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I possess an ensemble of signal observations:

$x_i[n]=s[n]*g_i[n]$, $i=1,2,....,N$ where $N$ is a very large number compared to individual signal lengths (signal lengths are identical). Here, $s[n]$ is a time localized signal (e.g. a positive deflection) while $g_i[n]'s$ are arbitrary signals.

I have two intentions:

  1. Recovering the original signal, $s[n]$.
  2. Recovering 'some information' for each $g_i[n]$. I am specifically interested in how spread these signals are. So, rather than exact $g_i[n]'s$, some statistics about their temporal centrality would be more than enough.

I am eager to hear about possible techniques for achieving these. I should note that, since these will be a part of an academical work, I prefer neat and structured techniques. Lastly, I prefer computationally inexpensive ways since I will perform this operations multiple times.

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    $\begingroup$ Hi! Is "$*$" the convolution or the multiplication operator? What do you know about $s$? What do you know about the ensemble of all $g_i$? Is it, for example, zero-mean for every $n$ when you add them all up? $\endgroup$ Commented Dec 31, 2019 at 9:57
  • $\begingroup$ Hi! It is a convolutive model. You can think s[n] as a half-cycle sine (a single peak) at the center of the signal and zero elsewhere. We do not have much information about gi[n]'s, unfortunately. Also, there is no zero-mean condition among samples. $\endgroup$ Commented Dec 31, 2019 at 10:59
  • $\begingroup$ But we do know that $s$ is discrete and limited in time and we know its length, right? couldn't we get $g_i$ as the convolution of the "last" sample of $s$ and $g_i$? $\endgroup$ Commented Dec 31, 2019 at 18:34

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