Estimating the underlying signal waveform, noise covariance and synchronization jitter from unsynchronized measurements

Abstract

In this paper a new synchronization technique is presented for applications using repeated measurements or experiments with periodically excited signals. The objective with repeated or periodic measurements is often to retrieve an estimate of the (noise reduced) signal and its uncertainties. However, these measurements need to be synchronized to obtain accurate estimates. Existing synchronization techniques are limited to specific signal and noise conditions, such as white Gaussian noise or narrowband signals, to achieve good performance. The proposed method, not limited by these conditions, extracts statistical information regarding the underlying signal and the noise contained in the measurements, to obtain good synchronization (asymptotically optimal). The Cramér-Rao lower bound (CRLB) is derived for the synchronization problem, including bounds for the underlying signal waveform and the covariance of the measurement noise, both considered unknown. The method, which is shown to be the maximum-likelihood estimator (MLE) in both white and colored Gaussian noise, is compared with the CRLB along with standard sub-sample estimation and aligning techniques using Monte Carlo simulations. The results show significant mean square error (MSE) improvements compared to standard synchronization techniques. Synchronization results using the proposed technique are presented for repeated ultrasonic measurements, to validate the method in a real measurement situation, and to experimentally support theoretical results.

Compensating distortion effects in repeated measurements under non-stationary conditions

“…The linear effects τ m and γ m are separated to emphasize that variations in phase delay (τ m ) can also be introduced in stationary conditions by non-environmental effects, e.g., synchronization trigger variations (jitter) from the measurement equipment addressed in [2].…”

“…(co)variances, are required to: optimally weight objective functions when fitting models or parameters to measured data; obtaining uncertainty bounds for estimated models and parameters; detecting model errors; and for experimental design [22]. In [2], the filtering effect was investigated in detail for the special case of variations in time-of-flight only, β(ω) = exp(−jωτ ), for different probability density functions of τ . The result showed a destructive low-pass filtering effect even for very small variations in time-of-flight (τ < T s ).…”

“…If the measurements are highly unsynchronized, i.e. |τ m | > T s , a useful step is to pre-align the measurements to the closest whole sample using standard cross-correlation techniques to avoid local likelihood maxima [2].…”

“…The Fisher information matrix (FIM) [20] for this compensation problem does not have full rank [2]. The reason for this is that the perfect reference point is unknown and the objective is only to minimize the dynamical differences relative to each other.…”

“…For example, repeated acoustic or mechanical wavepropagation measurements under small pressure or temperature variations, the dominant effect is the change in time-of-flight (the speed of sound), resulting in phase-shifted unsynchronized measurements. However, for large variations or drifts, the visible effect is not restricted to pure-phase delays that can be compensated using standard synchronization techniques [2,3], but also to changes in the signal's spectral content. A more detailed study of these effects can be found in section 2.2.…”