Time Delay Estimation for Sinusoidal Signals. H. C. So. Department of Electronic Engineering, The Chinese University of Hong Kong

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1 Time Delay stimation for Sinusoidal Signals H. C. So Department of lectronic ngineering, The Chinese University of Hong Kong Shatin, N.T., Hong Kong SP DICS: -DTC January 5, Abstract The problem of estimating the dierence in arrival times of a sinusoid received at two spatially separated sensors is considered. Given the sinusoidal frequency, a simple delay estimator using the phase dierence of the discrete time Fourier transforms DTFTs of the received signals is devised. With the use of periodogram, the algorithm is extended to estimate the delay when the frequency is unknown. The minimum achievable delay variances for the cases of known/unknown frequencies and constant/rectangular envelopes are also derived. The eectiveness of the method is demonstrated by comparing with the performance bounds for dierent frequencies, envelopes and noise conditions. Contact Information: H. C. So Department of lectronic ngineering The Chinese University of Hong Kong Shatin, N.T., Hong Kong Tel : Fax: hcso@ee.cuhk.edu.hk Sonar and Navigation and is subject to Institution of ngineering and Technology Copyright.

2 I. Introduction stimation of the time delay between two noisy versions of the same signal received at two spatially separated sensors has important applications such as direction nding, source localization and velocity tracking []. The mathematical model of the discrete-time sensor outputs is given by r n =sn+q n r n =sn ; D+q n n = ::: N; where sn is the signal of interest, q n andq n represent additive noises which are independent of sn, D is the dierence in arrival times at the two receivers and N is the number of samples collected at each channel. Without loss of generality, the sampling interval is assigned to be unity second. Many methods have been proposed for time delay estimation in the past two decades []-[8]. Generalized cross correlator GCC []-[4] isaconventional approach to the problem and its delay estimate is found by locating the peak of the cross correlation function of the ltered versions of r n andr n. When sn, q n and q n are all uncorrelated Gaussian variables, it has been proved [] that the GCC can provide maximum likelihood ML delay estimation. However, implementation of the GCC requires a priori statistics of the received signals and thus in the absence of these information, the desired performance is dicult to achieve in practice [4]. Moreover, it fails to work if the noises are impulsive [5] or spatially correlated [7]. In the presence of impulsive noise modeled as alpha-stable random process, accurate delay estimates can be attained using the fractional lower order statistics FLOS based techniques [5]-[6]. On the other hand, the higher order statistics HOS approach isan eective solution for correlated noises when the signal is a non-gaussian random process and the noises are Gaussian distributed [7] or vice versa [8]. The aim of this paper is to devise a simple and accurate time delay estimator when the source signal is deterministic, specically for a pure sinusoid that commonly occurs in radar, sonar and digital communications [9]-[], although the GCC can be employed in deterministic signal condition [3]. In our study, we consider that sn is analytic and has the form sn =anexpj! n + where an,! and [ represent the real envelope function, radian frequency and unknown constant phase of the sinusoid, respectively. It is assumed that q n and q n are uncorrelated zero-mean complex white Gaussian processes with variance q and D ;=! =! toavoid ambiguous delay estimates. The paper is organized as follows. Based on the discrete-time Fourier transforms DTFTs [] of the two sensor outputs, a computationally ecient delay estimation algorithm for sinusoidal signals is developed in Section II, assuming that! is known. The estimator variance in the case of constant Sonar and Navigation and is subject to Institution of ngineering and Technology Copyright.

3 envelope is also derived. Section III modies the proposed method for unknown frequency with the use of periodogram. The Cramer-Rao lower bounds CRLBs [] of the delay estimates for the cases of known/unknown frequencies and constant/rectangular envelopes are derived in Section IV. Section V evaluates the estimation accuracy of the DTFT based approach via comparison with the performance bounds, and nally, conclusions are drawn in Section VI. II. The Proposed Method In this section, a simple delay estimator for sinusoidal signals with known frequency is developed. For ease of analysis, we assume that the signal has an unknown constant envelope with an = A, n = N ;, although the proposed method can work for other envelopes. The DTFT of r n is given by R! = N X; n= r ne ;j!n = Ae j+! ;!N ;= sin! ;!N sin! ;! X N ; + n= q ne ;j!n 3 In the time domain, the signal-to-noise ratio SNR of r n, denoted by SNR, is equal to A = q. On the other hand, it can be easily shown from 3 that the SNR of jr! j, SNR,hasavalue of NA = q, which isn times larger than SNR. It is because the signal power is concentrated at! =! while the noise power equals N q for all frequencies in the DTFT domain. Motivated by this fact, the DTFTs of the sensor outputs at! =! are used to estimate the time delay as follows. For SNR >>, R! can be approximated as [3] R! = NAe j + N X; n= q ne ;j! n NAe j e j=fx! g 4 where X X! = N ; NA n= q ne ;j! n+ 5 and =fxg denotes the imaginary part of x. The phase angle of R! isthus of the form 6 fr! g + =fx! g 6 whose mean value is and has a variance of =SNR. Similarly, the phase of the DTFT of r n at! =! is given by 6 fr! g;! D + =fy! g 7 Sonar and Navigation and is subject to Institution of ngineering 3 and Technology Copyright.

4 where The quantity 6 6 fr! g. X Y! = N ; NA n= q ne ;j! n;d+ fr! g has the expected value of ;! D and its variance is identical to that of 8 Using the phase dierence of R! andr!, the delay estimate, denoted by ^D, is computed as where represents the conjugate operation. ^D = 6 fr! R! g! 9 Let = <f P N ; n= q ne ;j! n g, = =f P N ; n= q ne ;j! n g, 3 = <f P N ; n= q ne ;j! n g and 4 = =f P N ; n= q ne ;j! n g where <fxg represents the real part of x. Notice that these four terms contribute to the random components of ^D and they are independent toeach other and have zero means. As a result, the variance of ^D, denoted by var ^D, is fully due to the noise terms,, 3 and 4 and it is given by [4] var ^D = i i =f i g A f i g where is the expectation operation. This expression can be simplied and modied to Appendix I var ^D = min 3!! NSNR which is a constant ifnsnr < 3=, and is inversely proportional to!, N and SNR, otherwise. III. xtension to Unknown Frequency It is well known that [5] the periodogram will give the ML estimate of frequency for a single complex sinusoid in white noise. With the use of periodogram, an iterative procedure is proposed to nd D when! is not known, as follows.. Use periodogram to get an initial estimate of!,^! : where ^! = denotes the periodogram of xn. arg max! fp r!g + arg max! fp r!g P x! = N N X; xne ;j!n n= 3. Compute the initial delay estimate as ^D = fr ^! R ^! g ^! 4 Sonar and Navigation and is subject to Institution of ngineering 4 and Technology Copyright.

5 3. Construct a N-length sequence zn fromr n andr n of the form zn = 8 >< >: r n n = N ; r n ; Ne j ^! ^D+N n = N N ; which can be considered as a noisy sinusoid with frequency!. Since the variance of the ML frequency estimate is asymptotically proportional to one over the cube of the observation length [5], zn is employed to nd a more accurate estimate of! : 5 ^! = arg max! fp z!g 6 4. Repeat steps and 3 for a few times until convergence. In the simulation examples in Section V, at most ve iterations are required for the parameters to converge. The delay variance of the proposed method in the case of unknown frequency is derived as Appendix II var ^D =min 3! 3D! NSNR +! NN ; SNR Notice that the dierence between 7 and is negligible particularly for suciently large N. 7 IV. Derivation of Performance Bounds We rst derive the CRLB of the delay estimate for known!. The key is to group! and D as one variable, say, =! D, and combine r n and r n to form a N-length sequence fwng = fr r N ; r r N ; g. The probability density function PDF of wn isgiven by [] pw = N 4N q exp ; q N X; n= X N ; jr n ; Ae j! n+ j + n= jr n ; Ae j! n;+ j where =[A ] is the unknown parameter vector to be estimated. The 3 3 Fisher information matrix has the form [] I= ln pw It is clear that the matrix is symmetric since the order of partial dierentiation can be interchanged. The log-likelihood function is ln pw = ; ln N 4N q ; q " N ; X 3 7 5! jr nj + jr nj +A ; r nae ;j! n;+ ; r naej! n+ ; r nae ;j! n;+ ; r naej! n;+ Sonar and Navigation and is subject to Institution of ngineering 5 and Technology Copyright. # 8 9

6 The derivatives are easily found ln pw ; 4N ln pw ; 4NA ln pw ; NA ln = NA q 3 ln ln = 5 The CRLB for, CRLB, is obtained from the inverse of I [] and has the expression CRLB = Using [4] and 6, the CRLB for D, denoted by CRLBD, is evaluated as NSNR 6 CRLBD =! NSNR 7 Assuming the delay is uniformly distributed between ;=! and =!, the composite bound for the delay is identical to which means that the proposed method provides the optimal delay estimation performance for sinusoidal signals with known frequency and constant envelope. In the case of unknown frequency, the parameter vector will become =[A! ] and the size of the corresponding Fisher information matrix is 4 4. The following partial derivatives are needed and they are calculated straightforwardly ln = ; A NN ; N ; 3 8 = = ; A NN ; q ln pw = ; A NN q Sonar and Navigation and is subject to Institution of ngineering 6 and Technology Copyright.

7 Using -5 and 8-3, the CRLBs for! and are computed as and CRLB! = CRLB = 3 NN ; SNR 3 NSNR 33 The CRLB for D is only dependent oncrlb! and CRLB [4] and can be shown to be CRLBD =! NSNR 3D + 34! NN ; SNR As a result, the composite performance bound is identical to 7 and thus the proposed method also gives the minimum delay variance for the unknown frequency case. When the source signal is a gated sinusoid, that is, an =A for n = L; and equals otherwise, it can be shown in a similar way that the performance bounds of the delay estimate are given by and 7 for known and unknown!, respectively, with the substitution of L = N. V. Simulation Results xtensive computer simulations had been done to corroborate the theoretical derivations and to evaluate the performance of the proposed approach for estimating the time delay between sinusoidal signals. The mean square delay errors MSFs for the cases of known/unknown frequencies and constant/rectangular envelopes were investigated. The tone parameters A and were assigned to be and., respectively, while the time delay was set to.6s. Dierent SNR swere produced by properly scaling the noise variance. Unless stated otherwise, the sinusoidal frequency! and the observation length N had values of :345 rad/s and 3, respectively. Five iterations of 4 and 6 were used when the value of! was not available. All simulation results provided were averages of independent runs. Figure shows the MSFs of the proposed method for a constant envelope sinusoid with known frequency at dierent N versus SNR. It is seen that except in the no information region, the MSF decreased as the observation interval increased. Furthermore, the delay variances agreed with the theoretical values very well particularly for SNR ;5dB for all cases. The delay estimation performances using the sinusoidal signal at dierent known frequencies are compared in Figure. We observe that the MSFs decreased as the frequency increased and they were close to the corresponding performance bounds. Figure 3 illustrates the MSFs when the source signal was a gated sinusoid with dierent lengths of the rectangular envelope. As expected, the accuracy of ^D increased with L. In addition, it can be seen that MSFs were above the performance bounds by approximately.3db and 3.7dB at L =:75N and L =:5N for SNR ;5dB, respectively. This implies that the optimality of the proposed method will degrade as the width of the signal envelope decreases. Sonar and Navigation and is subject to Institution of ngineering 7 and Technology Copyright.

8 The variances of ^D when sn was a pure sinusoid with unknown! are plotted in Figure 4. It is observed that the proposed method met the performance bound and performed very similar to the known frequency case for SNR ;4dB. Below the threshold SNR of ;4dB, the MSFs became much larger than the composite bound which was due to the occurrence of anomalous ^! in the nonlinear frequency estimation process [5]. The test of Figure 3 was repeated for unknown! and the results are shown in Figure 5. We see that the MSFs were close to those of Figure 3 when the SNRs were greater than the threshold SNRs of ;4dB, ;3dB and ;db for L = N, L =:75N and L =:5N, respectively. VI. Conclusions An DTFT based method has been derived for estimating the time dierence of arrival between sinusoidal signals received at two separated sensors. When the tone frequency is available, the delay estimate is given by the phase dierence of the DTFTs of the received signals over the frequency. An iterative delay estimation procedure using the periodogram is also developed for the case of unknown frequency. It is proved that the variances of the delay estimates can meet the performance bounds for known and unknown frequencies when the sinusoid has a constant envelope. Numerical examples are included to validate the theoretical analysis and to demonstrate the eectiveness of the proposed approach. Appendix I The derivation of is given as follows. First, the real and imaginary parts of R! R! are of the forms and <fr! R! g = N A cos! D+NAcos 3 + NAcos ;! D + NAsin 4 +NAsin ;! D =fr! R! g = N A sin! D ; NAcos 4 ; NAsin ;! D + NAsin 3 +NAcos ;! D ; Let = =fr! R! g=<fr! R! g and noting that 6 fr! R! g = tan ;, the partial derivative of ^D with respect to at i = f i g =,i = 3 4, is then computed i = =! i = i = Sonar and Navigation and is subject to Institution of ngineering 8 and Technology Copyright.

9 N A cos! D ;NAsin ;! D ; N A sin! D NAcos ;! D N A cos! D = sin ; A: NA! Similarly, the partial derivative of ^D with respect to, 3 and 4 can be shown to be i 3 i 4 i = = cos NA! = sin ;! D NA! = ; cos ;! D NA! From A.-A.4 and with the use of fi g = N q =fori = 3 4, we have A: A:3 A:4 var ^D = A:5! NSNR Assuming that D is uniformly distributed in ;=! =! and combining A.5 yield the composite bound for the variance of ^D which is. var ^D = min 3!! NSNR Appendix II The variance of ^D for unknown! is derived as follows. Let the steady state frequency estimate of 6 be ^! =! + 5 where 5 is the zero-mean estimation error of! and f 5 = sin 5 N ;== sin 5 =. It has been revealed that the variance of ^! or f 5 g is equal to CRLB! of 3. In this case, var ^D is fully due to,, 3, 4 and 5.Following Appendix I, the real and imaginary parts of R ^! R ^! are evaluated as f 5 3 <fr ^! R ^! g = A cos! Df 5 +Acos + 5N ; +A cos ;! D + 5N ; f 5 + A sin +A sin ;! D + 5N ; and f f N ; f 5 4 =fr ^! R ^! g = A sin! Df 5 ; A cos + 5N ; ;A sin ;! D + 5N ; f 5 + A sin + 5N ; f 5 3 +A cos ;! D + 5N ; f 5 ; Sonar and Navigation and is subject to Institution of ngineering 9 and Technology Copyright.

10 Denote ^ ==fr ^! R ^! g=<fr ^! R ^! g. It can be shown that the partial derivatives of ^D with respect to,, 3 and 4 at i =fori = 5, are given by A. to A.4, respectively, 5 i = = ; D! A:6 The variance of ^D is then computed as var ^D = = i i = A f i g 3D! NSNR +! NN ; SNR Combining the results, we obtain the composite delay variance which is equal to 7, with the assumption that D is uniformly distributed in ;=! =!. References [] G.C.Carter, Coherence and Time Delay stimation: An Applied Tutorial for Research, Development, Test, and valuation ngineers, I Press, 993. [] C.H.Knapp and G.C.Carter, "The generalized correlation method for estimation of time delay," I Trans. Acoust., Speech, Signal Processing, vol.4, no.4, pp.3-37, August 976. [3] J.C.Hassab and R..Boucher, "Optimum estimation of time delay by a generalized correlator," I Trans. Acoust., Speech, Signal Processing, vol.7, no.4, pp , August 979. [4] K.Scarbrough, N.Ahmed and G.C.Carter, "On the simulation of a class of time delay estimation algorithms," I Trans. Acoust., Speech, Signal Processing, vol.9, no.3, pp , June 98. [5] X.Ma and C.L.Nikias, "Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics," I Trans. Signal Processing, vol.44, no., pp , Nov [6] P.G.Georgiou, P.Tsakalides and C.Kyriakakis, "Alpha-stable modeling of noise and robust timedelay estimation in the presence of impulsive noise," I Trans. Multimedia, vol., no.3, pp.9-3, Sept. 999 [7] C.L.Nikias and R.Pan, "Time delay estimation in unknown Gaussian spatially correlated noise," I Trans. Acoust., Speech, Signal Processing, vol.36, no., pp.76-74, Nov [8] Y.Wu and A.R.Leyman, "Time delay estimation using higher-order statistics: a set of new results," Proceedings of 997 International Conference on Information, Communications and Signal Processing, vol.3, pp.397-4, Singapore, September 997. Sonar and Navigation and is subject to Institution of ngineering and Technology Copyright.

11 [9] A.J.Weiss, "Bounds on time-delay estimation for monochromatic signals," I Trans. Aerospace and lect. Sys., vol.3, no.6, pp , Nov [] S.M.Kay, Fundamental of Statistical Signal Processing: stimation Theory, nglewood Clis, NJ: Prentice-Hall, 993. [] S.M.Kay, Fundamental of Statistical Signal Processing: Detection Theory, Upper Saddle River, NJ: Prentice-Hall, 998. [] A.V.Oppenheim and R.W.Schafer, Discrete-Time Signal Processing, nglewood Clis, NJ: Prentice-Hall, 989. [3] S.A.Tretter, "stimating the frequency of a noisy sinusoid by linear regression," I Trans. Inform. Theory, vol.3, no.6, pp , Nov [4] A.Papoulis Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, 984. [5] D.C.Rife and R.R.Boorstyn, "Single-tone parameter estimation from discrete-time observations," I Trans. Inform. Theory, vol., no.5, pp , Sept Sonar and Navigation and is subject to Institution of ngineering and Technology Copyright.

12 Figure Captions Figure : Mean square delay errors of a pure sinusoid with known! =:345 for dierent observation intervals N Figure : Mean square delay errors of a pure sinusoid with known frequency for dierent! at N =3 Figure 3: Mean square delay errors of a gated sinusoid with known! =:345 for dierent rectangular envelopes Figure 4: Mean square delay errors of a pure sinusoid with unknown! =:345 at N =3 Figure 5: Mean square delay errors of a gated sinusoid with unknown! =:345 for dierent rectangular envelopes mean square delay error db proposed N=64 performance bound N=64 proposed N=3 performance bound N=3 proposed N=6 performance bound N= SNR db Figure : Sonar and Navigation and is subject to Institution of ngineering and Technology Copyright.

13 mean square delay error db proposed ω =.345 π performance bound ω =.345 π proposed ω=.34 π performance bound ω=.34 π proposed ω=.3 π performance bound ω =.3 π SNR db Figure : mean square delay error db proposed L=N=3 performance bound L=N proposed L=.75N performance bound L=.75N proposed L=.5N performance bound L=.5N SNR db Figure 3: Sonar and Navigation and is subject to Institution of ngineering 3 and Technology Copyright.

14 mean square delay error db 4 - proposed unknown ω performance bound unknown ω proposed known ω performance bound known SNR db ω Figure 4: 4 mean square delay error db - proposed L=N=3 performance bound L=N proposed L=.75N performance bound L=.75N proposed L=.5N performance bound L=.5N SNR db Figure 5: Sonar and Navigation and is subject to Institution of ngineering 4 and Technology Copyright.