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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
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
P
Manuscript received June 21, 2010; revised October 09, 2010; accepted
November 29, 2010. Date of publication December 30, 2010; date of current
version March 09, 2011. 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 2010.
The authors are with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (e-mail: [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2010.2103069
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
(1)
is the (unknown) complex-valued source signal and
where
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
and
,
TDOA, respectively. The received signals are
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
1053-587X/$26.00 © 2010 IEEE
YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION
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 asymptotic1
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
1In 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].
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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
, etc.,
the discrete-time sampled versions (with
)
and with
(2)
denotes the sampled time-shifted
where
source signal. Defining the respective vectors
(3)
we observe that the relation between
and can be approximated using the discrete Fourier transform (DFT), as
, where denotes the
unitary DFT matrix (the
denoting the conjugate transpose), and where
superscript
is a diagonal matrix, such that
(4)
with
(5)
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
), the approximation error becomes negligible.
length (
mostly exists in the
Moreover, in some cases the signal
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:
(6)
where
.
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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
, namely
of , and the “actual” parameters
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
with mean
Complex Circular Gaussian vector,
and covariance
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
block. More specifically, we identify
the upper-left
(the inverse of the bound on )
the respective implied FIM
as the Schur complement (e.g., [5, p. 472]) of the lower-right 4
4 block, namely
(7)
where we have used the relations
and
.
We now need to obtain explicit expressions for . To this end,
, and note that
let us define the matrix
where
and
denote the
covariance matrices of
and , respectively, and where
. Note
is always a unitary matrix:
that
(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])
(12)
(13)
, we have
Consequently, since
(14)
We, therefore, obtain
as
(15)
(8)
Forming the derivative of
(Jacobian) matrix
where for shorthand we have used
w.r.t. we get the
(16)
(9)
where
denotes the derivative of w.r.t. (to simplify the
on and
notations, we omit the explicit dependence of
, denoting this matrix simply as ; Likewise, we shall omit the
on , denoting this matrix simply
explicit dependence of
as ).
Consequently (exploiting the unitarity of ), we have
(the delayed and frequency-shifted version of ) and
(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).
. For the first row
We now turn to obtain the elements of
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 imis block-diagonal, such that
plies, as could be expected, that
the upper-left element, which accounts for the estimation of ,
. In other words, since
is decoupled from the other three
we are not interested in , we may concentrate on the lower-right
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3 3 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 asis zero-mean, WSS Cirsumption that the source signal
. With our
cular Gaussian, with a known power-spectrum
white-noise assumption and under the assumption of high SNR,
, can be expressed by
the respective FIM in [20], denoted
(20)
is
.
The CRB on unbiased estimation of
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
), through the full expression for
also , which is
. 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
, which
on . A similar elimination can be applied to
completes the proof.
Before concluding this section, we briefly address the case
and
are unknown. In that
in which the noises’ variances
-long parameters vector should be supplecase, the
mented 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
, and
discrete-time sampled signal
(23)
(under the assumption that
is bandlimited at
,
for all
).
namely that
In Appendix I, we show a rather reassuring relation between
and : under these model assumptions (and asymptotically,
equals the mean
for sufficiently long observation intervals)
. We emof , taken with respect to different realizations of
phasize, 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
, especially when
is not a realization of a stafrom
tionary 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
, which reflects averaging over
to the bound obtained from
both signal and noise realizations).
is the
The main significant difference between and
, which decouples (namely, implies
block-diagonality of
decorrelation of) the estimation errors of and . While such
decoupling truly occurs when
is WSS Gaussian, it may
is a chirp-like
be strongly violated in practice, e.g., when
signal. Therefore, the resulting (WSS-based) bound in such
cases may be significantly different from our signal-specific
bound, as demonstrated here.
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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
only) in [1], which we denote
,
(with respect to
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
[see (12)],
the Schur complement of the respective block in
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
; Wax’
; and
three bounds: our signal-specific
. The experiments were run for two
Fowler and Hu’s
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 becan protween the TDOA and FDOA estimates, our
vide 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
and
their respecreference). Likewise, we denote by
tive 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
(27)
The true TDOA for the th time-interval is therefore given by
(28)
(26)
,
and
.
with
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
.
fits the data far better than
As could be expected, our
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
(29)
where
(30)
are unit-vectors in the directions pointing from the respective
sensors to the target.
2Approximately
299,792,458 m/S for electromagnetic signals in free space.
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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)
where
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
and
). A possible remedy is
with different units (
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-macan be substituted with interval-dependent weights
trix
, 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
taken from different intervals, the
TDOA-FDOA pairs
optimal3 weight matrices (under a small-errors assumption)
for the WLS criterion are well-known to be the inverse covari. Moreover, under asymptotic
ance matrices of these
conditions (sufficiently high SNR in our case), the ML estiin each interval can be considered a
mation error
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
3In the sense of minimum mean square errors in the resulting location estimates.
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Fig. 3. The operational scene: target ‘o’, sensors ‘3’. Sensor 1 is static, sensor
;
= , signal intercepted every 10 s.
2 is traveling at v
= [070 70][m 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
(33)
The 2 2 derivative matrix (Jacobian) of
can then be easily shown to be given by
with respect to
(34)
where
(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
, then the resulting
are chosen as
covariance matrix of the WLS location-estimate is given by
(36)
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,
as the TDOA-FDOA
2) A weighted scheme using
covariance for each interval;
as the TDOA-FDOA
3) A weighted scheme using
covariance for each interval;
as the TDOA-FDOA
4) A weighted scheme using our
covariance for each interval.
Our operational scenario is the following (see Fig. 3):
(marked ‘o’ in the
• A target is positioned at
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
segments are intercepted), and the pominute (namely,
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 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
as in (25);
filter
• In the second scenario a different chirp-like pulse
and
parameters]
[cf. (26), with randomized
is generated (directly in its baseband sampled verand
sion) at each time-interval. The values of
are drawn independently and uniformly in the
of
and
(respecranges
and
tively), where
.
The generated signals are time- and frequency-shifted according
to the geometrical model (and the sampling-rate), and then indeand
pendent white Circular Gaussian noise signals
with variances
are added, forming the reand
, from which the ML estimate of
ceived signals
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.
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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
(24) for computing the
the expressions for (20) and for
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).
target signals were generated, and
In each trial the same
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. Superconfidence-elimposed on the resulting scatter-plots are
lipse, 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
-weighted
provide a close fit. The distribution of the
results is also quite similar to the other distributions. However,
bound undersince (as we have already seen) the
estimates the covariances of the intermediate TDOA-FDOA
-based confidence ellipse is too
estimates, the resulting
small.
for weighting is
The significant advantage of using
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) TDOAFDOA 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-
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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.
-based weight matrices corIndeed, in this case the
rectly 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.
-based weighting are
On the other hand, the UWLS and
unaware of this correlation, and are therefore unable to apply
is
similar improvement to the resulting localization.
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
-based weighting, but also for the
case, not only for the
-based weighting. Only our
provides
UWLS and
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
, rather than
interto 5 s, thereby obtaining
mediate 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).
YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION
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(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
expression.
for substitution into the
V. CONCLUSION
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).
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
can be expressed, under the assumppower-spectrum
tion 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
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
is also
between the four weighting schemes, since our
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
to
lower SNRs—due to the sensitivity of the computed
the presence of residual noise in the estimated signal. Note, however, that, as can be easily deduced from the structure of in
(38)
(39)
We shall now show that if the observed segment
is a re, then
alization a WSS process with power-spectrum
.
asymptotically (in the observation length )
We denote the discrete-time Fourier transform (DTFT) of
over the observed interval as
(40)
and we shall use
(41)
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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.
for each element of
We proceed by showing that
, which
this matrix (ignoring the leading term
is common to both). Beginning with the (1,1) element
as
is valid
where the approximation of the sum over
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
; However, since the (1,1) eleexactly vanish, but equals
ment is proportional to and the (3,3) element is proportional
, the (1,3) element is much smaller than the square root
to
), and therefore its effect on
of their product (proportional to
the FIM is negligible (asymptotically).
The remaining term is the (2,3) element. Note first that
(42)
where we have used Parseval’s identity for the second transition.
Turning to the (1,2) element
(48)
where we have eliminated
and
from
and the unitarity of
the diagonality of and
fore, have, for the (2,3) element
due to
. We, there-
(49)
(43)
where we have used Plancherel’s identity and the definition of
(17) for the second transition. Similarly, for the (2,2) element
denotes the correlation matrix of the
where
denotes the trace operator. Under
random vector , and
(and therefore also
) is a WSS
the assumption that
takes a Toeplitz form, and is therefore (asymptotiprocess,
cally) 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 , and
consequently
(45)
Thus, turning to the (3,3) element we have
(46)
(51)
where in the last transition we eliminated
and
from the
product due to their diagonality and unitarity and due to the
and . Moreover, due to the diagonality of
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,
is even the sum over does not exstrictly speaking, since
actly vanish, but the residual term is negligible in the FIM).
for all of the
Exploiting the symmetry of both and
remaining elements, the proof is complete.
YEREDOR AND ANGEL: JOINT TDOA AND FDOA ESTIMATION
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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 2001.
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.