Title: Robust statistics from multiple pings improves noise tolerance in sonar.

Transcription

1 Title: Robust sttistics fro ultiple pings iproves noise tolernce in sonr. Authors: Nicol Neretti*, Nthn Intrtor, Leon N Cooper This work ws supported in prt by ARO (DAAD ), nd ONR (N--C- 96). N. Neretti is with the Institute for Brin nd Neurl Systes nd Physics Deprtent, Brown University, Providence RI 9. N. Intrtor is with the Institute for Brin nd Neurl Systes nd Physics Deprtent, Brown University, Providence RI 9, nd is on leve fro Tel-Aviv University. L. N Cooper is with the Physics Deprtent, Institute for Brin nd Neurl Systes nd Neuroscience Deprtent, Brown University, Providence RI 9. Edics: -DETE - Detection, Estition, Clssifiction, nd Trcking; -NPAR - Pretric nd Nonpretric Sttisticl Signl Processing. Corresponding Author: Nicol Neretti, Box 843, Brown University, Providence, RI 9 Tel. (4) , Fx (4) , E-il: Nicol_Neretti@brown.edu Abstrct The bility of sonr to detect objects is strongly influenced by the operting signl-to-noise rtio (SNR). As sound plitude decys very fst in wter this sensitivity reduces the effective sonr rnge. It is well known tht the rnge ccurcy decys for incresing levels of noise until brekpoint is reched fter which ccurcy deteriortes by severl orders of gnitude. In this pper we present robust fusion of tie-dely estites fro ultiple pings tht reduces the SNR corresponding to the ccurcy brekpoint. We show tht siple verge of the tiedely estites does not shift the brekpoint to lower SNR. The ethod cn iprove the resilience to noise of sonr syste, hence incresing its potentil rnge of opertion. I. INTRODUCTION The theory of optil receivers studies the design of pulses nd receivers to obtin optil detection in the presence of noise. Considerble work on the theoreticl ccurcy of rnge esureents hs been done in the pst strting with the Woodwrd eqution, which hs been derived using different ethods. A coprehensive description of erlier work cn be found in []. The vlidity of the Woodwrd eqution depends on vrious ssuptions in prticulr the ssuptions of very low signl-to-noise rtios (SNR), therefore it ust be reexined for the cse of low SNR's. The theory of optil receivers shows tht the tched filter receiver xiizes the output pek-signl-toen-noise (power) rtio [, 3], nd is the optiu ethod for the detection of signls in noise. Infortion bout the distnce of the trget is extrcted by coputing the tie t which the crosscorreltion between the echo nd replic of the pulse is xiu. This dely is converted into distnce by ens of the sound velocity in the prticulr ediu in considertion (e.g. wter or ir). This type of receiver is generlly referred to s coherent receiver.

2 The clssicl theory of optil receivers describes the rnge ccurcy of sonr syste vi the wellknown Woodwrd eqution, which cn be derived by using vriety of ethods [, 4-7]. For sll SNR's, one of the preters in the clssicl eqution i.e. the bndwidth hs to be odified, nd the receiver is then clled seicoherent. In [8] it ws shown tht the trnsition between the two types of behviors occurs t different SNR's depending on chrcteristics of the pulses such s bndwidth nd center frequency. With this observtion, novel syste bsed on n dptive choice of the pulse ws proposed [8]tht cn iprove ccurcy in the cse of reltively low SNR, when biguity in the choice of the correct pek of the cross-correltion function cnnot be voided. Due to the nonliner nture of the tie-dely estition proble, when the SNR drops below certin criticl vlues threshold effects tke plce. Threshold effects cn be chrcterized by shrp deteriortion of the tie-dely estitor vrince. Severl sttisticl bounds hve been used in the pst to describe the ccurcy of tched filter receiver perfornce in between the boveentioned SNR criticl vlues. These include the Crer-Ro lower bound [9-], the Brnkin bound [], nd the Ziv-Zki bound [3-6]. Such bounds hve been pplied both to the proble of tie-dely estition [5, 7-3] nd of frequency estition [4-6]. In prticulr, the Brnkin bound hs been used to define the SNR brekpoints corresponding to the chnge in behvior of the optil receiver s the SNR decreses for the cse of single ping nd single echo [7, 9, ], single ping nd ultiple echoes [, 3], nd ultiple pings nd single echo [8, ]. In this pper we nlyze the signl-to-noise brekpoint, studying the probbility of choosing the correct pek fro the noisy cross-correltion function with ethod siilr to the one in [5], where the threshold effect ws relted to the existence of highly probble outliers fr fro the true tie-dely vlue. This will enble us to extend the result to the cse of ultiple pings without priori knowledge on the tie-dely itself. This pproch is different fro tht used in previous work on the ultiple pings nd single echo cse [8, ], where the ultiple echoes for single object re obtined rtificilly vi ultiple receivers nd unique ping. In fct, the bounds found in [8, ] re vlid only if the noise t the different receivers is totlly uncorrelted or if the distnce between trnsducer nd receiver is constnt, both conditions difficult to relize in prctice. Section II introduces two working odels for the tie dely estition tht re used to copute the probbility of error for single observtion. In section III we show tht while verging of ultiple observtions iproves ccurcy, it does not increse noise tolernce. In section IV we derive theoreticl bound for the probbility of correct tie-dely estition within certin ccurcy tolernce bsed on the ode of the observtions. Section V describes set of Monte Crlo siultions confiring the theoreticl predictions. II. SINGLE PING BREAKPOINT A. A siplified odel for the utocorreltion function: δ-function. We first present siple odel where the noiseless cross-correltion function of the sonr ping is δ-like function tht is zero everywhere except fro t = where it is equl to the pulse energy (Fig. ). This odel requires pulse with infinite bndwidth; therefore it is not relizble

3 3 in prctice. However, its study will give soe insight on the echnis leding to the shrp decy in rnge ccurcy experienced by tch-filtered receivers. When white Gussin noise is dded to the returning sonr ping, then the cross correltion vector hs Gussin distribution with ultidiensionl centers t zero for ll vlues tht re outside of the bin t=, nd center t =E > for the vlue of the cross-correltion t t=, where E is the energy of the noise-free echo (see Appendix B for discussion on the vlidity of this pproxition). Let x i be the vlue t ech bin of the cross-correlted signl nd x the vlue t t=. We then hve: r r x v σ N e r T p( x, x, K, xn ) =, v = [,,, K,] () ( σ π ) Using the Gussin distribution, we cn clculte the probbility for the event x > xi, i, which corresponds to the correct echo dely estition, for given noise level σ. Denoting this probbility by α, we hve: (, ) x x x P X > X i = α = dx dx dx L dxn N e ( σ π ) i 3 + ( x ) x x σ σ = dx e e dx ( σ π ) N ( x ) N σ = e + erf( x) dx N + + π [ ] r r x v σ N () Note tht this probbility depends only on / σ, which is proportionl to the signl to noise rtion (SNR). β = α denotes the probbility of error, i.e., the probbility tht the plitude of t lest one of the points outside of the correct bin is lrger thn the plitude correct pek. B. A ore relistic odel for the utocorreltion function For ping with finite bndwidth it is ore relistic to odify the odel in eqution () to include the finite extension in tie of the utocorreltion function (Fig. ). In this odel the utocorreltion function is pproxited by piecewise constnt function, with n plitude equl to within the centrl intervl I of length, nd zero elsewhere. In this cse, we need to consider the width of the priori window of the cross-correltion. The width of the window corresponds to the sonr rnge. While the potentil error in dely estition is reduced when the width is reduced, so is the sonr rnge. If the priori window hs length of L nd the spling frequency is f s, then there will be N = N + N = L fspoints, N = f s of which will be within the centrl bin ( correct bin ), nd outside the centrl bin but within the priori window. Without loss of generlity, we consider the cse where nd re integers. N N N

4 4 We define rndo vector such tht the first N rndo vribles correspond to the plitudes of the points within the correct bin, while the lst N correspond to the plitudes of the points outside. As ws derived bove, for white Gussin noise, the joint probbility density function for the vector of n rndo vribles is given by: r r x = r σ p( x ), v =, x, K, xn N e,,,,,,, ( σ π ) 443 K 443 K N N T (3) The desired probbility of tie dely estition within the correct bin of the center of the crosscorreltion function is given by N P rg x { X j} I = P Xi > k, j N i= = N + ( X k i) x x x i i i dx k L dx dx dx N e N ( σ π ) i k k i= r r x v σ N ( x + ) x ( y ) x y σ σ σ = N dx N e ( ) e dy e dy σ π + N ( x σ ) = N e + erf x N π + erf ( x σ ) N ( ) [ ] N dx (4) Thus, for given noise level, the probbility tht the correct bin is selected is: + N ( x ) N N σ ( σ ) x N e erf [ erf x ] dx π ( ) (5) α = + + C. Accurcy brekpoint Let T be the R.V. whos probbility distribution is given by Eq. 4. For SNR vlue tht is bove SNR, we hve seen (Eq. ) tht the probbility for flling in the correct bin α is bove α. Let σ be the STD of the distribution of the echo loction in the centrl (correct) bin, nd let σ be the STD of the distribution which is outside the centrl bin. Now, suppose we sple fro the originl distribution T whose cuultive function is given by eqution (5). For n observtions, frction of α of the n observtions flls in the correct bin on verge, while βn fll outside. The stndrd devition of the distribution will then be give by: std ( T) = α std ( T ) + β std ( T ) = ασ + βσ (6)

5 5 We define the brekpoint s the level of noise for which the contribution of T to the totl error becoes doinnt. Thus, the RMSE will be significntly lrger thn the one given by the unifor distribution on I lone when σ α < σ + σ (7) We then define the probbility brekpoint to be: ( σ σ ) ( σ σ ) α = (8) + it is possible to find the SNR brekpoint s the SNR vlue for which eqution (5) equls the vlue in (8). III. WHY DOES THE MEAN FAIL? The clssicl wy to fuse infortion fro ultiple observtions is to verge. As the observtions re independent nd identiclly distributed, the centrl liit theore (CLT) iplies tht the stndrd devition (error) of the verged R.V. should be n ties sller thn the error de by ech of the n observtions seprtely. This is indeed the cse before the brekpoint. However, this process does not iprove the sitution fter the brekpoint nd, in prticulr, does not shift the brekpoint to lower SNR s. Thus, while verging iproves ccurcy, it does not increse noise tolernce. Below we provide theticl nlysis which explins why the brekpoint does not chnge. The esureent process described bove is equivlent to spling for unifor distribution on the intervl I with probbility α nd fro unifor distribution on the intervl I, with gp corresponding to I, with probbility β. Suppose we sple n ties to obtin T, T, K, Tn nd use the sple en s our estite for the dely: T n i n i = = T (9) δ δ δ On verge, α n vlues will be in the correct bin, T, T, K, Tα n, while β n will be spled fro the unifor distribution, T u, T u, K, u. Then the sple en cn be decoposed into two prts: T T β n = + n αn β n Ti Ti i= i= ()

6 6 If σδ is the stndrd devition of the δ-like distribution, nd σ u is the stndrd devition of the unifor distribution, then, pplying the centrl liit theore to the two sus in eqution () we obtin: αn std Ti = αnσ i= β n std T = βnσ i= i () The root-en-squre error will be significntly lrger thn the one given by the δ-like distribution lone when βnσ > αnσ, i.e. when σ α < σ + σ () It is observed tht this bound does not iprove with the nuber of pings nd is equl to the bound found for single ping in eqution (7). This explins why verging the echo dely estites fro ultiple pings hve not been found useful, nd probbly led to the wrong conclusion tht no iproveent cn be chieved fro ultiple pings dt. It should be noted tht it is possible to shift the brekpoint by estiting the tie dely for the verged cross-correltion functions of ll the observtions, process tht would require n extreely good lignent of the cross-correltion functions. This is bsiclly equivlent to reducing the noise level by verging the echoes [4] (see lso discussion in Appendix A), which is not relistic in sitution where the sonr is not copletely still with respect to the trget, or where it is not prcticl to store the entire echo wvefors for off line processing. However, successful ipleenttion of this verging cn be chieved by using ultiple receivers [8, ]. IV. USING THE MODE Suppose we divide the priori window into intervls of length equl to the size of the correct bin, to obtin = L/ intervls B, B, K, B, with B = I representing the correct bin (Fig. 3). If p, p, K, p re the probbilities for n estite to fll in ech of the intervls nd Y, Y, K, Y re rndo vribles representing the nuber of estites flling in ech intervl, then

7 7 j= j= Y p j j = n = (3) The joint probbility distribution for the nuber of estites in ech bin is given by the ultinoil distribution n! k PY k Y k Y k p p L p (4) k k ( =, =, K, = ) = k! k! Lk! The probbility of choosing the correct bin using the ode is the probbility tht the nuber of estites flling in the correct bin, is greter thn the nuber of estites flling in ny other bin k, i : i k n! k k P = P( Y > Yj, j ) = p p L p correct k, k, K, k k! k! Lk! k > kj, j k+ k+ L+ k = n k (5) The su in eqution (5) cn be decoposed into two prts: ) the probbility tht ore tht hlf of the n points flls into the correct bin; ) the probbility P <5% tht even if less thn hlf of the n points fll in the correct bin, the nuber of points in it is greter thn tht of ny other bin. P >5% ( correct bin) = 5% +P 5% P P > < (6) The first ter in (6) cn be written s: P k k > 5% k, k, K, k k! k! Lk! k > n/ k + k + L+ k = n k > n/ = n ( ) p k ( p p 3 L p k ) n! = p p n k L p k (7) The probbility of n estite to fll outside the correct bin is unifor over the priori window with probbilityβ, so tht the probbility for it to fll in ny intervl of size is β ( ) :

8 8 p p = ( α) ( ), j j = α (8) Substituting (8) into eqution (7) we obtin P > 5% n k ( ) ( ) n k k α α, (9) k> n/ = whereα is function of the SNR through Eq. (5). The coputtion of is ore coplicted. However, it is possible to derive n upper bound ( SNR >5% ) on the SNR brekpoint for the tie-dely ccurcy coputed by using the ode of n estites, s the SNR for which P> 5% = α, where α is given in Eq. (8): n k ( ) ( SNR> ) ( SNR> ) P <5% n k k α α 5% 5% = α () k> n/ The tighter bound corresponding to the totl probbility of choosing the correct bin given by Eq. (6) will lwys be lower thn the one derived fro Eq. (): SNRBP SNR > 5%, () thus, the brekpoint which the ode cn chieve will be for lower SNR thn the one clculted bove, which is lredy significntly better thn the brekpoint chieved by either single ping or by verging of the echo dely estites of ultiple pings (see Figure 4). V. EXPERIMENTAL RESULTS To test the theticl results presented in the previous sections, we developed set of Monte Crlo siultions using cosine pcket. We first nlyzed the histogrs of the errors in the dely estite of the idel receiver for different SNR's (figure 5). For high SNR ( db) ll the errors re sll nd follow the Woodwrd eqution tht corresponds to vlues within the centrl bin in figure 5. As the level of the noise increses, lrge errors in the estites pper. The errors re uniforly distributed over the entire -priori window, nd the reltive rtio between the correct estites (centrl bin) nd the level of the unifor distribution decreses with SNR (figures 5b, 5c, nd 5d). However, even for high levels of noise the centrl pek is significntly lrger tht the rest of the distribution. Figure 6 shows the perfornce of idel receiver for single ping. For high SNR the ccurcy follows the Woodwrd eqution corresponding to coherent idel s expected fro the theory of optil receivers. The perfornce breks for low SNR round 7 db. Figures 6b, 6c nd 6d

9 9 show the nlysis of the ccurcy brekpoint for different nuber of pings, 5 nd respectively. The blue line describes the optil ccurcy tht cn be chieved using crosscorreltion fro ultiple pings. Its breking point represents the optil breking point tht could hve been chieved using sttionry sonr nd trget, nd tht could be predicted by using the Brnkin bound s in [8, ] (see lso discussion in Appendix A). This breking point however is not ttinble, s it relies on creful registrtion of returns fro different pings. Such creful registrtion cn only be done if the distnce between object to trget is kept constnt, or if it is known for ech ping in dvnce. It cn be seen tht robust fusion of ultiple pings bsed on the ode (light blue, nd gent lines) iproves noise resiliency while retining close to optil chievble ccurcy under ultiple pings. In generl there is no significnt iproveent in the resiliency to noise when siple en of the observtions is used due the strong contintion of the distribution fro outliers (red lines). This confirs the theticl result presented in section III. Figure 7, shows sury of the results for the different ethods. The brekpoint for the verged cross-correltion function (blue squres) follows the idel curve obtined by reducing the level of the noise x explined in Appendix B (solid blue line). The brekpoint of the estite obtined fro the en does not substntilly chnge s the nuber of pings in incresed. A ore robust sttistics such s the edin iproves the resiliency to noise s the nuber of pings increses (green tringles). The best results re obtined by using the ode of the estites fro the ultiple pings (gent dionds). APPENDIX A In this section we show tht estiting the tie-dely fro the verged cross-correltion functions shifts the SNR brekpoint s if the noise level ws reduced by verging the echoes. We odel ech echo s su of two coponents: y r = u r + r η () r where u is n ttenuted replic of the pulse, nd r η is white noise with stndrd devition equl to σ η. We define the signl to noise rtion to be: E d = (3) N where E is the energy of the noise-free coponent u r esured in Ws, nd N is the spectrl density of the noise esured in W/Hz. If f = / dt is the spling frequency, then: s

10 r η dt σ N = = η fs fs r r u E = u dt = fs (4) nd the SNR cn be expressed s: d = r u σ η (5) The SNR cn be expressed in db s: SNR( db) = log d = log r u σ η (6) When verging n echoes, the stndrd devition of the gussin process generting the noise for ech echo is reduced by n due to the centrl liit theore. Thus, the signl to noise rtio becoes: r u SNRn( db) = log dn = log = ση n r u = log log n= SNR( db) log n ση (7) If BP is the brekpoint corresponding to single ping, then the brekpoint for the verge of n echoes will be: BPn = BP logn (8) APPENDIX B If x( t) is the pulse nd yt ( ) = ut ( ) + η( t) is the echo, then their cross-correltion φ xy cn be expressed s:

11 φ ( τ) = xt ( ) yt ( + τ) dt= xy = (9) = x( tut ) ( + τ) dt+ xt ( ) η( t+ τ) dt = φ ( τ) + φ ( τ) xu xη Moreover if the noise is white then the utocorreltion of the noise: R ( t, t ) = σδ( t t ) (3) ηη η If we tret x( t ) s filter cting on the noise η( t) to generte filtered version of the noise φ() t = φx η, then the utocorreltion R φφ is: η η ( ) ( ) ( ) R ( t, t ) = R t α, t β x α x β dαdβ = φφ ηη + + η xx ( ) ( ) ( ) = σ δ t β t + α x α x β dαdβ = ( ) ( ) ( t) = σ x β + t x β dβ = = σφ (3) Thus, the noise on top of the cross-correltion function between the pulse nd the noise-free coponent of the echo is not white, since its utocorreltion function is not δ-function. However, since the support of φ xx is sll copred to the length of the -priori window, we pproxite R φφ with σδ( t), nd consider the noise to be white. η REFERENCES [] M. I. Skolnik, Introduction to Rdr Systes, st ed: McGrw-Hill Book Copny, 96. [] M. I. Skolnik, Introduction to Rdr Systes, 3 rd ed: McGrw-Hill,. [3] D. O. North, "An Anlysis of the Fctors which Deterine Signl/Noise Discriintion in Pulse-crrier Systes," RCA, Tech. Rept. PTR-6C, June [4] A. J. Mllinckrodt nd T. E. Sollenberger, "Optiu-pulse-tie Deterintion," IRE Trns., vol. PGIT-3, pp. 5-59, 954. [5] D. Slepin, "Estition of Signl Preters in the Presence of Noise," IRE Trns., vol. PGIT-3, pp , 954.

12 [6] P. M. Woodwrd, Probbility nd Infortion Theory, with Applictions to Rdr. New York: McGrw-Hill Book Copny, Inc., 953. [7] M. I. Skolnik, "Theoreticl Accurcy of Rdr Mesureents," IRE Trns., vol. ANE-7, pp. 3-9, 96. [8] N. Neretti, N. Intrtor, nd L. N. Cooper, "Adptive pulse optiiztion for iproved sonr rnge ccurcy," IEEE Signl Processing Letters, 3. [9] H. Crer, Mtheticl ethods of sttistics. Princeton, NJ: Princeton Univ. Press, 946. [] C. R. Ro, "Infortion nd ccurcy ttinble in the estition of sttisticl preters," Bull. Clcutt Mth. Soc., vol. 37, pp. 8-9, 945. [] R. A. Fisher, "On the theticl foundtions of theoreticl sttistics," Phil. Trns. Roy. Soc., vol., pp. 39, 9. [] E. W. Brnkin, "Loclly best unbised estites," Ann. Mth. Stt., vol., pp , 946. [3] S. Bellini nd G. Trtr, "Bounds on error in signl preter estition," IEEE Trns. Coun., vol. COM-, pp , 974. [4] D. Chzn, M. Zki, nd J. Ziv, "Iproved lower bound on signl preter estition," IEEE Trns. Infortion Theory, vol. IT-, pp. 9-93, 975. [5] L. P. Seidn, "Perfornce liittions nd error clcultions for preter estition," Proc. IEEE, vol. 58, pp , 97. [6] J. Ziv nd M. Zki, "Soe lower bounds on signl preter estition," IEEE Trns. Infortion Theory, vol. IT-5, pp , 969. [7] I. Reuven nd H. Messer, "A Brnkin-type lower bound on the estition error of hybrid preter vector," IEEE Trns. Infortion Theory, vol. 43, pp , 997. [8] S.-K. Chow nd P. M. Schultheiss, "Dely estition using nrrow-bnd processes," IEEE Trns. ASSP, vol. ASSP-9, pp , 98. [9] R. J. McAuly nd E. M. Hofstetter, "Brnkin bounds on preter estition," IEEE Trns. Infortion Theory, vol. IT-7, pp , 97. [] R. J. McAuly nd L. P. Seidn, "A useful for of the Brnkin lower bound nd its ppliction to PPM threshold nlysis," IEEE Trns. Infortion Theory, vol. IT-5, pp , 969. [] J. Tbrikin nd J. L. Krolik, "Brnkin bounds for source locliztion in n uncertin ocen environent," IEEE Trns. Signl Processing, vol. 47, pp , 999. [] A. Zeir nd P. M. Schultheiss, "Relizble lower bounds for tie dely estition: Prt - Threshold phenoen," IEEE Trns. Signl Processing, vol. 4, pp. -7, 994. [3] A. Zeir nd P. M. Schultheiss, "Relizble lower bounds for tie dely estition," IEEE Trns. Signl Processing, vol. 4, pp. 3-33, 993. [4] L. Knockert, "The Brnkin bound nd threshold behvior in frequency estition," IEEE Trns. Signl Processing, vol. 47, pp , 997. [5] D. C. Rife nd R. R. Boorstyn, "Single-tone preter estition fro discrete-tie observtions," IEEE Trns. Infortion Theory, vol. IT-, pp , 974. [6] B. Jes, B. D. O. Anderson, nd R. C. Willison, "Chrcteriztion of threshold for single tone xiu likelihood frequency estition," IEEE Trns. Signl Processing, vol. 43, pp. 87-8, 995.

13 3 Figure Cptions Fig. - Siple odel where the noiseless cross-correltion function of the sonr ping is δ-like function tht is zero everywhere (within the priori window of length L) except fro t = where it is equl to the pulse energy (top). The botto figure shows the probbility of selecting given tie loction in the cross-correltion function predicted by this odel. Fig. - A ore relistic odel for the noiseless cross-correltion function includes the finite extension in tie of the utocorreltion function. In this odel the noiseless cross-correltion function is pproxited by piecewise constnt function with n plitude equl to within the centrl intervl I of length nd zero elsewhere (within the priori window of length L). The botto figure shows the probbility of selecting given tie loction in the crosscorreltion function predicted by this odel. Fig. 3 Model used in the coputtion of the probbility of correct bin choice in the cse of ultiple pings. We divide the priori window into intervls of length equl to the size of the correct bin, to obtin = L/ intervls B, B, K, B, with B = I representing the correct bin. Fig. 4 Probbility of king the correct choice s function of SNR for different nubers of pings. The rrows indicte upper bounds on the SNR brekpoints. Fig. 5 Histogrs of the errors in the dely estite in Monte Crlo siultion for different SNR's. For high SNR ( db) ll the errors re sll nd follow the Woodwrd eqution tht corresponds to vlues within the centrl bin in figure (). As the level of the noise increses, lrge errors in the estites pper. The errors re uniforly distributed over the entire -priori window, nd the reltive rtio between the correct estites (centrl bin) nd the level of the unifor distribution decreses with SNR, see figures (b), (c), nd (d). However, even for high levels of noise the centrl pek is significntly lrger tht the rest of the distribution. Fig. 6 RMSE s function of SNR nd nuber of pings,, nd respectively for the Cosine Pcket. Notice how the SNR brekpoint for the verge of ultiple pings (red line) does not decrese with the nuber of pings. Fig. 7 Brekpoint in db s function of nuber of pings for different ethods. The solid line corresponds to the noise reduction due to verging s discussed in Appendix C.

14 4 Figures L α α N L Fig. L α α N L Fig.

15 5 B B 3 B 4 B 5 B Fig. 3.8 Ping (BLACK) Pings (RED) 5 Pings (BLUE) 5 Pings (GREEN) PROB. CORRECT SNR (db) Fig. 4

16 6 () (b) (c) (d) Percent (%) Error in Pek Loction (s) Fig. 5

17 7 Ping Pings Pings Pings -4-4 RMSE (s) -6-8 Ave. XC Men Medin Mode SNR(dB) Fig. 6

18 8 5 Men Brekpoint (db) 5 Ave. XC Medin Mode Idel Nuber of pings Fig. 7