nomenclature t unknown location of the emitter tdoa based

TDOA Based Direct PositioningMaximum Likelihood Estimatorand the Cramer-Rao Bound

NARESH VANKAYALAPATI

STEVEN KAY, Fellow, IEEE

QUAN DINGUniversity of Rhode Island

The maximum likelihood estimator (MLE) and its performancefor the localization of a stationary emitter using a network ofspatially separated passive stationary sensors is presented. Theconventional approach for localization using multiple sensors is tofirst estimate the time differences of arrival (TDOAs) independentlybetween pairs of sensors and then find the location of the emitterusing the intersection point of the hyperbolas defined by theseTDOAs. It has recently been shown that this two-step approach issuboptimal and an alternate direct position determination (DPD)approach has been proposed. In the work presented here we take theDPD approach to derive the MLE and show that the MLEoutperforms the conventional two-step approach. We analyze the twocommonly occurring cases of signal waveform unknown and signalwaveform known with unknown transmission time. This papercovers a wide variety of transmitted signals such as narrowband orwideband, lowpass or bandpass, etc. Sampling of the received signalshas a quantization-like effect on the location estimate and so acontinuous time model is used instead. We derive the Fisherinformation matrix (FIM) and show that the proposed MLE attainsthe Cramer-Rao lower bound (CRLB) for high signal-to-noise ratios(SNRs).

Manuscript received August 15, 2011; revised November 30, 2011, May23 and June 11, 2012, and February 6, 2013; released for publicationApril 4, 2013.

DOI. No. 10.1109/TAES.2013.110499.

This work was supported by the Air Force Research Lab, Rome, NYunder Contract FA8750-09-2-0236.

Refereeing of this contribution was handled by W. Koch.

Authors’ address: Dept. of Electrical, Computer and BiomedicalEngineering, University of Rhode Island, 4 East Alumni Ave., KelleyHall, Kingston, RI 02881. E-mail: ([email protected]).

0018-9251/14/$26.00 C© 2014 IEEE

NOMENCLATURE

(xT , yT ) = Unknown location of the emitter(xi, yi) = Location of sensor i2N – 1 = Number of non-zero Fourier coefficients

N0

2= Noise spectral density

τi = Unknown time of arrival of signal atsensor i

A = M × 1 vector of the unknown attenuationfactors

A′ = M – 1 × 1 vector of the unknown relativeattenuation factors

IN = An N × N identity matrixAi = Unknown attenuation factor at sensor i

ai, bi = Fourier coefficients of the signalc = Propogation speed of signal

dmax = Distance between the farthest pair ofsensors

F0 = Fundamental frequency of the Fourierseries

Fs = Sampling frequencyM = Number of sensorsR = Distance from the emitter to a sensor

ri(t) = Signal received at sensor i

s(t) = Transmitted signal waveformT = Length of observation intervalt0 = Unknown transmission timeTs = Non-zero length of the signal waveform

wi(t) = Additive Gaussian random process atsensor i

εSi = Signal energy at sensor i

Iθ = Fisher information matrix of the unknownparameter vector θ

η = 3 × 1 vector of the unknown emitterlocation coordinates and the transmission timeφ = 2N — 1 × 1 vector of the unknownFourier coefficientsτ = M × 1 vector of the unknown TOAsτ ′ = M – 1 × 1 vector of the unknown TDOAs

h(t) = 2N – 1 × 1 vector as defined inAppendix III

ASNR = Average signal-to-noise ratioCRLB = Cramer-Rao lower bound

FIM = Fisher information matrixMLE = Maximum likelihood estimatorSNR = Signal-to-noise ratio

TDOA = Time difference of arrivalTOA = Signal to noise ratio

I. INTRODUCTION

Passive localization has been used for many years andhas always been an important topic of research [1–4].Localization can be performed using one or more of theemitter location dependent properties of the signal such asangle of arrival (AOA), time difference of arrival (TDOA),frequency difference of arrival (FDOA), or the energy ofthe received signal. Over the years the general approach to

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localization using the TDOAs, commonly referred to asthe TDOA technique, has been to first estimate thedifference in the times of arrival of the signal at a particularpair of sensors and then use these TDOAs to estimate thelocation of the emitter. Knapp and Carter [5, 6] proposed ageneralized correlation method for the estimation of theTDOAs for stationary and relative motion cases. Theymodeled the signal as a stationary Gaussian randomprocess. Stein [7] on the other hand modeled the signal asdeterministic but unknown and derived the MLE for thedifferential delay and Doppler for a two-sensor case.Under similar assumptions for the signal, Yeredor andAngel [8] have derived the Cramer-Rao lower bound(CRLB) for the TDOAs. After the TDOAs are estimatedthey are used to estimate the location of the source [9–12].Quite often, due to network capacity and computationalconstraints, not all sensor pair combinations are used.Fowler [13] addressed the problem of optimal selection ofa subset of the sensor pairs. Torrieri [1] proposed a linearleast squares estimator where the nonlinear relationbetween the TDOAs and the emitter location is linearizedby expanding it in a Taylor series about a reference pointand retaining the first two terms. This is an iterativemethod which requires some kind of a priori informationin order to obtain an initial guess. Alternatively, Chan andHo [2] use an intermediate variable, which is a function ofthe emitter location, in order to linearize the nonlinearequations. They use a two-step weighted least squaresalgorithm. Additionally, when the signal waveform isknown, localization may be performed from the times ofarrival (TOAs) instead of the TDOAs [14, 15]. In suchTOA-based techniques the unknown transmission timeoccurs as a nuisance parameter which will have to beestimated. Sathyan et al. [16] have shown the theoreticalequivalence of the TOA and the TDOA, taken against acommon reference sensor, based position fixingtechniques by comparing the Cramer-Rao bounds. Thisresult is complemented by Kaune’s [17] results whichshow the equivalence using Monte Carlo simulations. Theabove techniques may be called two-step techniquesbecause the TOAs/TDOAs are first estimated at the localsensors and these TOA/TDOA estimates are used in asecond step to compute the location of the emitter.

Weiss and Amar [18–21] have shown that the two-stepapproach is suboptimal and proposed a direct positiondetermination (DPD) approach. Weiss had derived themaximum likelihood estimator (MLE) for the sourcelocation for the case of a stationary narrowband radiofrequency transmitter using multiple stationary receiversin [18]. He uses a continuous time model and quicklyconsiders the sampled version without discussing theeffects of sampling on the emitter location estimate. Heshows that the MLE of the emitter location is obtained bymaximizing a quadratic form of the signal samples whosecoefficients are functions of the emitter location. Heconsiders the two cases of signal known and signalunknown but leaves out a more important case - signalknown but transmission time unknown which is most

likely to occur in real-world situations. He does notdiscuss the CRLB for this problem. There is an inherentambiguity in the commonly used model and Weiss uses aconstraint on the signal samples to resolve the ambiguity.No discussion is provided on the generality of theconstraint as to why it is an appropriate constraint, how itresolves the ambiguity and whether it reduces theperformance. In [19] Amar and Weiss extend the approachto a multiple emitters case. In [20] they address theproblem of localization using only the Doppler frequencyshifts and in [21] they consider the case of a singlestationary emitter and moving receivers. In all these casesthe results are similar, i.e., the MLE for the emitterlocation is obtained by maximizing a quadratic form of thesignal whose coefficients are functions of the emitterlocation. The derivation of CRLB is attempted in the laterpapers but is not sufficiently simplified. The effect ofsampling the signal is not discussed in any of the papers.Similar constraints are used to resolve the ambiguity in thepapers [18–21], but no discussion is provided on theeffects of the constraint.

In this paper we consider the case of a single stationaryemitter and a network of stationary receive sensors. Weaddress many of the shortcomings of [18–21]. We use acontinuous time model and provide a straightforwardderivation for the MLE of the emitter location for the twocases of signal waveform known with unknowntransmission time and signal waveform unknown withunknown transmission time. Our model is valid for eithernarrowband or broadband signals, lowpass or highpasssignals. We discuss the effect of sampling the signal on theemitter location estimate. Using simulations, we comparethe MLE against a conventional TDOA technique. Weshow that the variance of the MLE is two to three orders ofmagnitude lower than the conventional TDOA technique.

A more difficult problem is deriving the CRLB. If thesignal waveform is assumed unknown along with the TOAand the attenuation factor, then the commonly used modelhas an ambiguity. This ambiguity comes to light whenderiving the CRLB. Because all the unknowns in themodel cannot be uniquely resolved, the Fisher informationmatrix (FIM) becomes singular. We address this ambiguityin detail and derive the necessary steps to remove it. Thenwe derive the nonsingular FIM. The inverse of the FIM isthe CRLB. CRLB gives the theoretical lower bound on thevariance of any unbiased estimator. An importantapplication of the CRLB is in deriving an optimal sensorconfiguration. The performance of a location estimatordepends on the placement of sensors. A particularconfiguration of the sensors is called optimal if itoptimizes a norm of the FIM. A quite common result [22,23] is to place the sensors around the emitter in anequi-angular configuration. But when the sensors aregeographically constrained the problem becomes muchmore difficult. We will investigate the problem of optimalsensor configuration in a future paper.

In Section II we provide a detailed description of theproblem. In Section III we give the CRLBs and the MLEs.

VANKAYALAPATI ET AL.: TDOA BASED DIRECT POSITIONING MLE AND THE CRAMER-RAO BOUND 1617


Fig. 1. Physical placement of sensors (for M = 4) and emitter positionused for simulation.

Here we analyze the case of signal waveform unknownand the special case of signal waveform known, both caseswith an unknown transmission time. In Section IV we useMonte Carlo simulations to compare the performance ofthe MLE against the conventionally used TDOAtechnique. We show that at higher signal-to-noise ratios(SNRs) the variance of the MLE approaches the CRLB.Conclusions are provided in Section V. Most of themathematics are provided in the appendices. In Appendix Iwe derive a compact expression for the FIM. Appendix IIhas the derivation of the MLE. Appendix III presents theproperties of a matrix we use in the model. In Appendix IVwe discuss the transformation of the parameters and theconstraints used in order to remove the ambiguity in themodel.

II. PROBLEM STATEMENT

Suppose that a stationary emitter is located at anunknown location (xT , yT ) and a network of M sensors arelocated at known locations (xi, yi), i = 0, 1, . . . , M − 1as shown in Fig. 1. For simplicity we are assuming atwo-dimensional case. Extension to the three-dimensionalcase is straightforward. The sensors are all synchronizedin time and each of the sensors intercepts the signal withinthe time interval (0, T ). The emitter transmits a signalwith unknown waveform s(t) for an unknown durationTs < T starting at an unknown time t0 < T . We assumethat the transmitted signal waveform s(t) is real. It can benarrowband or wideband, lowpass or bandpass. Afterinterception, the signal received at sensor i in the presenceof noise can be written as

ri(t) = Ais(t−τi) + wi(t), 0<t <T, i =0, 1, . . . , M − 1

(1)

where wi(t) is a zero mean wide sense stationary additivewhite Gaussian random process with spectral density

N02 , Ais are the unknown attenuations due to propagation

loss, assumed real, and the τis are the unknown TOAsgiven by

τi =√

(xT −xi)2 + (yT −yi)2

c+ t0, i = 0, 1, . . . , M−1

(2)

where c is the propagation speed of the signal. In (2) thefirst term

√(xT − xi)2 + (yT − yi)2/c is the propagation

delay and the second term t0 is the unknown time oftransmission. We assume that the noise at a sensor isindependent of the noise at any other sensor, i.e., wi(t) andwj (t) are independent for i �= j and that the noise spectraldensity at all the sensors is equal to N0

2 . If the noise doesnot satisfy these conditions then the problem becomesmore complex. For example, if the noise spectral densityis different at each sensor but known, then the noise termdoes not factor out as in (6) but instead exists in each term.A more difficult problem is when the noise spectraldensity is different at each sensor and unknown, in whichcase, the noise spectral densities at each of the sensorsneed to be estimated as well. To keep the derivationssimple we assumed the above conditions for the noise.Notice that here we do not assume as in [7], that τi � T .Instead we just assume that max

i,j(τi − τj ) < (T − Ts).

That is, we are only assuming that the observation intervalis large enough so that, within the observation interval, thesignal reaches both the nearest and the farthest sensorsfrom the emitter. Based on the sensor geometry it ispossible to find a sufficient condition on the length of theobservation interval. If dmax is the distance between thefarthest pair of sensors, then the observation interval mustbe greater than dmax

c.

Sampling the signal in time has a quantization-likeeffect on the estimate of the emitter location. This isbecause if the signal is sampled, then the TOA estimatesare integer multiples of the sampling interval and hencequantized. For example, if the signal is sampled at afrequency of Fs samples/s, then the estimate of τi isquantized with a maximum quantization error of 1

2Fs. This

can introduce a maximum quantization error of c2Fs

in therange estimate. Therefore, it is possible that the signal may have to be sampled at a rate much higher than the Nyquist rate in order to achieve a desired precision in thelocation estimate. We have used a continuous time model to avoid this problem in our analysis and to allow futurestudies of errors due to time synchronization effects.

Fig. 2 shows the signals received at the four sensors shown in Fig. 1 from an emitter located at (130, 75)km

transmitting a Gaussian chirp. The propagation loss was modeled as a 1 attenuation in the amplitude of the

R

received signal, where R is the range. So, the farthestsensor has the largest TOA and smallest amplitude. We are assuming that the signal lies inside the observation interval (0, T ). So, we can assume that the unknown signalis periodic with period T and write it in terms of its



pc

Highlight

pc

Highlight

Fig. 2. Signals received at four sensors when Gaussian chirp is transmitted by emitter located at (130, 75) km.

Fourier series as

s(t) = a0√2

+∞∑

n=1

(an cos 2πnF0t + bn sin 2πnF0t) (3)

where F0 = 1T

and the Fourier coefficients are given by

a0 =√

2T

∫ T

0s(t) dt, an = 2

T

∫ T

0s(t) cos 2πnF0t dt,

bn = 2T

∫ T

0s(t) sin 2πnF0t dt. (4)

We are using a0√2

for the DC component instead of thestandard a0 because it simplifies certain terms in thederivation of the CRLB. For a band-limited signal only afinite number of the Fourier coefficients are non-zero. Ifthe signal is a lowpass signal, there exists an integer N

such that the Fourier coefficients are all zero for n ≥ N

and if the signal is a bandpass signal, then there existintegers N1 and N2, N1 < N2, such that the Fouriercoefficients are zero for n < N1 and for n > N2. So wecan approximate the lowpass signal s(t) as (for a bandpasssignal the summation is from N1 to N2)

s(t) = a0√2

+N−1∑n=1

(an cos 2πnF0t + bn sin 2πnF0t).

This is an important step as it allows us to model anyunknown signal and reduce it to a parameter estimationproblem. Now, if we let φ = [a0 a1 · · · aN−1 b1 b2 · · ·bN−1]T be the 2N − 1 × 1 vector of Fourier coefficientsand h(t) = [ 1√

2cos 2πF0t · · · cos 2π(N − 1)F0t

sin 2πF0t · · · sin 2π(N − 1)F0t]T

then we haves(t) = hT (t)φ. This reduces the uncountable unknownparameter set {s(t) : t ∈ (0, T )} to a finite countablenumber of unknown parameters φ. Therefore, we canrewrite the model in (1) as

ri(t) = Ai hT (t − τi)φ + wi(t), 0 ≤ t ≤ T ,

i = 0, 1, . . . , M − 1. (5)

Let τ = [τ0 τ1 · · · τM−1]T , A = [A0 A1 · · · AM−1]T ,and φ = [a0 a1 · · · aN−1 b1 b2 · · · bN−1]T . Letθ = [τ T AT φT ]T be the (2M + 2N − 1) × 1 vector ofunknown parameters. If we let η = [xT yT t0]T andα = [ηT AT φT ]T then, using (2), we can write the TOAvector as a function of η as τ = g(η). So, the problem canbe stated as, given the observations ri(t), i = 0, 1, . . . ,

M − 1 estimate the vector η. We are only interested in theparameters (xT , yT ) and the rest of the unknownparameters are nuisance parameters.

III. CRLB AND MLE OF THE EMITTER LOCATION

A. Signal Unknown with Unknown Transmission Time

For the continuous time model in (1) the log-likelihoodfunction [24] for sensor i is given by

l = − 1

N0

∫ T

0(ri(t) − Ais(t − τi))

2 dt. (6)



pc

Highlight

Since the noise at different sensors is independent, byusing (5) we can write the joint log-likelihood function as

l(θ) = − 1

N0

∫ T

0

M−1∑i=0

(ri(t) − Ai h

T (t − τi)φ)2

dt (7)

The (2M + 2N − 1 × 2M + 2N − 1) FIM [25] for thismodel is given by (see Appendix I-A)

Iθ = (T/2)

(N0/2)

⎡⎢⎣

(2πF0)2φT LLT φ(diag(A))2 (2πF0)(φT Lφ)(diag(A)) (2πF0)(A � A)φT L

(2πF0)(φT LT φ)(diag(A))2 (φT φ)IM AφT

(2πF0)LT φ(A � A)T φAT (AT A)IM

⎤⎥⎦ (8)

where � represents the element by element product (Hadamard product), diag(A) is an M × M diagonal matrix with ithdiagonal element as Ai, IM is the M × M identity matrix, and the (2N − 1) × (2N − 1) matrix L is given by

L =

⎡⎢⎣ 0(N,N)

[0(1,N−1)

diag(1, 2, . . . , N − 1)

]

− [0(N−1,1) diag(1, 2, . . . , N − 1)]

0(N−1,N−1)

⎤⎥⎦ .

This FIM must be inverted in order to find the CRLB forthe unknown parameter vector θ . By the form of thematrix in (8) it is easily shown (see Appendix IV) that thematrix is singular with rank equal to two less than fullrank. Weiss [18] uses an ad hoc method to overcome this.We, however use the exact transformation of theparameters [26] that is required to eliminate the singularityof the information matrix. The singularity arises becausethere is an ambiguity in the model. It is not possible touniquely determine all the unknown parameters in themodel in (1). This is because of the relationship betweenthe transmission time, attenuation factor, and the signalwaveform. Suppose that in (1), Ai and s(t) are the truevalues of the gain and the signal waveform that generateri(t). Then the pair of values (Ai/γ, γ s(t)) for anynon-zero constant γ also generate the same ri(t). So, fromthe observation ri(t) it is impossible to determine the truevalues of Ai and s(t). A similar relationship existsbetween the unknown transmission time and an unknownsignal waveform. Suppose that the transmitted signal sT (t)is generated by an unknown signal waveform s(t)transmitted at an unknown transmission time t0 so thatsT (t) = s(t − t0). Now the same transmitted signal sT (t)can also be generated by the pair of values((t0 − γ ), s(t − γ )) for any constant γ . This is moreclearly demonstrated in Fig. 3. Notice that given atransmitted signal, it is not possible to determine whetherthe signal waveform is s1(t) with transmission time t1 ors2(t) with transmission time t2. This causes the informationmatrix to be at least rank two deficient, as shown inAppendix IV. The overparameterization can be resolvedby applying an appropriate transformation that satisfiescertain constraints [26]. As shown in Appendix IV,

the appropriate transformation for this model is

τ ′ = [ (τ1 − τ0) (τ2 − τ0) · · · (τM−1 − τ0)]T

A′ = (1/A0)[A1 · · · AM−1]T

φ′ = A0

⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag(h(−τ0))φ

(9)

and can be shown to be the least restrictive constraint foridentifiability. Here we are using the (∗)

′notation to

represent the new parameters resulting from thetransformation. Notice that the transformed parametervectors τ ′ and A′ are TDOA and relative gain factor withrespect to sensor 0 and are each reduced by one parameterfrom τ and A, respectively, while the φ′ is simply theFourier coefficients of r0(t). Using these transformedparameters the model in (5) can be rewritten as

r0(t)= hT (t)φ′+w0(t) 0≤ t ≤T

ri(t) = A′i h

T (t−τ ′i )φ

′+wi(t), 0≤ t ≤T , i = 1, 2,. . . ,M−1

(10)

where A′i = Ai

A0and τ ′

i = τi − τ0. So, the effect of thetransformation is that the signal at sensor 0 is made thereference signal and the signals at all the other sensors aremodeled relative to this reference signal. Although (10)seems intuitively obvious, by arriving at it from (5) usingthe transformation in (9), we have mathematically verifiedthat (10) is indeed the correct model to use for the problemof localization under the unknown signal case. Weiss[18–21] uses (10) directly without this rigorous argument.This is a subtle but important result which is overlookedby Weiss. Now, let θ ′ = [τ ′T A′T φ′T ]T be the(2M + 2N − 3) × 1 vector of the unknown transformedparameters, η′ = [xT , yT ]T and α′ = [η′T A′Tφ′T ]T .Notice that the TDOA vector τ ′ is a function of only(xT , yT ), i.e.τ ′ = g′(η′). The unknown transmission timet0 does not appear and thus is not a nuisance parameter.The problem is now, given the observationsri(t), i = 0, 1, . . . , M − 1 estimate the vector η′. Thelog-likelihood function for this model with the



Fig. 3. Ambiguity when signal waveform and transmission time are both unknown.

transformed parameters is given by

l(θ ′) = − 1N0

∫ T

0

(r0(t) − hT (t)φ′)2 dt − 1

N0

∫ T

0∑M−1

i=1

(ri(t) − A′

i hT (t − τ ′

i )φ′)2 dt. (11)

As shown in the Appendix I-A, the CRLB for thistransformed parameter vector is

I−1θ ′ = HI

†θ HT (12)

where H is given in (50), or equivalently the FIM is,

Iθ ′ = (T/2)

(N0/2)

⎡⎢⎣

(2πF0)2φ′T LLT φ′(diag(A′))2 (2πF0)(φ′T Lφ′)(diag(A′)) (2πF0)(A′ � A′)φ′T L

(2πF0)(φ′T LT φ′)(diag(A′))2 (φ′T φ′)I(M−1) A′φ′T

(2πF0)LT φ′(A′ � A′)T φ′A′T (1 + A′T A′)I(2N−1)

⎤⎥⎦ . (13)

The FIM for the corresponding vector α′ is given by [25]

Iα′ =(

∂θ ′

∂α′T

)T

Iθ ′

(∂θ ′

∂α′T

)

=(

∂θ ′

∂α′T

)T

(HI†θHT )−1

(∂θ ′

∂α′T

)(14)

where the Jacobian(

∂θ ′∂α′T

)is given in (39). In

Appendix II-A we show that the MLE for the emitter

location is obtained by maximizing over (xT , yT ), themaximum eigenvalue of the M × M cross-correlationmatrix B′ = Y′Y′T =∑M−1

i=0 y′iy

′Ti where

Y′ = [ y′0 y′

1 · · · y′M−1 ] with

y′0 =

∫ T

0r0(t)h(t) dt and y′

i =∫ T

0ri(t)h(t − τ ′

i ) dt,

i = 1, 2, · · · M − 1. (15)

That is,

η′ = arg max lmax(B′)η′ (16)

or equivalently,

(xT , yT ) = arg max(xT ,yT )

lmax(B′) (17)

where lmax represents the maximum eigenvalue. Thematrix B′ is real symmetric and positive definite and so themaximum eigenvalue is real and positive.



B. Signal Known with Unknown Transmission Time

Quite often in practical situations it is possible that thesignal waveform is known but the exact transmission timet0 is unknown. In this case the number of unknowns isreduced to 2M . Let ζ = [τ T AT ]T be the 2M × 1unknown parameter vector. Similar to (7) thelog-likelihood function is given by

l(ζ ) = − 1

N0

∫ T

0

M−1∑i=0

(ri(t) − Ai h

T (t − τi)φ)2

dt (18)

where φ is known. The FIM for this model is given by (seeAppendix II-B)

Iζ = (T/2)

(N0/2)[(2πF0)2φT LLT φ(diag(A))2 (2πF0)(φT Lφ)(diag(A))

(2πF0)(φT LT φ)(diag(A)) (φT φ)IM

].

(19)

This matrix is not singular because, for this case, theunknown parameters in the model can be uniquelydetermined. Therefore, there is no need to transform theparameters as in the case of the unknown signal.Although, the unknown transmission time t0 is stillretained here as the nuisance parameter. For this model itis shown in Appendix II-B that the MLE for emitterlocation and the unknown transmission time is given by

η = arg max φT Bφ

η(20)

where B = YYT and Y = [y0 y1 · · · yM−1] withyi = ∫ T

0 ri(t)h(t − τi) dt, i = 0, 1, · · · M − 1. B is afunction of (xT , yT , t0). Using the fact thathT (t − τi)φ = s(t − τi) and η = [xT yT t0]T we canrewrite (20) as

(xT , yT , t0) = arg max(xT ,yT ,t0)

M−1∑i=0

(∫ T

0ri(t)s(t−τi) dt

)2

. (21)

Equation (21) is simply the correlation values between theknown signal waveform and the observed signal at each ofthe sensors, summed over all the sensors. The emitterlocation that yields the values of the TOAs that maximizethe expression in (21) is the MLE of the emitter location.

IV. SIMULATION RESULTS

In order to evaluate the performance of the MLE wehave run some simulations and compared the performanceagainst the CRLB and against a typically used TDOAapproach. The TDOA approach was implemented as atwo-step algorithm where, in the first step, the TDOAestimates �

τ ′i were obtained by cross-correlating the signal

at each sensor with the signal at sensor 0. Then a 1 km× 1 km region around the true emitter location was splitinto 100 × 100 grid points and for each emitter locationon the grid point the TDOAs were computed using the

formula

τ ′i =

√(xT −xi)2+(yT −yi)2

c−√

(xT −x0)2+(yT −y0)2

c,

i = 1, 2, . . . , M − 1.

Next the least squared error (LSE) between the estimatedTDOAs and the computed TDOAs was calculated as

LSE =M−1∑i=1

(�τ ′

i − τ ′i )

2.

This LSE is a function of the emitter location (xT , yT ). Theemitter location that minimized the LSE is the estimate ofthe emitter location. Since we know that the LSE is a2-dimensional parabolic function of the emitter location,we improved the accuracy by fitting a parabola throughthe 10000 points at which the LSE was computed. Thenby using the analytical formula for the minimum locationof a 2-dimensional parabola, we computed the minimum.Fig. 4(a) shows a realization of the LSE function. For thesimulation we have used 4 sensors placed at thecoordinates shown in Fig. 1 and the emitter was placed atthe coordinates (130, 75) km, also as shown in Fig. 1. Theintegrations in (4) and (15) are approximated usingsummations with δt = 0.33 ns which means the samplingfrequency is Fs = 300 MHz. The reason for choosingsuch high sampling frequency is that at this frequency, theposition quantization error due to sampling is on the orderof (c/Fs) = 10−3 km. A Gaussian chirp defined by

s(t) = exp

(−1

2σ 2

F

(t − Ts

2

)2)

sin(2πmt2)

was used as the unknown transmitted signal waveform.Fig. 5 shows the transmitted signal waveform. Notice thatthe signal is assumed to be approximately zero for t < 0and for t > Ts . We set Ts = 5 μs and σF = 0.2π MHz.The observation interval at each of the sensors was takento be T = 0.2 ms. The unknown transmission time of thesignal was set to t0 = 0.07 ms. With this configuration themaximum TDOA is 0.0988 ms. The frequency spectrumof the Gaussian window is given by |S(F )| = (

√2π/σF )

exp (− 2π2F 2/σ 2F ) [25], and the bandwidth of the chirp is

BW = 1.5 MHz. The rate of change of frequency for thelinear chirp was chosen to be m = BW

Ts= 3 × 108. A plot

of the Fourier coefficients of the signal is shown in Fig. 6.A value of N = 2 × BW × T = 600 was used to have atotal of 2N − 1 = 1199 unknown Fourier coefficients.Notice that the Fourier coefficients are almost zero forn > 600. To measure the levels of the zero mean additivewhite Gaussian noise, we used a metric called theaverage signal-to-noise ratio (ASNR). The ASNR is theratio of the average signal power to the noise power ateach sensor averaged over all the sensors, i.e., ifPsi = |Ai |2 1

T

∫ T

0 |s(t)|2 dt is the average power of thesignal at the ith sensor and N0

2 is the noise spectral densityat the ith sensor, then the SNR averaged over M sensors is

given by 10 log(

1M

∑M−1i=0

Psi

(N0/2)(Fs/2)

)dB. We set the



pc

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pc

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pc

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pc

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pc

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pc

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pc

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pc

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pc

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Fig. 4. (a) Realization of LSE at ASNR = –10 dB. (b) Realization of likelihood function at ASNR = –10 dB.

ASNR at –20 dB and ran a total of 300 Monte Carlosimulations to generate the scatter plot and thecorresponding 95% error ellipse which are shown inFig. 7. This is also called the 95% confidence ellipse. Thatis, if this estimator is used a large number of times forlocalization, then around 95% of those times the truelocation of the emitter will lie within this ellipse. Tocompute the MLE the maximization was performed usinga grid search. We used a grid of size 1 km × 1 km with

the grid points 0.01 km apart to have a total of 100 × 100= 10000 points. The grid is shown with a dotted line in thefigure. At –20 dB the variances of the MLEs of (xT , yT )were (0.0021, 0.0006) km2 and the respective CRLBswere (0.0012,0.0003) km2. The figure does not show 300points because some points lie on top of the others due toposition quantization induced by the finite number of gridpoints in the grid search for maximization. Due to thecomplex nature of the likelihood function (see Fig. 4(b)),



pc

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Fig. 5. Transmitted signal waveform.

Fig. 6. Fourier coefficients plot.

it is not possible to use any curve fitting techniques toreduce this quantization effect as in the case of theconventional TDOA approach.

Fig. 8 shows the comparison of the variances of theMLE and the typical TDOA approach against the CRLBfor different SNR values. Notice that for the ASNR valuesbelow –30 dB, the variance of the MLE remains flat. Thisis because of the restriction imposed by the finite grid size.As the ASNR increases above –30 dB, the variance of theMLE reduces rapidly to approach the CRLB at around–10 dB. Due to the nature of the conventional TDOAapproach it breaks down for the ASNR values below–17 dB. So this figure has the variances of the TDOAapproach only for the average SNR values above –17 dB.On the other hand, for this particular setup, the results for

the MLE are reliable for the ASNR values as low as–30 dB. It is quite obvious from this figure thatperformance of the MLE is very much better than a typicalTDOA approach. In this case the MLE performs as goodas a typical TDOA approach for an ASNR value of about10 dB less than that for the TDOA approach. Also noticethat at around –10 dB, the variance of the MLE is almosttwo orders of magnitude less than that of the TDOAapproach. We have noticed that for certain sensor-emitterconfigurations, particularly when the emitter was veryclose to the sensors, the variance of the MLE is up to threeorders of magnitude less than that of the TDOA approach.Therefore, under low probability of intercept (LPI)scenarios where a conventional TDOA technique cannotbe reliably used, the MLE can be used.



Fig. 7. Scatter plot and corresponding 95% error ellipse of MLE for ASNR = –20 dB.

Fig. 8. Comparison of variances for emitter location estimate using MLE and typical TDOA approach against CRLB for different ASNR values.

V. CONCLUSIONS

We have derived a direct positioning estimator foran emitter location. This is the MLE. We have shownthat for an unknown signal case, the model that isconventionally used has an inherent ambiguity and so allthe unknown parameters cannot be uniquely determined.We derived an appropriate transformation of theparameters and reparameterized the model to remove theambiguity. We have shown that for the special case of aknown signal with unknown transmission time, there is noambiguity in the model. We derived the MLE and the FIMfor the model. The performance of the MLE wascompared against a typical two-step TDOA-basedlocalizer and against the CRLB. The performance of theMLE is significantly better than a typical two-stepTDOA-based localizer.

APPENDIX I. CRLB

We derive the CRLB for the emitter location estimate.First, we show that the FIM for the model used forunknown signal with unknown transmission time case issingular. We then use a transformation of the parametersin the model and derive the CRLB. Letτ = [τ0 τ1 · · · τM−1]T , A = [A0 A1 · · · AM−1]T , andφ = [a0 a1 · · · aN−1 b1 b2 · · · bN−1]T , where

a0 =√

2T

∫ T

0s(t) dt, an = 2

T

∫ T

0s(t) cos 2πnF0t dt,

bn = 2T

∫ T

0s(t) sin 2πnF0t dt.

Let θ = [τT AT φT ]T . The TOAs τi are a function ofthe emitter location (xT , yT ) and the signal transmission



time t0.

τi =√

(xT − xi)2 + (yT − yi)2

c+ t0

where c is the propagation speed of the signal. If l(θ ) isthe log-likelihood function, then the FIM is given by

Iθ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−E

{∂2l(θ )

∂τ∂τ T

}−E

{∂2l(θ)

∂τ∂AT

}−E

{∂2l(θ )

∂τ∂φT

}

−E

{∂2l(θ )

∂A∂τ T

}−E

{∂2l(θ)

∂A∂AT

}−E

{∂2l(θ )

∂A∂φT

}

−E

{∂2l(θ )

∂φ∂τ T

}−E

{∂2l(θ)

∂φ∂AT

}−E

{∂2l(θ )

∂φ∂φT

}

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(22)


From (7) we have the log-likelihood function as

l(θ )=− 1

N0

∫ T

0

M−1∑m=0

(xm(t)−AmhT (t−τm)φ)2 dt (23)

where h(t) is as defined in Appendix III. Partialdifferentiation with respect to (w.r.t) τi gives

∂l(θ)

∂τi

= − 1

N0

∫ T

02(ri(t) − Ai h

T (t − τi)φ)

×(

−Ai

∂hT (t − τi)

∂τi

φ

)dt (24)

and w.r.t Ai gives

∂l(θ)

∂Ai

= − 1

N0

∫ T

02(ri(t) − Ai h

T (t − τi)φ)

× (−hT (t − τi)φ)

dt (25)

for i = 0, 1, . . . , M − 1. Partial differentiation w.r.t φ

gives

∂l(θ)

∂φ= − 1

N0

∫ T

0

M−1∑m=0

2(xm(t) − AmhT (t − τm)φ

)× (−AmhT (t − τm)

)dt. (26)

Next we evaluate the second derivatives. Partialdifferentiation of (24) w.r.t τi gives

∂2l(θ )

∂τ 2i

= − 2

N0

[∫ T

0(ri(t) − Ai h

T (t − τi)φ)

×(

−Ai

∂2hT (t − τi)

∂τ 2i

φ

)dt

+∫ T

0

(−Ai

∂hT (t − τi)

∂τi

φ

)2

dt

].

Taking the negative of the expected value of both sidesand using (48) gives

−E

{∂2l(θ)

∂τ 2i

}= 2

N0

T∫0

(Ai

∂hT (t − τi)

∂τi

φ

)2

dt

= A2i

(N0/2)φT

⎡⎣ T∫

0

∂h(t−τi)

∂τi

∂hT (t−τi)

∂τi

⎤⎦φ

= (T/2)(2πF0)2

(N0/2)(φT LLT φ) A2

i

where the 2N − 1 × 2N − 1 matrix L is as defined inAppendix III. Since ∂2l(θ )

∂τi∂τj= 0 for i �= j, we have

−E

{∂2l(θ )

∂τ∂τ T

}= (T/2)(2πF0)2

(N0/2)φT LLT φ(diag(A))2 (27)

where diag(A) is an M × M diagonal matrix with ithdiagonal element as Ai . Partial differentiation of (25) w.r.tτi gives

∂2l(θ)

∂τi∂Ai

= − 2

N0

⎡⎣ T∫

0

(ri(t) − Ai h

T (t − τi)φ)

×(

−∂hT (t−τi)

∂τi

φ

)dt

+T∫

0

(−Ai

∂hT (t−τi)

∂τi

φ

)(−hT (t−τi)φ) dt

⎤⎦ .


−E

{∂2l(θ)

∂τi∂Ai

}= Ai

(N0/2)φT

⎡⎣ T∫

0

∂h(t − τi)

∂τi

hT (t−τi) dt

⎤⎦φ

= (T/2)(2πF0)

(N0/2)(φT Lφ) Ai.

Since ∂2l(θ )∂τi∂Aj

= 0 for i �= j , we have

−E

{∂2l(θ )

∂τ∂AT

}= (T/2)(2πF0)

(N0/2)(φT Lφ)(diag(A)). (28)

Partial differentiation of (26) w.r.t τi gives

∂2l(θ )

∂τi∂φT= − 2

N0

⎡⎣ T∫

0

(ri(t) − Ai h

T (t − τi)φ)

×(

−Ai

∂hT (t − τi)

∂τi

)dt

+T∫

0

(−Ai

∂hT (t−τi)

∂τi

φ

)(−Ai h

T (t−τi)) dt

⎤⎦ .




−E

{∂2l(θ )

∂τi∂φT

}= A2

i

(N0/2)φT

⎡⎣ T∫

0

∂h(t−τi)

∂τi

hT (t−τi) dt

⎤⎦

= (T/2)(2πF0)A2i

(N0/2)φT L.

So, we have

−E

{∂2l(θ )

∂τ∂φT

}= (T/2)(2πF0)

(N0/2)(A � A)φT L. (29)

Partial differentiation of (25) w.r.t Ai and using (47) gives

∂2l(θ)

∂A2i

= − 1

(N0/2)

T∫0

(hT (t − τi)φ)2

= − 1

(N0/2)φT

⎡⎣ T∫

0

h(t − τi)hT (t − τi)

⎤⎦φ

= − (T/2)

(N0/2)φT φ.

Taking the negative of the expected value on both sidesgives

−E

{∂2l(θ )

∂A∂AT

}= (T/2)φT φ

(N0/2)IM. (30)

Partial differentiation of (26) w.r.t Ai gives

∂2l(θ)

∂Ai∂φT=− 2

N0

⎡⎣ T∫

0

(ri(t)−Ai h

T (t−τi)φ)(−hT (t−τi)

)dt

+T∫

0

(−hT (t − τi)φ)(−Ai hT (t − τi)) dt

⎤⎦ .


−E

{∂2l(θ)

∂Ai∂φT

}= 2Ai

N0

T∫0

hT (t − τi)φhT (t − τi) dt

= 2Ai

N0φT

T∫0

h(t − τi)hT (t − τi) dt

= (T/2)

(N0/2)Aiφ

T

and so

−E

{∂2l(θ )

∂A∂φT

}= (T/2)

(N0/2)AφT . (31)

Partial differentiation of (26) w.r.t φ and using (47) gives

∂2l(θ )

∂φ∂φT=− 2

N0

T∫0

M−1∑i=0

(−Ai hT (t−τi)

)(−Ai hT (t−τi)

)dt

= −M−1∑i=0

A2i

(N0/2)

T∫0

h(t − τi)hT (t − τi) dt

= −M−1∑i=0

(T/2)

(N0/2)I(2N−1) A2

i

and so

−E

{∂2l(θ )

∂φ∂φT

}= (T/2)

(N0/2)I(2N−1)

M−1∑i=0

A2i = (T/2)AT A

(N0/2)I(2N−1).

(32)

Putting (27), (28), (29), (30), (31), (32) back in (22), wehave

Iθ = (T/2)

(N0/2)

⎡⎢⎣


(2πF0)(φT LT φ)(diag(A)) (φT φ)IM AφT

(2πF0)LT φ(A � A)T φAT (AT A)I(2N−1)

⎤⎥⎦ . (33)

The CRLB matrix for the unknown parameter vector θ isthe inverse of the matrix Iθ . But in Appendix IV it isshown that the null space of Iθ is not empty and so it is notinvertible. This is because the log-likelihood function isnot uniquely defined by the model in (7). To eliminate theoverparameterization we use the followingtransformations.

τ ′ = [ (τ1 − τ0) (τ2 − τ0) · · · (τM−1 − τ0) ]T

A′ = (1/A0)[A1 · · · AM−1]T

φ′ = A0

⎡⎢⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎥⎦ diag(h(−τ0))φ.

(34)

Let θ ′ = [τ ′T A′T φ′T ]. This is a function of θ . Let

H =(

∂θ ′∂θ

)be the Jacobian. If H has row vectors that are

linear combinations of those eigenvectors of Iθ that havenonzero eigenvalues then the CRLB of θ ′ is given byHI†

θHT [26]. The † is used to represent the generalizedinverse. This condition is verified in Appendix IV.Therefore,

I−1θ ′ = HI

†θ HT . (35)



Alternately, the log-likelihood function for this model withthe transformed parameters is given by

l(θ ′) = − 1

N0

∫ T

0

(r0(t) − hT (t)φ′)2 dt

− 1

N0

∫ T

0

M−1∑i=1

(ri(t) − A′

i hT (t − τ ′

i )φ′)2 dt. (36)

So, computing the derivatives to find the FIM as donepreviously yields the FIM for the transformed parameter

vector as

Iθ ′ = T/2

(N0/2)

⎡⎢⎢⎢⎣

(2πF0)2φ′T LLT φ′(diag(A′))2 (2πF0)(φ′T Lφ′)(diag(A′)) (2πF0)(A′ � A′)φ′T L

(2πF0)(φ′T LT φ′)(diag(A′)) (φ′T φ′)IM−1 A′φ′T

(2πF0)LT φ′(A′ � A′)T φ′A′T (1 + A′T A′)I(2N−1)

⎤⎥⎥⎥⎦ . (37)

We have verified numerically that (35) is equivalent to (37). The elements of the TDOA vector τ ′ are given by

τ ′i = (τi − τ0) =

√(xT − xi)2 + (yT − yi)2

c−√

(xT − x0)2 + (yT − y0)2

c

so that the new parameter vector τ ′ is a function of only the emitter location (xT , yT ). So, if we let η′ = [xT yT ]T andα′ = [η′T A′T φ′T ]T we have

Iα′ =(

∂θ ′

∂α′T

)T

Iθ ′

(∂θ ′

∂α′T

)=(

∂θ ′

∂α′T

)T

(HI†θHT )−1

(∂θ ′

∂α′T

). (38)

The Jacobian is given by the (2M + 2N − 3, M + 2N) matrix

(∂θ ′

∂α′T

)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(∂τ ′

∂η′T

) (∂τ ′

∂A′T

) (∂τ ′

∂φ′T

)(

∂A′

∂η′T

) (∂A′

∂A′T

) (∂A′

∂φ′T

)(

∂φ′

∂η′T

) (∂φ′

∂A′T

) (∂φ′

∂φ′T

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎣

(∂τ ′

∂η′T

)0(M,M) 0(M,2N−1)

0(M,2) IM−1 0(M,2N−1)

0(2N−1,2) 0(2N−1,M) I(2N−1)

⎤⎥⎥⎥⎥⎥⎥⎦

(39)

where the (M − 1) × 2 matrix

(∂τ ′

∂η′T

)= (1/c)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(xT − x1)

d1− (xT − x0)

d0

(yT − y1)

d1− (yT − y0)

d0

(xT − x2)

d2− (xT − x0)

d0

(yT − y2)

d2− (yT − y0)

d0

......

(xT − xM−1)

dM−1− (xT − x0)

d0

(xT − xM−1)

dM−1− (yT − y0)

d0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(40)

where di, i = 0, . . . , M − 1 is the distance between thesensor i and the emitter.


From (18) we have the log-likelihood function as

l(ζ ) = − 1

N0

∫ T

0

M−1∑i=0

(ri(t) − Ai h

T (t − τi)φ)2

dt. (41)



Here the 2M × 1 unknown parameter vector isζ = [τ T AT ]T . So, the FIM is given by

Iζ =

⎡⎢⎢⎢⎣

−E

{∂2l(θ )

∂τ∂τ T

}−E

{∂2l(θ )

∂τ∂AT

}

−E

{∂2l(θ )

∂A∂τ T

}−E

{∂2l(θ )

∂A∂AT

}⎤⎥⎥⎥⎦ . (42)

Using (27), (28), and (30), we have

Iζ = (T/2)

(N0/2)[(2πF0)2φT LLT φ(diag(A))2 (2πF0)(φT Lφ)(diag(A))

(2πF0)(φT LT φ)(diag(A)) (φT φ)IM

].

(43)

APPENDIX II. MAXIMUM LIKELIHOOD ESTIMATOR

Here we derive the MLE for the two cases of signalunknown with unknown transmission time and signalknown with unknown transmission time.


The log-likelihood function with the transformedparameters is given by

l(θ ′) = − 1

N0

∫ T

0

(r0(t) − hT (t)φ′)2 dt

− 1

N0

∫ T

0

M−1∑i=1

(ri(t)−A′

i hT (t−τ ′

i )φ′)2 dt. (44)

Partial differentiation w.r.t φ′ gives

∂l(θ ′)∂φ′ = − 1

N0

∫ T

02(r0(t) − hT (t)φ′) (−hT (t)

)dt

− 1

N0

∫ T

0

M−1∑i=1

2(ri(t)

− A′i h

T (t − τ ′i )φ

′) (−A′i h

T (t − τ ′i ))

dt.

In order to find the maximum, we equate the above partialderivative to zero, which gives∫ T

0r0(t)hT (t) dt − φ′T

(∫ T

0h(t)hT (t) dt

)

+M−1∑i=1

A′i

(∫ T

0ri(t)h

T (t − τ ′i ) dt

)

− A′2i φ′T

(∫ T

0h(t − τ ′

i )hT (t − τ ′

i ) dt

)= 0.

Using the properties of the vector h(t) as shown inAppendix III, we have∫ T

0r0(t)hT (t) dt − (T/2)φ′T

+M−1∑i=1

A′i

(∫ T

0ri(t)h

T (t−τ ′i ) dt

)−(T/2)

M−1∑i=1

A′2i φ′T = 0.

If we replace the integrals with

y′0 =

∫ T

0r0(t)h(t) dt and y′

i

=∫ T

0ri(t)h(t − τ ′

i ) dt for i = 1, 2, . . . M − 1

then we have

y′T0 − (T/2)φ′T +

M−1∑i=1

A′iy

′Ti − (T/2)

M−1∑i=1

A′2i φ′T = 0.

So, the MLE of φ′ is

φ′ =(2/T )

(y′

0 +∑M−1i=1 A′y′

i

)(1 + A′T A′)

.

Putting this back in (44), we have

l(θ ′)=− 1

N0

M−1∑i=0

∫ T

0x2

i (t) dt

+ 1

N0

(2/T )(y′T

0 +∑M−1i=1 A′

iy′Ti

)(y′

0+∑M−1

i=1 A′iy

′i

)(1+A′T A′)

.

Maximizing l(θ ′) w.r.t A′ and τ ′ is equivalent tomaximizing the second term. So, let

f (A′, τ ′) =(

y′T0 +∑M−1

i=1 A′iy

′Ti

) (y′

0 +∑M−1i=1 A′

iy′i

)(1 + A′T A′)

.

If we let Y′ = [y′0 y′

1 · · · y′M−1] be the 2N − 1 × M matrix

then the maximum value of f (A′, τ ′) w.r.t A′ is fmax(τ ′)is equal to the maximum eigenvalue of Y′Y T . LetB′ = YY′T . The matrix B′ is a function of τ ′ = g′(η′)which is a function of the emitter location (xT , yT ). So, theMLE of the emitter location is found by maximizing themaximum eigenvalue of B′(xT , yT ). i.e,

(xT , yT ) = arg max(xT ,yT )

λmax(B′(xT , yT )). (45)


The log-likelihood function is given by

l(ζ ) = − 1

N0

∫ T

0

M−1∑i=0

(ri(t) − Ai h

T (t − τi)φ)2

dt. (41)

Partial differentiation w.r.t Ak gives

∂l(θ)

∂Ak

= − 1

N0

∫ T

02(rk(t)−AkhT (t−τk)φ

)(−hT (t−τk)φ)

dt = 0

for each of k = 0, 1, . . . , M − 1. Equating this to zero tofind the maximum value gives



φT

(∫ T

0rk(t)h(t − τk) dt

)

−AkφT

(∫ T

0h(t − τk)hT (t − τk) dt

)φ = 0.

If we replace the integral with yk = ∫ T

0 rk(t)h(t − τk) dt

and use the properties of the vector h(t) as shown inAppendix III, we have

φT yk − (T/2)AkφT φ = 0.

So, the MLE of Ak is

Ak = φT yk

(T/2)φT φ, k = 0, 1, . . . , M − 1.

Putting this back in (41), we have

l(ζ ) = − 1

N0

M−1∑i=0

∫ T

0r2

1 (t) dt − 2

(φT yi

(T/2)φT φ

)

×(∫ T

0ri(t)h

T (t − τi) dt

)φ

+(

φT yi

(T/2)φT φ

)2

φT

(∫ T

0h(t−τi)h

T (t−τi) dt

)φ

= − 1

N0

M−1∑i=0

∫ T

0r2i (t) dt + 1

N0

M−1∑i=0

(φT yiyT

i φ

(T/2)φT φ

).

Maximizing l(ζ ) w.r.t τ is equivalent to maximizing thesecond term. So the MLE for η is given by

η = arg maxη

M−1∑i=0

φT yiyTi φ = arg max

η

φT Bφ (46)

where B = YYT and Y = [y0 y1 · · · yM−1] withyi = ∫ T

0 τi(t)h(t − τi) dt, i = 0, 1, . . . M − 1. B is afunction or (xT , yT , t0).

APPENDIX III. PROPERTIES OF h(t)

The time dependent vector h(t) that was used formodeling the problem in equation (5) has some interestingproperties which simplify the derivation of the CRLB andthe MLE. These properties are derived here. We have

h(t − τi) =[

1√2

cos 2πF0(t − τi) · · · cos 2π(N − 1)F0(t − τi) sin 2πF0(t − τi) · · · sin 2π(N − 1)F0(t − τi)

]T

.

Differentiating both sides w.r.t τi , we get

∂h(t − τi)

∂τi

= 2πF0[0 sin 2πF0(t − τi) · · · 2 sin 2π2F0(t − τi) · · · (N − 1) sin 2π(N − 1)F0(t − τi)

− cos 2πF0(t − τi) − 2 cos 2π2F0(t − τi) · · · − (N − 1) cos 2π(N − 1)F0(t − τi)]T .

Let

L =

⎡⎢⎢⎣ 0(N,N)

[0(1,N−1)

diag(1, 2, . . . , N − 1)

]

−[

0(N−1,1) diag (1, 2, . . . , N − 1)]

0(N−1,N−1)

⎤⎥⎥⎦ .

So, we have the partial derivative of h(t − τi) w.r.t to τi as

∂h(t − τi)

∂τi

= (2πF0)Lh(t − τi).

Next we compute the integral∫ T

0 h(t)hT (t) dt . We have

h(t)hT (t)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1√2

cos 2πF0t

...

cos 2π(N − 1)F0t

sin 2πF0t

...

sin 2π(N − 1)F0t

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[1√2

cos 2πF0t · · · cos 2π(N−1)F0t sin 2πF0t · · · sin 2π(N−1)F0t

].



pc

Highlight

Let us compute each of the integrals in this2N − 1× 2N − 1 product matrix separately. Integral ofthe first element is∫ T

0

1

2dt = [t]T0 = T

2.

Integrals of the elements on the diagonal are given by∫ T

0cos22πkF0t dt = 1

2

∫ T

0(1 + cos 4πkF0t) dt

= 1

2

[t + sin 4πkF0t

4πkF0

]T

0

= T

2and ∫ T

0sin22πkF0t dt = 1

2

∫ T

0(1 − cos 4πkF0t) dt

= 1

2

[t − sin 4πkF0t

4πkF0

]T

0

= T

2for k = 1, 2, . . . N − 1. Integrals of the rest of theelements are given by∫ T

0cos2πkF0t sin 2πnF0t dt

= 1

2

∫ T

0sin 2π(n + k)F0t dt + sin 2π(n − k)F0t dt

= 1

2

[−cos 2π(n + k)F0t

4π(n + k)F0− cos 2π(n − k)F0t

4π(n − k)F0

]T

0

= 0

and∫ T

0cos2πkF0t cos 2πnF0t dt

= 1

2

∫ T

0cos 2π(n + k)F0t dt + cos 2π(n − k)F0t dt

= 1

2

[sin 2π(n + k)F0t

4π(n + k)F0+ sin 2π(n − k)F0t

4π(n − k)F0

]T

0

= 0

and∫ T

0sin2πkF0t sin 2πnF0t dt

= 1

2

∫ T

0cos 2π(k − n)F0t dt − cos 2π(n + k)F0t dt

= 1

2

[sin 2π(k − n)F0t

4π(k − n)F0+ sin 2π(n + k)F0t

4π(n + k)F0

]T

0

= 0

for k, n = 1, 2, . . . , N − 1. Therefore, we have theintegral of h(t)hT (t) as a scaled identity matrix given by

∫ T

0h(t)hT (t) dt = T

2

⎡⎢⎢⎢⎢⎢⎣

1 0 · · · 0

0 1 · · · 0

......

. . ....

0 0 · · · 1

⎤⎥⎥⎥⎥⎥⎦ = (T/2)I(2N−1).

Since h(t) is periodic with period T , for any τ ,∫ T

0h(t − τ )hT (t − τ ) dt =

∫ T

0h(t)hT (t) dt

= (T/2)I(2N−1). (47)

Now, we compute the integral of the cross-product of thepartial derivatives of h(t − τi) w.r.t to τi∫ T

0

∂h(t − τi)

∂τi

∂hT (t − τi)

∂τi

=∫ T

0(2πF0Lh(t − τi)) (2πF0Lh(t − τi))

T dt

= (2πF0)2L[∫ T

0h(t − τi)h

T (t − τi) dt

]LT

= (T/2)(2πF0)2LLT (48)

and the integral of the cross-product of h(t − τi) with itspartial derivative w.r.t to τi is∫ T

0

∂h(t − τi)

∂τi

hT (t − τi)

=∫ T

02πF0Lh(t − τi)h

T (t − τi) dt

= (2πF0)L[∫ T

0h(t − τi)h

T (t − τi) dt

]= (T/2)(2πF0)L. (49)

APPENDIX IV. TRANSFORMATIONOF THE PARAMETERS

In Section III we discussed the relationship betweenthe unknown attenuation factors and the unknown signal,and between the unknown TOAs and the unknown signal.Here we show that the FIM given in (8) is rank twodeficient. Then we show that the transformation given in(9) satisfies the conditions given in [26]. From (8), we have

Iθ = (T/2)

(N0/2)

⎡⎢⎣


(2πF0)(φT LT φ)(diag(A)) (φT φ)IM AφT

(2πF0)LT φ(A � A)T φAT (AT A)I(2N−1)

⎤⎥⎦ .



If ν1 = [1TM AT − (φ + (2πF0)LT φ) ]T and

ν2 = [1TM − AT (φ − (2πF0)LT φ) ]T then it can be

verified that Iθν1 = 0 and Iθν2 = 0. Therefore ν1 and ν2

are in the null space of Iθ . Also, Iθ+(1/2)ν1νT1

+(1/2)ν2 νT2 is nonsingular. This means that ν1 and ν2 are

the basis vectors for the null space of Iθ and so the matrixIθ is rank two deficient. The Jacobian of thetransformation is given by

H =(

∂θ ′

∂θ

)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂τ ′

∂τ

∂τ ′

∂A∂τ ′

∂φ

∂A′

∂τ

∂A′

∂A∂A′

∂φ

∂φ′

∂τ

∂φ′

∂A∂φ′

∂φ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (50)

Now, we compute each of the derivatives in theJacobian matrix. The elements of the first subcolumn aregiven by

∂τ ′

∂τ= ∂

∂τ([−1M−1 IM−1]τ )

= [−1M−1 IM−1]∂τ ′

∂A= 0(M−1,M)

∂τ ′

∂φ= 0(M−1,2N−1).

The elements of the second subcolumn are given by

∂A′

∂τ= 0(M−1,M)

∂A′

∂A= ∂

∂A

(([0(M−1,1) IM−1]A

) (eT

1 A)−1)

= (1/A0)[−A′ IM−1]

∂A′

∂φ= 0(M−1,2N−1)

where e1 is the first column of an M × M identity matrix.The elements of the third subcolumn are given by

∂φ′

∂τ= ∂

∂τA0

⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag (h(−τ0))φ

=

⎡⎢⎣A0

⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag

(∂h(−τ0)

∂τ0

)φ 0(2N−1,1) · · · 0(2N−1,1)

⎤⎥⎦

=

⎡⎢⎣A0

⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ (2πF0)L diag (h(−τ0))φ 0(2N−1,1) · · · 0(2N−1,1)

⎤⎥⎦

∂φ′

∂A= ∂

∂A

⎛⎜⎝(eT

1 A)⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag (h(−τ0))φ

⎞⎟⎠

=

⎡⎢⎣⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag (h(−τ0))φ 0(2N−1,1) · · · 0(2N−1,1)

⎤⎥⎦

∂φ′

∂φ= ∂

∂φ

⎛⎜⎝(eT

1 A)⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag (h(−τ0))φ

⎞⎟⎠

=

⎡⎢⎣A0

⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[IN−1 IN−1

IN−1 −IN−1

]⎤⎥⎦ diag (h(−τ0)) 0(2N−1,1) · · · 0(2N−1,1)

⎤⎥⎦ .



For the transformed parameters to have finite variance, therow vectors of H must be equal to the linear combinationsof those eigenvectors of Iθ that have nonzero eigenvalues[26]. In order to show that the row vectors of H are linearcombinations of those eigenvectors of Iθ that havenonzero eigenvalues, it is enough to show that the rowvectors of H are orthogonal to the null space of Iθ . That is,it is enough to show that Hν1 = 0 and Hν2 = 0 Now,

Hν1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂τ ′

∂τ1M + ∂τ ′

∂AA − ∂τ ′

∂φ(φ + (2πF0)LT φ)

∂A′

∂τ1M + ∂A′

∂AA − ∂A′

∂φ(φ + (2πF0)LT φ)

∂φ′

∂τ1M + ∂φ′

∂AA − ∂φ′

∂φ(φ + (2πF0)LT φ)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Substituting the partial derivatives that we computedpreviously and further simplifying gives

Hν1

=

⎡⎢⎢⎢⎢⎣

[−1M−1I(M−1)]1M+0(M−1,1)−0(M−1,1)

0(M−1,1)+(1/A0)[−A′ I(M−1)]A−0(M−1,1)

A0P(2πF0)Ldiag(h(−τ0))φ+A0Pdiag(h(−τ0))φ

−(A0Pdiag(h(−τ0))φ+A0Pdiag(h(−τ0))(2πF0)LT φ)

⎤⎥⎥⎥⎥⎦

where

P =

⎡⎢⎣

1 0(1,2N−2)

0(2N−2,1)

[I(N−1) I(N−1)

I(N−1) −I(N−1)

]⎤⎥⎦ .

Using the fact that Ldiag (h(−τ0)) = diag (h(−τ0))LT ,

we have Hν1 = 0. Similarly it can be shown that Hν2 = 0.

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Naresh Vankayalapati received his B.Tech degree in electronics and communicationsengineering from the Jawaharlal Nehru Technological University in 2003. He received aMaster’s degree in electrical engineering in 2005, a second master’s in mathematics in2011, and the Ph.D degree in electrical engineering in 2012 all from the University ofRhode Island, Kingston.

He worked at the MathWorks from 2005 to 2007. His research interests are in thefield of statistical signal processing with emphasis on inference techniques, bothBayesian and frequentist.

Steven Kay was born in Newark, NJ, on April 5, 1951. He received the B.E. degreefrom Stevens Institute of Technology, Hoboken, NJ, in 1972, the M.S. degree fromColumbia University, New York, in 1973, and the Ph.D. degree from Georgia Instituteof Technology, Atlanta, GA, in 1980, all in electrical engineering.

From 1972 to 1975, he was with Bell Laboratories, Holmdel, NJ, where he wasinvolved with transmission planning for speech communications and simulation andsubjective testing of speech processing algorithms. From 1975 to 1977 he attendedGeorgia Institute of Technology to study communication theory and digital signalprocessing. From 1977 to 1980, he was with the Submarine Signal Division,Portsmouth, RI, where he engaged in research on autoregressive spectral estimation andthe design of sonar systems. He is presently a Professor of Electrical Engineering at theUniversity of Rhode Island, Kingston, and a consultant to numerous industrialconcerns, the Air Force, the Army, and the Navy. His current interests are spectrumanalysis, detection and estimation theory, and statistical signal processing.

As a leading expert in statistical signal processing, Dr. Kay has been invited to teachshort courses to scientists and engineers at government laboratories, including NASAand the CIA. He has written numerous journal and conference papers and is acontributor to several edited books. He is the author of the textbooks, Modern SpectralEstimation (Prentice-Hall, 1988), Fundamentals of Statistical Signal Processing, Vol. I:Estimation Theory (Prentice-Hall, 1993), Fundamentals of Statistical SignalProcessing, Vol. II: Detection Theory (Prentice-Hall, 1998), and Intuitive Probabilityand Random Processes using MATLAB (Springer, 2005).

Dr. Kay is a Fellow of the IEEE, and a member of Tau Beta Pi and Sigma Xi. He hasbeen a distinguished lecturer for the IEEE Signal Processing Society. He has been anAssociate Editor for the IEEE Signal Processing Letters and the IEEE Transactions onSignal Processing. He has received the IEEE Signal Processing Society EducationAward “for outstanding contributions in education and in writing scholarly books andtexts. . .” He has recently been included on a list of the 250 most cited researchers in theworld in engineering.



Quan Ding received the B.S. degree from Shanghai Jiao Tong University, Shanghai, in2006, and the M.S. and Ph.D. degrees from University of Rhode Island, Kingston, RI,in 2008 and 2011, respectively, all in electrical engineering.

Since 2011 he has been a Postdoctoral Research Associate at the University ofRhode Island. His primary research interests include statistical signal processing,detection and estimation theory, sensor fusion, spectral analysis, and biomedical signalprocessing.



nomenclature t unknown location of the emitter tdoa based

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