EFFECT OF MULTIPATHS ON N-ARY TIME REVERSAL CLASSIFIERS

Foroohar Foroozan andAmir Asif

Computer Science and EngineeringYork University, Toronto, ON, Canada

Tel. No.: (416) 736-2100X70128Email: {foroozan, asif}@cse.yorku.ca

ABSTRACT Virtual Array I(single reflection

from the

The paper derives time reversal (TR) algorithms for N-ary top boundary)classification problems and studies the impact of multipathson the performance characteristics of these algorithms. TheTR classifiers have the ability of detecting the presence orabsence of a target in a rich cluttering environment with mul-tipaths and of further classifying the detected targets either as dan unknown enemy target or one of the known friendly tar- 0

gets. Compared to the conventional classifiers, the proposed Pyal/TR classifier provides a gain of over 5dB at low SNRs. An an- Inhomogeneous Mediumalytical multipath model, based on geometric optics approx-imation with strong total internal reflection from the bound- Virtual Array Iaries of the medium, is used to compare the effect of multi- from the

paths on the proposed classifiers versus the conventional al-gorithms.

Index Terms- Time Reversal, Detection, Neyman Pear- VirtualArrayIII K>son test, Maximum Likelihood Function, and Multipaths. doblrflectihe

bottom boundary) /

1. INTRODUCTION Fig. 1. Analytical multipath model used to quantify the effectof multipaths on time reversal.

Compared with radio telecommunications, signal process-ing in underwater acoustics is more challenging because oflower bandwidth, high clutter, random channel characteris- further classified as an unknown enemy target or one of thetics, and significant distortions due to Doppler shifts and other known friendly targets. No prior information is assumed to bespreading effects. In addition, the low speed of sound causes known about the enemy target, i.e., it is assumed passive withmultipath propagation to stretch over time delay intervals of an unknown signature. Our classification algorithms use thetens of milliseconds, which further affects the performance principle of TR, where an antenna array illuminates the acous-of the signal processing algorithms. By using the equivalent tic medium with a broadband probing signal. The backscatterof an embedded phase conjugation equalization technique, of the signal is observed, time-reversed, energy normalized,TR [1]- [3] compensates for distortion introduced by mul- and retransmitted into the medium. The TR classifier uses thetipaths, reduces the impact of the Doppler phenomena, and reflections of the time reversed signal in the classification al-mitigates the need for a priori knowledge of the channel gorithms. The TR classifier uses the reciprocity theorem [4]characteristics. such that the position of a point source and observation site

The paper derives TR algorithms for N-ary classification can be reversed without altering the observation itself even inproblems, which are capable of detecting the presence or ab- a lossless inhomogeneous medium.sence of targets in a rich cluttering environment with mul- A 'eodcnrbto ftepae sa nltclmlitipaths and random velocity profiles. The detected target iS pt rmwr o tdigteefc fmliah ncn

This work was supported In part by the Natural Science and Engineering ventional and TR classifiers in a simulated computational en-Research Council (NSERC), Canada under Grant No. 228415-2007. vironment. The framework is based on the analytical mul-

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tipath model [5], illustrated in Fig. 1, which assumes strong the recorded signal such that the processed signal is given bytotal internal reflections from the boundary of the mediumaccording to the geometric optics approximation. Each re- Vq = O,... Q - 1 and m = 1,... , Mflective multipath, considered in Fig. 1, is treated to originate G"qm) = F(wOq)Hti (wq) + Vi(m) (wOq) (2)from an additional virtual array outside the boundaries of the (medium under consideration. Richer multipaths with L re- Following the principle of TR, the recorded signal Gnm) (wq)flections, therefore, lead to an increased number of (2L + 1) is energy normalized, time reversed (equivalent to phase con-arrays. Ofthese (2L + 1) arrays, 2L arrays are virtual and lie jugation in the frequency domain), and retransmitted backoutside the domain boundaries. In array processing terminol- into the medium. After subtraction of clutter, the backscat-ogy, the TR array behaves like a lens with an effective aper- tered TR signal at the transceiver is given byture that is (2L + 1) folds larger than that of the aperture ofthe physical array. Increasing the number of multipaths mag- Vq = O,... , Q - 1 and m = 1,... , Mnifies the aperture, thereby, enhancing the achievable super- Pi(J)(inm)F*(bJq) Ht(iJ* 2 (3)resolution focusing associated with TR. In comparing the ef- - Wq) C Ffect of multipaths on TR classifier with that of the conven- + cjm)Ht, (wq)V(m) (wq) + WJiTn) (wq)tional classifier that uses only the forward propagation path,the final observations of both the conventional and TR classi- where Win)(wq) CJVf(O, (72) denotes the observationfies are normalized to have unit energy before being used by noise. Stacking Gim) (w.q), Pi m) (q), H (wq), V(m) (wJq)the respective detectors. (m)q) ti(

and W. (q) for all discredited frequencies wq, for (o <Our N-ary classifiers are based on the following hypoth- (1

esis: ~~~~~~~~~~~~~~~~q< (Q - 1)), respectively, in G$Tm), P(mn) Ht, V$mT) andWN(Tpm), Eqs. (2) and (3) are expressed as

HSO No target is present.HXi: Friendly target i, (1 < i < (N - 2)), is present. G(m) FHt, + V(T) (4)

THN-1 Enemy target, i(N -),Gispresent. ) m)* (5)

for which the the conventional and TR classifiers are pre- where F is a diagonal matrix of order Q with F(wq) arrangedsented in Section 2. The analytical multipath model is de- along its diagonal and symbol 0 denotes the Hadamardscribed in Section 3 and the effect of multipaths on the con- product. Given that V$m) CJV(O, cr2IQ) and W$m)ventional and TR classifiers is compared through experiments , i

CA\ (0, 0r2 IQ), where IQ iS the identity matrix of order Q, thein Sections 4. Finally, Section 5 concludes the paper. p' ~~~~~~~~~~pdf'sfor Gi and Pi under hypotheses HEi are given by

2. N-ARY MAXIMUM LIKELIHOOD CLASSIFIERS M 1 G) (6)Pr(GiTHi) H7FQ(2)Q e (6)In this section, we introduce the notation used in the paper andpresent the N-ary maximum likelihood conventional and TR Pr(17i )2 1 (7)classifiers. During forward propagation, a single transceiver Hi) 7Q('72)Qlocated at Xs = (Xs Ys) probes the channel with a widebandsignal F(wq), where Wq denotes the discretized frequency and where observations m, (1 < m < M), are assumed indepen-equals 2q7/Q, for (0 < q < Q - 1). The channel is probed dent of each other. For hypothesis Hol, the target responsemultiple (1 < m < M) times such that the m'th recording Ht, = 0. Notation 1 11 denotes the Frobenius norm. Inof the backscattered signal at the transceiver from a target at Eq. (7), we have accumulated the effect ofnoise vi to wi in the.rp =(xp, yp) is given by final observation P.i) and all normalization constants ci's

are assumed to be equal to c. Theorems 1 and 2 present theVq =O,. , Q - 1 and m =1, , M, N-ary conventional and TR classifiers for the ideal case when

R$m) (wq) F (wJq) (Hti (wq) + Hc (w)q) + Vi(m) (wOq)(I) target signatures Ht, (wOq), (1 < i < (N - 1)), are known. Ingeneral case, where the enemy target response HtN1 is un-

where Hc(wOq) and Hti (wJq) are, respectively, the frequency known, Theorems 3 and 4 derive the N-ary conventional and-4:'+]-- _A -T-(T)~- 1- TR classifiers. To save on space, proofs for Theorem 1- 4 areresponses of the clutter and target i. V(m)(wqz) models zerot.q . . . not included in the paper. Interested readers are referred tomean, circular white Gaussian observation noise with van1- [6fothsprfsadutereai.

ance oJ. The probablity density function (pdf) of the noiseis denoted as Vi(m) CJVg(O, os2). We use the background Theorem 1. The test statistics £i(Gi) of the N-ary con-subtraction process [7] to subtract clutter F(wq)Hc(wq) from ventional class.ifier based on the observation vector Gi is

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given by In terms of R;, the GLR test for the TR detector is given by

((FHtL)H z;;-G1 m max

fi(Gi) = s ((F Fm=H G ) (8) HLNl {Pr(R THN 1)} (19)

which is linear with thefollowing conditionalpdf's.To save on space, we express F(wq) as F, Gi(wq) as Gi, and~HtN (wOq) (estimate of HtN (wOq)) as HtNl1 in the follow-

V(i #O), £i(Gi) Hi CJV(,ii,M/2) (10) ing theorem.

where mean pi equals M FHt, /0v. Theorem4. Thegeneralizedloglikelihoodratiotest £GLR(Ri)Theorem 2. The linear test statistics £i (Pi) ofthe N-ary TR for the TR classifier is given byclassifier based on observation vector Pi is given by GLR (Ri HN 1) (20)

e (Pj) =((FHt,)* oHt )H M= 1 [{oQ1 m)2T Gm) 2 + 2F 2} H 2

(72(72 Lw vw

with conditionalpdf's 2

£i(Pi)Pl0o CAV(O,M/2) (12) o G HtNit£i(Pi) ThI~~ CJVF(vi,M/2) (13) whereHtN-l is given by

where vi equals Mc (FHt,)* oHt /law. M G(Tm)( )F(q)* c(m)G(m)(bJq)P(m)* )Since the effect of noise vi is accumulated to wi for the TR E W 2 W + (wq)2classifier, we note that the forward reflected signal G(m) Ht=Nm=lL7equals FHt M F(wq) 2 Gin) (q) 2 )1

If the channel response HtN1 for the enemy target is not 2 + Jknown, then we use the generalized likelihood ratio (GLR) m=l

max The Neyman Pearson test computes the thresholds i as-AGLR (Gi) = HtNl {Pr(Gi) H1N-1} (14) sociated with hypothesis Hi, for (1 < i < (N-1)), by fixingANj(G1 ) Pr(GiPo)° the probability of false alarm

in the conventional classifier. Theorem 3 presents the result.

Theorem 3. The generalized log likelihoodratio iGLRj7(Gi) PF .. Pr( ,.... , £N-1 THo)df1 ... d NA(2 1)In(AG/LR(Gi)) for the conventional classifier is given by

where Z = Zi U ... U ZN-1 with Zi being the observa-

£t(GL)G IIN1 z j (WmG q) 2 tion space associated with hypothesis Hi corresponding tofN 1(Gi)lHN-1l= E Ma2/2 (15) the presence of target i. For a 3-ary detection problem, theq=V expression for the probability of false alarm reduces to

with the conditionalpdf 3 1F( / \

GLR2(GP I=EeNl X2 (6) F - [erf -11) erf( -1)(erf 1 )erf( 1j)

where 2M Eq=O1 F(q) 2 HtN1(WJq) 2/V2 The esti- 2 /qz 2matefor the enemy target response is given by where erf(x) 2V e dt, x e I. (22)

Ht F~~~~~~~~~~~~~~~~~~~~~~~~~~~~iHNl1 F lM E G(T) (17) 3. ANALYTICAL MULTIPATH MODEL

m=1

For the TR classifier, we stack the conjugate of the for- To model the effect of multipaths, we use the analyticalwardobservations(Eq.(4))withtheTRobservations(Eq (5)) model [5] that is based on the geometric optics approxima-in (2QM x 1) vectorR( as follows tion with strong total internal reflections from the boundaries

of the medium. The analytical multipath model is shown inR;~ [FGi] (1I8) Fig. 1, where the bottom boundary of the channel is assumed

[Pijg aligned with the i-axis and the positive y-axis is oriented in

[i®@(FHJ*) + [(iHF> ~ W Ii the upward direction. The width ofthe channel is denotedby

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(xxy )/

prV (xp,yp) /

(a) Direct path.p rv(xp yp)

(c) Bottom-boundary, single reflection"pah

(a)Drectpath;(b)efletion f (t bo,2nday a

(/~~~~~U 12(c-,

p rV(Xp y,)\sTbht (x5,v-(2yYye))

(b) Top-boundary, single reflection path. (d) Top-bottom, double reflection path.

/ v(xs: Y')

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plt a-enFig.2.iutptsbtentermofanscingetrsansduetreleet by ansecon relcto fro th AoTobunayad(eR-(an Drealty each; trasdce elmnRoudbeelaeyaflection from the bottoboundary; folwdbyascnd re-YV\V(corecl oretdatenera.flection from the toptoboundary. b2

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Locations 4. EXPERIMENTSMultipath Transceivers (x A) Targets (x A)

1 (220, 30) (270,10) A single transceiver probes the channel with pulse(220, 50), (220, -30) (270, 70), (270, 10) f(t) -272V2(t I/V)C_,2V2(t_ /V)2 (23)

5 (220, 110), (220, -50) (270, 90), (270, -70) - ,7 (220,130), (220, -110) (270,150), (270, -90) for (0 < t < 0.7,us). Symbol v represents the carrier fre-

(220,190), (220, -130) (270,170), (270, -150) quency, which equals 3MHz. The probing pulse is transmit-11 (220, 210), (220, -190) (270, 230), (270, -170) ted into the medium according to the wave equation

Table 1. Locations of virtual transceivers and targets, eachsubsumes the respective virtual transceivers or targets for the 1 02bp0(y sz, t) f(t>5(S-) (24)lower order, i.e., the virtual transceivers for (m = 7) includes YA ' (c(-))2 &t2the virtual transceivers for (m = 5) and (m = 1).

where i' denotes the coordinates of the observation site and y

the coordinates of the probing site. c(z) is the random veloc-ity of the medium, which has a Gaussian distribution with amean of co equal to 1500m/s and a standard deviation of 1000simulated by a direct propagation from the virtual transceiver mea ofc.qa o1Omsada tnaddvaino0

simulated bya dret propagaCtion fromthrget. vrultncie about the mean. Based on the CFL conditions, the time steplocated at (' -(2YC - Ys)) to the target. \At is set to 5 x 10-8s.Bottom-Top, Double Reflection Path: is similar to the top- The channel is modeled as a two dimensional domain Rbottom, double reflection path except that the first reflection with discretization steps Ax = Ay = A, A being the wave-takes place from the bottom boundary. As shown in Fig. 2(e), length of 0.5mm. The discretized channel has finite dimen-the bottom-top, double reflection path is modeled by the sions of (0 < i < I - 1) and (0 j < J - 1), where i, jvirtual transceiver located at (Xs,2Yc + ys) that transmits are spatial indices representing site s= (iAx, jAy). Symboldirectly in a straight line to the target. k is the time index. The wave equation in (24) is discretized

The second column of Table 1 uses the aforementioned using the following finite difference approximationanalytical model to specify the locations of the virtual array 02_ _i,j,k±l - 20i,j,k + Oij,kl (25)when a single transceiver located at coordinate (220, 30)A is Ot2 =transmitting. The width Yc of the channel is 40. Therefore,only the real transceiver located at (220, 30)A is needed to with respect to time t- The second order derivatives of themodel the (m = 1) multipath. Notation (m = 5) includes field b with respectto

fand y are discretizedusing similar re-

the five multipaths included in subplots (a)-(e) of Fig. 2. lationships The finite difference representation of Eq. (24) iSSuch a five multipath model includes the real transceiver lo- Oi,j,k+l = 20i,j,k + k 2(±i+l1jk + <i l,j,k -20ij,k)(26)cated at (220, 30)A as well as four virtual transceivers locatedat (220, 50)A, (220, -30)A, (220, 110)A, and (220, -50)A. + ky(bij+1,k + <)i,j-l,k - 20ij,k) - <ij,k-1,Similarly, notation (m = 7) includes the real transceiver, the where kx = cAt/Ax and ky = cAt/Ay. In our simulations,virtual transceivers for (m = 5) multipaths, and additional the coordinates of the transceiver and target are (220, 30)Avirtual transceivers located at (220, 130)A and (220, -110)A. and (270, 10)A, respectively. The width of the channel isThe m multipath propagation of the probing pulse F(wOq) 40A. To quantify the effect of multipaths on time reversal,from the transceiver to the target is modeled by m direct the setup introduced in Section 3 is used. In our simulations,propagations, one from the real transceiver and the remaining we implement 3-ary TR and conventional classifiers with 5,(m - 1) from the (m - 1) virtual transceivers whose loca- 7, 11, and 21 multipaths. The 3-ary detection problem corre-tions are specified in the second column of Table 1 to a single sponds to determining if a target is present and further dif-target. ferentiating the identity of the target as friendly or enemy

The above procedure can also be used to model multi- target. The locations of the scatterers (unwanted elements)paths for the backscattered signal reflected from the target to are randomized during the simulations when the receiver op-the transceiver. The m multipath propagation of the reflected erating characteristics are derived. In our experiments, wesignal from the target to the transceiver is again modeled by m use a single transceiver as a transmitting/receiving antenna sodirect propagations, one from the real target and the remain- that the TR gain is attributed only to the multipaths, the fac-ing (m -1) components from the (m -1) virtual targets. tor that we want to highlight in this paper. The locations ofFor the real target located at (270, 10)A, the locations of the real and virtual transceivers and targets are specified in Ta-virtual targets used to model backscatter signal reflected from ble 1. We use the Neyman Pearson test with the likelihoodthe target to the transceiver for different values ofm are listed functions specified in Theorem 1 for the conventional classi-in the third column of Table 1. fier and Theorem 2 for the TR classifier for the friendly target

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21 Paths . . . 5. CONCLUSION11 paths

0.9 7 paths5 paths We derive the N-ary TR and conventional ML classifiers andDirect Path da T /: and0.8 _ ;j<< present an analytical multipath model to study the effects

0.7 ..... // of multipaths on our proposed classifiers. Our analyticalmultipath model is based on geometric optics approxima-

0.6 _ ,X7/ tion, which assumes strong total internal reflections from the

0.5 boundaries of the medium. While multipaths have a negativeimpact on the performance of the conventional classifiersbased on the backscatter of the probing signal, the perfor-

I_______________________________ _______ ___i___ mance of the TR classifier that uses backscatters of both the-30 25 20 1 -10 -5 0-30 -25 -20SNR(5dB) -10 _5 0 probing signal and the time reversed signal improves with

(a) Conventional Classifier. increase in the number of multipaths. In array processing ter-minology, the TR array behaves like a lens with an effective

,I aperture that is several folds larger than that of the apertureof the physical array. Increasing the number of multipathsmagnifies the aperture by a similar factor, thereby, resulting

0.8 _. ............. . ............. _ in performance gains for the TR classifier of about 1/4 dB0.7.................. .....per additional multipath.

6. REFERENCES

0.4 . . [ [1] M. Fink, "Time Reversal Acoustics," Physics Today,vol. 39, no. 5, 1997, pp. 555-566.

-30 -25 -20 15( 10 5 0 [2] L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berry-SNR(dB)

(b) TR Classifier. men, "Imaging and Time Reversal in Random Media,"Inverse Problems, vol. 18, 2002, pp. 1247-1289

Fig. 3. Probability of Detection of the enemy target versus [3] H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal,SNRs for different multipaths: (a) Conventional classifier (b) and C. Ferla, "Iterative Time Reversal in the Ocean Re-TR classifier.'TRclassifier. versal," J Acoustic Society ofAmerica, vol. 105, no. 6,

1999, pp. 3176-3184.

and GLR functions given in Theorem 3 for the conventional [4] Mathias Fink, "Time reversal of Ultrasonic Fields - Partclassifier and Theorem 4 for the TR classifier for the enemy I: Basic Principles," IEEE Transactions on Ultrasonics,target. Thresholds q11 and 172 in the Neyman Pearson test are Ferroelectrics, and Frequency control, vol. 39, no. 5,computed by setting the probability of false alarm to 0.01 in pp. 555-566, 1992.Eq. (22). [5] A. Asif, Q. Bai, and J. M. F. Moura, "Time Reversal

Fig. 3 plots the probabilities of detection PD for the en- Matched Field Processing: An Analytical Justification,"emy target as a function of SNR. This figure shows the effect in Proceedings of ISSPIT, Vancouver, 2006.of multipaths on the conventional classifier (top subplot) andTR classifier (bottom subplot). With an increase in the num- [6] F. Foroozan and A. Asif, "N-Ary Maximum Likelihoodber of multipaths, the performance of the conventional classi- Target Detection With Time Reversal," in Proceedingsfier deteriorates substantially. For PD= 0.5, there is a loss in of the IEEE International Symposium on Informationperformance of about 5 dB between the direct path (m = 1) Theory (ISIT), Toronto, July 2008, pp. 1873-1877.versus (m = 21) multipaths. On contrary to the conventionalclassifier,~~~~~ ~ ~~th.efracfteT casfe mrvswt [7] J. M. F. Moura and Y. Jin, "Detection by time reversal:classifier, the performance of the TR classifier improves with sigeatna"EETas o inlPoesn,vlsingle antenna,"IEEE Thans. on Signal Processing vol.an increase in the number of multipaths. For PD= 0.5, the 55 2007 187g201(m 21) multipath model provides a gain of 5dB over the 5 2 ppdirect path where no multipath is considered. On average,each additional multipath provides a gain of about 0.25 dB inthe TR classifier at low SNRs, while there is a loss of almostthe same amount in the gain of the conventional classifier permultipath.

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