
3.3.1 Alignment of Noisy Signals
K.J.Coakley and J.Wang
P. Hale and T. Clement
D. DeGroot
As part of a program to characterize photodiodes, we characterize the impulse response function of high speed oscilloscopes. To improve the signaltonoise ratio, we average many independent measurements. Due to instrumental drift, the independent signal measurements are misaligned. Thus, before averaging, we must align the measured signals. We study the relative performance of various methods for aligning noisy one dimensional signals. In each method, we estimate the relative shifts of a set of signals which are translated with respect to each other. We simulate signals corrupted by both additive noise and timing jitter noise The simulated signals have complexity similar to NIST data. For an example of a simulated signal, see the upper part of the attached Figure. In one method, we estimate the relative shift of two signals as the difference of their estimated centroids. We present a new robust algorithm for centroid estimation. In a second method, we estimate relative shifts from the analysis of level crossings of the signals. In a third method, we estimate relative shifts from crosscorrelation analysis. In the naive implementation of the crosscorrelation method, for a set of N signals, relative shifts are estimated from crosscorrelation analysis of N1 pairs of signals. In the complete implementation of the crosscorrelation method, estimates are based on crosscorrelation analysis of all N(N1)/2 distinct pairs of signals. In the adaptive implementation of the crosscorrelation method, relative shifts are estimated from 2(N1) pairs of signals. For various noise levels, we simulate a set of 100 misaligned signals. For all noise levels, the complete implementation of the crosscorrelation method is the most accurate method. For all noise levels, the robust centroid method is more accurate than the level crossing method. The relative accuracy of the robust centroid method and the adaptive implementation of the crosscorrelation method depends on the choice of noise levels. The relative accuracy of the robust centroid and the naive implementation of the crosscorrelation method depends on the choice of noise levels. In most all cases, the adaptive implementation of the crosscorrelation method is more accurate than the naive implementation of the crosscorrelation method.
In one approach,
we estimate
the relative shift of two signals
as the difference of their centroids.
Our robust estimate of
the centroid
of a signal,
where In a Monte Carlo study, given knowledge of the actual relative shifts, we can find the optimal value of the threshold by minimizing mean square prediction error. For real data, we do not know the actual relative shifts. Hence, we can not compute the mean square prediction error. As an alternative to the Monte Carlo selection rule, we provide an empirical threshold selection rule. The empirical estimate of the optimal threshold maximizes the total power of the average of the aligned signals. In simulation studies, we find that the empirical selection rule performs well. Figure 14: Top: Simulated signal. Bottom: Empirical and Monte Carlo selection rule estimates of optimal threshold for centroid estimate of relative shift.
Date created: 7/20/2001 