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X L TN HIU S NNG CAO

Ni dung bo co

PHP BIN I WAVELET

GIO VIN HNG DN: TS. Nguyn Ngc Minh.

NHM 9: on Minh Qun,

Nguyn Kim Dung,

Nguyn Hu Trng,

H Th Lan Anh.

LP: CH10 T3

H ni, thng 05- 2011

PHN 1

CC KHI NIM C BN V TNG QUAN V L THUYT WAVELET

Mc d khi nim Wavelet ra i cch y 10 nm, nhng c rt t bi bo hay cun

sch no vit v n, v ch yu ch l cc nh ton hc vit ra, vi rt t s tham kho

hay tr gip, v n l hon ton mi.

Trc ht chng ta cn bit ti sao phi bin i v bin i thc cht l g? Trong ton

hc, php bin i ln mt tn hiu l c c cc thng tin khc, m tn hiu ban

u (hay cn gi l tn hiu th) khng c. Trong phn nghin cu ny, ta gi thuyt tn

hiu min thi gian l tn hiu th, cn tn hiu c bin i qua cc cng c ton

hc l tn hiu c x l. C rt nhiu php bin i c p dng, song php bin i

Fourier l php bin i c ng dng rng ri nht.

Hu ht cc tn hiu m chng ta o c u l tn hiu trong min thi gian,v khi

chng ta biu din ln th, th lun c mt trc l thi gian, cn trc kia l ln.Tuy

nhin trong x l tn hiu th cch biu din khng phi l ti u. V trong nhiu

trng hp, th thnh phn tn s li l quan trng phn bit cc tn hiu vi nhau,

ngi ta dng ph tn s biu din cc thnh phn tn s c trong tn hiu.

Ta hy xem xt hnh v di y biu din 3 tn hiu tng ng 3 tn s khc nhau

Vy lm th no o c tn s v lm th no tm ra cc thnh phn tn s trong

tn hiu? Cu tr li chnh l php bin i Fourier. Php bin i FOURIER cho ta bit

ln tn hiu trong mi thnh phn tn s.

Xc nh thnh phn tn s c ngha quan trng trong k thut, v d trong y hc, da

vo thnh phn tn s o c trong nhp tim, m ta bit c ngi c khe hay

khng?

Tuy nhin c rt nhiu php bin i c p dng trong k thut v ton hc, nh bin

i Hilbert, bin i Fourier thi gian ngn, phn b Wigner , bin i Radon, Mi

php bin i u c nhng vng ng dng ring vi nhng u nhc im khc nhau.

Php bin i Wavelet m ta ang nghin cu cng khng l ngoi l.

bit s cn thit ca php bin i Wavelet, chng ta hy xem qua php bin i

Fourier. FT l php bin i 2 chiu gia tn hiu th v tn hiu x l. Ta s khng th

bit c thi gian trong tn hiu x l, v cng khng th bit c tn s trong min

tn hiu th. Vy mt cu hi t ra l ta c cn bit n c tn s v c thi gian cng

mt lc khng? Nu i vi cc qu trnh dng th vic ny l khng cn thit, v qu

trnh dng, thnh phn tn s l khng thay i theo thi gian. Ta hy xem v d di

y:

y l bin i Fourier ca n:

Khc vi tn hiu hnh 1.5, ta xt tn hiu khc khng dng c minh ha di y:

Li xt tip mt v d khc c 4 thnh phn tn s 4 khong thi gian khc nhau, do

y cng khng phi l tn hiu dng.

V bin i FT ca n c dng:

y c nhng on gn sng l do s thay i tn s t ngt. Ta thy tn hiu

thnh phn ph cao th c bin ln, cn thnh phn ph thp th c bin nh.

So snh qua hnh ta thy rng c hai tn hiu khc nhau min thi gian li tng t

nhau min tn s. Do , FT khng ph hp i vi cc tn hiu khng dng.

Bin i Wavelet khc phc nhc im ny, n cho ta mi lin h gia min tn s v

min thi gian ng thi.

Gi s ta cho tn hiu qua mt h thng cc b lc thng cao v b lc thng thp nh

m t s H1.1 di y:

Gi s ta c tn hiu c tn s ln ti 1000Hz, sau khi i qua h thng nh s trn, ta

s thu c 4 vng tn s l 0-125 Hz, 125-250 Hz, 250-500 Hz, v 500-1000 Hz. Nh

vy, ta thu c mt tp cc tn hiu con c bng tn khc nhau t mt tn hiu ban u.

Nu ta biu din chng trn th 3D th s c thm mt trc thi gian cho tng tn hiu

con. Ch rng ta s khng bit c thi gian tc thi, nhng ta bit c khong thi

gian ca tng tn hiu con .

Bin i Wavelet a ra gii php linh hot nh sau: thnh phn tn hiu tn s cao s

phn gii tt hn trong min thi gian, cn thnh phn tn hiu tn s thp, s phn gii

tt hn min tn s.

Chng ta hy xt s li di y:

f ^

|******************************************* continuous

|* * * * * * * * * * * * * * * wavelet transform

|* * * * * * *

|* * * *

|* *

--------------------------------------------> time

L gii s nh sau: pha trn cng ca trc tn s, ta c nhiu mu tn hiu, tng

ng vi nhng khong thi gian nh, hay ni cch khc l thnh phn tn s cao s

H1.1: H thng cc b lc

phn gii tt hn min thi gian. Cn di y trc tn s, ta c rt t im tn hiu,

do s kh phn gii tt trong min thi gian.

^ frequency

|

|

|

| *******************************************************

|

|

|

| * * * * * * * * * * * * * * * * * * * discrete time

| wavelet transform

| * * * * * * * * * *

|

| * * * * *

| * * *

|----------------------------------------------------------> time

Trong trng hp thi gian ri rc, cng tng t nh trn. Tuy nhin ch rng

nhng thnh phn tn s cao, th khong cch gia cc chm im cng nh hn.

Di y l v d v bin i Wavelet lin tc ca tn hiu hnh sin c 2 thnh phn tn

s hai thi im khc nhau:

Bin i Wavelet lin tc ca tn hiu trn c dng nh sau:

Ch l trc tn s c biu din bi nhn scale. nh ngha scale s c ni r hn

phn sau, v trong trng hp ny th scale l nghch o ca tn s. Mc scale cao

tng ng tn s thp, mc scale thp tng ng tn s cao.

PHN 2

PHP BIN I WAVELET LIN TC

Trc ht ta hy ni ti cc hm c bn trong chuyn i tn hiu.

Trong bin i Fourier, chui Fourier lng gic l mt cng c cc mnh c s

dng trong c hai trng hp ri rc v lin tc nhng cng c nhc im ng k,

l cc hm c bn

cos sinikte kt i kt

xc nh v lin tc trn ton on ; , do khng thch nghi tt vi cc tn hiu

c a phng ha, trong ngha ca d liu ch tp trung trong min tng i

nh. Tht vy, v d trng hp hm Dirac ( )t c gi tr tp trung ti 0t . Do ta c

cc h s Fourier

1 1( )

2 2

iktkc t e dt

v chui Fourier tng ng

2 21 1 ... 1 ...2 2

ikt it it it it

k

e e e e e

lm mt hon ton tnh cht a phng ch tp trung gi tr ti 0x ca hm Dirac.

V vy cn xy dng mt h cc hm trc giao c cc tnh cht tt nh h cc

hm lng gic Fourier, ng thi chuyn ti c tnh cht a phng ha ca cc tn

hiu. H cc hm cn tm l cc hm wavelet.

Ging nh cc hm lng gic, cc hm Wavelet c bn sao ri rc nhn c

bng cch ly mu. Php bin i wavelet ri rc c th tnh ton mt cch nhanh

chng, do d rt thun li khi x l cc tn hiu phc tp v cc d liu nh nhiu chiu.

Chng ta bt u vi 4 hm wavelet c bn c Alphr Haar (nh ton hc

Hungary) gii thiu nm 1910.

Hm Haar wavelet th nht gi l hm scaling (scaling function), xc nh nh sau

Hnh 2.1: Bn hm Haar wavelet

1( ) ( ) 1t t , 0 1t ,

Hm Haar wavelet th hai gi l wavelet m (mother wavelet)

2

1 0 1/ 2( ) ( )

1 1/ 2 1

tt t

t

Gi tr ca hm ( )t ti nhng im ri rc khng quan trng lm, nhng tng t

trng hp khai trin Fourier ta quy c cho ( ) 0t ti cc im 1

0, ,12

t .

Hm Haar wavelet th ba v hm Haar wavelet th t l dng nn ca hm wavelet

m, c gi l cc hm wavelet con (daughter wavelet), xc nh nh sau

3

1 0 1/ 4

( ) 1 1/ 4 1/ 2

0 1/ 2 1

t

t t

t

4

0 0 1/ 2

( ) 1 1/ 2 3/ 4

1 3/ 4 1

t

t t

t

Hm scaling ( )t v wavelet m ( )t c m rng ln ton b tp s thc bng

cch cho nhn gi tr 0 bn ngoi khong c bn:

1 0 1( )

0

tt

nu ng c l i

1 0 1/ 2

( ) 1 1/ 2 1

0

t

t t

nu ng c l i

T cc hm c bn, ta i ti php bin i Wavelet, c nh ngha nh sau:

y l hm s ca 2 bin: s v , trong ,

x(t) l tn hiu cn phn tch

s l vit tt ca scale, tm dch l bin phn bc hay bin t l. d hiu ngha ca

bin ny, ta c th so snh vi bn , nu t l cng ln th s cho ta ci nhn tng quan,

nhng khng chi tit, cn nu t l nh s tng ng vi ci nhn chi tit. Bin s trong

ton hc c trng cho nn v gin. s cng ln, tc l gin tn hiu, s nh tc l nn

tn hiu. Tuy nhin trong cng thc bin i Wavelet, s nm di phn mu s, v th

nu s1 tng ng nn tn hiu Tn s f l nghch o ca s,

mc tn s thp tng ng vi ton b tn hiu, cn mc tn s cao tng ng mt phn

ca tn hiu trong mt thi gian ngn.

(translation) l nh v ca ca s khi ca s dch chuyn sut tn hiu, n mang

ngha thng tin v thi gian trong khng gian chuyn i.

(t) l hm chuyn i, c gi l Wavelet m, hm ny c coi l hm ca s

nguyn bn ca mi ca s khc trong x l. Cc ca s khc c th l nn, gin hoc

dch pha ca wavelet m. Ngoi ra c th dng cc hm khc nh hm wavelet Morlet,

hm m Mexican.

Gi tr 1/sqrt(s) l m bo tn hiu sau khi bin i c cng nng lng vi tn hiu

ban u

Thng thng, cc tn hiu u c bng tn gii hn, nn ch cn tnh ton CWT trong

mt di xc nh

Ta hy xem v d v tn hiu cosin trong tng t l s khc nhau di y, tt c tn hiu

hnh trn u bt ngun t mt tn hiu cosin, s=0,05 l t l nh nht, v s=1 l t l

ln nht:

Hu ht cc ng dng thc t u c tnh cht trn tc l thnh phn tn s cao khng

ko di trong sut di tn hiu, v tn s thp thng ko di n ht tn hiu.

Qu trnh tnh ton c th minh ha qua hnh v nh sau:

T cc gi tr thi gian v s ta s c cc im th trn mt phng thi gian-t l.

Di y ta s xem xt mt v d c th: tnh CWT ca mt tn hiu khng dng gm 3

thnh phn tn s: 5Hz, 10Hz, 20Hz v 30Hz.

Bin i CWT ca tn hiu trn c dng:

Lu l trc translation tng ng vi thi gian

Ngoi ra, cn c wavelet m khc c s dng trong phn tch wavelet m Mexican,

c nh ngha t hm Gaussian:

l

V Wavelet Morlet c nh ngha nh sau:

trong , a l tham s iu ch, cn l bin t l, nh hng n rng ca s.

khi phc li tn hiu ban u, ta c cng thc bin i ngc ca CWT:

Trong C l hng s, ph thuc vo wavelet no c s dng.

Php bin i CWT ngc tn ti khi tha mn iu kin:

Trong , l bin i Fourier ca .

Bin i CWT c mt im ln l phn gii linh hot m ta s trnh by di y.

Phn gii min thi gian-tn s.

y l u im chng ta s dng bin i Wavelet ch khng phi l bin i STFT.

Hnh 2.2 di y minh ha vn ny. Tt c khi hp u tng ng vi gi tr bin

i Wavelet trong min thi gian-tn s. Ta khng bit chnh xc mi im c th,

nhng ta bit n thuc khi hp no. Cc hp ny c din tch bng nhau, song chiu

di v chiu rng khc nhau. trong vng tn s thp, chiu cao ca khi hp cng

thp, do m tn hiu s c phn gii tt hn min tn s. Cn vng tn s cao,

th chiu rng ca khi hp nh hn, tc l min thi gian c chia thnh nhiu

khong nh hn min tn s, do m tn hiu c phn gii tt hn min thi

gian.

Hnh 2.2

Sau y ta s a ra mt v d c th thy c tnh ng dng ca CWT trong thc

t. Hnh v di y l in no ca mt ngi bnh thng v mt ngi mc bnh

Alzheimer (tm thn)

Nhng tn hiu ny c th ni rt kh phn tch v nh gi s khc bit, tuy nhin khi

ta cho qua php bin i CWT th s phn tch tr nn d dng hn nhiu. Di y l

th ca tn hiu sau khi qua CWT

V y l t mt gc nhn khc r hn:

Cn ca ngi bnh th tn hiu chuyn i c dng sau:

Hay t mt gc nhn khc:

PHN 3

PHP BIN I WAVELET RI RC

gim vic tnh ton, m vn hiu qu cho vic phn tch v tng hp tn hiu ban

u, ngi ta s dng php bin i Wavelet ri rc (DWT). DWT c bit t nm

1976, khi Croiser, Esteban v Galand ch to ra k thut khi phc tn hiu ri rc.

Phn gii tn hiu l php o cc lng thng tin chi tit trong tn hiu, v c th thay

i lng tin bng cch lc tn hiu. Ly mu tn hiu c th tng hoc gim thng qua

tc ly mu tng hay gim n ln.

DWT ly mu ti cc thi im s0 = 2, 0 = 1, hay s=2j v =k*2j, tn hiu lc ny s l

mt chui x[n], trong , n l s t nhin. Chui ny c a qua b lc thng thp

(bng na bng tn ca tn hiu) c p ng xung l h[n] v b lc thng cao c p

ng xung l g[n], khi ti u ra ca b lc thng thp, tn hiu thu c l kt qu

tch chp ca x[n] v h[n] nh sau:

Ch l tn s f c chun ha v khng ly n v l Hz, c ngha thnh phn tn

s ln nht ca tn hiu l rad, tng ng tn s ly mu l 2 rad (theo nh l

Nyquist). Sau khi qua b lc thng thp, tn hiu thu c c tn s ln nht l rad.

Lc ny, s im ly mu s gim i mt na, tn hiu c gin gp 2 ln, ta gi l

subsample. Qu trnh subsample sau khi lc khng lm nh hng n tng hp li tn

hiu ban u. Ton b qu trnh trn c lp li, v tn hiu thu c u ra l y[n]

c tnh nh sau:

M t u ra ca b lc thng thp v thng cao c vit nh sau:

Vic ta gim na s lng mu tn hiu, cng tng ng ta chia nh min tn s. mi

mc, vic lc v subsample lm gim i s lng mu tn hiu (tc l lm cho vic

phn gii min thi gian khng tt), song lm phn gii trong min tn s tt hn.

Hnh 3.1 m t qu trnh trn, bng tn ca tn hiu ti cc mc c k hiu l f.

Gi s tn hiu ban u c 512 mu, vi tn s t 0 n rad. Ti mc u tin, tn hiu

qua b lc thng cao, do , s im ly mu ch cn li 256 im ng vi bng tn

gim cn mt na t /2 n rad. 256 im mu cn li a ti tip b lc thng cao

v thng thp, do s im mu ca tn hiu cn li l 128, ng vi bng tn t /4

n /2 rad. Qu trnh c tip tc n khi no ch cn li 2 mu tn hiu.

Nh vy ta gim i mt lng tnh ton ng k. Tuy nhin, iu quan trng cn lu

trong DWT l mi quan h gia p ng xung ca b lc thng thp v thng cao.

Chng c mi quan h nh sau:

Trong , L l chiu di ca b lc (s im tn hiu).

Cc b lc tha mn iu kin trn c dng ph bin trong x l tn hiu, v c gi

l cc b lc i xng vung gc (Quadrature Mirror Filters). Hot ng lc v

subsample c m t ton hc li nh sau:

Hnh 3.1

Vic tng hp li tn hiu ban u rt n gin v cc b lc na bng tn u c tnh

trc giao, qu trnh tng hp tng t nh trn nhng theo th t ngc li. Khi ,

cng thc tng hp tn hiu ti mi mc c m t nh sau:

Php bin i Wavelet c ng dng rt nhiu trong x l hnh nh, bi hnh nh c

phn gii rt cao, v rt tn khng gian b nh. Kt hp vi m ha, Wavelet t c

hiu qu cao trong nn d liu, v d nh JPEG 2000.

Bin i Wavelet bt u c p dng trong truyn thng tin m Wavelet OFDM l

mt cng ngh mnh c hng Panasonic p dng trong HD-PLC. Wavelet OFDM

c IEEE 1901 coi l mt chun.