sorry for the vague question (as I'm not even quite sure what I want to do is possible), just asking for some general direction to take my research.

For a brief description, my signal resembles exponentially distributed noise, bandlimited, with a triangle shaped spectrum around the baseband. I'm looking to remove additive white gaussian noise within the same frequency band as the signal, ideally without affecting the signal bandwidth.

Below are some simulated examples of a section of my signal and noise:
[![Time representation of signal and noise][1]][1]


and their bandwidth

[![Frequency representation of signal and noise][2]][2]

So the question is whether I can, somehow, attempt improve the SNR, given that they share the same frequency band, and my signal is not known beforehand (though it differs from the noise in distribution and spectral shape).

**EDIT** Averaging successive acquisitions is not an option, as what we want to measure is a local time delay of the blue trace. As such, just averaging would interfere with the measured delay.

  [1]: https://i.sstatic.net/s3ogx.png
  [2]: https://i.sstatic.net/Lzr6S.png