3.3.1 Alignment of Noisy Signals

K.J.Coakley and J.Wang
Statistical Engineering Division, ITL

P. Hale and T. Clement
Optoelectronics Division, EEEL

D. DeGroot
Radio-Frequency Technology Division, EEEL

As part of a program to characterize photodiodes, we characterize the impulse response function of high speed oscilloscopes. To improve the signal-to-noise 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 cross-correlation analysis. In the naive implementation of the cross-correlation method, for a set of N signals, relative shifts are estimated from cross-correlation analysis of N-1 pairs of signals. In the complete implementation of the cross-correlation method, estimates are based on cross-correlation analysis of all N(N-1)/2 distinct pairs of signals. In the adaptive implementation of the cross-correlation method, relative shifts are estimated from 2(N-1) pairs of signals.

For various noise levels, we simulate a set of 100 misaligned signals. For all noise levels, the complete implementation of the cross-correlation 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 cross-correlation method depends on the choice of noise levels. The relative accuracy of the robust centroid and the naive implementation of the cross-correlation method depends on the choice of noise levels. In most all cases, the adaptive implementation of the cross-correlation method is more accurate than the naive implementation of the cross-correlation 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,

$$\hat{C}= \frac{ \sum_ {j}\vert s( t_j ) \vert H(t_j,\alpha) t_j } { \sum_ {j}\vert s( t_j ) \vert H(t_j,\alpha) }$$

where

$$H(t,\alpha)=\left\{ \begin{array}{ll} 1 & \mbox{ if $\vert s... ...t > \alpha$ } \\ 0 & \mbox{ otherwise.} \end{array} \right.$$

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
Last updated: 7/20/2001
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