Goras, Liviu - Semnale circuite si sisteme

urauTfqo aTPunr8s enop Jolac € eT{€Ia-ro3olnp ap PTli}rI\tr r}u?Tri:i[8J

(It) os*{€*q f,= 1ulrf ! onrrate f,= 1u1x

:1u14 1i [u]x arTpoT":ao epluas Bnop rrpraprsuoo

{.[u)x,[=[u]x TaTnprJcJ uJoJuo'o{9'.9tr)u=eqHa:loeJluBerp[r??ua":BIcsfT'po:fepumontre1s$etm$eauur1-I{dqo ee

{0r } ssrrar$ f,= 1a )x

puJo; ap aXs? Jarrnod arJas qI BJ€lIoAzap JoJFJ e ae';fiolaur

"lTpc_r.red 6fariuras Joun €aJeuc}tr1uuSa u-r:d u-g$qo os eTe{ru€s ap {a}SsV r;

'rr:clg.rnSpJu! e EaJ nc gpeog:ad Tgeaaee nc ac'rp*1:e<3

rcu agnul{qo alaJJsTp sTaTeuflras 'a:euo11ue*a ap rapec"rT-rad 1e nTdT'+Tnrir
a?se p3rp$TJad ar€+lq-rngggu3 ne a?af,csrp aTaTBulrras uT{qo as pJs:!;F]:

BaJerroT;utl$a u1"rcl acT6oTeus :ofaT€uuas apeolrad aJ€o uT Tn?pp uI
' ar1potr:adau' TT"rslplngelu.!

pptsoTJad nc pTTqeJnsuauocuT alsa aJpuoTlupSa ap pppoTJad a.zea fi{

Tnz€: rlT 'nes 11:o1ern6F1ug TappoTJad 1e n1dr11nu epeol:ed na acrpt;tr:e,?

TJ ?CId oTg 'eaJeo1p.:rn€93u1 ne FpEoTJBd r€paapB nc acrBoTJad :Bspsarr
pors uT luns nu eclpot.rad aJEolEJnSeluI pug^p alarcsTp sTaTpurag

enTp0rJad a$olBJn$elq n0 aler0slp aF{rra$ ["fl

'zI TnTlEds uI lPrlrJor.lolJo ?as rrn ezparirri"';

{6€ i ldCL iantdI
:a6?rnnas ap €TTTlrpJ pc uaur{qo 'g=u umund (gg} elXeta:
au-JlIelrrp{$u3ladeupog':drexaialrTrrp.r:Tl.tprttueuudaasuuopIdnx'oJapparElJuBaocrcieuTz!o€JTconu}zoEEXaenufpnIuaPp"ll1tprsoTa]lcPTuA6n"r;[arsrFuzJoaeuppr€ln€11?dsarrr1xoa13lnduurou-n'

{8S) ztd #lifi:fl nd.ae,g)=",,-0,r" T *fatt1a6see.e=

*rytn== 1*rn11{ra{4a{..d* p I [u+:y].rf ' [,{i x> = lulo z

:aTBUU9F PnOp
eJT€pTupaus"sTEnnsrorppoJ:doTneeceele6aa1{ae|1seo:oec=ou}nnp:1auapde'1.r{ee1unna;t1E:peTdncuTgBf',a:e*
JoIBc Tp
ar13il1J

{rs} r46s{= fuilf Ia?I'sB= [u]x

z{ fluflfs tt I0uYd$ IT-ZZ llNll$ls I$ lll00[il '[T$iH[$

$mn[[, cncullr st $t$tBtx 22-r, $ttlIllt D[ $tIilll It

{- a;b^er4a ng-ta=zpn pez (12).
(431
r-'(YrLn't-)=.lE_E_

['-"-o ms-Ja,*z'n

nmctia de autocorelatle se nai poate scrie

fr .'lnT=*1,:#E h" ale -rrcr ^o"er'F r"-*r
)=

*:zimpE E alb'e'*rn*k) -rrar

gau ntrz (44)

(- F--
r' xY tLn"Jt =lE E aib4r*z#eJtrs'+'rnr' n$-Ju,*epm
condigla de
n$-Ju*zgn

Observdn ctr, pentr-u n=0, relatia (44) furnl-zeazI

ortogonalltate a celor doui s€nnale.

2.{ Transfonatele z gi lourier ale functiilor de auto- $i
inter- covariantl

tn anunite conditii functiile de auto- gi inter- eovarlantS pot fl

sennale care sX a&ritl transfornate d respectiv z cel putin in anumite
donenii ale planului conplex" Funcliile corespunzitoare de auto- gi
inter- corela!,ie nu pot avea transfornate z dec6t in cazul valorilor
nedli nule c{cl altfel ele sunt diferite de zero pe toatd axa,deci nu
adnLt transformate d sau z. tn cele ce urneaz6 ne vom referl nunai la
transformata z gi la cazul partlcular aI transforaratei Fourier.

Conform definitiel, transformata z a functiei de atrtocovarlantl k**
a unui sennal x[n] este

x*,(zl =E *,* lnl z-" (451

Fonnula de Lnversiune se scrie ({5)

k*lnl =h{**rzi zE lez

Functiile de inter- gi auto- covarianti gi corelatie satisfac rr,laqiile

'elTqeuJols{rsxX 1un6 nu TTpFE roITxotrBA e:eolgzundseroc e?u€lsuoa
eTelerres gr 1n1n1de1 FlTxoXBp JeTJnog TcTu Tcap Tg z alpuro;srrpJ?
ne nu altelsroo -oln€ Tg -JaTuT ap e111fcml 'Efnueu eTpefl erEoTpn
nr a?eJJsTp roTaTprnns TnzEO rr$ 'raTrnqtr Tg Z e?EuroJsusJj p?eTx6
6JpJ n:?usd ap sfTTfcun] ap s?pbaT
luns sns TEq g$ue1:enoa -o1ne 16 -raltrg 'I
olTutlsp exelnd Bp eTErlceds alTfplTsueo ffJtnUfSq

(zs) CIr,' ("rc)&)r{?= [o]*ry I opo"ge (,r"t*;l+= o{

[u]

:puglrnzaJ

(xgl ao6-e [u]ery 3= (*ra]ex

:amg{cera1u1

sp rT.rarnd e gre.rt"adE Ea?€?Tsueo Jr*u*r"o *" Jo?Pu?wsp pou ur

**l(0s, F= [o] *ry=fo
sF (*,rs )

"[u]x laJcETp Tn1nTerirsBs e e:aXnd ap g1e:pads B?s?Tsu€p algaPun3u9XaTsnz*oaXS

1.*;oPworo t"re1 at; ?= ttl *tr (*) uop-a [u]*rf 3= (or€)*x

:Jalrnog

Ta?Er[roJsrrp"r1 spundgaroa erpe zpc 'oJ83 ed ace; a?eod aE ea:e:6a1u1
z T€treurolsucrl Te gTt3uuen6l:€an'rueo1cnce1p1rIendTnuruIa'u-}uo/pI=T*ruuo1T5ra-lguTalul:Tncgrletpc
e6
elelTrm Tn xe) pu€J ep

TFXEPXTgttru[Tae6preaJnpuTon3sr'atJpTeTuEnoT-uroaJuToIplxogTJce?,guTlTunTzPeIJ: pTeTsiezTraoT: aalPlssleloJJssueuJT?c

t8r) tf l-fx- {z)kx tf lTx- (z)ox

op;sTtres g{ue;:enoe

-JsluT Tg -olne sp roTTTlJunJ aT€ z aTa?EnJoJsup:l 'rolgzundsa:o3

trll lu-lY:f= [jun]lee::f I [u-]1:f= [ulo:f
I [u-]{-r=
, [u*]f:= [u]s:

sI lIfiflls It lnII[{$ *I.ZZ flntt$t$ Is flIllt0ilc 'lIrult$

$g&$i1,8" f;I8. i{i},g $I F}$rEl.l !3_f"-, $94I1tJ!, 6g $$HFAr,s 1'

e" !9 ma.m Frin defirrif,i-e, sernnai.ui. s[.n] av6nd rnerli.a nu]6 9i

&:ryxm

fr:r:r:Sfe d.e eut,*e*relafie pr*por:f,i*na1A crii 6[n] o k*_[n]=o*"{lIn] se
nr,isee$te uq'Jcp1.]i aLh {facterruL ele proocrrSi"+natitats este uai:i_anla rl_'*
!c,&e{iek-***i=[{c}j=;.s*'}. 3}ensj-tatea sper;t.ra}.X eie pl:tere a zg,xnat-u. ri.!. .r.-1.!: e.*{:;*

3. gX$svgemffrin&mtrdpsii-E.9pRm*.nffai lfefli";eKfrlegzrrx.rr$l"Rtat&e fiinAu[r:arrsa*casi-rcu[l"su,rlKuRi 'a".uL[:scR-.Bg"Ti-*Fr,fnxEters-r
e,*,r"el"atieJ si c:sli:*xiar'![ei, represlnt6 o mXxur6 r, ase"mFn6r:i-i_ unrri-a el"isrtre
ele <:ri replii:ii-e degrl*eate ale ceJ.uil"aLt, i.uft.ntl sall il!! in c,-:ll.qj.dpre.i:.*

u;al-"1r.1"1-e !#edti". Af:i-rmagia este val.al:ii"6 Ci i.n i:'azqrl. ci:nvc,3li.ti-el, aj.rJ-
fi-j"*d fu:s€ v$rha de asem&narea cl] repJ"ic:lJ"e Jn?"sr;sdf--:s 61 el*p)_a.sate. rl
fa"*rcgie d* *e;ne3aSj.e eare seaeXe l-en? l,e uer* eAnd nl ti.r.lqle ?* pr}"r.a* l:ut

I*- mi"nu,s "i-sifi.ni-t 1me11e& * ase.p'lXnal'e mar-e a spxuraiel;:x d*pies,*le, de+3
La$r l:1&r*rter l:+latj"v l"ent de vax'j.aSie a e*est*r,n; + suE{ias* i'ilpi.rJ&
ind3.*6 r,or,r i.s.6i'3 r"apd"de a smnaSelnr de*areca repijc.i.)-e de[:rl.asate
se.*m$n$ puf."rel qg:,r+*3rusul scala;: tturde ]e e,*rol. ftl sf&rgit., *aras.rteruJ
peri*dte a1" futaef3ei d* corelagie j"ndi*5 p*rloctrie5-tnte c'-! eritsedg;

perSemr$b a s*u*na3alor {:ar:e xe e*mgrarX. Ti-n-fu:d eear-cla *e j,r{:erp::et.X::{J"e
futi*.$:i"e:i d+ eore,]-atis, etrgft <l* agt.*p*aB ca ti:ansf,cr.natele Fr.,le :.e.r {r:a;,:,

pa-r:t"'[.cu].nr. d* t1x',:nsJ:*meflA ai "qS" F.un6 ii"* ewj.'rte.ri:rtfi argr*rie ).e<;r't'.e ite

rr.,4"ter"a eie vex'i.qt'ie e sewnn].e,1.+r ,r.:lj.n tr'=.

?"5 #mmru'*3.x*&fa ryi c*netragfa wnon $*ffiF!fitre aSin ffu atfril"in$nq$

ffmrsr;is pelerffi spagfall 3P

Exlst6 siluaNJl e&nrl se"mlale d:i-n spagl,ul f" pct fi pr*1ur::.r_.ate prtrr
convoS-ug:i* :3i,,/,qa.t crlretratie f*1$$1nd fpr:mule .e pr.*rJu,euJ.r.r"i ss,$.Inn
vreanlaabrci.ai,heiL?"{.n1.ir,'+:ti"ct-rc3e..s3.t:u'ui,rat?ipilJ-t&dtqe-r.pl rfeiiJn"dilcurXnrsi esmen*}redif.enr-6i.*.l.*a
+pi.ie,l&i*

i: cerensn
seanaLelc,I: *1-r-:: tl5 prfn etst,eme diserete li.niare invarl*n$:* in timp"
Astfel" de *i"et::rffi p:t fi deeertre pr3-n funegia pnndere liln]" r:6s6xl.nsm1
la un seffimal el*: in?::p"l's e[nl f"{tn-* eonvolu},ia dintre frrnrt:i.,q F$i.rrlqy_ft

gi. exettatie

[n/v:r :;!] - $*eq e {kl"-'"'-." h I.1 { 53}

convoLt![ia fiind de.finit"H 1n sansu]" spat{'.lt'.!i 13.
tn sele ee ul:p*az& "dn:m anal".i.x,,* retas*i)-e {:&r's existX f.ntre

functlil,e Si. pa::a;$elrj-j" care carast.*r:-1.rqaz& *sqmira..l u} de int-rare {dl"n
Ct) $1 fran*fl:iii*r 9L per,-*netrJ.f r:* -* q.:r. e !:e;ri zr:nzE s*rqna]rr} ,le i.eg j.-le
(.de asemenea tli-l i:11.

&-{sii:}:iE.4v!riijrj+.1;-} d.rairu',.r'ri f i'r:j,:;!J{i; €f irr tIl-F J€T€38 TnTnsft't]*fd

:is'16Jfi3{i?n€ a1.*'a t$jli ep{jt1

ie"ii.j iLt"*; rir"n., rJ;l; "'.- -{i Tl{rilf s- s., *,,.d

J.J .3- lJ + 3 [a"..V]*', K

{e6} "!l,'nir r::i*;ii;7t.,',, .# t34'i*S}& : *** _t_fi - t
-.b _ lff -"i**-ri.;a;Jr{r

: aurgqc u6;

Ci t rLrri; &....t [* cil* ,w'q;.i2ii[.;".i *{u:" .l*- :iili:: .i *

I1,," 't { !*rio; +L*-*lt -

d*c F "-il.i,,, *ryfi:,toi'"o;
', .tLr! [".lO*" {ilr'i}.
: l . i^ r, :, Ji -i

rhis:;E €,{b.diu'ii&{}i}.i&i}td .t*3 ti-{hffiu&:H.ffia&, .E

v

ce,gl 4"',,-'ii"i i. l- ) H.::";ii

P.? g'*TrluBr

:..-,iJ

r-q-i $ a.)r\!i{ilq {

&

'ffil;i '{T"rgsgTdap
wi *l{:1rj!..1i'rrlsi;. fr-r:-:r,F r.'iitl?i.rr- .€# '!"'*fii'i-E-icajgrs !j-i;p€tt* €} Tft3{i€J ?EziTET?n €*g}

," :i rit,ilir 4-iJ'

: ,., ;i ',i i.{rtr ?''{'{-" 'l iq, ;:T* '-

l1;qr ,, h+3I 'i "-y I,;,:j,t ..-^i

:. i_,;, l ,{ 4 r,ri=";i;

: ,| w

..i.l$rlai !:,'.i,

{*$} ;- , ,1 ]*.,J... o{.r

*-, i*i-r,, :j.s'irie iirrsa.tr 6"p -ti:iTnrerjuafi

e'gpaU; i.* +'{.8,r,}:i.i? ?!: !.:irr,i'g'€{.r:!b.w-i$ S'{.pa$ €.??,{19 g;} iyj'*fegu*ifi€Li Wn@H ""6

r I H'l-9sf;$.f 38 ]lcr,r,ri! : ] *-f *.{ i,i',!1&iii$ !$ $;i,!0$tr3 '$Tv"gi{F$
{,. $,

$Enrd[fl, l]![l:il1?B 5i 5r$?EriH 22-to spATiril Dt $g!,lFAt f, I ?

fiinl = f ir trnlJ! [,r+m] =I [a] ti [-n] {60}
fiil-a

( e ilabol ul * ::eirre z lnt. X convr:lut ia corespunz6 ?'onre produsuJu j scal.a.r din

1=) .

tn particuiarn covarianta r6spursrrlui esle ieqatH de covari-zulti.
intrE::il Fr:"n relatj"a:

k*lnJ=k"*["n] *.[[n] {6x }
{6?}
Confe'rm propriei6gil-er t"ransforrnatei" e,

Hqzi =n1z) ni.11

z

a\reffi:

t{w{zl =fr{zi Kss\z) (63)
{64}
respectii', irl !:rarrsformatd Fourier,
l,o.1+1/ - 1ln""t1 p.1r1z* tr 651
'k-'t':1 r a1lt*1 =fi i o:lw\ K f e"io1
'*

fn panticrrJ"*r;

g

{'3"= =21n- Jf,ri'g t " I 't ilr (uu { sJa'1 ;i0,1

5e dsnnons'1.::*az* u6or $1. formrrl.a intercorelatiei dintre s*mnalul de
lntrare:
te$fu:e g: l:;:'1. cie

r*[nl *X & t.k] r"o [n-k) =,h In] sr,* [n] {66}

{convolugia er -.1 l-uatE in sensul spatiul-ui 1') , tn transformatd Z
respectlv Fou{:,;*l:, relafia devine ise consider:d numai functiile de
covarlanlX c6c:i r-'ele de corelatie nu artmit transformate dacX valori"le
nedli sunt nenul-e):

K lvi =H( z, K ! zt K f ,qJol -KIc-r6l x'rriedr! f p-roI t6?i
t^?*'*

3. T*ncs*En Ecrw{yru&ux arfi pntt srstgt{& nr$fln*?E particu"i,aria&nd
relaf:i"a {61} ei lran,sfnr.matele saie z 5"i- Fcurier al'eln. in cazttl.

gepnalelor rie int.rare de tlp zgo$ot ai.b, i:e1a!"111e:
h*.In] =S[n] *&,.Inl =sgfr[."1

respectlv

'lpruolsusrl rnTue[op uI eTTqPcrP[Er

olelTnzer .rotm EeJsuT{qo slTuf,ed erec 1de; 'reT.rnod nes z e?PuuoJctlerl

leTe1Tq1m3nr3pdzeeBonce13llBpcrdoeoIdlalTrrcea$sSdpPual'eep"r3ToI TauquslpIT!usteru'{TBImTFTqcsefEo8ea1cdJxaesBlrur[€reg!anJeegotTalu1pslee1epTeeTuaTcaor[aEod1rsozqee-uIeJP€T&o)lp3u1Iosut'Bprx[ uTaJTo{cndB1TTspnoa-oToAr1tolp:unaododEsJoeu1aTTP6appT{upeaeauT1TGpe{TTutISPTl^eutacruomeucBg]

'roTrelu" Enporluf rPTEos lnlnsnpord ezeq ed cseulJep Bs ea?secv
'ulfnloauocolnu nea pTielerocoXne pugluaze:de: eTEu[€ts 16 uncaxd

(pnq?uoc eluaaodroc) pTp'€eouTerglaeuleTaBleEot[dIoousfETxnzlPI utrplpouTsuuIEaecsnTpnuuTn erPo

tulz{tul "" Tryifi{=,ta luti'' [u]x>
rPIEcs lnsnpord nc

lrode: u! (TTqEJpdeaau) ?xBqTTI{ nTleds elss lTuTJap Tg}1BP TnTfPdsqns

-;,1 1u1xl try.ift
iv

clfelar JsJsTlPs a-rPc aT€uu€'s au1{uoc zI eTeutr€ts ep TnT{edsqns

,l,ffinu xu '[

(t9) TeTtr€Isr eeJEzTr€pc11red uTrd PlnuTlqo

tro 1"ge;H= (ogelrr"

: srlPTaJ

pugzTfT$ elPTncIPo TJ lod srepuod eTl3un; TS raJsuef,l ap e1{cmg
'olelTtm Tntrao ad reJsuexl
BTnrTBt eleod ss 11r1ge1 P aT{PTerocolnP
ep lalfrmJ TntprtEd ep
qIP lcxro6z srsrluT e1 pugcllde 'erP&rn uTrd
ETtcunJ puguturclep Tg

r;orf 1"pe;rl= Gre)Mx

cI lltttl3 il l0lll{s Lt-zz ililsts I$ tIIll3lI0 'llilItg

BIBLIOGRAIilE

BALABANIAN,N., Biokut, T,A. Electricd Netwo* Theory, John Wiley &

Sons, 1969

BERLEKAMP, E.R, Algebmic Coding Theory, Mc Crraw-Hill, 1968

BOSE, N.K., Dig*d Filten. Theory od Applicaions, North Holland, 1989
BOWRON, P.,.Stephenson, F.W, Active Filterc for Communicdians otd

Instumentdion, Mo Graw-I{ill (UK), 1979

BROWN, W.M., Anatysis of Liner Time-Invriott Systems, Mo Graw-HilI"

1963

CARTIANU, Gh., Sivcsou, M., Constantin\ I., Stanomir, D'Semnale, circuite

gi sisleme, Bucuregti, EDP, 1980

CHUA, L.A.,Intrcductton to NonlinetNetwo* Theory, Mo Graw-HilI, 1969
COULON, F. dc, Thdorie et Tmitment des Signun, Pregses Polyteohniqucs

Romandcs, 1990

DICKENSON, 8.W., Systems. Analisys, Design otd Computation, Prentioe
Hell, l99l
FRANKS, L., Sigtsl Tlnory, Prcntioe Hall, 1969
GELFAND, tr.M., Vilenkin, N.T., Funclii genetdizde, Apltc4tt de ordtzet
antonice, Bucuregi, Ed. gtiintifioi gi Enoiolopcdiod, 1985
GIACOLETTO, L J., Eleeaonics Designen' Hutdboo*, Mo Graw-H ill, lg17
GONOROWSCHI , 1.5., Rdio Circuits otd Sigttds, Mir Publish€rs, Moscova"
l9El
HOMEN'ICOVSCHi, D., Funclii complexe ca ryltcqtt in Stiinld Si tehnicd,

Bucruagti, Ed. Tchni,ori, 1986
IONESCU, Y., Teoria siskmelor, Bucuregti, EDP,I9S5

KALMAN, RE., Falb, P.L., Arbib, M.A., Topics in Mahemaicd System

Tlrcory, Mo Graw-Hill, 1969

KtI& G.1., Trcnds in Generd System Theory, John Wilcy & Sons, 1972

IVlanualul inginorului, Brnrcgti, Ed. Tehnid 1986
MATEESCU, A., Semnde, citr.uite gi sisteme, Buourogti, EDR 1984
SIEBERT, MaC.W., Circuits, Signds otd System,s, MIT PrcsC 1986

MESSIAH, A., Mdcadque Quortique, Dunod, Paris, 1969

MITRA, 5.K., Andysis od Synthesis af Lincr Active Netwo*s, John Wiley

& Sons, 1968

I

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1961 'dueduo3

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p'tulssacan4 paQts ptdrc fo rc1p41ddy rttoeql ''g 'ploD lt'"I 'uIINIgnt

986t t1cg truo6nmg topfruvec o ?oyaunu nnnngard

4 cpopht'1 euo; "y nmegql;ry ''n fruodry ''1 !$uogp51 ''g 'dgd
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