I have a time-signal sampled over a one-dimensional spatial domain $x$. I know it is made of 3 components:

  • a right traveling non-periodic wave (characteristic time width $T_{R,NP}$, mostly 0 outside this width). This is prescribed at the left end of the domain.
  • a left-traveling non-periodic wave (characteristic time width $T_{L,NP}$, mostly 0 outside this width). This is prescribed at the right end of the domain
  • a periodic signal (right traveling in my case, of period $T_P$)

enter image description here

While traveling, all these signals slightly change shape due to non-linearities. Again changes are not drastic and fairly small. The periodic signal starts as a single harmonic but then more harmonics show up, which have small energy and the peak period is still $T_P$. Here is an example of the signal at a specific location (purple), together with my best guesses (I drew them by hand) of the two non-periodic signals. Two signals were prescribed on the left and traveled right, and one was prescribed on the right and travel left.

enter image description here

Which is the best way to tell apart the 3 components at different locations of my domain? In particular:

  1. Is there a method that can take advantage of the known information about the direction of travel?
  2. Sometimes (but not always) $T_{L,NP} > T_{R,NP} > T_P$. Would this help? I would prefer a general algorithm that does not rely on this information, but if a general algorithm does not exist, solving for these special cases would still be something.
  • $\begingroup$ This is very similar to this question: dsp.stackexchange.com/questions/68602/… The OP found a solution there, looking for a better one. $\endgroup$ – Cedron Dawg Jul 27 at 1:50
  • $\begingroup$ Thank you. It is a bit different. He knows the phase. I don't beforehand. But I can get an estimate of the speed and so the phase as long as the signals are separated, and estrapolate it into the area where they merge. When overlapping the speed can change though, cause one affect the other non linearly, but would not change drastically, higher order corrections. Also my signals deform slightly, although not drastically. His signals seem to maintain their shape $\endgroup$ – Millemila Jul 27 at 16:32
  • $\begingroup$ Okay. This is a problem that interests me as well. I have a bunch of band recordings I made with a binaural mic arrange, aka French Orchestra, so that each instrument is coming in at a slightly different delay. I'll be able to work with fixed delays (and echo cancellations) by analyzing sections where the instruments are the only ones playing. I am still far from tackling it, but I'm still thinking about these "sub problems" of which this is one and the link another. $\endgroup$ – Cedron Dawg Jul 27 at 17:27
  • $\begingroup$ Can you provide some more details? Do you have a single frame captured? Likely hopeless. If multiple frames, how many, how often? How consistent are the shapes? Finding the best fit sinusoid and removing that is the easy part. If you have multiple frames, even better. Is this a personal project or work project? $\endgroup$ – Cedron Dawg Jul 27 at 20:28
  • $\begingroup$ Those are actual solutions of PDE so I can sample wherever and whenever. It is for some research I am carrying out at my university. $\endgroup$ – Millemila Jul 28 at 4:35

If you can sample at any point in time, this shouldn't be that difficult. The first step is to collect as series of frames in time.

In each frame you should be able to find a best fit sinusoid, or set of harmonic sinusoids, to your periodic signal. Since this signal is assumed to have the same parameters across time, except the phase values which would need to be time adjusted, you can find an "average" value across all your frames for the periodic signal.

Subtract the best fit periodic signal from your frames. This should leave the other two behind. From your drawing, it appears that you have regions where one hump will b e zero, and others where the other hump is zero. Those are the locations to read your humps at.

For more accuracy, once you have your hump signals estimated, go back and subtract them from your frames. This should leave the periodic signal behind. Once again, find the best fit sinusoid assembly. This time your results should be more accurate.

Rinse and repeat.

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