Time of Flight Estimation in the Presence of Outliers Part I-Singe Echo Processing

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1 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < Te of lght Estaton n the Presene of Outlers Part I-Snge Eho Proessng Alexander Apartsn, Leon N. Cooper, and Nathan Intrator Abstrat When the Sgnal-to-Nose Rato (SNR) falls below a ertan level, the error of the Te-of-lght (To) Maxu Lelhood Estator (MLE) nreases abruptly due to the wellnown threshold effet. Nevertheless, operatng near and below the threshold SNR value ght be neessary for any reote sensng applatons due to power-related onstrants. These onstrans ay nlude a lt on the axu power of a sngle soure pulse or a lt on the total power used by ultple sgnals transtted durng a sngle easureent. or narrowband sgnals, the threshold effet eerges ostly due to outlers ndued by loal axa of the autoorrelaton funton of a soure sgnal. ollowng the prevously explored path of bosonar-nspred eho proessng, ths paper ntrodues a new ethod for To estaton n the presene of outlers. The proposed ethod eploys a ban of phase-shfted unathed flters for generatng ultple based but only partally orrelated estators (ultple experts). Usng ahne learnng tehnques, the nforaton fro the ultple experts s obned together for provng the near-the-threshold To estaton fro a sngle eho.. Index Ters Sonar, Threshold Effet, Bosonar, Te-of- lght Estaton T I. INTRODUCTION he lassal Te of lght (To) estaton ethod eploys a athed flter[] for the estaton of the two-way travel te. Sne the returned sgnal s usually orrupted by an addtve nose, the resultng estator produes a value whh s only an approxaton for the true two-way travel te. The average agntude of the squared estaton error depends on the aount of addtve nose and a shape of the soure wavefor [3]. or oderate levels of nose and for soure pulses wth suffently wde bandwdth, the Mean Square Error (MSE) of the lassal estator losely follows Manusrpt reeved Mat,. Ths wor was supported n part by the U.S. Ary Researh Offe A. Apartsn s wth the Blavatn Shool of Coputer Sene, Tel -Avv Unversty, Israel(phone: ;fax: ;e-al: apartzn@tau.a.l) L.N. Cooper s wth Insttute or Bran and Neural Systes and Physs Departent, Brown Unversty, Provdene, USA (e-al: leon_ooper@brown.edu) N. Intrator s wth the Blavatn Shool of Coputer Sene, Tel -Avv Unversty, Israel(e-al: nn@tau.a.l) the Craer-Rao Low Bound (CRLB) whh s the best ahevable perforane for any good estator []. In the theory of optal reevers, these ondtons orrespond to a reever beng n the oherent state []. However, n any pratal ases, atve sonar or other reote sensng deves are eployed under ondtons that no longer ensure a oherent reepton and estaton. An applaton ght pose power and bandwdth onstrans on the soure pulses generated by a transtter. or nstane, the axu power of an ndvdual pulse ould be restrted by the ltatons of eletron or aoust equpent eployed at the transtter. or ltary applatons, t s soetes portant to eep the soure power low to avod deteton by the adversary. The low auray of an ndvdual estatons ould be soehow opensated by averagng (fusng) ultple ndependent easureents. However, the usage of ultple soure pulses for a sngle estaton ght be lted by onstrants posed on the total easureent te (or, equvalently, on a nuber of easureents) and by onstrans on the total eployed power. The later s rual for easureents that are done by a oble robot sensor wth lted autonoous power soure (a battery) [8]. Moreover, there are also ght be onstrants on a shape of the transtted soure sgnal. or nstane, underground exploraton by low powered ses pulses eploys lowbandwdth soure wavefors sne the hgh frequeny harons are attenuated rapdly n the ground [7]. The lassal approah based on the applaton of athed flter and on a sple averagng of ultple easureents s not adequate for these onstraned ases. It has been shown that, f the level of nose rses above a ertan level, the ean squared error of the onventonal To estator abruptly dverges fro CRLB produng the well-nown threshold effet [5]. Moreover, the threshold effet ntensfes when a soure pulse wavefor has low bandwdth spetru as t s soetes the ase n the underground exploratons. The lassal approah does not provde an effent way for tang the Mean Square Error (MSE) of near-the-threshold To estaton and t does not address the energy-related onstrants n a proper way. Therefore, the alternatve approah s requred for provng the effeny of To estaton that s arred out n the presene of the threshold effet and wthn the posed power ltatons. In [] [7], an alternatve approah for To estaton s desrbed. The To estaton s arred out usng a verson of the Kalan flter. The shape of the pulse envelope s odeled usng a paraetr odel and the paraeters of the odel and the To are jontly estated on-the-fly by an teratve or reursve proedure. The ethod desrbed n ths paper does

2 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < not assue any paraetr odel for the pulse shape or ts envelope. Usng the ahne learnng approah, the estator s aoodated for any spef pulse wavefor by tranng fro the sulated data. It appears that soe of the deas for provng the auray of near-the-threshold To estaton ould be borrowed fro nature. Many eholoatng anals use ehanal waves (e.g. arborne ultrasound or underground nfrasound) for ounatons and navgatons. Naturally, bologal systes have any onstrans slar to those desrbed above and yet, the eholoatng anals deonstrate strng ablty for the effent extraton of nforaton fro the very nosy sgnals []. Moreover, a bosonar proessng syste eploys ndvdual eleents (e.g. ells) wth nsuffent te-frequeny senstvty for relably estatng the sgnal paraeters dretly. However, organzed n assvely parallel sensory and oputatonal networs, these eleents an funton together wth astonshng auray [5]. A bologally nspred approah to the To estaton tres to extrat addtonal nforaton fro the reeved sgnal by eployng a ban of flters nstead of a sngle athed flter as n the lassal theory. The partal nforaton obtaned by the applaton of ndvdual flters s obned together usng robust fuson ethods n order to produe an estator that outperfors the lassal athed flter. There s a sgnfant body of researh fousng on analyzng and ng bosonar systes. In [5], a flter ban of obned band pass and low pass flters was nspred by the eholoatng haratersts of bg brown bat. The nforaton fro these flters was obned usng three fuson tehnques nludng average-le and votng-le ethods. Ths wor has shown that n hgh nose settngs, auray oparable to that of the lassal approah ould be aheved by ntellgently fusng nforaton fro ultple bandwdth-onstraned flters. In [9], the two-glnt resoluton of onstraned sonar was analyzed usng slar approah that nluded a flter ban of 8 low-bandwdth flters followed by a for of teplate athng. The resoluton aheved by ths approah was far beyond the resoluton of ndvdual proessng eleents (flters). In ths wor, we ontnue wth a bosonar-nspred approah by analyzng the threshold effet for sple narrowband soure sgnals. Identfaton of the ajor soure of the rapd degradaton of auray aused by the threshold effet otvates the desgn of a faly of full bandwdth phaseshfted flters. These flters obned together provde ore nforaton opared to the athed flter alone. Ths extra nforaton an be used for tgatng the threshold effet and, thus, for proveent n the auray of a sngle easureent. We eploy ahne learnng tehnques for extratng the nforaton fro the vetor of responses generated by the applaton of these ultple flters. We onstrut a lassfer to assgn a label to a easureent based on the vetor of flters responses oputed on the retuned sgnal. Usng ths assgned label, the lassal estate (obtaned usng the athed flter) s orreted to aount for the expeted bas that appears due to the threshold effet. In our prevous wor, we used oputer sulatons to deonstrate the effetveness of phase-shfted unathed flters for onstrutng a SNR-dependant Te-of-Arrval (ToA) estator that proves estaton perforane n the presene of the threshold effet. In a reent paper [4], a sple regresson ethod was used for onstrutng a ore robust ToA estator whh does not requre a pror nowledge of the operatonal SNR value. We extend ths proposed approah by provdng an analytal treatent of the phase-shfted unathed flters and by ntrodung a ore robust fuson tehnque whh s based on wea outler lassfers. The wor s organzed as follows. In seton II, the foral fraewor for the two-way travel te estaton proble s establshed and then ths fraewor s used for analyzng the ajor ause of the threshold effet for narrowband sgnals. In seton III, the phase-shfted unathed flters and assoated based estators are ntrodued and dsussed. The ahne learnng-based tehnque for fuson of the nforaton fro the based estates s outlned n seton IV. In seton V, we report sulaton results whh deonstrate the effetveness of the desrbed ethods. The wor s onluded by the suary seton whh also dsusses possble generalzatons of these ethods and possble dreton for future wor. II. TIME-O-LIGHT PROBLEM AND THE THRESHOLD EECT In ths seton, we provde an analytal treatent of the threshold effet assoated wth Te of lght (To) Maxu Lelhood Estator. We derve an approxaton for the probablty of an outler event usng the fundaental propertes of a narrowband soure sgnal, naely ts entralzed bandwdth to entral frequeny rato. In the next seton, we wll extend the analyss for the based estator and show that t s possble to devse a faly of based estators whh are not opletely orrelated. The Te of lght estaton (To) proble also nown as Te of Arrval (ToA) estaton proble arses n the ontext of Radar, Sonar and other reote sensng applatons. In dsrete forulaton, the proble ould be stated as follows []. An aoust soure ets a sgnal wavefor st () and, then, the refleted sgnal ontanated by Addtve Whte Gaussan Nose (AWGN) nt () s ped up by a reever after a te delay t. Then, the reeved sgnal s gven by: r( t) s( t t ) n( t) () Where s an unnown fator that aounts for a sgnal loss due to the attenuaton n the edu. Sne, n ths wor, we onsder the estaton under unnown SNR, wthout loss of generalty, we assue = for the rest of the presentaton. The estator that we desrbe below s nsenstve to the average power of the reeved sgnal and, thus, larger attenuatons orrespond to a stronger nose or to a saller SNR n our odel. In prate the Addtve Whte Gaussan Nose assupton does not always hold due to frequeny-seletve attenuaton and host of other eda-dependent effets. However, sne our goal s to present an proveent for the To estaton usng low-frequeny narrowband soure sgnals, the reeved sgnal s less affeted by the frequenyseletve attenuaton.

3 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3 The goal of To estaton s to reover a value of unnown te delay t based on the reeved sgnal rt () and on the nown soure wavefor st (). We usually see an estator t ˆ () r that nzes the MSE ost funton. Sne nose saples nt () are..d. rando varables wth a N (, ) probablty densty, the ondtonal probablty densty funton for reeved sgnal gven the te delay s Pr ( rt ( ) ) rt N () exp r( t) s( t ) t where N s the nuber of wavefor saples reorded durng an observaton nterval. Assung no pror nowledge on values of te delay (unfor dstrbuton) and usng Bayes rule, the a posteror ondtonal probablty densty s gven by: Pr rt ( r ) Pr( ) Pr tr ( r) Pr( r) (3) K exp r( t) s( t ) t where K s a noralzaton fator whh aounts for parts of the ondtonal densty that do not depend on t. The Mnu Mean Square Estator (MMSE) s the ondtonal expetaton of a te delay gven the observaton: t ˆ E [ r ] Pr ( r ) MMSE (4) Sne the evaluaton of ths su nvolves oputatonally expensve and nuerally unstable steps, the Maxu Lelhood Estator (MLE) an be oputed nstead []. The MLE s obtaned by axzng log-lelhood funton: l( ) log Pr ( r ) rt t r C( ) K r( t) s( t ) K t Agan, the unertantes about the attenuaton onstant are enoded by the unertantes n the nose power,.e. unnown SNR. The axzaton of the log-lelhood funton s equvalent to axzng the ross-orrelaton C() (the output of the athed flter) between the reeved sgnal and the soure sgnal []. Therefore, the Maxu Lelhood Estator of te delay s tˆ argax C( ) (6) MLE urtherore, the ross-orrelaton ould be expressed n ters of the autoorrelaton funton of the soure wavefor and an addtve fltered nose: (5) (7) C( ) s( t t ) s( t ) n( t) s( t ) R( t) W ( ) W s zero-ean, the sgnal R s the expeted value of the ross- C when averaged over all possble nose Sne the fltered nose () autoorrelaton () orrelaton () saples (the enseble average lelhood funton). As wll be dsussed later n ths seton, the ultodal shape of ths enseble-averaged log-lelhood funton plays a rtal role n appearane of outlers when the nose level rses above a ertan threshold. The aount of nose n the reeved sgnal s haraterzed by pre-fltered and post-fltered Sgnal-To-Nose Rato (SNR) defned by SNR SNR post pre Ps P P P R W n t st () N R( ) t s() t Where Ps, Pn, PR, P W are the power of the soure sgnal, the power of the nose, the power the autoorrelaton of the soure sgnal and fltered nose respetvely. When the SNR level s hgh, the fltered nose Wt () auses the MLE to be near the global axu of the autoorrelaton funton at t SNR (8). When the SNR falls below a ertan threshold level thresh, the fltered nose ay oasonally ause the global axu of the reeved sgnal to be deteted far away t. fro the true value or general autoorrelaton funton the wrong pea ould eerge alost anywhere outsde the vnty of the global axu of the autoorrelaton funton. But for a speal lass of sgnals and, ore generally, for a speal lass of lelhood funtons, the outlers wll be lustered around ertan loatons whh are haraterzed by the loal axu of the enseble-average lelhood funton. The narrowband sgnals whh are frequently used n sonar and radar applatons are good exaple for suh lass of sgnals sne the autoorrelaton funton of narrowband sgnals has sgnfant loal axa (sde lobes). Usng dsrete osne representaton, a DC-less narrowband soure sgnal ould be represented by (9) s( t) a os f t The total (fnte) sgnal energy, the entral frequeny, and ean square bandwdth of a sgnal n the above representaton are gven respetvely by E a a f, E B a f E () The entralzed ean square bandwdth of a sgnal s gven by B a f B () E We assue that ost of the sgnal energy s onentrated around ts entral frequeny or, equvalently, the sgnal s

4 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4 entralzed bandwdth s uh saller than the sgnal entral frequeny, naely B or f f for all suh that a Under ths assupton, the autoorrelaton funton of a sgnal has loal axa (sde lobes) loated at a dstane T fro the pea at t. or the rest of ths seton we onsder only the two ost sgnfant sde lobes orrespondng to values of although n our sulatons even ore dstant sde lobes are taen nto aount. The probablty of outlers s the probablty that a loal axu of the sgnal autoorrelaton funton beoes the global axu of the ross-orrelaton funton. Below we approxate the probablty of ths event (an outler event) usng a narrowband sgnal representaton. Wthout loss of generalty, we assue to as we are nterested only n relatve postons of the autoorrelaton axa. We also onsder the sgnal and nose values only n the vnty of the global axu and near a sngle hghest sde lobe of the ross-orrelaton (two-pont odel). Then, usng ths odel, the outler probablty s Pr( outler) Pr C() C () Pr R() R W W Usng Taylor expanson the left hand sde of ths nequalty s approxated by f R() R a ( os Denotng f a ( os V a f E B B (3), the dfferene between values of autoorrelaton funton at ts peas ould be expressed as EV. It s lear fro the representaton that the dfferene dereases when the entral frequeny s larger (as peas are ore losely spaed) or f the sgnal entralzed bandwdth s lowered, (the sgnal envelope beoes less sharp). Asyptotally, f the sgnal bandwdth s ade very sall (tends to zero), the resultng sgnal approahes a snusod wth zero dfferenes between peas heght. Now we turn our attenton to the dfferene between fltered nose saples. ltered nose saples are lnear obnatons of zero-ean noral varables and, therefore, are dstrbuted as zero ean noral rando varable wth varane E. The ovarane between fltered nose w saples at dstane s CovW (, ) E W () W R (4) Agan, usng Taylor expanson the value of the autoorrelaton funton near the sde lobe ould be approxated by: f R a os E V (5) Therefore, the orrelaton oeffent between fltered nose saples at neghborng axa s E V W, V E (6) The dfferene between two orrelated norally dstrbuted zero-ean rando varables s also zero-ean noral wth varane: W E E V 4EV (7) Therefore, the probablty of the outler beoes R W Pr( R W) Pr EV EV (8) E E Pr Z V QV Where Z s a rando varable wth standard noral dstrbuton and Q s the tal probablty of the standard noral dstrbuton. Therefore, the probablty of outler beoes large when ether sgnal-to-nose rato E bandwdth to entral frequeny rato V s sall. or an outler easureent, there wll be a bas of or entralzed ndued by left and rght sde lobes of the autoorrelaton funton. Sne the probabltes of left and rght outler are equal, the overall bas due to outlers wll dsappear. However, sne the pat of an outler to the overall error s proportonal to the bas squared, an outler event wll serously affet MSE of the estaton. The resultng MSE ould be

5 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5 deoposed nto parts refletng the pat fro nler and outler events separately: MSE Pr( outler) Varane Bas outler outler (9) ( Pr( outler)) Varanenler Assung that pat to the MSE fro the expeted addtonal varane due to outler Pr( outler) Varane Varane outler nler () s sall relatve to the pat by the bas squared ter, the MSE s approxated by: MSE Varane Pr( outler) Bas () nler outler Therefore, the ost of outler for the overall MSE s proportonal to E QZ V () The analyss above s onsstent wth the well-nown s- shaped urve desrbng dependeny between the MSE of MLE and the Sgnal-to-Nose Rato (SNR) [3]. or large values of SNR, the probablty of outler s sall and, therefore, the MSE approahes the CRLB urve and the syste s sad to be n the oherent-state. As soon as SNR drops below a ertan threshold value, the MSE nreases abruptly due to the presene of outlers and, aordngly, the syste s sad to be n the se-oherent state. or very low SNR values, the sdelobes produed by the strong presene of the entral frequeny n the sgnal spetru do not play a sgnfant role n the loaton of outlers and the syste s sad to be n non-oherent state. In the analyss above, we have negleted the pat to the MSE by the addtonal varane due to outlers. Ths s justfed by the fat that the probablty of outler and the dfferene between nler and outler-ndued varanes are nversely onneted. or the probablty of outlers to be sgnfant, the heghts of the an lobe and the seondary lobe need to be oparable, ang the dfferene n outler and nler varanes sall. Conversely, for the expeted bas oponent of the MSE, the bas square part s ndependent fro the probablty of an outler event. Nevertheless, n forthong part II paper we address the varane parts of the MSE by obnng estaton fro the ultple easureents (pngs). or the rest of the paper, we are anly onerned wth the se-oherent area and the ethods for provng the MSE by redung the effet of outlers on t. We onlude ths seton by notng that the entral frequeny appears n both nonator and denonator of the expresson above. Therefore, although the sgnals wth hgher entral frequeny have greater probablty for outler, the pat on overall MSE of eah outler event s saller. III. PHASE-SHITED UNMATCHED ILTERS In ths seton, we ntrodue the Phase-Shfted Unathed lters and the assoated based To estators. We extend the analyss fro the prevous seton to show that although eah of the based To estators has larger estaton error, the errors of these estators are not opletely orrelated. Ths property, whh we denote by Se-orrelated Estators (SME) enables a fusng of ultple based To estators nto a sngle robust To estator usng ahne learnng algorths whh are desrbed n the next seton A Phase-Shfted Unathed lter s generated fro the soure wavefor st () by shftng the phase of all of ts spetru ontent by a sae value. The phase-shftng operaton an be easly perfored usng the Hlbert Transfor [] that ould be oputed nuerally. Contnung wth prevously used notaton, the phase-shfted sgnal s represented as (3) s ( t) a os f t The ross-orrelaton between the reeved sgnal and the unathed flter ould be splt nto a su of two ters as n the prevous seton: C ( t) R ( t) W ( t) (4) The frst ter s the ross-orrelaton between the soure sgnal and the unathed flter. The seond ter represents an addtve fltered Gaussan nose. The ross-orrelaton between the soure sgnal and unathed flter s gven by (5) R ( t) a os f t The assoated To estator s obtaned by loatng the global axu n the unathed flter output, that s tˆ arg ax C ( t) (6) or sall values of phase shft, the global axu wll be loated near the pea of autoorrelaton funton. Therefore, the poston of the global axu of R () t ould be estated by settng the te dervatve to zero and usng Taylor expanson around zero phase and te R () t a f sn ft (7) t a f E B t sn f t a f f t (8) and, therefore, the poston of global axu s M t B (9) V The value of the ross-orrelaton funton at ts global axu ould be approxated usng lnear and quadrat ters of the Taylor expanson

6 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6 M R( t ) a os f B a f B Expandng and substtutng we obtan: a f B E E B B V V E (3) V R() E V The heght of the pea of the ross-orrelaton between soure sgnal and phase-shfted unathed flter s lower opared to the pea of the soure sgnal autoorrelaton. Therefore, the unathed flter does not axze pea sgnal-to-nose rato as the athed flter does. The dfferene V E depends on a spef value of the phase shft. V Sne the phase shft appears n quadrat for, the dfferene s the sae for postve and negatve phase shft of the sae agntude. The heghts of two losest sdelobes of ross-orrelaton funton whh are loated at dstanes T fro ts an pea ould be estated usng Taylor expanson: M R t f a os f B ( ) f a os f B R t M ( ) f a f B (3) (3) (33) R t M V E R V (34) Therefore, the dfferene between the heghts of sdelobes and the an pea s M M R R ( t ) R t V R E V (35) EV V Thus, the dfferene between the heghts of sdelobes depends on ther loaton relatve to the an pea. or postve phase shft, the dfferene between the an pea and the rght sdelobe s greater than dfferene between the an pea and the left sdelobe (fgure ). Sne the fltered nose has the sae autoorrelaton funton as n ase of the athed flter, the probablty of a left outler for ˆt s greater than the probablty of the rght outler. or nstane, the probablty of the left outler generated by the larger sdelobe s therefore Pr ( outler ) Pr W R left EV V Pr Z EV E Pr Z V V E QV V (36) It sees that the probablty of at least one lass of outlers s greater for a sngle phase-shfted estator. However, the relaton between probabltes of left and rght outler s reversed f we onsder the estator generated by phase shft wth the sae agntude but dfferent sgn. Therefore, the expeted bas of two estators wth opposte phase shft s aneled out and ther average onsttutes an unbased estator for To. tˆ ˆ t tˆ (37) We have reported the proveent aheved by ths sple average n []. The gan n perforane obtaned by suh estator depends on the SNR level and on a seleted phase

7 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 7 shft value. The eployed value of phase shft defnes the tradeoff n MSE between the nler varane and the expeted square of outler bas. The optal value of a phase shft ould be albrated to ath the underlyng SNR level and the soure sgnal paraeters [3]. Unfortunately, ths ethod requres the nowledge of the operatng SNR level whh s not always avalable n prate. Ths requreent ould be avoded by obnng nforaton fro several pars of syetrally phase-shfted unathed flters. Sne eah phase-shfted unathed flter perturbates the nose sgnal n a slghtly dfferent way, eah estator derved fro a dfferent phase-shfted unathed flter ontans soe addtonal nforaton on the nterrogated sgnal. or nstane, for two dfferent values of phase shft, the ovarane between nose saples at global axu s M M Cov ( W ( t ) W ( t )) w M M R ( t t ) R ( ) V V E V (38) And, therefore, the orrelaton oeffent (assung sall value of phase shft) V V M, (39) Thus, for low SNR (presene of outlers) the phase shft of the soure wavefor generates a faly of unathed flters, fro whh a faly of based se-orrelated estators an be onstruted. Ths faly of SMEs an be fused nto a sngle estator wth proved estaton auray. Sne the orrelaton between outler events s dereasng as the dfferenes between phase shft values s nreasng, t sees that, fusng a par of estators orrespondng to a greater dfferene between orrespondng phase shft values should ae sense. However, as the underlyng unathed flters nreasngly dverge fro the shape of the athed flter, the error of the ndvdual based estators nreases. Therefore, sne the probablty of outlers nreases wth a derease n the SNR, the estators wth larger phase shft dfferenes should be assgned a greater weght n the SMEs fuson as the orrelaton between the beoes ore portant. Ths suggests that the fuson of the SMEs depends on the SNR. In addton, sne often an unbased estator s requred, a fuson ethod should soehow redue the obned bas of ultple nput estators to produe an unbased output estate. In [4] we have ntrodued a ethod for fuson of ultple based estators usng regresson. The Deson Trees based regresson [] was traned usng a x of saples for a range of SNR values n the se-oherent zone and, thus, the obned estate dd not requre a pror nowledge of the operatonal SNR value. The ethod provded up to % proveent n the estaton auray opared to the onventonal Maxu Lelhood Estator. In the next seton, we ntrodue an alternatve fuson shee that reles on phase-shfted unathed flters for a wea lassfaton of To MLE outoes nto lasses orrespondng to peas of the sgnal autoorrelaton funton. Usng the labels produes by the lassfer, the expeted bas ntrodued by an outler easureent s redued fro the MLE value, resultng n up to a 3% nrease n the estaton auray. The addtonal nforaton generated by the lassfer ould be used for devsng an effent ethod for fuson of ultple ndependent To easureents n the presene of outlers. Ths and other applatons of the desrbed ethod wll be addressed n forthong papers. IV. MACHINE LEARNING USION O BIASED ESTIMATORS Ths seton desrbes a novel ahne learnng ethod for provng the MSE of the Maxu-Lelhood To estator at the se-oherent regon. The ethod eploys a sngle wea lassfer that s traned to label the MLE value aordng to the sde lobe of the autoorrelaton funton that ndued the pea n the output of the athed flter. The lassfer reles on a ban of phase-shfted unathed flters to generate an nput feature vetor for tranng and lassfaton. Based on the label produed by the lassfer, an expeted bas due to outlers an be oputed usng the estated pror probabltes of dfferent outler types and the estated onfuson atrx of the lassfer. The resultng estator s obtaned by subtratng the expeted bas fro the To MLE value. Below we desrbe the ethod n a general for to deonstrate ts applablty to slar estaton probles. Consder an estator ˆx of a paraeter x ontanated by addtve rando nose : ˆx x. We odel outlers by assung that the rando nose s sapled for dfferent dstrbuton (slar to a xture odel). These nose probablty dstrbutons are haraterzed by the vetor of eans (... ) and the vetor (... ) of ther varanes. There s also a vetor of pror probabltes p ( p... p), p suh that the probablty of a nose saple to be seleted fro dstrbuton s p. We defne true (unnown) labelng funton Nx ( ˆ). We say that N( xˆ ) f nose at easureent ˆx s generated usng the probablty dstrbuton wth ean and varane b The expeted bas of the easureent s of unbased estator u p. Therefore, we an opute the error xˆ xˆ b as

8 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 8 e ˆ ( xu x) ( b) p p bo (4) If the expeted bas s zero as n the ase of syetr outlers ndued by an autoorrelaton funton, the last ter dsappears and the total error beoes: (4) e p p Assue we an tran a (wea) lassfer that assgns labels L( xˆ )... to an estate ˆx suh that Pr( L( xˆ) N( xˆ) j) g, j whh s the probablty for a easureent to reeve a label by the lassfer gven that the easureent s nose s generated by the j-th dstrbuton. These ondtonal probabltes for the onfuson atrx G of, the lassfer: { }.Than we an express G g, j, j posteror probabltes for a saple to be labeled as lass as: q Pr( L( xˆ ) ) Pr( L( xˆ ) N( xˆ ) j) Pr( N( xˆ ) j) j or n a atrx for q j g, j probabltes, the lelhood atrx p j G p. Gven ths posteror lass C { } Pr( N( xˆ) L( xˆ) j), j g j, p q j j,, j, j Pr( L( xˆ) j N( xˆ) ) p q (4) beoes: (43) Based on ths lelhood atrx, we an also opute the ondtonal bas of a easureent gven the lassfer has labeled t wth lass. ( ˆ ) b x x (44) L( xˆ ) L( xˆ ) j j, j Therefore, we an redue approprate bas b(x) fro the estate usng addtonal nforaton obtaned fro the lassfer usng: b( xˆ) b f L( xˆ) (45) The expeted value of ths funton s equal to the bas of the orgnal estator as b( xˆ ) q b j j, j j o j j q p b (46) Ths allows onstrutng the odfed unbased estator xˆ xˆ b( xˆ ). The MSE of the new odfed estator s e ( ˆ ) ( ( ˆ x x b x)) qb eo bo qb o e q b q b (47) The proveent of the MSE depends on the dfferenes between the last two ters whh s always non-postve due to Jensen s nequalty [4] qb qb (48) or the ase of the syetr outlers b q b p and for a perfet lassfaton o q p ahevng, the bas square part of the MSE s opletely redued, resultng n MSE ontanng only the varane part e p (49) To deonstrate the auray gan obtaned by ths approah, we onsder a sple exaple nvolvng two syetr outlers (alled left and rght outlers) whh our wth pror probablty po eah. Therefore, the vetor of the pror probabltes beoes: p ( p, p p ) ( p p p, p ) (5) L M, R O, I O O The outlers are syetr, thus, the vetor of eans s (,, ) (,, ). The varanes of outlers L R O O are dental and are dfferent fro the varane of the nler (,, ). Assue we have anaged to tran a O I O lassfer whh produes a onstant slassfaton rate on all lasses. That s g ˆ ˆ, j Pr( L( x) L( x) j), j and g ˆ ˆ, Pr( L( x) L( x) ). Under these settngs, the bas b of the estator ˆx s zero and the error beoes e pi I po O poo (5) The vetor of posteror lass probabltes an be shown to be q ( q, q, q ) ( q, q, q ) L M R O I O ( p ( 3 p ), p O O I p, p ( 3 p )) O O O (5)

9 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 9 An fnally, the ondtonal expeted bases gven the lass labels are b L bm OpO br ( ) (53) q Then the proveent n the MSE s equal to p e e e q b O o ( ) qo L po ( 3 ) p ( 3 ) O (54) Agan, for a perfet lassfer wth, the dfferene aheves ts axu at e p and, therefore, t O opletely anels out the bas square part n the estator error. or rando guess lassfer wth 3, ths dfferene beoe zero and no gan n the auray ould be aheved. The above algorth requres paraetr odelng of outler lasses to obtan estates on, and the pror probabltes p. These paraeters ould be obtaned fro labeled tranng data by varous ethods (e.g. EM algorth). The sae tranng data ould be used to tran ultlass lassfer to assgn labels aordng to outler lasses. In addton, the lassfer perforane needs to be estated and suoned n a onfuson atrx whh ght be done usng the evaluaton data set. In the next seton, we desrbe the tranng and estaton proess along wth sulatons that are used for evaluatng the ethod. The sulatons show that the ethod aheves a sgnfant proveent (up to 3%) n the auray of the se-oherent To estaton. We onlude ths seton by notng that although the desrbed ethod tres to redue the bas square part of the error ntrodued by outlers, t does not attept to aount for hgher unertanty whh s usually assoated wth outler easureent. The nluson of the varane of the easureent as a easure of ts unertanty requres fuson of nforaton fro ultple ndependent easureents whh s onsdered n the forthong paper. V. THE ESTIMATION PROCESS AND RESULTS The sulaton proess (gure ) onssts of two phases. Durng the preproessng phase (blos P through P4 n the dagra), a large data set of randoly generated saples s reated and, then, dfferent portons of ths data set are used for tranng the lassfer and for the evaluaton of the lassfer perforane. Durng the estaton phase (blos E through E4 n the dagra), the To estaton for a new sulated saple s oputed usng the lassfer s predtons. We start wth the desrpton of the preproessng phase. About data saples eah representng a reeved sgnal are sulated by addng rando Whte Gaussan Nose (WGN) to the seleted narrowband sgnal (P at the dagra and eq. ). The nose s produed usng ten post-fltered SNR values orrespondng to the se-oherent range (3-8dB). The saples of all SNR values are xed together for all further proessng. A feature vetor for eah saple s extrated usng a ban of n phase shfted flters (slar to the feature extraton step perfored durng the estaton phase). The flters n the ban are generated usng orgnal soure sgnal wavefor (eq.3). or an odd n, the values of n phase values are seleted fro the nterval a: a aordng to a a where,..., n (55) n, eanng that the ddle Thus, n and n eleent n the feature vetor orresponds to the Maxu Lelhood Estator. An n-valued feature vetor t ( t,..., t ) s fored by oputng the loatons of the n global axu n n ross-orrelatons: t arg ax C ( t ) (56) where C () t s the ross-orrelaton of th flter wth a sulated saple (eq.4). The value returned by the Maxu Lelhood Estator (whh s obtaned usng the zero phase-shfted flter as n eq. 6) s subtrated fro all eleents of t. Therefore, a orreted feature vetor s oputed as t ( t,..., t ) ( t t,..., t t ) n n n ( t,..., t,, t,..., t ) n n3 n n (57) The subtraton of the MLE value s neessary as we are nterested n dentfyng an outler by ts loatons relatvely to the loaton of the global axu of the autoorrelaton funton. Consequently, n-valued feature vetors always wll have zero as ther ddle eleent and therefore they atually represent total of n- non-trval features. The MLE values are also used for oputng the saples true labels aordng to the dstane between the MLE and the true value of te-delay. One out of possble lass labels s assgned to a saple aordng to ts dstane to one out of seleted peas of autoorrelaton funton. Then, about a half of the labeled data s used to tran the deson trees lassfer (blo P3 n the dagra) whh aept n-valued feature vetor (eq. 57) and produes one out of possble lass labels. The seond half of the data s used for evaluatng the perforane of the lassfer, that s for oputng the onfuson atrx and the posteror lass probabltes (eq. 4, blo P4 n the dagra).

10 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < of The results presented n ths paper are obtaned usng values a and n,7,3. The nuber of lasses - s set to fve wth four outler lasses orrespondng to four largest sdelobes of the autoorrelaton funton (two at eah sde of the an lobe) and the nler lass orrespondng to the an lobe of the autoorrelaton funton. After the preproessng phase, the To estaton for a new saple proeeds n straghtforward anner. The feature vetor for a saple s extrated by applyng n phase-shfted flters and oputng orrespondng based estates (blo E n the dagra). After the feature vetor s orreted by subtratng the MLE value (eq. 57), the vetor s fed nto the lassfer whh produes one out of possble lass labels (blo E3 n the dagra). The expeted bas s then oputed as desrbed n the prevous seton usng the lassfer s onfuson atrx and posteror lass probabltes (eq. 44). The oputed expeted bas s then subtrated fro the MLE resultng n a bas-orreted To estate. or oparson purposes, the spler fuson of ndvdual based estates s oputed usng a edan statsts as t edan{ t,..., t } (57) edan The resultng Root Mean Square Error (RMSE) noralzed by the sgnal entral frequeny s shown n the fgure 3. The flter ban of n= flters s used n ths experent. ro the graph, t s lear that the resultng error s sgnfantly redued (up to 3%) by orretng the MLE estator usng the expeted bas (provded the estaton s arred-out n the se-oherent area). The Craer-Rao Lower Bound (CRLB) s shown at ths fgure at red. As expeted, the To estators devate rapdly fro the CRLB as SNR approahes the threshold value. gure 4 shows RMSE urves obtaned usng dfferent szes of flter bans. Generally, the ore flters are eployed, the ore delate nterrogaton of the sgnal s possble. Sne saples wth all SNR levels are xed together for tranng, a fner analyss of the sgnal results n a larger generalzaton error. Ths s llustrated n gure 4 by the n= lne havng better perforane over the n=7 lne n the regon of hgher SNR levels. The relaton s reversed as ore oarse n=7 ase gves saller generalzaton error for lower SNR values. The effet of the soure sgnal bandwdth on the estaton auray ould be seen at gure 5. In aordane wth the analyss n the seton II (eq. ), the probablty of outlers dereases when the fratonal bandwdth s nreased. Therefore, the overall error and the proveent aheved by the desrbed ethod are saller for estaton perfored usng a soure pulse wth lower bandwdth. The nrease n the pulse s entral frequeny leads to a sharper an lobe and, thus, to an proveent n estaton auray for nler events. However, the sdelobes are also beong hgher whh ause an nrease n the probablty of the outlers. Therefore, an proveent n the estaton auray for low SNRs s stll ahevable by bas reduton as ould be seen at the gure 6. n VI. SUMMARY AND CONCLUSIONS In ths paper, we have ntrodued a ethod for provng a near-threshold To estaton. Sne the threshold effet n ths proble eerges due to the ultodal shape of the enseble-average lelhood funton, t s possble to redue the threshold effet through usage of a wea lassfer. After the lassfer has been traned, ts output ould be used for oputng and, subsequently, subtratng fro the MLE the expeted bas ourrng due to these outlers. The seond ontrbuton of ths wor ntrodued phase shfted unathed flters as a eans to reate a olleton of based estators. These estators were used to generate a feature vetor that haraterzes the axa of lelhood funton. The sulaton results showed the obned effet of these two approahes on near-threshold MSE of Te of lght MLE. The proble of se-oherent To estaton eerges when there are varous power and spetru onstrants posed by the applaton on the soure sgnal. In partular, the desrbed ethods proves the auray To estaton arred out usng pulses wth low entral frequeny and of a very lted bandwdth. In the planned sequel papers, we address an addtonal onstrant whh lts the total power spent on ultple pulses and the assoated proble of provng effeny of easureent per unt of the nvested power. IGURES gure : Two syetrally phase shfted unathed flters and the soure sgnal (top); the te-algned ross orrelaton between the soure sgnal and unathed flters (botto)

11 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < gure : The sulaton proess onssts of the preproessng phase (P- P4) and the estaton phase (E-E4). Durng the preproessng phase the tranng and the evaluaton data sets are generated (P) and, then, are labeled by approprate nler/outler lass labels (P). The tranng data set s used for tranng a wea lassfer (P3) and the evaluaton data set s used for estatng the lassfer s onfuson atrx (P4). Durng the estaton phase, the feature vetor s extrated fro the saple (E) and fed nto the lassfer (E3). The predted lass label s used for oputng the expeted bas and subtratng t (E4) fro the prevously oputed Maxu Lelhood Estator (E). gure 5 : The RMSE urves for dfferent values of fratonal bandwdth. or hgher fratonal bandwdth and the sae entral frequeny the probablty of outlers s lower, thus, the overall error and the proveent aheved by bas reduton are saller gure 6: The RMSE urves for dfferent values of the entral frequeny but the sae bandwdth. As the frequeny nreases, the an lobe beoes sharper, leadng to an nrease n the estaton auray. However, the probablty of outlers s also nreases due to hgher sdelobes. Therefore, a sgnfant proveent s stll ahevable by bas reduton. Referenes gure 3: The RMSE of the To MLE (blue), the bas-orreted estator (green) and the Craer-Rao Low Bound for the To estaton. The basorreted To estator proves the lassal MLE by about 3% gure 4: The RMSE obtaned usng flter bans of dfferent szes. The n= (red) gves better perforane at hgher SNR as ore delate nterrogaton of the sgnal requred. On the other hand, t results n larger generalzaton error opared to ore ourse estator for n=7 (green) [] E. Alpayadn, Introduton to Mahne Learnng, MIT Press 4 [] L. Angrsan, A.Bagalup et al. A Measureent Method Based on Kalan lterng for Ultrason Te-of-lght Estaton IEEE Transatons on Instruentaton and Measureents, vol. 55(), 6 [3] A. Apartsn, LN. Cooper, N. Intrator. Bosonar-nspred soure loalzaton n low SNR, In Pro. Internatonal Conferene on Bo-Inspred Systes and Sgnal Proessng (BIOSIGNALS),, pp [4] A. Apartsn, LN. Cooper, N. Intrator. SNR-dependant flterng for Te of Arrval Estaton n Hgh Nose, n Pro. IEEE Mahne Learnng for Sgnal Proessng (MLSP),, pp [5] A. Apartsn, LN. Cooper, N. Intrator. Se-oherent Te of Arrval Estaton Usng Regresson. The Journal of the Aoust Soety of Aera, vol. 3(),, pp [6] E.W. Barann. Loally best unbased estates. The Annals of Matheatal Statsts, vol., 949, pp [7] C. Chapan. undaentals of Ses Wave Propagaton. New Yor: Cabrdge Unversty Press, 4. [8] A. Elfes. Sonar-based real-world appng and navgaton. IEEE Journal of Robots and Autoaton, vol. 3(3), 987, pp [9] B.M Sadler, R.J. Koz. A Survey of Te Delay Estaton Perforane Bounds, n ourth IEEE Worshop on Sensor Array and Multhannel Proessng, 6, pp [] S. M. Kay. undaental of Statstal Sgnal Proessng. New Jersey: Prente Hall, 993, 595 pages.

12 > REPLACE THIS LINE WITH YOUR PAPER IDENTIICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < [] S. S. Keerth, C.J. Ln. Asyptot behavors of support vetor ahnes wth Gaussan ernel. Neural Coputaton, vol.5 (7), 3, pp [] T. Kh, M. Reshef, J. Terel. Evdene for the use of refleted self-generated ses waves for spatal orentaton n a blnd subterranean aal. Journal of Experental Bology, vol. 8(4), 5, pp [3] R.N. MDonough, A. D. Whalen. Deteton of Sgnals n Nose. San Dego: Aade Press, 995,495 pages. [4] N. Nerett, N. Intrator, and LN. Cooper. Adaptve pulse optzaton for proved sonar range auray. IEEE Sgnal Proessng Letters, vol. (4), 4, pp [5] N. Nerett, M.I. Sanderson, J.A Sons, and N. Intrator. Tefrequeny oputatonal odel for eho-delay resoluton n sonar ages of the bg brown bat, Eptesus fusus. Journal of the Aoust Soety of Aera, vol. 3(4), 3, pp [6] D.R. Pauluzz, N.C. Beauleu. A oparson of SNR estaton tehnques for the AWGN hannel. IEEE Transaton on Counatons, vol. 48(),, pp [7] A. Pnus, J. Tabran. Barann bound for range and Doppler estaton usng orthogonal sgnal transsson, n Pro. of IEEE Radar Conferene, 6, pp [8] M. I. Sanderson, N. Nerett, N. Intrator, and J. A. Sons. "Evaluaton of an audtory odel for eho delay auray n wdeband bosonar." The Journal of the Aoustal Soety of Aera, vol. 4, 3, pp [9] J.A. Sons and R. A. Sten. Aoust agng n bat sonar: Eholoaton sgnals and the evoluton of eholoaton. Journal of Coparatve Physology, vol. 35(), 98, pp [] M.I. Soln. Introduton to Radar Syste. New Yor: MGraw- Hll, 96, 77 pages. [] GP. Su, G. Prado, R. Gapert, and TK. Pedersen. Probles n Ses deteton and Trang, n Pro. of SPIE, vol. 44, pp [] Harry L. Van Trees. Deteton, Estaton and Modulaton Theory. New Yor: John Wley & Sons, In., 4, 7 pages. [3] L. R. Varshney and D. Thoas. Sde lobe reduton for athed flter range proessng, n Pro. of IEEE Radar Conferene, 3, pp [4] P.D. Wasseran. Advaned Methods n Neural Coputng. New Yor: Van Nostrand Renhold, 993, pp [5] P. Woodward. Probablty and Inforaton Theory, wth Applatons to Radar. New-Yor: MGraw-Hll, 953,4 pages [6] L. Yu, N. Nerett, N. Intrator. Multple png sonar auray proveent usng robust oton estaton and png fuson. The Journal of the Aoust Soety of Aera, vol.9, 6, pp [7] [8] ZJ. Yao, QH. Meng, M. Zeng, Iproveent n the auray of estatng the te-of-flght n an ultrason rangng syste usng ultple square-root unsented Kalan flters, Revew of Sentf Instruents,8. () Nathan Intrator reeved hs PhD n appled atheats fro Brown Unversty n 99. He s a Professor of Coputer Sene at Tel Avv Unversty and an adjunt Professor at the Insttute for Bran and Neural Systes at Brown Unversty. Professor Intrator s researh fouses on statstal and ahne learnng ethods for sgnal proessng and data fuson fro ultple experts. He has been studyng bosonar extensvely and the onneton between ahne learnng expert fuson and ultple deson fuson n bosonar. Hs earler wor nluded objet representaton and reognton and densonalty reduton va exploratory projeton pursut ethods. He developed varous ethods for estatng predtor onfdene whh an be easly used for sensor fuson as well as lassfaton and deteton based on ultsale representatons. He urrently apples hs ethods n varous boedal sgnal proessng applatons. Alexander Apartsn s a Ph.D. student at the Blavatn Shool of Coputer Sene at Tel-Avv Unversty. Hs researh fouses on probles whh requre synthess of ahne learnng and sgnal proessng tehnques. Before jonng Tel-Avv Unversty, Alexander Apartsn was at Wezann Insttute of Sene where he had reeved hs M.S. degree for studyng ultsale segentaton algorths. Leon N Cooper s presently the Thoas J. Watson, Sr. Professor of Sene and Dretor of the Insttute for Bran and Neural Systes at Brown Unversty, Rhode Island, USA. He was a Meber of the Insttute for Advaned Study and a professor at varous unverstes n the USA and Europe. He was a o-repent of the 97 Nobel Prze n Physs. Hs researh nterests nlude nulear physs, quantu feld theory, superondutvty, theoretal neurosene, synapt odfaton, and the bologal bass of learnng and eory storage, neural networs, foundatons of the quantu theory, algorths for oparson of gene data sets and tests of the lnear-no-threshold hypothess for low-level radaton daage.