# A Delay Between Two Filtered Chaotic Signals

it is a common practice to use a shift of cross-correlation peak to evaluate a group time delay of two signals (chaotic signals are included). Can one synchronously and equally filter these chaotic signals (one is a shifted copy of the other) and then get a cross-correlation between filtered copies of them? What will happen with a group time delay after that filtration? I made a numerical experiment with two simple delta-correlated signals (one is a shifted copy of the other, fft, fft filter and inverse FFT to get a cross-correlation) which indicated that absolutely nothing happens with a delay time after filtration. The same result one can get for correlated signals (I applied a moving average to simple delta-correlated signals and again - nothing happens) The delay time is the same after filtration (with different bandpasses of filters). But when I come to real signals the picture had changed dramatically. The value of the delay time depends on the filter's cut-off limits and the size of the spectral window. Any ideas why? Why do we suppose that we evaluate a delay time correctly when all spectral components are taken into account and can not be sure in the same estimations after filtering?

I assume that your experiment where you saw no effect was with signals that underwent a linear phase vs frequency process in their delay (constant group delay). Under many situations the phase of the channel in the frequency domain is non-linear, causing different frequency components to experience different delays (group delay variation). When you frequency select portions of such signals, you will then see the result of the delay that is dominant for those bands.

This is the same reason it is not best practice to simply use cross-correlation peak to evaluate the group delay. See further details specific to the challenges of doing this, and the better solution for those cases, at this posting here:

How determine the delay in my signal practically

And here:

How to reliably compute the group delay of a comb filter

At the bottom of the first post is a plot of group delay that illustrates my first point. This is an extreme case of delay variation where we see as duplicated below there are frequency selective regions where the delay is as much as 15 samples, while generally elsewhere it is only around 2-3 samples. If we further filtered this signal (with a linear phase filter to not cause further delay variation), we change the weighting between a 15 sample delay and a 2-3 sample delay.

• Hi, Dan! Thank you very much for sharing my concern. I'm afraid the resources you cited do not relate to the task. The problem lays in the fact that I meant the usage of two equal filters for two signals (one of them, is supposed, to be a delayed copy of the other). In this particular case, it doesn't matter how these filters work as they produce equal changes in signals that coross-correlated then. Commented Feb 10, 2020 at 15:40
• Dan, I found two discussions about practical techniques of filter group delay. It's a well-known problem which bears a significant similarity to that of one could find in quantum physics. I mean a tool which changes the entire system. As for me, it is almost the same problem. Because the filter in question changes the signal itself (in particular, it shifts phases of different harmonics of the signal differently). Therefore if one is going to use the output of such a filter, then he or she must take into consideration the corresponding difference in signals. Commented Feb 10, 2020 at 16:51
• I dare to suggest that it doesn't (in this particular case) because the changes in signals are equal, and signals are (almost) equal. What is more, the differences in time delay I encountered while came to real signals was in the order of several times... Commented Feb 10, 2020 at 17:08
• data: savepice.ru/full/2020/2/10/… . delay time within bandpasses (number of harmonics are in the legeng): cdn1.savepice.ru/uploads/2020/2/10/… delay time with a smaller spectral window: cdn1.savepice.ru/uploads/2020/2/10/… Commented Feb 10, 2020 at 18:03
• Yeah! Thank you for the discussion! It was very useful. (I told you at the beginning that it would be interesting and not about Wiener-Hopf equation, not only about it, more precise ;-))) Good luck! Commented Feb 24, 2020 at 16:31