Joint TDOA and FDOA Estimation: A Conditional Bound and Its Use for Optimally Weighted Localization Arie Yeredor, Senior Member, IEEE, and Eyal Angel

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1 1612 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011 Joint TDOA and FDOA Estimation: A Conditional Bound and Its Use for Optimally Weighted Localization Arie Yeredor, Senior Member, IEEE, and Eyal Angel Abstract Modern passive emitter-location systems are often based on joint estimation of the time-difference of arrival (TDOA) and frequency-difference of arrival (FDOA) of an unknown signal at two (or more) sensors. Classical derivation of the associated Cramér-Rao bound (CRB) relies on a stochastic, stationary Gaussian signal-model, leading to a diagonal Fisher information matrix with respect to the TDOA and FDOA. This diagonality implies that (under asymptotic conditions) the respective estimation errors are uncorrelated. However, for some specific (nonstationary, non-gaussian) signals, especially chirp-like signals, these errors can be strongly correlated. In this work we derive a conditional (or a signal-specific ) CRB, modeling the signal as a deterministic unknown. Given any particular signal, our CRB reflects the possible signal-induced correlation between the TDOA and FDOA estimates. In addition to its theoretical value, we show that the resulting CRB can be used for optimal weighting of TDOA-FDOA pairs estimated over different signal-intervals, when combined for estimating the target location. Substantial improvement in the resulting localization accuracy is shown to be attainable by such weighting in a simulated operational scenario with some chirp-like target 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 estimation of time-difference of arrival (TDOA) and/or frequency difference of arrival (FDOA) of a common target-signal intercepted at two (or more) sensors. The target is usually assumed to be fixed (with some exceptions e.g., [9]), whereas the sensors are moving, and their positions and velocities are known (with some exceptions, e.g., [4]). The TDOA and FDOA are then caused, respectively, by different path-lengths and by different relative velocities between the target and the sensors. When a narrowband source signal modulates a high-frequency carrier, the Doppler effect induces negligible time-scaling on the narrowband modulating signal, but significantly modifies the intercepted carrier frequency, giving rise to 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; accepted November 29, Date of publication December 30, 2010; date of current version March 09, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jean Pierre Deimas. The material in this paper was presented at ICASSP The authors are with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel ( arie@eng.tau.ac.il; angeleyal@gmail.com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TSP usually cannot be phase-locked to each other, and since the medium might be wideband-dispersive (causing deviation of the carrier s phase-difference from the anticipated product of the TDOA with the carrier-frequency), the resulting phase-difference between the received carriers at the different sensors cannot be directly associated with the TDOA and FDOA, and therefore has to be considered as an additional unknown parameter. Therefore, after downconversion to baseband, the continuous-time observation model is given (for the basic case of two sensors and a single target) by where is the (unknown) complex-valued source signal and and are additive, zero-mean, statistically independent complex, circular Gaussian noise processes. The nuisance parameters and are the (unknown) absolute relative gain and phase-shift, respectively, of the second channel, and and are the (unknown) parameters of interest the FDOA and TDOA, 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 and FDOA estimation has attracted considerable research interest (e.g., [2], [11], [14], [15], [20]), with renewed interest in recent years (e.g., [1], [3], [4], [8] [10], [17], and [18]). Most classical approaches model the source signals (as well as the noise) as Gaussian stationary random processes [2], [7], [20] (and probably also [14], although this is not stated explicitly in there). However, while the assumption regarding the noise is generally justified, in some applications the assumption that the source signal is Gaussian and/or wide-sense stationary (WSS) may be strongly violated. Moreover, bounds derived under an assumption of a stochastic source signal are associated with the average performance, averaged not only over noise realizations, but also over different source signal realizations, all drawn from the same statistical model. It might be of greater interest to obtain a signal-specific bound, namely: for a given realization of the source signal, to predict the attainable performance when averaged only over different realizations of the noise. Such a bound can relate more accurately to the specific structure of the specific signal. The difference between the two approaches is particularly significant in predicting how the specific signal s structure induces inevitable correlation between the resulting TDOA and FDOA estimates. Under the Gaussian stationary signal model, the Cramér-Rao bound (CRB) for these estimates predicts (asymptotically) zero (1) X/$ IEEE

2 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 TDOA may be used interchangeably to explain the differences between the received signals: Consider, for example, a chirp signal with a linearly-increasing frequency. Neglecting end-effects, a delayed version of that signal would look the same as a negative Doppler-offset version of the same signal, since at each time-instant the instantaneous frequency of the delayed signal is smaller (by the same absolute offset) than that of the original signal. Thus, a time-delay and a frequency-offset cannot be well distinguished from each other, and their estimation errors must be correlated. Nevertheless, the deterministic-signal model has not seen as 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, Fowler and Hu [1] were the first to offer a new perspective on the CRB for joint TDOA/FDOA estimation, by considering the deterministic-signal model, elucidating the essentially different (non-diagonal) structure of the resulting bound. However, their derivation implicitly assumes that the source signal, as well as the phase-shift, are known. This assumption considerably simplified the exposition in [1], but is rarely realistic in a passive scenario, and consequently (as we shall show), the resulting bound in [1] is too optimistic (loose) in the fully passive case. Thus, in this paper we derive the CRB for the deterministic signal case, regarding the signal, as well as the relative gain and phase-shift, as additional nuisance parameters. We show that the associated Fisher information matrix (FIM) can be reduced in size to accurately reflect the bound on,, and alone. We compare our bound to Wax stochastic bound [20], as well as to Fowler and Hu s bound [1]. Besides the theoretical value, an important practical implication of knowing the correct bound (namely, the asymptotic 1 covariance of the ML estimate) is the ability to use this covariance for properly weighting the estimated TDOAs and FDOAs when calculating the resulting estimated target location. As we shall show, the use of the correct bound (covariance) for such weighting can attain a significant improvement in the resulting target localization accuracy, relative to the use of uniform weighting or of weights derived from the classical bounds. In addition, the resulting estimates of the confidence ellipse (for the estimated target location) become much more reliable when the correct bound expressions are used. In a related conference-paper [19], the basic expressions for the resulting bound were presented. In the current paper we substantiate these expressions with explicit derivations and comprehensive discussions. In addition, we offer a detailed example illustrating the attainable improvement in geolocation accuracy and 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 section we define the discrete-time model and derive our signal-specific CRB. In Section III we compare our CRB to some existing bounds (developed under different model-assumptions), both analytically and by simulation. In Section IV we address 1 In the context of our deterministic signal assumption, asymptotic efficiency of 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 of TDOA-FDOA pairs taken over different intervals, comparing different 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 the Nyquist rate to yield their discrete-time versions. To simplify notations, we shall assume unit sample rate (implying that the continuous-time signals are bandlimited between 1/2 and 1/2). Any different sampling-rate (and bandwidth) merely imply expansion or compression of the timeline, which in turn implies rescaling of and of, as well as of their estimates, and hence of their estimation errors and associated bounds. We thus obtain the discrete-time sampled versions (with, etc., and with ) where denotes the sampled time-shifted source signal. Defining the respective vectors we observe that the relation between and can be approximated using the discrete Fourier transform (DFT), as, where denotes the unitary DFT matrix (the superscript denoting the conjugate transpose), and where is a diagonal matrix, such that with This approximation essentially replaces the linear time-shift with a circular time-shift (such that the part shifted-out from one edge of the interval is shifted-in from the other edge). However, under the common assumption that the shifts (namely the TDOAs) are very small with respect to (w.r.t.) the observation length ( ), the approximation error becomes negligible. Moreover, in some cases the signal mostly exists in the middle of the observation interval, and tapers to zero towards the edges which can serve as further (partial) justification for this approximation. Given this relation, we may express our discrete-time model (2) in vector form as follows: where. (2) (3) (4) (5) (6)

3 1614 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011 We are now interested in the CRB for estimation of all unknown (but deterministic) parameters of the problem. To this end, we begin by defining the vector of all real-valued parameters, composed of the real-part of, the imaginary-part of, and the actual parameters, namely is a vector. Recalling that is a deterministic (unknown) vector, whereas are independent Complex Circular Gaussian vectors, we observe that the concatenated vector is also a Complex Circular Gaussian vector, with mean and covariance where and denote the covariance matrices of and, respectively, and where. Note that is always a unitary matrix: (the identity matrix). To simplify the derivation, we shall employ the common assumption that the noise processes are white, with variances and, respectively, namely and. For further simplification of the exposition, we shall regard the noise variances as known parameters, thus excluded from.we shall comment on the more realistic case, in which these variances are unknown, at the end of this section. The FIM for estimation of a real-valued parameters-vector from the Complex Circular Gaussian vector, when only the mean depends on, is given by (see, e.g., [6, p. 525]) (7) (8) inverse of. This block is particularly convenient to obtain from a matrix of this form, by exploiting the four-blocks matrix inversion expression and taking advantage of the diagonality of the upper-left block. More specifically, we identify the 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 relations and. We now need to obtain explicit expressions for. To this end, let us define the matrix, and note that Consequently, since We, therefore, obtain as, wehave (13) (14) (15) Forming the derivative of w.r.t. we get the (Jacobian) matrix (9) where for shorthand we have used (the delayed and frequency-shifted version of ) and (16) where denotes the derivative of w.r.t. (to simplify the notations, we omit the explicit dependence of on and, denoting this matrix simply as ; Likewise, we shall omit the explicit dependence of on, denoting this matrix simply as ). Consequently (exploiting the unitarity of ), we have (17) (which can be loosely termed the derivative of up to a factor of, since for a continuous-time signal, multiplication by in the frequency-domain is equivalent to differentiation in timedomain). We now turn to obtain the elements of. For the first row of this matrix, we observe the elements (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 actual parameters, given by the lower-right 4 4 block of the (18) Evidently, the last three are imaginary-valued, and would therefore vanish in when the real-part is taken [in (12)]. This implies, as could be expected, that is block-diagonal, such that the upper-left element, which accounts for the estimation of, is decoupled from the other three. In other words, since we are not interested in, we may concentrate on the lower-right

4 YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION block of, which we shall simply denote. The relevant elements (in this block) are easily obtained from (15) (19) where we have used the unitarity of in obtaining and. We observe that all of these terms except for are real-valued (regarding, note that, which is real-valued). Therefore, substituting in (12) (for the lower-right 3 3 block only) we obtain covariance, which does not depend on the other parameters, it is straightforward to show (using the complete Slepian-Bang expression for the FIM, e.g., [6, p. 525]) that the resulting FIM would be block-diagonal, with the block accounting for these variances decoupled from the block(s) accounting for the other parameters. This means that the CRB on the estimation of the parameters of interest is indifferent to the knowledge (or lack of knowledge) of the noises variances. Of course, this does not imply that the same accuracy can be obtained in both cases in general; 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 assumption that the source signal is zero-mean, WSS Circular Gaussian, with a known power-spectrum. With our white-noise assumption and under the assumption of high SNR, the respective FIM in [20], denoted, can be expressed by (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 as on the unknown actual parameters. However, under good signal-to-noise ratio (SNR) conditions, reliable estimates of the signal and of the parameters may be substituted into this expression to obtain a reliable estimate of the estimation-errors covariance. Moreover, as we shall show in the sequel, this information can be used for proper (near optimal) weighting of estimated TDOA-FDOA pairs obtained from different segments when used for estimating the target location. It may be interesting to note, that although the bound depends on the true TDOA, it is in fact independent of the FDOA.To observe this, note that the only terms in (20) which may contain dependence on the TDOA or FDOA are those involving (and also, which is ), through the full expression for. But observe that in all of these terms the resulting involvement of is through an expression of the form (in elements,, and )or (in ). Now (21) (exploiting the diagonality of both and in the second transition and the unitarity of in the third), which only depends on. A similar elimination can be applied to, which completes the proof. Before concluding this section, we briefly address the case in which the noises variances and are unknown. In that case, the -long parameters vector should be supplemented with these two parameters, forming a vector. Note, however, that since the dependence of the probability distribution of on these variances is only through the (22) where the second equality follows from our unit sample-rate assumption, such that is the power-spectrum of the discrete-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 between and : under these model assumptions (and asymptotically, for sufficiently long observation intervals) equals the mean of, taken with respect to different realizations of. We emphasize, however, that this averaging effect is exactly what we aimed at eliminating in developing our signal-specific bound: per any given realization of, might be significantly different from, especially when is not a realization of a stationary process. The bound obtained from is more informative regarding the intrinsic properties of the observed signal, averaging only over different realization of the noise (as opposed to the bound obtained from, which reflects averaging over both signal and noise realizations). The main significant difference between and is the block-diagonality of, which decouples (namely, implies decorrelation of) the estimation errors of and. While such decoupling truly occurs when is WSS Gaussian, it may be strongly violated in practice, e.g., when is a chirp-like signal. Therefore, the resulting (WSS-based) bound in such cases may be significantly different from our signal-specific bound, as demonstrated here.

5 1616 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011 The conditional CRB developed by Fowler and Hu in [1] implicitly assumes prior exact knowledge (to the estimator) of the source signal, as well as of and. In practical situations of passive FDOA and TDOA estimation, these assumptions are usually not very realistic. Although the assumption that is known 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 taking the Schur complement of the respective block in [see (12)], due to the assumption that is known. A more significant difference is the absence of the -related row and column of due to the assumption that is known. As evident from the expression for, the lower-right 2 2 block may be strongly coupled with the upper-left element. By ignoring this coupling, the resulting bound matrix becomes significantly smaller (has smaller eigenvalues) and differently oriented (has different eigenvectors). In Fig. 1 we demonstrate the fit (or misfit) between the empirical distribution of the ML estimates of and, and the 90%-confidence ellipses (in the plane) computed from the three bounds: our signal-specific ; Wax ; and Fowler and Hu s. The experiments were run for two essentially different specific signals, each of length (shown on top in the figure). The first signal (representing a stationary case) was a sample-function of an auto-regressive moving-average (ARMA) process, generated by filtering a sequence of white Gaussian noise with the filter for the stationary Gaussian signal. Note that is considerably more optimistic (loose) than the other two (in both cases), due to the inherent assumption that the targetsignal and the phase-difference between the received signals are known to the estimator. IV. COMBINING TDOAS AND FDOAS FOR TARGET LOCALIZATION Besides the theoretical characterization of the correlation between the TDOA and FDOA estimates, our can provide considerable improvement in the accuracy of TDOA- and FDOA-based target localization, by prescribing the proper (or even optimal) weighting. When estimated TDOA-FDOA couples (taken from different intervals of intercepted signals) are combined to form an estimate of the target location, the introduction of proper weighting into the process can result in significant improvement in the resulting accuracy. In addition, more reliable confidence-ellipses for the estimated location can be obtained if the true error distributions (covariance matrices) of the intermediate measurements are used. To be more specific, let us derive the weighted estimate of target location based on TDOA-FDOA couples. Assume that the target signal is intercepted in different time-intervals (segments, or pulses) and that the TDOA and FDOA are estimated in each time-interval (using ML estimation). We denote by and the positions of the first and second sensors in the th time-interval (for simplicity we assume a two-dimensional model in this context see Fig. 2 for reference). Likewise, we denote by and their respective velocity vectors. Let denote a (static) target position. The distance-vectors between the target and the two sensors in the th time-interval are thus denoted and, whereas the respective (scalar) distances are denoted (25) The second signal (representing a chirp-like case) was a Gaussian-shaped chirp pulse, generated as The true TDOA for the (27) th time-interval is therefore given by (28) (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 scatterplots represents the ML-estimation errors of and in one trial. The ML estimates were obtained using a two-stages fine grid search over and, assuming the deterministic (unknown) signal model with unknown and and additive white Gaussian noise as proposed, e.g., in [15]. The true values of and were and. As could be expected, our fits the data far better than the other two for the chirp-like signal, and is comparable to the where denotes the propagation speed. 2 Denoting by the carrier frequency of the target-signal, the Doppler-induced FDOA for the th interval is given by where (29) (30) are unit-vectors in the directions pointing from the respective sensors to the target. 2 Approximately 299,792,458 m/s for electromagnetic signals in free space.

6 YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION 1617 Fig. 1. Empirical distribution (400 trials) of ML estimates of and with 90%-confidence ellipses derived from the three bounds. to normalize each term by a respective variance (with the same units), thereby combining unit-less terms (32) Fig. 2. A general scene: target o, sensors 3. Both sensors in this scene are moving at a constant speed. Given estimates of the TDOA-FDOA pair in each interval, an estimate of the target position can be obtained by minimizing the least-squares (LS) criterion (31) with respect to, and taking the minimizing value as the LS estimate. However, this unweighted version of the LS criterion obviously entails an inherent flaw, because it combines terms with different units ( and ). A possible remedy is where and are some prescribed variances, and can be regarded as a weight-matrix. We term this the uniformly-weighted LS (UWLS) criterion. Obviously, if the estimation variances over different time-intervals are different (and known), then the constant weight-matrix can be substituted with interval-dependent weights, reflecting different weighting of results taken from different time-intervals. We term the resulting criterion a weighted LS (WLS) criterion and the minimizing the WLS estimate. Assuming statistical independence of estimates of TDOA-FDOA pairs taken from different intervals, the optimal 3 weight matrices (under a small-errors assumption) for the WLS criterion are well-known to be the inverse covariance matrices of these. Moreover, under asymptotic conditions (sufficiently high SNR in our case), the ML estimation error in each interval can be considered a Gaussian zero-mean random vector, with covariance given by the interval s CRB. Under these conditions, taking the inverse CRB for each interval as the respective weight-matrix would yield the ML estimate of the target location with 3 In the sense of minimum mean square errors in the resulting location estimates.

7 1618 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011 Fig. 3. The operational scene: target o, sensors 3. Sensor 1 is static, sensor 2 is traveling at v =[070; 70][m=s], signal intercepted every 10 s. respect to the estimated TDOAs and FDOAs. Under a small-errors assumption, the resulting WLS estimate would thus attain the CRB for estimation of the target location from the TDOA-FDOA estimates. Furthermore, as a by-product of using the measurements covariance for weighting, we can obtain (under a small-errors assumption) the covariance of the resulting location estimate,as follows. Define In the following experiments we explore the differences in the obtained localization accuracies and in the reliability of the respective confidence-ellipses, resulting from the use of four different weighting schemes. 1) A uniformly weighted scheme, assuming that the variances of the TDOA and FDOA estimates are constant over all intervals, and prescribed by the mean (over all intervals) of the WSS-case bound, ; 2) A weighted scheme using as the TDOA-FDOA covariance for each interval; 3) A weighted scheme using as the TDOA-FDOA covariance for each interval; 4) A weighted scheme using our as the TDOA-FDOA covariance 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 at and travels with a constant velocity vector. A 2 ms-long segment of the transmitted target signal is intercepted by the two sensors every 10[s] over a period of one minute (namely, segments are intercepted), and the position of the second sensor when intercepting the th segment is (33) The 2 2 derivative matrix (Jacobian) of with respect to can then be easily shown to be given by where (34) (35) are projection matrices onto the directions perpendicular to and to, respectively (here denotes the 2 2 identity matrix). Then, denoting the covariance matrix of as, if the weight-matrices used for the WLS criterion are chosen as, then the resulting covariance matrix of the WLS location-estimate is given by (36) (37) The target signal s carrier frequency is. The sampling frequency (after conversion to baseband) is, and therefore the number of samples in each 2 ms 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 segment is generated (directly in its baseband sampled version), by filtering a white Gaussian sequence with the same filter 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 version) at each time-interval. The values of and of are drawn independently and uniformly in the ranges and (respectively), where and. The generated signals are time- and frequency-shifted according to the geometrical model (and the sampling-rate), and then independent white Circular Gaussian noise signals and with variances are added, forming the received signals and, from which the ML estimate of the TDOA and FDOA is obtained (using a fine grid-search over and in our model see, e.g., [15]) for each interval. The estimated target location is obtained from the UWLS or WLS solution (using an iterative Gauss-Newton method see, e.g., [12]). integrating the TDOA and FDOA estimates from all intervals.

8 YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION 1619 Fig. 4. The case of stationary target-signals: Empirical distribution (400 trials) of differently weighted LS estimates of the target-location. Superimposed are 90%-confidence ellipses computed from the inferred estimation covariance (36). For obtaining the different weight matrices, the respective CRB matrices were computed based only on the observed data, and not on any other prior knowledge regarding the underlying signal models except for the noise variances and, which were assumed known. The ML estimate of and immediately yields, as a byproduct, ML estimates of the unknown and parameters, as well as of the noiseless target-signal. All estimated parameters (and signal) were thus plugged into the expressions for (20) and for (24) for computing the signal-dependent and (resp.). For computing we used the empirical periodogram of the estimated in lieu of the unknown power-spectrum (note that, strictly speaking, in the nonstationary case of a chirp-like signal, such a power-spectrum does not exist; nevertheless, the periodogram provides a reasonable tool for computing the resulting bound and weight matrices for the purpose of comparison to the other bounds). All the bound expressions were normalized by accounting for the nonstandard sampling-rate in this example. In Fig. 4 we present the results for the stationary signals scenario, whereas in Fig. 5 we present results for the chirp-like signals scenario (both showing the four weighting-schemes in four subplots). In each trial the same target signals were generated, and only the additive noise signals were redrawn (independently). The resulting estimated target location (which also represents the estimation error, since the target is located at ) for each trial appears as a dot in the respective figure. Superimposed on the resulting scatter-plots are confidence-ellipse, calculated from the respective inferred localization-error covariance (36) (averaged over all trials). Note that this covariance expression is based on the underlying assumption that the true covariance matrices of the TDOA-FDOA estimates are described by the respective CRB matrices so, evidently, whenever this assumption is breached, the resulting ellipse would not fit the empirical distribution. The results for the stationary case show that (as could be expected) in this case the uniformly-weighted, the -weighted and the -weighted results have a generally similar distribution, and the respective confidence ellipse provide a close fit. The distribution of the -weighted results is also quite similar to the other distributions. However, since (as we have already seen) the bound underestimates the covariances of the intermediate TDOA-FDOA estimates, the resulting -based confidence ellipse is too small. The significant advantage of using for weighting is revealed in the scenario of different chirp-like signals. It is important to realize, that optimal weighting is not only about attributing weak weights to the poor (high variance) TDOA- FDOA estimates and strong weights to the good (small variance) estimates. The more subtle geometrical interpretation of the weighting of two-dimensional (TDOA-FDOA) data makes a far more significant difference in this context: the optimal weighting is able to capture and exploit the directions of high and low variances in the TDOA-FDOA plane. For example, for a chirp-like pulse with an increasing frequency, the TDOA and FDOA estimation errors would be strongly positively-cor-

9 1620 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011 Fig. 5. The case of chirp-like target-signals: Empirical distribution (400 trials) of differently-weighted LS estimates of the target-location. Superimposed are 90%-confidence ellipses computed from the inferred estimation covariance (36). related. Conversely, for a chirp-like pulse with a decreasing frequency, they would be strongly negatively-correlated. Therefore, although the variances of the two TDOA (or FDOA) estimates taken from two such intervals might be similar, proper geometrical weighting (as provided by the two 2 2 weight matrices) would account for (and exploit) the different directions of smaller variance: perpendicular to the different common-mode directions in both cases. The resulting weighted estimate would then enjoy the benefit of both, realized by the smaller variances along both of these directions. Indeed, in this case the -based weight matrices correctly identify (and, therefore, provide correct weighting for) the strong correlations, resulting from the chirp-like structure, between the TDOA and FDOA estimation errors in each segment. On the other hand, the UWLS and -based weighting are unaware of this correlation, and are therefore unable to apply similar improvement to the resulting localization. is generally aware of the possible correlation between TDOA and FDOA estimates, but since it is implicitly based on the assumption that the source signal and phase-difference are known, the deduced covariance strengths and directions (eigenvalues and eigenvectors) are inaccurate for our scenario, and the resulting weights are suboptimal. We note in addition, that the deduced localization covariances (leading to the confidence-ellipses) are too optimistic in this case, not only for the -based weighting, but also for the UWLS and -based weighting. Only our provides a reliable confidence ellipse for this type of signal. As mentioned above, for the purpose of computing the bounds to be used for weighting, we use the estimated, rather than the true parameters (including the estimated target-signal). When the SNR is not sufficiently high, the estimated target signal might contain some significant residual noise, which would generally make the signal appear wider (in frequency-domain) if the noise is white. This residual noise might induce errors on the computed bounds, which might in turn cause some degradation in the weighting scheme. In Figs. 6 and 7 we present the localization performance of the four weighting schemes (for both signal types) versus the SNR, for SNR values varying from 20 to 40 db. Although these SNR values are usually considered high, the time-bandwidth product of the very short (2 ms) target signal pulses in our specific scenario is relatively low, and has to be compensated for by good SNR. We present the accuracy in terms of the long and short axes lengths of the 90% confidence-ellipses derived from the empirical 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 from the analytically predicted localization-error covariance matrices (36)). For these figures we slightly varied the operational scenario by shortening the time-intervals between pulses from 10 to 5 s, thereby obtaining, rather than intermediate results (otherwise the ellipses for the lower SNRs for the chirp-like signals become significantly larger than the distances from the target to the sensors, thereby severely violating the small-errors assumption).

10 YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION 1621 (20), the correlation coefficient between the estimation errors of and in our does not depend on SNR. Therefore, the weaker relative improvement at the lower SNRs is to be blamed exclusively on the errors in estimating the signal and parameters for substitution into the expression. Fig. 6. Long and Short axes lengths of the 90% confidence-ellipses computed from the empirical localization-error covariance for each of the four weighting schemes (for the case of stationary target-signals). V. CONCLUSION We considered the problem of joint TDOA-FDOA estimation for passive emitter location. Assuming a deterministic (unknown), rather than a stochastic signal model, we obtained an expression of the signal-specific ( conditional ) CRB for this problem. The most prominent feature of this bound, as opposed to the classical bound (for stochastic, stationary Gaussian targetsignals), is the existence of possibly significant nonzero off-diagonal terms especially when the target signal has a chirp-like structure over the observation interval. Such off-diagonal terms reflect nonzero correlation between the TDOA and FDOA estimation errors. We proposed using the computedsignal-specificcrbforweightingtdoa-fdoameasurements taken over different time-intervals in the estimation of the target location. Accounting for the TDOA-FDOA correlation in the weighting enables to take advantage of the diversities in chirp structures (increasing/decreasing frequencies) between intervals, so as to attain significant improvement in the localization accuracy. In our simulation examples with chirp-like signals, the use of proper weighting reduced the scatter area of the localization results by a factor of 25 under good SNR conditions. APPENDIX I THE RELATION BETWEEN AND The FIM developed in [20] for the case where the sampled target signal is a WSS Gaussian process with a known power-spectrum can be expressed, under the assumption of white noise and high SNR, as given in (22) and repeated here for convenience Our signal-specific FIM (20) is repeated here as well (38) Fig. 7. Long and Short axes lengths of the 90% confidence-ellipses computed from the empirical localization-error covariance for each of the four weighting schemes (for the case of chirp-like target-signals). As evident in the figures, in the case of stationary target-signals (Fig. 6) there are no essential differences in performance between the four weighting schemes, since our is also nearly diagonal in such cases. However, in the case of chirp-like target-signals (Fig. 7), the overwhelming relative improvement (more than fivefold in each axis) in using the signal-specific bound for weighting at the high SNRs is seen to become somewhat less pronounced (down to about 25% improvement) at the lower SNRs due to the sensitivity of the computed to the presence of residual noise in the estimated signal. Note, however, that, as can be easily deduced from the structure of in (39) We shall now show that if the observed segment is a realization a WSS process with power-spectrum, then asymptotically (in the observation length ). We denote the discrete-time Fourier transform (DTFT) of over the observed interval as and we shall use (40) (41)

11 1622 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011 For shorthand, we shall implicitly assume asymptotic conditions (namely, that is sufficiently long ), by regarding (41) as an equality without the lim operator. We proceed by showing that for each element of this matrix (ignoring the leading term, which is common to both). Beginning with the (1,1) element (42) where the approximation of the sum over as is valid for 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 not exactly vanish, but equals ; However, since the (1,1) element is proportional to and the (3,3) element is proportional to, the (1,3) element is much smaller than the square root of their product (proportional to ), and therefore its effect on the FIM is negligible (asymptotically). The remaining term is the (2,3) element. Note first that where we have used Parseval s identity for the second transition. Turning to the (1,2) element (48) where we have eliminated and from due to the diagonality of and and the unitarity of. We, therefore, have, for the (2,3) element (43) where we have used Plancherel s identity and the definition of (17) for the second transition. Similarly, for the (2,2) element (49) where denotes the correlation matrix of the random vector, and denotes the trace operator. Under the assumption that (and therefore also ) is a WSS process, takes a Toeplitz form, and is therefore (asymptotically) diagonalized by the Fourier matrix, namely (50) (44) is a diagonal matrix. We, therefore, have Before turning to the other elements, recall that is a delayed and doppler-shifted version of consequently, and Thus, turning to the (3,3) element we have (45) (46) (51) where in the last transition we eliminated and from the product due to their diagonality and unitarity and due to the diagonality of and. Moreover, due to the diagonality of, the matrix is a Toeplitz matrix, and therefore the last trace expression in (51) is simply the sum over (from to ), multiplied by a constant value (the value on the main diagonal of the Toeplitz matrix ). Since the sum vanishes, so does the entire term (again, we note that, strictly speaking, since is even the sum over does not exactly vanish, but the residual term is negligible in the FIM). Exploiting the symmetry of both and for all of the remaining elements, the proof is complete.

12 YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION 1623 REFERENCES [1] M. L. Fowler and X. Hu, Signal models for TDOA/FDOA estimation, IEEE Trans. Aerosp. Elect. Sys., vol. 44, pp , Oct [2] B. Friedlander, On the Cramér-Rao bound for time-delay and Doppler estimation, IEEE Trans. Inf. Theory, vol. IT-30, pp , May [3] K. C. Ho and Y. T. Chan, Geolocation of a known altitude object from TDOA and FDOA measurements, IEEE Trans. Aerosp. Electron. Syst., vol. 33, no. 3, pp , Jul [4] K. C. Ho, X. Lu, and L. Kovavisaruch, Source localization using TDOA and FDOA measurements in the presence of receiver location errors: Analysis and solution, IEEE Trans. Signal Process., vol. 55, no. 2, pp , Feb [5] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, [6] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, [7] C. Knapp and G. Carter, The generalized correlation method for estimation of time delay, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-44, pp , Aug [8] J. Luo, E. Walker, P. Bhattacharya, and X. Chen, A new TDOA/FDOA-based recursive geolocation algorithm, in Proc. Southeast. Symp. Syst. Theory (SSST), 2010, pp [9] D. Mušicki, R. Kaune, and W. Koch, Mobile emitter geolocation and tracking using TDOA and FDOA measurements, IEEE Trans. Signal Process., vol. 58, no. 3, pp , Mar [10] R. Ren, M. L. Fowler, and N. E. Wu, Finding optimal trajectory points for TDOA/FDOA geo-location sensors, in Proc. Inf. Sci. Syst. (CISS), 2009, pp [11] D. C. Shin and C. L. Nikias, Complex ambiguity functions using nonstationary higher order cumulant estimates, IEEE Trans. Signal Process., vol. 43, no. 11, pp , Nov [12] H. W. Sorenson, Parameter Estimation. New York: Dekker, [13] A. Renaux, P. Forster, E. Chaumette, and P. Larzabal, On the high SNR conditional maximum likelihood estimator full statistical characterization, IEEE Trans. Signal Process., vol. 54, pp , Dec [14] S. Stein, Algorithms for ambiguity function processing, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, pp , Jun [15] S. Stein, Differential delay/doppler ML estimation with unknown signals, IEEE Trans. Signal Process, vol. 41, pp , Aug [16] P. Stoica and A. Nehorai, MUSIC, maximum likelihood and the Cramér-Rao bound, IEEE Trans. Acoust., Speech, Signal Process., vol. 38, pp , Dec [17] R. Ulman and E. Geraniotis, Wideband TDOA/FDOA processing using summation of short-time CAF s, IEEE Trans. Signal Process., vol. 47, no. 12, pp , Dec [18] G. Yao, Z. Liu, and Y. Xu, TDOA/FDOA joint estimation in a correlated noise environment, in Proc., Microw., Antenna, Propag. EMC Technol. Wireless Commun. (MAPE), 2005, pp [19] A. Yeredor, A signal-specific bound for joint TDOA and FDOA estimation and its use in combining multiple segments, in Proc. ICASSP, [20] M. Wax, The joint estimation of differential delay, Doppler and phase, IEEE Trans. Inf. Theory, vol. IT-28, pp , Sep Arie Yeredor (M 98-SM 02) received the B.Sc. (summa cum laude) and Ph.D. degrees in electrical engineering from Tel-Aviv University (TAU), Tel-Aviv, Israel, in 1984 and 1997, respectively. He is currently with the School of Electrical Engineering, Department of Electrical Engineering-Systems, TAU, where his research and teaching areas are in statistical and digital signal processing. He also holds a consulting position with NICE Systems, Inc., Ra anana, Israel, in the fields of speech and audio processing, video processing, and emitter location algorithms. 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 IEEE TRANSACTIONS ON SIGNAL PROCESSING. He served as Technical Co-Chair of The Third International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP 2009), and will serve as General Co-Chair of the 10th International Conference on Latent Variable Analysis and Source Separation (LVA/ICA 2012). He has been awarded the yearly Best Lecturer of the Faculty of Engineering Award (at TAU) six times. He is a member of the IEEE Signal Processing Society s Signal Processing Theory and Methods (SPTM) Technical Committee. Eyal Angel received the B.Sc. degree in physics from Tel-Aviv University (TAU), Tel-Aviv, Israel, in He is currently pursuing the M.Sc. degree in electrical engineering, mainly working on passive timedelay estimation in multipath environments. His main areas of interest are in statistical signal processing. He also holds the position of System Engineer with the Process Diagnostics and Control (PDC), Applied Materials Ltd., Rechovot, Israel.