Mc lc Danh sch hnh v ......................................................................................................................................... 2

Li ni u. ................................................................................................................................................... 3

A.Tm tt ti ............................................................................................................................................ 4

B.Ni dung ..................................................................................................................................................... 5

1.Gii thiu v Compressive sensing. ....................................................................................................... 5

2..L thuyt c bn v Compressive sensing ............................................................................................ 7

2.1.Tnh tha v biu din tn hiu ....................................................................................................... 7

2.2. Ma trn o( Measurement matrix) ................................................................................................ 9

2.3.iu kin khi phc li tn hiu trong Compressive sensing ........................................................ 11

2.4.Phng php khi phc tn hiu ................................................................................................... 12

2.5.Applications of Compressive sensing. ........................................................................................... 18

3.ng dng ly mu nn cho tn hiu ting ni trong h thng thng tin di ng ................................ 20

3.1.Gii thiu tng quan h thng ...................................................................................................... 20

3.2.Trin khai thc hin ...................................................................................................................... 20

C.Kt qu t c v cng vic tip theo ................................................................................................. 25

TLTK ............................................................................................................................................................. 26

Danh sch hnh v Hnh 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 dung

1.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 sensing

L 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 Sensing

2.1.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 1 2 3{ , , ,... }N vi N l di ca tn hiu ,th x c th c biu din

bng mt t hp tuyn tnh ca K ( K

Vi in

, {1,2,3,...., }in N . Cho 1 2 3[ , , ,..., ]T

N l vector h s ca tn

hiu x trong .Php o ngu nhin tn hiu x c th c biu din nh sau :

.y , : M x N , K < M

Hnh 2.2 :Ly mu s dng Compressive sensing

(a) Biu din trong min thi gian ca tn hiu gm 300 mu

(b) Ph Fourier ca tn hiu c m ha

(c) Khi phc li ph Fourier dng l2 minimization

(d) Khi phc li ph Fourier dng l1 minimization

Ngc li,nu khi phc dng l1 minimization cho kt qu gn nh l hon ho. Ta

c th thy r bng vic so snh hnh 2 (b) vi hnh 2(d) .Tm li, k thut ti u da trn

l1 minimization s c s dng khi phc li tn hiu trong Compressive sensing.

2.2. Ma trn o( Measurement matrix)

Phn ny s a ra im nhn quan trng biu din tn hiu vi c s ri rc. Qu

trnh o tuyn tnh c miu t trong hnh 3 tnh ton M < N tch trong gia x v tp

hp vec-t 1{ }M

j c ,T

j jy x vi j = 1,,M . T

j biu th cho ma trn chuyn v

ca j v .,. biu th cho tch trong. Cho mt vec-t y c kch c Mx1,trong k hiu ma

trn vec-t y thu c t biu thc :

y x (2.3)

Vi l ma trn o MxN ,mi hng l mt vec-t o Tj v l h s vec-t vi K

thnh phn khc 0.Mt s ma trn o c th c dng trong bt k hon cnh no, ch

cn chng c lp vi c s c nh nh Gabor,sin hay wavelets.Qu trnh o

Compressive sensing vi vec-t x tha K (K-sparse) c miu t trong hnh 3:

Hnh 2.3 . Phng php o Compressive sensing [4]

Ma trn o ng vai tr quan trng trong qu trnh khi phc li tn hiu gc. iu

ny t ra mt vn th v : Lm sao thit k mt ma trn o v c bn l tp hp

ca N vec-t K chiu ?. Trong Compressive sensing,chng ta c hai loi ma trn o c

th s dng: ma trn o Random v ma trn o xc nh trc .Nu mt tn hiu x

gm N mu l tha th tn hiu c th khi phc li bng vic dng

log( / )M O K N K php chiu tuyn tnh ca x ln mt c s khc. Hn na, x c th

khi phc hon ton s dng cc k thut ti u khc nhau. Nu l mt ma trn cu trc

ngu nhin, th cc hng ca ma trn ngu nhin c lp v chng c ngu nhin to ra

t cng mt vec-t con ngu nhin. Ma trn ngu nhin c chuyn v v trc giao ha.

iu ny s c tc dng to ra mt ma trn biu din mt c s trc giao.Nu ma trn o

l ma trn xc nh trc, ma trn c th c to ra bi cc hm nh hm Dirac v hm

Sin .Trong trng hp ny, tn hiu c nhn vi mt vi hm Dirac ti cc im khc

nhau thu c vec-t quan st.Sau tn hiu ting ni c th c khi phc bng

phng php l1 normalization s dng vec-t quan st v ma trn o xc nh trc.

Lp trnh tuyn tnh l mt th tc khc ng vai tr quan trng trong vic khi phc

li tn hiu gc. l mt cch tip cn ton hc c thit k c c kt qu tt nht

trong mt m hnh ton hc cho trc,l trng hp c bit ca lp trnh ton hc. Lp

trnh tuyn tnh c th c din gii theo mt s nguyn tc sau :

Cc i cTx sao cho .A x b (2.4)

Vi x biu th gi tr s c xc nh, c v b l vec-t cc h s v A l ma trn ca

cc h s.Biu thc trn m c th cc i ha hay ti thiu ha c gi l hm mc

tiu v phng trnh .A x b xc nh nhng hn ch m hm mc tiu c th ti u ha

c.Cui cng,vic khi phc li tn hiu ting ni ph thuc vo vec-t quan st v ma

trn o.

2.3.iu kin khi phc li tn hiu trong Compressive sensing

2.3.1 .Restricted Isometric Property (RIP)

Pht trin gn y trong l thuyt tn hiu ch ra rng tn hiu tha l mt m hnh

hu ch trong mt s lnh vc nh : thng tin, radar v x l nh.V th gi thit rng mi

tn hiu c th biu din dng tha gip cho vic nn tn hiu c quan tm.S

khi phc li hon ton tn hiu x ph thuc vo ma trn o v vec-t o y. L thuyt

Compressive sensing ni rng khi ma trn tha mn iu kin gn trc giao Restricted

Isometric Property (RIP) [8] th n c th khi phc K h s quan trng ln nht t mt

b log( / )M O K N K php o y cng kch c. iu kin RIP ni rng cho vec-t x

bt k c K h s khc 0, vi 0 th cn tha mn iu kin sau :

2 2 2

2 2 2(1 ) (1 )x x x (2.5)

Kt qu l ,tn hiu tha c th khi phc li bi mt s k thut ti u khc nhau

nh k thut ti u l1 norm .K thut ti thiu ha u tin c dng khi phc li tn

hiu l l1 minimization

(P1) min 1l

x sao cho x y (2.6)

Cn c gi l thut ton ui khp c bn (P1).Mc ch ca k thut ny l tm

nhng vec-t c l1-norm nh nht

11

n

ili

x x

(2.7)

L1-norm cn c bit n vi tn gi Taxicab norm hay Manhattan norm. Kt qu

thu c trong [9,10] ch ra rng, nu mt tn hiu x tha ,tn hiu c th c khi

phc li da trn thut ton ui khp c bn (P1) . Cc k thut ti u khc gi l ti u

convex (cvx) s gii quyt cc vn c quy m nh v va .Dng cvx cc tiu ha tn

hiu khi phc li tn hiu gc [11].

2.3.2.Incoherence (iu kin c lp)

Trong khi iu kin RIP cung cp s m bo cho vic khi phc li tn hiu tha K

(K-sparse), vic kim tra li ma trn A c tha mn hay khng l mt bi ton t hp ph

tp,v mi trng hp cn xem xt nk ma trn con.Trong nhiu trng hp,c th tnh

ton mt cch d dng hn m bo khi phc li tn hiu nh iu kin c lp.Gi

tr lin kt ca ma trn A , ( )A , l gi tr tuyt i tch trong ln nht gia hai ct bt

k ai , aj ca A :

1 i < j n2 2

,( ) max

i j

i j

a aA

a a

(2.8)

C th d dng nhn thy gi tr lin kt ca ma trn ( )A nm trong khong

,1( 1)

n m

m n

. Ch rng khi n >> m ,cn di c th xp x rng ( ) 1/A m .

iu kin c lp c pht biu nh sau[86].Nu :

1 11

2 ( )k

A

(2.9)

Th vi mi vec-t o my R lun tun ti nhiu nht mt tn hiu x tha K (K-

sparse) sao cho y Ax .

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 y x

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 minimization

Chng 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 :

0

minl

x x (2.10)

sao cho y x

y 0

# : 0ilx i x .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

2

1

min minN

ilx xi

x x x

(2.11)

sao cho y x

Phng php ny cho p n dng gn ng T T -1= ( ) yx . 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 :

11

min minN

ilx xi

x x x

(2.12)

sao cho y x

Thut ton ny c th khi phc chnh xc tn hiu tha K s dng M php o

tuyn tnh vi log( / )M cK N K .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 :

11

min W min wN

i ilx xi

x x x

(2.13)

Vi iu kin :

0

(0)

0

(0)

0.4857.

256

l

l

x x

x m

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 :

1

( ) ( )argmin Wl ll

x x vi iu kin y x .

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

( 1)( )

1w li l

ix

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

(1)

0 0.240lx x c

x

ln

lp li u tin, 0

(1) 256l

x m , 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 (1)0 0.2407lx x

, 0

(1) 256l

x m , 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 Pursuit

Orthogonal 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 :

y x

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 m x n1( )n

i iA a R 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 1 1

2min min

o

k k

c i c ica r ca r vi mi i.

8. thit lp 1 0{ }k kS S i .

9. tnh 2

x argmin Axk x y vi iu kin supp x = Sk.

10. tnh rk = y Axk.

11. ti khi 2

kr .

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 imaging

Phng 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 ng

3.1.Gii thiu tng quan h thng

Ly 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 hin

Trong 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 t

C.Kt qu t c v cng vic tip theo Trong 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.

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Mathematicians, Madrid, Spain, 2006, vol. 3, pp. 14331452.

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2010.

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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.