Time delay estimation in unknown spatially correlated gaussian noise using higher-orider statistics

A novel approach is plesented to estimate the difference in arrival times of a signal at two senson. '& sensor measurements are assumed to be corrupted by spatially correlated Gaussian noises whose covariances are unknown. We propose an optimization approach where the time delay between the two signals is estimated by minimizing a mean fourth cumulant criterion. The proposed criterion is unaffected by the correlated Gaussian noises. It is shown that the criterion yields the vue time delay under the assumption that $e signal is non-Gaussian with non-zero founh cumdant at zero lag. The proposed approach also yields the correct time delay in the presence of spatially independent non-Gaussian noises. Some variarjons of the proposed method are also discussed. Simulations are presented in support of the proposed approach. I. INTRODUCllON The estimation of time delays between received signals at two (or more) sensor locations is an impomt problem in several fields [11-[51.[111.[12] such as sonar, radar, biomedicine, and geophysics. For example, in passive sonar, the time delay is used to estimate the position and the velocity of a detected acoustic source [ I ] Various methods have been proposed and implemented over the years for time delay estimation: see. e.g.. [1]-[5]. The conventional methods based on crosscorrelation techniques [11-[31. 151 do not work well when the noises at the two sensors are correlated 141. To alleviate this problem when the noises are jointly Gaussian and the signal is non-Gaussian, Nikias and Pan 141 recently proposed a novel approach to time delay estimation by exploitation of "higher-order cross-ctunulants." Both parametric and nonparametric approaches were considered In Uus paper u e propose an alternative paramctric approach that alsu rxploils higher.ordcr tulislics of the signal as In (41. hut also w o k s uhcn the noiws arc spatially independent non-Gaussian pmccsrs. unlikc the approach of 141. 'lhc approach of .41 doeS not offur my optimi?.rtion cntcnun: rather. a cross-iumulmt extension of the usual crurs-conrlanon approach IC proposed in [4]. Whcrras the croswonrlalion appmach follows from the I..MS (least mean squaw) ermr cntcnon. the crosssumulant approach docs not follow from any optimizauon critrnon We propose extrcmi2ing 3 mean founh.cumulant criicnon (LIFC) lo eslimate the lime delay. AS such. our prop.& criterion may k vieued as an Cxtcnsion of the LMS criterion. Hinich et al. 161 have shown via bispectral analysis that whereas the ambient man noise is indeed Gaussian. Ihe ship radiated noise is non-Gaussian. Thus, use of higber-order statistics for time delay estimation in passive sonar is well motivated. The problem at hand can be cast in the framework of a plant (system) identfication problem where signal at one sensor is the (noisy) input and the signal ai the other sensor is the noisy output. The non-standard aspect of the problem lies in the fact that the plant noise is comlated with the input. We can also interpret the given problem in the general framework of enom-in-variables models [IO]. The use of higher-order autoand cross-cumulant spectra for general linear system identification dates back to Akaike [81,[91. More recently, use of higkr-order slatistics for identification of ermrs-invariables models has been suggested by Deistler and Anderson [IOl. To the best of our knowledge. the use of the fourth-cumulant criterion has not been suggested before. The paper is organized as follows. The mathematical model is described in Section 11. The mean fourthcumulant criterion is presented and analyzed in Section Ill. Some extensions of the proposed approach are described and analyzed in Section IV. A simulation example is presented in Section V. II. MODEL ASSUMPTIONS Let (rl(t)) and (rz(t)) denote the two sensor measurements of a (2.1) (2.2) where D is a fixed time delay (advance). The following conditions are assumed U1 hold. (H1) The signal process (s(t)) is ("founh-ordet") stationary and (2.3) (H2) The noise processes (n,(t)] and In2@)) are both independent of the signal (s(t)). Furthemore, (nl(t)] and Inz@)) are jointly Gaussian with unknown covariances and may be mutually "elated. AS noted in Section I, Hinich et al. [til have shown that ship signal s(t) (tis discrete time, for convenience). given by (1) = s(t) + nt(0 r2(t) = s(t-D) + n2(t). nonGaussian with 74. := E(s4(t)) 3 [E(sz(t))l2 # O . radiated noise is indeed nonGaussian and it satisfies (2.3). The objective here is to estimate D given the sensor measurements (rl(t). ISEN) and (rz(t). lStsN). 21 I 23ACSSC-12/89/0211 51.00 01989 MAPLE PRESS IIL OPTIMIZATION CRITERION Fint recall from [I31 (see also [SI) that the "basic" (no prefiltering) cross-comlator approach to time delay estimation is equivalent to choosing a delay d which minimize the mean-square ermr criterion Ii(d):=E([ri(t-d)-r~(t)l2). (3.1) In tetms of the data records and sample averaging, (3.1) may be rewritten as jlN(d) :=NI [ri(t-d)-r2(t)?. (3.2) N ,=I Then the estimate of the time delay, b, is dehned as 6 := arg( Il(d) 1 Motivated by the ahove approach. we propose the following cost function for the case of spatially wmlatcd Gaussian noisc: Jdd):= C W I r lW)r2 ( i ) I (3.3) CUM,,(w) :=EIw') -31Elw21 1'. (3.4) where In terms of the data rccords and sample averaging. (3.3) may rewnacn .U H t i l i.m := K' 1 1 ri(1-d) 40) 1'