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Show An Efficient Algorithm for Joint Estimation of Differential Time Delays and Frequency Offsetst Ahmad R. Naghsh Nilchi and V. John Mathews Department of Electrical Engineering The University of Utah Salt Lake City, UTAH 84112 ABSTRACT This paper introduces an efficient algorithm that jointly estimates differential time delays and frequency offsets between two signals. The approach is a two-step procedure. First, the differential frequency offsets are estimated from measurement of the autocorrelation functions of the received and transmitted signals. The time delays are estimated from estimates of the higher-order statistics of the two signals involved. The major advantage of the new approach is its remarkably reduced computational complexity over traditional approaches. The experimental results indicate that the algorithm performs better than the traditional methods in most cases of interest in spite of its reduced computational complexity. I. INTRODUCTION Estimation of differential time delays (OTOs) and differential frequency offsets (OFOs) between two signals generated by the same source is important !n many applications. Consider two signals represented In the following form (1) Here X(n) is the source signal and is assumed to be at least locally stationary in the derivations, and W1 (n) and W2(n) are zero mean, white, additive noise signals uncorrelated with each other and with the source signals. M is the number of the targets, {am, m=1,2, ... , M} are the complex attenuation factors and {8m, m =1,2, . .. , M} are the initial phase shifts. 8m and 8k are assumed to be uncorrelated with each other for m:;t:k. Furthermore, it is assumed that each 8m is uniformly distributed in the range [n, on). Om and wm are the time delay and frequency offset parameters, respectively, that are associated with the mth target. t This work was supported in part by a University of Utah Research Support grant. V-309 Our objective is to estimate the differential time delays Om and differential frequency offset wm between the "transmitted signal" St(n) and the "received signal" Sr(n) from a time-limited segment of the signals. Traditional approaches to the joint estimation of OTOs and OFOs between the transmitted and received signals involve use of the complex cross-ambiguity function of the two signals [1,5,7). The complex crossambiguity function of St(n) and Sr(n) is defined as [6) +00 • -jan A('r,w) = L St(n) Sr(nH) e , (2) n=-~ where • denotes complex conjugation, and 't, and ware the time lag and frequency offset variables, respectively. It is well known that E{A('t,w)} peaks at the true values of the (time delay, frequency offset) pairs and therefore one can estimate the OTO and OFO parameters by finding the values of 't and w for which the magnitude of the complex cross-ambiguity function of the two waveforms peaks. Even though the above approach is conceptually simple, it is computationally extreme~y complex. We will present an approach that IS computationally much more efficient and whose performance is superior to the traditional approach. II. ALGORITHM DERIVATION A. Estimation of Differential Frequency Offsets Consider the autocorrelation function of the received signal defined as (3) Substituting the expression for Sr(n) given in Eq. 1 into Eq. 3 and making use of the fact that the random ph~se shifts are uncorrelated, and that the source and nOise signals are also mutually uncorrelated, Rrr('t) reduces to the form 0-7803-0532-9/92 $3.00 © 1992 IEEE (4) where Rxx{'t) is the autocorrelation function of the source signal X(n) at lag 1:.0; is the variance of the white noise sequence W2(n) that corrupts the received signal. and 0(1:) is the Dirac delta function. Note that Rxx(1:) is related to Rn(1:) as 1\(1:) = Rxx(1:) + <{,0{1:). (5) 2 where °1 is the variance of white noise sequence W1 (n) that corrupts the source signal X{n). and Rtt{1:) is the autocorrelation function of the transmitted signal. Let T) (1:) denote the ratio of the autocorrelation functions of the received and transmitted signals. Dividing Eq. 4 by Eq. 5. we get 2 M 2 2 M °2 -L laml °1 ~ 2 jCl\n't {mz1 } T)(1:) = f;r I~I e + --1\-(0-) -- 0(1:). (6) The right-hand side of Eq. 6 has the same functional form as the expression for the autocorrelation function of M complex sinusoidal signals in the presence of additive white noise. Recognition of this fact immediately reduces the problem of the frequency offsets estimation to one of estimating the frequencies of M complex sinusoids embedded in additive. white noise. There are several available methods that can be used to estimate the frequencies of sine waves in additive white noise. including many high-resolution algorithms. We used the root-MUSIC algorithm [2] for estimating the frequency offsets. B. Estimation of Differential Time Delays Let the estimated values of the frequency offsets be ro1 . C\ ..... roM . We will use these estimates in our approach for estimating the unknown time delays. Let us define the product signals Yp{n) as (7) for several integer values of p. Consider the autocorrelation function of Yp(n) evaluated as (8) It is easy to see that M 2 •• jCl\n't RypYp (1:) =LJ~I E{ X(n)X (n-Dm-p)X (n-'t)X(n-Dm-p-'t) }e 11'1=1 for 1:~1. (9) Let ~(n) = X(n)X(n-'t). (10) The fourth-order correlation function in Eq. 9 can be expressed as an autocorrelation function of ~(n) . That is. This implies that we can rewrite (9) as (12) Now. we can easily estimate the time delays by realizing the fact that the autocorrelation function Rz z (p+Dm) attains its peak value at the time lag p ~-Dm .m=1.2 .... M. One can solve for lam/2Rz.z.{p+Dm) from (12) if one knows RypYp (1:) . RypYp (1:) can be estimated from the measured data. V-310 III. EXPERIMENTAL RESULTS Several experiments were conducted using a narrowband and a broadband versions of linear-FM waveforms as the source signal. In the experiments. the statistical expectations were replaced with the corresponding time averages. It is not very difficult to see that the results in Section II hold at least approximately even when the statistical autocorrelation functions are replaced with deterministic autocorrelations defined using time averages. The source signal has the functional form given by [3] jn:bnZ X(n) = a(n)e • (13) where a(n) is the envelope of the signal and b is a constant that is designated as the slope of the instantaneous frequency of the FM waveform. Note that if a(n) is confined to a time duration T. the instantaneous frequency of the waveform will sweep over the bandwidth of bT radians/sample. The narrowband and broadband source signals were created by selecting b to be 10-5 and 5x 10-4 • respectively. The estimates of the time delays assumed that the unknown time delays were integer multiples of the sampling period in all cases. While this is not a realistic assumption, it considerably reduced the complexity of the simulations. In practice one will have to interpolate between samples of the crosscorrelation estimates to find the peak values. All the results presented are averages of fifty independent experiments. All the experiments made use of 4096 data samples. Two cases were considered. In the first case, there was only one unknown (OFO, OTO) set. The unknown OFO was 1 radian per sample and the unknown time delay was 5 samples. In the second example, there were two unknown (OFO, OTO) pairs given by (1 radian/sample, 5 time units) and (1.2 radian/sample, 50 time units). The attenuation factor associated with the components of the received signal corresponding to both (time delay, frequency offset) pairs was one. Table 1 presents the mean and meansquared deviation from the actual value of the frequency offset estimates obtained using our approach and the direct method that involves ambiguity function calculations for the first case. The corresponding results for time delay estimates are shown in Table 2. Tables 3 and 4 display the results involving multiple (OFO, OTO) sets. The results show that the new method of frequency offset estimation performs better than the direct method for both narrowband and broadband source signals in all SNR environments. Using the broadband source signal results in poorer performance than using the narrowband signal in all experiments. This is to be expected since computation of l1('t) in equation (6) is more noisy when the broadband signal is used than when the narrowband signal is used. This is so because the autocorrelation function of the broadband signal decays rapidly and the estimation noise may dominate the estimation of RttCt) that appears in the denominator of (6) especially for large values of 'to The results for time delay estimation show that our approach for OTO estimation perform as well as the direct ambiguity function method for both narrowband and broadband source signals in high SNR environments. However, at low SNR, the time delay estimates of the new approaches are somewhat poorer when compared with the ambiguity function method for narrowband situation. Again, note that using the broadband source signal results in poorer performance than using the narrowband signal in all time delay estimation experiments. A large number of additional experiments involving a variety of situations have been done and documented in [4]. v. CONCLUDING REMARKS In this paper, we presented a new and efficient approach for estimating differential time delays and differential frequency offsets. The major advantage of the new approach is its remarkably reduced computational complexity. Besides, the experimental results indicate that the new method is capable of estimating the frequency offsets more accurately than the traditional approaches, especially when the signalto- noise ratio is very poor and when a narrowband source signal is used. The performance of the time delay estimator is slightly worse, but comparable to the traditional schemes for good signal to noise ratios. These aspects of the new approach should make it very useful and attractive in practical applications. REFERENCES [1] Elliot, o. E., Handbook of Digital Signal Processing: Engineering Applications, pp. 838-843, Academic Press Inc. San Diego, Calif, 1987. [2] Haykin, S., Adaptive Filter Theory, 2nd Ed., pp.368- 371, Prentice-Hall, Englewood Cliffs, NJ, 1990. [3] Mitchell, R. L., Radar Signal Simulation, pp.51-58, Artech House Inc., Dedham, Massachusetts, 1976. [4] Naghsh-Nilchi, Ahmad R., Joint Estif1]ation of Differential Time Delays and Frequency Offsets, M.S. Thesis, University of Utah, Salt Lake City, Utah, December 1990. [5] [6] [7] V-311 Stein, S., " Algorithms for Ambiguity Function Processing," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-29, No.3, pp. 588-599, June 1981. Ville, J., "Theorie et application de la notion de signal analytique," Cables et Transmission 2, pp. 61-74,1948. Woodward, P. M., Probability and Information Theory With Applications to Radar, Pergamon Press, Oxford, 1953. Narrow Band Broad Band Ambi- New Ambi- New , gulty Method ! guity Method 20 dB J.I. 1.004 1.000 1.007 0.999 MSD 1.87SE·5 1000E·8 4.43&E·5 1.904E·S 10 dB J.I. 0.994 1.000 1002 0.998 MSD 3.35ZE·S 2.S60E-8 9.S26E·5 3.920E-& o dB J.I. 0.982 0998 1.021 0.997 MSD 3.258E·4 2.341E-fl 4.452E·4 S.8S0E-a ·10dB J.I. 1.032 1.003 0.9S0 0.994 MSD 9.99SE·4 S.fl56E-6 1.5a3E-a 3.74SE-5 Table 1.-The mean and mean squared deviation of the frequency offset estimates in the first example. The actual frequency offset was 1.0 radian per sample. Ambiquitv New A )proach 20 dB J.I. 1.003 1209 1.001 1.201 MSD 9.923E-& 7709E·S 2.132E·6 1.254E-6 10 dB J.I. 1.006 1213 1.002 1202 MSD 3.9S6E-S I.S63E-4 2.8S6E-6 5.617E-6 o dB J.I. 1.009 1.170 1.004 , .203 MSD 8.354E·S B.92BE·4 1552E-S 6.970E-6 ·10 dB J.I. 1.012 1.251 1011 1204 MSD 1.553E·4 2.621 E-3 1.272E·4 I 246E·5 (a) Ambiguity New A )proach 20 dB J.I. 5.000 50.000 5.000 50.000 MSD 0.0 0.0 0.0 0.0 10 dB J.I. S.ooo 50000 5.000 50.000 MSD 0.0 0.0 0.0 0.0 o dB J.I. 5.000 50.000 5.080 49.900 MSD 0.0 0.0 3.S00E-3 1.000S·2 ·10 dB J.I. 5.120 36.580 5.180 49.240 MSD 1.44OE·2 11.695 3.240E·2 5.77flE·l (b) Table 3. The mean and mean squared deviation of the (a) frequency offsets and (b) time delays estimates in the second example when the narrow band source signal was used. Narrow Band Broad Band Ambl- Higher Ambi- Higher !guitv Order ! guity Order 20 dB 11 5.000 5.000 5.000 5.000 M::;U 0.0 0.0 0.0 0.0 10 dB 11 5.000 5.000 5.000 5.000 M::;U 0.0 0.0 0.0 0.0 o dB 11 5.000 5.000 5.000 5.020 M::;U 0.0 0.0 0.0 4.000E·4 -10 dB 11 5.000 5.0S0 4.740 4.940 M::;U 0.0 3.S00E·3 6.760E·2 3.600E·3 Table 2. The mean and mean squared deviation of the time delay estimates in the first example. The actual time delay was 5 time units. V-312 AmbiQuitv New A lProach 20 dB J.I. 1.009 1.239 1.004 1.198 MSD 7.709E-5 1.557E·3 1.927E·6 2.866E-6 10 dB J.I. 0.988 1.247 1.010 1.208 MSD ',390E·4 2. 173E·3 '.098E·. 7.089E·5 o dB J.I. 0.963 1.308 0.987 1.170 MSD 1.369E-3 1.187E·2 1.069E·3 8.952E·4 -10 dB J.I. 1.069 1.314 1.060 1.262 MSD 4.821E·3 1.309E-2 3.649E-3 3.B39E-3 (a) Ambiquity New A lproach 20 dB J.I. 5.000 50.000 4.940 50.080 MSD 0.0 0.0 3.600E·3 4.000E-4 10 dB J.I. 5.200 50.120 5.140 50.020 MSD 4.000E·2 1.440E·2 1.960E-2 6.400E-3 o dB J.1. 15.120 52.440 5.240 51.820 MSD 102.414 5.954 6.760E-2 3.312 -10 dB J.I. ·13.000 24.920 5.660 44.640 MSD 324,000 629.004 4.356E·1 28.730 (b) Table 4. The mean and mean squared deviation of the (a) frequency offsets and (b) time delays estimates in the second example when the broad band source signal was used. |
