{erado TuTJep -tod es 'Beua[asE

{*T{"Ts-rcr 'e:lnloauoc 'lnsnpo:d 'pura} suJa?ur
;a*Tc'l'eic-.Treea?cTIlorFl'aE{fBfTiTfpinBpT3TTns-TwpJotoTEe[rul!:T]IBeJpEB?1JBToTrdnoxecEsn€fer,plTs?€TuTorr^lBzsespalPpnoJaldutra${detIrnPpulueuToe1-suJalesupef)a'naldlwdlpgu{ua1oxnaau
o s?ss {'Fx$?exdo nBs TTlpredo n3 B.rolsaop eaf,E:?sezuI) eTprruBg
ap roTrw.r'lTr'tr n€r*-rn?rllJ?S f,TnNuflL)filiJs of,LLfud l3vJ.r1{utro "T

TJoJPJado

g$ pgflex#o irH TTtTlpr alp ailqrsd TrgrnXcul$ Z'l

'1a1{eu.:og:uT e psTz-nTrdord

h?g-vtset€A xE$p t3 'legldeoar tr6 TeTsTmo eg{g111epou TG aTal?Jnp
'elesueuos F?rrpxrcxlilT ?rms nu :e-redp nu dugl TnJolcBJ 'nTduBXa
isBse rJ'I ',€,a mo:eueq 'gJT?Jerd u! EunpaploluT gTdurglrr1 es nu eJcJ

n-r-"rlT "s3B:iITuapT T!lJ.:r:l€-redo aT; ps 1n:o1e.redo ptTpp slueupJ? Tat

nnTnglETFsT$offir€r$yiilsl,{qseu3Tur,-!r{o1r1IpdacT€aJuTnn?Ts€uunE€osJnep3alTreJcop.roaleprea<tlrsoeTn'rtnm61esaeuonl{'rtet{
rJT-rd '3i.']p!€atlJs arBpaa ap lcund uTp 'glpTapoE elsa ee '{saJe?uT
sp aTaTBT$eaB a?s{}? @p!.rTrdno B n:lued eJpu ap lusTeTJns

qs stuq8J? hi) rX uTt! T€uutBs Tnrm TT:B?TEEupJ? epundssJoo .roldac"TaJrl

{sn €{ :rG?**lTHB ffir rtr ap *6e:1uI rPtrnu Tn(m E€JalTususJ,l nffd'E(]l 's

'{T-H'" "f '0} paqlTnu 'fnTuauopo3 JpT f PTpaJ

strR p?ie$:! TnTuasnop ptiBAE elEzTltrenc aclbor€tfe JoTeTPtms eBurydTnu

ni 'eicr_:'5e arXul 'pJ:ouclzT a?sa rn oT"uras ep Be[FlTnH (€)
"J?e lalp?TTpe.I uI F?sTxe nu! nEs

qt tlus?$ 1{ YtHii16l N.ET lnlsts I5 llttuill 'tlnt]s

$:gt E!:!ii{* ttf sl[ilill tl

{:,. ::,! i;iil4i.r: li-ilt:i:e cftr€ unul
i: .i:: qi jits:i1ir.l 1_1j i in aCeSt
t: ,: r t- +i"':: i {i ti;] f ! e def" initfi

i r .'': ',-Ir! Ftiilr.lill i.n alt
+:!r"l :-i.p.l- j i:ate Cu
{.,'1, .'..
o f .i. r-:on$idel ati
r; a' :i i' 'r {r Tt. pri?let ele
yir,'t i ej iirt j e ?.I1tre
.:
i:: ri *l+rin !il'"ne. a riOue
:,' /. , .i.,..rtr;l Strmrra,]-e
!.: :
,,,1,-i rt i'*jqr.t pll.f $1
:' t'S,t j ,tO"
., . :. j-, 'l1r i i? S
5.i-i1r1 i..n;,L L
,..i'.,i' .. .; | ". 1.rc,l- r1 ib{,
ABU'.., t
rapi::I i.. i1: {rq-,r n.:1

&C4:g:.,i. {?)
SSlPilaI I I
ij
j
irr .t ij'i l iti ':"iji s mu t.i.me;r !11 este
4. ;;
iaiFhl {3)
':i'
. ".' lr,j'-:.vt'f i {4}

ivrriiii:;:l tr{r}ii:.req"^i, tr1r'sd!}Elr,I1 m()du]o M.
jfr''; 16 1rr1'- p1*4 sd fie iftehlsl in
i L ii. s;l x:*r.ui..Ile r*'!€_l ., e onfOfm
'i:ri:g":;ejj,-r"en mil
aJntr''1 Ei !tr.l ,:Af e J"ucren Sunt
e..l i,j ' ,;i- i. gi. ri.perfi"t,j,:L exter:ne) "

ria

j illu:iii:ij 1j: -,1 , este
:*,',^xlr',:1 !3!.[l t]..1ft4?e.

!:r .'.,.r .i iX.*rrit):

:€umus 16 1e:11dtuoc 1eu
?.saJE uI '(nu),{9 Ez€alou
eoal as (Tua)TJssuas?TxaoTFerf"|lpTpncTn"ud1ue'gaJ:1Pf1ruunuuTfpzJcoc ap BTsualxe alseunu as

€s Tg
aaq1n)seestEsfaiaoeJElueJmonoXupJganTXJnlsszapRJsooufTuI:IPE1zT€TE9O31es1JqdT1n'sAdoTud?gaeo?u'sJwaogIaurTdTeurnguTrua?nsTua?pEtn1nlcfoed[ugsIffE,aEupArrPUoUX]IJgeoSpJ g '9

BcBp 'rETncTlJ€d utr 'T€TJol3aA ntr{eds suTAep TnTnpo{ 'ugrd elsa I
11a1rd
'(q) Inpou ec Tl eleod (P) TeuT un 'Z'6Td

'ri ,,;^.
// \\

{o'o'1-r'r'-'r'ol
leurqul ;gm:do moP| 1au1

Ba*s*xa ape*ad* n ,*, fl..uftil?;)
\'*/
e61glug ailerada e t-*"?'m-f-ifIs'"l'l
&*q n:ad dsrtrl StSsR

lndnr6 g{ueplae '"ro1;;r'sT€3s Te s"goTES (lnnd)gugoc'txT1geIito{ u€TaEuleuIeds Bp
$g T€J1rE es-nprgund { as

^TTn1TT4RpedB 'zBc 1s6D€ uT 'aJeu.m uTJd '(H),{9 ?e?ou eTtr Es e?gaTnu$Tqo eB
TS sToTe^J duryo alSaruru as url:d H Tnun JolPzundsExoa XTUTtr IndEgC
eocTai'draeeropc€fxlpneu(ndu3gryta'xpau?eesTsmtdsJladendeo:6run) sops1lTJ:o3ird?leodJauAugTcn;Tl{TeeudsssxAaAeaLuprTTAqaapJJ€fon'TTpTnoJurEeTTen€f;ngsl}ufTaunuT{aaleuTTpa
{t ee trs lsnsug T€ ed 183€p TJozTATp aI€ nu) ETrd
,H eouFllnu e:res uI tftzer uI gfift&flts flg trTLEds H3 qfluvlfiftlJnu,l,g '9
igwtu e?sa !{

'(dugc else

roTTJ€TEos TnTeuT) ltrTJd e1se ri 'Rlduaxe ap 'geep TeTJo?*aA nTlBdB alsa
raneITplrasuuTlaoTJlr{{€eepFparu:uTT{u{u{s"TJ!Iumls-T'fpueo3TJaTeTlnspJaTeaapnTAdpFdoJdammsnp.RiTl:l6oncu}TwlT€alnTud€pTpaoerruurn€Ti'*u$lpsHnaBprt3g€pmpu€€JmaAPus?LollIspl:oogaJAFuuXliot'aT{J{UlZgx''T{6aa}HT$1,aBI'dt1}ta;{l1e}dlss:naEeerdnlpaolHluuoPT1ua?p{qETTToreanu}Tpa[eTgPausseuurloneaasdp
{9} IPOPI s (€CIP '8) = (e '0) 'e= [P] 'e

tt ilf;c]s !g tllltlfft o-sr 15lt$t$ I5 IilllSlil '1iiltfl]s

$tffg$i.i, fjrf:::i .rr i.r:i,i ;l}rlratfi 0t $[iilf;tI ;l $tiltLt,

. "!,,:: ,:-.::rrtrrc].li.S. f :..ec&r'u:r €l*mFl;-. r di^fi )GF iF,lil rm pcl.i'nom r(x) Pr
-.i.' . ::,i:iti-::;.i.:iir1: r:,-I-r ilFlp] ag1,:f*j- ktt:At I'{*t;",9t
-, i::.r: .,i'..j{!F?.* ,jis ;rt:-;.i,tJr:ffi twtsttvt.i";*r *rtrile 91 ireduct.lbll pix} de pozlti
i;;',rri ii,.r.,-:,'i=irj'-::i**ii iil chmptil GFin) {s"rc dem*nstreaz6 cd exlstE
.-i-:iiir.l-,::l,l-r.a':: ,i..t B:li..l!:,.r. pO1.1..nrL6si g.i. g* rief Lne$te S,frfrilt,*Ul a dOU* ut111z
:i.e poate

*i"#t*t:1 ;: | , s.t " ., i', r;Firc'l i '.-;x iri-li::i *l-em,rgntr.iJ. r: efi.rr"r-la if Nu
*iri 1+:.;t):liir-!i: i, I i : rr. ;li:;:!.i .i:1 i :l ! i-; i.W ) ttd I I i I "
dacl d
;, :.':,i"- ;,,' ,:,i,..l"ui.E ill$ YE{}&T& F{{WER. :,LIR Intre

$-t-uiit,il], :ii+,r;';$!:;i..ei.',i tii-.* muitfun'r;: *{' se lr&ze*"[,'i ps t"ltili-zarefl ilnor ce dou
*i.*?: L.Li(1;;::j.1Y- ij$.|- i:j tre pfgeg;:tflUt Sp. (4,27 )
f8!.1it-1 :,ei i.r i.i' ,.; na{gl:3. :1}][lI li{) i:*
B. Cr,
(:C.gj ;,' l-.rii,:. ::r il
Rc:
?,; :t*q:rr#i: iirir, 1rup$'rgLrit nuwryr*i*r ?lltr*qi
este (
l,'!!i 'i i:..:.i.:, r:. f:fi.-.;;trj jj i;ii'{a"-l 6i. :1,.'Stri-,:.;.i j e ':-:,tl: fgg, +Xj"Std .O
i ,':,: :itir-1"*.11i ,; S j ; ll,u,it-f e.l ,tlrr.:l!t;: echlva
g - i,lr,r;"if.r:i'r .l
::.: t"rq 4 :_ iJ:{f': ri:. lnr
;ir4:i: r:,r:I:ii" .l ::.!r3r.r i j"j-'i j. j.u:, .* .l ,i il -'ii:;: i $sir!! ,r'$,Ef tiI *+rl .r'ee-l.dtltlJ tsto,1

Ufit':t' r'1 '::,:,. i: ..i ..i.!!ir'" ii; r.."tjii-"ir';. ;; :- Lr:j. r*. ,!* r:i" j..3-1zeaz6 notaliil* nultln
ii.ir; :-t .;&';".i".i.l.elr ;1 rij.'ritls pS ri pasgiectfv r.J divj.de pe a"
}.u..i. $ srodulo
"n1i.:,:1. ,.:i:,t.;:rri. iii:rl*.r,:;li. qi:--nr:acg'i g::r:1, 1. t"i1 4d *,gte mwlar prim, lnch1s
modulo
blrt, q:* , n . rrirr r':i.r{i:+i,*.; rcsrBxrfie g:i. p*{. fi t+tise ,tuh fr.tma: resp€c
devlne
S?t ;l.r1ii.,, :
fir.]6gg *'i.3 . 11,1,11, 4. Ps

*=ilT *;' {?} ll=.

J prln:

unde m, ;l:\.i;ri d, ,. ': cii'': i=l!i{!r ti h:t' .gt-ln'L nu,mgre lint.;:egi Elca$.tlvs. care r
Teoremd .f *i;r-i;tsi*:; ' ::r., r' ,.; :':]':l "i.:netlcti gtahileste f *pttll c& exBrrlmarea unui. tuturo
nrgr&f * i* ,.{.;i..:i:.:: ril'.tllinil:.r *tr fcir-sle {:J}, e.gf-.q arr;ic6"
EJr
2" Tnq*neye .l ,',=,,' i . i::.'::e n1il*tr]r€ ini:r*g1 * ,91 h diferit* rle zero,
eg.tsf& ii:i ii, .::.*|i;-ii.i-.i. d.:{i, l:qua:lt E+:,i p;e.i mar..* divi.zof {:otrliit} {emnele} prine'
a eler
al egl{:l: {'ii- i...':; i:i,riiri:i:::, ';ri,i. p:.*FU Lr.:t. .i.:1e+;
aia&"pi 3.i.iu:,!tii:,.r,.;riir,liei ilirl,i'i.er+$::te*ri;::nis.l:*:rr{:jiji;i blject
1) df
2) dl 1

Se lii.j..i ::ed; scrle

relatl

aste

EoE=u {pa-:p1nillJEd} qrm€ uzu*iu E}Tftz*q 'H>lrs--trs ru-T'H o:T*;:5txt-:i::
ae1e8?endErTec-qss '{ITuq=TlSuOadrrdrrnelo1s8Bq6rn3b€r.erlasco'Op=X{{s,rrx-taUf}Opp=a(uu**lulu :Bolp1TIB6s aTJ3s
t
n:lued pcBo

.t{<*t{:83 patrpe[1g

a1{cuny o plulzarder pppcennpscfe"p:ee{::pgTe{d1Tn[n=aunfrufrf;T'eT?]6JJp?aasue€npld$JsTnvanstterzrlyrsc{a1nlusueorToenTpsu?eguuee'ruteegTqa:dp
arelnurad o acnpo":d
"tt uTp roTaluamTe Iorn?R?
^TlpTeJ lurrs !i 1S e

Brdnse p m s-z€sTTd'flTnfl 6p 1riTrt^role;ado ea.realldu u1:d p11nz€J aJec

{8) tsOPsEI

: u1:d

olTuTJap r FXuereTe H EatrT$q{i 'HBp Ss:I?u3 un nJ {T*t{'".''I'S}=fi
uTp rB Aueur€Te sJB.)aTS lc aTnpeu eBsTTdT1Tnu ?:'e6 fryJ.{tgtgd "t

t{ sew'{{Tnm 'E?Ern?*n^z?s (€rrsri{i H?€1ou TJ s1e$d 1S Jaap euTrrap
?Tnu$gqo gnlnsnBc:d ntr1oadse.:
Tsg?sv 't{ E'f
aXlnu€lqei Tswttg TSjr'SJH'Sw$ 1"1r"3$B.t rt* eTBSa trnxs asatffix$ Unop P n aTnpou

lnsnpo:dl 1ts rww s*B?Rsep 6?T1buT€ru etr'E1te":edo ns l.rodv"t ug psTi1lu3
a?ss n peryEflTn!,{ 'e AT:}3*driisJ r€ s R} €}c}u t!r0A 6T ami] ad K oTnp()ul
sTEnTsp:sEoTrrntdT?nfgfuLSr?nBJFrTn?nlBsg€:Jo$seutrggluTT5{a5€:paXzdes**6"wr;qexJ$psu.#iat{bgepe,r$n:T'n+nuqwT*esuToTlJpJodu-tao:onz.Tlgs'l{as{a€-dHTset".:Bi'e."rno3T1Z{T;' uXnrny'0{}1$ndu
{bh+e=q I ggrl} = {suq I zlN}=P;
iEtu"T€ATrIJs
€B asatl i.rT g rtr:udug {r"d'Z il1 g{ua1u*1q*e ep a1{e1e: 0 B?s6
{r{*e } lx c"'> ildu
u"grd R+'fuT+Bp gxgp8
ETSETaU 'Zeq's TS ?i.;r-lr J*finxu iJri ggt{ "I'e H$m['W&lrI3s s6 fltrl3 "€

' ,t-=1,.? I.7=1 16eP 1=(L4'Vl
nlduara eq "aw6;d *?s'aTu*$su *;: qun* nu d1e ailug aui-rd B:ftwnu Enop et
rilrpX ug 6{8 sry.r1 eur6"ed 1S iuepl;Aa -"gurur aw;:d sJaumu Enop : aTa aJlitg
eq.:d araunu T€ as'p.td eJEs&* e.lt$T s;Tqasoap els3 uoA'I={qrp}=F tcep
ilEfrd A$?eTs.r arilHnu nes 6tB e:trug awr-td psfrurftu as q Tg p stsJaamlq
'.r64;rqu'g ; w'!E 'qu + €u = p
:gwJs; {iE sTJpS eletld
6s q TS s roTe:srtrru ls p i";piiaiis Ell 'plTang TnT- Tnu?Tro6p pugz1113n
T'onr.ErEnEunTu:lu6o?gsB{r!qBT}ESrsp€TASa*Jg?!.IETrloepJua€tftfbii€u *d eBtr*trp sf ajrps uryxeu ATXTzod
qnep s -']pmfia 'asetum ut:4

rI ntff!$ t0 v3$tilftft f"t-._{*E'}:" 1$t.r$t$ I$ liifi3xit'ltHf{l$

Ii l.i: ! ,. .i .,1 ffi;Ji ttF{A li{ liqH&ir fi1
i _i:'15
i.tir I .: .:i
, ;,1 :,.-.i ''.;,r- . :;r:i' rr .dr-..u& nffFei".'a $e fa{:e

'.. '. 11 .,,-.
;^. ii-"i.1 l'i: t,:i':n. .!.-.i. cr:nEitS in ssent$ f"l;
,j:i^r'.- '. / t:1.. - s-l ti -. 4r.ritg iugiler* ?ntreqj. :
' 'l'rt .i,'

E:::,;, ,l+;

1- "*j., rt . r,
i

i:f,:{?., i, -i.lr.,

:!

,- -.i,.r.,- ," '
):
!'-i r=:r.J-- rt - ,

* srr' .+ 'i
'i
:.t
)-',. i ! {ej

3i 1';,

$tj,, rii::.: -.r . i. Ltiiii-i. t;gst" f::.i"f e}:i.t- r,:4 ;{rx!r; $ilii,} irrsiijd{-: i:!.i. r!i!!s{t[e.l.{-r r i
:tr it'tii- $ijg .$B gif ti. grl:':iji.* $:r;!rh,{:"i. i { .i.n f *:. ti;,-,;
S.l l: .fil
{}4r

, ) i:,i: i:atr1 i i'i,;: ittreiJL;ttli * iil.{i'.i I i.:m'; i.ttiti. -i rli $ir-l1L-i ai ,3r;i.1;,1 i;14:f,13! r;;i
i.". l:"1:1 ::: r' r- :;'t,".:,'.:,i ,iti i.tli f, c*":i!..i] ii i:i.:r?i-::r:iliil;i-c I-i..r-l:.11 1 :,. i ri.".ll- I!"tlfl!fl].i
.i: ,: .i..-:... . :' . ': 'l r, ij";'rtt+:"r;l.t! ,:.tr,r:li: j,l:r,ti, ii:]A'1*11i.], ir.,: .xj{i "*itfil{l tr}ipf :'!mA eA
{:} E!,i.+'rj: rl ....' ri ri...if i.itl,1";1._., :

x'[. I -{ ;1, il i =!!';;rrrii}, li1, ci/,FH

m &!j. ii -i,l!..,i.i: l ::;r:'r'l rj;,,r9 j-:j!.il r fl* r"i" ,

" iJ-J*' 5t ':1 if '1.{t':.:i1 ia"$J*i- r}sj"ef6 d{lf.i; r-}i?,-qlsl-*: "intrnE:' .fft Si .fl

esti*,l .ri:.',i . .l:..; rj.i-i+.

' . r:2"2 ,.' r.qrj;,!} Chi.ng;*SC

"tre{:r'*r"iile .i:q$a :l :ii r--rh:ineue6{ se r'ef,e.,:* la pr:stb-i}j"t.fit,€E t'*prefie.nl&r'1J

Ltliul f[:*r-]:e€ F]-r:t.'r -: .,:;j",jr,;r.ti::.1* ireeturi:ile] 3.mp,S-l ti.r'i.j. se*st.!.iis l-r rn sat
-Sl"ri.r*tir*e:"gi1:;ri*t'i,:'rl'i+ ,'.-.i.;ili."."r,r F j,slj- lntre ei,
de :'i.:Fi*i:**ia::e a numgie-l.i:.v'

;:::-j.n re;.idui[i:i. perffiit*

cal{:iili:-: 9ri;-ilt:.i.:r':i-1- fl:.:r{!i"!se1*:: pl ,u.r:,ri$t&,{4}..'i.i:€spe{:tlv n:nglt.1l.:-l"i"gnre*
rezi-du:.tf',i.":.r'r ;,;-,i: 1rt.1..1.l?-;etr€ii "t"t:,}ili,:rri:{,,i:..it.:,', " ilo+ l: rifi r€u 1{truu La

C( Id-"!-c].r r -

RepIe:i.L''..:,,i:,j. ilL:..'i.11 I'#:i"r.-*1]ari ff:[.i.3:'i.$:-A bnC* r"irlOf t.r:lrftj"Cl, de r:a"l.rjnitr
napld ut i i .: ::.ttl-:., r"'r r:: :;.;469;.;x3"e* s$:Fj!ti:ii fi-i;.,i'

irs

!olf::.1*i iJ=:9r>.iiF: . iFlli,irll if * -=.irligi
'+ 1 i. Ti.i.;1-"4-i ;leTF"i.rl1)s
ie a1'*l*ua*g

ir. : l' ,r "1:r I'lf" tj rir "I *,.

li;i l. j !f . _:-_-l-., ?
il''-
t4t 1

' le&ftl] ,jl, $ iTl.ix"r p{}L!i

.." *!i .$ " " ' |i!tt ,:1f,1icr1ff ii;:.1T.!::{.r '3-1,P$1*Tfi uT ?,i{ " { r''1x.l *;elrlf *lx
Ij"; { :T;;nlSc+.,l ; a:l -,:,liriFF l?+ i i}FB$lI1U !}!:i
a 14 l?c:{'r$t'

{,-iv po'.*r,, ir! i

T''.e 'ir"" i,, 'l{-fl i
i.rgt,

-i€+ +. 'g BftilTP riB.SFJ!:'i iftlJ.-T4}li,, at 'F l.!r{r:f eil{!:: 1$r.rF '11. :: ii:i!{i

" t'".t{ .,::i h' '*,. .i
;
I: -r- ii: i , '1-$i !.1; . -
'\ru I :

: ar-'.1:t'tr!*-j' i!.iif, €{.irrBre":,*fi es f s ?if}isl&rlli

,. J'::t r
'l.;i

.!a: ti. r,J|t!! a
.f
i l

:.Ftiffic:;

!,i.: rl .I$n'nv.t #i-)T-T{.} :}gfig$K:X.ffi1 T . t'{f}r.s'+{ KHsitfrfl T T
tij

,, - ? ? +- t{li)tJ{ .r:l -ifi.I -r ! ' ' 'T=1 ''l ji$ ri'iA:fqrfii1$

,.

rt"i ;.: i 'ffi*v ' 'g= 1-r" ' rut-]
; iitsr tl,1 +:.lilisii[.l ?.|1z:g 1,* {r. {}liJE} i+EllIll

€.,$-- ds i:*J_..

n"j"- .il:r!:11'!ii'..iJIEl.itFp I q E"l{r;

r, | - -'- :i- :!jtr,it iri ;:if,3$1.1 '?lHsii :;
-.d 9.

gEfi$trt, *{fi{ti:}t il iitit$t ir i.. Br.!ii?$iq $s 5[8$61[ H1
+ ..,t i i

-:,r ; { 1e}
-' i 'i.'

A, n T'*{iaf.l\},f $i:j.;"'.i, :!tsr;E*rFr j\,ru,a, r,;
$CHCIm$tqATJS

!1i;.Fi;.$-t: ,,,1,i Iid?l

I

;

nu er€ factfryj. ilf,ig{i}tt!i i3i ISi"'j r-!,-li':j}-1..i.& 9.*.1'!-.,i.l r:i...5 $i6;dri'i.f-:,i$ !.ii1-i surri.
prltrs 5,ntr* el*. 1,itJ.-i j;:ii,t'ir-:i slq{:}i'1l.Lreir,J- }u: fitr,::l j..L'i, $e i:{iaf-s r:Lqt.art'in'tlie
ru? ntrmAl t- , *r;t,f e.l 1.1-r+& t. r
'''-iII;:-"'1r{-"t:,!'r..,i
j {?[i
i

ti r: di .,

e**rs-td*: Ewr !t!l-*x&r t! i ii{!-..j : $I i,,.
,t,; .l f)l"ur;,f
,ix5t,' ,.,".., * l l:: I {fJi
I
1.i:
tI"tr':tl,fu

ju$ile .r siffit. a"€eL{"}..'r'i3,* 5,rilg:rX:r:'5 .1;.i i. ,Lr il l.r rrr,,..j . &f,*gi.,* i.€jui-ilr j- g*.llir-
unifle m*dii-l,i.l H. friLg.,-ii*ev$r', iiritd,'" ;.iI j.!i
r€Elt''ezsjft1,B!:i ;iP j:*r'-.S* {,1?i i:ri x.'ri:-:;1-i:t.j. i.;., fr{:,:,i:,':d. ajr r};rj:3T-& don,E
tgrpfu*$t j-i l:r.!-1.,:'-'!: dr.:*S;*Ertt,"r€r,-tfia. T,,rd[*,,, ;:$jst-u].
$.1 '.r;j:
I ia frr,-.i r*;l,e ti*-t- rrr,lr*ai-ale L:rr $i r.or
c6el" t{-}fi r:ei.)..-:.. 11.i i"ei''r}iirrri1. $ri:tt i-i.t.;:ig}hj-1 i: g;ii.i:: Hi.;*j Aerui.lx ;A
fegtt"lril.5j tr'*h::;i* gi+ t- lr:,,j{.}:}!i!.

3,$*5,;i.-al,:r}d:'':,'.)j..'6'#':,,i',j']-..#.-: ;":u t13:itrr!.twi i!!re&r
lui *, dg $ $.!'s,*. tE-eit,.tii 5,:ii;:'5.ig.i.l',ij l-;: i,l.r''';: *ij8.i { ;i3 }

l,! ";;ilf j; I .,,."u o

fftt,le -i.e.t.g;:'* .1. J..'*ilr€j:iiilt.E t"llt .!,;l!,lei"1 ritrtr. rir lrl:! 1,;i;rr:,* ti Pr il': u*T{}nre,
pr*dus?.t.1 "s.s., ;ii* ir{:jeis +xl,}}.';*-in* 3;i l,-i,T?tia;
sig,'.'{ i.tr,}t:, L; ii'irl,1:ir-iI'It i,1;ul': i'*t ir-r;f;'i3*
5nr',' .t ' ., ,' ..,. ...., i74 )

i ieSAl-ll-eLS'e. s-ii..nN.r'e: 6 i-, r"r.:;i'i'i i.-!,iiriii .J:l.tl eri.'*iii.s-{r^ri t*rit: *} ir.l r i.
ttl nL,'l *vi.*uiiL $ j i:i-,i:il !.r*i:l'i {:l,i .i-*,j {"s- *1 s j '.:*rrL.lfi i- i &i:ast€l
co,irrni!:-i !,*!i fx*'r.,lrr: r lui l.li .
ii.lt fuiti rei.nt.1e . .*$ ', -,, r13 i.i:. i.111 ; .E?.
: I .,Ir- .:,,1,; r.,,., ,, I

ri_ii:tl. , -i:'!

rl

.r llr::.:

I iriittl4.?i

. ;, i i':::i::.

. :r.i u iit l;. ' ii" -l
.; :t l:,:::t4i! : .:.:, *1

I :. . I tr r; i :.::a. l'"._i I

,. i.:;. i : 1i

j j : l: :il :

sillilt, encuft st st$rtff 15-r= t0tllttt Dt Stttttt tl

6. 0c,8BmfrB Reztduuril€ 11 ears apar in expresla teorenel restulul
chl.nezasc pot fi exprlnate sub forra:

r#i|.l - {26}

'r=E

Astfel, exprssla tllenel restulul chlnezesc devlne:

,'r=lr r"'m-{..f1\1i 4r.",r,''lljl*, (22 7

W

tn aceast6 expresle rezlduurlle E-au expri^nat prin dezvoltErj.le
nodulo nr1: nunerele r-" reprezlnt[ coeficlen?il reprezentirtlor
nj-sre ale rezlduurllor. Aeest"e nunere se obgin prtn irnpirtlri
gucceslve ale rezlduurilor la n, gl satisfac relalla Frr(nr'. Av@

astfel:

IcJ-1 r: ymi=rJo+t j;n.1+x1{ni+. . . +rJ(o:t,r.t*it t {28}

rr=
JlF

se conrestsattni ilmimedpiaitrtdlrii"r*lueiste{rrre-rsrtou)llmInrpdlart"[j:ri,i 1u1 r, la m'

rr, E.-=, restul
lmBErgir:-l lui [(rr*rro],/mr-r:"j/m: Ia n, etc. ].1a1 observiat c5,
datorit6 restrictilfor la eare sunt supuse reziduurlle 111, su$€l€
g produsele dln expresia trul r, pot f1 luate obi$nuit sau modulo
nr'i {sl, desigur modulo lti.

7. 0atrvn cwrl{Tr D&spns srstrggfiLr DE fiEpnEzgffinng A auHERHxln

Reprezentarea unui mr,rn6r N intr*o bazi B prin juxtapunerea

si"nbolurllor inumerelor) bo in formal .b-r*-rib-,

N=bo,rbr-rbn-r. " "brbo,b-rb-t,. {es1

s;LmbolizeazX umritoarea egalitate : (30)
/V=bo -r8t', n hn_.zBo-a n. . . +!rB+$* +

+h,.tB L+b,28-?+. . , +!_ tn-t)E (rl-x) +b-mg n

unde su$a se faee in mod obignuits. Prj"ntr-o scalare corespunzitoare,
se poate renunta la vlrguld sau, eeea ce este echlvalent, se poate
considera virgula inalntea primului sfunbol semniflcatlv (asa numlta

usetilriiza'8l,tTidrsainnshsemrmoinlsaidaLuegpdi6.eisnisrrinenbEcoirsol tnsraaizruaerapearsapjJ-e-nernlbto,rulupraie-lousrrel de mai .eus $e Pot face
bua de clate
(du$'igu{
fa*e oopara:
ea irrLr e

"cuvlntele" transmise),

' Tnluauodxa ' g ,1Pi ;j;\li"I ' e i:rl:it- irlliiir at'.itl
H apull g€lxH=ll euIJoJ sp i]'lsa B?u??or] P[n6tr4 u! r]-afef uG :';de* fi

{I-l't'''"0}=H eaurt{1nu uTp Tlrpd Teun E TJnToqwts sX"Tp nc paxelou
nJ pluaTe^Tqsa a16e anT?P6eu JoTaJaulnu paJErapTsuoc ',puoJ uI
ix=r IOuII1P ',[n'O]aX
'AT1e6AU tslse n-X=T TCrmlE ',ln',I+n]ax EcBp TnJpunu EOBp :Tallse
"roll.rnnpTzal
ea1e1a1:doJd are I: ro?pzundsaroc x
aoe] es psazauTqe TnTnlser pureJoa? pLITsoToJ 1 ;'anA1nT6lPaF:1suutsruplauJnsscetfa: gapun

(IE} n,,,,,7.=' t Io'n-]3r 3"IT'l'r!o-""rlfr]l=I r,
[{7 'O]
:rtfe1a.r JolaJ€o?pmJn ezeq sd (If,'""'lr)

T:nnpTzal ep EluaA3es o EJ alepoc TJ ?od (gbBeJXu! valted alse elda:p
azalupred JoTaurlltn e-1Jeo1gluuas) [e/ fi-nt]*n uagpuI:n
aTaJsurnsi 'H TnT a1e trlu alnpour X aTao nc lrode: In'n*jaT 16ar1ug

aTTrnnpTzsJ uTxd

JTun pou u! lppoo r; aleod {T-tt'''''g} eeuTlfnu uTp r?tmu eeTJo
'rsazaufqc Taue:oal uro1uo) xAl;'Effifl IS InIJ"IzOd '6
TnTnlsal
" z .'' rtrooo' IrT0I=0ter' "*Ot{
Ez ?) p?Tnzau
. . . ,99,o+T=ZxVg' O , Vg, 0+I=Z*Z6' O

'26' l'+r=2x96'o'95'0+0=zxg?'0'Bt'o+q=7xgg'g' bz,' o+e.zxzr'e rc
' lII'f0T=01€? <= T+qxZ-T'0+Ix7=e'l+Zxe=g'T+gxZ=IT'I+IIxe=0leZ (I

: TeJlEp

trrppeoo:d e pzpq uI 0lZT'€,2 TnJEunu pmt:dxa B tr:?ued rn413p11 '8

eltu:ad g nc 'z-q 'l'q JoITJnIogqs e 'eu1p.ro ug 'eareulu:e1ep

Rn!8ar3nB Earr lTnuu! 9r€rroTl3pr*,r."1.*,U,onl1;":r.'r,.1*,

pT acnpuos g eT gATsacons ea:r{:gdtry 'RbBaJluT paged nJluad '(T
:IaJ?s€ e3B] e6 (g
ap p6:ecarfi
Tezeq eaJequr,rqcs) TnXIB BpTaeac"rp'sxTuceazJad:daT:eeupjeuler?BsdTsuuInp€oTc eJpceTJ n:?ued

'eeaTpJo alsa E:alTp 6c
EtSlTreeJTrapuanlusaaaJe?nudTeplxpnt[ruaall:pnuToodqsFat:o83pu'aTElToTroaJlTppJolTsaTclpEJeTlltTrrupluuaaszaprduag^.zp
rep TrnToqurs lSealace op Ezpazn erec T:glueze:da: uTp nPlsuoc eTg
'eJapal ap alcund axTumue uTp aseofplrlpAp sre?uazerda: ap enTlpure?Te

Xdarp aXe?aJdraXuT TJ Xod arec Trnpoo BlTtaJTp 18 unce:d TnfnrPdrnu

TnTnuuas eareluaza:da: nrluad rtlua^uoc €?TJaJrp pzpezTfTln as
a0 'Z=E a:apgcerd nc pzpazTlTln as Tncfpc ap aTauslsTE uI
'Eeueuesp ap gleuopJo g{uaaces o
'{I*g'""0} eew.r{1nu uTp €rartrnu
nc

pluaT€ATr{Ja a'lsa g Rzsq o-JXrrT JoTs.raunu EBJBluazsJda: ge uIPcJPtlEtu
'( ,Xxrl PTttbtYn ug atvxuezezdat

lt lilfits l0 uiltlllt ot-sT tMstS I5 llt(lSilt ']]Wns

i ;: i:i;i $ti{l

iri;:. rr.,r; j i ."- r-ir
,r! , ia
IR r-r "
i ri.......i ':i 't t, 'itl iJ.,! ;tr i ,;l . ,1 -il{:F) :l.i..i j ).1.j.'1. iJ ?.i! its|;,i Li.i{iesi
pe:
ij' , i:rt.:i,ii.L:: l r.ri.l-,: i. .r,i Slst g evi,deil t:
tti r.::1.;:i! H--:!i{l , i]1.1 s'i pri,f!.. 'A
i :,.:. : ,i! ',i.. 1 .:i1::i €'ll:.4 i. l- {:,::,1 r. $}liitli.de ci*zLl.}u;i
.11:,:: i'r i-: i :r ti i I r.ri at: i.:
..::+*,i :.:i' :1ir .-fti41 '
..r !,J ; r., j el
;i,:- . ,i-, :, :riir! i. .:l
:irr!tr1t t:. i:r ,1 t,,14.i;T.lir,1 rir 6-
ri';;lii :r-ti";i!:- : .
r rr, ! iri.r jr f ':i .i f-tA d*n6!t ii,
i, .il'
r.jt,;,j .:!"i.-! r.tii a1. 4.fl.|"ef -'Le1l
$'i!tr{!i+ !s: -

:.. i il ;',lr

:i, .r'

!i

f't

',11 rl : I :i 1' :.:i 1i'; lr i r't F*r i *irde
iii ,,iji; ii i;L'r

, ii.i.:'. .i11!l::dl ;J.{.'ilit.;A'f F}ii+

;r - llli

4'

1:. :ri r]ita .:ri:ii::j.j. l't'lJli i'-:li.l"':'ii .ll',: .::::r"'r - 4rr1; I a lrt: i i:: f4- n li4j':i.r*Jrll.fi

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fa;ls€ gi1 eau,rJlnr ::u1'wytd JWU N ptepFsuol
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pTipr8do ns E?s;?sezuT TI eqeod $il{ a'f€uuas ap eaul{p1 ff.i,qugd0 "e

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ri*.ri, F:ii.r *.i- sj.*lpLu, prin at"6tura::e* slsib*lr-trllor. Opera$il}e lntre
,lr:ali:r:i:. $ih t4=#F{M} se vor nota cu * pi, o. Oeslgur, i-n esenSd, E

t t's,t.!r il.{:'q:-..iI.-:. i.l.s sunt mr:r-iuio F{, 5.
cii
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i"n F t"q "2 **:g*r:&nrhl--s* faptul cfi vaLci il.e nunerice ale
rl)&*.'tfi*nt* a r,:rtr;i seffina.N *par'$in&.rrrJ :nul$imli Hll sunt orlcbrel fc
limitate
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ac*#'trea f::ij.n':i r.$ir;'ezsentate de*a lungul eerq:ul-r-li- Hlara al" torului. N
f i"r,:'*nif eriet*le ,;er-:cri::ileir care Eenereaz6 tr:rul , Fle care sint. CA
re-prezenie.t* ctxpone*tele, se presupr.in a fl- impArfite l"n 14 p5rt: egale,
un
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{ir:t;ii .: r ::r}qeil t.+ *i:nf lr:m re l-at j-e j-J :

qxlr:l ,ylnl n *ffi ,r.i.f I *ytil i lzl

"{.c

Produsui" i.iri:.er.r! astfel rietlnit. este c func$ionald biliniarX:

<a:.: i,r i -t l:yl lrl, z I n ] >= ( aocx I n j, z I n I ] ] e { bCI<yI n ], z In ] > ]

<x i r.i I . l=r".i n j +bz I n ] >= { 66{x I n ], yI n j > ] n { bo<x I n.], z I n ] > ]

gi sl*ret: .l.rX;

<xInj,yln]>=<yIn],xlnl>.

Dam& semnale sinl Ci yln] care satisfac re]a!.ia <x[nl,y[n]>=0 sunt

ortog*nal"e
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ii ir i I r;r -ji:.,-:r i ret ,:i: :jiiti !. aj:X.iiif-.{:e

!il '. r, :.i:. I r i ii::,r,.r:, .jr:: r.J!:!rn:i iit t.if' .;iliqSl.';4

'i.:" , . r:!; _r .i il i:i.;*.rr r::,-!,4 Si ttrS.,i;J *_detg lr

ri:, i .i;";{i 1 i:li f.';1-!'}*'{f.l
:' , r.i.;:11i 1 i:!- iii '.r.: -,rrti {-l{ritrJql;U$t.*i
-,
,: : :ii,,-';1.i ;;;'- +:i.i.,, t i.. t- s*tiIET,*.j-i.f ,l
1i.i;i:l'!.i

:...r:'- I Ll'.,,a,.i.i. jr!;t 1 L,&:*: 1{t$iif.!-l_.$

r. i. : . :,,:,,,,i-:, :j*r::i:i,.,:i, dit Sjlsrditi*.!"r"l,!"Ul
,',: r:, rt.,:r';i.iii ai.ittli*j-"* i:li

ji':r .'; :.ll i : i).iii,l: i. ! I i.: FdjAllJrrdlu).t!-j,
l,-:i:.ii: r :rii, r:
11 rrli a]" yi :! est*

i:!i :r..+{.'ix ;, i i,g}j 1.,';$ ag

,.-,-i

;ti{,1"i j-r+.d{:'+p {?jllFr3.i /-'i:r'liil'tf .}--.rirl r.i; '1;1.::r.j-i,:ji:i1'1 ?ji.r:, r-l1it i?,:;: r'!'i'9F.:ljiliil
AJA'I'i:,t-I]:J T,TSg*T,'liiir ii', !d i:j;li:li. lit!, ::'.-.r: . *.! I 1i ; i.j rie].!tlil l

Ai!!Flrj:ii::iir,]dTr:."f:ltft:.rf;irii1ii:.,i;l$,;'s.'-''' , .lliiit-:i:l ::i, '.1f '.ii;" r-.i!rJ:. 1'-r"'+Lir;l .,i r."*f.:-l::Ap
-;::.: rr'aii!".;,i:!!.!,:i:1 ir:l i-f ll,!-it1.l:t,lilr-: jl; l,lS,HeA{B$-fi

iiI i 'i -itt ' ' -;,.1I.l irl,i l*r.J l:-ir'l.ll
irij.,{ ii ..&ijr i;:*r;1::!
II Ii_-i;I {t 1*l '{
I
I'

!

ii-'r i i 1fi,] { i!11.'. irl'{ itli,

..,r ir::i1., iIi+ 17.;r' ir.11
' i,:r: ir.t,'i i0l,'t

I I 1" 'l::'i ,': :e r-.i f

l ii.ljL inI

IU]f' r:,.: t:'-r.r rri;r ..itirT*:{,.,r:' ;?ir ilillt:iiit:litfi1 ..1r-r'l.ti","l.lfrdE+.*fr; 1l'r.f*:le.-l+d6

' "/11?{:?'}i1lJl{J:} S1}!ift

-€i'f i1.1 ii ;,,il 1.RTaI{,,i:}7i.1i'if'tr ,ct.i 1-I:i.t;ir}:r*..{,5 'ir iYJTi.:?i,l}g griFF,l{ji!lr.?f.i:II*L-l sfi "ItIi]l
::;r.i ii.ii'i.r ir.{l!? i}InF:rlxr +.rfirrg:1,r: r'r o.:wspTdq:.r e1 tr:T.*+:-r qs frle qJ.}r.t!'c}{.1"8?ijr
*ia;:;rrp".t",11 +Lr.lautduafl;_i -1;1rr.rp ili:tig ?}:1E) .le$q.!e-q: UA A?S?i lU jg 'l1tTp?it}.j"$
{;i?ii}(.:.',Jr 1ri 1?'IFJG.1.T6}1iT al,r p!1i:.4.rfl.^ri 'ii:.i e..;q14 *.-:, ir ltrl:4 TxlTtrTFli!$*s s Ir.tji;
"tii:J!,i&qa€ ia ,-r: ti!;rdli$ :itt"qT+.3i?11(J?'{r1i iili +i:,3r:r:i3 r4 Jtjld. TS ir.tl,X :ifltrr:1n!t"WeS ts
ii-t j;--: 111 1'.;;;riiTi-.r i.r'{'!-ll-1it$r ;'i].riF:t::.ri tiir'":ri,t ; ;,f,: :.i:TLir:JS-1 t_x$.r.r:.r T1?U! €l\ e* ;ClX

{c.:r i i i: j i'i.*$.'i; l #"{ itrlE i.i.i'sf'.ftl.'i'' [r.{J.-rr} t' " '
;rrl .tt'{ioiu'i:iri'l ','lti j"'}.i ii,:1 q'.. 1rt1 d''
ltfl.r}=

r.. !. ?- r tI i di!-l i l: I :r {'d}= 1,',ri

iFi

: iuls +Trl.ii*ui 'g#e,{ i nr:
uirl:g;r ei, g:r;:*;tgc.,{,: iulrr Eti:*a3Bg ;i; lrrjx rali,ri}AJ#s i€ iri'*r'?Lin l'li'snr6r-?if

ilift?, t "*;rjiit:i3l-tt{iri3 ;:1€*l r.r_{;i.lrlG ltritsr_jii_las rji}-* I u i"{ [ii | fi jx {31{_{'*A;:res
elr{)i; Fl il.r'lr: i-Ortu,** !.1!,r, +il}F.,:l{ij 4*r{el-4.:;;"t.: Cp il:Xtlpc-td li*Si 1?pfuia;e.r;
.' " t-i.r.-".rq,,, ji;, ii; ql} gr.!g,grgr'r qn!$flr,t{xdfi 'F
fdY:' "t ,4r.&Ff;# $:.t f,fl
{}

it;. il:.t l'l j,4 f i-.ff1i i{.*!\r}t; i.T-/fl .ti'i

i,.i r:i1.'i iL:1,'l "{ '{i ? -ff j { e- fJl ',,{i
i. i"- rYl I
{9ri irj.q I r'; -,-r
l,l ; ,!' t I: .,r- {.v} n
I r l,d i;.1 "1
i"il ,{ ir"tl"1 ii-r'{jr l*lx i rJ.'{

B?i .{1Ff il:l_{ 1!{ :':ii'i,i: 't"I T .'Tl iT ..,4I1,'i ,1,\ f

{r f "'s"i i{t{€t$f i lii !it$3$r3 'i'!.iHqFl

$EKIAr,B, f{&f,il1T8 $tr $i$?gllB 3-S-rt $tiTIuI ln sxrnllr nt sB!

pot ac$lona chiar asupra elementulul eare i-a generat, ac(

co::espunz[tor autoeonvotrugiei sau autocorelafiei. ne(
ge(
{3i Existd secvenfe x[n] Sl y[n] pentru care convolutla cielieE ca!
este nul&" Aceasta lnseannd cI x[n] este ortoEonal atAt pe y[]i*nl
{=y[-nl] c6t gi p€ tr-,61* replicile acestula deplasat* r:icli-c cu 6.
grt*e k, ylH-(n*k) 1=y[-n+kl.
se
5. fl Sr C0&VOLUTTA 1JNOR SECV&rfE pAETrCUrAnE e1t
iISRETSTTA
0 particuJ-aritate regrarcablld a spa!,lului # constd in existenla xir
unor pereehi de elenente pentru care cor:elalia (sau convolutia)
{cJclle6 Eau modul"o un elenent preciea9} este nuifl, tn cazul unl
cernvalutlei clcllce:
y[]f-?l . . . yt.zJ ytll all
y l0l ")y'[[AoI-]xi y[al-l] . . " y[31 xlitl ol
y [] I y[?] ol POI
xx [[?1]] sex
;'1121 .i/[:.] -],lol "..yIAJ yt3l n-.'ltil tle)
7,
l"rtn*el yt/r-31 y[N^aJ .", .yrtl0rl] yh/-1l x [ir*z] ,I
y[.fiI-3] " .. ol a
iyflr-r: y[]r-el .]"t0.i x [Ivt-1]
oJ dr:

(x este ortogonaj. pe tsate rep).lclle deplasale ciclic gi inversate ale la
1ui y).
I
ne aseinenea, exi;rt& el"emente x a c&rclr autoconvsluFie {ciclic6 sau l
modulo un element precizati furnizeaz6 tot el"enentul" x" fn cazul-
autr:e*nvn-i.llt iei ci"c1lce : It.
xr'Ar.-3lxt.M*?l xt?l x[1]
IIIxxtot-_l]i II II xxttor]l i
lxtzl rlol rlnr*l] ".. x[3] x[2] -':'xx I[r{J}] I
x{rl x[0] "lr[4] x[3]
x[2] t
l_l
{20} bi(
ttt i
x["lf-?j x[ff*3] x[$*a] ,,, x[0] xUrt-11 xUv-z1 ll"irv-zr nu.
x[J/*1] xlff*z1 x[]I-31 x[1] x[0] 8.
x [t\i-l] J [xilw"r)

Ceeilr'1 partlculare remarcabl.le de corelat,le seu cr:nvol.utie {nu
neparat elciie6) a doul secvenle sunt aceLea pentru care ce1" putin
ulti"rneie k *sant.ioane ale uneia dintre secvente si cel pulln ulLiurele

ri-k-r esantrsane "llxi;iiii$g:3ssssl;n,I

l"':il*:i,,*,",,,*,1,

fn acaastd situagle corelatla sau convolutia ciclicA cc rduc la

7 pq:u l.tille egantioanelor nuLe gl al.c celor nenuLe p,ll f I

.inv*rsate
"

Er"uTq BT{Ere,ls ss B?TuTJ6p T3 a1€od palT{Tnuu! rn6Tsap l:epu11.:ed
E:osleurpa:datoral.nrdun'#eatu"rnTgp $e:1rualduerorTpe
BXuarraTa pnop e ear.rfpuug utrepTsuoo
pnop € lnlnenpo:d ?uepTAe eptrndseran
PiueA3es o n3 a:r{1nm1 ep leuobelp 1n:o1e:adg IIItAU5ST 'g

"ug.:elngq6unlde:p €J?6EaJaJ,t o aXso JoAEJBdo ap IaJIFE un 'aTnu
pUTTJ roT€1u€iEeTB TnlseJ p€?€lTrm no afc6a a?ueuaTs nc lceduoc colq
un eulf,uoc aleod 1n1n:o1e:ado TBsT-r?pu eleuobelp ':e1nc11:ed u1

' I r-x-u3 p 1 1-H]xI r-s]p+''' +[ T-u]gt T I x[ r ]p+[u]9[o]xt0lp=[u]I
re?sa Iujx TnlnT€uril€s pJdnsp rnTnJolp:ado 11un1{ce Tnlelfnzau

rIrt-rv,t1tva'1l If
LftL-lrrlr^:vfr

pI e-i'r1x
[ L-,tirr'.ll a6ll r l-.lI Js UUV 6a
1g-nr;
el , 00
rz_,'vt*l 0 lz-lul
:ll{ee} lI= I

Irjx Ir] F I rlorllxx I I 0 0 [r]B 0
I 0 0 0 [0]p
Io1x IoJ e I I

'aTnrr aluaumfa alTETafeJ Blpo:l TS pT€uo6eTp

ad alrumee e"r?uTp BTn{m aTa?txaur€Ta pugAp Tpuobelp "ro1e:edo gntrn e

sunTtJp eJ P?TAT:d S$ $?€ed TEus'as ?Ts un ns r€uumF rnun sa;TxTnuful

rwxQ9q1fr [JIlrJur{ tro rsffcslru ruol"nr4dO " t-

'[u]x TnTnTpuues
sndpsJp pT E?esp{dap ro11c11da: e 16 nc Alerapuod
*TE [n]e uoTJnpgo?gsTuor]os
ttlS TnTnJo1eiado eeunl{ce 's.:euJn uTJd '[u]B
EaipwnE
oTnptur 'l$ed q ns eldnarp eT €srEs€Tdap u1 Elsuoe [u]x lueurste Tnun
E'[Judn]xs€TVllTtTtTnpTunw"raos?€E:ad[uo]€B {nu) T:o I ep a?p?€de"r TTunTiJp
TnXJsIg [u]x
stnpou e"]dperp eT psJ€sETdap ug B?suoc

TnTntruaKeTE eJdns€ W TpsT.rlsrl BaunTto€ zEil ?saJp utr '[u]E TnTnluausT€
tgXe€eXe ES*TJXFTE a?sa easTJlnn pugo Toun?p :nldwgs gzeale:d"reluT ag
[u]x TnT e:dnse (v]S xnln:o1e:edo eauntrfce puec olJenlTs o ElsTxg

1r:1xr-*g"r-d$+' " '+ it,I1xryr6+ iu;x06* tulx(v)6
eaun*lf{icgu"r:ao1-reg:H-onds6soT"I1rrnrilsrmgast1u6gd,nF'Ts{B'-v[Nu]txgrurTTu€XoueErEcrapusetdl:n6ouanHpouir'{{drprno}sd6erTf(lnufEl]n6ouETTnTfltorndmJoa?.Tte.{3:atd"o9

'a.rede nu 11":gseldap Tp ,,ll oTnpou* Trr:aloesea
TaSASU 'S eT Eugd '€Tnu TxoTEdl na saJ€?aTduoo TS BfpT{TuT JoTaluaAoes
ellq6uerg ru: 1.:odu.r u3 e$eu ep ?ueTsTlns N Tnun ea:eldope u1:d a{uances

?nop e xJpTuTT sTfnTo.x{.!o3 EUTTPaJ uaxnd ?c ulEl€lsuoJ 'P3Tt3T3au

uereseldepr nc Btr{!€TpATErs a'+se EoTToTJ ea:eseldep :Jo?puJn 1n1o11dec

uT E?TrI't"#sp {q#TTsTsau} Pr?TuTl e1$nlonuoc n3 1p?Tnza; lselece

It tlvIfll$ !{ lnt}$d$ zr-97 fl!{Eist$ r$ [H0]&ic 'rlYtirfls

$g.{f:r,,tr, .;. ,,.; ... ,':i-;iii : i;-r : t;! lilil lti lF$c1ifl $!

I l.il! i'e- !lirti",1 nrr.:i:r, !i i + ii_!'r ir iiB;tt i. ? $ ii.,]r []1, {r{{!.r'sil.i F
;1t'"i i *$: i- l" +:ir-j .

iii ' i,$J$ irlsgtrAl,E Fril{1ti$r.ti $xi'.i?rit ai

i l-r:'.&ir,3.r:
s i+1.': 1r
irt;r'"1; rtei-L:r-i'r. +p:e1;1!.i.riq1.]q. .f ,i!;'rli lJ?:Fjflt. ,:.!t

{.'<.7 t

l.!i,i l 4e!.f{l,i *u.' i' :1:iiitj= a ir!i:?i g,.3.qL,+,i rlt; i {:ilt};1"':.
,:-r.:,1:-'':arlE;j ,1y. ql1"1:.,r!.u,t-r,i,.*J.f;.i:,'i-rl.l. ,!g 111r;'rr 1:Jl:t;a.t-i t;t-r+ fi.titij
1U_1! U!::;ti;14 I i :': i;li:li-.!":i :,rin? *1" Z[:',: :
pf'":;4r'r.;l.l.i rl ir:

t I i,:1 ,.:.,:,,i r;l ).rii:rj -, .{ii;j .": t,'i.i.ll !: iili _: {p,$i

:::r i ri,?,, i,. .l .:, ::.,:rrl :r't:t.:":,i,i.,:.,.:.,;11:i.1., ".",1 it.;1r'r,:1,1 t-:iit:;.::.rrrr:rl :.:._iJ::. -4"._,",.t:
E:l]ri!..1 i. i.t.rir-::, : I ',',..ii 1...:t"ir-", Llt .,.ri,:i _..:,'i:. .:"r',i;!2.ill].,

d
"Je

i131,1;1l:';-",','r-: :: - ' ,.;r,r;.:;,'.:r. ,,r.'.rfji l'i:i.r':ir-:ir.'i+..:::i.-r1r.t*:r:;Si.,;,.:ftli,-i-g1.:.r+sfe;Filr.rrle;lieIill_rett .t-L r:-riab:j_1.:.t*
sif+$iUiu-i ffi.{"
: :: , t.:..:ir.tirl.+

t. iiiatl.n" ;ri. A i$$t_[T ;Hhg"itil], t"r.i.F uff.,if i;,t. i- ".ik,i1 il€i.r.r A..tS.[t1nAn{i

ei:i!i. 1.,.,;. I ;;.:. i::.:'."i.iri!;rLie 3l r.r4s j.3.i1j.]s ,l aVji*$,.Adpesfi$.tsfri,,i+drrj.t.,ea1h';".if.-lLl..1.S.e{},
pt'-i-fi; Si {i',rri I}r16.q.-!.rn .$.-
e&gBptu lj r )',: 4l, ,:

r.,.'{.r(]}:€il,1p+'li ali,::lr! . : i:.:'.:rr, i:.i l_i.t

rf i l! ,' i" ;ri.i ji':lr:r- +j-l-;;'l.,"ii,. xlH-il*ii;.-.1{+1l .i,.=> { ?5i
! ,. . Fiir-1,.:t:,
.:' . .l ') !'- ; r,i:'

Frj,.j iir:nil;-t. lr i i : i{i iri.rjr4:fii:8 !.J;&!tE{ArWAiA ej 4? .€gj8l;fi]Ul-t"ii X{::.!.
Vf; 3..i.,)hj.J.iir r-i'l l.irr.., r' , :;;rji. "*rdli "lar:.iilLri.Le d€ ,:;'<f$,,;.,;r.s, xi:,lrll s:i.:ql:ryli-z&nci
fapt:-r"i ii ';i r ..:'+:::,:..,: : :Frrtri.ti1::.r.1:i. crJrsBFliln?s.it':irri e $r.i?ilmgntr.il r)l.i t *ste x i i" i,
COef i.{.:-;ji:'i.1.: . c'".,til :,,,r: i :.r'r.i. :iisl ilH=inlh?.:l:isi:* leln*i i l " Tr,gngf$rr-,xei,,q d
.fr frI'ni:.3l;'',11 :, 1 :.;,::'r',: ;1;'! ;'1 ..;g.xi:il:ri d esi.tr .!:.. i:..'ji.!1.if,i 4i1 .;fad ma.{j.jfr N*t eq
sr{a1....r != r,!, il,f. j j j t".{!t,*"} XtitrgduCerli
*oef,J"c j.{'ir i,'i i.ir i:i,"qp{il. q!€

t:a!'rsf,::ltl,:t.,,,,. t: :1..ttai ti8i.:uiFrlig.l iiF'{:.:" I +1;-::i_3 ial niar€ siFi$ut'& de
{.'*fFfr ;l8g. i:iirr;:':. { i ;' i-l :i* i,.,':r gi*i"r:.: i
,;.i i:.ti..r+ *ps{,3t,{ Iii d* degr}.anar*,

: ',

{:,i)n1ft-},ltit i * .t ., ,.:' :r,:.1 tt.j..,: t):. 5-:.*r,r:.'{ij,r.:tp Er'i:l..i.i:iEtexrq.l r:r r€lp.rs.ze*.talxl

. ..,,! :!tirii.ri.-lp{$l.q

:,' . : i '.: i.?gl"r' l-.-ii i*{:1
- ";;:
l:t , filil/":,i e't4'i;i;'r

.tlE

' r,r! I r, , !rqq,.. f l: : ri
,:. ,-:,.. ,i.. r'ilk:!iil$

: : .1 ;r lr. i{:ltilE!*$ qf;j.l

!: . i4 ljii. 1 i',:;+1.i.

!...,i,t iej -i*;,
;:. :lii:: i L {,i4i.!dil!i.'i.:lj}l

4 i;l,;i r;i I :,'i{'l rrj:!111

::\

|,r:r.l t:ii::- i

.I rr irti I

i;l

$FS{*q,8, {1tri'{j;Tq frt ii$?tFre -8ff)-,' $PtTilrt !t $Br[,ti,I sl 58il

$,e demrx;:;Lr*es& !:R*rlj.at e& p*i-!.r:c*iui r:arecleristlc al operatorulut 1n
sa
atagat, seu,fiai"ul.rai aIn] eete e4;a1 *'.: f;-+,.i"i-;lo:ltu1 *tagat aceJ,u.fapi sen:ra.L:
t4
.S:* idi *po icr) *,.j4'-A { cJi { ro\
pr
'*in saest" &srt3v, erper'':r?,or:uJ aiep*t'*fiui.semn*,3. alnj se tra nota fie fa
pri$ &ainl fj* prj-+ nltal,l), :d tl ) f .ii.nd g:elJ"nomr:-) atagat semnah.lluj" a[n],
pc
&daptan*a i:i,:<.;"i [.'*nsfsr-'m'1ri a$emer]ca px.i..n matri"ce M potrlvite
p€B:*i].t{* sehler!;,*r*a f,crruq.i. ma'[::ir:ei i-ll1glale f &r{ moeiif iearea el
stic.po)'1n*wgX.rr1. q-:lu:*"q:1n:r i,istfe-1., ",ln operator & pcoate fi t,ransforuat
intr*un operat*:r %,ii,*i afag;rf el-ecnr',ifltu,.!"r-li" a.In] " gU

B" F*i"xwx l)g &..'.{i.:tr*isft.e &x, qwg? l{q?Errin {otoo**} A esre ordce ir
p*r3":in*+.r r- i,jl {e:::* :+ +r;:tJ-e-qri1 r:irrrJ d +.,qte in}sciril r:,.t A adtcd satisfaee
Pr
retrat1;: i:{,g1;f
uI)I-t
4. Y.scrpffi{a i.]+ul"lsu_.*ffie$'rl,?trffi
ir
PoS.j.ncmul ".s a:r"':sr'j..$t.i,* e.l. unei. s:af-ri.r:e & E*ete pnJ-j.nnm de anulara
;.r*ntr'u &. 't.

sesrans,L:at jre: {*9**)'1 = [a*i {'iT*e}]/pr id,} }T
Fi id).?'''iti.t '&.],r'di {ei-{,-'tri :r- Pr {&}r=i&-'6}adi idX**t*O

5. F*:,lxtge F;:tii.T".?iA:, flJ *inel- matr{{:e & aste pslln*n*ul" nsonic de Erad

pinj-;r:.t;u,id);,'errg,r* r:&.r'e siF{ai=it. PolinomuL minima} est"e monle gi

repr€iiirli.,r ii:i ..i:;:rci..i-rr: prnl:r'r=l *:r:i.r:+ pcl.$.nom ale anrllane, in particular
pentru prlij.r:i.--.:rr,:.r 'r,t{i'i}ar1.el:isi't:i-* Fr{d} fare p{rate fl e"l" insu65. pollnom

minimal"
"

6" ffnrcfi{e gnsl 1 ;{'ffi $**,nrm pn}l.neunuj. earacterlst,l.c al matrteci & ,
det{r!.[*Frt .;r; se sehJrnh& daea asupra matricet df*A se efectueaz6
opera$-:.;r e,ls$,.,.,,,:,r::'r i?;f*e .I:tlil-r sau es-trpafip" Operaglile elementare

sunt:

* Tirtsi.:sqtri,*lflu-"airea n ,:iou& i.in11 {soloane ) ;
- trlirr'.llti.rr+ rmei "i:i.rli. {*r:),r.rane} cu un scalar nenul di.n cflnpul^

scala*rl 1{1I, r";::f.q::&tei l":isil {ecloaile} inmulgi.tb cu un pollnom la o
&duriarea
altd l"1r:irt iccrl+a,t i;] .
, fiLIti-i:r:*i'rr* ,J,i,rsiri.1i..i-1. oi.eslsntare mfit"r: r:""*rs c'15*S poate ae!ilsA la

f.oram cagpg.,,a;'i{ .stur.i e* :

snpo":d €s rS 'TlqET.r€AuT Trolts] TireTTafr T€ TsffifuTu T,:nnu6o1ffsTeTogd eJluI
es JTlsTJeicEJsJ InuouTTod
aTras oleod I

r

t*Ip E€sTrleur 'arpluEffaTa TrFluo:lgsuTEnT-nrlJouleTrorddo sJpo?pzundea:oo

fndffiXil - L

'TJ€?usw€Is roTTJozT,lTp are a?usTrenul r-rgdu:6 pzBazrTEal TtusTrpfluT
TTJoXaed '{a:eceeld ap U*np TaJT-rXsfl e areuFuasp
u;:d ee:retu:olsuerX ep
truaJsJTpuT T6€o€.Je a?se q?Tins pcTuorres euuol) paueuas€ -roTTrguroJslrEJ?
uuun uT EquTqJs as nu araa aureou'fl,loTdrplluuen8erT1a{u.urc1,T:eTarouzgTATlTproplJpsdnpo,ld

ec'19$:g3e u3'T6 (ZE) ErrroJ uT'a1:ed plTE Bp ad,{93):euJoJ uT

e1:ed o ep ad a1:os aleod es oT?sTJa?JpJ€J 1nuou11od sJeurn urJd
'TrgXnd Treu rpt[ eTec pug^E Treluemele TTrozTArp luns
pTl9acq 'TTedTJuTrd T-rozTrlTp ossunu es TE{r[uTtr rnlnuoullod rTro?rpd

{rcl *, (P)n'd* {p){d
;su-ro3 egr eueoullod ltms 1$ fJelaeunTe

TJtJZTftYp 3saunu 6s 1{uugre*u1 .roTT.roT*eJ Tp TTTqT?rnparT rT.rolc€J

{ss} CI= iff) rt
:egde1e": BJpJsTlEs a.rpl u1ruTur pe;6 op 1nuoul1od

EsTp€ U TnTruole.r6do Ts rffirurs 7nrwu|7od elsa {p)[i={p)i TmorrtTod

T"f

{uq) fPirfiU=iP)orf

: r{uerreaul JoTTr0XcsJ
Fwnp0rd a+s6 E TaJTJ?su T€ oT?sTJalps.rpo Trurioutrlod luaplaa pou uf

'es?€?Trlfi na aleba TJ jod eueoullod tH>{) { alaug:d
':r1n:;1:ed ug '(roT;sdns aJTprrT no lnuouglod epT^Tp uouTTod eJpcaTJ)

trs) "'T'0)3tr',A {pit'rdl (p)r4l
1S ecluou lunr
ggtwysvazry r-r{3*"rBJ arTurnu' (p} Ii "

"r#i;rT;":ffiT;

tJO

{s€) 0 (p) zs 0

0 CI (B) x _l=,

[x flTYllrt$ [6 Iilrt{$ 9I-97 [n[I$t$ 1s xtifl[ilt'[TYgfl[s

$[xlAtx, ctE{t]]T[ $t $tsltrl 16-', $Plutll u sttrnl,l t'l

[oor11]. 11.l
rr o o o orio
;;;;;;;ilo ol
r r r
lo rl

.=i; (35)

tlo o o o o 0 o tl
0 0 0 0 1
l0 1l

[ooooooloJ

{x={0,l}=GF(2}} se poate scrle:

l[ororoooooo o o 0
o o 0
0
ds-A =l;;;;;;I o 0 0 (36)
0
110000r0.10n0 0 0

;
loooool+d o
looooo o d(l+d) 0

0
[I ooooo o o d(t+d)z (r+d+d2l

Frin urmare, pollnonuJ caracterlstic este

polinomul mi"nimar r+d){ { t+d+d2 ) '

factoril ""lf(Ad=)o='d(( l+d) 2 ( 1+d+d2 ),
lnvarlanti suntr
!, i., !, L, L, (1+d), d(1+d), d(l+d)2(1+d+d2)
lar divi-zorli elementari sunt:
d, d, (1+d), (1+d), (f+a;2, (t+d+d2).

g- fonxE cAxCInrcx neTrorArx A urur opERAToR Ori de cate orr
dcasirusieeaebgdlmsonpueinaaJcnNattt€rllilaaubltc,liapelarearalll)nlsafucosplnrpebmaualtoallslEucoidlu)nputrapaeligcrllmolanifinutiaiocangl rdlesiSaai;pnl(o.pnfslaroaeDetucea,placuusf6rlineeboisplnabopedulcorneuuaecvstrol6icdrar,6eec(pn!nc1d!raa.oeii-rtnenrbpelcteraeroez-sdcoa6tleuaatargraiasofsnuonenspemsafnrotletaunrramelduseaauunsrcpnueaeutl UT

La forna canonic1 raS,ionalf.
Forna canonici ratlonald, a unei matrice A avand
fr(d) rezultf, printr-o divlzorli
eLementari transf,ormare asenenea cu
o natrlce

pqEtT uTp arecnp€J? ) JoTa{ralsT8 ETJ'To?e6JeJ'tne3Tnogd',P' gJTu'qqe3pJB,z',p'Vg'',.(IPzer6ua
d

ereoullod Fnop.(B(p()J8p,lu(paJ)s) )u=n((&po)sr,o(zpT)AvT)=p(epJ)8E[pTTplpulofauDgz'eEreuzTrrTnTlnu1e:gs
tplle.= 101.; (p) ls rs (p)vl (pi ts
(cprnrrl tp)g I rozT^Tp) (p)sl(p)e rs (p)ul(p)s (z

(unsoc (r

: oITiplaTrdord
h3 (tputrc) rmuK)o -rozTATp arpu TpE Tac lTuru (p)S cTun uouTfod un
R sTxa 'o:ez op alTra]Tp (p)S Tg (p)H eueouTlod eclro nJlued (Z)

p'(zPp)setoouFopso(up'()(psp'l()vp(vp€TTr)6nnvTT>eTTdn(ppnep)TpuTfr,z^opeTzrTep^nrT(e6pps;:)sllsTgun'n=r(lupss(a)lpu(rrp)a(v)+R]0ll()s(upTpE)sr)seo()1((ppp')(T)u)psns)lIug=s'ocl(=(p(6p(p)1p)0)c8)0au'(a(pFup)Bc)vp0oglu(aTppr)ouedn

ep EoTm aqrared o pXsTxe (p)s TC (p)v edEouTlod eclro nJlued (I)

ucrrslrtsof,rrod uEr^Lugdrg rn $€Eofl& 'r

JolaupoffiI0d pTJ001 ap eluaual0 pAOlBJ z'[

'dwgc un-r1q 1{ualclgeoc no Jo1aupoullod all{91e1:do:d e1
a.r€o?TraJsJ amlqord Enelgo pXsTAa: ur uecaJtr PzpaHlrn eJ aTas uI

'3T1sT.ral3pJe3

plnuo,uglod TE TJpluaoaf€ JoTTJo?oBJ TiPgBts ;1ro1e:ado p?uTza,rdal Ts

.; :.:lt-''tr T
n o nl

tB€) zs o ... o r ol= =rr
o "' o
r[ o rl

oD o "' o o o"|

ErLroI €p puTTl pTeuo6eTp ed ep aleeeTrlslr sTTJnJoIq

i't ::: : "l_1 =
'..o)A
{r€} o "' o o It
o
"'t.|
alse 1s SBsPeT€ 11a1r1od I

tl [TYIflt$ l{ l0Itf{s 8I-9T rrrrsr$ rs ilrflc{m '[IYrfl[$

$BilIlt[, crtcili?t st 5r$TxttI t6-tn $lAlrut m srlrnr trfi SBilT

Ald) Ai 8{dl este poJ"lnomul de grad naxin care le dlrrlde pe anindoud. pe(
Csmdc f/dl se paate scrie in forna
ine.
Eid) = u(ct)A(d) + N(d)B(d),
(Acr
Pol"ino*n'e1e A(d) Sl B{d) se nrmesc pri:ne intre ele sau relativ
asti
prlne dac6 frid)=1.
mic
z. lurArconrnnn fucr"m Ds DETERuETaRE A cHHDc A DouX POLI}NOAUE trJ\t)
Consider6n pollnoamele eo(d) Si A1(d), primul avdnd grad nal nare
sau egal cu al, doilea : unde
Se formeazt relat"ille:
rel,a
A6 (d)=P1 1d) .41 (d)+Az (d) grad Ar(d)<grad A1 (d)
uncie
A, (d)=P2 {d).AZ (d)+A3 (d) grad 43{d)<srad A2{d)
lnca
-- ltir:i,,.iiii:;; ;q,11.e* ra) Erad Ag(d)<grad Ak-l (d)
CU nl
A1-r {d)=Pg (d}Ak (d)
grad
Ai (di= emrleflfu (di ,Ar {d) ] .

Algorltmul kli euclld este echlvaLent cu dezvoltarea in fracile
contlnu[ a raprtului polLnoamelor Ag (d) Si A1(d] :
, /d\
L
ffi=Pr(d) + 4td)+
F, (d) + (3s)

Pr_i (d) +-- 1
.Fa(d)

-s,u,blolin}itenma*cL* _algAokri{ldm}uel esstete cmmdc al polinoamelor Ao{d) gl Altd}.

valabil penr-ru polinoaine avAnd coeflcienli
in o:ice sAmp {in particular I{=GF(U) , Z, A sau C).

3. Teonp{A REswLlm cnranzEsc Analog verslunll valabire pentru

nunere, teorema reetului chlnezasc referitoare la polinoame permlte
firn€upttprre-dzuerngntlarciraiemaapcpeodsleintoonraaunmelaLeorreod, einsgrearfdluemnadcxilemlepprdoeleicniozraaemtsgetui parrlvflauennde{rceionzeitdrfulecuieralrnleel}l.i
consideratille care urmeazd sunt intr-o analogle perfectE cu
cele
prezentate in capltoJ"utr anterior.
He referim tra pollncla$e av&nd coeflcienfi intr-un
partlcular, M=GF(M) {p.1=prlm} . Consider[m pol_inonuJ. cA$p, in