Mc lcDanh sch hnh v2Li ni u.3A.Tm tt ti4B.Ni dung51.Gii thiu v Compressive sensing.52..L thuyt c bn v Compressive sensing72.1.Tnh tha v biu din tn hiu72.2. Ma trn o( Measurement matrix)92.3.iu kin khi phc li tn hiu trong Compressive sensing112.4.Phng php khi phc tn hiu122.5.Applications of Compressive sensing.183.ng dng ly mu nn cho tn hiu ting ni trong h thng thng tin di ng203.1.Gii thiu tng quan h thng203.2.Trin khai thc hin20C.Kt qu t c v cng vic tip theo25TLTK26

Danh sch hnh vHnh 2.1 : S khi c bn Compressive sensing.Hnh 2.2 :Ly mu s dng Compressive sensing.Hnh 2.3 . Phng php o Compressive sensing.Hnh 2.4 : Khi phc li tn hiu tha bng phng php lp li l1 trng s ha.Hnh 3.1 :S khi h thng s dng ly mu nn trong h thng thng tin di ng. Hnh 3.2 :Tn hiu ting ni u vo. Hnh 3.3 : Ph nng lng ca tn hiu ting ni u vo. Hnh 3.4 :Ph FFT sau khi qua ca s ngng. Hnh 3.5 :Khi phc ph FFT bng l1-minimization.Hnh 3.6 :So snh li gia ph khi phc v thc t.

Li ni u.Compressive sensing (ly mu nn) l mt cng ngh ang ni ln v mang tnh t ph da vo s tha tht ca tn hiu. Trong Compressive sensing ,tn hiu c ly mu nn tha bng cch ly mt lng nh php chiu ngu nhin ca tn hiu m cha hu ht cc thng tin quan trng.Gn y,Compressive sensing c ng dng trong nhiu lnh vc nh :thng tin di ng,x l nh,h thng radar v h thng sonar.L thuyt v ly mu nn c nhng bc i tuy cha th ni l hon ton thnh cng v ng dng thc tin c trong cuc sng,nhng cng ghi nhn nhng c gng ca nhng ngi tham gia nghin cu. Thc tp tt nghip l giai on quan trng cho sinh vin hiu bit thm v thc t, gn kt thc t vi l thuyt chuyn ngnh, l tin cho vic thc hin n tt nghip.Trong t thc tp tt nghip ln ny , em s tp trung nghin cu v kh nng trin khai ng dng ly mu nn (compressive sensing) trong h thng thng tin di ng v nh hng ca n ti tc d liu.Khi thc hin ti ny, em xin gi li cm n n cc thy c trong b mn K thut thng tin ,Vin in t-Vin thng, trng i hc Bch Khoa H Ni,c bit l thy Nguyn Hu Trung nhit tnh gip em trong qu trnh xc nh,tm hiu v thc ti. Em cng xin gi li cm n ti c Nguyn Minh Hin cung cp thm cho em kin thc v gip em gii p nhng thc mc,a ra li khuyn gp cho ti ca em.Sinh vin thc hin:Nguyn c Nam.Lp:K s ti nng-in t Vin Thng K54.

A.Tm tt ti Trong mt h thng thng tin di ng thng thng, cc tn hiu quan tm c ly mu theo tn s Nyquist. nh l ly mu ca Shannon/Nyquist ni rng m bo khng b mt thng tin v c th khi phc li hon ton tn hiu th phi ly mu tn hiu vi tn s ly mu cao hn t nht 2 ln bng tn ca tn hiu.Tuy nhin,y khng phi l phng php hiu qu nht nn tn hiu, v n t nhiu gnh nng ln tn hiu gc trong khi ch mt t l nh h s bin i l cn thit biu din tn hiu.Kt qu gn y trong ly mu nn cung cp mt phng php mi ti to tn hiu ban u vi mt lng rt nh cc h s. Trong Compressive sensing ,cc thng tin quan trng v tn hiu/nh c thu nhn trc tip, hn l thu nhn cc thng tin quan trng khc ri cui cng s b i..Mc ch ca ti ln ny l xut mt h thng thng tin di ng mi s dng Compressive sensing nn tn hiu ting ni bn pht v gii nn bn thu. Kt qu mong i t h thng xut s lm tng tc truyn d liu ca cc h thng ny. m phng Compressive sensing c p dng th no,mt tn hiu ting ni c dung lng nh s c ghi trong Matlab. Trong trng hp ny,tn hiu u ra s c nhn vi ma trn o gm cc thnh phn l cc s c to ra ngu nhin. Ma trn o c chn bng mt cch m cc tn hiu tha s c khi phc mt cch chnh xc bn thu bng cch s dng mt thut ton ti u c sn khc nhau.Mt khi tn hiu tri qua qu trnh ly mu nn ,n sn sng truyn i thng qua h thng di ng.Tn hiu c truyn i sau c khi phc bi b thu t mt s lng nh cc mu quan trng bng cch s dng bt k k thut nhn ti u ha c sn.Thut ton s c m phng trong MATLAB.Kt qu ch ra rng nu mt ca s ngng c p dng cho tn hiu ting ni c truyn v di tn hiu c gi nguyn th tc nn ca tn hiu ting ni s c tng ln.

B.Ni dung1.Gii thiu v Compressive sensing.Thng tin di ng l mt trong nhng lnh vc nghin cu quan trng nht trong lnh vc truyn thng. Ngi ta d on rng trong mt vi thp k na,mt s lng d liu ng k cc kt ni ging ni v d liu s mt phn hay hon ton truyn bng khng dy.Mt trong nhng cc thch thc ra cho h thng khng dy l cung cp tc d liu cao hn trong mi trng di ng. t c mc tiu ny ,mt k thut ly mu mi gi l Compressive Sampling c th c s dng thay th k thut ly mu truyn thng.Compressive sensing (CS) c th c trin khai trong h thng di ng bi hu ht tn hiu trn th gii hin nay c mt s biu din dng tha di cch bin i min nht nh.Trong mt h thng thng tin lin lc in hnh,tn hiu c ly mu t nht bng hai ln tn s cao nht cha trong tn hiu .Tuy nhin,iu ny gii hn cc cch hiu qu nn tn hiu ,v n t mt gnh nng ln trong vic ly mu tn hiu gc trong khi ch c mt lng nh cc h s bin i l cn thit cho biu din tn hiu [1].Mt khc,ly mu nn cung cp mt phng php mi khi phc li tn hiu gc t mt s lng nh cc h s bin i. CS l mt m hnh ly mu cho php chng ta i xa hn gii hn Shannon bng cch khai thc cu trc tha ca tn hiu. N cho php chng ta thu thp v biu din cc tn hiu nn mt tc nh hn tc Nyquist mt cch ng k.Vi CS,bc ly mu l kh nhanh v n s dng php chiu tuyn tnh khng thch nghi c th gi c cu trc ca tn hiu. Sau ,tn hiu c khi phc li t cc php chiu ny bng cch s dng cc k thut ti u khc nhau. Trong qu trnh ly mu nn ch cc thng tin quan trng v tn hiu mi c thu thp, thay v thu thp cc thng tin khc ca tn hiu s b loi b pha thu.Bng cch trin khai nguyn l ny,mt h thng thng tin di ng mi c xut,trong ly mu nn c s dng nn v gii nn tn hiu ting ni bn pht v bn thu lm tng tc d liu.Trong nhng nm gn y,nhiu kch bn ly tn hiu khc nhau c pht trin trn th gii.Vetterli v cc ng nghip [2] trnh by mt phng php thng nht ly mu tn hiu lin tc mt cch ng u, v d nh non-uniform splines hay stream of Dirac. Tuy nhin,phng php ly mu nh vy rt kh trin khai v cn phi c y thng tin v b phn li khi phc tn hiu trc khi ly mu n.Trong khong thi gian ,l thuyt v CS ni ln [1,3] ch ra rng mt tn hiu ri rc ly mu khng u c th khi phc li mt cch hon ton vi t l thnh cng cao bng cch s dng cc k thut ti u khc nhau v bng cch xem xt mt cch t hn cc php chiu ngu nhin v php o so vi chun Nyquist.Cc yu t chnh cn c gii quyt trc khi s dng Compress sensing l : lm th no tm min bin i trong tn hiu c dng biu din tha, v lm th no ly mu tn hiu mt cch tha c hiu qu trong min thi gian v cui cng,lm th no khi phc tn hiu gc t cc mu bng cch s dng cch k thut ti u.Tm li,s lng ln d liu cn c ly mu tn s Nyquist,c bit l tn hiu ting ni,nh v video thc y vic nghin cu Compressive sensing nh mt gii php kh thi cho h thng thng tin di ng tng lai.Tn hiu tha c nh ngha l tn hiu c th biu din bng mt s cc im d liu gii hn trong min bin i.Nhiu tn hiu thc c th c phn vo loi trng hp ny bng cch s dng mt min chuyn i thch hp.V d nh,nu tn hiu x l mt hnh sin ,r rng n khng tha,nhng min bin i Fourier ca n v cng tha. Vic thu thp mt lng ln d liu,cng thm chi ph nn c th ci thin bng cch s dng Compressive sensing. Kt qu l s c rt nhiu tim nng trong vic tit kim nng lng,b nh v qu trnh x l.Trong cch tip cn xut, mt tn hiu ting ni c ghi li v ly mu nn s dng ma trn o. u ra ca thut ton CS l mt vector quan st c truyn n bn thu. bn thu, phn tn hiu c khi phc li t mt lng nh cc mu quan trng bng cch s dng cc k thut ti u khc nhau nh l1-norm hoc convex optimization. M phng MATLAB c thc hin nn tn hiu ting ni vi tn s nh hn Nyquist v khi phc li n s dng mt trong cch k thut ti u khc nhau c sn m khng lm mt bt k thng tin quan trong no.

2..L thuyt c bn v Compressive sensingL thuyt v compressive sensing c pht trin bi Candes cng cc ng nghip [3] v Donoho [1] vo nm 2004. N bao gm vic ly cc php chiu ngu nhin tn hiu v khi phc li t mt s lng nh cc php o s dng thut ton ti u.Trong nh l ly mu truyn thng, tn hiu c ly mu s dng tn s Nyquist,trong khi vi s h tr ca Compressive sensing tn hiu c ly mu tn s thp hn tn s Nyquist. iu ny l c th bi tn hiu c bin i sang min m n c dng biu din tha.Sau tn hi c khi phc li t cc mu s dng mt trong cc k thut ti u khc nhau c sn. S khi c bn ca h thng s dng CS c th hin trong hnh 1

Hnh 4.1 : S khi c bn Compressive Sensing2.1.Tnh tha v biu din tn hiu

Biu din tn hiu v tnh tha ng mt vai tr quan trng trong Compressive sensing. Cho x RL biu din mt tn hiu thc ,gi s rng tn hiu x l tha trong c s trc giao vi N l s chiu ca tn hiu ,th x c th c biu din bng mt t hp tuyn tnh ca K ( K m ,cn di c th xp x rng .iu kin c lp c pht biu nh sau[86].Nu :

(2.9)

Th vi mi vec-t o lun tun ti nhiu nht mt tn hiu x tha K (K-sparse) sao cho .

2.4.Phng php khi phc tn hiu

Khi phc li tn hiu ng vai tr quan trng trong l thuyt Compressive sensing khi tn hiu c ti to hoc phc hi t mt s lng ti thiu cc php o.Bng cch s dng nhng k thut ti u ha bn thu,chng ta c th khi phc li tn hiu m khng lm mt thng tin.Vi phng php ly mu nn,sau khi thu c tn hiu th bi ton t ra l tm li tn hiu x t cc gi tr y.Ti nay, c mt lot bi bo lin quan n vic khi phc li tn hiu t thng tin khng y mt cch gn chnh xc nht.2.4.1.Thut ton khi phc l1 minimizationChng ta cn khi phc li x,tc l tm li chnh xc cc gi tr x[n] , n=1,2N khi m ch c M php o y.Tuy nhin do M < N tc l s phng trnh thit lp c l nh hn s n cn tm, do s c v s cc nghim tha mn, v tt nhin nu khng cho thm bt k thng tin g v nghim cn tm th ta s khng th tm c nghim chnh xc.Tuy nhin,trong trng hp ny,tn hiu m chng ta cn khi phc l bit v mt cu trc ,n l tn hiu tha K hay tn hiu c th nn c.V mt ton hc, di gi thit tn hiu x l tha, chng ta c th khi phc li tn hiu x bng cc phng php minimization. S dng l0 :

(2.10)

sao cho

y .Phng php ny c th cho php khi phc chnh xc d liu bng cch kim tra tng d liu tha mn phng trnh trn,tuy nhin tc tnh ton ca phng php l chm,do thut ton ny t c s dng trong thc t v khng s dng trong ly mu nn. S dng l2:

(2.11)

sao cho

Phng php ny cho p n dng gn ng . Tuy nhin, l2 minimization gn nh khng bao gi tm ra mt nghim tha K. Phng php ny khng khi phc ng d liu S dng l1 :

(2.12)

sao cho

Thut ton ny c th khi phc chnh xc tn hiu tha K s dng M php o tuyn tnh vi .Phng php ny s dng trong Compressive sensing cho vic khi phc d liu.Nghin cu gn y ( thng 10 nm 2007), Emmanuel J.candc,Michael B.Walkin v Stephen P.Boyd ci tin phng php ny cho php khi phc tn hiu chnh xc hn gi l phng php L1 minimization c trng s ha ( Reweighted L1 minimization ). Phng php ny khi phc tn hiu bng phng trnh sau :

(2.13)

Vi iu kin : y ma trn W l ma trn cho vi w1,w2,,wn l cc trng s dng nm trn ng cho,cc trng s cn li bng 0.Cc trng s dng ca ma trn W ny c tnh ton bng cc bc thut ton sau y:1.Thit lp l = 0 v wi(0) = 1, i = 1,,N.2.Tnh :

vi iu kin .3.Cp nht cc trng s : vi mi i = 1,2,..,N :

4.Kt thc tnh ton nu l hi t hoc l t ti mt gi tr cc i lp i lp li lmax.Nu khng, tng l v quay tr li bc 2.

Tham s c a ra bc 3 m bo s n nh v chc chn rng nu c gi tr x(l) = 0 khng ngn cm vic c lng khc 0 bc tip theo.S dng thut ton lp i lp li xy dng trng s (wi) c xu hng cho php c lng thnh cng hn v tr cc h s khc 0. Mc d cc ln lp li u tin c th cho c lng tn hiu cha chnh xc,nhng cc h s tn hiu ln nht c kh nng c nhn dng khc 0 . Mt khi cc v tr ny c tm thy ,chng c nh hng lm gim trng s tng cng nhy tm nhng h s khc 0 cn li c gi tr nh.

Hnh 4 minh ha s linh ng ny bng mt v d phc hi tn hiu tha. Hnh 2.4(a) l tn hiu gc c chiu di n =512 vi 130 nh khc 0. Chng ta dng m = 256 php o vi ma trn l ma trn chun thng thng c lp. Thit lp = 0.1 v lmax = 2. Hnh 2.4(b)-(d) v biu phn tn h s theo h s, vi h s tn hiu gc x0 vi gi tr khi phc li x(l). Nu dng lp li khng trng s ha ( hnh 2.4(b)), chng ta thy rng tt c h s ln ca x0 c xc nh chnh xc khc 0 v ln lp li u tin, , vi 15 nh khc khng ca x0 c khi phc li bng 0 v 141 nh bng 0 ca x0 c khi phc li l khc 0.Sau bc lp li trng s tip theo ( hnh 4(c),kt qu c ci thin vi , , 6 nh khc khng ca x0 c khi phc li bng 0 v 132 im bng 0 c khi phc li khc 0 .S c lng tn hiu c ci thin khi phc tn hiu mt cc hon ton bc lp li trng s th hai ( hnh 2.4(d)).

Hnh 2.4 : Khi phc li tn hiu tha bng phng php lp li l1 trng s ha.(a) Tn hiu gc x0 n = 512 vi 130 nh.(b) Biu phn tn h s theo h s, ca x0 vi tn hiu khi phc x(0) dng l1 minimization khng trng s.(c) Tn hiu khi phc x(1) sau ln lp li trng s u tin.(d) Tn hiu khi phc x(2) sau ln lp lp trng s tip theo.2.4.2.Thut ton khi phc Orthogonal Matching PursuitOrthogonal matching Pursuit (OMP- Thut ton ui khp trc giao) l mt gii thut tham lam(Greedy Algorithm) kinh in cho xp x tn hiu tha.Gii thut tham lam l mt thut ton gii quyt bi ton theo kiu khm ph, tm kim la chn ti u a phng mi bc i vi hi vng tm c ti u ton cc. Chng ta c th la chn gii php no c cho l tt nht thi im hin ti v sau gii bi ton con ny sinh t vic thc hin la chn va ri. La chn ca thut ton tham lam c th ph thuc vo cc la chn trc . Nhng n khng th ph thuc vo mt la chn no trong tng lai hay ph thuc vo li gii ca cc bi ton con. Thut ton tin trin theo kiu thc hin cc chn la theo mt vng lp, cng lc thu nh bi ton cho v mt bi ton con nh hn. Gii thut tham lam quyt nh sm v thay i ng i thut ton theo quyt nh , v khng bao gi xt li cc quyt nh c.

Nhc li bi ton ca chng ta, vi i din cho ma trn kch thc M x N ( m thng M M > K ) tha mn :

Bi ton li tr v ging nh tm cc t trong mt quyn t in ghp thnh cu c ngha no . V chng ta s thut ton OMP ( Orthogonal matching pursuit) thc hin iu ny.u vo :

Ma trn v vec-t xRn. Ngng li .Thut ton :1.Khi to k = 0.2.Khi to nghim ban u x0 = 0.3.Khi to gi tr d ban u r0= y Ax0 = y.

4.Khi to gi (tp con m ti gi tr hm s khc 0) ban u S0 =supp x0= .5.Lp li cc bc 6.thit lp k = k + 1.

7.chn i0 sao cho vi mi i.

8.thit lp .

9.tnh vi iu kin supp x = Sk.10.tnh rk = y Axk.

11. ti khi .u ra :Nghim gn ng xk.

Mt trong nhng c im ng ch ca OMP l tnh n gin. OMP l thut ton c tnh cnh tranh cao v hiu sut xp x.2.5.Applications of Compressive sensing.2.5.1.Compressive imagingPhng php ph bin nht trong h thng nh k thut s l c th thu c cng nhiu pixels cng tt v sau nn nh chp bng phng tin k thut s [12]. Nn c mong mun lm tng dung lng lu tr v tng cng qu trnh thng tin lin lc. K thut nn khai thc s d tha v mt th gic thng thng ca con ngi ti vic cm nhn hnh nh. Sau khi chp nh quang hc v p dng nn d liu,nh c biu din bng mt s lng nh hn cc pixel so vi nh gc.Vic gii nn hnh nh phi tha mn cht lng hnh nh m ta mun.iu ny gi ln cho ta mt cu hi : c cn thit phi thu ton b mu nh v ri nn n? .Cu tr li chnh l s pht trin gn y trong l thuyt ly mu nn. tng c bn cho vic thc hin Compressive sensing l khi phc li mt nh khi c th thm ch vi t hn s lng php o hn s lng pixel trn danh ngha.2.5.2.Medical imaging.Mt trong nhng ng dng ha hn nht dng ly mu nn l ng dng hnh nh trong y t. My scan MRI ly mu cc dng vi k-space. Ly mu mi dng mt thi gian v nng lng a vo bnh nhn [13] . Vi cng ngh MRI c ci tin, ngi ta mong mun hn l s dng cng trng cao hn phn tch cc b d liu ln hn,nh nh 3D hay nh ng. Mt s k thut lm gim s lng mu c s dng nh K-t BLAST , K-t SENSE v K-t vDUST khai thc mi tng quan tuyn tnh trong cc chui hnh nh. Ngc li ,Lusting v ng nghip [13] c nhng kt qu s b cho thy c th khai thc tnh tha tht theo thi gian ca d liu.2.5.3.Chuyn i tng t sang s.Analog-to-digital converters (ADC) c s dng trong cm bin v thng tin do s tin b trong s l tn hiu s.Qu trnh ADC da trn nh l ly mu Nyquist ,ly mu u tn hiu vi tc ti thiu bng 2 ln bng thng c th khi phc tn hiu hon ton.ng dng ni ln nh pht hin radar hay truyn thng bng siu rng ang y mnh cc gii hn ca ADC. Nhng pht trin gn y trong lnh vc ly mu nn gip cho vic thit k cc b ADC c th thu c cc mu tn s ly mu thp hn.2.5.4.Radar nn.L thuyt mi ca compressive sensing c th dng trong h thng hnh nh radar c thit k xc nh khong cch , cao, hng v tc di chuyn cng nh cc i tng c nh [6]. Tn hiu radar thu c c th c khi phc li vi s lng php o t hn bng cch gii mt bi ton ngc thng qua mt chng trnh tuyn tnh hay thut ton tham lam. Vi vic trin khai compressive sensing trong h thng radar,s cn thit cho vic nn xung kt hp vi b lc pha ngi nhn v b chuyn i ADC hot ng tn s Nyquist cao c th xa b. Kt qu l, s phc tp cng nh chi ph phn cng bn c th gim c mt cch ng k.2.5.5. Ly mu nn trong h thng truyn thng di ng.Mc ch ca n lc nghin cu ny l thc hin ly mu nn trong h thng di ng.Bng cch s dng k thut ly mu nn, tn hiu ting ni c m ha trc pha pht c th gi n c bn thu thng qua knh truyn khng dy.Kt qu l, mt lng nh cc mu c truyn ,v iu ny lm tng tc truyn d liu khi so snh vi h thng truyn thng hin ti .Trong h thng truyn thng xut,tn hiu ting ni c ly mu vi tn s thp hn tn s Nyquist bng cch dng Compressive sensing. Ph nn sau uc truyn thng qua h thng khng dy v khi phc thnh cng bn thu m khng lm mt i bt k thng tin quan trng no 3.ng dng ly mu nn cho tn hiu ting ni trong h thng thng tin di ng3.1.Gii thiu tng quan h thngLy mu nn trong h thng thng tin ng ra i vi mc ch tng tc d liu truyn i vi mng di ng hin nay v c th p dng cho cc th h mng di ng tip theo. Bng cch s dng k thut ly mu nn, tn hiu ting ni c m ha trc pha pht c th gi n c bn thu thng qua knh truyn khng dy.Kt qu l, mt lng nh cc mu c truyn ,v iu ny lm tng tc truyn d liu khi so snh vi h thng truyn thng hin ti. bn thu,tn hiu c khi phc hon ton t mt s lng nh cc php o bng cc k thut ti u khc nhau nh l-norm hay ti u convex.S khi c bn ca h thng xut :

Hnh 3.1 :S khi h thng s dng ly mu nn trong h thng thng tin di ng.3.2.Trin khai thc hinTrong giai on u tin ca d n, mt tn hiu ting ni c to ra s dng b to tn hiu ngu nhin Laplace ( hnh 3.2).Vic quyt nh s dng b to tn hiu Laplace m hnh ha tn hiu, bi v cc loi tn hiu ting ni thng thng c phn b Laplacian [14].Tn hiu ting ni c m hnh ha c nh x vo min tn s ri rc s dng FFT. Kt qu thu c t php bin i ny c biu din trong hnh 3.3. Trong giai on th hai ,trc khi p dng Compressive sensing cho tn hiu. mt ca s ngng c s dng loi ra u l h s quan trng vi tn hiu. Ni cch khc, tt c cc h s vi bin nh c nhn vi 0.

Hnh 3.2 :Tn hiu ting ni u vo

Hnh 3.3 : Ph nng lng ca tn hiu ting ni u vo.Trong hnh 3.4,chng ta c th thy biu din ca ph FFT sau khi ca s ngng c p dng. Mc ch ca ca s ngng l m bo rng ph FFT l tha.

Hnh 3.4 :Ph FFT sau khi qua ca s ngng.Trong giai on th ba, ph ngng c nhn vi ma trn o l ma trn c to ngu nhin.u ra ca thut ton ly mu nn c chuyn i sang tn hiu s bng cch s dng b chuyn i ADC c th truyn qua h thng di ng. bn thu,mt tn hiu d on ban u c to ra bng cch s dng ma trn o v vec-t quan st (vec-t tn hiu) ,c dng gn vi tn hiu ting ni u vo. Cui cng, tn hiu ting ni c khi phc li bng cc s dng k thut ti u c sn.Tn hiu c khi phc u ra ca module ti u c biu din trong hnh 3.5. S khc nhau gi tn hiu thc t v tn hiu khi phc c tnh ton quan st li gia hai tn hiu . Li ny c th hin trong hnh 3.6.

Hnh 3.5 :Khi phc ph FFT bng l1-minimization

Hnh 3.6 :So snh li gia ph khi phc v thc tC.Kt qu t c v cng vic tip theoTrong giai on thc tp tt nghip, em hon thnh c cng vic la chn ti lm n tt nghip l ng dng ly mu nn (Compressive sensing) vo h thng thng tin di ng.Vi vic nm chc c l thuyt v Compressive sensing,em ra c k hoch v xy dng c m hnh thc hin bi ton dnh cho h thng theo ba bc :1.To tn hiu ngu nhin Laplace cho bn pht.2.Dng ca s ngng lm tha ha tn hiu v p dng ly mu nn cho tn hiu.3.Khi phc tn hiu bn thu dng l1-minimization.Cng vic trong giai on tip theo l m phng h thng trn Matlab mt cch n gin v chnh xc nht c th, sau s so snh vi cc phng php nn khc nh :Wavelet,..

TLTK[1] D.L. Donoho, "Compressed Sensing," IEEE Transactions on Information Theory, vol. 52, pp.1289-1306, 2006.[2] M. Vetterli, P. Marziliano, and T. Blu, Sampling Signals with Finite Rate of Innovation, IEEE Transaction on Signal Process, vol. 50, no. 6, pp.1417-1428, 2002.[3] E. Candes, J. Romberg, and T. Tao, Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information, IEEE Transaction on Information Theory, vol. 52, pp. 489509, 2006.[4] Compressive Sensing, a New Frame for Imaging, http://www.cs.jhu.edu/~misha/ Reading Seminar/Papers/Baraniuk06.pdf.[5] E. Candes and J. Romberg , Practical Signal Recovery from Random Projections, Processing SPIE International Symposium Electronic Imaging, pp. 7686, vol. 5674, 2005.[6] R. Baraniuk and P. Steeghs, "Compressive Radar Imaging," Radar Conference, 2007 IEEE, doi:10.1109/RADAR.2007, pp.128-133, 2007.[7] Compressive Sensing, http://www.ricam.oeaw.ac.at/people/page/fornasier /CSFornasier Rauhut.pdf.[8] E. J. Cands, Compressive sampling, in Int. Congress of Mathematicians, Madrid, Spain, 2006, vol. 3, pp. 14331452.[9] E. Candes and T. Tao. Near-optimal Signal Recovery from Random Projections and universal encoding strategies, IEEE, vol. 52, pp. 5406 5425, 2004.[10] Disciplined convex programming, http://cvxr.com/cvx/cvx_usrguide.pdf, 2010.[11] S.M. Kay. Fundamentals of statistical signal processing. Prentice Hall, 1998.[12] Compressive Sensing, http://www.see.ed.ac.uk/~mdavies4/Research/CS/[13] MATLAB CENTRAL, http://www.mathworks.com/matlabcentral/ fileexchange/7309.[14] FFT Tutorial, http://www.phys.nsu.ru/cherk/fft.pdf.10