Maximum Likelihood Time Delay Estimation Research Articles

Approximate maximum likelihood time-delay estimation for two closely spaced sources

The study of ground reflections of Global Navigation Satellite System (GNSS) signals, as in GNSS Reflectometry (GNSS-R) can lead to the receiver height estimation. The latter is estimated by comparing the time of arrival difference between the direct and reflected signals, also called path separation. In ground-based scenarios, this path separation can be very small, inducing important interference between paths, which makes it difficult to correctly obtain altimetry products. The path separation estimation can be obtained by a brute force dual source maximum likelihood estimator (2S-MLE), but this solution has a large computational cost. On the other hand, the path separation is so small that a number of approximations can be done. In this study, a third order Taylor approximation of the dual source likelihood criterion is proposed to reduce its complexity. The proposed algorithm performance is compared to the non approximated 2S-MLE for the estimation of the path separation, and to a standard single source processing for the estimation of the direct signal time-delay. These results, along with the corresponding lower bounds, prove that the proposed approach may be of interest for two applications: ground-based GNSS-R altimetry (or radar with low elevation targets) and GNSS multipath mitigation.

Time delay estimation in reverberant and low SNR environment by EMD based maximum likelihood method

In recent times, time delay estimation (TDE) has received significant practical importance in sonar, radar, GPS, and various other fields. In a passive sonar system, estimation of time delay for low frequency and low signal to noise (SNR) acoustic source is a difficult task. If the source and receivers are kept inside the reverberation environment, time delay estimation becomes more difficult because of sound source echo’s. This research work proposes a new TDE approach named empirical mode decomposition maximum likelihood time delay estimation (EMD ML TDE) method, for the low-frequency and low SNR underwater machinery acoustic signal in a reverberant environment. EMD ML TDE method is based on maximum likelihood (ML) method, and it is using empirical mode decomposition (EMD) denoising technique to estimate acoustic sound signal and noise from the noisy reverberant signal. The experimental results are provided that this new approach is better to estimate time delay in reverberant environments.

Eigenvalue-based time delay estimation of repetitive biomedical signals

The time delay estimation problem associated with an ensemble of misaligned, repetitive signals is revisited. Each observed signal is assumed to be composed of an unknown, deterministic signal corrupted by Gaussian, white noise. This paper shows that maximum likelihood (ML) time delay estimation can be viewed as the maximization of an eigenvalue ratio, where the eigenvalues are obtained from the ensemble correlation matrix. A suboptimal, one-step time delay estimate is proposed for initialization of the ML estimator, based on one of the eigenvectors of the inter-signal correlation matrix. With this approach, the ML estimates can be determined without the need for an intermediate estimate of the underlying, unknown signal. Based on respiratory flow signals, simulations show that the variance of the time delay estimation error for the eigenvalue-based method is almost the same as that of the ML estimator. Initializing the maximization with the one-step estimates, rather than using the ML estimator alone, the computation time is reduced by a factor of 5M when using brute force maximization (M denoting the number of signals in the ensemble), and a factor of about 1.5 when using particle swarm maximization. It is concluded that eigenanalysis of the ensemble correlation matrix not only provides valuable insight on how signal energy, jitter, and noise influence the estimation process, but it also leads to a one-step estimator which can make the way for a substantial reduction in computation time.

Joint maximum likelihood time-delay estimation for LTE positioning in multipath channels

This paper presents a joint time-delay and channel estimator to assess the achievable positioning performance of the Long Term Evolution (LTE) system in multipath channels. LTE is a promising technology for localization in urban and indoor scenarios, but its performance is degraded due to the effect of multipath. In those challenging environments, LTE pilot signals are of special interest because they can be used to estimate the multipath channel and counteract its effect. For this purpose, a channel estimation model based on equi-spaced taps is combined with the time-delay estimation, leading to a low-complexity estimator. This model is enhanced with a novel channel parameterization able to characterize close-in multipath, by introducing an arbitrary tap with variable position between the first two equi-spaced taps. This new hybrid approach is adopted in the joint maximum likelihood (JML) time-delay estimator to improve the ranging performance in the presence of short-delay multipath. The JML estimator is then compared with the conventional correlation-based estimator in usual LTE conditions. These conditions are characterized by the extended typical urban (ETU) multipath channel model, additive white Gaussian noise (AWGN) and LTE signal bandwidths equal to 1.4, 5 and 10 MHz. The resulting time-delay estimation performance is assessed by computing the cumulative density function (CDF) of the errors in the absence of noise and the root-mean-square error (RMSE) and bias for signal-to-noise ratio (SNR) values between −20 and 30 dB.

Time-Delay Estimation in the Multi-Path Channel based on Maximum Likelihood Criterion

To locate an object accurately in the wireless sensor networks, the distance measure based on time-delay plays an important role. In this paper, we propose a maximum likelihood (ML) time-delay estimation algorithm in multi-path wireless propagation channel. We get the joint probability density function after sampling the frequency domain response of the multi-path channel, which could be obtained by the vector network analyzer. Based on the ML criterion, the time-delay values of different paths are estimated. Considering the ML function is non-linear with respect to the multi-path time-delays, we first obtain the coarse values of different paths using the subspace fitting algorithm, then take them as an initial point, and finally get the ML time-delay estimation values with the pattern searching optimization method. The simulation results show that although the ML estimation variance could not reach the Cramer-Rao lower bounds (CRLB), its performance is superior to that of subspace fitting algorithm, and could be seen as a fine algorithm.

Fast Iterative Maximum-Likelihood Algorithm (FIMLA) for Multipath Mitigation in the Next Generation of GNSS Receivers

In this paper, we efficiently solve the maximum likelihood (ML) time-delay estimation problem for GNSS signals in a multipath environment. Exploiting the GNSS signal model structure and the spreading code periodicity, we develop an efficient implementation of the Newton iterative likelihood maximization method by finding simple analytical expressions for the first and second derivatives of the likelihood function. The proposed fast iterative ML algorithm (FIMLA) for timedelay estimation, which uses the correlation function of the received signal with its local replica, is shown to be an attractive technique for mitigation of closely-spaced multipath arrivals. For the future modernized GPS and the European Galileo signals based on binary offset carrier (BOC) waveforms, the correlation function has multiple positive and negative peaks leading to potential tracking ambiguities. Instead of the standard crosscorrelation, we propose an implementation characterized by a different choice of the local replica so as to cancel the sub-carrier phase, thus eliminating ambiguities. The asymptotic performance of FIMLA is analyzed by deriving the corresponding Cramer-Rao bound (CRB). Representative simulation examples are included to illustrate the FIMLA is performance for delay estimations in the presence of multipath for both C/A code and BOC signals.

New perspectives for maximum likelihood time-delay estimation

This paper introduces a new realization for maximum likelihood time-delay estimation (TDE) that illuminates the relationships between maximum likelihood TDE and other methods. We obtain the result by deriving the likelihood function using a fundamental method that appears to be new to the field of array processing. This method is a natural complement to the generalized Karhunen-Loeve expansion.

Maximum likelihood time delay estimation in non-Gaussian noise

In a non-Gaussian noise environment, it is theoretically possible to design a delay estimator that performs significantly better than the conventional linear correlator. We study the maximum likelihood estimator for passive time delay in non-Gaussian noise. We show that the form of the best estimator depends strongly on signal-to-noise ratio (SNR), and the estimator optimal at low SNR can fail catastrophically at high values of SNR. The paper uses simulations to examine this sensitivity problem and proposes an ad hoc instrumentation that is reasonably robust over a wide range of SNR values.

Asymptotic receiver structures for joint maximum likelihood time delay estimation and channel identification using Gaussian signals

The problem of jointly estimating the relative time delay and the impulse response linking two received discrete-time Gaussian signals is addressed. Using two different methods, possible structures for the joint maximum-likelihood (ML) estimator are proposed, when the observation interval is long compared to both the delay to estimate and the correlation time, of the various random processes involved. These structures generalize the cross-correlation method with prefiltering that implements the ML estimation of pure time delays.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A lower-bound for the error-variance of maximum-likelihood delay estimates of discontinuous pulse waveforms

A new lower bound is developed for the error variance of maximum-likelihood time-delay estimation when the received signal is a square pulse, corrupted by additive white Gaussian noise. The bound is generated by combining concepts previously developed for a special class of stochastic processes, induced by the signal model. For moderate signal to noise ratios, the new bound is significantly tighter than previously known ones.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exact maximum likelihood time delay estimation for short observation intervals

An exact solution is presented to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T>> tau /sub c/, mod d mod made in the derivation of the conventional asymptotic maximum likelihood (AML) time delay estimator (where tau /sub c/ is the correlation time of the various random processes involved and d is the differential time delay) is relaxed. The resulting exact maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators is compared by means of computer simulations. >

Abstracts

The conduction velocity v of a nerve fibre is calculated from the time delay Δ of a propagating action potential between two recording sites along the fibre. However, the conventional method of determining Δ cannot be applied to short nerve fibres. Therefore several linear signal analysis methods for the estimation of Δ have been compared with regard to the reproducibility of their results obtained from pairs of simultaneously recorded action potentials at several small inter-electrode distances. It was found that, estimating Δ from the cross-correlogram and as a second one a variant of this method (maximum likelihood time delay estimation) give the most reliable values of v in short nerve fibres.