Performance analysis of ETDGE - an efficient and unbiased TDOA estimator

H.C. So P.C. Chi ng

Indexing terms: Time dqference of arrival, Auhptive signal processing

Abstract: A computationally efficient estimator is proposed which can provide instantaneous delay measurement for many practical applications such as radar, sonar and geolocation via satellites. It consists of an adaptive FIR filter to model the time difference of arrival (TDOA) between the received signals from two sensors and a variable gain to adapt to the changing signal-to-noise ratio environment. The TDOA is obtained directly on a sample-by-sample basis and no interpolation is necessary. The proposed algorithm gives an unbiased delay estimate, and its performance for both stationary and nonstationary conditions is evaluated rigorously. It is also proved that the least-squares realisation of the estimator attains the Cram&-Rao lower bound.

1 Introduction

The topic of accurate estimation of the time delay between two noisy versions of the same signal received at two spatially separated sensors has attracted much attention in the literature [l]. One important applica- tion is to locate the position of a signal source in sonar and radar systems [2] . Recently, time delay estimation (TDE) has also been used in global positioning systems [3]. In contrast to other geolocation methods [4], this technique does not require disruption of the normal satellite operation since delay estimation is obtained passively. Furthermore, additional spaceborne hard- ware is not needed, which makes the TDE approach more cost-effective. By making use of the differential delay measurements of an uplink signal received by three or four geostationary satellites, the location of the target transmitter can be determined. Three satellites are needed when the emitter is on the earth’s surface and four if its altitude is not known CI priori [3].

Given the two received discrete-time sensor outputs

z ( k ) = s ( k ) + n ~ ( k ) and y(k) = s ( k - D ) + m ( k ) (1)

0 IEE, 1998 IEE Proceedings online no. 19982431 Paper first received 28th November 1997 and in revised form 12th May 1998 H.C. So is with the Department of Electronic Engineering, City Univer- sity of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong P.C. Ching is with the Department of Electronic Engineering, The Chi- nese University of Hong Kong, Shatin, N.T., Hong Kong

where s (k) is the unknown source signal, n, (k ) and /zZ(k) are the uncorrelated white Gaussian noises, and D is the time difference of arrival (TDOA) to be deter- mined. Without loss of generality, the sampling period is set to unity. I t is assumed that the signal and the cor- rupting noises are bandlimited between -0.5 and 0.5 because the received signals are usually lowpass filtered before sampling.

In this paper, an efficient and unbiased algorithm, called the explicit time delay and gain estimator (ETDGE) [5], is proposed to find the time delay between the two spatially separated sensor outputs. The ETDGE consists of an adaptive FIR filter for explicit delay measurement and a variable gain that provides a scaling factor for optimal filtering. It is sim- ple to implement and involves relatively few computa- tions. I t can also track nonstationary delays under conditions of moving source andlor sensors. Based on the Wiener solution, the adaptive ETDGE is derived. The unbiasedness of the algorithm is shown by examin- ing the global minimum of the performance surface, while its computational efficiency is illustrated by com- paring it with the conventional least-mean-square time delay estimator (LMSTDE) [6]. Performance analysis for both static and linearly time-varying delay is given as well. It is proved that the delay variance of an opti- mal realisation of the ETDGE achieves the CramCr- Rao lower bound (CRLB) for all signal-to-noise ratios (SNRs).

2 TheETDGE

In this Section, a cost function based on the Wiener solution for passive time delay estimation is developed to provide unbiased delay estimates which are free of interpolation error due to finite filter length. An LMS- style realisation of this method, ETDGE, which con- sists of an adaptive delay element to model the time difference between x ( k ) and y (k ) and a variable gain for optimum filtering, is then devised. We examine the convergence behaviour and mean square error of the delay estimate for both static and linearly time-varying delays. A continuous updating equation for the gain parameter is also established.

Consider the received signal x(k) to be filtered using a noncasual filter with impulse response { w ( k ) } , k = -P, -P + I , ..., P. This filter is designed in such a way that the output is as close to y (k ) as possible in the mean-square-error sense. For ease of derivation, it is assumed that the source signal s(k) is an ideal lowpass process [7] with Gaussian probability density function. For passive localisation, this assumption is not imprac-

325 1EE Proc.-Radar, Sonar Navig., Vol. 145, No. 6, December 199X

tical, although there are applications that the source signals are narrowband and nonwhite. We shall later on demonstrate the applicability of the algorithm for nonwhite signals. Denote the signal power by a,’ and the variance of the corrupting noises by 0;. Differenti- ating the cost function J(w) = E { b ( k ) - C; w(i).u(k - i))’} with respect to w(i) and equating the resultant expression to zero, we obtain the minimum mean- square optimum filter coefficients or the Wiener solu- tion for time-delay estimation,

t i ~ O ( z ) = osinc(z - D ) (2) where the gain CI !a?/(o? + 03 = SNR/(l + SNR) and sinc(v) 2 sin(m)/(m). To find nonintegral delays, it is necessary to interpolate the filter weights [6, 81 to obtain the delay estimate, D, which is given by

- P 5 i 5 P

I { z$P w0(2)sinc(r - 2 ) 1 (3) D = argmax

If the filter length P is infinite, then will be ex_actly equal to D. However, it has been shown [8] that D is a biased estimate of D for finite filter length, although the delay bias decreases as the filter length increases. For example, when P = 5, the largest possible error is 8.2% and reduces to 4% when P increases to 10. In this paper we attempt to devise an accurate delay estimation method which is not only free of interpolation error but is also computationally efficient. To do so, we first introduce another cost function J( 6, D) as follows:

J ( & , D) f / P

z=-P ’ I (4)

where 6 and D denote the estimates of the gain CI and the delay D, respectively. Note that ~ ( i ) is now being decoupled into two variables, namely 6 and sinc(i - D ) , which are explicitly related to the SNR and delay parameters. Expanding eqn. 4 gives

E’

~ ( 6 ~ 0 ) = 0: + 0; + ii2(a: + 0;) sinc’(i - 0) z=-P

P

- 2&,“ sinc(i - D)sinc(i - D) 2 = P

(5) Apparently, the performance surface J( e , d) is multi- modal. Parfial differentiation of J ( 6 , D ) with respect to I3 and D yields

2=-P P

- 203 sinc(i - D)sinc(i - 6) z=- P

(6) and

a J ( & D) aD

326

I’

= --2h2(0: + o:L) sinc(i - D ) , f ( i - D) 2= f’

E’

+ 2 6 ~ : sinc(i - ~ ) f ( i - 6) z= f’

( 7 ) Setting eqns. 6 and 7 to zero, it can be shown that the global minimum occurs when

6 = Q ancl D = D (8) That means the global minimum of J(w) and J ( 6 , d) are essentially identical. However, the former is a non- linear function of D, while the latter provides a direct delay measurement. Since no interpolation is needed, the delay estimate here, as compared with the conven- tional methods [6, 8, 91, is unbiased and is free from modelling error.

/i

Y(k) G Fig. 1 System block diugrum of e-xplicit time deluy and guin estimutor IETDGE)

Using the stochastic gradient descent technique, an adaptive realisation for minimising J( 6, D ) , namely the ETDGE, is suggested and its system block diagram is depicted in Fig. 1. Basically, it consists of a variable gain and an explicit time-delay estimator (ETDE) [lo, 111. The ETDE is employed to compensate the actual time difference between ~ ( k ) ?nd y ( k ) . The fiIter coeffi- cients given by {sinc(i - D ( k ) ) } , -P 5 i 5 P,-are expressed as a function of the delay estimate, D(k) , only. The variable gain, &(k) , is introduced to provide a scaling factor as close to q,’/(q,2 + a,A as possible so that optimal solution can be achieved. In the ETDGE, the output error, e(k), is computed from

P

e ( k ) = y(k) - ~ ; ( k ) sinc(i - D ( k ) ) z ( k - i) (9)

By differentiating e2(k) with respect to 6 ( k ) and d(k), stochastic gradient estimates which are similar to those in the LMS algorithm are obtained. The gain parameter and the estimated delay are adapted independently and iteratively to minimise the instantaneous square output error, according to the following equations:

a=-p

and

P

= B ( k ) - p D e ( k ) f ( i - f i ( k ) ) . r ( k - i )

(11)

i=-p

where pa and p D are positive scalars that control the convergence rate and ensure system stability, whilst the function f is defined as f ( v ) (COS(JIV) - sinc(v))/v. Com- paring the ETDE and ETDGE side-by-side, we can see that the updating equation of the delay estimate is the same for both methods, but the gain estimate & ( k ) in eqn. 9 for the ETDGE is fixed to unity in the case of the ETDE. It has been shown [5] that the ETDGE pro- vides a more accurate delay estimation than the ETDE, particularly at low SNR, although it requires one more addition and three more multiplications to compute the error function and &(k) . It is worthy to note that eqns. 10 and 11 are similar to the multipath gain and delay updating rules, respectively, of the adaptive multipath canceller (AMC) [12] which is derived for estimating the interpath delay of a source signal in a multipath environment at high SNR. However, the AMC cannot provide accurate delay estimates when the multipath is not highly resolvable or when the SNR is low.

To reduce computation, values of the sinc and f func- tion are retrieved from prestored tables [IO]. It is shown that ( 6 P + 4) additions and (6P +, 8) multiplications are required to calculate d ( k ) and D(k) at every sampling point. While the LMSTDE needs (24P + 2) additions and (24P + 13) multiplications for the same filter length, assuming that the delay estimate has a resolu- tion of 0.001. Notice that the computational load of the ETDGE is significantly less than the LMSTDE because it does not involve interpolation of filter weights.

It is required to examine the learning rates of the gain parameter and delay estimate of the ETDGE algo- rithm by evaluating their" expected values at time k, that is, E{ & ( k ) } and E{ D(k)} , respectively. Assuming that both d ( k ) and d(k) are uncorrelated with the sen- sor outputs and the filter length P is chosen sufficiently large, substituting eqns. 1 and 9 into eqn. 11 and tak- ing expectations will then give

E { @ k + 1))

= E { D ( k ) }

I 1

P

- pLna;E { sinc(i - D ) f ( i - f i ( k ) ) i=-p

+ P D ( d + 02)

x E a ( k ) sinc(i - f i ( / c ) ) f ( i - i j ( k ) ) P

21-p

(12) {

Since 2 L . p sinc(i - 0) f ( i - d(k)) = f ( D - @k)) when the filter length P tends to infinity and J(0) = 0, eqn. 12 can be approximated by

E { i j ( k + 1)) Y E { f i ( k ) } - p & E { f ( D - 6 ( k ) ) } (13)

Using Taylor's series to expandf(D - d(k)) up to the

IEE Proc-Rudur, Sonar Nuvig., Vol. 145. No. 6 , Deremher 1998

first-order term and as long as the condition 0 < p D < 6/(0,,2$) is satisfied, the learning characteristics of the delay estimate can be expressed as [lo]

E { i ) ( k ) } N D + (B(0) - D ) (1 - p 0 5 2 7 r 2 )i. (14)

where d(0) is the initial delay estimate. When d(k) converges to D , we can use eqn. 10 to obtain the learn- ing trajectory of E{ & ( k ) } , which is of the form

k E{ci(k)} = a! + ( q o ) - a!) (1 - pa (a? + a i ) ) (15)

where 0 < pcz < 1420: + 20,,2). The initial gain value d ( 0 ) can be chosen arbitrarily between 0 and 1. When the delay estimate is unbiased, the delay variance, denoted by var D, is an important performance meas- ure of the ETDGE algorithm, and it has been evalu- ated as [ 131

From eqns. 14 and 16, we see that the delay conver- gence time increases as the step size pD decreases, while the delay variance is directly proportional to pD. This shows that it is impossible to attain both fast learning speed as well as small variance for the delay estimate at the same time. However, a compromise between the two contradicting requirements can be reached by either splitting the delay estimate equally in both chan- nels for convergence speedup as in [14] or using a time- varying p D ( k ) [15, 161, which aims to provide faster convergence at an early stage of adaptation while ensuring smaller final error

Comparing var D with the delay variance of the ETDE, which is given by pDo,z(1 + SNR)/SNR2 [lo], it can be seen that they are identical and have a value of pno,?" when SNR >> 1. However, when SNR << 1, var D equals pD0,2i(2sNR), which is only half of the ETDE variance. Similarly, the variance of the gain & ( k ) , denoted by var &, can be shown to be

(17) pea:, SNR >> 1

- { pcYc:/2, SNR << 1

Replacing the actual delay D by D(k) = D + /3k in eqn. 13, where /3 represents the Doppler time compres- sion, yields the tracking behaviour for a linearly mov- ing delay [l I ]

The second term is a transient factor which converges to zero as k goes to infinity while the last term, 3f3/(pn0,,'$), contributes to the time lag at steady state. I t can be seen that this time lag is directly proportional to /3 and inversely proportional to p D and 0,'. In the presence of bias, the mean square delay error, denoted

327

by ~ ( d ) , is an important statistical average for evaluat- ing the performance of the ETDGE. It is equal to the delay variance in eqn. 16 plus the square of the time lag [17], which is given by

(19) / L I ) C T ? ( ~ + 2SXR) 9 / P +- N -

2SKR2 / L j ) C T ! 7r4

The first term of eqn. 19 is directly proportional to p, but the second term is inversely proportional to p;. Thus, p, must be selected appropriately to achieve the best performance. As a rule of thumb, when noise dominates, a smaller value of p, shodd be used. Oth- erwise, a larger value of p, is preferred, particularly when the delay is changing rapidly with time.

4 Simulation results

Computer simulation had been conducted to validate the performance of the ETDGE for both static and nonstationary delays. In our tests, white Gaussian processes and other bandlimited signals were used as different input sources, while the corrupting noises were all uncorrelated white sequences. The white sequences were generated by a random number genera- tor with Gaussian distribution. The noises had unity power and different SNRs were obtained by proper scaling of the source signal. Without loss of generality, the initial values of the gain and delay parameter were set to 1 and 0, respectively. To demonstrate that the ETDGE is unbiased even for a short filter length, P was chosen to be 3. The results provided were averages of 400 independent runs.

3 Comparison with the CRLB

I t is useful to compare the optimum delay variance of the ETDGE with the CRLB [ls], wl-ich gives a lower bound on the variance attainable by any unbiased esti- mator using the same input data. When both signal and noises are white, the CRLB for passive time-delay estimation is given by [2]

(20) 3(1+ 2SNR)

7r2NSNR2 CRLB(D) =

To contrast the performance, we use :qn. 4 to develop an optimal realisation of the ETDGE that minimises the least-squares cost function J L s ( & , B), which is of the form

C J l , S ( & D)

) 2

N P

= (y(i.:) - ii sinc(i - ~ j ) x ( k - i ) k=1 z=-P

(21) where N is the observation time. Denote the global minimum of JLs(&, 6) by 6'' and 9, which are the estimates of the gain a and delay D , respectively. Con- sidering ergodicity holds, the least-squares parameter estimates will be given by eqn. 8 as N - W. To calcu- late the delay variance, var d", we assume & * = CL and d* is located at a reasonable proximity of D and is given by [6]

In practice, the delay variance will be larger than in eqn. 22 due to joint estimation of th? gain a and time delay D but the difference will become negligible as 6" approaches the optimum value. Following a tedious analysis procedure (see the Appendix), eqn. 22 can be simplified to a value which is approximately equal to eqn. 20. Therefore, the least-squares estimator is asymptotically efficient because it provides an unbiased delay estimate with the minimum achievable variance. It is noteworthy that for a white signal and white noises, this least-squares estimate is equivalent to the maximum likelihood delay estimate [:I.

3 2

" . I 0.61

SNR=-IOdB

0 2000 4000 6000 8000 no. of iterations

Fig.2

~. . thcoreticdl value from eqn. 14

Dduj esiiiiiute f iw U srutic ttekuj ~~ ~~ estimated delay

0 2000 4000 6000 8000 no. of iterations

Fig. 3 . . . . . . . . . theoretical value from cqn. 15

Guir? estiniute in U izunstutionury SNR ~.m.ironnieizr actual gain

ustimated delay

In the first trial, the source signal was a white sequence. The delayed signal s(k - D ( k ) ) , where D ( k ) is the true delay at time k , was produced by passing s(k) through a 61-tap FIR filter whose transfer function was given by Z,i" 3! sinc(i ~ D(k))z-'. The convergence characteristics of D(k) and & ( k ) for a constant delay D = 0.5 are depicted in Figs. 2 and 3, respectively. The SNR was initially set at lOdB and then step-changed to -1OdB at the 5000th iteration. The sJep sizes were cho- sen such that the time constant of D(k) and & ( k ) were kept identical under both SNR conditions. At SNR = lOdB, p, and pa were assigned with values of 0.00004 and 0.0004, while ,U, = pCr = 0.004 was used for SNR = -1OdB. In Fig. 2 it can be seen that the delay estimate converged to its optimal value after - 4000 iterations. As expected, the learning curve fluctuated after the 5000th iteration due to a sudden change in SNR. Since

IEE Proc.-Ruiiiir. Sonur Nurig.. Vol. 145, No. 6, Dewinher IY9N

eqn. 14 is derived by using the first-order approxima- tion of the f function, there was a slight discrepancy between the theoretical and estimated transient behav- iour of D(k), but it was not significant. Fig. 3 shows that & ( k ) converged to 0.91 and 0.09, which were exactly the desired values, at roughly the 2000th and the 6000th iterations, respectively. It is observed that at SNR = -lOdB, the convergence dynamics of the gain variable agreed very well with the theoretical value as given by eqn. 15. However, at the beginning of the adaptation, the learning trajectory of & ( k ) damped and converged at a slower rate than the predicted curve because of inaccurate estimation of D during tran- sients.

2000 4000 6000 8000 no of iterations

Fig. 4 ._

_ _ estimated delay

Delay estimate for a bneurly time-varying deluj - actual delay ( D ( k ) = 0 5 + 0 0001k)

Fig. 4 illustrates the tracking performance of the ETDGE when the actual delay was a linearly time-var- ying function which was given by D(k) = 0.5 + 0.0001k. The values of the SNR, pD and pa were set according to the previous tests. The learning curve of d ( k ) is not shown here since it is fairly similar to Fig. 3. It is noted in Fig. 4 that after 3000 iterations, the delay estimate converged and lagged D(k) by 0.094 at SNR = lOdB and by 0.11 at SNR = -10dB. Evaluating the steady- state time lag using eqn. 18 gives a value of 0.076, which is smaller than both measurements. The differ- ence is mainly due to the approximations made in deriving eqn. 18.

The mean-square errors of d(k) and d ( k ) for the above two cases are tabulated along with the theoreti- cal values in Table 1. It can be seen that for both sta- tionary and nonstationary delays the simulation results for var d are fairly close to those derived from eqn. 17. The measured variance of the TDOA also conforms to the theoretical value when the original delay is a constant. Furthermore, these variances are not far from the CRLBs. From eqn. 20, the CRLB for the high and low SNR conditions are given by 1.28 x

and 7.30 x 10 3, respectively, which means that the performance of the ETDGE is only a few dB worse than the optimum variances. On the other hand, the

Table 1: Mean square errors of b (k ) and d (k )

result for ~ ( d ) is less satisfactory when the delay is time-varying. This is mainly due to the difference between the actual and theoretical time lags as illus- trated in Fig. 4. At SNR = lOdB, it is noted that the value of E(D) was dominated by the time lag. In this case, a larger value of pD might be used to attain a smaller mean-square delay error. It is noteworthy that the uncorrelation assumption of the delay and gain estimate are confirmed because both the estimates and the mean-square errors agreed with the theoretical cal- culations.

0.6

0 2000 4000 6000 8000

Dduy estimute for nontvhite source signuls no. of iterations

Fig. 5 ~ sum of indepcndent white noises

~ ~ sum of sinusoids . . . white noisc (theoretical value)

Since the delay of a bandlimited signal can be mod- elled by an FIR filter whose coefficients are samples of a sinc function [8], the ETDE will still provide a proper time shift even to a nonwhite signal. This means that the proposed algorithm can also give accurate delay estimates for bandlimited signals of different spectral shapes. To illustrate this, the first test was repeated with two nonwhite input processes. One of the signals was given by s(k) = w(k) + w(k - 2), where { w ( k ) } was an independent white sequence while the other signal consisted of a sum of sinusoids and had the form of s(k) = E t , sin(0.2nik). As in the case of a white signal source, the sum of correlated noise sequences was delayed by using a sinc filter and appropriate phase values were added to the sinusoids to generate the required time-shifted versions. The adaptation dynam- ics are plotted in Fig. 5, and it is observed that the delay estimates did converge to the optimal value while the convergence behaviours were similar to the theoret- ical trajectory of a white source signal, although the learning speed for the tone signals was slightly slower. Therefore, eqn. 14 can be used to approximate the learning behaviour of the ETDGE delay estimate even when the source signal is non-white.

5 Conclusions

In conclusion, an adaptive system (ETDGE) for esti- mating the differential path delays is presented which

SNR = 10dB SNR =- lodB

Measured value Theoretical value Measured value Theoretical value

var( b ) 4.5 x 10-5 4.2 10-5 2.7 x 2.4 x D = 0.5

var (2) 4.3 x 1 0 - ~ 4.0 10-4 2.2 10-3 2.0 10-3 D(k) = 0.5 E( b ) 8.8 x IO” 5.8 x 10-3 4.4 x 10-2 +O.OOO1k var ( & ) 4.1 x 4.0 x 10-4 2.3 10-3 2.0 10-3

3.0 x

IEE Proc.-Radar, Sonar Navig.. Vol. 145, No. 6 , December 1998 329

finds applications in source localisation by radar, sonar and satellites. The ETDGE, whose design is based on the Wiener solution for white signals and corrupting noises, consists of an explicit time delay estimator and a variable gain. It is shown that the algorithm is com- putationally efficient and is unbiased for a short filter length. The performance of the estimator is analysed for both static and linearly time-varying delay, and the effectiveness of the algorithm was demonstrated by computer simulations for different bandlimited source signals.

6 Acknowledgment

rhis work is partly supported by a research grant awarded by the Hong Kong Research Grants Council.

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References

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8 Appendix

The delay variance of the least-squares estimator is derived as follows.

The first derivative of JLS(cr, d) with respect to D is given by

~ J L s ( Q , D) aD

330

I’

= -2aC y(k) - a .r(k - i)sinc(i - 13) k = l N ( 2= f’

P

x z(~; - j)sinc’(j - 13) J = P

To calculate E{(dJLs(cr, d)/dD)’} at D = D, we use the following approximation [ 131:

N 4 o ‘ S o : ~ ~ (1 + 2SNR) - 3SNR‘

Differentiating dJLs(a, D ) / d D with respect to D again, we have

a2 J L S ( Q , D) a2 D

P \

\

/ n \ 2 )

It can be shown that

and

Thus

(24) Dividing eqn. 23 by eqn. 24 yields the delay variance,

.. 3(1+2SNR) var(D*)

7r2NSNR2 which is eqn. 20.

IEE €‘roc.-Radur, Sonar iVari&. V(J/. 145. N o . 6. Drcrmher I Y Y K