Low observable target motion analysis using amplitude information

Low Observable Target Motion Analysis Using Amplitude Information

T. KIRUBARAJA�, Student, IEEE Y. BAR·SHAJ_OM, Fellow, IEEE University of Connecticut

In conventional passive and active sonar systems, target

amplitude information (AI) at the output of the signal processor

is used only to declare detection. and provide measurements.

We show that the AI can be used in passive sonar systems, with

or without frequency measurements, In the estimation process

itself to enhance the performance in the presence of clutter where

the target·originated measurements cannot be identified with

certainty, i.e., for "low observable" or "dim" ( low signal.to.noise

ratio (SNR)) targets. A probabilistic data association (PDA)

based maximom likelihood (ML) estimator for target motion

analysis (TMA) that uses amplitude information is derived. A

track formation algorithm and the Cramer-Rao lower bound

(CRLB) in the presence of false measurements, which is met by the

estimator even under low SNR condition., are also given. The

CRLB is met by the proposed estimator even at 6 dB in a cell

(which corresponds to 0 dB for 1 Hz bandwidth in the case of a 0.25 lIz frequency cell) whereas the estimator without AI works

only down to 9 dB. Results demonstrate improved accuracy and

superior global convergence when compared with the estimator

without AI. The same methodology can be used [or bistatic radar.

Manuscript received October 4, 1994; revised April 20, 1995.

IEEE Log No. T-AES;32/4/08006.

This work was supported under ONR/BMDO Grant NOOOI4-91.J.1950, NUWC/ONT Grant N66604-92-C-1386, and AFOSR Grant F49620-94·1..Q150.

Authors' address: Dept. of Electrical and Systems Engineering, University of Connecticut, U157, Rm. 312, Engineering III, 260 Glenbrook Rd., Stom, CT 06269-5157.

001 8-9251 !96i$5.00 © 1996 IEEE

I. INTRODUCTION

The standard target motion analysis (TMA) consists of estimating the position of the target and its constant velocity from bearings-only measurements corrupted by noise [12]. Although this motion model is very restrictive, it is widely used in underwater passive target tracking. This is because the model bears close resemblance to the actual scenario: it is common for an underwater target to maintain constant speed and course for considerably long durations and for the sensor, which is on a possibly moving platform, not to use active sonar to obtain range information to avoid being identified.

Bearings-only tracking is a very ill-conditioned estimation problem due to the nonlinearity in the target motion-to-sensor measurements relationship. Due to this nonlinearity, the observability of the target and the accuracy of the estimates are very sensitive to the configuration of the scenario. To establish observability, the platform needs to have a relative acceleration with respect to the target [1]. Even if the target is observable, due to the low information content of the measurements, the estimates may effectively be useless: they are highly inaccurate. To make the problem more complex, the target-originated measurements are not always received and there arc false measurements at each sampling instant. In a real scenario these problems exist and any estimation algorithm has to take these aspects into consideration.

Bearings-only tracking has received much attention in the literature (see (8, references]). Recently, narrowband passive sonar tracking, where frequency measurements are also available, has been studied [10]. The advantage of narrowband sonar is that it does not require a maneuver of the platform for observability and it also greatly enhances the accuracy of the estimates. Howevelr, not all passive sonars have frequency information available. In both cases, the intensity of the signal at the output of the signal processor, which is referred to as measurement amplitude or amplitude information (AI) here, is used implicitly to determine whether there is a valid measurement. This is usually done by comparing it with the detection threshold, which is a design parameter. The AI is available in some cases to the operator to discriminate possible target measurements from clutter. The use of the sound level received by a single hydrophone to estimate target velocity and range was suggested in [6]. The method suggested in [6] exploits the change in sound intensity due to absorption at high frequencies.

In this work it is shuwn that the measurement amplitude carries valuable information and that its use in the estimation process increases the observability even though the AI cannot be correlated to the target state directly. Also superior global convergence

IEEE 1RANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO.4 OCTOBER 1996 1 367

�L_M_fil_alt_:_ed_...JHL.._En_de_::-_Iope_---lH Th.-. �= Fig. 1. Block diagram of amplitude receiver.

properties are obtained. Previously, AI has been investigated only for active sonar tracking [11]. Here we derive a scheme to estimate the target motion parameters and to form validated tracks using AI for narrowband as well as wideband sonar. In both cases we also derive the Cramer-Rao lower bounds (CRLBs) that specify how accurate the estimates can be in the presence offalse measurements. The same methodology can be used for bistatic radar [9].

Although accuracy is the main concern in any estimation scheme, there are many conflicting aspects to it: 1) the need to maximize the probability of accepting valid tracks (target detection) , 2) the need to maximize the probability of rejecting false tracks (clutter rejection), and 3) the need to minimize the number of measurement sets required for estimation (rapid estimation). Here we present simulation results for low signal-to-noise ratio (SNR) ("low observable" or "dim") targets to compare and contrast the performances of the estimation schemes with and without AI in these aspects.

To derive the estimator we use probabilistic data association (PDA) combined with maximum likelihood (ML) [2]. In Section II the problem consisting of the target, sensor, and measurement models, and the statistical assumptions used to derive the estimator is formulated. In Section III the estimator is derived and its numerical implementation illustrated. Also the CRLB based on the PDNML estimator is derived there. W hen an estimate is obtained, the track formed from the estimates should be validated. A track acceptance test is also presented in Section III. In Section IV, simulation results are presented and the performances compared.

II. PROBLEM FORMULATION

In this section, we define the notations used in this work and give the motion models for both bearings-only and narrowband sonar tracking. We also present the statistical assumptions made about the sensor and measurement characteristics.

A. Amplitude Information

First we illustrate how the AI is obtained for both bearings-only (wideband) and narrowband sonar.

Since a number of detections are made by the sonar at each sampling instant, we need to decide which measurements are likely to have originated from the target. Even then, more than one measurement

is likely to satisfy the validation criterion and this gives rise to a situation where the estimator has to operate in the presence of false alarms. T he signal from the target is passed through a matched filter and an envelope detector. The output is thresholded after which a detection can be declared and then one has a measurement corresponding to the particular bearing (and possibly frequency) cell under consideration [3]. The block diagram in Fig. 1 illustrates this.

The noise at the matched filter is assumed narrowband Gaussian. When this is fed through the envelope detector, the output is Rayleigh distributed. We denote the probability density function (pdf) of the envelope detector output a when the signal is due to noise only by po(a) and the corresponding pdf when the signal originated from the target by PI (a). If the SNR 1 is d, thc density functions of noise only and target-originated measurements can be written as

po(a) = aexp ( -�), a>O (1)

a ( a2 ) Pl (a) =

1 + d exp - 2(1 + d) , a> 0 (2)

respectively. 2 A suitable threshold, denoted by T, is used to

declare a detection. We define the probability of detection Po as

Po �P{a target-originated measurement exceeds the threshold T} (3)

and the probability of false alarm PFA by

PFA � P{ a measurement due to noise only exceeds the threshold T} (4)

where pr} gives the probability of an event. Both Po and PFA can be evaluated from the (pdfs) of the measurements. They are givcn by

Po = 100 Pl(a)da (5)

PFA = 100 po(a)da. (6)

Clearly, in order to increase Po one has to lower the threshold T. However, this increases PFA too.

lThis is the SNR in a resolution cell, to be denoted later as SNRc. 2This is a Rayleigh fading amplitude (Swerling I) model believed to be the most appropriate for shallow water passive sonar.

1368 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO. 4 OCTOBER 1996

Therefore, depending on the SNR we have to select T so as to satisfy two conf licting requirements.3

The density functions given above correspond to the signal at the envelope detector output. Those corresponding to the output of the threshold detector are

p�(a)= p�APo(

a)= p�Aa exp (- �2) ,

(7)

= ;D 1 : d exp ( - 2(1: d») , a> T (8)

where Po (a) is the pdf of the validated measurements that are due to noise only and PI (a) is the pdf of those originated from the target. In the following, a

is the amplitude of the candidate measurements. For Rayleigh distributed measurements Po and PPA satisfy

PD = exp ( - 2(lr� d») (9)

PPA = exp ( _ r:) . (10)

Finally we define the amplitude likelihood ratio p as

B. Target Models

pHa)

p= p�(a)'

(11)

We assume that n sets of measurements made at times t = tbt2, ... ,tn are available.

For bearings-only estimation the target motion is defined by the 4-dimensional vector

(12) where e(to) and 'I/(to) are the distances of the target in the east and north directions, respectively, from the origin at the reference time to. The corresponding velocities, assumed constant, are ( and Ij, rcspectively.

The state of the platform at ti (i = 1, ... ,n) is defined by

f:;. ' . I Xp(ti) =[ep(ti) TJp(ti) ep(ti) I}p(tj)]. (13) The relative position components in the east and

north directions of the target with respect to the platform at ti are defined by re(ti,X) and r,.,(tj,x) respectively. Similarly V{(ti,X) and v,.,(tj,x) define the relative velocity components. The true bearing of the

3For other probabilistic models of the detection process, different SNR values correspond to the same PD,PFA pair. Compared with the Rician model receiver operating characteristics (ROC) curve, the Rayleigh model ROC curve requires a higher SNR for the same pair (PD,PPA), i.e., the model considered here (Rayleigh) is pessimistic.

North

Ii '1(t,) T&l'let • t

iI.(t,) r.(t;)

�.(t,) O,(z) FbU'onn T(t,)

I Mt,) {(t,)

Fig. 2. Scenario at ti.

target from the platform at ti is given by

9i(x) � tan-1l'e(ti,x)/r,.,(tj,x)].

East

The above scenario is illustrated in Fig. 2. The range of possible bearing measurements

(14)

is [0,211'). Since scanning tlllis entire region is not efficient, we assume that a human operator is able to identify a subregion as the surveillance region for bearing, namely,

(15)

The set of measurements at ti is denoted by

(16)

where mj is the number of measurements at ti and the pair of bearing and ampli1tude measurements zj(i) is defined by

The cumulative set of measurements during the entire period is

(17)

(18)

In addition to the above, the following assumptions about the statistical characteristics of the measurements are also made [10].

1) The measurements at two different sampling instants are conditionally independent, i.e.,

p[Z(il),Z(i2) I xl = p[Z(il) I x]. P[Z(i2) I xl

\:I il", i2 (19)

where p[.] is the pdf. 2) A measurement that originated from the target

at a particular sampling instant is received by the sensor only once during the corresponding scan with probability PD and is corrupted by zero-mean Gaussian noise of known variance. That is

(20)

KIRUBARAJAN & BAR-SHALOM: LOW OBSERVABLE TARGET MOTION ANALYSIS 1369

where fij � N[O,O"�] is the bearing measurement noise. Due to the presence of false measurements the index of the true measurement is not known.

3) The false bearing measurements are distributed uniformly in the surveillance region, i.e.,

(3ij �lJ[Eh,82]. 4) The number of false measurements at

a sampling instant is generated according to a Poisson law with a known expected number of false measurements in the surveillance region. This is determined by the detection threshold at the sensor (exact equations are given in Section IV).

(21)

For narrowband sonar (with frequency measurements) the target motion model is defined by the 5-dimensional vector

It is also assumed that these two measurement noise components are conditionally independent. That is,

The measurements due to noise only are assumed to be uniformly distributed in the entire surveillance region.

I I I . MAXIMUM LIKEL IHOOD ESTIMATOR COMBINED WITH PDA

In this section we present the derivatiun and the implementation of the ML estimator combined with the FDA technique for both bearings-only tracking and narrowband sonar tracking.

(22) A. Derivation of Maximum Likelihood/Probabilistic

where "/ is the unknown emitted frequency assumed constant. Due to the relative motion between the target and platform at ti this frequency will be Doppler shifted at the platform. The (noise free) shifted frequency, denoted by "/i(X), is given by

( ) [ v�(tj,x)sin8j(x) + Vl/(ti,X)COSBi(X)] 'Y i X = 'Y 1 - -"--'----'----'---'---'-'---'--"----'--'c

(23)

where c is the velocity of sound in the medium. If the bandwidth of the signal processor in the sonar is [QI. Q2]' the measurements can lie anywhere within this range. As in the case of bearing measurements, we assume that an operator is ab1c to select a frequency subregion [fl, f2] for scanning. In addition to the bearing surveillance region given in (15), we define the region for frequency as

(24)

The noisy frequency measurements are denoted by fij and the measurement vector is

a,j],' (25) As for the statistical assumptions, those related

to the conditional independence of measurements (assumption 1) and the number of false measurements (Assumption 4) are still valid. The equations relating the number of false alarms in the surveillance region to detection threshold are given in Section IV.

The noisy hearing measurements satisfy (20) and the noisy frequency measurements fij satisfy

fij = "/i(X) + Vij (26)

where Vij � N[O,O"�] is the frequency measurement noise.

Data Association Estimator

If there are mi detections at ti we have the following mutually exclusive and exhaustive events [2]:

<Ai) :;

{ {measurement zj(i) is from the target},

{all measurements are false}, j '" 1, .. . ,mj

j=O (28)

The pdf of the measurements corresponding to the above events can be exactly written as

p(Z(i) I € j(i),x)

= {U�-�':i P�jj):ij rr��l p�(ail)' j = 1, ... ,mi U ITI=l Po (ail ) , j=O

(29) where U =:: U() is the area of the surveillance region.

Using total probability theorem we can write the likelihood function of the set of measurements at ti as

p[Z(i) I xl = u-m;(l_ PD) I1p�(aij)J.lf(mj) j=l

m; == u-mi (1- PD) II Po (aij)J.lf(mi)

j=l 1 m lnj u- iPDJlf(mi-l)

rr T + Po (aii)

mi i=l

. t �1_ exp (_� [f3ii - Bi(X)] 2) Pi . V21fCTe 2 CTe } }=1

(30)

1370 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC S YSTEMS VOL. 32, NO. 4 OCTOBER 1996

where Jlf(mi) is the Poisson probability mass function of the number of false measurements at ti.

Dividing the above by p [Z(i) I Eo(i),x] . Jlr(mi) we obtain the dimensionless likelihood ratio 1\ [Z (i), x 1 at ti. Then

. p [Z(i),x] <Pi[Z(I),X] = p[Z(i) I co(i),x]

PD m, 1 = (1- PD) + - I: --Pij A )=1 v'2i(J9 ( 1 [(3ij-ei(X)]2) ·exp --2 (To

where A is the expected number of false alarms per unit area.

(31)

Alternatively we can define the log-likelihood ratio 4>;[Z(i),x] at t; as

. { P[Z(i),X] } ¢,i[Z(I),X] = In p[Z(i) I Eo(i),x]

[ PD mi 1 = In (l-PD)+ - L --Pij

,\ j=! y'2-i(J(J

Using the conditional independence of measurements the likelihood function of the entire set of measurements can be written in terms of the individual likelihood functions as

n p[zn I x] = I1p[Z(i) I x]. (33)

;=1 Then the dimensionless likelihood ratio for the

entire data is given by n

<p[Zn ,x] = II <Pi [Z(i),x]. (34) ;=1

From the above we can write the total log-likelihood ratio ¢[zn,x] as

n 4>[Zn,x] = 2.:: MZ(i),xj

i;;;;:l

n [ p m; 1 = 2:)n (1- PD) + ; I: M':: p;j i=1 j=1 V 27r(Jo

( 1 [(3ij - fli(X)] 2)] .�p -- . 2 (Jo (35)

The maximum likelihood estimate (MLE) is obtained by finding the state x = x that maximizes the total log-likelihood ratio. In deriving the likelihood ratio it has been assumed that the gate probability mass, which is the probability that a target-originated

measurement falls within the surveillance region, is one. The operator selects the appropriate region.

We can use arguments similar to those given earlier to dcrivc thc MLE whcn frequency measurements are also available. Defining E j(i) as in (28) we can write the pdf of the measurements as

p[Z(i) I fj(i),x]

= {U�:i p�jj)�(fij)Pij Il7�IPOCaij), j == 1, ... ,mi

U Ilj=IPo (Uij), j=O

where U == UoU, is the volume of the surveillance region.

After some lengthy manipulations the total log-likelihood ratio is obtained as

<l>W,x] = L<I>i[7(i),xl i=l

n [ PD mi Pij = "" In (1- PD) + - ""--L... >. L... 27rCIOCI, j=1 j=l

(36)

.exp(-� [�j�:i(X)r - � [fjj:�j(X)r) l

For narrowband sonar, the MLE is found by maximizing (37).

B. Numerical Calculation of Maximum Likelihood

Estimate

(37)

The maximization of the log-likelihood ratio is done using a quasi-Newton (variable metric) method. This can equivalently be done by minimizing the negative log-likelihood ratio. Instead of using the exact Hessian to find the minimum point , an approximation to the inverse Hessian matrix , which is updated after each iteration, is used. In our implementation of the estimator, the Davidon-Fletcher-Powell variant of the variable metric method is used [14].

The Newton-Raphson method ith iteration is

(38)

In (38) the left-hand side gives the finite step taken at the ith iteration to reach the minimum. We have to know the exact Hessian H for the Newton-Raphson method. For a variable metric method, at ti we use an approximate inverse Hessian, Hi-1, which is updated after eaeh iteration and satisfies

1· HA -1 - H-1 .1m i - . ,�C>O (39)

An advantage of using an approximatc Hcssian is that the iteration is guaranteed to proceed downhill toward the minimum. This ensures that the Newton step is taken in a direction decreasing in value. By


contrast, when wc usc the exact Hessian there is no guarantee that the Hessian is positive definite. This may lead to a step uphill away from the minimum [5].

The log-likelihood ratio may have many local maxima, i.e., it has multiple modes. Due to this property the line search algorithm may converge to one of the local maxima if the search is initiated too far away from the global maximum. To remedy this, in [101 a multipass approach was suggested.

The basic idea is to maximize a modified likelihood ratio in K passes instead of finding the global maximum directly. For bearings-only tracking, at the kth pass, the modified likelihood ratio qi [zn ,x 1 is constructed as follows:

where ak satisfies ak+ 1 < ak for k = 1, . . . , K - 1 and aK == 1. In our implementation O'k == K - k + 1. A lso the penalty function c(x) is given by

with

( � I [V(X)-V]2

C x) ==--2 (J"v

v(x)� JX(3)2 + x(4)2

(41)

(42)

where v is a typical value of the target speed and (J"v is the standard deviation of the typical speed [10]. The penalty function c(x) is used only in the first K -1 passes and is set to zero at k == K.

To find the global maximum, an initial guess obtained by rough grid search is used for k == 1. The resulting maximum point is used as the initial guess for k = 2. This is repeated until k == K at which stage the global maximum is found. The reason for this approach is to prevent the maximization procedure from converging to an unrealistic local maximum. The modification gives a "broader view" to the estimator. In our implementation this multipass approach can be triggered optionally.

Similarly, for narrowband sonar, at the kth pass the following modified likelihood ratio is maximized:

k n � n [ Po mi 1

tjJ[Z,x]==l:ln(I-PO)+Tl: 2 Pij 1=1 j=1 27rO'k(J"6(J",

( 1 [(3lj - B;(X) ] 2 ·exp --

2 ak(J"O

_ ! [f;j -fl(X) ] 2) ] + c(x) 2 O'k(J",

(43) where c(x) is given by (41).

C. Cramer-Rao lower Bound for MLE

For an unbiased MLE the Cramer-Rao lower bound (CRLB) is given by

E{(x - x)(x - x)'} � J-1 (44)

where J is the Fisher information matrix (FTM) given by

J == ([vxlnp(Zn I x)][vxlnp(Zn I x)]'}lx=Xtru<' (45)

Only in simulations will the true value of the state parameter be available. In practice CRLB is evaluated at the estimate.

As expounded in Appendix A, the FlM J is given in the present ML/PDA approach for the bearings-only case-wideband sonar-by

where Q2(Pn,Avg,g) is the in/ormation reduction /actor that accounts for the loss of information resulting from the presence of false measurements and less-than-unity probability of detection [2] and the expected number of false alarms per unit volume is denoted by A.

In deriving (46) only the bearing measurements which fall within the validation region

at tl have been considered. The validation region volume (g-sigma region) Vg is given by

(48) The information reduction factor Q2(Pn,AVg,g)

for the present 2-dimensional measurement situation (bearing and amplitude) is given by

1 fi� lif(m-I) Q2(Po,AVg,g) == 1 + d V 1[ � (gPFA)m-llz(m,Po,g)

(49) where h(m,Po,g) is a 2m-fold integral given in Appendix A where numerical values of Q2(PD, AVg,g) for different combinations of Pn and ,\ v g are also presented. The derivation of the integral is based on [lOJ and [7]. In our implementation g == 5 was selected. Knowing Pn and A v g one can find PPA using

where Vc is the resolution ceIl volume of the signal processor (this is discussed in more detail in Section IV). FinaIly d, the SNR, can be calculated from Po and AVg using (50), (10), and (9).

(50)

1372 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO. 4 O CTOBER 19%

The rationale for the terminology information reduction factor can be seen by noting that the FIM for zero false alarm probability and unity ta rget detection probability, 1o, is given by [1]

From (46) and (51) it is clear that q2(PO,>'Vg,g), which is always less than or equal to unity, represents the loss of information due to clutter.

For narrowband sonar (bearing and frequency measurements) the FIM is given by

1 '} + "2[\7 x/i(X)][\7 x/;(x)] u, (52)

where, as shown is Appendix A, qz(Po,>.vg,g) for this 3-dimensional measurement (bearing, frequency and amplitude) case is evaluated using

1 00 2m-1f/,f(m-1) qz(Po,>.vg,g) = 1 + d :Z= (2Pl<'A)m-l h(m,Po,g).

m=1 g

(53) The expression for h(m,Po,g) and the numerical

values for qz(Po,>'vg,g) are also given in Appendix A. For narrowband sonar, the validation region is

defined by

V;(x) � { (f1ij,fij): [f1ij -:.:i(X) r + [Jij -(j�i(X) r � g2 } (54)

and the volume of the validation region Vg is

D. Acceptance of Maximum likelihood Estimate

(55)

Each estimate i obtained by maximizing the log-likelihood ratio is tested whether it can be used as an acceptable track. The test is necessary since due the multimodaI nature of the log-likelihood ratio the maximization procedure may converge to a local maximum resulting in an unrealistic track.

The test is formulated as a hypothesis testing problem between the following hypotheses:

Hi = {There is one track and i is the global maximum}

Ho = {There is no track}. (56) (57)

Before proceeding we ddine two quantities, namely

7r m � Miss probability

� P{ accept Ho I HI true} (58)

7r r � False track rejection power of the test

�P{reject HI I Ho true}. (59) The power of the test gives the probability that a track is rejected when it should be rejected.

The optimal decision to validate the estimates, according to Neyman-Pearson lemma, in the sense of maximizing 7r r for a given 7r m is obtained by considering the log-likelihood ratio [4]. For this purpose, at ti the log-likelihood ratio of the measurements that lie within the g-sigma region is used. We denote the log-likelihood ratio formed from the m; validated measurements by cp[zn ,x].

The test statistic to be lLlSed is

A � 2:7=1 CP;[Z(i), x] - E{ cp;[Z(i),Xtruel I HI} HtlHo- J

. L:7=1 Var{¢;[Z(i),Xtrue] I Ht}

(60) Defining

we get

f/,1 �E{¢i[Z(i),Xtrue] I HI} (61)

ui � Var{ ¢;[Z(i),xtrue] I HI} (62)

(63)

Assuming the Central Limit Theorem is valid for (63) means that the test statistic has a zero-mean, unit variance Gaussian pdf. Using this property, for a given 7r m, the most powerful test for track acceptance is

If AHt/Ho � e"", then HI is accepted (64) where e"", is calculated from the one-sided tail of the standard Gaussian and satisfies

7rm = � r�'" e-i12dy. v27r .1-00 (65)

Now we proceed to find 7r n the false track rejection power of the test. Using (64) we can rewrite (59) as

7r r = P{AHt/Ho ::; e"", I Ho}. (66) We define the following test statistic where there is

no target:

A � 2:7=1 tPi[Z(i),i] - E{¢i[Z(i)] I Ho} (67) Ho/H1- � V 2:7=1 Var{¢;[Z(i)] I Ho}

KIRUBARAJAN & BAR-SHAT DM: LOW OBSERVABLE TA RGET MOTION ANALYSIS 1373

which results in

where

A - L;'=l 1>i [Z(i), x] - nJ.lO Ho/H, - r.; Vn(JO

J.lo;;E{¢i[Z(i)] I Ha}

(J�;;Var{¢i[Z(i)]1 Ha}.

Applying the Central Limit Theorem to AHo/ H, results in

� P {A < C'Km(Jl + y'iiCJ.lI - Po) lu } 1[,,,,,, Ho/Ht_ 0'0

110

= erf r C1l'mO'l + :(J.ll - J.la) ] where erfC) is the error function given by

erf(x) = -- e-Y /2dy. 1 jX z v'2iF -00

(68)

(69)

(70)

(71)

(72)

The development of the test is similar to that in [10, Section IV]. In Appendices B and C the derivation of the above moments are given. If the true state parameters are not available, these moment� are evaluated at the estimates.

IV. RES ULTS

We implemented both the bearings-only and narrowband sonar problems with AI to track a target moving at constant velocity . The results for the narrowband case are given below. Bearings-only tracking with AI gives similar results. Following that, we present the advantages of using AI by comparing the performances of the estimators with and without AI.

In narrowband signal processing, different bands in the frequency domain are defined by an appropriate cell resolution and a center frequency about which these bands are located. The received signal is sampled and filtered in these bands before applying fast Fourier transform (FFT) and beamforming. Then the angle of arrival is estimated using a suitable algorithm [13]. As explained earlier, the received signal is registered as a valid measurement only if it exceeds the threshold 1'. The threshold value, together with the SNR, determines the probability and the probability of false alarm as given in (9) and (10).

The signal processor was assumed to consist of the frequency band [500 Hz, 1000 Hz] with a 2048-point FFT. This results in a frequency cell Cy whose size is given by

C, = 500/2048 � 0.25 Hz. (73)

As for azimuth measurements the sonar is assumed to have 60 equal beams resulting in an azimuth cell C(I

with size Co = 1800/60 = 3.00•

Assuming uniform distribution in a cell, the frequency and azimuth measurement standard deviations are given by4

O'''{ = O.25/m = 0.0722 Hz

0'0 = 3.0/ V12 = 0.8660•

(74)

(75)

(76)

The SNRc in a ce1l5 was taken as 6.1 dB and Po = 0.5.6 The corresponding SNR in a 1Hz bandwidth (SNR1) is 0.1 dB. These values give, using (9) and (10),

T = 2.64

PPA = 0.0306.

(77)

(78)

From PPA we can calculate the expected number of false alarms per unit volume, denoted by A, using

(79)

Substituting the values for Co and C, we obtain

0.0306

A =

3.0.0.25 = O.0407/deg. Hz. (80)

The surveillance regions for azimuth and frequency, denoted by U(I and U"'f' respectively, are taken as

Uo = [-20°,20°]

U, = [747 HZ,753 Hz].

(81)

(82)

We can also calculate the expected number of false alarms in the entire surveillance region and that in the validation gate Vg• These values are found to be 9.8 and 0.2, respectively, where the validation gate is restricted to g = 5. These values mean that for every true measurement that originated from the target there are about 10 false alarms which exceed the threshold.

The estimated tracks were validated using the hypothesis testing procedure described in Section IIID. The track acceptance test was carried out with a miss probability of 5%.

To check the performance of the estimator, simulations were carried out with clutter only, i.e., without a target and also with a scenario where a target was present and measurements were generated

4The "uniform" factor,;u corresponds to the worst case. In practice, (J'B and (J''''( are functions of the 3 dB bandwidth and of the SNR.

SThe co mmonly used SNR, designated here as SNR1, is signal strength divided by the noise power in a 1 Hz bandwidth. Our SNR, denoted as SNRC, is signal strength divided by the noise power in a resolution cell. The relationship between them, for C, = 0.25 Hz is SNRc = SNRl - 6 dB. It is believed that SNRc is the more meaningful one because this is what det ermines the ROC curve. 6The estimator is not very sensitive to an incorrect PD' T his is verified by running the estimator wi th an incorrect PD on the data generated w ith a different PD. Differences up to 0.15 are tolerated by the estimator.

1374 IEEE TRANSACTIONS ON AEROSPACE AND ELECmONIC SYSTEMS VOL. 32, NO.4 OCTOBER 1996

xlO"4 The Scenario

5.0

4.5

4.0

3.5

I 3.0

,§. 25 E

2.0 Z

1.5

1.0

0.5

0.0·

-1.00 -0.75 -0.50

F ---.� -

-0.25 0.00 East (meters)

0.25 0.50 0.75 x10'4

1.00

Fig. 3. 1.tajectories of target and platform.

Target-originated and false frequency measurements 753r---�-:��--,---�r-��--� __ � ____ � ___ �

752

� 751

El � I ::I " 750 Ei >. II " 8- 749 .!:1

748

I

: . . .

. :

Time

I

0 0 0

o - Target originated • - False alarms

Fig. 4. Frequency measurements.

accordingly. Simulations were done in batches of 100 runs.

When there was no target, irrespective of the initial guess, the estimated track was always rejected. This corroborates the accuracy of the validation algorithm given in Section IIID.

For the set of simulations with a target the following scenario was selected. The target moves at a speed of 10 mls heading West and 5 mls heading North starting from (5000 m, 35000 m). The emitted frequency was set at 750 Hz. The target parameter is thus x = [5000 m,35000 m,-lO m/s,5 m/s,750 Hz]. The motion of the platform consisted of two velocity legs in the Northwest direction during the first half and Northeast direction during the second half of the simulation period with a constant speed of 7.1 m/s. Measurements were taken at regular intervals of 30 s. The observation period was 900 s. Fig. 3 shows

the scenario including the target true trajectory (solid line), platform trajectory (dashed line) and the 95% confidence regions of the pOSition estimates at the initial and final sampling instants based on the CRLB (52). The initial and thc final positions of the trajectories are marked by "I" and "F', respectively. The purpose of the confidence region is to verify the validity of the CRLB as the actual parameter estimate covariance matrix from a llIumber of Monte Carlo runs [11.

Figs. 4 and 5 present the sets of frequency and azimuth measurements, respectively, in one fun. The target-originated and noise-only measurements are distinguished by denoting them by "0" and "*", respectively, for clarity. However, the index of the target-originated measurement is not known to the estimator. Fig. 6 shows a set of amplitude measurements.


Target-originated and false azimuth measurements 20 . . 15 t ·

10 I . !!l 5 5 I 0 Ii . . .:; = 8 ·5 �

• t . .

·10 . . .

• 15 . .

.200L---100�-'-L 20�O-":"""300L---=- 400�.!--:- 500"::-""":"-:'600=-.....o..:::70:::0--";;:80:::0--::'900

Time o . Target originated • - False alarms

Fig. 5. Azimuth measurements.

Target-originaled and false amplitude measurements 6.5

5.5 � § 5

I 4.5 " '0 ,g 4 1 3.5

3 � •

2.5

. I . I . ! !

. • i I I

• I I I .

t I 0 100 200 300 400 500 600 700 800 900

Time o - Target originated •. False alarms

Fig. 6. Amplitude me asurements.

Fig. 7 shows the 100 tracks formed from the estimates. It can be seen that in 94 runs the estimated trajectory endpoints fall in the corresponding 95% uncertainty ellipses.

In Table I, the numerical results from 100 runs are given. Here x is the average of the estimates, 8 the variance of the estimates evaluated from 100 runs, and

O'CRLB the theoretical CRLB derived in Section lIIC. The range of initial guesses found by rough grid search to start off the estimator are given by Xinit.

The efficiency of the estimator was verified using the normalized estimation error squared (NEES) [1] defined by

Ex �(x - X)' J(x - i) (83)

where x is the estimate, J is the FIM (52). Assuming approximately Gaussian estimation error the NEES is

chi-square distributed with n degrees of freedom where n is the number of estimated parameters. For the 94 accepted tracks the NEES was obtained as 5.46, which lies within the 95% confidence region [4.39, 5.65]. Also one can easily see that each component of x is within 28/ V100 of the corresponding component of Xtrue.

Similar results were obtained for the case where frequency measurements were not available. For the scenario shown in Fig. 3 the improvement in the target observability due to the use of frequency measurementsis presented in Table II. It can be seen that the use of frequency measurements is very beneficial, even when the platform is maneuvering.

From these results one can conclude that the performance of the estimator is commensurate with the CRLB based on the PDNML theory and that the

1376 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO.4 OCTOBER 1996

6X�1�0'�4 _______________ Tru __ e_a_nd_&_mm_· _a_re_d_�_�_·�_W_ri_� ________________ -,

I � z

2

+---____ -+-___ -I-___ --+_--=:c==-=-I--___ +-____ -+-____ -+ ___ � xlO"4 0.50 0.75 1.00

East (meters) Pig. 7. Es timated tracks from 100 runs for narrowband sonar with AI.

TABLE I Results of 100 Monte Carlo Runs for Narrowband Sonar with AI

Unit "'true :2'iinit X "'CRLB 0-

m 5000 -12000 to 12000 4991 667 821 m 35000 49000 to 50000 35423 5576 5588

mls -10 -16 to 5 -9.96 0.85 0.96 m/s -4 to 9 4.87 4.89 4.99 H. 750 747 to 751 749.52 2.371 2.531

Note: SNRc = 6.1 dB.

estimator can be used efficiently to estimate targets that do not maneuver.

A. Advantages of Using Amplitude Information

In this section we discuss the advantages of using AI. To compare the performance of the estimator, we consider the accuracy of estimates for a given

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1 0.2 0.3 0.4

TABLE II Improvement in CRLB Due to Use of Frequency Measurements

Parameter O'CRLB (unit) with frequ<:Rcy

(ltd (m) 689 !)(t,) (m) 5759 e (m/s) 0.88 fdm/o) 4.89

"'CRLB without frequency

3674 25634

5.58 18.20

set of data, its computational complexity, and the convergence/acceptance of the estimates.

T he most important aspect of the estimator is its CRLB indicating how accurately it can estimate. For the same scenario the difference in the CRLB of two algorithms is dictated by their information reduction factors qZ(PO,AVg,g): the higher the factor the better the algorithm, provided that it is efficient. Fig. 8 shows "constant SNR lines" of Q2(Po,AVg,g) at different

12 dB

I)·

.............

o.s 0.6 0.7 0.8 0.9 Probability of detection

Fig. 8. Q2(PD, Avg,g) with AI (- .. ) and without (- - - ) [or narrowband sonar versus SNRc.


TABLE III

Results of 100 Monte Carlo Runs for Narrowband Sonar Without AI

Unit �true Xinit x aCRLB "

m 5000 12000 to 12000 6395 689 8653 m 35000 49000 to 50000 41370 5759 23094

m/s -10 -17 t0 4 -9.86 0.88 1.21 mi· 5 -5 to 10 3.55 4.73 7.36 Hz 750 747 to 751 749.03 2.448 2.751

Note: SNRc = 6.1 dB, SNRj = 0 dB.

values of Po for bearings-only tracking. In the figure dotted-lines and the dashed-lines represent the q2 values obtained with and without the AI, respectively.

From Fig. 8, it is noted that using AI increases qz implying an increased accuracy. It has been shown experimentally that the estimator proposed here is efficient. Therefore we ean conclude that using AI will improve the estimation accuracy. The increase in the information reduction factor is significant and the proposed algorithm is more attractive under low SNR conditions. The improvement in accuracy was verified in Monte Carlo runs too. For the same scenario illustrated in Fig. 3 the results of the estimator without AI are given in Table III. The number of accepted tracks was 69 out of 100. The NEES was 60. Although Tables I and III predict only a marginal improvement in the CRLB when AI is used, the experimental improvement is very high: the CRLB is not met by narrowband sonar without AI and the variance of the estimates in 100 Monte Carlo runs is much higher than the values predicted by the CRLB. Therefore, we can conclude that at low SNR values, the information reduction factor dcrived in [10] for narrowband sonar without AI does not give a true measure of the quality of the estimates (the calculated lower bound is not met as before). Note that for the same SNR, the CRLB is still met by narrowband sonar with AI and thus the CRLB given in (52) represents the true reduction in information in the presence of false alarms.

For the scenario illustrated in Fig. 3, the lowest SNR in a cell down to which the estimator with AI can be used while meeting the CRLB is around SNRc = 6 dB. The corresponding limit for the estimator without AI is 9 dB. The value SNRc = 6 dB corresponds to SNR1 = 0 dB for a resolution cell of size 0.25 Hz and to -3 dB for a 0.125 Hz cell.?

The high accuracy and the superior global convergence of the estimates for narrowband sonar with AI, even under extremely low SNR conditions, are the major benefits of using AI.

7Theoretically, by refining the frequency resolution, one can further reduce the SNRj limit down to whlch the algorithm works.

TABLE IV Pertormanccs ot Estimators for Different K Values for Narrowband

Sonar

with. AI with.out Al accepted a.verage time accepted average time

K track. t"ken (.) track. taken (5)

3 95 3.19 71 3.57 2 91 2.68 60 2.63 1 82 1.97 18 1.69

Note: SNRc = 6.1 dB, SNRI = 0 dB, 5% miss probability acceptance test.

Another concern in the implementation is the computational complcxity of the algorithm. As noted in Sect�o� II�B, �he estimator was implemented using the modIfIed likelIhood function in multiple passes. The multipass approach can be triggered by setting K to a value greater than 1 . In our implementation the default value of K was 3. Obviously this approach would increase the time taken to find the estimate after the measurements are taken. However, reducing the value K would result in unacceptable tracks. Th check �heir performances in these conflicting aspects, we Implemented the estimators, with and without AI, for different K values. The average computation times for a single run from 100 Monte Carlo runs are given in Table IV. The computation time taken for estimation was calculated as the time elapsed between entering and leaving the maximization algorithm. The time taken to generate the measurements was not included. In a real scenario the measurements are available and the performance of an algorithm depends only on how fast the convergence is for that set of data.

The simulations were carried out for the same sccnario shown in Fig. 3 using the same initial conditions given in Thble 1. The computation times were on an IBM PC compatible with a 486 processor running at 33 MHz.

The above results show the superior global convergence of the estimator when the AI is also used in addition to bearing and frequency measurement�. The estimator with AI gives estimates that are acceptable almost 80% of the time even when a single-pass maximizer is used. For the estimator without AI, the percentage of acceptance drops to only 20%. Note that when amplitude information is not available, even during the third pass much improvement is made on the estimates (reflected by the time taken during the third pass) . The estimator with AI has converged mostly during the second pass and the third pass does not require much time. Similar results are obtained when frequency measurements are not available.

V. CONCLUSIONS

In this paper we have prescnted a new exact PDA-based ML estimator that uses AI to estimate the

1378 IEEE TRANSACTIONS ON AEl{OSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO. 4 OCTOBER 1996

position and constant velocity of a target. The AI is used not only to validate measurements, but also in the estimation itself. The estimator was derived for both bearings-only sonar and narrowband sonar, which includes also frequency measurements. We also derived an expression for CRLB to quantify the accuracy of estimates in the presence of false measurements. In both types of sonar the estimator was found to be efficient-the CRLB was met, even under low SNR conditions. The cell SNR limit down to which the algorithm is efficient is SNRc = 6 dB; this is 3--4 dB lower than the limit of the algorithm without AI.

Another advantage is improved convergence properties in comparison with the algorithm without AI. The percentage of accepted estimates is around 95 at SNRc = 6 dB. Under low SNR conditions the improvement is significant in the convergence/acceptance of the estimates. For a small reduction in the number of acceptable tracks the computational time can be reduced by almost 40% making it very suitable for finding an initial estimate quickly to be used in a recursive filter (e.g., an IMMPDAF [2]). This is a significant improvement over the estimator where amplitude information is not used for estimation.

In this work, some rather restrictive, but realistic, assumptions about the target motion and measurement models were made. The possible effects of the violation of any these assumptions have to be studied. Specifically, the change in the performance of the estimator if the target maneuvers slightly during the observation time, or if the emitted frequency varies slightly or if there are multiple targets within the observation window has to be studied further.

APPENDIX A. DE RIVATION OF FISHER I N FORMATI ON MATRIX

Now we derive the FIM for noisy measurements in clutter for both bearings-only and narrowband sonar when AI is also available.

The FIM can be written in terms of the individual FIMs 1; at ti as n

(84)

From (45) Ji is defined by

1; = E ( [Y'x Inp[Z(i) I x]] [Y' x In p[Z(i) I x] ]'} Ix=xt ... . (85)

When AI is used together with bearings-only measurements, p[Z(i) I x] is given by (30). Defining

I - m . 1 Po s = V ' r:.= -/-Lr (mi - 1) V 21f(J"o mi

(86)

(87)

we can rewrite p[Z(i) I xl as ( mJ

[ 1 ((3. ' _ (J. ) 2] ) p[Z(i) l x] = r + s f; Pij eXP - 2" �

mi . II pQ (aij) j=l

(88)

where Pij is the amplitude likelihood ratio as define in (11). For Rayleigh amplitude models described in (1) and (2) Pi} is given by

Pr<A ( aijd ) Pij = PD(1 + d) exp 2(1 + d) . (89)

Only the measurements that lie within the validation region (47) are mnsidered for the derivation of FIM. The number of such measurements is mi. In this case 1; can be written as the following 2m-fold integral

Ji = f. J . . . J Y'xp[Z(i) I xl

m; = l p[Z(i) I x]

. [Y';fi��? ;t]] ' p[Z(i) I x] dZ(i). (90)

The gradient of p[Z(i) I x] is

V' xp[Z(i) I xl = s it p� (aij) [i�>ij exp [-� e\r: 9; r] }=1 }=1

For notational simplicity the index i in Ii, mi , (Ji , /3ij , aij and Pij i s dropped in (92)-(106). This is reintroduced in (107).

Now we introduce a new variable � sue' that

Then J can be written as

co 100 100 jg jg [Y' xp [�, a I Xl] I = :L . . . . . . m=l r r - g -g p [�, a I x]

(92)

(93)

KIRUBARAJAN & BAR-SII ALOM: LOW OBSERVABLE TARGET MOTION A N A LYSIS 1379

The cross-terms vanish from the integral which then becomes

co

100 loo;g ;Ii

1 = L: . . . . . . m=1 r T -g -g 0"0 s2 rr;=l Po Caj) 2:;=1 pJ exp( -m�J

r + s 2:;=t Pj exp(-�J/2)

1 . 2 [\7 xO (x)] [,\?x(i (x)]' da d� (94) O"(}

00 100 100 jg jg = L: . . . . , .

m =I ' r T -g -g

0"0 S TIj�l Po (aj) 2:j�l PJexp( -�J)�J r / S + "'£;=1 pj exp(-�J /2)

1 . 2[\7 xO(x)] [\7 xO(x)]' dad�. (95)

O"(} Let

II (m, PD,g) � 100 • • • roo jg . . . jg T iT - g -g

. TI;=l Po (aj)

m�;=l Plexp(-�J)�J dad(

r / s + "'£j=l pj exp( -�7 /2)

On simplificatiun this becumes

100 100 ;g jg I1 (m, PD,g) = m . . , . . . r r - g - g

. TI;=I PO�j)PI eXp(-a)�? dad�. r / s + 2:j=l pj exp( -�J /2)

After substituting for Pj

It(m,PD,g) = m (PD�F: d) r 100 . . . ;00 1> ·1:

Another change of variables

aj = exp(-ay/2) results in

(96)

(97)

(98)

(99)

(HX1)

a -2dJ(1+d) exp(-e){2 . 1 1 1 dadE. PD(I + d)r "m -dJ(l+d) ( �; )

� L.., '-l a . cxp - -PFAS J - ] 2

Using these results

I = E sO"��:/iA� m .IaPFA . . .. laPPA .lag . . . fog

(101)

-2d/(l+d) ( 1'2),2 . al exp -':,1 ':,1 2 da d�

PDC1 + d)r "m -d/(l +d) ( �j ) P

+ L..,J'-1 aJ, exp --2 liAS -

(102)

We define another 2m-fold integral h(m, Po,g) as (FA .

. 1PFA 18 . . . 18 lz(m,PD,g) = Jo

a -2d f(l+d) exp( _(2)(2 . 1 1 1 dad(

PD(l + d)r "m -df(1+d) . ( �) + L.., · -l L>j exp - 2 PFAS ]-

Then the FIM is found to be 00 scrm 2m p1-m m

1 == ; �o(l ;Ad) lz(m,Po,g)

1 . 2 [vxe(x)J[vx8(x)]' O"e

which can be simplified to

(103)

(104)

(1 05)

(1 06)

where PFA and d can be calculated from Pd and AVg using (9), (10), and (79).

Reintroducing the subscript i and defining the information reduction factor Q2(PO, AVg ,g) as in (49),


TABLE V Values of Q2(PD, Avg ,g) of Bearings-Only TI-acker With AI for

g = 5 PD OJ) 0.8 0.7

'\vl 0.1 0.8837 0.7701 0.6571l

0.2 0.8718 0.7487 0.6283

0.3 0.8567 0.7260 0.5995

0.4 0.8436 0.7057 0.5745

0.5 0.8294 0.6861 0.5493

we get

The values of Q2(Pn,AVg,g) for different combinations of Pn and A v g are given in Table V

Using the same procedure with some additions, we derive the FIM for narrowband sonar with A I. In this case the FIM is given by

and Q2(PD , A Vg ,g) by

_ 1 00 2m-1/-Lr(m - l) Q2(PJ), AVg,g) - 1 + d L ( 2P )m - l lz(m, PJ) ,g)

m�l g FA (109)

where

(110)

T he numerical values are given in Table VI.

APPENDIX B. DERIVATION OF MOMENTS OF L I KE L I HOOD RATIO U N D ER TARG ET HYPOTHESIS

(H1 ) First we derive the moments for bearings-only

tracking with AI. T he kth moment of the log-likelihood ratio restricted to the validation gate at ti is defined in terms of the 2m-fold integral

E{ ¢i [Z(i) I Hit}

� f J . . . J ¢;[Z(i), X]kp[Z(i) I x] dZ(i) m, =O

(111)

TABLE VI Values of Q2(PD,AVg,g) of Narrowband Sonar With AI for g = 5

PD 0.9 0.8 0.7 AVt 0.1 0.8884 0.7779 0.6720

0.2 0.B815 0.7640 0.6489

0.3 0.8681 0.7467 0.6334

0.4 0.8584 0.7372 0.6148

0.5 0.8565 0.7176 0.5988

where p[Z(i) I x] is the pdf of the measurements under hypothesis Ho defined in (30), but restricted to the validation region. In this ease p[Z(i) I x] and ¢; [Z(i),x] are given by

m; p[Z(i) I xl = (Avs)-m' ( l - PD) IIp6'(aij)/l/ (mi)

i=l

� 1 > ( 1 [f3ii - (h(X)] 2) . . . � -- exp - - P,) . V2ir(1o 2 (10 ) =1

cfJi(Z(i),x) = In [(1 - PD) + P: f � Pij j�l v27fuo

( 1 [f3ii - O;(X)] 2) ] · ap - - . 2 (10

(112)

(113) Applying the same change of variables and using

the same manipulations as in Appendix A, we get

E{ 1Ji[Z(i) I JIlt} = (1 - PD)[ln(l -- PD)te->-vs

00 ( AV ')m e - '\Vg + � gP:A , In!

· lPFA . . · lPFA 19 " .18 . { In [(1 - PD) + fl (1 !��vg

. tajl/C1+d'expc_e/2)] } k )= 1

. [( 1 - PD) + (2 gPFA mc//(l+d) V 7r ( l + d)>"Vg 1

. exp(-d/2)] dad�. (114)


TABLE VII Values of ill Under Hj (or Bearings-Only TIacker With AI for

g = 5

PI> 0.9 0.8 0.7 "'Vg 0 . 1 27.2116 11. 1702 5.9981

0.2 20.9101 8.3350 4.3582

0.3 17.1854 6.6865 3.4210

0.4 14.5529 5.5518 2.7871

0.5 12.5364 1.6893 2.3014

TABLE VIIf Values of 0"1 Under HI for Bearings-Only 1racker With AI for

g = 5 PD 0.9 0.8 0.7

),v. 0 . 1 30.7654 14.0693 8.4459

0.2 24.1291 10.9399 6.4560

0.3 20.3617 9.1020 5.3048

0.4 17.6473 7 .7904 4.4792 0.5 15 .4668 6.7707 3.8388

The numerical values of the first two moments are given in Table VII and Table VIII for a set of FD and ). v g values.

When frequency measurements are also available, the expression for E {q); [Z(i) I Hdk} is

E{ ¢i[Z(i) 1 Hit} k AV N ( 2.\Vg ) m e - )..Vg = (1 - Po)[ln(l - Po)] e - g + "'"" ----Z-p -,-D g FA m . m==l 1PFA

. . · lPFA 19 .. . lg

. {In [(1 - PD) + 2(/� �;;\Vg . �at/(I+d) exp(-e/2)] r

. (1 - P ) + g FA mad/(1+d) [

2p o 2(1 + d).\Vg 1

(1 15)

The numerical values of the first two moments are as follows in Tables IX and X.

APPENDIX C. DERIVATION OF MOMENTS

OF LIKELI HOOD RATIO U N DER NO-TARGET

HYPOTH ESIS (Ho)

When there is no target the likelihood ratio restricted to the validation gate is defined as

TABLE IX Values of PI I Jnder III for Narrowband Sonar With AI for g = 5

Pn 0.9 0.8 0.7

AVg 0.1 34.5938 14.4344 7.8770 0.2 28.1 489 1 1 .5262 6 .1383

0 .3 24.4364 9.8471 5.1733

0.4 21 .7099 8.6007 4.4959

0.5 19 .6084 7.6834 3.9523

TABLE X Values uf 0"1 Under HI for Narrowband Sonar With AI for g = 5

PI> 0.9 0.8 0.7

AUg 0.1 38.4339 17 .7476 10.7593 0.2 3 1.9478 14.6118 8.7604

0.3 27.9389 12.7728 7.5968

0 .4 25.1926 1 1 .4284 6 .7564

0.5 23.2538 10.3894 6.0881

E{¢; [Z(i) I Han

� f 1 · . . 1 ¢; [Z(i), xtp[Z(i) l x] dZ(i). tni �O

(116)

Under no-target hypothesis the pdf of bearings-only measurements with Al is given by

e-)Wg m p[Z(i) 1 Hal = -,- IIp6(aj) (117) m. j=l

and the log likelihood ratio remains the same as in (1 13).

After similar manipulations we obtain the expression for E{¢; [Z(i) I Ho]k} as

E{r/>i[Z(i) I Hot}

. { In [CI - FD) + fI gFFA V '1r (1 + d»,vg

. ta�/(1+d) exp(-�J/2)] }k dad�. J�l

Using Monte Carlo integrations the numerical values for the first two moments were obtained as shown in Tables XI and XII.

(118)


is

TABLE XI Values of 110 Under Ho for Bearings-Only Tracker With AI for

g = 5

Pn 0.9 0.8 0.7 AVi 0 .1 -2.1693 -1.4824 -1 .0818 0.2 -2.0836 -1 .4009 -1 .0057 0 .3 -2.0113 -1 .3326 -0 .9447 0.4 ·1.9439 -1.2717 -0.8896 0.5 -1.8807 -1.2159 -0.8385

TABLE XII

Values of Uo Under Ho for Bearings-Only Tracker With AI for g = 5

PD 0.9 0.8 0.7 AVi, 0 . 1 0.6156 0.5913 0.5690 0.2 0.7319 0.6998 0.6656 0.3 0.8048 0.7662 0.7220 0.4 0.8596 0.8127 0.7596 0.5 0.9076 0.8489 0.7874

For narrowband sonar the corresponding equation

. rPM . . . rPFA r . . . rg 10 10 io io

. { In [(1 - Po) + 2(lg:P;)"vg

. t,a1" ''''' eXP(' (1/2)] }'

do d( (119)

and the numerical values are given in Tables XUI and XlV.

REFERENCES

[1] Bar-Shalom , Y., and Li, X. R. (1993) Estimation and Tracking: Principles, Techniques and Software. Boston, MA: Artech House, 1993.

[2] Bar-Shalom , Y., and Li, X. R. (1995) Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS Publishing, 1995.

[3] Bierson, G. (1990) Optimal Radar Tracking Systems. New York: Wiley, 1990.

[4] Casella , G., and Berger, R. 1.. (1990) Statistical Inference. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books and Software, 1990.

[5] Dahlquist, G., and Bjorck, A. (1974) Numerical Method5. Englewood Cliffs, NJ: Prentice-Hall, 1974.

TABLE XIII Values of {-Io Under Ho for Nal1rowband Sonar With AI for g = 5

PD 0.9 0.8 0.7 AVg 0.1 -2.2120 -1 .5223 -1 .1210 0.2 -2.1570 -1 .4703 -1 .0706 0.3 -2.1078 -1 .4255 ·1 .0303 0.4 -1.0641 -1 .3868 ·0.9945 0.5 -2.0221 - 1.3507 -0.9629

TABLE XIV

Values of Uo Under Ho for Narrowband Sonar With AI for g = 5

PD 0.9 0.8 0.7 AVi 0. 1 0.5115 0.4945 0.4772 0.2 0.6060 0.5840 0.5628 0.3 0.6723 0.6414 0.6121 0.4 0.7240 0.6831 0.6492 0.5 0.7645 0.7119 0.6756

[6] Dommermuth, F. M. (1992) Exploiting sound absorption for target motion analysis, Journal of the Acoustical Society of America, 91, 3 (Mar. 1992), 1545-1551.

[7] Fortmann, T. E., Dar-Shalom , Y., Scheffe, M" and Gelfand, S. (1985)

Detection threshold for tracking in clutter-A connection between estimation and signal processing.

IEEE Transactions in Automatic Control, AC-30 (Mar. 1985), 221-229.

[8] Holtsberg , A. (1992)

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A statistical analysis of bearings-only tracking. Ph.D. dissertation, Lund Institute of 'Technology, Sweden, 1992 .

Howland, P. E. (1995) Passive tracking of airlborne targets using only Doppler and DOA information. lEE Colloq. on Algorithms for Target Tracking Digest, 104, London, (May 1995).

Janffret, c., and Bar-Shalom, Y. (1990) Track formation with bearing and frequency measurements in clutter. IEEE Transactions on Aerospace and Electronic Systems, 26

(Nov. 1990), 999-1010. Lerro, D., and Bar-Shalom, Y. (1993)

Interacting multiple model tracking with target amplitude feature. IEEE Transactions on Aerospace and Electronic Systems, 29

(Apr. 1993), 494-509. [12] Nardone, S. c., Lindgren, A. G., and Gong, K. E (1984)

Fundamental properties and performances of conventional bearing-only target motion analysis. IEEE Transactions on Automatic Control, AC-29 (Sept. 1984), 775-787.

[13] Nielsen, R. O. (1991) Sonar Signal Processing. Boston, MA: Artech House, 1 991 .

[14] Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992)

Numerical Recipes in C: The Art of Scientific Computing. New York: Cambridge University Press, 1992,

[15] Shertukde, H. M" and B ar-Shalom , Y. (1990) Detection and estimation for multiple targets with two omnidirectional sensors in the presence of false alarms. IEEE Transactions on Acoustics, Speech and Signal Processing, 38 (May 1990), 749-763.


1384

Yaakov Bar-Shalom (S'63-M'66-SM'80--F'84) was born on May 11, 1941. He received the B.S. and M.S. degrees from the Technion, Israel Institute of Technology, in 1953 and 1967 and the Ph.D. dcgrcc from Princeton University, Princeton, NJ, in 1970, all in electrical engineering.

From 1970 to 1976 he was with Systems Control, Inc., Palo Alto, CA. Currently he is Professor of Electrical and Systems Engineering at the University of Connecticut, Storrs. His research interests are in estimation theory and stochastic adaptive control. He has been consultant to numerous companies, and originated a series of Multitarget-Multisensor Tracking short courses offered via UCLA Extension, University of Maryland, at Government Laboratories, private companies and overseas. He has also developed the interactive software packages MULTIDAT for automatic track formation and tracking of maneuvering or splitting targets in clutter, PASSDAT for data association from multiple passive sensors, BEARDAT for target localization from bearing and frequency measurements in clutter, FUSEDAT for multisensor tracking and IMDAT for target image tracking using segmentation.

Dr. Bar-Shalom was elected Fellow of IEEE for "contributions to the theory of stochastic systems and of multitargct tracking." Hc coauthorcd the monographs Tracking and Data Association (Academic Press, 1988), Multitarget-Multisensor Tracking: Principles and Techniques (YBS Publishing, 1995), the graduate text Estimation and Tracking: Principles, Techniques and Software (Artech House, 1993) and edited the books Multitarget-Multisensor Tracking: Applications and Advances (Artcch House, Vol. I, 1990; Vol. II, 1992). During 1976 and 1977 he served as Associate Editor of the IEEE Transactions on Automatic Control and from 1978 to 1981 as Associate Editor of Automatica. He was Program Chairman of the 1982 American Control Conference, General Chairman of the 1985 ACC, and Co-Chairman of the 1989 IEEE International Conference on Control and Applications. During 1983-1987 he scrvcd as Chairman of thc Confcrence Activities Board of the IEEE Control Systems Society and during 1987-1989 was a member of the Board of Governors of the IEEE CSS. In 1987 he received the IEEE CSS Distinguished Member Award. Currently he is an IEEE AESS Distinguished Lecturer.

Thiagalingam Kirubarajan (S'95) was born in Sri Lanka in 1969. He received his B.A. degrce in electrical and information enginecring from Cambridge University, England, in 1991.

While in England he worked for the Central Electricity Research Laboratories, Leatherhead, Surrey, as a research assistant. From 1991 to 1993 he was an assistant lecturer in Electrical and Computer Engineering at the University of Peradeniya, Sri Lanka. Sinee 1993 he has been a graduate student/research assistant at the University of Connecticut, Storrs, in pursuit of a Ph.D. His research interest� are in estimation and target tracking.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO. 4 OCTOBER 1996

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Low observable target motion analysis using amplitude information