A Proposed Warped Modified-B Time-Frequency Distribution Applied to Doppler Blood Flow Measurement

Transcription

1 0 In'l Conf. Bioinformaics and Compuaional Biology BIOCOP'5 A Proposd Warpd odifid-b Tim-Frquncy Disribuion Applid o Dopplr Blood Flow asurmn F. García Noci, J. Solano Gonzálz, E. Rubio Acosa Insiuo d Invsigacions n amáicas Aplicadas y n Sismas Univrsidad Nacional Auónoma d éxico Circuio Escolar, Ciudad Univrsiaria, éxico D. F., 0450, éxico Conacing mail: fabian.garcia@iimas.unam.mx Absrac - On of h main goals in ulrasonic Dopplr blood flow masurmn is h simaion of h man vlociy. Th Dopplr signal s insananous frquncy has radiionally bn usd o sima h man vlociy. In his wor, a nonuniform discr im frquncy disribuion is proposd: h warpd discr odifid-b disribuion (WTFD B ). Th proposd procdur simas h insananous frquncy by concnraing h frquncy rsoluion around h Dopplr signal s insananous frquncy. As a rsul, a br prcision is obaind in h spcral simaion by using a WTFD B for noisy signals whn compard o ohr mhods such as h Discr odifid-b Tim Frquncy Disribuion (DTFD B ) wih h insananous frquncy calculad as h cnroid of h spcrum. Kywords: Signal Procssing, Warpd Tim-Frquncy Disribuion, Dopplr Flow asurmn Inroducion I is nown ha h blood flow man vlociy hrough a vssl s cross scion is proporional o h insananous frquncy of h Dopplr ulrasonic signal. This wor is focusd on h accura compuaion of h insananous frquncy of a signal (Caroid Arry simulad signal) in h prsnc of nois. A classic mhod o sima h insananous frquncy of a signal includs h compuaion of is spcrogram using a Shor Tim Fourir Transform (STFT). Howvr, i assums ha h analysd signal is saionary and i compromiss is mporal and frquncy rsoluion. An alrnaiv mhod is o us h Cohn Class Tim Frquncy Disribuions ha ovrcoms h saionary assumpion. Howvr, in som cass, i is dsirabl o incras only is frquncy rsoluion around crain frquncy of inrs, for xampl, around h insananous frquncy. Tha can b achivd if h lngh of h analysd discr signal is incrasd bu i also incrass noably h compuaional cos. On h ohr hand, nonuniform discr Fourir ransforms ar availabl such as h Warpd Discr Fourir Transform, which is abl o achiv ha as wihou having o incras h lngh of h analysd discr signal. Th aim of h wor prsnd in his papr is o incorpora a warpd frquncy scal (non-uniform) o h Cohn class of im-frquncy disribuions. W focus on h dvlopmn of h warpd discr odifid-b im-frquncy disribuion, alhough h procdur can b asily xndd o ohr disribuions. In a prvious wor, a warpd discr Wignr- Vill im-frquncy disribuion has bn proposd []. Tim Frquncy Disribuions of h Cohn Class Th im frquncy disribuions of h Cohn class (TFD) ar dfind as follows []., b h disribuion rnl. Tha rnl drmins h disribuion and is characrisics of mporal and frquncy rsoluion. also, b h Fourir ransform of h disribuion rnl: Thn, l i,, d, R x x d b h drminisic gnralisd local auo-corrlaion funcion, whr x is a complx signal. Finally, l j TFD, d b h im frquncy disribuion, which is dfind as h Fourir ransform of h local auo-corrlaion funcion, whr () () (3)

2 In'l Conf. Bioinformaics and Compuaional Biology BIOCOP'5 is h im variabl and is h (angular) frquncy variabl. 3 odifid-b Tim Frquncy Disribuion p 0 p N p 0 p N pn N p whr p 0,..., and is lngh is N. Finally, no ha (7) can b wrin as: (9) Th odifid-b im frquncy disribuion (TFD B ) blongs o h Cohn class [9][0][]. Is rnl is: i i, Thn, h disribuion is dfind by: TFDB, f j f G x x d d cosh whr G and f is h frquncy variabl,. Th dirc discrizaion of xprssion (5) consius h discr disribuion (DTFD B ), ha is: N pn m n, G p j W pwp x m n pxm n p cosh m whr n is h im discr variabl, is h frquncy discr variabl, W n is a sampling window and x n is a discr complx signal wih suppor nn,..., N and lngh N. Now, h discr odifid-b im frquncy disribuion wih priodic xnsion (PTFD B ) is sad []. Th procdur ssnially consiss on h following. Firs h discr disribuion (6) is valud a n 0 : N 0, G j W pwp x m pxm p pn mcosh m Scond, h gnralisd local auo-corrlaion funcion is idnifid: pg (8) WpW p x m pxm p mcosh m whr p N,..., N p (4) (5) (6) (7). Now, h funcion p which consius h priodic xnsion of consrucd as follows: R, is j p p0 0, R p (0) A his poin, h following scaling in h frquncy axis is carrid ou. I consiss on rducing by half h frquncy rsoluion. Th rsul is dnominad a discr odifid-b im frquncy disribuion wih priodic xnsion (PTFD B ): B 0, p0 jp PTFD R p () 4 Warpd Discr Fourir Transform Th main characrisic of h warpd discr Fourir ransform (WDFT) [3][4][5] is ha i can concnra h frquncy rsoluion around a frquncy of inrs, sinc i posssss a non-uniform frquncy rsoluion. Conrary o h convnional discr Fourir ransform (DFT) ha posssss a uniform frquncy rsoluion. Th warpd discr Fourir ransform wih a firs ordr allpass filr is dfind as: j p WDFT x p () j p0 whr xp j is a complx paramr ha xn wih drmins h warpd frquncy scal, n0,..., is a complx discr signal wih lngh, and 0,..., is an indx rlad o h discr frquncy. Th warpd frquncy scal mapping is givn by: sin W arcan cos (3) whr wih is h convnional uniform frquncy scal. Th magniud of h paramr,, is slcd according o h prcnag of frquncy poins o b

3 In'l Conf. Bioinformaics and Compuaional Biology BIOCOP'5 concnrad insid h spcral lob rlad wih h frquncy of inrs. Th angl of h paramr,,is slcd according o h valu of h frquncy of inrs. 5 Warpd Discr Tim Frquncy Disribuions In his wor, a warpd frquncy scal is incorporad o h discr im frquncy disribuions wih priodic xnsion. Alhough his procdur is illusrad for h odifid-b disribuion, is gnralisaion can b dircly gnrad. Th procdur calculas h warpd discr Fourir ransform of h priodic xnsion of h gnralisd local auo-corrlaion funcion, insad of calculaing is convnional discr Fourir ransform. Thn, h warpd discr odifid-b im frquncy disribuion (WTFD B ) is: WTFD R p j B 0, j p0 p (4) whr 0,..., p is h priodic xnsion of h gnralisd local auo-corrlaion funcion (9). and h signal Th angl of h paramr is h insananous discr frquncy of h signal bu xprssd in radians. I is imporan o considr h rduccion by half of h frquncy rsoluion. Third, h warpd discr im frquncy disribuion of h signal is calculad according o (4). Thn, h dfiniiv insananous frquncy of h signal is h phas associad o h frquncy componn wih h maximum magniud ha is obaind. B 0 angl max WTFD 0, 7 Applicaion o Dopplr Flow asurmn (7) I is nown ha h man vlociy of h blood flow hrough h cross scion of a vssl is proporional o h insananous frquncy of a Dopplr ulrasonic signal. In his wor, a fmoral arry signal is considrd for his sudy. Th simulaion of ha signal is daild in [6][7][8]. Th horical insananous frquncy is shown in figur. 6 Frquncy Esimaion using h Warpd TFD Th procdur o sima h insananous frquncy of a signal wih a dominaing singl frquncy and a narrow bandwidh is as follows [5]. Firs, h spcrum of h signal is calculad, using a convnional discr Fourir ransform: S p0 j n x n (5) Scond, h valu of h paramr xp j is calculad. For his, h prliminary insananous discr frquncy of h signal (h cnroid of h spcrum) is calculad: i N N N N S S (6) Figur. Thorical insananous frquncy of h simulad signal (Fmoral arry). Th sampling frquncy usd is 800 Hz. Sampling windows wih 50% of ovrlapping and lngh qual o 56 ar usd. For hos condiions, h insananous frquncy is calculad using boh, h discr odifid-b im frquncy disribuion wih priodic xnsion, PTFD B (), and h warpd discr odifid-b im frquncy disribuion, WTFD B (4). Th procdur o calcula h insananous frquncy has bn dscribd in h scion 6. Finally, h RS rror rspc o h horical insananous frquncy is calculad.

4 In'l Conf. Bioinformaics and Compuaional Biology BIOCOP'5 3 8 Rsuls Figur dpics a graph wih h simad insananous frquncy using h warpd discr odifid-b im frquncy disribuion. alhough his procdur can b xndd asily o ohr disribuions. Th mhod is applid o sima h man vlociy of h blood flow hrough a vssl, which is proporional o h insananous frquncy of h ulrasound signal obaind in h procss of Dopplr flow masurmn. Th xprimns hav bn carrid ou using a Fmoral arry simulad signal. Th rsuls obaind by h warpd discr im frquncy disribuion ar compard wih hos obaind by h discr odifid-b im frquncy disribuion wih priodic xnsion. Rsuls show ha h disribuion wih priodic xnsion is asir o calcula sinc O Nlog N ar usd; FFT-li algorihms of complxiy whil h warpd disribuion uss algorihms which ar basd on h marix muliplicaion whos complxiy ar O N. Nvrhlss, h prcision of h warpd disribuion o sima h insananous frquncy is nooriously br in h prsnc of nois. Acnowldgmns Figur. Esimad insananous frquncy of h simulad signal using h warpd discr odifid-b im frquncy disribuion (SNR=0dB) Tabl shows h RS rrors in h simaion of h insananous frquncy for h simulad signal. Diffrn lvls of normalisd gaussian nois (SNR of 40, 30, 0 and 0 db) hav bn addd, wih a concnraion of frquncy poins around h insananous frquncy of 40%. Th prcision of h warpd disribuion o sima h insananous frquncy is nooriously br in h prsnc of nois. odifid-b TFD Wihou Nois SNR 40dB 30dB 0dB 0dB Wih priodic xnsion PTFD B Warpd WTFD B Tabl.- RS rror [Hz] obaind in h simaion of h insananous frquncy of h simulad signal 9 Conclusions This approach prsnd in his papr incorporas a warpd frquncy scal (non-uniform) o h Cohn class im frquncy disribuions. Paricularly, h warpd discr odifid-b im frquncy disribuion has bn dvlopd, Th auhors acnowldg projc DGAPA-UNAPAPIIT (IN03), projc Consorciado CYTED (P506PIC095) by h financial suppor. Also w wan o acnowldg o. Funs, J. Conrras, S. Padilla and. Vazquz for hir chnical suppor in h dvlopmn of his wor. Rfrncs []. Cohn, Tim-Frquncy Analysis (Prnic-Hall PTR, 995) [] B. Boashash, P. Blac, An Efficin Ral-Tim Implmnaion of h Wignr-Vill Disribuion, IEEE Transacions on Acousics, Spch, and Signal Procssing, ASSP-35(), 987, [3] A. arur, S.K. ira, Warpd Discr Fourir Transform: Thory and Applicaions, IEEE Transacions on Circuis and Sysms -I:Fundamnal Thory and Appicaions, 48(9), 00, [4] S. Franz, S.K. ira, J.C. Schmid, G. Dobling, Warpd Discr Fourir Transform: a Nw Concp in Digial Signal Procssing, Procdings IEEE Inrnaional Confrnc on Acousics, Spch, and Signal Procssing,, 00, [5] S. Franz, S. K. ira, G. Doblingr, Frcuncy Esimaion using Warpd Discr Fourir Transform, Signal Procssing, 83, 003, [6] J. Cardoso, G. Ruano, P. Fish, Nonsaionary Broadning Rducion in Pulsd Dopplr Spcrum asurmns Using Tim-Frquncy Esimaors, IEEE Transacions on Biomdical Enginring, 43(), 996, [7] J. A. Jnsn, Esimaion of Blood Vlociis using Ulrasound (Cambridg Univrsiy Prss, 996).

5 4 In'l Conf. Bioinformaics and Compuaional Biology BIOCOP'5 [8] P.Fish, Physics and Insrumnaion of Diagnosic dical Ulrasonic (John Wily Sons, 000). [9] B. Boashash, V. Sucic, Rsoluion asur Criria for h Objciv Assssmn of h Prformanc of Quadraic Tim- Frquncy Disribuions, IEEE Transacions on Signal Procssing, 5(5), 003, [0] Z. Hussain, B. Boashash, Adapiv Insananous Frquncy Esimaion of ulicomponn F Signals Using Quadraic Tim Frquncy Disribuions, IEEE Transacions on Signal Procssing, 50(8), 00, [] B. Bara, B. Boashash, A High-Rsoluion Quadraic Tim Frquncy Disribuion for ulicomponn Signals Analysis, IEEE Transacions on Signal Procssing, 49(0), 00, 3-39 [] E. Rubio, J. Solano, F. Torrs, F. García-Noci, A Proposd Warpd Wignr-Vill Tim Frquncy Disribuion Applid o Dopplr Blood Flow asurmn, Procdings of Biomdical Enginring (BioED), 006.