A Delay Between Two Filtered Chaotic Signals

Question

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?

Answer 1

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.