A Fast Algorithm for Nonstationary Delay Estimation H. C. So Department of Electronic Engineering, City University of Hong Kong Tat Chee Avenue, Kowloon, Hong Kong Email : hcso@ee.cityu.edu.hk June 19, 1998 Indexing terms : adaptive algorithm, time delay estimation Abstract : A computationally ecient algorithm for estimating time delay between signals received at two spatially separated sensors is proposed. The delay estimate is adapted directly to maximize the mean product of the rst sensor output and the ltered output of the second received signal. Convergence behaviour and variance of the delay estimate are derived. Itisshown that the proposed algorithm outperforms the explicit time delay estimator at low signal-to-noise ratio. 1
Introduction : Time delay estimation between noisy versions of the same signal received at two spatially separated sensors has important applications in direction nding, target localization and velocity tracking [1]. Generalized cross-correlation method [] is one of the conventional approaches for nding the time dierence of arrival but it requires a priori statistics of the received signals in order to obtain accurate delay estimates. Adaptive ltering techniques [3]-[4], on the other hand, avoid spectral estimation from nite length data and are also capable of tracking nonstationary delays due to either relative source/receiver motion or time-varying characteristics of the transmission medium. Based on maximizing the mean product of one of the sensor outputs and the time-shifted version of another received signal, an adaptive delay estimation method, called the simplied explicit time delay estimator (SETDE), is devised. It is shown that the proposed method is similar to the explicit time delay estimator (ETDE) algorithm [4] but it is more computationally ecient and provides more accurate delay estimates under very noisy environment. The SETDE : Let the sensor outputs be represented by r 1 (k) =s(k)+n 1 (k) r (k) =s(k ; D)+n (k) (1) where s(k) is the unknown stationary source signal, n 1 (k) andn (k) are the uncorrelated white Gaussian noises, is the attenuation factor between the sensors and D is the dierential delay to be determined. Without loss of generality, we assume that the signal and the noise spectra are bandlimited between ;0.5 Hz and 0.5 Hz while the sampling period is 1 second. When r 1 (k) is time-shifted by ^D to give r 1 (k ; ^D), we can easily show that the mean product of r 1 (k ; ^D) and r (k) will achieve its maximum when ^D = D. Using the
interpolation formula [5], r 1 (k ; ^D) can be expressed as r 1 (k ; ^D) = 1X i=;1 r 1 (k ; i)sinc(i ; ^D) () where sinc(v) = sin(v)=(v). An approximate version of r 1 (k ; ^D) isthus given by ~r 1 (k ; ^D) = r 1 (k ; i)sinc(i ; ^D) (3) where P should be chosen large enough to reduce the delay modelling error. Assuming that s(k) is a white process with variance, the expected value of ~r s 1(k ; ^D)r (k) is calculated as Ef~r 1 (k ; ^D)r (k)g = s sinc(i ; ^D)sinc(i ; D) ssinc( ^D ; D) (4) It is clear that an accurate estimate of D can be obtained by maximizing Ef~r 1 (k; ^D)r (k)g with respect to ^D. Using this idea, the SETDE is developed and its system block diagram is depicted in Figure 1. In the SETDE algorithm, the delay estimate ^D(k) is adapted on a sample-by-sample basis to maximize the instantaneous value of ~r 1 (k ; ^D(k))r (k) according to a least-mean-square (LMS) style algorithm as follows, ^D(k +1) = ^D(k)+ @~r 1 (k ; ^D(k))r (k) @ ^D(k) = ^D(k) ; r (k) f(i ; ^D(k))r1 (k ; i) (5) where is a positive scalar that controls convergence rate and system stability while f(v) = (cos(v) ; sinc(v))=v. To signicantly reduce the computational load of the updating equation, look-up tables of the sinc and cosine functions are constructed [4]. It is noteworthy that when r (k) is replaced by r (k) ; ~r 1 (k ; ^D(k)) in (5), the adaptation rule 3
will become exactly the ETDE algorithm. This means that the SETDE is more computationally ecient than the ETDE because (P + 1) additions and multiplications are saved for each sampling interval. Taking the expected value of (5) and with the use of (), we have Ef ^D(k +1)g = Ef ^D(k)g ; s Ef sinc(i ; D)f(i ; ^D(k))g Ef ^D(k)g ; s Eff(D ; ^D(k))g Ef ^D(k)g ; s Eff(0) + f 0 (0)(D ; ^D(k))g = Ef ^D(k)g ; 1 3 s (Ef ^D(k)g ;D) (6) because f(0) = 0 and f 0 (0) = ; =3. Solving (6) gives Ef ^D(k)g = D +(^D(0) ; D)(1 ; 1 3 s ) k (7) where ^D(0) is the initial delay estimate. Provided that 0 < <6=( s ), Ef ^D(k)g will converge to D with a time constant of3=( s ). By squaring (5), then taking the expectation on both sides and considering the steady state, the delay variance, denoted by var( ^D), can be shown to be var( ^D) = lim k!1 Ef( ^D(k) ; D) g ( 4 s + s n1 + s n + n1 n ) s (8) where and n1 represent thepowers of n n 1(k) andn (k) respectively. Simulation Results : Computer simulation had been conducted to compare the performance of the SETDE and ETDE for nonstationary delay estimation. The source signal s(k) and the corrupting noises were independent white Gaussian random variables and they were produced by a pseudorandom noise generator. The step size parameter and 4
the attenuation constant were set to 0.0006 and 1 respectively while P was chosen to 10 in order to maintain an acceptable truncation error. To provide a delay resolution of approximately 0.1% of the sampling interval, the size of the cosine and sinc tables were 104 and 513 x 31 respectively. Without loss of generality, the initial value of ^D(0) was selected to 0. The results provided were averages of 1000 independent runs. Figure shows the trajectories for the delay estimate of both SETDE and ETDE for a step change in D. The SNR was chosen to ;10 db and this was obtained by setting the power of s(k) to 1 while n1 = n = 10. It is seen that the SETDE delay estimate converged to the desired values of 0.4 and 0.6 at approximately the 000th and the 7000th iteration, respectively. We can also observe that the learning curve of the SETDE agreed with the predicted trajectory although (7) is derived by using the rst order approximation of the f function. On the other hand, the delay estimate of the ETDE attained the values of 0.30 and 0.46 upon reaching the steady state which implies that the ETDE cannot provide accurate delay estimation at low SNR. Upon convergence, the delay variance of the SETDE was measured as 0.0430 which conformed to the theoretical value of 0.0363 and was smaller than the measured ETDE mean square delay error of 0.065. Conclusions : A real-time algorithm for estimating the dierential delay between two sensor outputs is presented. Using an LMS-style algorithm, the delay estimate is adapted explicitly and iteratively. Theoretical analysis of the algorithm is derived and conrmed by simulations. It is shown that the proposed method can track time-varying delays accurately under very noisy environment and is an improvement to the ETDE. 5
References [1] CARTER, G.C.: 'Coherence and Time Delay Estimation: An Applied Tutorial for Research, Development, Test, and Evaluation Engineers', (IEEE Press, 1993). [] KNAPP, C.H. and CARTER, G.C.: 'The generalized correlation method for estimation of time delay,' IEEE Trans. Acoust., Speech, Signal Processing, 1976, 4, (4), pp. 30-37. [3] REED, F.A., FEINTUCH, P.L., BERSHAD, N.J.: 'Time delay estimation using the LMS adaptive lter - static behaviour,' IEEE Trans. Acoust., Speech, Signal Processing, 1981, 9 (3), pp.571-576. [4] SO, H.C., CHING, P.C. and CHAN, Y.T.: 'A new algorithm for explicit adaptation of time delay,' IEEE Trans. Signal Processing, 1994, 4, (7), pp.1816-180. [5] OPPENHEIM, A.V. and SCHAFER, R.W.: 'Discrete-Time Signal Processing', (Prentice Hall, Englewood Clis, NJ 1989). 6
r (k) 1 P -i Σ sinc(i-d)z i=-p r (k-d) 1 X r (k) Figure 1: System block diagram of the proposed time delay estimator 0.7 0.6 0.5 delay estimate 0.4 0.3 0. 0.1 actual delay theoretical calculation from (7) simulation result of SETDE simulation result of ETDE 0 0 000 4000 6000 8000 no. of iteration Figure : Comparison of SETDE and ETDE at SNR = ;10 db 7