1612 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL

1612 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011

Joint TDOA and FDOA Estimation: A ConditionalBound and Its Use for Optimally Weighted

LocalizationArie Yeredor, Senior Member, IEEE, and Eyal Angel

Abstract—Modern passive emitter-location systems are oftenbased on joint estimation of the time-difference of arrival (TDOA)and frequency-difference of arrival (FDOA) of an unknown signalat two (or more) sensors. Classical derivation of the associatedCramér-Rao bound (CRB) relies on a stochastic, stationaryGaussian signal-model, leading to a diagonal Fisher informationmatrix with respect to the TDOA and FDOA. This diagonality im-plies that (under asymptotic conditions) the respective estimationerrors are uncorrelated. However, for some specific (nonstationary,non-Gaussian) signals, especially chirp-like signals, these errorscan be strongly correlated. In this work we derive a “conditional”(or a “signal-specific”) CRB, modeling the signal as a determin-istic unknown. Given any particular signal, our CRB reflects thepossible signal-induced correlation between the TDOA and FDOAestimates. In addition to its theoretical value, we show that theresulting CRB can be used for optimal weighting of TDOA-FDOApairs estimated over different signal-intervals, when combinedfor estimating the target location. Substantial improvement in theresulting localization accuracy is shown to be attainable by suchweighting in a simulated operational scenario with some chirp-liketarget signals.

Index Terms— Chirp, conditional bound, confidence ellipse,frequency-difference of arrival (FDOA), passive emitter location,time-difference of arrival (TDOA).

I. INTRODUCTION

P ASSIVE emitter location systems often rely on estimationof time-difference of arrival (TDOA) and/or frequency

difference of arrival (FDOA) of a common target-signal inter-cepted at two (or more) sensors. The target is usually assumedto be fixed (with some exceptions—e.g., [9]), whereas the sen-sors are moving, and their positions and velocities are known(with some exceptions, e.g., [4]). The TDOA and FDOA arethen caused, respectively, by different path-lengths and bydifferent relative velocities between the target and the sensors.

When a narrowband source signal modulates a high-fre-quency carrier, the Doppler effect induces negligibletime-scaling on the narrowband modulating signal, but signif-icantly modifies the intercepted carrier frequency, giving riseto the FDOA (sometimes also termed “differential Doppler”).The TDOA accounts for different relative propagation times(time-shifts) of the received signal. Since the remote receivers

Manuscript received June 21, 2010; revised October 09, 2010; acceptedNovember 29, 2010. Date of publication December 30, 2010; date of currentversion March 09, 2011. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Jean Pierre Deimas. Thematerial in this paper was presented at ICASSP 2010.

The authors are with the School of Electrical Engineering, Tel-Aviv Univer-sity, Tel-Aviv 69978, Israel (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2103069

usually cannot be phase-locked to each other, and since themedium might be wideband-dispersive (causing deviation ofthe carrier’s phase-difference from the anticipated product ofthe TDOA with the carrier-frequency), the resulting phase-dif-ference between the received carriers at the different sensorscannot be directly associated with the TDOA and FDOA,and therefore has to be considered as an additional unknownparameter.

Therefore, after downconversion to baseband, the contin-uous-time observation model is given (for the basic case of twosensors and a single target) by

(1)

where is the (unknown) complex-valued source signal andand are additive, zero-mean, statistically indepen-

dent complex, circular Gaussian noise processes. The “nuisanceparameters” and are the (unknown) absolute relative gainand phase-shift, respectively, of the second channel, and and

are the (unknown) parameters of interest—the FDOA andTDOA, respectively. The received signals are and ,from which the parameters of interest are to be jointly estimated.

Over the past three decades the problem of joint TDOA andFDOA estimation has attracted considerable research interest(e.g., [2], [11], [14], [15], [20]), with renewed interest in recentyears (e.g., [1], [3], [4], [8]–[10], [17], and [18]). Most classicalapproaches model the source signals (as well as the noise) asGaussian stationary random processes [2], [7], [20] (and prob-ably also [14], although this is not stated explicitly in there).However, while the assumption regarding the noise is generallyjustified, in some applications the assumption that the sourcesignal is Gaussian and/or wide-sense stationary (WSS) may bestrongly violated. Moreover, bounds derived under an assump-tion of a stochastic source signal are associated with the “av-erage” performance, averaged not only over noise realizations,but also over different source signal realizations, all drawn fromthe same statistical model.

It might be of greater interest to obtain a “signal-specific”bound, namely: for a given realization of the source signal, topredict the attainable performance when averaged only over dif-ferent realizations of the noise. Such a bound can relate more ac-curately to the specific structure of the specific signal. The dif-ference between the two approaches is particularly significant inpredicting how the specific signal’s structure induces inevitablecorrelation between the resulting TDOA and FDOA estimates.Under the Gaussian stationary signal model, the Cramér-Raobound (CRB) for these estimates predicts (asymptotically) zero

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YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION 1613

correlation [20]. However, for some nonstationary signals, e.g.with a chirp-like structure, it is evident that FDOA and TDOAmay be used interchangeably to “explain” the differences be-tween the received signals: Consider, for example, a chirp signalwith a linearly-increasing frequency. Neglecting end-effects, adelayed version of that signal would look the same as a neg-ative Doppler-offset version of the same signal, since at eachtime-instant the instantaneous frequency of the delayed signalis smaller (by the same absolute offset) than that of the originalsignal. Thus, a time-delay and a frequency-offset cannot be welldistinguished from each other, and their estimation errors mustbe correlated.

Nevertheless, the deterministic-signal model has not seenas much treatment in the literature. A maximum likelihood(ML) estimate for this model was derived by Stein in [15],but without derivation of the bound. More recently, Fowlerand Hu [1] were the first to offer a new perspective on theCRB for joint TDOA/FDOA estimation, by considering thedeterministic-signal model, elucidating the essentially different(non-diagonal) structure of the resulting bound. However, theirderivation implicitly assumes that the source signal, as well asthe phase-shift , are known. This assumption considerablysimplified the exposition in [1], but is rarely realistic in a passivescenario, and consequently (as we shall show), the resultingbound in [1] is too optimistic (loose) in the fully passive case.

Thus, in this paper we derive the CRB for the deterministicsignal case, regarding the signal, as well as the relative gain andphase-shift, as additional nuisance parameters. We show that theassociated Fisher information matrix (FIM) can be reduced insize to accurately reflect the bound on , , and alone. Wecompare our bound to Wax’ stochastic bound [20], as well as toFowler and Hu’s bound [1].

Besides the theoretical value, an important practical impli-cation of knowing the correct bound (namely, the asymptotic1

covariance of the ML estimate) is the ability to use this covari-ance for properly weighting the estimated TDOAs and FDOAswhen calculating the resulting estimated target location. As weshall show, the use of the correct bound (covariance) for suchweighting can attain a significant improvement in the resultingtarget localization accuracy, relative to the use of uniformweighting or of weights derived from the classical bounds. Inaddition, the resulting estimates of the confidence ellipse (forthe estimated target location) become much more reliable whenthe correct bound expressions are used.

In a related conference-paper [19], the basic expressions forthe resulting bound were presented. In the current paper we sub-stantiate these expressions with explicit derivations and com-prehensive discussions. In addition, we offer a detailed exampleillustrating the attainable improvement in geolocation accuracyand reliability by use of the resulting bound-matrices for (near-)optimal weighting in the target location estimation.

The paper is structured as follows. In the following sectionwe define the discrete-time model and derive our signal-spe-cific CRB. In Section III we compare our CRB to some ex-isting bounds (developed under different model-assumptions),both analytically and by simulation. In Section IV we address

1In the context of our deterministic signal assumption, asymptotic efficiencyof the ML estimate is obtained in the sense of asymptotically high SNR [13],but generally not in the sense of asymptotically long observation interval [16].

the issue of target-localization based on several estimates ofTDOA-FDOA pairs taken over different intervals, comparingdifferent weighting schemes (based on different CRB models)in simulation. The discussion is concluded in Section V.

II. THE DISCRETE-TIME MODEL AND THE CRB

We assume that the received signals (1) are sampled at theNyquist rate to yield their discrete-time versions. To simplifynotations, we shall assume unit sample rate (implying that thecontinuous-time signals are bandlimited between 1/2 and 1/2).Any different sampling-rate (and bandwidth) merely imply ex-pansion or compression of the timeline, which in turn impliesrescaling of and of , as well as of their estimates, and henceof their estimation errors and associated bounds. We thus obtainthe discrete-time sampled versions (with , etc.,and with )

(2)

where denotes the sampled time-shiftedsource signal. Defining the respective vectors

(3)

we observe that the relation between and can be approx-imated using the discrete Fourier transform (DFT), as

, where denotes the unitary DFT matrix (thesuperscript denoting the conjugate transpose), and where

is a diagonal matrix, such that

(4)

with

(5)

This approximation essentially replaces the linear time-shiftwith a circular time-shift (such that the part shifted-out fromone edge of the interval is shifted-in from the other edge). How-ever, under the common assumption that the shifts (namely theTDOAs) are very small with respect to (w.r.t.) the observationlength ( ), the approximation error becomes negligible.Moreover, in some cases the signal mostly exists in themiddle of the observation interval, and tapers to zero towardsthe edges—which can serve as further (partial) justification forthis approximation.

Given this relation, we may express our discrete-time model(2) in vector form as follows:

(6)

where .


We are now interested in the CRB for estimation of all un-known (but deterministic) parameters of the problem. To thisend, we begin by defining the vector of all real-valued pa-rameters, composed of the real-part of , the imaginary-partof , and the “actual” parameters , namely

is a vector.Recalling that is a deterministic (unknown) vector, whereas

are independent Complex Circular Gaussian vectors, weobserve that the concatenated vector is also aComplex Circular Gaussian vector, with meanand covariance

(7)

where and denote the covariance matrices ofand , respectively, and where . Notethat is always a unitary matrix:(the identity matrix).

To simplify the derivation, we shall employ the common as-sumption that the noise processes are white, with variancesand , respectively, namely and .For further simplification of the exposition, we shall regard thenoise variances as known parameters, thus excluded from . Weshall comment on the more realistic case, in which these vari-ances are unknown, at the end of this section.

The FIM for estimation of a real-valued parameters-vectorfrom the Complex Circular Gaussian vector , when only the

mean depends on , is given by (see, e.g., [6, p. 525])

(8)

Forming the derivative of w.r.t. we get the(Jacobian) matrix

(9)

where denotes the derivative of w.r.t. (to simplify thenotations, we omit the explicit dependence of on and

, denoting this matrix simply as ; Likewise, we shall omit theexplicit dependence of on , denoting this matrix simplyas ).

Consequently (exploiting the unitarity of ), we have

(10)

where . Taking twice the real part, and denoting, we obtain the FIM, structured as

(11)Fortunately, we are only interested in the bound on the “ac-

tual” parameters , given by the lower-right 4 4 block of the

inverse of . This block is particularly convenient to obtainfrom a matrix of this form, by exploiting the four-blocks matrixinversion expression and taking advantage of the diagonality ofthe upper-left block. More specifically, we identifythe respective implied FIM (the inverse of the bound on )as the Schur complement (e.g., [5, p. 472]) of the lower-right 4

4 block, namely

(12)

where we have used the relationsand .

We now need to obtain explicit expressions for . To this end,let us define the matrix , and note that

(13)

Consequently, since , we have

(14)

We, therefore, obtain as

(15)

where for shorthand we have used

(16)

(the delayed and frequency-shifted version of ) and

(17)

(which can be loosely termed the “derivative” of up to a factorof , since for a continuous-time signal, multiplication by inthe frequency-domain is equivalent to differentiation in time-domain).

We now turn to obtain the elements of . For the first rowof this matrix, we observe the elements

(18)

Evidently, the last three are imaginary-valued, and would there-fore vanish in when the real-part is taken [in (12)]. This im-plies, as could be expected, that is block-diagonal, such thatthe upper-left element, which accounts for the estimation of ,is decoupled from the other three . In other words, sincewe are not interested in , we may concentrate on the lower-right


3 3 block of , which we shall simply denote . The relevantelements (in this block) are easily obtained from (15)

(19)

where we have used the unitarity of in obtainingand . We observe that all of these terms except for

are real-valued (regarding , note that, which is real-valued). There-

fore, substituting in (12) (for the lower-right 3 3 block only)we obtain

(20)

The CRB on unbiased estimation of is .Naturally, such a bound (often also termed a “conditional”

CRB) depends on the specific (unknown) signal , as well ason the unknown “actual” parameters . However, under goodsignal-to-noise ratio (SNR) conditions, reliable estimates of thesignal and of the parameters may be substituted into this ex-pression to obtain a reliable estimate of the estimation-errors’covariance. Moreover, as we shall show in the sequel, this in-formation can be used for proper (near optimal) weighting ofestimated TDOA-FDOA pairs obtained from different segmentswhen used for estimating the target location.

It may be interesting to note, that although the bound dependson the true TDOA , it is in fact independent of the FDOA . Toobserve this, note that the only terms in (20) which may containdependence on the TDOA or FDOA are those involving (andalso , which is ), through the full expression for

. But observe that in all of these terms the resultinginvolvement of is through an expression of the form(in elements , , and ) or (in ). Now

(21)

(exploiting the diagonality of both and in the second tran-sition and the unitarity of in the third), which only dependson . A similar elimination can be applied to , whichcompletes the proof.

Before concluding this section, we briefly address the casein which the noises’ variances and are unknown. In thatcase, the -long parameters vector should be supple-mented with these two parameters, forming avector. Note, however, that since the dependence of the prob-ability distribution of on these variances is only through the

covariance , which does not depend on the other parameters,it is straightforward to show (using the complete Slepian-Bangexpression for the FIM, e.g., [6, p. 525]) that the resulting FIMwould be block-diagonal, with the block accounting for thesevariances decoupled from the block(s) accounting for the otherparameters. This means that the CRB on the estimation of theparameters of interest is indifferent to the knowledge (or lackof knowledge) of the noises’ variances. Of course, this does notimply that the same accuracy can be obtained in both cases ingeneral; However, it does imply that asymptotically (in the SNR[13]) the same bound can be attained in both cases.

III. COMPARISON TO OTHER BOUNDS

The CRB for was derived by Wax in [20] under the as-sumption that the source signal is zero-mean, WSS Cir-cular Gaussian, with a known power-spectrum . With ourwhite-noise assumption and under the assumption of high SNR,the respective FIM in [20], denoted , can be expressed by

(22)

where the second equality follows from our unit sample-rateassumption, such that is the power-spectrum of thediscrete-time sampled signal , and

(23)

(under the assumption that is bandlimited at ,namely that for all ).

In Appendix I, we show a rather reassuring relation betweenand : under these model assumptions (and asymptotically,

for sufficiently long observation intervals) equals the meanof , taken with respect to different realizations of . We em-phasize, however, that this averaging effect is exactly what weaimed at eliminating in developing our signal-specific bound:per any given realization of , might be significantly differentfrom , especially when is not a realization of a sta-tionary process. The bound obtained from is more informativeregarding the intrinsic properties of the observed signal, aver-aging only over different realization of the noise (as opposedto the bound obtained from , which reflects averaging overboth signal and noise realizations).

The main significant difference between and is theblock-diagonality of , which decouples (namely, impliesdecorrelation of) the estimation errors of and . While suchdecoupling truly occurs when is WSS Gaussian, it maybe strongly violated in practice, e.g., when is a chirp-likesignal. Therefore, the resulting (WSS-based) bound in suchcases may be significantly different from our signal-specificbound, as demonstrated here.


The conditional CRB developed by Fowler and Hu in [1] im-plicitly assumes prior exact knowledge (to the estimator) of thesource signal , as well as of and . In practical situationsof passive FDOA and TDOA estimation, these assumptions areusually not very realistic. Although the assumption that isknown does not affect the resulting bound, the assumption that

and are known bears a significant effect. Indeed, the FIM(with respect to only) in [1], which we denote ,is given (when put in terms of our conventions and notations) by

(24)This expression is similar, up to scale, to the lower-right 2 2

block of . The scaling difference is a consequence of not takingthe Schur complement of the respective block in [see (12)],due to the assumption that is known. A more significant differ-ence is the absence of the -related row and column of due tothe assumption that is known. As evident from the expressionfor , the lower-right 2 2 block may be strongly coupled withthe upper-left element. By ignoring this coupling, the resultingbound matrix becomes significantly smaller (has smaller eigen-values) and differently oriented (has different eigenvectors).

In Fig. 1 we demonstrate the fit (or misfit) between the em-pirical distribution of the ML estimates of and , and the90%-confidence ellipses (in the plane) computed from thethree bounds: our signal-specific ; Wax’ ; andFowler and Hu’s . The experiments were run for twoessentially different specific signals, each of length(shown on top in the figure).

• The first signal (representing a “stationary” case) was asample-function of an auto-regressive moving-average(ARMA) process, generated by filtering a sequence ofwhite Gaussian noise with the filter

(25)

• The second signal (representing a “chirp-like” case) was aGaussian-shaped chirp pulse, generated as

(26)

with , and .Only the additive white Gaussian noise signals were redrawn(independently) at each trial, with noise levels

. We ran 400 independent trials, and each dot in the scatter-plots represents the ML-estimation errors of and in onetrial. The ML estimates were obtained using a two-stages finegrid search over and , assuming the deterministic (unknown)signal model with unknown and and additive white Gaussiannoise—as proposed, e.g., in [15]. The true values of andwere and .

As could be expected, our fits the data far better thanthe other two for the chirp-like signal, and is comparable to the

for the stationary Gaussian signal. Note thatis considerably more “optimistic” (loose) than the other two(in both cases), due to the inherent assumption that the target-signal and the phase-difference between the received signals areknown to the estimator.

IV. COMBINING TDOAS AND FDOAS FOR TARGET

LOCALIZATION

Besides the theoretical characterization of the correlation be-tween the TDOA and FDOA estimates, our can pro-vide considerable improvement in the accuracy of TDOA- andFDOA-based target localization, by prescribing the proper (oreven optimal) weighting. When estimated TDOA-FDOA cou-ples (taken from different intervals of intercepted signals) arecombined to form an estimate of the target location, the intro-duction of proper weighting into the process can result in signif-icant improvement in the resulting accuracy. In addition, morereliable confidence-ellipses for the estimated location can be ob-tained if the true error distributions (covariance matrices) of theintermediate measurements are used.

To be more specific, let us derive the weighted estimate oftarget location based on TDOA-FDOA couples. Assume thatthe target signal is intercepted in different time-intervals (seg-ments, or pulses) and that the TDOA and FDOA are estimated ineach time-interval (using ML estimation). We denote by

and the positions of the firstand second sensors in the th time-interval (for simplicity weassume a two-dimensional model in this context—see Fig. 2 forreference). Likewise, we denote by and their respec-tive velocity vectors.

Let denote a (static) target position. The distance-vectorsbetween the target and the two sensors in the th time-intervalare thus denoted and ,whereas the respective (scalar) distances are denoted

(27)

The true TDOA for the th time-interval is therefore given by

(28)

where denotes the propagation speed.2 Denoting by the car-rier frequency of the target-signal, the Doppler-induced FDOAfor the th interval is given by

(29)

where

(30)

are unit-vectors in the directions pointing from the respectivesensors to the target.

2Approximately 299,792,458 m/S for electromagnetic signals in free space.


Fig. 1. Empirical distribution (400 trials) of ML estimates of � and � with 90%-confidence ellipses derived from the three bounds.

Fig. 2. A general scene: target ‘o’, sensors ‘�’. Both sensors in this scene aremoving at a constant speed.

Given estimates of the TDOA-FDOA pair ineach interval, an estimate of the target position can be obtainedby minimizing the least-squares (LS) criterion

(31)

with respect to , and taking the minimizing value as the LSestimate. However, this unweighted version of the LS criterionobviously entails an inherent flaw, because it combines termswith different units ( and ). A possible remedy is

to normalize each term by a respective variance (with the sameunits), thereby combining unit-less terms

(32)

where and are some prescribed variances, andcan be regarded as a weight-matrix. We term

this the “uniformly-weighted LS” (UWLS) criterion.Obviously, if the estimation variances over different time-in-

tervals are different (and known), then the constant weight-ma-trix can be substituted with interval-dependent weights

, reflecting different weighting of results taken fromdifferent time-intervals. We term the resulting criterion a“weighted LS” (WLS) criterion and the minimizing —theWLS estimate.

Assuming statistical independence of estimates ofTDOA-FDOA pairs taken from different intervals, theoptimal3 weight matrices (under a small-errors assumption)for the WLS criterion are well-known to be the inverse covari-ance matrices of these . Moreover, under asymptoticconditions (sufficiently high SNR in our case), the ML esti-mation error in each interval can be considered aGaussian zero-mean random vector, with covariance given bythe interval’s CRB. Under these conditions, taking the inverseCRB for each interval as the respective weight-matrixwould yield the ML estimate of the target location with

3In the sense of minimum mean square errors in the resulting location esti-mates.


Fig. 3. The operational scene: target ‘o’, sensors ‘�’. Sensor 1 is static, sensor2 is traveling at �� , signal intercepted every 10 s.

respect to the estimated TDOAs and FDOAs . Under asmall-errors assumption, the resulting WLS estimate wouldthus attain the CRB for estimation of the target location fromthe TDOA-FDOA estimates .

Furthermore, as a by-product of using the measurements’ co-variance for weighting, we can obtain (under a small-errors as-sumption) the covariance of the resulting location estimate , asfollows. Define

(33)

The 2 2 derivative matrix (Jacobian) of with respect tocan then be easily shown to be given by

(34)

where

(35)are projection matrices onto the directions perpendicular to

and to , respectively (here denotes the 22 identity matrix). Then, denoting the covariance matrix of

as , if the weight-matrices used for the WLS criterionare chosen as , then the resultingcovariance matrix of the WLS location-estimate is given by

(36)

In the following experiments we explore the differences in theobtained localization accuracies and in the reliability of the re-spective confidence-ellipses, resulting from the use of four dif-ferent weighting schemes.

1) A uniformly weighted scheme, assuming that the variancesof the TDOA and FDOA estimates are constant over allintervals, and prescribed by the mean (over all intervals)of the WSS-case bound, ;

2) A weighted scheme using as the TDOA-FDOAcovariance for each interval;

3) A weighted scheme using as the TDOA-FDOAcovariance for each interval;

4) A weighted scheme using our as the TDOA-FDOAcovariance for each interval.

Our operational scenario is the following (see Fig. 3):• A target is positioned at (marked ‘o’ in the

Figure);• A static sensor (marked ‘ ’) is positioned at

, with a velocity vector.

• A mobile sensor (marked ‘ ’-s) is initially positioned atand travels with a constant velocity

vector .A 2–ms-long segment of the transmitted target signal is inter-cepted by the two sensors every 10[s] over a period of oneminute (namely, segments are intercepted), and the po-sition of the second sensor when intercepting the th segment is

(37)The target signal’s carrier frequency is .The sampling frequency (after conversion to baseband) is

, and therefore the number of samples in each 2ms interval is .

As before, we considered two types of target-signals:• In the first scenario a different realization of a segment of

a WSS process is generated at each time-interval. The seg-ment is generated (directly in its baseband sampled ver-sion), by filtering a white Gaussian sequence with the samefilter as in (25);

• In the second scenario a different chirp-like pulse[cf. (26), with randomized and parameters]is generated (directly in its baseband sampled ver-sion) at each time-interval. The values of andof are drawn independently and uniformly in theranges and (respec-tively), where and

.The generated signals are time- and frequency-shifted accordingto the geometrical model (and the sampling-rate), and then inde-pendent white Circular Gaussian noise signals andwith variances are added, forming the re-ceived signals and , from which the ML estimate ofthe TDOA and FDOA is obtained (using a fine grid-search over

and in our model—see, e.g., [15]) for each interval. The esti-mated target location is obtained from the UWLS or WLS solu-tion (using an iterative Gauss-Newton method—see, e.g., [12]).integrating the TDOA and FDOA estimates from all intervals.


Fig. 4. The case of stationary target-signals: Empirical distribution (400 trials) of differently weighted LS estimates of the target-location. Superimposed are90%-confidence ellipses computed from the inferred estimation covariance (36).

For obtaining the different weight matrices, the respectiveCRB matrices were computed based only on the observed data,and not on any other prior knowledge regarding the underlyingsignal models—except for the noise variances and , whichwere assumed known. The ML estimate of and immedi-ately yields, as a byproduct, ML estimates of the unknownand parameters, as well as of the noiseless target-signal .All estimated parameters (and signal) were thus plugged intothe expressions for (20) and for (24) for computing thesignal-dependent and (resp.). For computing

we used the empirical periodogram of the estimatedin lieu of the unknown power-spectrum (note

that, strictly speaking, in the nonstationary case of a chirp-likesignal, such a power-spectrum does not exist; nevertheless, theperiodogram provides a reasonable tool for computing the re-sulting bound and weight matrices for the purpose of compar-ison to the other bounds). All the bound expressions were nor-malized by accounting for the nonstandard sampling-rate

in this example.In Fig. 4 we present the results for the stationary signals sce-

nario, whereas in Fig. 5 we present results for the chirp-like sig-nals scenario (both showing the four weighting-schemes in foursubplots).

In each trial the same target signals were generated, andonly the additive noise signals were redrawn (independently).The resulting estimated target location (which also representsthe estimation error, since the target is located at )for each trial appears as a dot in the respective figure. Super-imposed on the resulting scatter-plots are confidence-el-

lipse, calculated from the respective inferred localization-errorcovariance (36) (averaged over all trials). Note that this covari-ance expression is based on the underlying assumption that thetrue covariance matrices of the TDOA-FDOA estimates are de-scribed by the respective CRB matrices—so, evidently, when-ever this assumption is breached, the resulting ellipse would notfit the empirical distribution.

The results for the stationary case show that (as couldbe expected) in this case the uniformly-weighted, the

-weighted and the -weighted results have a gen-erally similar distribution, and the respective confidence ellipseprovide a close fit. The distribution of the -weightedresults is also quite similar to the other distributions. However,since (as we have already seen) the bound under-estimates the covariances of the intermediate TDOA-FDOAestimates, the resulting -based confidence ellipse is toosmall.

The significant advantage of using for weighting isrevealed in the scenario of different chirp-like signals. It is im-portant to realize, that optimal weighting is not only about at-tributing weak weights to the “poor” (high variance) TDOA-FDOA estimates and strong weights to the “good” (small vari-ance) estimates. The more subtle geometrical interpretation ofthe weighting of two-dimensional (TDOA-FDOA) data makesa far more significant difference in this context: the optimalweighting is able to “capture” and exploit the directions of highand low variances in the TDOA-FDOA plane. For example,for a chirp-like pulse with an increasing frequency, the TDOAand FDOA estimation errors would be strongly positively-cor-


Fig. 5. The case of chirp-like target-signals: Empirical distribution (400 trials) of differently-weighted LS estimates of the target-location. Superimposed are90%-confidence ellipses computed from the inferred estimation covariance (36).

related. Conversely, for a chirp-like pulse with a decreasing fre-quency, they would be strongly negatively-correlated. There-fore, although the variances of the two TDOA (or FDOA) es-timates taken from two such intervals might be similar, propergeometrical weighting (as provided by the two 2 2 weight ma-trices) would account for (and exploit) the different directions ofsmaller variance: perpendicular to the different common-modedirections in both cases. The resulting weighted estimate wouldthen enjoy the benefit of both, realized by the smaller variancesalong both of these directions.

Indeed, in this case the -based weight matrices cor-rectly identify (and, therefore, provide correct weighting for) thestrong correlations, resulting from the chirp-like structure, be-tween the TDOA and FDOA estimation errors in each segment.On the other hand, the UWLS and -based weighting areunaware of this correlation, and are therefore unable to applysimilar improvement to the resulting localization. isgenerally aware of the possible correlation between TDOA andFDOA estimates, but since it is implicitly based on the assump-tion that the source signal and phase-difference are known, thededuced covariance strengths and directions (eigenvalues andeigenvectors) are inaccurate for our scenario, and the resultingweights are suboptimal.

We note in addition, that the deduced localization covariances(leading to the confidence-ellipses) are too optimistic in thiscase, not only for the -based weighting, but also for theUWLS and -based weighting. Only our providesa reliable confidence ellipse for this type of signal.

As mentioned above, for the purpose of computing thebounds to be used for weighting, we use the estimated, ratherthan the true parameters (including the estimated target-signal).When the SNR is not sufficiently high, the estimated targetsignal might contain some significant residual noise, whichwould generally make the signal appear wider (in frequency-do-main) if the noise is white. This residual noise might induceerrors on the computed bounds, which might in turn causesome degradation in the weighting scheme.

In Figs. 6 and 7 we present the localization performance of thefour weighting schemes (for both signal types) versus the SNR,for SNR values varying from 20 to 40 dB. Although these SNRvalues are usually considered high, the time-bandwidth productof the very short (2 ms) target signal pulses in our specific sce-nario is relatively low, and has to be compensated for by goodSNR. We present the accuracy in terms of the long and short axeslengths of the 90% confidence-ellipses derived from the empir-ical covariance matrix (estimated over 400 independent trials)of the localization errors obtained with each weighting scheme(not to be confused with the confidence ellipses derived fromthe analytically predicted localization-error covariance matrices(36)). For these figures we slightly varied the operational sce-nario by shortening the time-intervals between pulses from 10to 5 s, thereby obtaining , rather than inter-mediate results (otherwise the ellipses for the lower SNRs forthe chirp-like signals become significantly larger than the dis-tances from the target to the sensors, thereby severely violatingthe “small-errors” assumption).


Fig. 6. Long and Short axes lengths of the 90% confidence-ellipses computedfrom the empirical localization-error covariance for each of the four weightingschemes (for the case of stationary target-signals).

Fig. 7. Long and Short axes lengths of the 90% confidence-ellipses computedfrom the empirical localization-error covariance for each of the four weightingschemes (for the case of chirp-like target-signals).

As evident in the figures, in the case of stationary target-sig-nals (Fig. 6) there are no essential differences in performancebetween the four weighting schemes, since our is alsonearly diagonal in such cases. However, in the case of chirp-liketarget-signals (Fig. 7), the overwhelming relative improvement(more than fivefold in each axis) in using the signal-specificbound for weighting at the high SNRs is seen to become some-what less pronounced (down to about 25% improvement) at thelower SNRs—due to the sensitivity of the computed tothe presence of residual noise in the estimated signal. Note, how-ever, that, as can be easily deduced from the structure of in

(20), the correlation coefficient between the estimation errors ofand in our does not depend on SNR. Therefore, the

weaker relative improvement at the lower SNRs is to be blamedexclusively on the errors in estimating the signal and parametersfor substitution into the expression.

V. CONCLUSION

We considered the problem of joint TDOA-FDOA estima-tion for passive emitter location. Assuming a deterministic (un-known), rather than a stochastic signal model, we obtained anexpression of the signal-specific (“conditional”) CRB for thisproblem. The most prominent feature of this bound, as opposedto the classical bound (for stochastic, stationary Gaussian target-signals), is the existence of possibly significant nonzero off-di-agonal terms—especially when the target signal has a chirp-likestructure over the observation interval.

Such off-diagonal terms reflect nonzero correlation betweenthe TDOA and FDOA estimation errors. We proposed using thecomputedsignal-specificCRBforweightingTDOA-FDOAmea-surements taken over different time-intervals in the estimation ofthe target location. Accounting for the TDOA-FDOA correlationin the weighting enables to take advantage of the diversities inchirp structures (increasing/decreasing frequencies) between in-tervals, so as to attain significant improvement in the localizationaccuracy. In our simulation examples with chirp-like signals, theuse of proper weighting reduced the scatter area of the localiza-tion results by a factor of 25 under good SNR conditions.

APPENDIX ITHE RELATION BETWEEN AND

The FIM developed in [20] for the case where the sam-pled target signal is a WSS Gaussian process with a knownpower-spectrum can be expressed, under the assump-tion of white noise and high SNR, as given in (22) and repeatedhere for convenience

(38)Our signal-specific FIM (20) is repeated here as well

(39)

We shall now show that if the observed segment is a re-alization a WSS process with power-spectrum , thenasymptotically (in the observation length ) .

We denote the discrete-time Fourier transform (DTFT) ofover the observed interval as

(40)

and we shall use

(41)


For shorthand, we shall implicitly assume asymptotic conditions(namely, that is “sufficiently long”), by regarding (41) as anequality without the lim operator.

We proceed by showing that for each element ofthis matrix (ignoring the leading term , whichis common to both). Beginning with the (1,1) element

(42)

where we have used Parseval’s identity for the second transition.Turning to the (1,2) element

(43)

where we have used Plancherel’s identity and the definition of(17) for the second transition. Similarly, for the (2,2) element

(44)

Before turning to the other elements, recall thatis a delayed and doppler-shifted version of , and

consequently

(45)

Thus, turning to the (3,3) element we have

(46)

where the approximation of the sum over as is validfor large values of (asymptotic conditions). Likewise, for the(1,3) element we get

(47)

Note that due to the assumption that is even, the sum does notexactly vanish, but equals ; However, since the (1,1) ele-ment is proportional to and the (3,3) element is proportionalto , the (1,3) element is much smaller than the square rootof their product (proportional to ), and therefore its effect onthe FIM is negligible (asymptotically).

The remaining term is the (2,3) element. Note first that

(48)

where we have eliminated and from due tothe diagonality of and and the unitarity of . We, there-fore, have, for the (2,3) element

(49)

where denotes the correlation matrix of therandom vector , and denotes the trace operator. Underthe assumption that (and therefore also ) is a WSSprocess, takes a Toeplitz form, and is therefore (asymptoti-cally) diagonalized by the Fourier matrix, namely

(50)

is a diagonal matrix. We, therefore, have

(51)

where in the last transition we eliminated and from theproduct due to their diagonality and unitarity and due to thediagonality of and . Moreover, due to the diagonality of

, the matrix is a Toeplitz matrix, and thereforethe last trace expression in (51) is simply the sum over (from

to ), multiplied by a constant value (the valueon the main diagonal of the Toeplitz matrix ). Sincethe sum vanishes, so does the entire term (again, we note that,strictly speaking, since is even the sum over does not ex-actly vanish, but the residual term is negligible in the FIM).

Exploiting the symmetry of both and for all of theremaining elements, the proof is complete.


REFERENCES

[1] M. L. Fowler and X. Hu, “Signal models for TDOA/FDOA estimation,”IEEE Trans. Aerosp. Elect. Sys., vol. 44, pp. 1543–1549, Oct. 2008.

[2] B. Friedlander, “On the Cramér-Rao bound for time-delay and Dopplerestimation,” IEEE Trans. Inf. Theory, vol. IT-30, pp. 575–580, May1984.

[3] K. C. Ho and Y. T. Chan, “Geolocation of a known altitude objectfrom TDOA and FDOA measurements,” IEEE Trans. Aerosp. Electron.Syst., vol. 33, no. 3, pp. 770–783, Jul. 1997.

[4] K. C. Ho, X. Lu, and L. Kovavisaruch, “Source localization usingTDOA and FDOA measurements in the presence of receiver locationerrors: Analysis and solution,” IEEE Trans. Signal Process., vol. 55,no. 2, pp. 684–696, Feb. 2007.

[5] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:Cambridge Univ. Press, 1985.

[6] S. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[7] C. Knapp and G. Carter, “The generalized correlation method for es-timation of time delay,” IEEE Trans. Acoust., Speech, Signal Process.,vol. ASSP-44, pp. 320–327, Aug. 1976.

[8] J. Luo, E. Walker, P. Bhattacharya, and X. Chen, “A newTDOA/FDOA-based recursive geolocation algorithm,” in Proc.Southeast. Symp. Syst. Theory (SSST), 2010, pp. 208–212.

[9] D. Musicki, R. Kaune, and W. Koch, “Mobile emitter geolocation andtracking using TDOA and FDOA measurements,” IEEE Trans. SignalProcess., vol. 58, no. 3, pp. 1863–1874, Mar. 2010.

[10] R. Ren, M. L. Fowler, and N. E. Wu, “Finding optimal trajectory pointsfor TDOA/FDOA geo-location sensors,” in Proc. Inf. Sci. Syst. (CISS),2009, pp. 817–822.

[11] D. C. Shin and C. L. Nikias, “Complex ambiguity functions usingnonstationary higher order cumulant estimates,” IEEE Trans. SignalProcess., vol. 43, no. 11, pp. 2649–2664, Nov. 1995.

[12] H. W. Sorenson, Parameter Estimation. New York: Dekker, 1980.[13] A. Renaux, P. Forster, E. Chaumette, and P. Larzabal, “On the high

SNR conditional maximum likelihood estimator full statistical charac-terization,” IEEE Trans. Signal Process., vol. 54, pp. 4840–4843, Dec.2006.

[14] S. Stein, “Algorithms for ambiguity function processing,” IEEE Trans.Acoust., Speech, Signal Process., vol. ASSP-29, pp. 588–599, Jun.1981.

[15] S. Stein, “Differential delay/Doppler ML estimation with unknown sig-nals,” IEEE Trans. Signal Process, vol. 41, pp. 2717–2719, Aug. 1993.

[16] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and theCramér-Rao bound,” IEEE Trans. Acoust., Speech, Signal Process.,vol. 38, pp. 2140–2150, Dec. 1990.

[17] R. Ulman and E. Geraniotis, “Wideband TDOA/FDOA processingusing summation of short-time CAF’s,” IEEE Trans. Signal Process.,vol. 47, no. 12, pp. 3193–3200, Dec. 1999.

[18] G. Yao, Z. Liu, and Y. Xu, “TDOA/FDOA joint estimation in a cor-related noise environment,” in Proc., Microw., Antenna, Propag. EMCTechnol. Wireless Commun. (MAPE), 2005, pp. 831–834.

[19] A. Yeredor, “A signal-specific bound for joint TDOA and FDOA esti-mation and its use in combining multiple segments,” in Proc. ICASSP,2010.

[20] M. Wax, “The joint estimation of differential delay, Doppler andphase,” IEEE Trans. Inf. Theory, vol. IT-28, pp. 817–820, Sep. 1982.

Arie Yeredor (M’98-SM’02) received the B.Sc.(summa cum laude) and Ph.D. degrees in elec-trical engineering from Tel-Aviv University (TAU),Tel-Aviv, Israel, in 1984 and 1997, respectively.

He is currently with the School of Electrical Engi-neering, Department of Electrical Engineering-Sys-tems, TAU, where his research and teaching areas arein statistical and digital signal processing. He alsoholds a consulting position with NICE Systems, Inc.,Ra’anana, Israel, in the fields of speech and audioprocessing, video processing, and emitter location al-

gorithms.Dr. Yeredor previously served as an Associate Editor for the IEEE SIGNAL

PROCESSING LETTERS and the IEEE TRANSACTIONS ON CIRCUITS AND

SYSTEMS-PART II, and he is currently an Associate Editor for the IEEETRANSACTIONS ON SIGNAL PROCESSING. He served as Technical Co-Chair ofThe Third International Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009), and will serve as General Co-Chairof the 10th International Conference on Latent Variable Analysis and SourceSeparation (LVA/ICA 2012). He has been awarded the yearly Best Lecturerof the Faculty of Engineering Award (at TAU) six times. He is a member ofthe IEEE Signal Processing Society’s Signal Processing Theory and Methods(SPTM) Technical Committee.

Eyal Angel received the B.Sc. degree in physics fromTel-Aviv University (TAU), Tel-Aviv, Israel, in 2001.

He is currently pursuing the M.Sc. degree in elec-trical engineering, mainly working on passive time-delay estimation in multipath environments. His mainareas of interest are in statistical signal processing.He also holds the position of System Engineer withthe Process Diagnostics and Control (PDC), AppliedMaterials Ltd., Rechovot, Israel.

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