Maximum Likelihood Time Delay Research Articles

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.

Time Synchronization of Turbo-Coded Square-QAM-Modulated Transmissions: Code-Aided ML Estimator and Closed-Form Cramér–Rao Lower Bounds

This paper introduces a new maximum likelihood (ML) solution for the code-aided (CA) timing recovery problem in square-quadrature amplitude modulation (QAM) transmissions and derives, for the very first time, its CA Cramer–Rao lower bounds (CRLBs) in closed-form expressions. The channel is assumed to be slowly time varying so that it can be considered as constant over the observation interval. By exploiting the full symmetry of square-QAM constellations and further scrutinizing the Gray-coding mechanism, we express the likelihood function of the system explicitly in terms of the code bits’ a priori log-likelihood ratios (LLRs). The timing recovery task is then embedded in the turbo iteration loop, wherein increasingly accurate estimates for such LLRs are computed from the output of the soft-input soft-output decoders and exploited at a per-turbo-iteration basis in order to refine the ML time delay estimate. The latter is then used to better resynchronize the system, through feedback to the matched filter, so as to obtain more reliable symbol-rate samples for the next turbo iteration. In order to properly benchmark the new CA ML estimator, we also derive for the very first time the closed-form expressions for the exact CRLBs of the underlying turbo synchronization problem. Computer simulations will show that the new closed-form CRLBs coincide exactly with their empirical counterparts evaluated previously using exhaustive Monte Carlo simulations. They will also show unambiguously the remarkable performance improvements of CA estimation against the traditional nondata-aided scheme, thereby highlighting the potential performance gains in time synchronization that can be achieved owing to the decoder assistance. Over a wide range of practical signal-to-noise ratios (SNRs), CA estimation becomes even equivalent to the completely data-aided scheme in which all the transmitted symbols are perfectly known to the receiver. Moreover, the new CA ML estimator almost reaches the underlying CA CRLBs, even for small SNRs, thereby confirming its statistical efficiency in practice. It also enjoys significant improvements in computational complexity as compared to the most powerful existing ML solution, namely the combined sum-product and expectation-maximization algorithm.

First-order bias and second-order variance of maximum likelihood time delay and Doppler shift estimates

Exact expressions for the bias and variance of a maximum likelihood time delay and Doppler shift estimate are difficult to obtain analytically. It has become popular in such nonlinear problems to compute limiting bounds on mean-square estimation error, such as the Cramer–Rao bound (CRB), since these are easier to obtain than the true variance. Recently it has been shown [Makris and Naftali, J. Acoust. Soc. Am. 106, 2 (1999)] that the first-order bias and second-order variance of a general maximum-likelihood estimator (MLE) can be computed analytically, using higher-order asymptotics. This approach is applied to the classic radar/sonar problem of estimating the time delay and Doppler shift of a deterministic signal in additive white Gaussian noise. The MLE is shown to be unbiased to first order for this problem. By evaluating the asymptotic expansion for the variance for three specific signal waveforms, Gaussian, LFM, and HFM, it is found that (a) the CRB yields an unrealistically optimistic variance estimate for SNR’s lower than 20–30 dB, (b) both first- and second-order time-delay variance terms decrease with increasing signal bandwidth, and (c) both first and second Doppler-shift variance terms of LFM and HFM signals approach constant values as bandwidth increases.

Performance Study of Time Delay Estimation in a Multi-Path Environment

A theoretical analysis of the performance of the maximum likelihood (ML) time delay estimate in a multi-path propagation is proposed. An expression for the probability of anomaly of the ML time delay estimate is obtained. Percentages of anomalous time delay estimates obtained through Monte Carlo simulation are shown to be in close agreement with theoretically predicted values.

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>

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.