XI International PhD Workshop OWD 2009, October Digital Signal Processing in Neurology: Picture and Sound

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Merilyn Bailey
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1 I Itratoal PhD Worhop OWD 9, 7 Octobr 9 Dgtal Sgal Procg rology: Pctr ad Sod Daa Krzow, Tchcal Urty o Lodz (3.6.9, pro. Zygt Cota, Tchcal Urty o Lodz) Abtract I th ral world o phycal qatt ar cto o o, two, thr or or argt. Atr thr coro to gr, a array o br obtad. Th cto ca b alzd drt way. I th ca o ag w ar dalg wth two-doal array, ad th oc gal ar both atrc o two ad o-doal. Th atrc, ad aaly o data wth g gal procg algorth, ar ow wdly d or dc. Th d to th larg pha o th o o-a dagotc thod ad to cra th accracy o dago ad otorg o patt. It ow bcog ar d to th larg coptatoal capablt o procor. I addto, w algorth ar dlopd or typcal dcal rqrt. O o th ara, whch both ag procg thod, ad th od rology. Whl roagg alrady d or ral dcad, t a coptr aaly o otg oly th plag tag, bca oc aalyzd oly by phoatrt. I th papr algorth, whch ca b d cclly th aaly o ag ad od, ar prtd, ordr to or accratly dty da ad to otor th progr o hoptalzato o patt.. Iag procg For th prpo o roagg agtc roac agg ad coptd toography d. Th rltg ag, graycal, a twodoal atr data. Ad t d aoal that tad ot ara or ara that d to tad ot [5]. Addtoally, t z ad locato hold b dtrd. Th a tool or th prpo ar, bd o ha y, th aro typ o ltrato ad dcopoto [6]. At th ott, th dcrt traor ca b t, alatd both a, or apl, orthogoal Forr traor (DFT) or co (DCT) [7]. r th tg at algorth or D traorato ar d. At th bgg ach l o ag ar traord ad agd a ctor o coct D traor pt plac a ctor aalyzd pl. Th th atr coct o th horzotal D FFT aalyzd rthr,.. t col ar bjctd to a r o D traorato. Sc th copl Forr pctr o D ral-ald gal ytrcal, ytry ca alo b obrd, or rddacy, a trcoctd D Forr pctr. I addto, D DFT calclato rqrd or copl br, ad th a rlt o traorato dclt to trprt bca th dcopoto o th bac atr o th actal ag copl. For th rao, th ag procg algorth th rqcy appl oly to th traorato o D DCT, whch ha bac cto o ral-ald. Wth rqcy coct, yo ca ody th. Th, atr a r o r traor th a pctr, wth o rqcy copot rod, or ltrd. A lar approach cold alo b carrd ot th plt way FI ltrg o ag,.. brg t dow to a r o parat ltrg all row, th a parat ltr or ach col. Ag that, a two-doal gal who argt ad ary cotoly, th coto Forr traor ca b wrtt (,. I tr, ( a dcrt gal o th argt, ad DT-FT( ad DFT(, th two-doal Forr traor wth cotat rqcy ( ad dcrt rqcy (,. Forr traorato o th coto- coto ca b wrtt a: (, ),, (, ) j(π + π j(π + π dd () Forr traor dcrt-coto dtrd by th orla: DTFT ( j(π ( / ) j(π ( / ) ) () pr pr pr / pr pr / pr / DTFT pr pr / (, ) jπ ( / pr ) jπ ( / pr ) 73

2 Th orla d to dg o-rcr pl rpo ltr to a g D rqcy charactrtc. By cotrat, Forr traor dcrt-dcrt wrtt a: DFT (, l DFT π j l (, π j π j π j l (3) Two-doal, coto ad dcrt Forr traor, pl ad oppot, aot to th pltato o two r o o-doal traorato, th cod oprat o th otco o th rt, bt alog a drt a. A a rlt o D DFT th ppr lt corr (DC) rltg atr th o all lt o th atr ad th coct o th col all rqcy dgatd or row all-rqcy. Th arthr away ro t, th rqcy bco hghr. To carry ot a ltrato atr o coct twodoal Forr traor yo t rbr ot to olat ytry D pctr. Othrw, a r o r traorato do ot rc a atr wth ral coct. A th ca o twodoal, dcrt Forr traor twodoal dcrt co traor D DCT lg two t o D traor th row ad col o th ag: DCT (, whr: l πl π ( co + α ( )co + DCT β (4) π πl (, α( )co + β ( co + α( ) β (,,,...,, l l,... (5) (6) I coparo wth D DFT th grat adatag o D DCT th act that t bac cto ar ot th copl al bt oly th ral al. Sgcatly rdc th coptatoal coplty ad rlt th abc o rddacy th DCT pctr. Th traorato ca alo b d to ltr th ag by rog ro th pctr l portat actor. Ually th do by zoal or thrhold thod. Crrtly, th ltr appl atrc D ltr wght. Th rt tp to dty th charactrtc o apltd-rqcy ltr. Sppo tally that th pl rpo ltr a D pl rpo ctor prodct o two D ltr. Thr lgth ar ally ch allr tha th do o th ag. Bca o parato o arabl dcrbg th z o th D ag, th ltr pctr ca b obtad a th prodct o ctor D pctra o ltr d D:, ) ( ) G ( ) (7) D( D D owr, th thod ca b d or dgg D low-pa ltr ad oly abot apltrqcy charactrtc hag a rctaglar hap. Low-pa plt D ltr ar cooly d cto o th t wdow, atr th oralzato o th o thr apl to a al o. It pobl to obta D ro o wdow D, ag th o D wdow o th a lgth. Th t th wdow ar ot oralzd. Sch D wdow ar ytrc wth rpct to th dpot o th atr. I th ca o Gaa wdow: w + σ σ σ D( w D( ) w D( πσ πσ Th pctr o th wdow: W πσ π σ ( ) (, ) + (8) (9) A t a coto pctr o D Forr D Gaa cto alo a D Gaa cto. Icrag th al o σ td th wdow ad a corrpodgly arrow t pctr. Th wdow thod at th bgg aalytcally dal pl rpo ltr LP, P, BP ad BS t b dtrd, by g r Forr traor lctd rqcy rqrt, ad th th cto ltpld wth th lctd t wdow. Ipl rpo ltr, P, BP ad BS ar cobato o pl rpo ltr LP ad Krocr pl. For ch a D ltr pl rpo ltr LP too log. A altrat olto to o away ro lar ltr or hort olar ltr, ad rplact o procg typ "th a or al" procg typ "to ach accordg to d": adapt ad local. A apl o ch procg ay adapt th ltr wght dpdg o th local ag cott. Aothr thod o dgg D o-rcr ltr a thod th rqcy doa. It hold a th atr, that pcc al o th apl, a atactory apltd-rqcy charactrtc o 74

3 th ltr, th traor t to a ld poto r Forr traor thod or D IDFT, ad th po a patal wdow o th rltg wght by rdcg th ocllato thr ba. Pay pcal attto to placg th apl th pctral bad ltr trato, bca t a t l tp, whch tr rdc th ocllato th pl rpo ltr. Copard wth th thod o th wdow th thod o rqcy ot t p a ltr pl rpo aalytcally, oly rcally: g a D IDFT o th atchg rqrt. Follow ro th codrato that rqr tp o th ltr th rqcy doa pl ry trog ad broad patal ocllato o th pl rpo. A a rlt, th o th coolto ry log. Th rlt that plac o occrrc o dcott ad dcoto chag th al o th apl th ag o th ltr trodc trog dtorto. At th a t d to th t z o th ag, ad probl wth g apl o t hor, yo ha to trapolat t ad th way trodc dcott, cag ocllato. Shortg th pl rpo by wdowg odrat oly partally ol th probl. d to ta dratc rdcto wdowg ad do o th atr ltr. Wth wghtg atrc o lar ltr, yo ca calclat th rqcy charactrtc. For th prpo, hold b copltd arod th atr o coct wth zro ch a way that th dpot o th ltr ta a poto o DC coct th a th wtch o qartr o atr ad traor th rlt to th rqcy doa g th D DFT.. Voc aaly Spch prograd th crbr. Ay daag o th crbr ca l or or otcabl chag procato. Phoatrt rarch coctratd aly o a tdy o th lary, whch th proc o artclato o pch od grator. Otpt o th grator th o-calld larygal to, th od o th prodc datal rqcy o t o z or th ba, a w hdrd z or trbl. Charactrtc o th od rch pctr occpyg th rqcy bad to a w z. It cota all th haroc - both ad odd. Acotc charactrtc o larygal to dpd o th arglottc prr ad ccy o tracto old oc. I phoatrt tgato do ot ha drct acc to th larygal to, o yo th rltg pch gal, whch ar, a w ow o th tr ocal trac. Cograto o ocal trac dtr th trattac o trato a odl o th phycal proc o artclato. Cat o th ocal trac ca th tc o th roac rqc, atd th pctr o th pch gal a a hghlght o o o th rqc. Th typ o ara ar calld orat ad ar ard qtally ro all rqc. Forat ar dcrbd by thr paratr: th rqcy ad ll. Forat rqcy th rqcy o th ddl o th bad. Chag aatocal trctr, rdcg actty o th lary, ad chag prograg gal ody th phycal paratr o th odl ad orc o larygal to rlctd a corrpodg chag th trctr o acotc oc [4]. c th coclo that, bad o aaly o acotc paratr, ca b d aatocal aoal th trctr ad ca otor actty ltato, a wll a dtr th wdprad o th caalty o th crbr [3]. A a rlt o acotc ttg ha bco a tgral part o th at phoatrt, wth th ot portat ar t-rqcy thod. Th prary aaly wa a aaly o th olatlty paratr F, whch ha bco a a tool or th alato o th lary, ad ha log b th ot alatd th l o prctag, th arag rat o dtrbac o th bac to, t alo l to aalyz th tattcal dtrbto o th prod o th bac to ad th prctag dtrbto t chag or t []. Th paratr F rlatd to jttr paratr d to tat th tablty o th larygal to. Th ba or all othr aaly pctrogra cary to calclat arag pctr thr alo a poblty to choc o th tatao pctr. Th oothd pctra allow obtag th orat paratr. O th ba o th ddal tatao pctra cptr o powr calclatd by org a total o cptrogra whch th ba or tatg th rqcy o larygal to. Cptral aaly carrd ot bad o th co traor: ˆ ( ) c (,5) π [l ( ] co () whr: ( - dcrt powr pctr - br o pctr bad, - a br o bad o aalyzd rqcy pctr - br o cptral actor. Spctr cptral oothd obtad ro th orla: ( c K ˆ c( ) co π () whr: K ordr o cptral oothg. Th obtad oothd pctrogra ad wth dato o paratr F calclatd paratr F jttr. Jttr ad hr ca b 75

4 qatd by qott th dato o th rqcy or jttr ad qott o th rgy dtrbac or hr [], accordg to (prtrbato qott): K K ( ) ( + ) K K K PQ % K K () K ( + ) K whr () - th qc o gl prod (or rqcy) lgth th ca o jttr (PPQ -ptch prtrbato qott), or a qc o gl prod rgy th ca o hr (EPQ - rgy prtrbato qott). I addto, th algorth calclat atocorrlato cto ca b d, whch ca b l a a ar o th gal prodcty ad rqcy (z ad locato o th a a). I paralll, aalyzg th t cor paratr hr obtad. Atocorrlato cto d a: ( )(,, ( (3) (, ) (, whr: - th lgth o ra apl, - th al o th -th apl ad +, - a. Atocorrlato cto o gal ca b wrtt a: yy p ( λ + σ ) + σ (4) p+ whr: λ - gal o th atocorrlato cto atr, - gctor o th atr, - br o gal o th atr. I th clacal Paro thod: ( z) p (5) + a z or pt gal d a: p a ) (6) Th o ha zro a ad o-corrlatd wth th gal; thror atocorrlato cto ha th or: yy a σ a (7) It ollow that th oght ctor o coct a th latt ctor o atr yy, aocatd wth t gal σ. Yo ca th thod dcd o Paro thod - USIC (ltpl Sgal Clacato. I th thod th rqc o th ba o argt o cto a ar tatd: P jω jω j( ) Ω [,,,..., ] T jω U( ), (8) p+ Alo th rqcy aaly o t-aryg otatoary gal g t-rqcy rprtato o gal ca b d. To do th, th Forr traor STFT (Short-T Forr Traor) ad th Wgr-Vll traor. Th rt ca b trprtd a o-dcrt t ad rqcy Gabor traorato. To adatag o th dcrpto: T STFT ( t, ) + F STFT ( t, ) τ) γ *( τ t) + jπt jπτ dτ ( Γ*( ) jπt d (9) whr: γ (t) a cto o th tporal wdow o obrato, Γ () th Forr pctr d a a wdow. I th ca o Wgr-Vll traorato hold b otd that prctly c lar chag o rqcy th T-Frqcy ara. By to: S S W ( V ) W ( V ) ( t, ) ( t, ) + + τ τ t + * t jπτ + * dτ jπt dt () Th rprtato charactrzd by th hght coctrato o rgy T-Frqcy ara, whch ha th bt rolto. 76

5 I addto, th tgato ca b dtrd traor o ocal trac ltr, who a o th charactrtc o T-Frqcy graph or orat graph. Th prpo o dltato o th datal rqcy ca b d to dtct th rt al o th a gal rqcy a ad trcato o hghr haroc. Th a ach tp ar dtrd aw at a pcd tral. r yo ca ta adatag o th atocorrlato cto, whch or gal o th pch tr ot to b a or accrat thod o th t ad th cptral thod o datal rqcy dtrg. 3. Coclo Udably odr gal procg thod ar trly l ad ot dpabl, dcal dagotc. rology, partclar, rqr rlabl thod, d to attd th ot t ha orga - th crbr. roagg rlt tll bg aalyzd by doctor, wthot th o gal procg tchq. Slarly, a carr ot aaly o th oc o patt. Dtly or objct ad or rlabl thod or coptr tchology, g th apl prtd th papr algorth. Athor: Sc. Daa Krzow Tchcal Urty o Lodz l. Wólczańa / Łódź tl. (4) al: darz@dc.p.lodz.pl Bblography [] Gabla Bra Charl: Aaly ad Syth o Pathologcal Vowl, PhD drtato, Urty o Calora, 3, Lo Agl [] rphy Kathar: Dgtal Sgal Procg Tchq or Applcato th Aaly o Pathologcal Voc ad oorphoc Sgg Voc, PhD drtato, Urdad Poltècca d adrt, 8, adrt [3] Godo J. I., Sáz., Oa V., Aglra S., Góz P.: A Itgratd Tool or th Dago o Voc Dordr, dcal Egrg & Phyc 8, 6 [4] Sáz., Godo J., Oa V., Góz P., Aglra S.: A thodology to Ealat Pathologcal Voc Dtcto Syt, procdg o AVEBA 5, 5, Florc [5] Fry., Laht A., o T., Datdar P.: Clcal Applcato o I Iag Procg rology, Itratoal Joral o Bolctroagt Vol, br, 999 [6] Sltch E. D., Dl O. G.: Iag Procg ad Data Aaly Coptr Toography, Urty o Bchart, 6, Bchart [7] oo Vctor: Bodcal Sgal ad Iag Procg, PhD drtato, Czch Tchcal Urty, 5, Prag 77

Comparisons of the Variance of Predictors with PPS sampling (update of c04ed26.doc) Ed Stanek

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