Adaptive phase transform processors for time delay estimation

This paper introduces two recursive realizations of the phase transform (PHAT) processor for time-delay estimation (TDE), using a simple one-pole low-pass filter and the least-mean-square (LMS) adaptive filter, respectively. It is shown that these adaptive methods are capable of tracking time-varying delay functions which correspond to moving sources or receivers, and are very effective in reducing the effect of interfering tonals which must be generated by the target as jamming signals to mask its movement. The performances of these methods are compared with those of other existing adaptive TDE algorithma via computer simulations.

Adaptive phase transform processors for time delay estimationa) Oae Hee Youn Department of Electronic Engineering. Yonsei University. Seoul. Korea Shen-Neng Chiou Department of Electrical Engineering. University of Southern California, Los Angles. California 90007 V. John Mathews Department of Electrical Engineering. University of Utah. Salt Lake City. Utah 84112 (Received 10 January 1984; accepted for publication 10 February 1986) This paper introduces two recursive realizations of the phase transform (PHAT) processor for time-delay estimation (TOE), using a simple one-pole low-pass filter and the least-mean-square (LMS) adaptive filter, respectively. It is shown that these adaptive methods are capable of tracking time-varying delay functions which correspond to moving sources or receivers, and are very effective in reducing the effect of interfering tonals which must be generated by the target as jamming signals to mask its movement. The performances of these methods are compared with those of other existing adaptive TOE algorithms via computer simulations. PACS numbers: 43.60.Gk, 43.30.Vh INTRODUCTION The problem of estimating the time difference of arrival of the same signal at two spatially separated sensors arises in a variety ofapplications of sonar, radar, acoustics, geophys­ics, and biomedical engineering where we need to locate the signal source. 1-5 Of interest in this paper are passive systems, in which, unlike the active systems, the source signal strength cannot be controlled. However, their covertness can be advanta­geous, since passive systems do not rely on self-generated energy that is reflected off the source or target. An important example of such systems is a passive sonar system which receives the signals generated by a source, possibly corrupt­ed by noise, at an array of spatially separated sensors. It is well known I that the location of the source can be deter­mined if the time delays between the arrival times of the signal at three sensors are available. We consider the two-sensor time delay estimation (TOE) problem, where the signals received at the two sen­sors are given by XI (k) = s(k) + WI (k) + p(k) (1a) and x 2(k) = s(k - D) + w2(k) + p(k - D), (lb) where k is the discrete time index, s(k) is the source signal, WI (k) and w2(k) are the additive noises at sensors 1 and 2, p(k) denotes interfering tonals which might be generated by a target as a jamming signal to mask its movement, and D and D are delay parameters associated with the signal and interfering tonaIs, respectively. Also, it is assumed that the source signal s(k) and additive noises WI (k) and w2(k) are mutually uncorrelated random processes with zero mean. oj Part of this paper was presented at the International Conference. on Acoustics. Speech and Signal Processing. San Diego. CA. March 1984. Most approaches for TOE have been shown to be relat­ed through generalized cross correlation (GeC) methods which involve prefiltering the received signals and estimat­ing the time delay as the time lag where the cross correlation function of the prefiltered signals R W(m) = p-I{W('l(f)eJ812(fl}, Iml<M (2) is maximum.6 In (2), p-l{.} denotes the inverse Fourier transform of { . }, we,l (f) is a weighting function in the frequency domain that is determined by the prefilters, and 812 (f) is the phase function of the cross-power density spec­trum (cross-POS) ofxl (k) and x2(k). That is, ei812(fl = [G12(f)]lIG12(f) I, (3) whereGI2(f) is the cross-POSofxl(k) andx2(k). If there are no interfering tonaIs in the received signals [i.e., p(k) =0 in (1)], the phase function in (3) is given by 812(f) = 211'/ D, which means that the phase function is di­rectly proportional to the delay parameter D. The frequency domain weighting functions of the GeC methods of interest in this paper are summarized below: W(Bl(f) = IGI2(f); (4a) BCC (basic cross correlation) method,2 (4b) Roth processor,7 (4c) . PHAT (phase transform).2 Recently, the BCC method and the Roth processor have been realized using a simple one-pole low-pass filterB-Io and the LMS adaptive filter,IO-I. respectively. The main advan­tages of these recursive time-domain implementations are that they track time-varying delay functions and also avoid the difficulties encountered in spectral estimation with finite record lengths. 188 J. Acoust. Soc. Am. 80 (1). July 1986 0001-4966/86/070188-07$00.80 @ 1986 Acoustical Society of America 188 The phase transform processor was proposed as an ad hoc method to reduce the effect of strong tonals by uniformly weighting the phase function ejfJll (f) in the entire frequency band.2 The purpose of this paper is to introduce two recur­sive methods which realize the PHAT processor. In these adaptive techniques the relevant GCC functions are updated using a simple one-pole low-pass filter8-10 and the LMS adaptive filter, 10-16 respectively. In Sec. I, adaptive realizations of the BCC and the Roth processors are briefly summarized, while Sec. II is devoted to the PHAT processor and its adaptive implementations. Experimental results and conclusions are presented in Secs. III and IV, respectively. I. SOME THEORETICAL BACKGROUND From (2) and (4a), the GCC function of the BCC method is given by the cross correlation function of the re­ceived signals without prefiltering. That is, R If)(m) =F-1{G12(f)} = Cdm), Iml<M, (5a) where (5b) and E{ . } denotes the statistical expectation of { . }. It has been shown8-10 that the cross correlation function of Xl (k) and x 2(k) can be estimated using a bank of simple one-pole low-pass filters as C12(m,k) = f3CI2(m,k - 1) + (1-f3)x l (k)x2(k + m), Iml<M, (6a) where Cdm,k) denotes an estimate ofCI2(m,k) in (5b) at time k and 0 <f3 < 1 controls the time constant of the low­pass filter whose transfer function is given by A(z) = (1-f3)/(1_f3z-1) (6b) when Xl (k)X2(k + m) is applied as its input. The time con­stant of the above low-pass filter can be approximated as9 • 10 'TA sd/(l - f3) samples. (7) From (5a)-(6a), we can see that taking the Fourier trans­form (FT) ofCdm,k) with respect to m yields an estimate of the cross-PDS ofxl(k) andx2(k) attimek. That is, G12(j, k) =F{C12(m,k)}, (8) where F{ . } represents the FT of { . } with reseect to m. The cross correlation function estimate CI2(m,k) in (6a) has been used to estimate the time delay param­eters, 8-10 and the approach has been referred to as the ABCfDE (adaptive basic cross correlation for TDE) algo­rithm. lO From (5a), (6a), and (8), we can see that the ABCfDE algorithm realizes the BCC method in a recursive way. From (2) and (4b), the GCC function of the Roth pro­cessor is given by R If)(m) = F-1{[Gdf> ]I[G22(f) n, Iml<M. (9) It is known that R If)(m) represents the impulse response approximates Xl (k) as a weighted sum of x2(k - m) for Iml<M. A class of adaptive filter algorithms has been developed to recursively update the optimum filter coefficients. 15-17 In this paper, we restrict our interest to the LMS adaptive fil­terlS. 16 since it is computationally very simple but still very effective. The LMS adaptive filter algorithm updates the fil­ter coefficients hI2(m,k) to minimize the mean-squared er­ror E{e2(k)} in Fig. 1, where Xl (k) andx2(k) are applied as primary and reference inputs, respectively, and the M-sam­pie delay is introduced to Xl (k) to make the system causal. The LMS algorithm is summarized in the following: hI2(m,k + 1) = h12(m,k) + lIte(k)x2(k - m), Iml<M, (lOa) where M e(k) = Xl (k) - L h(m,k)x(k - m). (lOb) m= -M In ( lOa), I-' controls the convergence rate and stability of the adaptive filter. The time constant of the LMS adaptive filter can be approximated as l5,l6 n~ l/lIt~, (11) where ~ is the variance ofx2(k). From (9) and (lOa), we can see that taking the Fourier transform of hdm,k) with respect to m yields A A H(j, k) ~F{h(m,k)}, Iml<M (12a) (12b) which is an estimate OfGI2(f)/G22(f) in (9) attimek. From (9), (12a), and (12b), we can see that the im­pulse response function of the LMS adaptive filter is an esti­mate of the GeC function of the Roth processor. This ap­proach has been referred to as the LMSTDE (LMS for TDE) algorithm. IO,13,14 e (kJ function h 12 (m) of the optimum (Weiner) filter which best FIG. 1. Block diagram of the LMS adaptive filter algorithm. 189 J. Acoust. Soc. Am., Vol. 80, NO.1, July 1986 Youn st s/. : Adaptive phase transform processors 189 TOE wo(k) D(k) ALGORITHMS FIG. 2. Block diagram for generating signals for simulations. II. THE PHASE TRANSFORM PROCESSOR AND ITS ADAPTIVE IMPLEMENTATIONS The phase transform processor was proposed as an ad hoc method to obtain a clear indication of the peak and to remove the effect of interfering tonals of the pertinent Gee function by weighting the phase function in (3) uniformly over the entire frequency band. Thus, from (2) and (4c), the Gee function of the PHAT processor is given by R If>(m) = F- I{[GI2if)VIG12if) I}, Iml<M =F-I{ei812(/)}. (13) Introducing a time index k in ( 13) yields the time-vary­in~ Gee function of PHA T as R Ifl(m,k) = F- I{[G12(f, k) VIGI2 (f, k)l} = F -1{eJ612 (f, kl}. (14) Now, using (8) and (12b), the time-varying GCe func­tion of the PHA T can be estimated using the one-pole low­~ filter in (6a) and the LMS adaptive filter algorithm in (lOa) and (lOb) as follows: and RCPIl(mk)=F-I{qI2(f,k)} Iml<.M (ISa) 12, IGI2 (f, k)1 ' A RCP2l(m,k) =F-I{ ~12(f,k) } IHI2(f, k)1 ~ =F-I{(GI2(f,kJ [I G12(f,k) I-I]}, G22(f, k) G22 (f, k) Iml<.M. (1Sb) The above approaches in ( ISa) and ( ISb) will be referred to as the APHAT-l and APHAT-2, respectively, when the TABLE I. Summary of the parameters used for the simulations. Case B(z) P(k)8 D(k) 1 O+Z-I)12 0 4 2 b 0 4 3 I 0 4 4 Z-I 3PI (k) + 2P2(k) 4 I -Z-I + 0.8z-:-2 time-delay estimate is given by the argument m = D(k), . where the relevant time-varying Gee functions R If ll (m,k) and R If2l (m,k) are maximum. In many passive sonar signal processing problems, the received signals often include strong tonaIs p(k) [see (I)]. One of the sources of the periodic components might be the engine or propeller of a target. Another important case of such signals can be encountered when the target transmits narrow~bandjamming signals to hide its location and move­ment. In general, there may be more than one tonal involved. Computing the cross correlation function of XI (k) and x2(k) in (l), we have C12(m) = C ... (m - D) + Cpp (m - D), (16a) where Css (m) = E{s(k)s(k + m)} (16b) and Cpp(m) =E{p(k)p(k +m)} (l6c) represent the auto correlation functions of s(k) and p(k), respectively. If no periodic components are involved in the received signals, (16a) becomes (17) and the time-delay parameter D can be estimated as the ar­gument m =D, where CI2(m) is maximum. However, in the presence of strong tonals, the cross correlation function CI2(m) might yield peaks at several different places to esti­mate incorrect delay parameters, since the cross correlation functions of periodic signals are also periodic. The PHA T processor in (13) is rather simple but per­forms very well in the presence of strong tonals when the source signal is white or broad bandlimited. If we consider the magnitude of the cross-PDS of XI (k) and x2(k) in the presence of strong tonals, the spectral components of the periodic signals are given by impulse functions at the rel­evant frequencies. Thus we see that normalizing the cross­PDS with its magnitude as in ( 13) or ( 15) produces an effect of de-emphasizing the strong tonals. . Now, consider the case of G12if) = 0 in some frequen­cy band (i.e., bandlimited source signal). Then the phase function in (3) is undefined in that band and the estimate of the phase is erratic. Thus normalizing the cross-PDS with its magnitude or weighting the phase function uniformly in the entire frequency range introduces errors in estimating the time delay. Therefore, this behavior suggests that the phase D(k) P p. 0 0.9998 SXIO-5 0 0.9998 Sx 10-5 0 0.9999 SXIO-5 9 0.9999 S.88XIO-6 5 I 2PI + P2(k) - 8 +0.OO2k 0 0.99 1.11 X 10-3 6 I 3PI (k) + 2P2(k) - 8 +O.OO2k 4 - O.OOlk 0.998 1.18 x 10-4 • PI (k) = sin (O.46·1/'·k-O.S) and P2(k) = sin (0.12·1/'.k-O.S). b B(z) for case 2 is the 6th-order Butterworth low-pass filter with cutoff frequency of 0.2 Hz, and sampling frequency of2 Hz. 190 J. Acoust. Soc. Am., Vol. 80, No.1. July 1986 Youn et at : Adaptive phase transform processors 190 function eJ812(f) be additionally weighted to compensate for the presence or absence of signal power as in the case of the Roth,' Scot,8 and ML (maximum likelihood)2 processors. Even though the APHA T algorithms, like the conventional PHA T, have the above problem, it will be shown that they are very effective when the source signal has broad band­width and when the received signals contain strong interfer­ing tonals. This property will be demonstrated in the next section via computer simulations. III. EXPERIMENTAL RESULTS The properties of the APHA T algorithms will be dis­cussed by comparing the performances of the APHA T -1 and -2 processors with those of the ABCfDE9,lO and LMSTDEl6-14 algorithms through computer simulations. The schematic diagram used to generate the received signalsx1(k) andx2 (k) is depicted in Fig. 2, where a white Gaussian random signal wo(k) is processed throughB(z) to generate the·source signal s (k). Also, the source signal s (k) and the periodic signalp(k) were passed through time-vary­ing filters with the transfer functions of e-M1D(k) and -30 1\ -30 -30 -30 " o o ~(p1) R12 (m,8000) ,AAAAA ,AA " Y,~ ~ ,~ v, o --(p21 R12 (m, 8000') .AA/.A.A ,A YV v.v o (a) m 30 (b m 30 (e) m 30 (d) m 30 FIG. 3. Estimated GCC functions for broadband low-pass source signal with additive white noise: (a) ABCfDE; (b) LMSIDE; (c) APHAT-I; (d) APHAT-2. 191 J. Acoust. Soc. Am., Vol. 80, No.1, July 1986 e-fl.,qD(k) to generate s[k - D(k)] and p[k - b(k)], re­spectively. 19 Here, D(k) and D(k) represent the time-vary­ing delay functions related to the source signal s(k) and in­terfering tonalsp(k), respectively. For all of the simulations, 61 coefficients of C12(m,k) and h12(m,k) were estimated (i.e., M = 30) and a Hamming window function with 61 eoints was applied before taking the FT of C12(m,k) and h 12 (m,k ), respectively. For all of the simulations except case 3, the source signals and additive noises were scaled to have unit variances, while the variances of s(k) and W, (k) are given by 0.1 and 0.9 for case 3(a) (i.e., 8NR = 1/9) and 0.0476 and 0.9524 for case 3 (b) (i.e., 8NR = 1/20). Other parameters for the simulations are summarized in Table I. The estimated Gee functions at k = 8000 for cases 1-4 are displayed in Figs. 3-6, where the delay parameter of interest is constant [i.e., D(k) = 4 samples]. Also, the estimated delay functions for cases 5 and 6 are presented in Figs. 7 and 8, respectively, where the delay function of the source signal linearly increases from - 8 to 8 in 8000 samples as indicated by a dotted line, and the delay parameter was computed ev­ery 20 samples, starting from k = 80 and ending at k= 8000. (a) (b ~~~~~~~~~~~m -30 0 30 -30 R~:1) (m, 8000) o R~:21(m, 8000) (d FIG. 4. Estimated GCC functions for narrow-band low-pass source signal with additive white noise: (a) ABCfDE; (b) LMSTDE; (c) APHAT-I; (d) APHAT-2. Youn sf at : Adaptive phase transform processors 191 -30 -30 -30 ~ h12 (m, 8000) o R~~1) (m, 8000) A 'v" ~ '" A, v,f \... o R~:21 (m, 8000) A " ",A f\... '\l ~ y.v o SNR = 1/9 (a) (b (e, m 30 (d) m 30 (e) ~ h12 (m, 8000) -30 o R~~11 (m, 8000) (g) ~~~~~~~~~~~~m -30 0 30 R~~21 (m, 8000) (h) ~~~~~~~~~~~~m o 30 SNR = 1/20 FIG. S. Estimated GCC functions for white source signal with additive white noise: (a) ABCTDE; (b) LMSTDE; (c) APHAT-l; (d) APHAT-2; (e) ABCTDE; (f) LMSTDE; (g) APHAT-l; (h) APHAT-2. IV. DISCUSSION Cases 1 and 2: The estimated GCC functions in Fig. 3 demonstrate that the APHAT-l and -2 algorithms perform as well as the ABCTDE and LMSTDE algorithms do, when the source signal has broad bandwidth. However, since the source signal for case 2 is narrow bandlimited, the phase information outside the frequency band of the source signal is not related to the time delay, but is given by a random­phase function. Therefore, uniformly weighting the phase function in the entire frequency range as in APHA T -1 and -2 results in emphasizing the frequency band where only spec­tral estimation errors exist, to yield noisy GCC function esti­mates as shown in Fig. 4(c) and (d). The results for cases 1 and 2 suggest that the APHAT-l and -2 are efficient methods to estimate time delay for the source signals with broad bandwidth, but fail to estimate correct delay parameter for narrow bandlimited source sig­nals. Case 3: The relevant GCC functions for the four adap­tive time-delay estimation algorithms are displayed in Fig. 5 when the source signals are white and for two different SNR's (Le., 119 and 1120). These results show thatthe per- 192 J. Acoust. Soc. Am., Vol. 80, No.1, July 1986 formances of the APHA T -1 and -2 are as good as those of the others. Here, the less noisy GCC function estimates for the APHAT-l and -2 are due to the Hamming window func­tions applied before taking the Fourier transform of C\2(m,k) and h\2(m,k), respectively. Case 4: The signals used in this set of simulations were obtained by passing white Gaussian signals through a sec­ond- order bandpass filter and then corrupting the output with interfering tonals as well as additive white noises, and the delay parameter of the source signal is given by D( k) = four samples. The result in Fig. 6( a) shows thatthe GCC function for the ABCTDE algorithm is maximum at m = 0, which is the delay parameter relevant to interfering tonals. Similarly, the GCC function in Fig. 6(b) for the LMSTDE algorithm peaks at an incorrect position, even though the effect of the tonals is less than that of the ABCTDE algorithm. However, the GCC function estimates of the APHAT-l and -2 are maximum at m = 4, and yield the correct time-delay estimate. We notice that the APHAT- 2 performs better than the APHA T -1. This is because the periodic components have been already de-emphasized and the bandlimited source signal is whitened in the process of LMS adaptive filtering. Youn 6t al .. : Adaptive phase transform processors 192 C'2(m, 8000) ~t -30 0 30 h'2(m, 8000) ~~rnltAm -30 0 30 ~ (pI) R'2 (m,8000) t~~m -30 0 30 R\~2)(m, 8000) b~~~Jm -30 a 30 FIG. 6. Estimated Gee functions for bandpass source signal with additive white noise and interfering tonals: (a) ABCfDE; (b) LMSIDE; (c) APHAT-I; (d) APHAT-2. :.: ['" ---:J~ -8.0 /?:. k 80 8000 ::j'" ::?I -•. o~. • 80 8000 ::j'" ?1'. -8.0-~._k 80 8000 FIG. 7. Estimated time-varying delay functions for white source signal with additive white noise and interfering tonals (case S): (a) ABCfDE; (b) LMSIDE; (c) APHAT-l; (d) APHAT-2. 193 J. Acoust. Soc. Am., Vol. 80, No.1, July 1986 D(k) 8.0 --,,-.. (8) 0.0 -8.0 - k 80 8000 8.0 D(k) 0.0 -8.0 k 80 8000 - J~~'~ 80 8000 ..... 8.0 D(k) 0.0 -8.0 80 k 8000 FIG. 8. Estimated time-varying delay functions for white source signal with additive white noise and interfering tonals (case 6): (a) ABCIDE; (b) LMSIDE; (c) APHAT-I; (d) APHAT-2. From the above results, we see that the APHAT-l and -2 are effective even when the source signal is narrow bandlimited but still has some power in a wide range offre­quency bands. Cases 5 and 6: The last two sets of simulations concern the problems of estimating time-varying delay functions, which correspond to moving source or receivers. 10-14.21-23 Here, the delay functions relevant to the source signals lin­early increase from - 8 to + 8 in 8000 samples, while those of the interfering tonals are constant [i.e., D(k) = 0] and linearly decrease from + 4 to - 4 in 8000 samples; D(k) = 4-0.001 k for cases 5 and 6, respectively. From the estimated delay functions in Figs. 7 and 8, we observe that the ABCTDE method estimates the delay func­tion relevant to the interfering tonaIs [i.e., D(k), see Figs. 7(a) and 8(a)], while the LMSTDE algorithm, APHAT-l, and APHA T -2 track the correct relay parameter relevant to the source signals [i.e., D(k)]. Also, the results show that the APHAT-l and APHAT-2 perform superiorly to the LMSTDE algorithm. V. CONCLUSIONS Two adaptive implementations of the phase transform processor, using a bank of simple one-pole low-pass filters Voun 61 al : Adaptive phase transform processors 193 (APHAT-l) and the LMS adaptive filter (APHAT-2), re­spectively, were introduced. It was demonstrated that these algorithms are more effective than the ABCTDE and LMSTDE algorithms in tracking constant and time-varying delay functions associated with broad bandlimited source signals in the presence of strong tonals. It was also shown that the APHAT algorithms should be used with caution when the source signals are narrow bandlimited. The APHAT-l algorithm is attractive because of its computational simplicity. However, as demonstrated in Fig. 6, the APHAT-2 processor may give more accurate time­delay estimates because of the inherent whitening of the source signals in the process of LMS adaptive filtering. [G. c. Carter (Ed.), Special issue on time delay estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP·29, 461-624 (1981). 2C. H. Knapp and G. C. Carter, "The generalized correlation method for estimation of time delay," IEEE Trans. Acoust. Speech Signal Process. ASSP·24, 320-327 (1976). 3W. R. Hahn, "Optimum signal processing for passive sonar range and bearing estimation," J. Acoust. Soc. Am. 58, 201-207 (1975). 4y. H. MacDonald and P. M. Schultheiss, "Optimum passive bearing esti· mation in a spatially incoherent noise environment," J. Acoust. Soc. Am. 46,37-43 (\969). 'G. C. Carter, "Passive ranging errors due to receiving hydrophone posi· tion uncertainty," J. Acoust. Soc. Am. 65, 528-530 (1979). 6K. Scarbrough, N. Ahmed, and G. C. Carter, "On the simulation ofa class of time delay estimation algorithms," IEEE Trans. Acoust. Speech Signal Process. ASSP·29, 534-540 (1981). 7p. R. Roth, "Effective measurements using digital signal analysis," IEEE Spectrum 8, 62-70 (1971). sN. Ahmed and S. Yijayendra, "An algorithm for line enhancement," Proc. IEEE 70,1459-1460 (1982). 194 J. Acoust. Soc. Am., Vol. 80, No.1, July 1986 90. L. Johnstone and O. L. Frost, "High resolution differential time·of· arrival and differential Doppler estimation," ARGO Systems, Inc., 1069 East Meadow Circle, Palo Alto, CA 94303. 100. H. Youn, "A class of adaptive methods for estimating coherence and time delay functions," Ph.D. thesis, Department of Electrical Engineer· ing, Kansas State University (1982). lip. L. Feintuch, N. J. Bershad, and F. A. Reed, "Time delay estimation using the LMS adaptive filter-dynamic behavior," IEEE Trans. Acoust. Speech Signal Process. ASSP·29, 571-576 (\981). [2F. A. Reed, P. L. Feintuch, and N. J. Bershad, "Time delay estimation using the LMS adaptive filter-static behavior," IEEE Trans. Acoust. Speech Signal Process. ASSP·29, 561-571 (1981). l3D. H. Youn, N. Ahmed, and G. C. Carter, "On using the LMS algorithm for time delay estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP·30, 798-801 (1982). [40. H. Youn, N. Ahmed, and G. C. Carter, "An adaptive approach for time delay estimation of bandlimited signals," IEEE Trans. Acoust. Speech Signal Process. ASSP·31, 780-784 (1983). ['B. Widrow, J. R. Glover, Jr., J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, Jr., and R. C. Goodlin, "Adaptive noise cancelling: Principles and applications," Proc. IEEE 63, 1692-1716 (1975). [6B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson, Jr., "Sta· tionary and nonstationary learning characteristics of the LMS adaptive filter," Proc. IEEE 64, 1151-1162 (1976). 17S. R. Parker and L. J. Griffiths (Eds.), Joint Special Issue on Adaptive Signal Processing, IEEE Trans. Acoust. Speech Signal Process. ASSP.29, 625-775 (1981); IEEE Trans. Circuits Syst. CAS·29, 465-615 (1981). [sG. C. Carter, A. H. Nuttall, and P. G. Cable, "The smoothed coherence transform (SCOT)," Proc. IEEE 61, 1497-1498 (\973). [90. H. Youn, N. Ahmed, and G. C. Carter, "A method for generating a class of time·delayed signals," in Proceedings of ICASSP, Atlanta, GA, 1981,pp.1257-126O. 20p. C. Chestnut, "Emitter location accuracy using TDOA and differential Doppler," IEEE Trans. Aerosp. Electron. Syst. AES·18, 214-218 ( 1982). 2 [C. H. Knapp and G. C. Carter, "Estimation of time delay in the presence of source or receiver motion," J. Acous(Soc. Am. 61,1545-1549 (1977). 220. M. Etter and S. D. Stearns, "Adaptive estimation of time delays in sampled data systems," IEEE Trans. Acoust. Speech Signal Process. ASSP·29, 582-587 (1981). 23y' T. Chan, J. M. F. Riley, and J. B. Plant, "Modeling of time delay and its application to estimation of non stationary delays," IEEE Trans. Acoust. Speech Signal Process. ASSP·29, 577-582 ( 1981). Youn et a/. : Adaptive phase transform processors 194