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The problem may be summarized as follows: I have a stochastic signals and I want to measure an increasing delay, as depicted in the following picture. The picture represents the signal as it is being delayed. I pick one acquisition as a reference (e.g. black signal), and then cross-correlate with later acquisitions (red and green) to find the delay.

enter image description here

As the signal is delayed, however, it progressively decorrelates with respect to the first measurement, which we take as reference. This deteriorates TDE performance (due to worse cramer-rao), leading to gradually worse performance and ultimately, to anomalous estimates.

The solution that we have currently implemented is to make a qualitative measurement of cross-correlation between two signals (checking the amplitude ratio between the main cross-correlation lobe and secondary lobes), and once it crosses a certain threshold we update the reference signal to the current one.

There are a couple of problems with this approach, which pose a trade-off: First, before updating, as the signal gets increasingly decorrelated, the performance of TDE is gradually worse due to decorrelation between both signals. On the other hand, updating the reference implies attributing to the last measurement a reference time-delay value, which will have its own associated estimation error. Therefore, every time the reference signal is updated, we are effectively integrating estimation errors, which ultimately present themselves as 1/f noise.

The question, in this case, is whether there is a smarter way to track the signal and minimize reference updates, or address the onset of 1/f noise in any way. Also, as a side-note, what would be a good measurement for correlation quality between two delayed signals?

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  • $\begingroup$ could you update your plot with actual labels. the amplitudes seem to be increasing with delay which makes your description confusing $\endgroup$ – Stanley Pawlukiewicz Mar 25 at 21:49
  • $\begingroup$ Sure. I will update it soon. The amplitude change you mention is the decorrelation I am describing. The signal is delayed and decorrelates with the original signal progressively it is delayed. The reason I removed the labels it's because in this picture the axis are already converted to other units (Time to position, namely, which might make the plot confusing). Either way, you can think of that as the black one being the first acquisition (reference), the red one one acquisition 1 second later which is some nanoseconds delayed, and decorrelated. $\endgroup$ – Luis Costa Mar 25 at 21:53
  • $\begingroup$ I would not expect a gain in cross correlation amplitude with decreasing similarity so I don’t understand your plot $\endgroup$ – Stanley Pawlukiewicz Mar 25 at 21:58
  • $\begingroup$ These are not cross correlation plots. These are the signals being cross correlated. In order to find the delay we would correlate the red signal with the black one, then the green with the black. In the case of updating the reference at t=1s, we would correlate the green with the red. I will attempt to clarify in the text, I don't have access to the picture at the moment. $\endgroup$ – Luis Costa Mar 26 at 0:01

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