The objective of the TDE is to estimate the value the time delay, D, from the knowledge of the x(n) and s(n) as well as some statistical assumptions about the reference signal and noises.

The TDE problem arises quite naturally in surface profiling using airborne ultrasonics. In the case of zero mean, stationary, uncorrelated Gaussian random processes representing the reference signal and the noises, the optimal time delay estimator is the generalised cross correlator. In recent years, a number of studies on the TDE based on the higher order statistics (HOS) has emerged. In these studies one generally assumes that the reference signal arises from a stochastic process (Gaussian or non-Gaussian) and the time delays, although not assumed explicitly so, tend to be rather large (in units of sampling time, T_s). One particular problem, common to all these methods, is the severe degradation of their performance at low SNR.

In this paper we consider the case of the reference signal being narrowband and deterministic while noises are zero mean, stationary, and uncorrelated random variables and the case of the time delays being very small (less than T_s) while the SNR can be very low. In this paper two new estimators (MSX and MXS) based on cross and auto-correlations are proposed as well as a further two cross-correlation based standard algorithms are presented for comparison both using simulations and experimental data.

Simulations

Simulation are carried out using MATLAB on unix based SUN workstations. In all the simulations to follow, the search space to locate the extremun, for both the DC and ASDF estimators has been reduced from all lags to the first few legs. Here the reference signal (ultrasonic echo) has been modelled as a 'decaying' cosine wave in the form of

where A is the amplitude, controls the rate of decay, and f₀ is the centre frequency. The choice of this form for the reference signal relates well to the applications mentioned earlier. The following parameter values have been used in simulations - A = 7X10⁵ V_s^-1, = 10¹⁰ s^-2, and f₀ = 10⁶Hz, T_s = 50X10^-9 s (corresponding to a sampling frequency of 20X10⁶ Hz), and 1000 samples (N=1000) for each echo have been generated.

Simulations are carried out with additive noises being zero mean, uncorrelated Gaussian noise for both the reference signal and the time delayed signal. Bothe gx(n) and gs(n) are uncorrelated noises but of identical' power spectrum. The SNR has been defined by

To assess the performance over a broad range of time delays (as will be the case in ultrasonic applications) 1000 time delays are uniformly randomly generated between 0 and T_s. Signals corresponding to 10 dB, 0 dB, and -5 dB SNR have been generated. Results are summarised in the table below. Each pair of numbers in the table for a particular estimator and a specific SNR represents the mean of the differences between the estimated time delays and the corresponding true time delays, and the standard deviation of these differences. These demonstrate the poor performance of the DC and ASDF estimators at low SNR, and the comparatively better performance of the proposed estimators over the corresponding low SNR.

Bias and standard deviation (in units of the sampling time, T_s) of time delay estimates obtained in simulations. Each pair of numbers is obtained from 1000 delays, uniformly and randomly, chosen between 0 and T_s

Concluding remarks

Two estimators for time delay estimation of narrowband signals, based on matching estimated cross-correlations with estimated auto-correlations, are proposed. These estimators have been seen to perform adequately over the SNR range used in simulations of -5 dB to 20 dB. Results from the analysis of experimental data will be presented.