- " ii.-... .. ,. '. "-u{-tift."-e€ '*a}"rr$'f-l^-8" r{1- pr s1r :r6
&; ' " I'
fF.,lu* 'f ]= tut's
EEF
i+t ;
rra arafua.Eare Eo[ TPrr ilBJeilnuB
*.trlT+Tr":trnir..a ;ir":.:iil?'r. g.{ .32;?i{ ;S?S"drJg € e?TqBgOAp TaIUE?fOdS,r gXTfOXg(l
-r,;:;i{--j^-* ri .Y Jff ';.-';1*/- t**
' * ,*=
{cr"} t',r**.f * gu*",,*?* [rs]{S* [:r'eri S
+;13{eu,*-i,; FF, *':i€?{rFge;ele; e.1sa igTeuo6oxsa) Rzeq llTp o
u1*Jls'i&: J*{.€ft*d E'r3fi$ 'plaJssTp JaIJn0J Pzsfl z'l
"*:li.tr?Tilr! €alTJSeu 'lu"pTAe 'a1sa pTnxsatp
IS1J1':,+iln' ";:i'::;r;:;r'i.ii-'rf,ii ' i{il 'Y Ti1{1]?'ig&}S n} A'rf{;*fU1 U1.rd 'eree PaCTI?e!{
}il" g;r'r'{i*!ii:1:'.. r::';::r.:rr j"Tr.h5B;{ G-I t}.F* T*Tn"IV}Sdg TTJP'ZT}e.Id apUndgSIOJ ATES
fsT€{irr-"I?1!?'r:.,,'r i}'lr:iiiiT-trJ: l{:.Td ?ntn1qqwae ee;reztpa:d 'e1ulnna aATE nJ
{f.$a }
tplsi:oJ ep a?sa qTp.r?o6d6 Eelr?IoAzep
'{,:41} j*+i' '1 i':::'J ii;i:c3rffi .lf.rT!!l-B.1iws$ eTIJ0TEA ep JPTIIS lelUeZardar
s?$a ?r::r'rr-;f.rigg11 1g?;.iii ir*r r.;r)q-q?-" vT oi) tJgp Tguuo8 Tnun rn:lcads
i i '*
{*r } ',i,; o "'" t?i fT* ,rv i, ^. ,nl_{n
"r:.:
{ r.l t:o{I*tfol.'"'T'0l=}ogeauT{1nu ad T{TuI}op
ep F?p?uaza.zdar aXsa EsTrprIEo szpg
gCITrr0llPc Ezsf; I'z
"3 3.lt**,{1; lc !6ilF i 5 -ff1 ilil$ts ts ]il$il!3 ']Iiln$
stHtt, ctmltlt sl slslllt 18-ro $ttltut Bt stffltt cp 5tfft
frr-r[n]=11, "J{ot}S, ..,, "'t*t''#} [lx*
neronstrtrE in continuare ortogonalitatea eleuentelor bazen
lj
dlscrrtate:
IIltvX-x'
lr.l -JhlE e"Jkrg1 *
#=I -rs_1 I
<0r [n] ,0rlrrl )=E O; [n] 0rt',1
lxt.
{tI'-o-l t' " =- r -r1-*eej,tf'-t-')) 3F --1f o' t:r,'
ii, dir
"J"'N* , $" + j=k
de
Prln urnare, baza reciproci bazei Fourier discrete este
tra
0 [n,.k] t*r lrr l22l
ace
=e'-- Ast
atu
D€zvoltarea spectrali a unui sennal in raport su baza Fourier eate:
tin
xlnl = J-1!F\rr-flixl"kl, r-e"'"-' $ t23l
\tot
Deternlnarea epEctrulul xtkl (notaile consacratd.') sennalului x[n]
in raport cu baza constltulte dln exPonentiale ee face ln nodul
oblsnult, thend seana de avantajul ortogonalltit,il:
X[k] =(S*[n] , xlnl > =fi *[rr] *-J*T (24)
sp€ctrul xIk] coreepunzEtor semnalului xIn] se num€Fte seri,e
discreti dsimgncarelutliu. iRxe[nna],rciiautrcpXerxe[ckhJeapxri[vnl]t<c-a> X[k] se
Fourler (sFD) a
nune$te pereche Fourier sennal,
apartine de as€nenea spatlului CF. Dece not&t:
wr=6tf7il (251
relatlil-e (algebrice) reprezentate de SFD se serlu natriceal in forma:
' x[k] corespunde, conform unor notatli Eenerale t tilizate
anteri.or, cu dt" sau a,""
'qds eTTlPlaTrdoJd PI TueAeJ roA
n3 aTpulrB8 ap InTlds Lr.rp 3z€q rolT'eg-replTTs(mPlpsTunAeTrgu(IXs/etranscaer?ergTdlTnrowrry)
RAT?T?*{ '}:Etsr/f,Jsupr1 o af?uT snpo:d :ETeurou alsa gsrpuJolsuErJ
(szl -"f =r-"
T3un1B
'x;=1 : [X]X <* [u]x TTJFuroJsrJprl BecTrlEr[ [ nc ugXou B3ep 'TaJlsU
'a?IpTTaTea e glebnfuoc eendsuerl nc BTpba 'ETpJs ap rolJEJ TcpTaJp
pT pugd 'a?sa sTar.{R B6Jal\r:T TcaC 'olT?TTeTaJ e P?p6n[uoo BFndsuEJl
Bun '!i/I e1H.g"rupg"6d)alupnTsn.TroT?regpuJfoelsTu'eaPrcul oeTrdadc'l1cTatJml:ausTaaezfceaqTuJeTloprrTuoo?3PJnareoAp1(apoTxaepXJocse;alpp
Jo?JeJ un
Tezpq TTJoI)eA luns TTJe[uo]slrs"r? TasTr?eu eTTTT.ET PJ qAJasqo
lLdl ' (r,ff) cfd ,, ",f'
["]'ltl(- [a]x [:r]'-es> og*t ,!^ ffil
I zN'A t!^
lI r;r< t)rlx [:rj's> G-ouln I
< I:rlx [:r] oo> ] (T-r$)It'i-'
TT
(ezl ,lr-*ft t, "lfh ,r-"F
< [:i]x [:lj'T0> ttItT-rv]xl HT
II t.i"I Ltor]jxx I N4 'fafa
< [)i]x t]rllQ> I rZ l{r)-- ' la
{ [)r]x t)rl l*>
Nu .II !,a
J {r-rv)"'
T
,l tlil$s t0 mtttd$ IT BT ]tll$fs Is 1lt0iln'tuitts
ttttt, etmltlt il stsl[tr I-8-tt snlrt tf ttilttt 0r
2,3 Bua Hartley discretE" Transfonata [artley discreti
Baza llarlley dlscretl esle utillzatf, la dezvoltarea setrnlelor
reaLe dln e'3, decl este o bazi ln Rr. Vectoril bazej- sunt datl de
rel-atla:
S [n, &] =0r [.n] =66s 1 2rg I , = (2e)
=cos [ ry] +aint#t , .k,n*Q,1, . ..Jrr*l
Expresla dezvolt6rii unui smna.L x[n] i-n raport cu baza Hartley
dl-screti este:
xtnl =fN-Fkiu-'[-k"l -e'a-sr' ?lF I ; .k=0, ] , .rr.-1 {30}
ry
Sp€ctrul sslalulul rea1, x[n], in raport cu baza ]Iartley dlscreti
ae deterrtrfi cu fornula:
rttl s*1 caaf tHf , k=0,1, . . .Jv-1 (31)
-ExlnJ
r"0
Uultinea valor{lor H[kl foraeazd spectruL Hartley dlscret al
sstaPluerleuci.hexa[nrllniltra<n->sfoHr[nka]tasetlsaprutlneeycdii"rsecprreetzi]i.ntE o pereche Hartlay
dlecreti.
Dsonatrfu in continuare ortogonalitatea bazel ltartl€y. Pentru
aceasta este nacaaar ei catculin urnitoarele stre:
{-:' (32)
,5r=1+coex+coszx+. . . +cos {li-1) x=f, cosnx
,.0
g-1
x-y
Sr=Binx+sin2x+ - . . +ein (lU-f 1 nH.0 einnx (33)
Aceste srne pot fl considerata ca pirtife reale respectiv inaginare
ale slnel:
g" +jg^=9oxJ+6rrJlg:lxJ*. . , +*r (n-:.lxJ=F edt (341
r-0
cars eate o progresle gcoetrlc[ ce ee scrle succesiv:
,: . i:r .t .'ri 'r .1.:.1, I':t,a;.1?ds
?'!i' ;t ;'tr' , ,;prr-r{i ltii ra;}
t'? - i, i. r-f,r r: I q i.S+J [3PS
I r i .ir:!_r, "ir:5rirF iJ'Ip {s
-r'1.; ;,.1i_t 1.+ :i1liit tr i.,:t, 1t!?;..!,
:i i' L3-I Li
.:,;:1s :3:,i:i ff-lt1 i111p
- a. i lS
I i,': :,.1t i i -'r i l,(iS
_ ..: !t Eq
l! f-i
:: '!- 1k4
i:\
::tii': ir"i .-
3i !: -Yi. ";i,t,
*l; l;i:ili'.
*.h
': Il
l-d r.d
;t.t i" ,ry
sO,
iff
.,,,,,;",, trl:-i, * i ir; lr{b r fllirS"
: A'3'[ ] IeF.t
lli 1:, I ii:: :i j::a1tr F i $fiititt UqiA3,.,[
" 1'' t lil
. a--
"' .b'
: i:r,1,.=-- ?{Tf]
i.{-[r :
: r:?e{}
| ,..:l ^t", lr?fl x-
i.:^1 --t -. ri.r:r ,i. r:Tg
' '.:j:,. t:' r.i ! t:s'.lili
', -_,: ', L. -'bI
ri
.::1 r::tl$il '1ltTtl:
silutt, encilIt st st$rtf,t l8-t * sttriul tf ttilrilt cr sIt
ro
2.{ Baza cosinus dismetfi" Transforsata cosinus discretl
Baza coslnus dtecreti este o bazd ortogonal"x real6 in c-. vectoril
bazel eunt:
S* [n] =ges 1?4*1] krE ,. A, n=Q , 1 , . . . Ai-l {37}
DenonstrEnn retatia de ortogonal.itale;
."o" @*t& , "**J?$fr*, =H *nr_G€:4SE*** !-zr,jl) +r =
="+\l;xl"f-vft"o* i?q$iejN$t+)t*"f#fi.o" l){k-:}r);lIJ.2,'"' _-^_.,_.j_-t=
)
=*H *"*{l&#-"r . tS;}ri.*p ."-4e4*. ,u;# *
)=
= f "ou S:!-E fi "." -U::#* - { s in J&d r ff s in -L&rF!r .
*f corJ&4-Efi *"*rk;+n-.f nin:3:Jrryfi uir, (ft- q'! *
=Ef *-^-oo(,-k-+t:)rn-e*i.o-l-&.1-J4*c]T$ls"{-'Ek+?i-}'"-{"ff--"-"li*--f*f-t-
?$
- 2tr -,- (&z+N:)o "t'""14*&ain*{"&l:J-l&fl-ry
uto (4:+l-ry
?Jf
* -] cos J&il r "" {=*g*q:Hl:i I &
2N
-t1"*s"1znN(rkc-!),ru:ff:."lffi=-
2Ar
li ji
:rrii. I'i:-i-i- br;". r
;ri$
't -:{ ': I.11 ;. '4 -.- + ;l-*
"
,li:t :: fi'..'nil{''". l{1
:'r - =: qi::',t '*.t-
.-
=,
r- rt
.^. {
a): o!. ".. r ilqi.tf:
J
- :!l;li
:. h !!.rlrt*.j.
*3 {"l@i:j j{ :t:j ,';
I
' r:; !i;r
:l I ii'i:'l
:
':= : ,,ri .:
!. :r
., r,
':-;j.l"i :, i r 1ii*jii!)'itf$*nii
Stiflti, elec$!1f 3f Si'!ii*{ ,E fd,ttr L* $Fulqt o{ sumtt cs stilr
I
iio+:rrr',: r:1 ai :r,,.:- ','*rlt.:. xip j in l::1.!!rrt cu t:aea c..:slnils elisergt6 se
f ace r:{*l'6 f r,.:rliruJ r l
#*.,.: -,i .- y^,?., lt '! 1k] vt&r.....,s {,rn*} kn . (3e)
.\; 2.1;
*ii)!' -.'j::i-,s s:ir:i,-;, Jr+r-ri ,--ii,1, ",.;,1t"-:t-,
i i-'
1af spe+:"tl.r.rtr {ti,'i iqi r.qtili,a,llnsijtr:-tq d:'i*cr:tlX} Ee ca1ctrleazi cli formula:
st rInl ,;'1,.= :,.-,J--{.a,jI*--j r:'l
,t
.e. ls:=1H'*{") ltt
nFn (40!
*,itl I',--l I gik) ..1" .ir ".r:" il-O,1.,, " .t{-1
d4^
{'i-l;rr}:,rilr'&,r, 13 61.'{.r.q::,tus di"nt:a."etft rie bazeazi pe proprietaiea unei
sacrjeliiif- r. |',.i,,i.:. *i.,,?] iie a se g'iltea eRpr'llnfi. fiunsi in f unctle de
cosil:u.xurl,, i:.r;]t:ri-i*r.i;;i! s€{:,;eii$a pa:4 de lutigime ?tl Si retinAnd nuaraj-
junEt e,t.e i j n ;:,.i r.iei; j s,i r:htino ll flnsf +r$ata Ccstllr.rs discretS .
COnsidelet:t-e aserirH-q{t,:.r,tqe .li! i."1gSi:1.rS:."r1 5rrlgi,11i1}ole }:e.1J-e i:npo"fe eondUC
Ia Cief j-ni.r.;,r.6 t rt.Srr$f ,rrq|rlt6j S;.1i!1S
?.5 Hagn 'ilir;,., i!j*'i;; +t'H. T'fAri'*,f*fnnta $inUS diSCfetil"
*az* .'1.;;::rl ir.,;s':;! ii i?st.a + a,l.ti, bd.zE e.!t-e'jnnal.i reali in Rt
Vector:r .; r
t, r:i 'SJn.lf .'s:nf {l--M$L'l tli-'Tj l (41}
I
O*on"tt5"r irr ; t;,::,rl"'r ,.-':-tr:gl.rqa.Li t.a.t.ea gi-gter,4\lltti- vectori,l-on bazei..
Avefir ' , -l{j9r'r- Bin -(n:._1tiL(I{ _11.1!uir., itr_1)G_!I =
(N-1.)
<*ul':J '4'
/* .t, jlx't{ (k*:)nl--.N-': \
[ r'
. L- c,A''d d I
'r- !: ': lt ,..1..1..--{n.+".l.}:n-i.r-..1.-.-.-l= J
--l 9
{*j:V:i':lr..':LrNJ' "f.t\"i
"iu##ffiil'*.*ffi"."-
(T+fr)u 6oc 1+.fiI s--o-c\ l-Tt$(3rt|++($l,*luv$:)ff)-iz,'1ur"e 7,
i.-lvITr+{J- ar ( .r*)rJ- T
6(rfif-r{
*(t"ry) z o-'*^^uldir_^4-.)^-.-i_*l1u'((rlr*fr:vf{)]i_uz re z
$ (T-ff) (r-ry) I-
)e+t f-T+Y:fT]F--"':-'-* tT+NlZ
l4df { F+:f )
I+AI Z
uT.r+Tl-eoct-
i,IT €I
- -u-TlTT.--fi"ilrrrft-r(zr{y.rr"-h)-:i+V-u,f!^idiu,--u6rTr*ynr;Jr-xej-:*:r:" fl r*,M _*_?, -
F:tl-ursT
r{mT1.4Ti_qure I+Al Z
- ----l:;'ri-L+lv,,G -*(t";rfufii;z il:G:TJ-eoc'=
uts
sTr -lv) tr -{ l ecli
=-di#Tr"t- 3,, ffi;"'uf n
t -!V
-;;ffi*", ffu6*€s); ."H{f"r" H;ffi.,r*i-
i+N *"" ts# t-ff z
uuEl-:t' 1*.u 4r-yi "o"T=
=id*T. ";,#fli."'g?-
rl I'WilIS t0 lflttud$ 1r-81 lnlsts Is lIlitutJ ']turuts
$t*gtt, ilpe$ETE 5l $l$I[rE 18-tt sF*ilM Bt tfiltiuf ex *!t *r
:'-*t-'-i*:'-l?kil#gff$#i - eee (k-:l) *e-+2s:sfi"fi,"i_6f!6{e.g6sF {1r+i) eal
PlgcutEm val-oarea expreslei- de pai eus in ussdtoarele cazuri: giL:
a) k*i=2p+1" k+1=2{p+1)+!., k*l Si k+l sunt sinultan impare:
css (k: j) 3*eos (k+i)-3 =6 ex,
?,1
pl expnesia este nu16.
b) k*i=Zp. 1"r1=2{p'*i), k*l gi k+i sunt slmr"a}tan pare; expresj-a LlU(
devine tiel
s- ir-i-rdyjl. u. io {Pjj}4tu un{
k={
_4I* -
e;r"ffi' t;;W 't i-l-)'p-r* val
{.-t )e-J est
*ts
* 3 q-31r.t { -1)u-* { -1}p":ur { -r.) e,r*p
Pentrll k=1" anaxn gueceslv:
t ***3ffi&fi *Hgs* [nJ #n*{4* [n] , S* t"r > =f *
j2-n.--.-1z"5nf=ru-i 'c"-oF.agff{dgli3*[=2J- 3 r-"*-uIf3+Ii9HH--"c'ial+s?L#L -
2
:Tffi"?:sffii*.-r--i'', *J2-
?tq&*l s-'-:-n. .:f&f+tr )' 23-e*-s* ?If,+ry1 s--i-n-?_(&ff+-l-)
IrI+1 #,
+. i -i- zkn .s-.-ln.'-o2-kl-;1-r].j=\trI."ftf"i_-T_S__.#* -
5u:.nn fr_r.
ZJ--tf *in& 1gss Ia4I6o"lsl3ru+L:j--s;"-*n"" r2\I-tr1["s"i"n"]rlEli-;1T)--r' =
;:;"T:-rryu.f-
-tu"+1-
'ges ruf;$$€xf p.xE{rewfi
3TTTtJffll$ qXsT"FsdsJ g=H ru?ssd pffrmpp'€H Ta3TJ?EG 6TTTU_T'{ I
:€pewg
ses rr"I!,lr#d prPffips!{ TT"rys-xosslr€J? E83T"1?€H 'T* nBs T Teffinu T"ToTs&
naT e?orDsTis prsffipsl{ ET-FTt3rlrld'T={I0T}e{T00}st00I}*{x,T> Ts TT**A
S=T rlJlued 'nlduoxe 6g 'mrrTq irg i{ ?*
'TOI*T /E=J ffi B '8=!t Tg €=X e } aTe:ermir 4rrns r"^I=[ Fx 16 Fx
t TTsTpiry Fw$Jdx€ erp3 nps epaxa
{I
''{s*} {'ryp'r } ffi " ffi { rryfr}tF i €l { Dry$0r i - t"gltt$- {ry' F >
!uTne[o] wi6rd Ef6ol=: RJ Ef"'
{:(-:}T5a?up noqgd** un*r?trT 4€?uazaJd'llfl d'$l sp} mT$sds W Eo ?T{JTStstp
B?ca tT*) E TBTJ 6s ssBJ e1 eeralnd pugluezerdal {rraluT Tn€np$Jd Bpim
{se}
:egle;a: q:d gn'prdxs 3s p.:pnisp'€H TneT;)yrN
w&affsTp p#sffispsH BlBers$$H'€rJ| "B?aJlsTp pfrErepsH s?pg g-r
sp 'u raT a{p r-ffi?nd nr eruels H TriT Bt€ }raren nxluad -1eil5#1}ffi$
'gS T$ soT"rsffiru 6TTJ*TSA TEen{J S?€a*-pTTWr gJ Trr?dES rry,:d aqeztr:a?i}e.xeJ
sXsssSTB s"€ct SAS?F3 E?ST*,€.X s1 "6p6J4 mo/i 'ts_rBtru??uop ils
{sr} ,r6.ffiW u$s [dI] x ffi,H- t*,"
{es} a*r,;"ff-*r{'-u-se rair K}ffi$* [*jx
:-sw]iis
sa$$3sTp sn$Ts TexB$il*fla's?.rpJ3 a.fF0J?zrrfirlsB-r*ir ?t'tlrrlf,O; gllrlB BtEilt
T+J{ f '\'!',-Ti: '1 8aJTrL1J4ffr"*;,l*-.*-*Tfl
ir*n)T**- E;*'iiTs E l+Jd ?
T+"IV il- ff*{T-.i-J*, ultr:f tr
ffuTs*"r ) **.*
*3 lTiltrl$ K lillltd$ ET-ffT Stlt$t$ I$ lita38t3'l"tufi$t$
sffri{"e, tif;tfii:i i! rtsi;sf tB-zo stAllut" tt sil$tt tr $[H
a,9
t'! 313. 111 tr. sl
:.r r"-1 1"-1 1-1
""3-1 1 1 -1-1 l.rr!.i
I -3,
I 13 or:i
l
t.
'o'e=i I -t "-i, "! 1 -1 -1 1 {16}
i fi x L -3*1-1*1
i -t _L -1 -1 ..1 .1 t L
It1-1 *1. 1" "r. 1" 3. -i,
i.ln e6wml ase's3,at cu epectml sbu fiadmarei constltuie <-" perachc
lg*dwrd.
2,T Basa FaSeg $fseretX, Transfornata Pa3ey dis*retH
Fur*?.iije P*l"e"d {Si decl rauelgrui, Pa}eyi eunt generete de formu"}.a:
.Fs**(*1")1J"}}i -J,k*0rL,...Jf:x {4?}
unde <j*"h-> se*aelif,dcfl produsul scalnr in GF{?r}, iar j'" reBrezinti
j.ndlceLe j exgr.'3m*t in bJ$ar cu ordtnea blgtlor invereatE" Pri.$ urnare
produaul se*lar {j*6;c} se exprleA in f*rsa:
<j-. k)* q;,u-rffirr,lffi{_jr:(*k1}ffi" . . {Jc,&, ri
(481
uj.ndePJr*lnSr$rs8".e.asre*l,qibLaeala"frFealSeeyreep'ex-teazerantpdr:e'1zl"eonr tabtiXnadree ala ntmereJ.or i 91
aranjate 5.rr *"?i& ;rx'<n.1ne.
funefi"iLe fladanard
Ur, s"il;sti43 s.siscEst cu sprcflru sEu Faley constituie o pereche Pale?.
2,8 Baza Wa3rh ffifseretH" Transfornata Halsh dlser*tH
Funegit)-e !{alah r*preaint5 un alt set de funclli ortogonatre in C*
Wde t=2'." eu ,;;:ux&r natura"l . Ele {ca gi nueleul WaLsh) sunt generate
de for*ula:
w3s* {-1} <s(J}'&); j,k*0, t, . . .}d-1
i4el
unde S{ji beicnmarnBifia.*&l"utiajpotubltincuXtdln produsul eealar &€ utllizeaz$
axprlnarea
a. pnln inversqrea bi.tj"Lor iea Ia transformata Faley) gl
b. prLn codessa €ray a rezultatulul.
codorul gi dee*delrul" Gray sunt sleteme llntare ln raport cu
eeil"ialel.e 61 operaSille din GF{2-}, grafu-rJ-le de eonversie urm&nd a fi
dlscutate s*.va mei. tfusiu. $pectrul- seilFnalulul x[ri] i.n raport cu baza
ttalsh se n'u!6€gtft ereensfo[Bata Wai-sh a emrna]"u]ui x[n] Eewralul Ai
spectnrl fssm&nd o s-nereche l{ai"str.
*F s;t ttTP lElruss Tfiml s "E€ss s{l'iif,{iri+ n;i ?**ctl"r lI{ eBiE?t$drass
s1:1-Yr,rj5-t9^ r--|* Jr"r "€-+zZWFr{* fir'E]:rq* [r] t-xeE
ZI
qglluns}EF (Z; lL--.t'r,., ou *uf,i -r6lgf* [$, g] :rr;* [rr] effr4
Pj-{:"j- ?.J}* to ,T+sB j rer{* [ti'S1rer;* [rr] E:er{
fln5fl fl J
?T5lu:s: o 9E--il* [.u '*+ ufi] rsr"f* {rf,'v}:rr4* [uJ trprl
6";r;u;s;,g f"F i [ru*. ', rc + 1{] r*q* fff'g] rers= fmi E:rri
if fp"J>.f;f=se'- "gil- I f.".ra rr'{* ldr u zJ;eq* e:ueg
[r"r]
f ""CI+tfli
: 4Elf!6i r;,ff*eff1rFf-eilrfx,.reersy.6-:rf€4ji-d5af.'.a;r.*;i]lx."g:vut"g*'1d'13*8+136t€e{'."$36$r*r'sx.t,e?qsgs6#'j[i6g$]6$'' 'tl{Jry}:3{'i}*TTq$g'*TT{[T3ud$]fetE#6:ax€;'gEs=msefJf6irilya6 "{
6+p wlpce
0*lI
dtg* lrrl'd6,U t*l" roq 3
{E6i T.S
ls3 s3 +€ffi;ii){Jil4.}{-\
e*a un guffiB*rse# {C TftT B F.ds*$d 'Si }s€E{ s?*l.3#{p T {:SJunS il sTefiTrd
t*€3 jU-j,9""'*'grftxilf f t{i" " u.g ,6*d' 91* [rrJ sa*q
gffe$J f f'-Sr"''-L'$*lf,
F
t{ t.*m} r,a rrf;', "t u * .affil} r.,*;.-r,i# t p*--[ -* [i-r'u] v-irel .rffi* [ei] arnE
7 tf .$rfr,.rg.6fr '*UJ
*ar ) .,a. "af, )
(f
l-cr1:lHidxrr "tiltBJ fig Tilt&xry T8.3Q
ptxws'$"-tsa{} "5sB*r"r3ap s* {:rlry'e* f-i16}t;}t!u Tgi J€*t{ s}BJpaTP aTTTteueu
HSSS*ST$ rs*ffi s?gff"Js$$U€X& .s?*J$STp S',psH prsg $-il
a3 3W$$ J€ l*lt$*$ Ttr-ffi il #]r$3$ f* 3tlfil*i3 '11Yffi3$
s[:$ili, ilBeulxf 5t 51s]Et[ "l-B-ra StiIIUt" 0t st;lftt cr
scrie in forea
jgr q sz)
x [n] =tu.fc xIkJ harr{"nJ
lar transformata (spectrtrl) Haar se deterninX cu forouLa: { 53}
l-1
Xlkl =/-ft x[n] har*[nl
l?.0
Penechea x[r:] <*> x[k] se ntme$te perecFre Flaar.
}2.10 $une&i1le oblice { ns}ant* diserete, fransfomata
*bIie6 diserett
iln alt *e":" de functli ortogonaLe este reprczerrtat de functiile
obllce ("slanto'i" t{ueleu]" nslantt'pentru N=2 este aeelagi" cu nucLeul
ttradanrard d* acelagl ondin, uruJ"tlplicat (tJ ?-t/=, adicd
=mj.lfi"^r
o l" 1 (s{}
-tJ
!{uc}eele $1. decl vectorli bazet pentru ordine nal narl (puterl a1e
Lui 2) se *Lrf.i"n util"lz&nd o farmuli de recurenlE
tAU IU
s",g F"rs -o* P*
r;*z o
o p0--o1- r.. ^ 1".. o I
rt-t o U
t;;['*.4a'.l=l llJTl 0 _Trff_ { 55)
Ioi-F.,
1
u*
unde I* este ilatrtcea unitate de ordinul k, 0 izero) narcheazE matri"ce
nule ale cXror ordi&e car€ rezult5 d1n etructuri, o"=10 lar *. gi B*
sunt scalarl" da}.l de ureitoarele fornule:
,"oa=-[ta3sI'-i:r J)+ I" "t'"--[f irt'-r i* {56)
4]r-1j
D1n forrnula d* recuren$E se obtl"ne, 1n cazul particrrlar l{=2, ci
uatrlcea ao este dxt6 de relatia:
'JO?pZlmdseJOs TaZEq TrrJO136A ep €?Eluezarder rpT{JOJTps
Fdnp roTs?ueuodroc a1p1pn1tr1dee luns alnulfqo aTerawrru :?ITctpTBA
aJpntrrXuoo uT a1sa a"re1e:dralur Tgpaacg "eair€slo+supJl EutrEJa?ep erpJ
rJToos?ruasfszfi;ts}i.B"pi JeJ,iH"+i'i:*ro?luFj*re€BETe8€T}uasrE'sr3LnrBrp?uosBJodUu:.zTnsnlaXzTpeUcaduTrnTeT:'1TTnfnf1TiJnnoT;dEaeu1udrreaspTuBan?dlpue6aapc?:Fop-iIeeaTle1BuTuoBoldTua{JnrumnTgJc
pqutr ze:da: T€ums Tnun eTa?enro} suerl er?u-Tp BTs:gJT Jo elaluauoduoJ
(8S) [{]x= { [ujx' [:f*u] S.[>)t-u]g
!TeflFV ,,pTl]aJTp,, gdnp .C
TfuaTaTpup;fotuTo*Tar{e?}?u{"ua{armu}JetaTeT*aa-uraonJTTlau{lcuTeepu'o{fdn[:uu{o-noua]eglTsTaauot6pT{urT?onTuqTpudoauueTaaTpzo?pEqEeczprprpl;Jen,lJxn€cegTTXnnlrlseraldlo€pduqEul}etr$ezu9leq
TlTTunana?TTurlnapeTwFTtT€aeTodar:s,d$Tl:posmsT-'praa€a?XoRlTugspppa"oAufcfirTIpJlT:mBaeaTpTrqElqJouJTagda?plnJussEsare€p3euITp'loSea3duTsc'o€1alluednrooxclnrToufe,Jnonuu;TBTnBlroJc:aorudpp?rn3aiEEedupAanBerTssa?llndpicg?oeSJEdue,?e?*lseu)E3aaeutE*rJaIJV SpT.ufFnengrzTpITlEsJtrpeuXadTorrdsJ3J
ad pleuortounJ aoTro p'pf.roolrse6oa?pTraaTJeaTrpn,pzsTeSg€gTTrqneTrBrdaasslJ?oreaqlTTuHrroJauIToTJ{EdE
aJ?rFp TffEsT3rowozT
alEuoTlsunjl I't
*3 t{I ffi&tuugdo I$ xTHil0lxil{ffi '[
'r?uafsr aqaaead o ?zeauJo} [u]x TnT € uXr,rpTs,r E?puro]suErl e?sB
[qJx apun [:{'[]{Xr]<x-T>r}[TuBJxftsltarpr{T)eFJrapd?sJtBtrrTo']es{lerp1"r;u?np1B{rlyszeaJEdBa"lrErlo[JluoIzsrlmrEdrJs,a:oc
:ueTtpJuo"1u€ep1e[uu]Txa?r*oql3TeTAJoulcneATeXuun?suBafJsp!,urpoaJcsrr.rrpl€x!?&
ar?uTp FnpoJd fruo?een
ep TecrJ?Eur aTTTuTT
s/ s/IItT-ci
I
t -e_ _sr_^ll|
IIrr r r rlgrd sr/' f lr ' -'{r.$ }rr- Y! iL to
tsf 19=rtpt
tl*€_
sy'
I
.i I'l!ff]s ll lntts*$ r.-8r Inl$t$ !5 lilfl3ilt 'llil$s
sFswl€- ft*{€r!f }i s!}igs{ ld{.tj l Iird, ig.t s[ .vitidai r'; 5tt
3,ft Wrmtmr.t elrt;e s'Bg;r.r*slent"At tts G xll!iiue $€ *r:si1l: d;.;.r u$
Utt *pfirat(}f,. pg *d mf-x"$.c*f e"rrclt, elmentc dln fr* **t"fel f;**&$, ac
&tsl*t:s sau *o]-sesieSe pr
awalul. {vs*torul} rezt*tat ca pr*dtrs i-ntre matrte€,* flg"'&F:&t'{}a:u}-qg.{. Si
!s*ct*ffi}} ecmlal pCIate fj" fnt"erpr*tat" ea t{*uplu eixdstiefr R ntffi*re 2.
reeultata d$.m a*?iunea q:clor: ffi funetrr;nale pu'er:tx*te cJe IcnltS* ba
c1i
ij* ha
*atrf*sl op*r*te,ralul , Smlete de c*niugara d1:r mesLtxtl"L al" dr:;lea eueli: pr
Rse*#&re penf-rru a respectn definltla pradueulul se&lfir.
li!Jr iI1n ] 1 an'{*ri I x[n] I
dl l fll I x[n] f *
1 i
axls:i = tl
II
lIlz;! 3 !iqJi
r, fur56wa i'ffiJg opffietffi egg$pp& t;ffiJ'd r*msst&.s" rirg ffts rt+3 un op*ira*c':"
? euncsest prln nndlrl" In eare [:'*r:sfery,* veet*s:i.{ baesl {presu$rrag{*
*r'8og*na1e) l
i >?r&* [:rJ *g, lnl " *ffi ts$s [r,i Lie*{*.ts[r:l , S, [*t] {$$}
Ac*ti.{tt*e nper:at*r"u"lui esrl*re uiirui #ffis&i ;i[*l e*'Le: {6}"}
i I -,t -r{tnj' l*Hi.o{[ ii 3rry l,,t - e}Hl#.{ti!*,i,,1
Y [r:j -*&.rri{1H "*-,,,, 'r '*;i/'F!r r{^,h! ir}!*un", t'csdr
l' ;[*t:ro
*di"c& tE *tr jyf _. iI-.l -. s-r'
*] - * j.0 si*x[rl - *.r.0 {$;l:ri , gu [;i] r,tIr {er}
Uri s]i p$nr;t da ve*s;'l;e,,s:'$st.R i.$ *aeec *& *;j':e {rpet"a&c}sr 'l{n eru ss
&'#d*t€ s6t i=e c& s su&A F;&dae.tic* de epsretori de ps:sri.ee$te:
uTrd p[$rdxa as ezeq TTou Brdnse 'q 'lo1e:edo Tnun paunT{oy
{6s} egrar S=xp
T,.lt
1a;: 1€ ezeq Ea{{}€n n-r l:e?oedi:eTJsw{IsTasTruun?ona-xd1suaadJl'uaTzepqeTltTeo;ua;:<lqlgfTznuznadusa:oa
C.{ 0"f |-t 0*:l
3t* ff,-rj*S{Fur
T-lf luix
3= wlrgiv == 1o1{${n f,=
I-tf
r-tr r-rv
: rar{eTal erol:uoo ro1eTsltups :o1a:1ca<ls p*:PcltfiTLtros aanpr:d
I,!-9i [ulx6T tt 0.x iu] tS
*{,=
T-tf
:s{ero3 ug al{3aA sap ep aT{;xinl uI qr.("Idxe as €r€3 'ezPq gnou o
ET peJala-:J, 'gieTlTuT pzeq g?utza.rda: 'T-N"""'l'0=[ '[u]rS apun
tee) lul l$trl o-t iri tqr
3=
: TAZeq
BTETsuwar gffiJcg${.raJ1 as aJEJ uT JoTaTtuuas s{aJqaads ulrd lnssouni
TT€cuanpT"r?-;solelle1nruH*s1[e:s.3*rsaud€orX'1ee?:seeuJ€a6e:uP31 'g3Tuouep pzeq TBOTqo aespp'euuq
uT Tnpo$ uTJd gzeazgca:d
TnTteds ug ro?e:ado Enm eaunltcq ffluus muvsItltffi r$ ruogtngq 'e
{se} .((r,tjt'rr^"*r'"*?tltg)g
:urAap el[{]x '[u]x TrrT :o1;T{aalord
aTT-rapficd : *a?€csoF.roog Bp etex€{ ad elnlsace aTa?lreuoduoc u1 ezeauol{ee
uTclqa EJdn$e Tn?riasaTa Jnp aJEO TTro?erado luns TTro?celord aprm
{*e} C.F il]t5t$ IS ililtilm 'ilut$$
,t llT$*J5 i$ IRItYd$ i [ujx)rarr s= ( lujxiJ
T-19
rr-87
$fKtf. fll[Ellf $l Sitl{lr ,$. *i.'z,i sp&TItlN- Ff srff&ta tc
16,Ini =H **,*ui"i -$ *,,tH ir'.^s,L,,j j* { i{}i
arr /w.r I
-I lH "-f*,m,n16, [nJ
iJfr 1tr"6 "" ',
s*ru pren
(n"t \ *t r,--r n",/*-: r i,.jil"r'',,'
S, =.ffi
iur ={E tr,$* rnr j=p,n,*E r*i0: fnj ip;,n,or*,js,
Egalfuid eele douX expresil rbtirlem
ff ro,*ru'H ,i,i,pf,.p1 { ?4,,
"k"0 k.0
*auu in ft:rx$ matrj.eeal6, {?3}
gftgW, #=*f-sgff
tn eoncluzle, matrl"cea operatorulul T in rag**;rl frll fi{]ql&. bea& s*
exprj.m6 tn funct"ie cle matrisea e'peraterului ln rapox'i cu baza v€elle o
pr:"n i"nteruedlul matrj.cel" sodafe H.
3.3 tr*lim*ffie de anuler*, fl&s&cterfstiee gf mir:imml* n}*
un*i matrice
fle gl ?,n eaz$1. spat{ului }d", nui*im poJi*om efe e*ulare aL u:il'+i nia'Lr1;-*
ft *ri.r:s p*]"j,nrm r{d} eare se &nu-leae& ciraei el este -1,nl*cilit i:l [&l ad:lcd
sati"sfa*e retre!.ia; r{ [4tr i=0" tn partleular. 3ro-I-ri*mul *araeteristis a3
taat*f matirlee ast€ defi-nit de reJ.a!,ia:
F{d} *,5*g {dtl'l - [.4] ]
{ 74}
gJ" e*;'i-e p*Iinm <ie anul.are pentru mat::!.ce* [Al conform $;ao:i-etnei C,iiy].ey.-
i&le"srqi!f1fielfi,laornp{odletrmnoexinnusltrpagir,iianaelstael ldentj.e6 cu sea de 3a *p*!tul *{*), ns
unel seatr1ce [Ai
+tste pclilrexnul ticuerX"c dc
,:i5:*fd &jrli.!,it p* {d} pentru care nn* { [A] ]=0.
i'l,.;.i.iu.s*ul miruima} eet"e ue;tllc St r*prez;i.rri;& ure f;Ect,tlr 5)e$t"rr"d elri.ss
i,*,i!i!frl* ik: &99.1*"*u iu giartict.llAr pentrti p*llno&Ul Caraeteristi-c F*{d}
rliiLi .tli.i*t.rj fi. el iilsug:j" p*linOm minj_mal",
Eq$-.a1 uTp erasnpeJl) rolauls?sTs eT."rTo?eSLsj'n{senTgo'pd3'Tnu'q{-a{ap&cpgE'v''T{qao el$sa
n:iue.d aBJ as aoT1uapT 11{eraplsuoc ':nSTsaC "T€uquTiu i\T?r}dsa: oJ€Tnup
ep '$T?gTJ€1JEJB3 TnfnuouTlod Tnpp:6 Xgcap f,Tu Teu per6 Ep uior.rtrXod
un*;1n'g:d a3TJX€E o ep 11{cmJ Ta.rEtTJo eereur[Jd:<€ ?Tuuad TpiirTriTur
nes aJ€Tnup ap 'JT4sTra?3s:eJ eTaueouTTod -g 'pn1n5{eds tnz€p ug g8 ep
{6rl ( [v] ) r= ( itrl )s
'3TlsTJa?aH,J€J TnuouTTod :alrnlgxnd
i.ry '[H.l Ts3Ttr?eu Tp aJBTnuB ep uoullod a?sa {p}6 E3Pp '8put'r Bp
{sa } (p)"r+ (p)E (P)F= (p)J glTnzaH 'tsl6 TnT
T?Iu"fp;c) ?gsap pTrrr TBr[ TnuTpJo pug^e qsBJ ruoullod m a?sa (p]J JpT d{B}e
gr*pria:mg1od sTTJnof,az ug g1e1nbef, poT?TT€ue ag{:un; o eese {p}b apu&
{r/"}
a"r€}a3po :aTr3s a?Pod ag 'ucrrTTd nn {p)s rs
? rgrr{ ,prouTulgTtt€*uupgacTf,JlmruJsrour{.Tuu-ttg{'g"l"'u0t}rIf<lr{-w{oIs-r}lot"d "'0} : tp}} aTd "e
ffiH rr;H":t"xry
"rrEprof scTuouec Tauuol pundsaros gpuobelp ad
ap €Tg"rncoTq '1e:auo6 uI 'ty-p EurJoJ paAp Xod (p)"d ':etrnoggred u6
{s{} ,r [ (s) rd1= 6116 tq[(z')td'] dtr= @)Tdt'1= fr,*
+i tp)rdJ= (p)ra try[ (p)td] t'I= (p) rat'I= (ptu
x, eerxe oleod exlsTrs?r'.rec riluourrod :ffiJr1li ilil#d;ffi*:,l?lJ:-;$
uE H?suoo sH TnTleds ap giel psf,Tqesosc '[u] Tacl:reu! Te 3T?sT"ra?oBJpp
1:rwmu11od alsa snpord f,orpJ Tp 'U nss C uf TiualcTlaot nc '(dlrd pru.sos
6p apTl*TJaXJpJEs arreouTTod Joun pundsaroc unduoc o aJpo cTacTJ?sH
tol{a 1nl 0
{sa } lclEa 1yl
00
:pareoXerfitn alsa Tg 6gspaTp 'gtrn1"l1od l{ a}T:}eu!
v0 i13 FA{IAtrASP ereuuoJsup.rl o-:1u1:d p?TnzeJ (p)"d T:pluamaTa
TT-xozTATp FUEAS H SJnrJVr{ rsHn xruHos.Lw usrwo$€3 ffiCI$ 'T
x3 }TY€SI$ 1[ l$lt{ds LZ-87 t$:i5t$ t$ lHft36l3 ':luilH]s
stH[t, etmuilt $i $tsTtfr[ X&*r* sttilu 0t s[ffiil.t {r ita
transf,orsata z. 91
p3&soi"sscfliIrebhlslnl)dmdeaersuuconrrzelxalla"ndtterrelancnoeEs-fnAllonamr{ntrroErixcprnel*}Rj"cdmlelnaonlga*sortneD[aL!.H,Lan6ga"dosenAtatfe*lelntzlfaneirlnenr,caaarpeuteanonup(tiametoruapntt€recarl;"narustcoLfSortnrrimdliHnee-i,r"aseaAtaetr{ de
&eerls6 de matrlcea & 4.
Y=Ax {80} !l
s€ urlXre$te deternlnarea unel (transf*rmfui) natri.ce H {nuslttr qatrj_ee r€
mdal6) care s;, stabflea,sci o gt
re]..af,i* de frlrsa 5.
x * .Fft' t81l b
;X
crr noile eleaente q (natri.cea M realizeazi o sehj,nbere de variabllE). d
$e pot scrfe suceeelv ral.atl"lle:
y * A}.83 r{-13r=$1-14169 * Ag u)
t82!
H
*0tend c1
z * l€'Lv {83} tl
rezuLtE z*Aq ;ti
{84}
ca€a ee reprezlntS un set de ecualil decuptate.
Yfo,re€e1ca,oz2loEa{m*naabltaenzicxnearectrcoiJelo-peerlnccei iocditnranlesep$}eetmf.ciueirnlretdeal)r.zrdcaorebafazc{,plaarrtle.fuertirie"}meenLtueLle
b{*a
xo
nac*
;4t* [u'e,.."&r4J (851
gi tr",rr,,.,rh rvel*rre(rzrirn-ttrl) baza reciprocX, se poa?e ecrle:
q+(r*ry) tlt+. . . +(rd, ylu, {86}
D&ir
" ,i-"\i
y = Mt6-Ly = [ei. ii?, . " iio]M-e;v =[ijr ir' ...u""J,1l.1'l (s?l
I
in i,J
SL" .y *cot4natirn+uuaarzea,+. ..+uaz* (f1r yTur+4,y",yyt4r.
= " ,.r€xrrr.l,)t!& {gg}
W}de yo zn ua gl ro sunt weetari- {gatrXce qoloEnAir l<ar :sr r.rdlt. r,r:;rl,r, !
rapreeent&nd aomponentela veetoratui a. R*ault,& q&
:r:'rJd axFsuTJap s$
p p plPmi[(tJsu8il
'Taf?sv 'n oTnpon cpJ Tprs as nu [u]x Tswres Tnrm
ec ilartqesoap no
acTlatrtTre aTTTI€rado
(T-.F TnmuTTd '?.pzTTT?n TEU 1ac dwrl TSeTeoe ug rS relrlpll.Ted ug 't{
pe:6 ap 3ruo& ruourlcd rm oTnpffi aoBJ as rofaeBouTTod lnsrpord) *H uT p
Te?et[JrJls{,rEJ1 TE orrurJep ap TnTnpou boTpuB e?sa R) uTp friums Tnun B p
IBTFalrO?rpeIaT€urposlAHsuTapsnlT?TJlpTadelsTappJTflu€rTolJu?aT"pu.reaaJpradT: nn1pnpo1-nos1'PrTdTeBec?8puaJgBE lJeaufEJUoa:lPspuzTe?Tr*TlW?urElmonppdoaJf1fluil{qTf1rso"d9
'p?€zTJe-,Id gzpq o
pTctrarplTcApTrndJTuJT?p.oTdrfruouTs:sa?luTEE
na lrode: u-.1 rurqcads e?sa eJe?erdra1u1
uT Teuuas un Jnp e:Pc Tro?Eredo
o3 rJTp
e1e1ueza.:d aTTde&ro;slrprl 'ul TnT lol alrgtredp *C uEp T€rmrss
Tmrn e '€zeq aorJo nc 1:ode.: uI 'Inr?Jedg eca:poeq fAlU ffigq) 'S
'nH fnT{Pds nt
EJnlpnaT Lr: sXe?u€ze"rd la1ac pFofeue eXse eBxp?ErJ' 'NJ uTp TTTqesJBffiB.r
1:o1e-redo ?$Ttrgc lrncs ad pXuazard Jo/! as pz€aurn aa aTa3 {rI
prrol 'zEc la63e uf Tg aITqETen ltms ( '31e r{lTt[S EOTuouEp
'1ledlcupd TrozTATp) oToJp eleluazard elaXdacuoc e?po$
poo'AXnnseraoprJopTIaoTJrlnutEerloaTdI:sqSsnXp.p?upe€sx.'eA"geJpa1unao!eJJTlseTr1Xonned1uoua1roXT1d?ueeaBucunreauTada!pouR1nu6?dens'lgeul'Jee1ozueoao:Ja6daue-aTrpe6peccauumdegpFau".rBruoe(uo,fnS??ppopou?TppT?ilaouoanr:rlsuusppg3
pTTqTsc.: a?se nu rolEredo Tnnn pe.rpzrTEuo6€rc ff$U*UXSq1 "$
(16) frg - 1fn rr;tr acarsoop
[<on,*:r>
:l:
lo{06} r = I
.'.<en'zt>
Io ol l= ["n" ' tn tn]
[l0 Ottt "trj
{6s! Br{lI T6
,!? illruI$ il ltiltds n^,:., ,l
"<t{, zz
rl''<A'Tz
67"*8T ]!ltst$ i$ ltiRlBt] 'jlffin$
stnilt, cnoilTt st stslfft l-B-ro sttllut 0I siltil.t tr 5tH
. .x+[nx]tk=lr6s1[sn1-0k][n+] .+.x.[+1x] t0rv[-n11-r0] +. . . (92',/, tra
[n-.v+r]
7ut
D{xlnl } =x(d) =f, x[nl d" (e3)
a[n
,.0 est
Prln urnare, toate proprletdtlle transfornatei d prezentate in 1ar
capltold. congacrat gpat,lulul X'rEn3n valabile gl in cazul spatiului
Cx, crr deoseblrea arlntltl. cir
7. fltlonrrlpl r\rEDcEDsmcnrrT5rBDsEzxEAIrcrrectEofA,ILcEEcinR$oERzrnrTrrrrtr{smDrp.Eou*r*EriHoD,rrrrRrI d1n
operatorll de aceet tlp sunt descr191 de natrice ldentlce cu cela Sl
prezentate in capltolul conaacrat spatlulul Xr" 10.
cor
8. Go.-*nr" rresa? urrr sExrAL Drx CN; DEpTasAREA lroD{rr6 o["] .
fel ca in cazul spatiului X', operatorul atagat unul elenent PAI
alnl=ao61o1+a16[n'1]+. . . +q"_,6[n-N+11 { 1n particular a[n]=[[n] ) este, aaE
natrlcea [A]-r-, (1n partlcular ca
prln deflnlt,le, operatorul degcrls de
un
IA]or-, ); t000 0 -€!ol
o 1-a,. c1c
I' o o o = too 0 ... 0 *1 ( e4) 1ar
-d. Acr"r 0 ...0 0
Aator = lo 1 0 |to 0...0 0 tra
It. l: i 0... 1 0 prE
-;,j[ooo
::: ; [o o
Actlunea unul astfel de operator asupra unul elenent x[n] dLn l{.
corerpunde deplasdriT 7a dreapta cu w pas nodufo eLerentuL aInJ adlcl
nodulo pollnonul (nonlc) dr+ax_1dr-1r&._rdr-2+. . .+ard+ao:
([Ax[]1,,]r,-xa[.nx]*r)=6-lano-xZ*lr0+.[n. ]. (x[c] -arxr-r) 6 [n-r] *
+ + (x*.-a"_rx*r)0 [n-lV+tl t,o",_,
+
tn cazul partlcular a[n]=-61n1, pollnonul este dr-1 gi ccrespunde
deplasdrTi circalare Ia dreapta:
lAI tro:xlnl = xUtI-110 [n] +xlol0 [n-1] *. . . +xht-2 j 6 [n-Jf+ t] (e6]
Toate conslderatllle referitoare la acest gublect fEcute 1n
capltolul eonsaclrat spat,lulul M' rEnAn valablle.
e. t[*]uCnadgehlllydipn[\nlrcro]amwzeutsrlnltAespDeaBAltelcuDmulO"e\uIrnAlotMuEdFrnit{,nrna'AcCLelEl*uunHarresOxaunsp'uInrOnafiuAtultLfno, upAelLreamlteoEm*rIoLedEneuAtf*oxSn[sBnnnaql oNyde.(.,i[nuA.H],enr-[rsr]el
EIPI 1Em|-1c€I3 Tt e?md nu p 'rErtpor TnTtEouTToal EerEzTcord
al3llro3ru"Il .p
t(pord rail TtTx or
TsTtpTaroc Tg TaTtrnTo u@ B lAT?Tntup eug6orrT o qtftrze"rd E.6Ta u3
TosnpoJd epmAreroc Tl 3f:ErctuDgs1oEd n1o3pqepceeocfTnTpcoTr rp1xao1T{nsllorlarouJmrureprTl
{oqr) t:flX,p -r [ur-uJx
e(EptdRI1-!€BdGr)pooTeTaE1lpBoganrpppenTecslle@eea:r)rogfTs{fuTreilrrre!.fTezTpr-tnf'lflgrdgjsTeT:rToTtc!ln?TepIT.aprfdTuoeerrTrd.teueEoesneoTeper1me{Tepear1se1oooTlpuTicl.Tgmcc
ilTnzar.t{ fRliE'edTETCetrBdTsteApluenlcrlureozTe."rrcd?urcoTec?J me6!o' Ter1ue11e{XeureeJpelpruTosucoo.BWsuqer8sE
tr'l'fBrposu[Ju[a]IA.oIdufs[ (A[gruTeiJt?BfArrefuodTnFn{lretuLlre2uGlp€rBTtTFrtrrer€asInBslo.crootno,elt[,rn|Erof1TifnlnT'mpJnrfoTnErTcttTrrppsadlfTsnslnnlToInomazpuucoorfucs,trvrsTerpxu6lnnmEdcCsclalr.I.eoodor
eq
u!
d{
eue?l:fEsnEll$orntrArTunnfomlTpperTurTlgdnlTrnceco-zTa*u1&6Trec1Te1ndpTeen"'rT3en[6s{teDEJdaolts?1T-sn6tT3r1g[?ur,y.r].ra.xsn,uI{,oou0dToresncro3mToe"srTxmltpu3ncTTopJpTeresFlJetEJs?3derBeldTeTenJrplwjxreT1pT6,
t ol,f t rl .{ lc -Nl'd Lz -t{14 lr-nl4
I 1 IJ' oJ : :J' rr"[? Is t?,-^rl-lifl
,
rtr(65)
I €l,f 'It i t *olnr{l/ r rli t ?)r I
t zlA il I
lr t ol,{
t I
rlrt E] I
I el T tz-ifl^{ lr-trl rf t 0lr( I
:IlEtnsr[3 sesTrlg! ep sTrsssp els€
[u]n TnluekTe nr e-r€TllsJT'Ic-rTpeTrinBTfnopuooroT:noTlerzrcuunTdloedetoEoTpJre.rpol,e[u:eJdgg=[u]p
TnTnluesle uereldopn ernrdnseld gJeTr:.r,4p eFfnferucr ,J?TnJT1rpd uf
t86) lplvpcccFlx. [p],u <-> Iul? po@[u]x* [u]d : [u]p TnT
Wiple 'lrnJs?nfpnop:odf TnNo.rTTad o1npod [u]r{ T6 [uJx TnT eT? p sfet€Lrofsu?r1
TelsEOe p E?Etrolsusxl n;rd
e.r+uFp eprmdseroc Tl TunTFe
te6) lrrJx ( r{l[ ttrl r{rf+ ' ' '+ lrlf [y] s,{* trlr [y] trf+ yt; orf ; =
= [u]x I ro1" [v] rrf- rr)a00r wrx* [u1't
:.,trd grtrdxe
rt tffit$ lt utltds n€-87 lnlsts Is lltililt'ilffns
t[Hrl, clFt*i?E 5p sl,ciiat L#-=t sp&Tls|- *{ g{Brui.t 485
xlnl +yIn] *tEr-l' x[k] y[n-Ft ** stdl rldl f xfll i
ftt
eLc'rjd"sre a dou& -ee<xrense" s€c-lrBn&a rezul?antH este s€sv€.n?* atre e#ref
e6wt'a,toane surEt srffia produseJ"or egantioanelor eare se af3,6 wne]"s srp-b
a-lte?e pl@stte'u faaeere d.trn&re eela ll pozj.til ale eil,indr:"l"er:.
-/\ " __""4 c
i- !q
F**fuk+"1 b
*- *{ltg:-.:-I F
--..1
f'- f
-.-s14
II i p
tr"="'_-wi grf,r*SQ r,i
_".'a.*-,'.f,..?. p
F"{9,3" 3xa.e.q$"me fntu$"t*w6 e eonwolugl"si ej"nsutere a du:r',E seevcnf;.e' ei
o,
Lry; eer, Flsc"etrffi3]"er de e.omwS"usie ei"eLles a d*u6 servemg"* ,*ste a*eJ-a
$resgf"'!: ep.ss e*3 ptc$l$l ultj-m€I"a h e$antir:ame aie unelm elj"rrtr* ssc.yec)g€ 1.
$l e-e"I,- r:+4.::x nrlts-mele w*k-} egantj-oene nl"e eele{tra}te s*elunm+ee *r*ll'f. fi
t.
rsaif,"*, d
*#**n-{tF***00CIO0CISSOSCI0SC ( k e$a&t"coe&e ftqd"ml dr
ur
**j?*****&******00000ss00 {bI*k*l apant:"oarae nntr*}
dr
fn aeeams6 .sg.tl.l€g,j"e *onvoJ.u!"ia clellch ennduee }e eqetr*sj. ::*esrltat
cu eonvoi"teg,""!-a t i-niar6" t)lt,
rr" vdftiam"nlawwhgf,lfge*-l'T}a*sgu.sirstehams*ce&rewebualrseoiapll*eeraxfdgscieimLeevtasns}agauelceoappienltroaartueoelr"lsLteoednaeszeaos#n!r$avrfltlrlustmgeim"genearm6lemeLr&oanrl
cqrlaxe, ewsls g"t. Bnodusele se f,ae fu nori oh*6nuft, 0 rle*eehSre
eeen$i"al"H eonst6 3-m aecea cdo ln Cx nu sal exletH sacv,enge x[n] gl
y[nJ p*ntrtr care soncro]ugia cicllcE poate fi nut6. Aeeas&a ee
datoregte fac,txxS.u$" ed 5n spagful {Hllhart.} effi nu exirsBE
dj"wj"rsrf
a1 lul zaro; pr*elusc&x m dou6 porinome in vanrabtln d se
&rr,ilenaffi
dac5 6t nq.maf dae# cele dctd polfnoama su$fl nsl*"
t?" &rgnrnms sxffiffiI mE H&rqrfg DTAGffil,ffi f$re*prLild. *semr{nitto: eil
spa$fuJ ffi o a*pe*mmiteler etf,atrsstrssE.
Te;3"l- l-rjti*YF..'1j :.! I ,::11'r |i.T?cl:i. i :'i .11,:l e! ij'if..j'ri"i" i1l: }!':Tt1:':1:}.?_:€x1{: *#"3$li
{$*E } I ,ry ,ir{l " 4' .::' = i iff]-xlr:ruf
i,:.r1..,': ..
i';-'*r.:i'-eis$ rir:r.- r'i*tS *.tFdi$'i TTAlgd PA:€n"[' *
'.tl^o"i n] +-'r-iJ,:i!i tr,_.i!{j ir.5.E-,} a{gusst+"rg; *n*g: alel *d ep
;ilTe;:,,a'i':,i'ln;!}.:i:.i lti !'i'ni.! r.1 i"trl.i..itgi{rt.-:::i:1.-|;:r,,i,Jij€A* *xs B}T,1"ggul * *p sTJ*sap
61ffiF ie3'!*ilt_ .t:::-::]i--i",i!jstas ;i};l';si.: i;';: ,:;.;;r.g iT-;5,{f ? a.TsnT aF $?Js?p:ced$
{ft$T !
''iI*-_i-.i ;1 '1'r,i):., I :':Tj!'.ii.ryiiirr 1.1ii-.;'.i.uTf i9.j.{r#'tt,1.1e{3 €BJen'{ *
&{ts"T'}Arii{ at; iir-ir:,.t.il *,'.',,
i-''r*i"s-1 :iri:-J'"" ij' !J,,g_*itaffiixs:i'{?,t:rijr?.n!n[€pBs}.{yr6?r6!€n(iT.t46}};€*?tru{i
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'!":l:]i{-i jq:: },,:"!r ,,j:riii'i I j 1..:,r ill[.]'] ..'tit;ifi'lr:: 1,rt4-.. :i;i'itirp F:{{juT-fitruT {}?.ryd €*,{F[]1 ...
;i';:i.: .3"{?:,}..i
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.ii .ge.$pfi{ .*
i'i 1 .ili, 1',i'. t::,.' . j.: J?:r"f F, +},ES.l :tll.Aqd
111,.1{'..r}l#?*rrll,, ;,r;Siii i
n i{S.q E!":i:i il,-il:i'l. l'+tj:J :r. ll:'l .lp.-{ jr., :.i'ir.r ..,{i 1'S
sE \3_TEEri4ii! :il::,.. '1 ' /-i -'.'. t,;ii:.1 ::..r f-iii.iiiii,,jrr:r jij?:e 'i1;1't t rl ri.:|li n}t+ : ; Ii:ve'i;{,i, $)j'j"#t i.t'4 ;.}&}
I".F! :g+]rli:ta::,;.: rt,arl..::iiji*? -;i],:. ?r'"i r'!.ti:t:i,r?
1ta"._r1 .;.rJ ',o:": s'i.g; .r{rrTt3?qlm€€s
.iii"{trifr.3"f€ws tH{r&k*1ryatrJ, .r$1H *i.
T€?:l?r:i iliIii3qF3'r.1 liii :r't 1..1i."t'i;ir.! .-;i 'i uj"s
j1r r, .'_r J!'ti:!nti r g :i : i l: t :+_td4."{ .+ jtFalisi:i.rj}1ddri.,Iil"l
Tl?fj{.!+1ri j{: altrjlr.' f1}r,j:j,!qi;: i, i.:. - i..; i tr, ',.:::1j't;+,:8.Ii..:!i.':iF-.,t;i'ifli;r..1! j,,,(;:a$i;lt]uj.l-j!Di-.r!-':ti1fr:jt.i-{i.r*t
'f jY ":
i :r'j lit .i'
+:lE* ;i{4.r,:l:i:i":) t ':!: i.1.,!:tql:i:: F::., 'f :jl: li 1 ,|::_irr+:i r, i:1i\f,t!Fil!:- q: Lq:j.'i',f ,', 6j f 1€ j,i-!ir
: ;-:r1T', ;
ri I ii :;.1:: i rj,irllt.:i:',-i'tsiljjtj ,:s,lljl&xra 11: fr:"}r.!tlt).i_.ri
, -:..t,.r,t.: + .r-,-"ji!r 1,{ L.
.: rr:r .-i,-r.1iij i.:I lei,{ i_* ttils ,1{{:;i. ;gr_rfi$3:j
:ii r,: i.:: 1i:;!a fTi-+i'ClrffiAC Ftr ?.1"{i.l.rd}'!it'.
,,:+r.
:
\ , :, ,:.
Irf#. +r., : l :,!', :? .i:rl.*aa.ld*.i
ir:i: -.!-rl.jc,+Ft.it*r$.i'r
4l--';,-u': ."i :q:"rr;ffir;:ji': ':.rri;.:;: lj : .rlrlgi+'i #hgs.Effi .tF.t&Klfffi..J' i=!
ij
.-i l:t'ii! r ! i r .
J'n, i]
"c[F$tit, t!fl{:il!tt $t sl5rtff l-B-.0 spiltlt Bf $ililtt cu $$tfiltr
core,spuRde unel natrlce Nxx avdnd elenentele nule cu exceptia ce-Icr RE2
douX dlagenale: etr€!ftentele dlagonalei principale sunt ega-le *u L/Z,iar
lii.
cra.l-e die pe dlngonala secundard an -L12. uzuale ,
Lesarcfu ci x[n]=p3r(xlnJ ]+rnp(x[n] ] ei ci xlg{-ki=x[-k] . pe de
4. frHIIIIMT adunEr r
Spailul de sanale d este constituit din sesnel.e discrele av6nd alt6 pa
un nunXr id, ftrrlt de conponente couplexe gi este spa?lu Hilbert finlt
dlme:rerional in raport cu produsul scalar prive$:
negh-ra
(xfr:l , ylnl !r*1 x- [r] y j] tinpul
dat, se
>""=E I mlc de
exislen'
cane induce norna gi netrlca. tn spatiul astfel deftrnlt s€ pot. adopta-
diferite baze ortonorilate care corespund unor transforroiri remareailil-e trans f c:
utiii:ate ?n prelucrarea seunalelor. Cea mal trurportantE transforlare mlicgora:
EBte $erJ.a Fsurler Dlscret6 care corespunde bazei formate din
exp*ren!1"ale conErleze. Alta baze Sl transfornirl re[Bascablle sunt I.. AM
bazele St transfornHrlle dlscnete: Hartley, costnus gi- sin::s, Hadanard, 1.1 Pri
Faley, Walsh, *Iaar, obllce"
De',
opeP'raretlcurcirbrerfpi"reezsemnntaaltei }porrindlnmeaptralctieul p€i3trcaotrlecsepu.nldnacptaiurt.niciuitru-naor,r
transfor-m&rl}e enmerate nai sus pot f,l conslderata aetiunj- a.!-e unor bazele r
.1.n care
operatortr" partisuLairl, pe baza obaervaglei" cX spectrul unui. samnal, *i-n
gEtect.ru,
Cn este tot sennal din Cn.
inumltlr
1n
ef,ectut:
de cal-.;
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ET &,vnfiEgu
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r rFeffs.ffJd
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f..Rf rl; tri..i : , .,,rtrrt.,. .:ri.;llii.'-i ij,;:.i,;1 .ih". {?i.ii i-J.1,::i.j:: gri.j-r,: :,q:..i--+il:;. i , ,:.1.?fii t: "i_Bl. f:d.-ie
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f.S&ittir:11 i,l.r r , ..:.! ". . ,.'a":r::i :, :::jt:!.i :-,- .t. ,tt ..!.:,1:.:ii i:?qir.Ai$::.lt.tl
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se f&ce luAnd prisnii H te;rx*g11. {i t} .qisrJ drept pni.ca *ecuyeng,al Si ulti.$j.i
t{ ternenl dretrJt rse
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Fig"7" ffraful t-ransfor:#at*1 Fct;r:ier r:.*Bi_de e;u decisryr-e 'in tisrp pentru
B=S"
fu ftr c<rntj.nuare xepa.r$xu egant:r.r:an*J-e p6re de *ela Lsp&rs (deeigare
frecvengXl
r[2il J{*1 v-r a.in+6r1 Jd-it {xlni +xin"fu1 } F,;H izt)
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{qI4tr8 Iq.ti4&$'EY T r-ffi.E :lq*,C$i$ it gliits€tli ?'lE.Elrt
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zlt
unde prl"eruI" ttp de paranteze corespunde innulfirtlor, i*r cei de al
d:s3Lfllul-elteuear,ioorhe' gadirs.d:lpopnrtdhrnrl&nil"rropdru"fi,"agnrffXi:egvaai1ndateen1&a*1lXe2ian: lgolori"ctnudi'eui-ilfplruetsrelrrtit--r&'iti1in1'umtnltd. riandi2"xo
tz r)
&stfe} ,3* algoritm5 na nuse*c l.'ad-tx r" '":";L w*i ut"i"lizat fij.nd
*Lgcrtteii). redlx 4 ir=4)3"
4. ErthEffiwRTTft HxprSsanee. satr:cea.16 e fi.tr.{i#rire1'aiuf de clecimar* in i
*i-mp s€ nf*Sj"ne di-n *s*a a atrEaritmul"ui" ricc-i-s5r:i.i in frecven&fi prin
{
transpurnerea prodtaselor de $atrfce c$r€spuitz,ltoare, seea ce revdRe" :
etrat,s;"1t6 s:ixetrl j-Iox ruatrj-e*fu:r ]a ,ectrjxhare* or'dinii de irurtrl"tlre"
c
a. R$fltrH&f
t
Tispu.l de eal"cr:l aj" oricEre"l f,rnnsf+r:rnaLe cllscrete este deteminat
ln esenfl* q.ie nuxrXrul, de md"tipli*6-r"X utj.3i"zat de nl6ori.txul utillzat. 5
Existr:ef* !.ulsr si-metr31 in ma'is:leele coreeptwaEtonre dlf errteLr)r C
tre*sfors"{r:i $}sate fl xp*euJ"a'L$ peutru a face qrup$nl care sE permlbX
ategerrer:** ::qgs5::re1"o"i*r *e t-nr*'ui-g1rl. Fr.lgorlt"riii raplzl se haeeae6 gre t
utilleareia l*Lleqrcart"liivx'drilaour relesltecepluotorali:5cJle-Hcaulre,rilu:ilan&u$md iptir'o"fdluusttudtrKsr:'*.neScckr'el*rreaaL
ccmpa*t;i a E
matricel"*r. C$tts$'& Lransforp6ri rapide 6u,nt transformaeele i{aciailard,
palev, idelsh, Hear, Fouri.er, Pent.a'r* tc'arisfarffiata Ferurier rapld& exj.st6 i
diferits vartsref"e de algprltmX rapi?i intre car"e eej cu dect.m*r€ in
tj..mp ffesi)*ct"lv i.n frec.rent&.
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t"Jmpu1"u1" de calcn:l
detelmlnF reduc*s:ea t"jsp',"l3r"rl de cai+ui. pe;te rru
eaunTlce 'oH TnTiEds nJ FoleuV 'TITCtBcJEEaJ TrolEJedo
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J TJETUT T
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.l:i q. i F-*F:t;r$?fli::i"v TLk !, atglncl" I
*,?r[n] +hrf'.p] {" } €3:i&j +f..,=v l4;l
{1}
3 iV;'! {Fi a$s.s"$p, ., 1{ 1. !{.& N & I}SEAE T& .tV T F.il$-i {3}
rlr;i X{f".ii xli, 1l ".." Errin,.i Tl 3f !
tnt:r.'-'ad+';.i.ll: t'r*-;"l s.x .-{ v;k{r"''ri '- {l{r
- *t'-r*=.';t*tr.
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tn par?:j"eili.i Jpdo*a-"t"e'- " ". e:i.? i, r:na Lrr
e r,. gl1ffinl€rltu1 xIn] d- *gni
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trang f $.q Frfit * 1 r:rl 6,r'.i ll $ES : {ljnyqsq}1,l.rii3e gr:t, f,rl_!..is1":. j.
1i
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sati.sfHqlrte d* t:;it*.irf*x"mehft d cSci-:
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e .! +melr.le u:al":q
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Ts Iq*jA <-> [u]6 ,e3:ed FtTs ep ad
r*TrT.gpTer glT:o?Ep
frt-TII } iw-tli'r* H
tctx iryi"r
.{_ff
gxu nJlnad 'H?Tnaa.: '{g) larfc1a"r e pluT3asuos
fff 3t.{ ^. 0.!,
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ug6
r-ltf " r-ilr rm ilqUOflf .t
Ee?ip3p1u*e TftiflSrcd
p,,*,1*i to.]x [z-n]x Ir*rr]x
[n,, *,ln$
(6) [t]x [z]x
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Ht.lt lr:/ir]x Io]x
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x'' x+*&{r-s}{f,u I r*ar-..*r] [ tt ] x"
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' '+ n***,*9'4 [r]x+u,*f,e {'e ;x* 1,,*1q{6 [rui*r tff-*r*r1n [:r]x
uaug {qt:
S.s
.,ffi
{r}
4e"tl lL{i-$ ,"S^= t4JX
t""v
;9p,4&{3gr$:aasr*1:J&e?{Trun*Us,rE'KT"+pB:udTsoT-:rdnqaueJaTTuTp'JrEooTotTeBqTt\JulJapi?uTpr
*--*3d nD €a-rTg X;'ar*r;_1 un1.xd '
FzeeJa$afi e-T€* '9ft{sr-rri}*arr,
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fu* 5[:r];i iryl-s'**sfi* [e1"{
t{,rjx T-tr
(E) i.'**1 fr*u 7-iq 0-s\ o*u
n[wix
/
r-r,f
313=ry{aos}fs
iel,4 f:-frit {mlx fxffii=*fiafii 1u-ui r-r
Ylttc$t0 tEIrfiOJ cxEss *-ffif, flHBTC I S t$ filmut '[1Ytf;lS
"qgIrffi8. ct${rJtlB $! $t$TF$fi 20-E $EEIl1 F&OI?IEg $I${:RBTA J
HFtatre$e y*[n] <*> Y*[kj (
dem*nstreaza i: , avesl I
js-t imedia =*-N-rF/H-1 -z" Iklxtej Ir?1 (
tnl xtral (
a" (
saur au e sctlj"rnhare de notatie, I
&1
.I('tmt ytml = 1" F,f-1 x- [&] :f Ik] { 3.S} j
9 -, "tr_ 0
.Nfu
rB*C
adicX
\t-x-t-[afJ] tlL--f*rl tl 1 {34}
Ir
rela*:i"e f;are,se lnterpreteazE ca o (lonserv&re {p&ciH La
csnstant,a b3i a pr*dusui"ui sealar a1 arigpin.*}eJ-*r gi e.l"
lnnaginil*r glrin SFP Si este o consecinfrX e c&racteru]-r-rl
cvasiunltar a constante j" 1,1N} al tr:n"p.sf ormFrJ.l.
{exi"stenf
f;n sf&rglt, e3"ae6 5:ax'tleularizdm yIn]=xIi:l] . *!rt'jr!]e!E
Ffr*' ;'r**riet-= ll{FHi,' trrkr t' {}s}
adieE { 16}
l- * Tta$dr-s*r[-.1{r] 12 -f- *r.rr-r $"
i"xJ f"'
5, Txonmm SIHEIRTEX ** s111 X[n] **
x[n] => f{x[".k] {1I}
fntr-adevKr, i-n"r.ersf;nd vanlabilele k gi n, irv*m
rrlni = ff 'f *t ";*=*H rrx[-k] w*i-tt ( re?
de unde resrr.l"t.6 ter:rema enuntatX.
nerrl:coexJS6llTCsuaaeafTcuudnoaTalafa€plpscsauaTarndaEsda^sauJuaeuTrJTaaBuoJsrTarB€EJTsleJcuJnpT'aseuupAan^ooncpusyTeeTlJupTasaeo€nd"ur3'rc1!relaeetalruaTl1sITudnarpueanuc:Ts:leTvqJJeu6'eTEdp1epczT:uI:pTdarIeTuITBsJla-sHar-N1aESsunn]JnxuauuTATlren$unUuaqac1Tnros:eEdEFrSaapen9JEclVlE1ppyo1spTrTluJsoaJa1plaT-A3rd;auaaeTcdppo
i.**lt eJuaacas nc a{uaAgas Taun TnenpoJd :JpIncTUpd zeC
i{-[:r].r* tryl4= td*rrl,{ fa:x
grz) E r{-D+dJ sm tot* rarxf 3" 3"-**!^# 3 pl" rot" 3*
f=*!^(g^pr"
\
tr-+nY-A ,*-ror*d+)H=*r, 1u1r 1u1x
r-ft- )t-N r-iy
: aTIeJ?suouac
t€z ) [:r]"r+ rrrt{- [ur-:r].r I'tx $+ -- p)r [u]x
:xff,rAcss rrxxr u ffrrnsmoud v 0JS 'g
3I g-"\ o_.e
(.z7l [:r-/v].x= [:t*],x=.l(;-n f"tx f'r,*l=/ *-I-or [u],x
aT{Brlsuouaq
(rz) [:f-U]-f -* [u]*x r [:f].r ** [u]x
:srwxuxoo rrlrgr^lrss u 0JS 'L
0.r
3[:f*rrr]x* try-lx=dn turlx
(ozl r\!.8 t*o.o i.;
= (.-r*)rld [u]x fr=*ra [u-P]x f,
r-lv
:eerpolpurJR alsa ET{pJlsuouaq
{6r) [:f-f]x. I{-lx ** [u-]rlx-:r[fu,J-.H]xslutxlXr lrxx-l*m*r4[tuas]xu qJS '9
tlilcslt ult0ot vlffis s:02 tHtlsts ts tIIncilc'[tvlffis
silltlt, 0Ile$tIB $t st$TBflB 20-u $Illl I00lIlt lI$crnr
ifxlnlwff *- 't", ,ff*i*=x[ft-tr] (25)
RemarcEn simetria cu proprietatea referltoare Ia
deplasarea secventelor.
s. SFI) r ro*oryro"*t r[*]'r[*-ll: (261
x[n] -xln-:"1 ]4 (1-w"t;;J31
Dual, (r-ro?)xlnl *- xt.kl -xtk-l1
l"7l
ro. $FI) n snirf,
PF12 xtrnl -4 xi4l (28)
t-rJr
Denonstraf ie: n
rtnJ =p xlnl i y[n-tl =l!_r. rtr] , (nt
ytn] -y{n-11 =xlnJ (r-srri}) rlkl =;14
iPsinnurttoerg-RdreupeurnmreseaaustraclzpHricaemeotilunocudrrcnpoaondniao1vteugaorEilgful.!fkludenaatcdipn?aiilnrldicndoolnnnnteeundxlgltouuiunll risletipnmal{upg0l,i.uf.rl.ueA.icc,vCNeeFa-,nlsc}ft.ili
Prln dual-ltate se ohftne:
xlnl -+ f$fi ",r, {30)
L-wi"
1,1 Transfonatele Fsurier discrete ale unor semale
rerarcabile din #
gfD a elenentelor bazei canonice dln cN
0[n] <-) 1i 6In-M) <-> yio
{31}
SFD a sennalulul constant dlscret
1 {-} ff ";*= 1-g;?=.{o{nl - {$f:3 {32}
TJOTEn nJlrrad 'Zlli*=u Bt ruEoTula€4d?nud'lne1aun?slIusaaeuIPnP6aJJIEnTmuaRtpTunaeJqne1zls1euJ6pacu
PZEaJabns Ea TiRaluleeiAJcoaJs
puezTrTln'EuTurralap es,.;!t"1,1i;' $;11':'n""t":'u;::
(5c) r+lzoT=fZl{=$t\-'Iof- t".r)i=-l*tr*-fy,r--r-;
(B€) =,*"f'\t*{,t, =** ,T "- J]l,: ft: j=,",."
$zTpzltpJr, TnTnTpuuas E oJsl
r*rffi-r-$tn€* t *-frre*t f,=*f.';p.'*" go'a *' ofn'.
;ffiiffi*(q-^)F*a
T-N' ' ' 'l'0=ri -*6=[u]x TnlnTpuuras € OJS
- [tz{ Fz(e€)
( tr{*:rl9: [Fr+y] g") -- _*_ i\r ,__-
1"""j_JilT=ryE?urs
(Nlut{ltz)uTs }eJOsTp IEpTosnlrTs TnInfEu$as e ogs
--(ss) ( tr-ryl9+ [n+ry] g)# {**Fu;$e) f =wfl-uo"
(l{luHrrZ)soo lsr3sTp TepTOsnuTsoJ TnfnTEuuIas p CJS
gtr= ,5(r€) r .*t6
[H-ry] [f-n] uTs ir_rrl-*+
-*-_nI^':*;t= o-u o'a
({-r)ff' L .*_*rf;d t3{l=*I^,# r-N --
*^*H Tnlnreuuas
3 *(rs) =
s cJs
vliltslt trlt00l YHts L-OZ ttfltsts Ic rut[ctIt'nYtlts
$BTII[$, CICCUITB $I $I[?BI{B 20-" $TIIA TOUIIBI TISCIIIA
os[-n] = O:z=0,...,I --{ (I-r ar- k=2i (m)
n=!r*tr...,JV-1 | -2 -J. k=o
[r.tn# k=2i+I
SFD a sennalulul olnl+6[n-N/2] (4x)
a tn--f;] *- ( -1) t
{ Q k=Zi
's-rl.n-ll +.rorL--l?-IV-l , c{ 1I {*=o (421
fr +icw4 ff k=2ii+1
SFD a sennalulul olnl-o[-n]
Prln scXderea relatlllor stablllte anterior,, obtlnem:
o [n] -o [-n] -* {-rirr+r=l='rr., k=0,1t,r...t'l-1 {43}
Prln dualltate, se demonstreazE exlstenta peerechii:
lioo<k<4
n=paz 2
-n=imoar -- *k--tu0';
[rir[ti=j{tr-2f cDt"qrnsn xtk] i (4d)
=j0 #.*."
l.,
STD a f eresf,rei rectangulare de lxtime 2l'l+F11 centr atX in
orlg 1ne
p"r,,i -{l H;f -- P,([,kt=-,=ifflil,nlrtl-
IIn (45)
1,2 ProprietHli ale $erlel I'ourier Discrete trntru cdteva
cLase reffireabile de sernale
1. Sennale Lnvarlante Ia luarea pXrtli reale ( semnale
redle), Re{xIni]=xIn]; transformata trourler dlscretX este
parl in raport cu mlJl"oeul se*reesrtef;
,r|.c4Flltffi"t!;Elt 3,Jlll"rit rj'i I " irii
= {#arri:o'-ol [u]xl+*farut:drcal tul4] q0-r'
r-f
= (d,,i lullx+$ynr"l*) ,$ rlrl*o1I{ Ift*qx$.*,, r"l* $
;i-" o.rr o_r
fr,{r**Ia tulx r3-s=*!,t lulx
=n!a Lulx 3= Irr]x
r-{t
: (-:ed g uaundnsard) a1{e:lsuouag
tzs ) : BTeeJ a?se qdg[ui-l(Va]J"€xd=aT[Euu]uxras ) Ta4uaacas 1nco1[1ru
1:odB: ug a:ed TT{Jpd eaJEnT Ef alupTJEAuT afpuuas "t
nr
f{rs} ( t{-rl ,z- b{1zl = frl^
pugnl 15
{0s} tulf|+[u]x-fu)z Iu]z axafduoc
Iulrt TS Iu]x
TatruaAras p :u1:d elTuTIap
a{u1casuo3
gJS pugTnrfpr areJ aleod as
aTpaJ afuancas Rnop B qIS EereuTuJalaq
:
0.r
{5r} [{-}I].x-= [:f-],X-=orf,r [u]x 3r-=" irflx
:aT4pJxsuouag
{sr} :laNuanca[s{-Tfin]exo-T=[1uHnlxc 1:ode-r u3 gredul alsa
elaJcsTp aJaJETuJnToE.p{fieTleTuTl$JJoFldsuppaJJpl ni TIuB]Tx=a?[uup]xTJ]upnlu'(TaaJpfEuTu6uEruals:n.Zd
€IEuuras)
{r}} [{-lrt],x= 15g-1"x=*],r 1u1x fr,{=r 1ry1y
: a1{e:lsuouaq
€r) [:f-n]x = [:f]x
lIiltSIr rlil0or tIu3 6-AZ tnl$I$ l$ flIIncilc 't'tvilts
SBitat[, cllculTs $I $I$TBHE *ff-io $BtIA t0unllt lI$cutA
lt.
?-a
lx[n] leas 1 ?@-6ggxlnl )
=2aHF-0
{" semnale lnvari.ante la luarea p$r}il Iml:are ln rasrorE
cu niJ locul secvenpel ( semnale impare ) ; sFI] ', j,itl'ts pur
lmaginarE:
x[n] = -x'[f{*n] (52)
Denonstra$ie (presupunem N Par):
x-!. Jz-r-
xtkl =fEr4x[n] !f-,oxtnl *;*o s-l
E
nrud* ."-[r:] w!"k=
&--*-fTu
rv -. t_" ro
f.ou-.0*t't *;**-h- 2 tx[n] w**-x* trl wff) *
,r.0 xtn*-ft *;'"*g'u* tf.0,
Tty "_.
f, tl*t"t lel'rtrlw;d-l"xlnl le-arrtnlwff) =
u"
q#7-L
-argprlnl )
=*2j rE.0 lxtnl lsin r
5" Serynele lnvarlante la luarea slmultanE a pXrtli pare
gi a pErtl1 reale: rxeIna]l=Ex*e[nl-np]=axrIEN*(nr]emiseamrcnaeflel $]ere gl
rea]"e ) ; sI'D este
ordlnea ]
Xa(slkepJsfHl=nnXartLn*ie[rlNodi*"imnkdp]p*uaxaarlerNle,-rkseglealm"lean)a;pleEslerrDtLltensvrtaeeratpalaenntEexIng1]al;-plxuu*a[rNreim-ana]=sgl-imnxEalNlrlit*"annI5
6" semnale lnvariante l"a inmul9irea eu diferite ferestre:
a- semnale lnvarlante 1a innul$irea eu fereastra
recbangularX de lItlne M gt centru m.
b- sennal"e lnvarlante Ia inmtllllrea cu semnaltll treaptH
clrcularE
ffiutc(ss) 1ar1rx ttA\9fJ(+Z4an-= [u]'x
T_F
(r.s) ffiurc1w1'x$-+ = rurrx
:aTrrlefal cpJsT?Es pcTpp XJaqfTH EXpr[Jo]supJl
uTrd a1e6a1 luns aTe laluapuadapul luns nu
ap Taf1sp Tnun aTp Iu]'x e:eu16eur1 TS Iu]-x Efeal af1pluf:ugaas
'(o)N" " ' r+z/N'?/N--\ nJ?uad 0=[>t]x :ET{TpuoJ aJPJsTlps
JaTJnoJ plpltlJo]sueJl TnJpc e Tpuuas un alsa '()t)
FlaJ3sTp TnTuauop uI ,,Tpznpo' fpuuas un 'alelTfpnp uTJd
giuaneaJJ 'psranuT
TrlpTeJ
lJaqTTH alalsurroJsuprl puTnXTlsuoo pTngoEptcaaJTTape
'?JaqTTH aqcaJed o aTnXT?suoc [>t]tX tS [:t]-X afeTBuurog
ta1fur.o-tr
(es ) [,f]'x=ffiUp twjTx H +
rrdrf.E-{
3(ss) [:r]rx=ffia:o twt,x + plBar eITtJpd
:cgs aTe [{{]"IXnTuEaJuEoUpTFuE!UaTpTuSndIsIa]'JXoc u TnTueuop uI
aJlu! TJnlpEaI Taun
nfpzneJ, InJaXcpJec ap gsndul e1{c1:r1sal p3 plfnzar apun ap
fu6(rsl lcr:z-) I'f x g+
[:r]x= ( " :eauauresE ap 'a1:ls aleod aS
(ss)' [{]x= ( trf-l (- t)rl g). t{lxf -- [u]x= ( [u-] o- [n] o) ' [u]x
:aITjplTXuapT aTJcs 1od as eJ uBlElsuoc
au'{?TupprTTJnTETrATuulTaTunlouurpnus!uIu,maT3ep1' z{Iunn-leo]cno'-uao[Tuca]ofaepufnuuelnpsdus'u(a0a:o)sNcnt Tc" Ie' a'Tu:1+{Zln/TNtnu'TZuu/egNu=opupI
nrluad pu0T=TtrulxBc elfe1a-r 3EJsTlEs aJpc Jolaleuuas
esEIc raTpznpsr JoraTEuuas PsPTc ilTuTJiaq
slalcslp uaqITH PIP!,roIsup$ g'l
llil0sMil[00r rruls ,r-02 Itrlsts Is ill00lIc 'tlttns
t - #-, ACffitF@r, -
8IT!II,B, OIICUIIE $T $T$fBilB 20-tz s[nrt ?outIlt DI$cl3ll
1"{ fornule dc Filua]"itafre ti-W-frecvea&H'r
FconsfderXm Perecheat": JIu-r *r"* {5e}
x
0 tn-rnPrl
!!O a-0
uefonlerdnme6eHne/tNcehleeivspateelernelntcdthr.etAlgvd.eenDn, oapreilmsduessEopaPoraailttrfetuen,fl c& flecare dtnfre
expnimat[ intr-o
tx. 'i* =-, *g=ffirre-i.#j (60)
Ji
.ar0
tar gcroendseldaerlteHntep'adretes,impretntrlea,naelsotgeleIn(dseifsernrueln" te)xE: ronentului '
d1n
-x.f !t-' #* { 61)
0,"-**.oE
Aslnmoebtr$lalnu1toar,sotfedluaurlintXattoeartenltere feirmule care indj.cd, Prln
domeniul n pt k:
#-Atr-r x-1 Jdr-
O"[nl ,=f,61n*nt4 =+E
!-0 (62)
HaL observ$n cH 5e Poate scrie: Trt --,
niffi=E *ff (63)
tr-O .t
N1uttAccfneoS"l.)seenotrccafvf'l"Eaodnewocrrs)ollemee,uans(gcncpdcierlto.ounaoinnonnngnnslddrdseuatureonaAlDrcnlneruiek,.tesaana,vln"ralnooelMmuufrfSuulinsetonLlrneeslbDEtmltsr$er&lesn1lnvneamnu*aeefsnitlDivue:lacleEsclllu6nzveDp$itaercueeen&1tvrH&$ndcelre.ro1unon1lum9ddgpXg1lrleenecaelsslnfx5ungnploleelrtcMf{serbsvaesLeetnldcvmi.etvjd'"nteeevoaccnfnlsushutalotxlmvustr"eete":aPl'ce't6velrreelr*netlsn[rutn}trsteat.l'll